THE NEARNESS OF GRACE A PERSONAL SCIENCE OF SPIRITUAL TRANSFORMATION Arnold J. Mandell HOUSE_OVERSIGHT_013501

Table of Contents Acknowledgement ................ 00. ccc cece cece eee e nce n acces ene e eee eneensenaeenes 3 Chapter 1: In Search of the Miraculous .......................ccccceee eee ene ees 4 Chapter 2: Doesn’t Everybody ............... 0.0... ccc cc cee e cece ee ee ee eneeeenens 22 Chapter 3: Transmogrifications Of Energies .........................cceceeeee 42 Chapter 4: Sensual In-Between Entropies .........................2.eceeeee ees 64 Chapter 5: Some Entheogenic Entropies .........................cceeeee ee ee ee 87 Chapter 6: Pentecostal Phase Transitions ..........................ccseeee eee 122 Chapter 7: Amphetamine Roll-Up And Splitting ..........................008 144 Chapter 8: Faith And Rationality ..............0... 0.0 cccc cece cece eee e eee ee nes 168 Appendix: An Intuitive Guide to the Ideas and Methods of Dynamical Systems for the Life Sciences ............... 186 HOUSE_OVERSIGHT_013502

ACKNOWLEDGEMENTS Appreciation is expressed to the Fetzer Institute for their support of this work. Particular thanks are due their imaginative Vice President, Dr. Paul Gailey, who shared my vision and hope that these somewhat disparate themes could be blended into a meaningful whole. Time and the reading by others will tell whether this idea was realized. The Fetzer Foundation and Dr. Gailey have facilitated exploration into blends of science and spirituality, particularly in the context of personal meaning. They also have a history of supporting serious work in this era’s most powerful and rigorous exercise in holism as represented by the mathematical and applied mathematical fields of modern dynamical systems theory. Fetzer very special environment and years of dedication have encouraged the variety of personal meanings within science to emerge and be recognized as legitimate and important parts of the research enterprise. It would be difficult to imagine a more propitious context for this effort. The book is dedicated to my daughter Buna, and to my intellectual and creative companion, Dr. Karen Selz, whose deep and lovely mind wrote much more of this book than is formally acknowledged. HOUSE_OVERSIGHT_013503

CHAPTER 1: IN SEARCH OF THE MIRACULOUS More than a half-century of naive persistence and driven search for unity in the biophysics of mind and personal spirituality as the basis for healing transformation has led me into many laboratories. The motivation may have been genetic. My father said that we were descended from several generations of Jewish mystics, none of them able to attain the salaried status of rabbi or cantor. These ecstatic men lived lives of peripatetic eccentricity, stirring congregations with provocative insights and uncomfortably personal inquiry. But only for a little while. Soon they were asked to leave the synagogue and often their Eastern European Jewish townships called shtetels as well. My father, in the first generation of our family without rabbis in over a Century, was a businessman-musician, who in the early mornings studied Talmudic commentaries. He taught me about why it was that most interpretations of the book by the rational, physician, lawyer, philosopher, Moses Maimonides, called Guide for the Perplexed, were in error in their assumption that man cannot understand God’s nature with his mind. He took issue with the opinion that the union of a person’s intellect and Spirit with Him was not possible as long as a person was living. Ibn Tibbon, Maimonides’ best-known early translator and interpreter, relegated the cognitive, analytical, physical and alchemical transformational sciences to the earthly, not spiritual realm. My father disagreed. He espoused the work of the 13" HOUSE_OVERSIGHT_013504

Century proponent of a school of Jewish ecstatic mysticism, Abraham Abulafia, whose interpretation of the Guide and his own Commentary on the Secrets taught that the human mind, if transformed into a “state of active intellect,” could become one with Spirit, realizing the Kingdom of God in rational mystical experience in a state of excitement with new ideas. The new consciousness achieves deep knowledge of both the “upper” and “lower” realms of what he called “reality” both spontaneously and directly. He said that without personal transformation, this knowing is not possible. Abulafia’s lesson was that the mundane intellect of man has the potential for transformation into another kind of mind in a spiritualization of thought. This occurs via developmental stages that begin with intellect and imagination and culminate in what he called prophetic emanations. The exercises leading to this transformation are to be strongly willed and practiced with regularity. This work results in ascension to an ecstatic state accompanied by great intuitive powers, which Abulafia called “prophesy.” Ibn Adret, the Chief Rabbi of Spain at the end of the Thirteenth Century, banished Abulafia from the Country, a Century before the Spanish Inquisition ousted all the Jews. Following what my father said was required in the practice of Kabbalah, a 13" Century tradition of esoteric and mystical interpretations of the Scriptures, | learned the secret meanings of each of the twenty-two letter Hebrew alphabet. Much like the Platonic view of mathematics, that it existed before the physical universe, these symbolic equivalences were believed to be eternal in the transcendental realm. One of the rare written accounts of this oral tradition is in the thirteenth-century Hebrew Book of Splendor called the Zohar which describes the Hebrew alphabet as the heavenly code of the cosmos. | learned that the Tegragrammaton’s repeated letter Hei, being fifth in the Hebrew alphabet, represents the number five. In the Kabbalistic tradition, He/ implicates the functional five-partition of the human inner self or soul. The five parts are: nefesh, instinctual drives; ruach, mood, affect and emotions; neshamah, cognitive activities of the mind; chayah, efforts to understand and _ attain transcendence; yechidah, experiencing the world as a cosmic unity. Later in life as HOUSE_OVERSIGHT_013505

a psychoanalytical neuroscientist with a computational bent, the partitions divided thoughtful, forewarning forebrain from automatic and stereotyped hind brain, the signal analyzing thalamocortical system from the emotional and impulsive brain stem-limbic, the symbolically logical left from _ intuitively geometric right hemispheres. We divide the neurotransmitter moods of dopamine aggression from the transcendentally erotic serotonin and the organized dynamical states of periodicity and quasi(multi)periodicity from the real world complexity of chaos. | learned that it is comforting to divide an unknown whole into two or more unknowable parts. The Jewish guru and Hebraic tutor of my childhood, Rabbi Isadore Kliegfeld, smiled when | told him about my sudden loss of panic during nighttime Hebrew letter meditations. He said that | had had received personal evidence that these powerful symbols could call forth the transformational powers of God. He said that | had been given a blessing, in Yiddish, a nachas. Maybe panic is not that far from the transcendence of an activated mind. In my tenth summer, behind closed door in a hot back bedroom, first by accidental touch and then by more systematic chaffing, | evoked a pleasurably urgent and yawning feeling that began in the lower part of my abdomen and back. It filled me with thought emptying fullness that a sudden involuntary burst of pelvic contractions found resolution in an hour or two of an unexplainable sadness. | had been struggling to understand my father’s well warn copy of William James’s Varieties of Religious Experience and | wondered if | had been visited by one of the altered states he described. Was this what he meant by a _ transformative experience? A few months later, a late night meditation produced physical evidence, a thick, sticky, salty sweet stuff that by morning stuck my sheets together. Later that year, in my father’s library, | found a translation of the 1500 BCE Egyptian Book of the Dead. It contained a creation myth of two Gods in which “rubbing with my fist, my heart came into my mouth and | spat forth Shu and Tefnut.” Psalm 23, read rather regularly in Sunday school, began to make me wonder about the meanings of*...rod and staff that comforts...” and what was meant by “...my cup runneth over.” Among the ten regions of the Zohar, connecting the inner world of HOUSE_OVERSIGHT_013506

man to the upper world, is the tree of ten sefirot in which Yesod , the phallus, occupies a central place. Now we’re allowed to know that G-spot stimulation of the para-urethral glands in the female can result in spurt as well as a cup that runneth over. Other occasions of the temporal disappearance of the self-conscious | occurred while doing the theorem and proof work of high school geometry. Axioms and the rule bound processes of deduction created difficult journeys from that which was given to what should be found. Rocking back and forth in a desk chair for hours, chewing on fingernails, cuticles and pencil ends, time disappeared in a none self aware state of work-a-day well-being. Sri Aurobindo’s Bhagadvad Gita described this state as one of the rewards of karma yoga. Abulafia’s Kabalistic School emphasized the importance of hitbodedut, detachment and seclusion in concentrated thought, as a technique for the attainment of spiritual “intensification.” Stacks of lined yellow paper piled up full of blind alleys as | lived in humbling dumbness. One of my teachers of mathematics described it as the working mathematician’s dark night of the soul. A breakthrough to a route from premises to proof brought an expansive rush. Engagement in a struggle to fuse two differing contextual worlds may be transporting. Geometric visions can be used to do imageless algebra in a brain state that feels like intuition. The brain does something like this: Let the number of a sequence of unit squares, each side of measuring 0 to 1, be the denominator of a series of fractions, say fifths. Now put five of these boxes in a row. Then the sequence of all possible fifths, 0/5,1/5, 2/5,..5/5, is inscribed by cutting the vertical sides of the five sequential squares with a diagonal from the lower left of the first one to the upper right corner of the last. This line cuts each sequential square’s front boundary with vertical lengths, 0.0, 0.2, 0.4...1.0 in a series of decimal fractions equivalent to the sequence of all possible fifths, the proper fractions 1/5, 2/5...5/5. It was Abulafia’s kabalistic belief that symbolic, (algebraic), operations in (geometric) spaces can unify the “upper” and “lower” worlds in the eternal tensions between the body and soul, the inner world and the cosmos, the conflict making the global system both sensitive and stable. The geometric-topological approach to HOUSE_OVERSIGHT_013507

modern dynamical system’s theory describes a convolution of the expansive motions (as in the upper world) and contractive motions (as in the lower world) embedded naturally in the curved time and space geometries of what are called hyperbolic spaces. Each point in this space can be visualized as a little saddle in which orbital flows from pommel and back flow down to the seat, bringing points together in contracting motion, and flows away from seat down along the sides are expanding the distance between nearby points.. In the middle of the saddle, simultaneously expansive and contracting orbits demonstrate hyperbolic stability composed of intersecting destabilizing and stabilizing influences. Loss of this countervailing hyperbolic dynamical stability results in global system transformations called bifurcations and/or phase transitions. Transformation as a loss of stability is a theme of a recent poetic translation of portions of the Zohar called Dreams of Being Eaten Alive by David Rosenberg. He writes that at some time in the difficult journey through the often- incomprehensible Zohar, in order to gain entrance to the kabalistic cosmos, there arose what he called “heartbreak.” “No matter how much intellectual study is involved, the reader cannot understand the text unless he or she has offered his heart to be broken on the altar of poetry...and prayer.” Surrender may be the source of the strange, uplifting feeling of worked through dumbness. My mother, once a conservatory teaching assistant in piano, sat beside me while | practiced almost daily, weekends included, from the age of two until the midteens. Her quiet analytic counter-point sounded mathematical, “You can hear that that this harmonic progression goes through intervals of fourths of dominant seventh chords.” | felt the persistent lack of harmonic resolution as growing tension in my groin. “If you transform each of the 12 notes in a chromatic scale, multiplying it by five (in what mathematicians call) mod 712 (the numbering system goes from one to twelve, not ten, before it repeats), one can recover the circle of fourths, the commonest harmonic chord progression in music.” Though her computational talk supported rational thought, in my adolescent heat, the addition of Charley Parker’s flatted fifth and ninth to the dominant seventh chord led suddenly somewhere else and she knew it. Hearing my arrangement of a Beethoven piano piece become a HOUSE_OVERSIGHT_013508

mix of classical and modern jazz themes that | called “How High the Moonlight Sonata,” she laughed lasciviously as though tickled by this sensual violation of musical canon. A boogie-woogie Bach two and three-part invention brought more excited disapproval. Mysterious are the conditions of attentive (preoccupied) and none attentive, (fugued out) disappearing time. | found a musical way for it to happen when improvising: continue to shuffle a small set of notes that stay within the melodic field of the tonal center of an unchanging tonic chord. In contrast, most melodies and their chords leave the tonal center to which they return in harmonic and melodic progression. We can call these conventional tonal centers unstable fixed points. They are attractive repellers of melodic and harmonic expectation. It has been mathematically proven that these hyperbolic systems are globally stable. In contrast, a melody that remains stuck in the tonic chord, a purely contracting stable fixed point, is technically a chant. Paradoxically, it can be shown that this kind of fixed point is globally unstable. Rigid things can more easily fracture. The rich, altered states of consciousness that emerge while hearing the beat of Tibetan monks meditating, the Sufi chant-dances of Rumi and the John Coltrane and McCoy Tyner’s endless, single chord, tenor/piano dialogues exemplify the bifurcation to hallucinatory new stuff arising spontaneously from the experience of unchanging repetition. Constant repetition of the conditioned (expected) stimulus drove Pavlov’s dogs, especially those with “nervous temperaments,” into frozen, catatonic states. Abulafia’s 1280 book on ecstatic techniques, Hayyei Ha’Olam HaBa, recommended the recitative rearranging of a finite set of Hebrew letters, frontward and backward, many times, using prayer melodies, until “...the heart will suddenly become aware of the intellectual, divine and prophetic...” and hitbodedut will rest upon him. The instructions were “...combine letters (and associated musical notes)... reversing and rolling them around rapidly until one’s heart begins to feel warm.” It was in my freshman year at Stanford University when | met Michael Murphy, later to co-found Esalon, the California center for mystical pursuits and naked mud bathing. He is the author of Golf in the Magic Kingdom and with George HOUSE_OVERSIGHT_013509

Leonard, Integral Transformative Practice. | watched him go through a dramatic personal transformation after participating in Professor of Asian Studies, Frederick Spiegelberg’s seminar (with meditation lab) about Sri Arubindo’s interpretation of the Hindu Bible, the Bhagavad-Gita. Shortly after the semester, he climbed into an abandoned tower on campus to continue his meditation. He remained there for several months, refusing to come down even after the Stanford Student Health Service sent a medical school psychiatrist to investigate. | was more than curious about how it was that this hard drinking, and like his brother Dennis, all night poker playing, Phi Gamma Delta party boy, had suddenly become a transcendent ascetic. My girl friend Mary and | signed up for Spiegelberg’s seminar in Indian Religions. We were made breathless by his accounts of administering a Rorschach Test to the Indian Saint, Swami Sivananda. He recounted discussions about God with the artists Paul Klee and Max Ernst and the philosophers Rudolph Otto, Paul Tillich, Martin Heidegger and Martin Buber. As homework, Mary and | practiced breathing awareness mediation twice a day. During the year, Spiegelberg sponsored a visit by the aging but still very lively Aldous Huxley to our seminar. He also brought us Alan Watts and several lecturers from the Jung Institute of San Francisco. Shortly after hearing Huxley talk about the spiritual power of a particular exercise of will and loving thoughts, Mary and | began the daily practice of karessa, some Call it coitus reservatus. | was eighteen and she was nineteen. We found that withholding an orgasm in order to achieve nirvanic extinction of all desires and passions was difficult. We spent hours in karessa meditation, trying to experience the detachment described in the Bhagadvad Gita. This biblical explication of karma yoga told how it was that the warrior, Ardjuna, instructed by God Krishna in the form of his charioteer, was able to detach sufficiently to do his assigned job of killing without emotional involvement. Ken Wilbur, a modern, self proclaimed pandit, an academically oriented articulator and intellectual justifier of the dharma, the spiritual work of Hindu and Buddhist practice, contrasts the nirvana (literally “end”) composed of emptiness in time and space, dharma Kaya in which “...no objects are n arising...” with the lesson of the Bhagavad-Gita. Its message involved realizing 10 HOUSE_OVERSIGHT_013510

ones spiritual unfolding within the stream of real time and space, finding emptiness in the world of form and inaction in the world of action. We worked at karessa so ardently that there was barely enough time left to do our assignments in biology and chemistry. In a darkened room, Mary and | lay legs locked, lying on our sides, moving slowly and rhythmically, humming Om and waiting for our ascension. We worked at making the journey through Sri Aurobindo’s soul planes of higher mind, illumined mind, infinitive mind, over mind and finally, the supermind of infinitely empty no mind. This somewhat unusual way to study for a three credit course in Asian Studies at Stanford grew naturally out of the central message of Spiegelberg’s seminar that whereas “...deriving a universal theology is not possible, having the universal experience is required for an understanding of any of the world’s theologies.” The controversial Bishop of the Episcopal Diocese of Newark who teaches that Christian forms continue to evolve, John Shelby Spong, D.D. says, “...every biblical word represents an attempt on the part of our ancestors in faith to make sense out of a God experience in their time and place. The experience ...is eternal and real. The explanations will never be eternal and real. They will last only as long as the (cultural) mind-set that created them.” Mary got an A+ grade, topping Spiegelberg’s class with a final examination essay, which, in literary detail, described her episodes of samadhi, yoga’s state of unity with the creator. Her 25 page blue book contained accounts of walking fugues, spontaneously strong genital sensations, changes in tastes and smells, sudden feelings of rising spinal-abdominal kundalini, middle of the night dreams of oceanic orgasmic fusion with God. She failed to mention that she was describing her usual pre-menstrual state. During these college years, | learned about two Isaac Newtons The first | met at elementary physics lectures; the unit was about how things worked called mechanics. Logically and computationally consistent but taken on faith, | learned about an invisible field force between masses called gravity that decayed in strength like the inverse of the square of their distances apart and operated in my intuitive world like an electromagnetic spirit. Less occult were the expressions of gravitational fields as contact forces, computed for the tension in the string of a HOUSE_OVERSIGHT_013511

pendulum or the pressure of the floor on a weight resting upon it. Faith in this realm came from exercises in physical object visualization followed by manipulation of self-consistent algebraic symbols. | learned about experiments attesting to the “reality” of these ghostly fields (that now include electric, magnetic and strong and weak nuclear forces), and yet it was the physicists that already believed them who designed the machines to demonstrate them. It was Gregory Bateson, Margaret Mead’s houseboy, lover, photographer and social anthropologist who said, “Newton didn’t discover gravity, he invented it.” One college summer | found a second Isaac Newton, perhaps not so estranged from the first. He appeared in the form of a marble bust in the chapel of Trinity College at Cambridge University, holding the prism he had used to explore the polychromatic properties of light like a talisman. In his essay called Newton, the Man, the early 20" Century Cambridge Don and economic theorist, John Maynard Keynes, said that the Newton of the chapel followed “...certain mystic clues which God had laid about the world to allow a sort of philosopher’s treasure hunt to the esoteric brotherhood.” Michael White’s biography, called Newton the Last Sorcerer, described his work as an attempt to integrate the magic of the Old World with the science of the New Age. Newton’s awe over what he saw as the wonders of the universe maintained him in private theological study throughout his life. Arthur Waite’s Alchemists Through the Ages describes how Newton’s alchemical orientation toward the earth’s fundamental substances such as fire, air, wind and water, their powers and potential for transformation, was joined imperceptibly with his metaphysics and physics. In his hands, experimental observations involving gravitation, celestial mechanics and optics, though motivated by esoteric alchemical theories, generated experimentally accessible phenomena and testable ideas. The French mathematician, Jacque Hadamard, in his The Psychology of Invention in the Mathematical Field, said that mystical preoccupations were never far from the minds of most of the English and European mathematicians and physicists of the 18" and 19" Centuries. This orientation served as an impetus for them to pay attention to the almost imperceptible whispers of their emergent thoughts. E.T. Bell, the historian of mathematics and mathematicians said even 12 HOUSE_OVERSIGHT_013512

Descartes, the essential Enlightenment rationalist, was responsive to his “...call of the Spirit...” Napier the inventor of logarithms wrote an exegetical commentary on the Book of Revelations. The mathematician and physicist, Pascal, believing that contact with a religious relic had cured his terminally ill sister, wrote long tracks about whether or not the Devil could work miracles. The great mathematician, Cauchy, was known for his persistent efforts to convert fellow mathematicians to Roman Catholicism. Gauss, who was not particularly religious, said that a difficult to prove theorem did not result from hard work but “...the grace of God.” In letters between Liebniz, who along with Newton was the inventor of calculus, and a member of the family of great mathematicians, John Bernoulli, used scriptural quotations and biblical diagrams as part of their theoretical correspondence. Perhaps the greatest mathematician of the 18" Century (or ever), Euler, in his Letters to a German Princess, discussed the functional characteristics of spirits and tt the connections between body and soul. Bell said Euler “...never discarded a particle of his Calvinist faith.” It was to the working out of a law of mechanics called “the principle of least action” that Ernst Mach attributed the beginning of the separation of physical mechanics from formal theology. The flavor of this change is captured in his 1893 The Science of Mechanics that stimulated Bridgeman’s 1936 more formal philosophical analyses of physical theory, from a position that came to be called operationalism: the restriction of physical concepts to those definable in terms of the experimental operations required to demonstrate or prove them. Mach said that these events marked the move of formal metaphysical thinking about mechanics and the physical sciences more generally into the personal and private realm of belief and meaning. Maupertuis, an eccentric friend of Frederick the Great and president of the Berlin Academy, proposed the principle of least action as evidence of the infinite wisdom of the Creator. As an early psychopharmacologist, Maupertuis recommended the use of opium to facilitate creative thought and was famously parodied for doing so by Voltaire in his 1752 story in which he is portrayed as the naively foolish Dr. Akakia. The physical law of least action belongs to a set of ideas HOUSE_OVERSIGHT_013513

that are called variational analysis. They involve the natural (or miraculous) selection of maxima or minima in quantifiable physical processes. Of all possible two-dimensional shapes with the same perimeter, the circle contains the greatest area; in three dimensions, it’s the sphere. In his Principia, Newton reports his work determining the optimal shape of round solids, with circles of revolution having the same effective cross section, in order to minimize frictional resistance to gravity in a medium. The principle of least action says that imparting energy; say by a kick, to a physical body on a rigid two-dimensional surface like the earth, results in it taking the shortest route possible from its initial to final position. The related 1650 Fermat’s “principle of least time” is about light. As Feynman explains in his Lectures in Physics, “...out of all possible paths that light might take from one point another, light takes the path that requires the shortest time.” Feynman, using elementary relations from high school geometry, proved that the /east time principle could lead directly to Snell’s law of the refraction of light at the interface of two different conducting media such as air and water. His analogy was the optimal choice of the path to take in order to rescue a pretty girl drowning in the ocean. Whereas the shortest distance to the girl leads directly into the water, faster running along the beach to the point that minimizes the distance required for the intrinsically slower rate of swimming increases the distance traveled but reduces the time required to reach her. Euler attributed the optimization principle to an expression of the meaning and purpose of a loving God. Infused with this spirit, he developed mathematical methods describing smooth variations in position of an object in motion, the Euler differential equation, in which differential coefficients are varied to prove the principle of least action for mechanical motion. He gave the law Maupertuis’s name. Mach quoted Euler’s conclusion, “As the construction of the universe is the most perfect possible, being the handiwork of an all-wise Maker, nothing can be met with in the world in which some maximal or minimal property is not displayed.” Such faith based mathematical formalisms were rejected by Joseph Lagrange, an early 19" Century mathematician, who, among many other things, proved that every natural HOUSE_OVERSIGHT_013514

number could be expressed as the sum of at most four squared numbers. It was his strongly held opinion that metaphysical speculation was both foreign and inimical to the conduct of mathematics and science. His work in the ca/culus of variations led to the development of a system of algebraic manipulations seeking the value of constants, Lagrange multipliers, in place of solving Euler’s differential equations. It makes it possible to immediately write down a computable expression for the maximum of a mathematical equation. The technique is now routinely taught to high school students and with no mention of the role of belief in the perfection of God in its discovery. | was a fortunate freshman medical student. After a visit to his office and a stimulating discussion about some of the correspondences between the ideas of psychoanalysis and neurobiology, Robert Heath, Tulane Medical School’s Gary Cooper-like charismatic chairman of the psychiatry department, offered me a place in his animal and human neurophysiological laboratory. Between classes, evenings and weekends, | used a Horsely-Clarke apparatus, one of the world’s first stereotaxic devices. It allowed the precise placement of electrodes into functionally specific regions of a cat’s brain. The electrodes were cemented to the skull in place and their wires connected to a device by which the frequency, amplitude and wave shape of the electrical stimulation could be oscilloscopically monitored and electronically controlled as the conscious cat walked around the room. | spent hours observing and recording changes in spontaneous behavior that followed activation of various nuclei in the cat’s brain with small electrical currents. Deep in the part of brain that resides in the upper neck, called the lower brain stem, the region thought to regulate functions such as breathing, heart rate, blood pressure, gastrointestinal motility and global states of consciousness such as wakefulness and sleep, | found stimulus sites that, after 15 seconds of electrical activation, led to several minutes of hissing and objectless rage. One cat attacked an empty chair. These regions when activated also inhibited spinal reflexes such as HOUSE_OVERSIGHT_013515

the knee jerk of the standard neurological examination. Such phenomena were already well known in the late 1930’s in what W.R. Hess and later John Flynn, following electrical stimulation of cats in the lateral hypothalamus, called “hypothalamic rage.” In the late 1940’s and 1950’s, work by National Institutes of Mental Health’s Paul MacLean attributed it to the actions of parts of the emotional “limbic” brain, particularly the fear-rage-attack coloring of experience by the temporal lobe’s amygdaloid nucleus. Modern imaging studies in man have shown that this source of emotional coloring is activated by new information, even before the more rational parts of the neocortical brain processes it. How we feel about something new arises before what we think about it. These survival-oriented states of fight or flight are known to be biologically universal and demonstrable in even single cell organisms. A greater contribution to my brain metaphysics followed observations that after several seconds of stimulation of other brain stem sites, the cats became alert but quiet, staring into space for several minutes. Then, they circled slowly and curled up on the ground. This was followed by several minutes of grooming and loud purring. Difficult to handle cats became transiently tame, some coming close for petting. | found that these same sites also increased the amplitude and reduced the threshold for the cat’s knee jerk reflex. Responsiveness increased with calmness. Particularly interesting was the finding that electrical induction of this purring state could immediately stop on-going stimulation-induced episodes of hissing rage. | referred to these experiments with my friends as my neurophysiological studies of Old Testament vengeance and New Testament forgiveness. It seemed that the hissing rage would produce eye for an eye and a tooth for a tooth hypertension, the talon principle of the Old Testament and Koran. New Testament forgiveness would yield low blood pressure health and Jesus was a healer. It was about this time in the early 1950’s that Northwestern University social psychologist, Jim Olds, found that rats could be trained to push levers to obtain current delivery via electrodes in various parts of their brains. Shortly after, Joseph Brady, then of the Walter Reed Army Institute of Research, showed that squirrel monkeys would do the same. With depth electrodes attached to wires running to a HOUSE_OVERSIGHT_013516

miniaturized electronics box strapped to their belts, some of Robert Heath’s schizophrenic patients spent hours pressing their switches with beatifically expectant smiles. It was after several months of cat experiments that Professor Heath suggested that we spend some time interviewing a hospitalized, chronically ill female patient, Donna, before and during the time she was being studied with recording and stimulating depth electrodes in the human _ neurophysiology laboratory. Donna, bony thin in a lose fitting green hospital gown and sandals, had dark red toenails, blonde hair and eyes shadowed darkly. In her mid-thirties, she had never married and, when she could, worked as a beautician. She told us that since her menarche at 13, she increasingly often had episodes of spontaneous ecstatic rushes along with sudden visions of strong white light. She attributed these experiences to visitations of “...an unseen Christ.” She showed me a stack of notebooks filled with hand written accounts of her religious experiences interspersed with biblical quotations and difficult to follow discussions of what she called the Christian ideals underlying the Civil War. She read parts of it to us. One of her memorable stories was about being invited to a Children’s Crusade that had begun in Georgia, led by a great grandson of Stonewall Jackson. “We were trying to find the Lord to see if He would part the waters and open up an escape route from General Sherman’s march to the sea.” From a relatively poor family of Southern Baptists in rural Louisiana, she had lived in a state psychiatric hospital for almost three years. Her diagnoses ranged from borderline schizophrenia to temporal lobe epilepsy. The collateral interviews with her mother from several years before had been placed in the hospital chart. They recounted that in the patient’s middle to late teens she had become suddenly promiscuous, frequently approaching strange men in city parks. Obsessed with fellatio and swallowing sperm, she told her mother that she was receiving a holy sacrament. More recently, the increasing incidence of ecstatic episodes and compulsive note taking coincided with the complete loss of interest in sexuality in any form. Her talk was now full of moralizing detail about the shoulds and should nots of daily living. She referred to herself as a non-Catholic nun who was married HOUSE_OVERSIGHT_013517

to Christ. The brain waves recorded from electrodes deep in her brain demonstrated transient episodes of spiking in a midline limbic structure called the septum and in the right hippocampus, deep in the temporal lobe. Paul MacLean and others since have shown that electrical stimulation of these and related brain regions could produce pleasure and grooming reactions in cats and prolonged penile erections in squirrel monkeys. Many years later, | spoke about Donna with the Harvard professor of neurology, Norman Geschwind. He took me to his twice a week epilepsy clinic. In an effort to demonstrate what is now known as the Geschwind Syndromes of between seizure, inter-ictal personality changes in patients with temporal lobe epilepsy, he stood in front of the patients’ waiting room. In a loud voice, he asked that all people keeping diaries and personal notebooks please stand up. Several did so, some displaying their notebooks in outstretched hands. The pages that | saw were filled mostly with religious writing, biblical quotations and exclamation points. Gathering the positive responders together, he asked them in turn what religion they were. Several answered the question with the question, “When?” It turned out that many reported having several experiences of religious conversion. Geschwind called them “Jamesian Episodes” after William James’ Varieties of Religious Experience. He then asked when was the last time they engaged in sexual activity. For most of them, including those that were married, it had been years. Thought the men said they were not impotent, experiencing early morning spontaneous erections, they claimed a complete loss of interest in sex though feeling warmly affectionate toward people generally. As he anticipated, the patients were emotionally intense and unstoppably loquacious, needing to speak at length about their moral philosophies. They persisted in following us around the clinic waiting room, several speaking at once. In his lectures and papers, Geschwind called this last feature, difficulty in separation, interpersonal “stickiness.” First reported by the French electroencephalographer, Henri Gastaut, a history of multiple ecstatic religious experiences, increasing emotional intensity and lability, hyposexualilty (not impotence), moralizing religiosity, compulsive and frequently poetic writing and tendency to cling to people is now called the Geschwind Syndrome of temporal lobe HOUSE_OVERSIGHT_013518

epilepsy. Some say it is relevant to the likes of Apostle Paul, Sister Teresa and Joan of Arc. One evening in the human neurophysiology laboratory, | was invited by Dr. Heath to join him and several other brain scientists behind a two-way mirror to watch an interview with Donna while electrical current was being put through her recording electrodes. We watched and listened as a psychiatrist interviewed her about her past. The patient was speaking about her childhood. Unseen by the patient, the neurophysiologist, with us behind the mirror, was intermittently pushing the button evoking brain stimulation with very low current applied to the septum. Dr. Heath told me to listen for subtle changes or discontinuities in the flow of the on- going conversation that he said might reflect alterations in her thoughts and feelings. . “The first time we were allowed to take a break from Sunday school for the church service and | got to hear the choir and the pipe organ, | suddenly got a feeling of happiness that | hoped would last forever. My Sunday school teacher told us how much Jesus loved us and that’s what the music made me feel like. For the first time in my life | felt completely safe.” Though the two way mirror | saw the psychiatrist nod silently. “When | learned about the real meaning of Christmas and Easter, it was frightening and beautiful.” Within a few seconds after the neurophysiologist, behind the mirror and unseen by the patient or her interviewing physician, pushed the switch on the stimulus generator, the patient stopped talking. After a little more silence, her interviewer encouraged her to continue, “You were talking about how beautiful the holidays were. Tell me in what ways?” “| don’t want to talk about that anymore.” She blushed and looked very uncomfortable. The neurophysiologist's hand remained on the switch. She continued to speak with her psychiatrist. “| have to ask you a favor and | don’t know why. | hope you don’t get upset. The thought won't leave me alone.” She seemed embarrassed even as her body relaxed against the back of the chair languorously. 19 HOUSE_OVERSIGHT_013519

“Of course not, Donna. You know that with me you can say anything.” Her face reddening further, she stuttered something unintelligibly and then was silent. “Pardon me, Donna, | didn’t hear what you said.” “Would you mind if | rested my legs on your shoulders?” Further Readings for In Search Of The Miraculous The Hebrew Alphabet, A Mystical Journey, Edward Hoffman, Chronical Books, San Francisco, 1998 The Book of Letters, A Mystical Alef-bait, Lawrence Kushner, Jewish Lights Publishing. Woodstock, Vt., 1990 Studies in Ecstatic Kabbalah, Moishe lIdel, State University of New York Press, Albany, N.Y. 1988 Beyond the Human Species, The Life and Work of Sri Arubindo and The Mother, Georges van Vrekhem, Paragon House, St. Paul, MN, 1997 Bhagadvad Gita, Sri Aurobindo, Lotus Press, Twin Lakes, WI, 1995 Play of Consciousness, Swami Muktananda, Syda Foundation, South Fallsburg, NY, 1978 Alchemical Psychology, Old Recipes for Living in a New World, Thom F. Cavalli, J.P. Tarcher/Putnam, NY 2002 Studies in Schizophrenia, A Multidisciplinary Application to Mind Brain Relationships, Robert G. Heath, Harvard University Press, Cambridge, MA 1954 20 HOUSE_OVERSIGHT_013520

Role of Pleasure in the Brain, Robert G. Heath, Harper-Row, N.Y. 1964 Psychiatric Aspects of Neurological Disease, D. Frank Benson and Dietrich Blumer, Grune and Straton, N.Y. 1975. Mathematics —The Music of Reason, Jean Dieudonne, Springer-Verlag, N.Y. 1991 Mathematics for the Liberal Arts, F. Richman, C.L. Walker, R.J. Wisner and J.W Brewer, Simon and Schuster, N.Y. 1998 The Feynman Lectures on Physics, R.P. Feynman, R.B. Leighton and M. Sands, Addison Wesley, Reading, MA, 1963 21 HOUSE_OVERSIGHT_013521

CHAPTER 2: DOESN’T EVERYBODY Varieties of religious experience and the potential they bring for personal change are embedded in and perturbative of our unique and common personalities. The obsessive compulsive may have an easier time with the rigid restrictions of Fundamentalism or be more resistant to the flagrancy of none rational mystical experience. The hysteric may find subjective evidence for the Holy Ghost more accessible and rules of behavior beside the point. The potential for double-jointed multiplicity in personal styles and quick transitions between them characterize what is called the borderline personality. |It is in these ways that temporary and permanent brain styles in us and important others supply much of the ground for the possibility of spiritual transformation and the often attendant alterations in personality. How can we think about this facilitator and source of resistance to new spiritual practice? A skinny, knobby kneed, small breasted, mousy haired, bright-eyed psychotherapy patient of mine at UCLA’s Neuropsychiatric Institute Outpatient Clinic was among the highest priced Santa Monica call girls serving Beverly Hills. Answering my unaskable question about her thousand-dollar fee, she explained that she was living proof that, in her profession, what was more important than physical beauty was “griv sense.” She explained that by her middle twenties, she had 22 HOUSE_OVERSIGHT_013522

developed the ability to anticipate the most highly prized but often embarrassing-to- say longing for a particular sexual act without being asked. She told me that she had to “empty out my personal sex manual” to feel the cravings of her clients. What the john most wanted appeared suddenly in her mind in the form of a cartoon. A university criminologist later explained that the word “griv” was probably derived from what pick pockets call grift sense, the ability to intuit who was likely to have enough money in their billfold to justify the risk, even if they appeared in the worn clothes and dated cars of old money. In his 1913 Dernieres Penses, Henri Poincare’, France’s seminal theorist in nonlinear dynamical systems theory, described intuition as a mental faculty which allows us to “...immediately see the end from afar...” In the context of mathematical epistemology, the instantaneous images of a geometer contrast with the labored sequential logic of the mathematical analyst. Poincare’ claimed that inclinations toward one or the other of these two cognitive styles and their associated mathematical tools arise from different kinds of minds. He contrasted the 19" Century German mathematicians, Weierstrass, who he said reduced his general tt theory of functions to “...a prolongation of arithmetic...without a single (pictorial) n figure in any of his books...” with Riemann who called geometry to his aid in describing functions. He created “...an image that no one can forget... once he understood it.” Experiencing the behavior of others, we create a set of anticipations about whom and how they are that align with parts of ourselves. Aware of one aspect of a person, we imagine the others. With a small amount of initial information, we connect the dots, fitting features we have seen and heard to personality configurations stored by informal category in our brain files. Our conclusions about them “being one of those” can both facilitate and impair our perceptions. Eastern metaphysicians, Western mystical religionists, socially liberal secular humanists, Shannon information theorists and today’s students of dynamical systems in brain and behavior can, in different ways, make the case that the content of these stereotypes reflect a pattern of constraints, our personal limitations resulting from the rutted roads of worldly experiences. Baba Muktananda, the Hindi Saint from the 23 HOUSE_OVERSIGHT_013523

Indian village of Ganeshpuri, called them our samsara. These limit the formlessness of anticipation that underlies sensibility. Our samsara reduces the uncertainty that could serve as grounds for new perceptions and understanding of others. Pre- emptive distortions reduce the bandwidth available for new information. They impair the range of empathic relations with others as well as ourselves. These restrictions in possibilities and choices are expressed in enduring patterns of behavior, thinking and feeling that mental health practitioners call personality and character. When confronted with these constrictions, the self justifying and diagnostically revealing thought about a feature of one’s personality is, “...doesn’t everybody? “ This pride in our shape contrasts with the teachings about emptiness of one of Baba’s favorite Indian holy men, Zipruanna, who sat all day, loin clothed naked in a garbage dump, instructing his students and followers about knowing and being nothing. We quantitate deficiencies in formlessness using statistical measures of entropy. They characterize the system’s behavior as a distance from the state of highest entropy also known as maximal randomness. Professor Karen Selz of Emory University did a study in which her human subjects, after taking a battery of personality inventories, were asked to remove as many dots as possible from a computer screen full of them in three minutes. They were to do so by left clicking on each of them with the mouse key. Two seconds after a dot was removed, it reappeared and became subject to removal again. As they went about the dot removal task and unbeknown to the subjects, the orbit inscribed by their dot removing mouse travels was recorded for later graphic representation and quantification. Most subjects with the usual broad mixture of personality traits inscribed a wide variety of orbital line styles: little wiggles, big wiggles, large and small loops, little smooth slides and big and little jumps. The counter-intuitive coupling of stylistic rigidity and whole system instability (as in non-hyperbolic fixed points described in the previous essay and below) is in evidence at the personality and graphical extremes of her subject group. A fastidious, rigidly organized, severely obsessive-compulsive subject repeatedly removes the same dot, only occasionally moving to a neighboring one to do more repetitious left key mouse clicking. Very little of the large computer screen 24 HOUSE_OVERSIGHT_013524

of possible mouse travels is occupied. All the action is centered on a small set of points. When such a minimal entropy person is injured and feeling helpless, their stuckness can grow bizarre. Ruminative fixation in self-critical and persecutory ideas extend into poisoned food anorexia, circular pacing, weight loss and middle- of-the-night, worried insomnia. Suffused with sin, they ask forgiveness for soiling the chair by their sitting in it or smelling up the room with their body odor. At the high entropic extreme, the mouse orbits of the seductively dramatic, new reality-creating hysteric includes big jumps, disorganized whorls and large and small restless and short attention span scribbles that tend to fill up the entire screen. The fragility of fixation at this end manifests itself in breakdown into impulsively out- of-control and floridly dramatic displays. Their decrease in contact with reality precipitates social chaos around them. The Montreal behavioral neurologist, Pierre Flor-Henry, using electroencephalographic and psychological test data, described the difference between these two extreme forms of personality expressions as the overly dominant expressions of one or another of the /eft obsessional or right hysteric hemispheric emotional styles. As examples, Flor-Henry said that a left half brain depression feels like hopeless and agitated indecision and the depression of the right brain is an experience of emptiness like homesickness. Left-brain happiness is being exactly correct and right brain joy rushes like being especially chosen. The church going obsessional resonates with the sermon of the punitive priest who invokes the tension and relief of sin and salvation. The practice can result in a life long addiction to the transient high of this temporary forgiveness. In other churches, the hysterical character gets spiritual respite in disassociative visitations of the Holy Ghost and attendant signs and wonders. At Wednesday night healing services, new hope arises from personal surrender in a floor hitting, backward collapse called dying in the Lord. Both of these antipodal personalities contrast with the more receptive state of in-between entropy (with enough entropy available to form messages) which predicts more flexibility and higher potential for undistorted information processing. Relatively style-less and ego-less people are more open to hearing a variety of Gods in themselves and others. High alertness 25 HOUSE_OVERSIGHT_013525

without presupposition, ecstatically aware and selfless, it is God’s gift realized, a joyfully awake and nonjudgmental empty state of transcendence. As we sit, we work at feeling this in the brain of the enigmatically smiling stone Buddha. The externally inactive state of high internal activity, the Bhagavad-Gita’s formlessness in the world of form, inaction in the world of action, has a natural mathematical representation in the simultaneously expanding and contracting motions of hyperbolic dynamics and its associated entropic descriptors. How can this kind of formlessness equip us for almost instantaneous knowing? In a resting state of uniform hyperbolicity that only looks like randomness, accurate impressions of others can arise quickly and from only a few data points of observation. In the late 1960’s, University of California mathematician, Rufus Bowen, proved the now famous shadow theorem. This says that in dynamical states of hyperbolicity, directly observable on the screen in computer simulations, the first few points of the on- going wild dynamical dance that appears to jump randomly from here to there on the computer screen, counter-intuitively will quickly outline the entire skeleton of its future global shape, its geometry, though more time of observation is required to realize this structure in full detail. The contracting motions on the stable surface of action, called a manifold, “iron down” all the points onto the unstable manifold that serves to outline the shape of the attractor of all starting points. In such a system, observation of just the first few points outline the whole. Intuition, anticipatory knowing and that which some call prophesy, may be expressions of the hyperbolic brain’s mind doing dynamical shadowing. To review briefly, hyperbolic brain flow is made up of three decomposable components: (1) The apparently predictable one along the main road of the action, going straight ahead and round and round on a throughway called the center manifold—analogous perhaps to what might be a sequentially logical development; (2) Intersecting the center manifold transversally is a field of influence moving the action away from the center manifold with out-of-the-box motion, exploring side paths of unpredictably new, creative possibility called the unstable manifold, we might think about inspired risk-taking, impulsive associations in thought; (3) Another transversally intersecting field of influence, which conservatively, rationally, “irons 26 HOUSE_OVERSIGHT_013526

down” the expansive flow back onto the road, the entire constrictive field called the stable manifold. This influence herds points into shadowing the main road of the dynamics, like the hair of the dog that stay close to the real body of the animal in motion. It is in this way that just a few often slightly off the mark points nonetheless shadow the real (called fiduciary) orbits of the attractor, outlining its global geometry with just a little information. The intuitive reason shadowing works Is built into these natural countervailing tendencies of hyperbolic dynamics, which on one hand tends to spread out nearby initial points and brings disparate others together. The latter inclination is the one that smoothes down the escaping points onto surfaces of actions that mathematicians call manifolds. However, the details of the orbital paths don’t look that orderly due to the mixing of the sequence of points in hyperbolic motion. The mixing process on manifolds has been analogized to that of the bundled pink loops of the stretching (expanding) and folding (contracting) taffy puller at the carnival candy stand. The process gets sequences of small particles of candy out of sequential order while maintaining the taffy’s overall geometrically ovoid shape. Disorder is local with the entropy being generated by the repeatedly shuffling of the line up of the original orbital sequence. This results in the impossibility of any point- to-point prediction for more than a few points even though the over all shape is maintained. Exactly what minute a habitually late sleeper awakes can’t be predicted. On the other hand, the skeletal manifold of the global structure is entirely in evidence from almost the beginning. Late risers remain late risers even without a precise, minute-to-minute, predictable schedule. It is also interesting that a uniformly hyperbolic dynamical system, unlike the fixed-point attractors of stylistic fixation, resist perturbation-induced changes in global dynamical form. In an apparent paradox worthy of metaphysical allusion, the dynamically hyperbolic kind of formlessness has structural stability. The global geometric predictability of this point-to-point, completely unpredictable system can be both the subject and object of Zen frustration and thoughtful meditation. During weekly professorial rounds at Los Angeles’s Neuropsychiatric Institute, | assigned a standard exercise for psychiatric residents on clinical rounds, 27 HOUSE_OVERSIGHT_013527

which involved limiting their contact with a patient to five minutes. This was followed by detailed discussion of everything we’d seen and heard. |’d ask them to predict what we’d find in the many pages of personal interviews and nurses observations in the clinic chart. The student psychiatrists with the most street smarts, called emotional intelligence by Daniel Goleman, were particularly quick at shadowing and thus predicting the patient’s global dynamical pattern. Do personality patterns exist? Evidence from biometric studies of the hereditary aspects of personality style in animals and humans suggest that relatively few global component properties underlie a variety of complicated-looking manifestations of behavioral style. Primary colors are the source of all hues. Harvard psychologist, Jerome Hagen, has reviewed the history of this idea in his book, Galen’s Prophecy. While there are differences among personality research programs, almost all rating scale and questionnaire-based studies result in clusters of traits that reflect statistically associated properties which when taken together are called temperament. This idea is close to what we mean by personality. These relatively few response clusters are given descriptive names such as introversion, extroversion, neuroticism, impulsivity, sociability, task persistence and tolerance of ambiguity. As defined by psychological inventories, studies of families show that these styles are heritable in the range of 60%. Hans Eysenck, in over four decades of work and more than 5000 published papers from London’s Maudsley Hospital, derived common global factors of personality using questionnaires. The best known was called the Eysenck Personality Inventory. His studies resulted in evidence for only a few fundamental behavioral axes, behavioral manifolds, which describe extremal properties of personality types analogous to stable and unstable manifolds: introversion- extroversion, shyness-sociability, low and high activity level and emotional constriction versus impulsivity. To make the issue of personality as dynamical system more realistically complex, we can call on some examples of the rich history of behavioral genetic studies using animals such as the mouse. They can be selectively bred for underlying personality factors, such as dominance, fear, aggression or exploratory 28 HOUSE_OVERSIGHT_013528

courage. Not surprisingly, social interactions, as configured by the mouse’s own personality style, contributed significantly to their behavioral patterns. As an example, the C57BL strain of laboratory mouse has strong tendencies toward impulsively wild behavior. To be anthropocentric and using Hagen and Eysenck-like behavioral dimensions, we could describe the C57BL mouse as exhibiting high psychotocism, P, energetic sociability, high energy, E, and low emotionality, low neuroticism, N. The C57BL also loves alcohol and will dominate the low E, shy, low P, retiring, alcohol avoidant, high N, emotional, anxious, frequently defecating albino BALB strain of mouse when they are placed together for a limited time in a novel situation during the daylight hours. Over a more extended time, however, the BALB mouse comes to dominate the C57BL, beginning with attacks in the dark and finally as the persistent and patient survivor over days of aggressive fighting. BALB’s low E, social fear eventually turns into rage and aggression. The C57BL is quick to mate and ejaculate but very slow to recover sexually, so that the less post-orgasmically refractory BALB also wins in long term sexual competition in a cage full of fecund females. Modern social psychological approaches to human personality are beginning to approach the interactions of genetic brain proclivities and collective social dynamics in this way. Employing Eysenck categories of personality characteristics, similar results about style as influenced by genetic selection can be seen in humans. The correlations between factor scores based on B. Loehlen’s studies using the California Personality Inventory in twins demonstrated as much as threefold higher correlations among identical twins for extroversion (E) and neuroticism (N) factors compared with matched fraternal twins. The primacy of some of the in-born biological roots of these personality styles is suggested by G. Methany’s finding of higher correlations between identical as compared to fraternal twins when studied at the age of two months. The similarities in personality and temperament measures included activity level, regularity, approach-withdrawal, intensity, persistence, distractibility and adaptability. More recent familial studies of the heritability of personality characteristics included childhood shyness, neuroticism, depressive symptoms, aggressiveness, 29 HOUSE_OVERSIGHT_013529

behavioral inhibition and anxiety, behavioral flexibility, narcissism, deviant motor activity levels, novelty seeking, harm avoidance and reward dependence. These studies were conducted by R.R. Crowe, J.F. Rosenbaum, A. Methany, and J.L Robinson and indicated familial congruity of these characteristics among first and second degree relatives in the range of 40-50%. This level of heritability in genetically unrelated family members was found to be less than 20%. Low entropy fixations of personality can also evolve developmentally. Experiments in young animals have shown that stress-induced high levels of adrenal hormones exaggerate the normal developmental process of trimming back unused neural connections, called pruning, the normally complexly over-grown sprouting pathways. The pruning actions of the pituitary-adrenal stress hormones come to dominate sprouting actions of neural growth factors and their protection of neuronal axonal branching and connections during development. The research program of Bruce McEwan of Rockefeller University and others document nerve cell loss resulting from the neurohormonal concomitants of stress. This reduction in neuronal connectivity and neuronal cell content has been conjectured to contribute to the pathological simplification of neuronal projections and neural network complexity, reducing information processing capabilities. The still intact machinery underlying the global patterns of neurological activity, such as those that underlie personality styles, is arranged around these pruned, unoccupiable holes of lost brain possibility. If this range of potential behavior is extremely reduced, the behavioral syndrome is often called a personality disorder. Those that have one are the predictable Johnny one notes of response to perturbation: thrash out, lie without reason, get drunk, binge on promiscuity, steal unneeded things from department stores, or withdraw into interpersonal isolation. A more abstract and quantifiable way of representing the pathological simplification-induced emergence of low entropy, stereotypical personality style is inscribed on the head stone of the post-suicidal grave of Ludwig Boltzmann. This father of modern statistical physics expressed the idea in the form of a transformation: the (maximal) entropy, S, of a system is the logarithm of the number,Q, of its available ways of being, (i.e., S = log ). That is, one way a 30 HOUSE_OVERSIGHT_013530

reduction in the dynamical entropy of a system can occur is by reducing the number of its available states. As the repertoire of ways of personal responding, log Q, is reduced, so is the brain system’s entropy, S. Reality constrained patterns of behavior, as in successfully adaptive personalities, lie in some optimal in-between place between the maximal and minimum measures of entropy. The dynamical state that is postulated to yield in- between-valued entropies is called nonuniform hyperbolicity. This is best seen when the values of the experimental observations are plotted in a two dimensional phase space with each point represented by two values: along the x-axis is plotted the value observed, along the y-axis is graphed the change in the value from the last observation. The signatory motions of these observations plotted in phase space are irregularly varying in rate of expansion (near by initial values are separating in time) and contraction (greatly differing initial values are coming together in time). Values are not fixed, rhythmically varying nor in random motion. These nonuniformly hyperbolic motions are seen in speeded up, talking head videos showing bursts of hand gestures and in normal neuronal activity. Silences have widely varying lengths and bursts of hand movements and neuronal discharges are irregular in duration and character. The statistical pattern of neuronal inter-burst intervals is not the convergent Gaussian distribution of |.Q. or heights but the nonconvergent, long tailed, Levy distribution of flood incidences and, according to Mandelbrot, stock market crashes. The labored logic and inscrutably compact mathematical formalisms of the Nobel Prize winning physicist, Ilya Prigogine, and his Belgian school, explain the thermodynamics of these long lasting niches of restricted variation in our personal style as energy requiring dissipative structures. Compulsive nail biting, driven promiscuity, readiness to be suspicious are seen as a persistence of deviations from the maximum entropy of formless, flexible, receptive end states. The system is trapped in possibility reduced, energy requiring, samsaric niches of what Prigogine called minimal entropy generation. We unique and oddly shaped and entropy leaking balloons maintain our characteristic distortions through energy-requiring, 31 HOUSE_OVERSIGHT_013531

persistent efforts at insufflation. The maintenance of neurotic defenses and eccentric habits can be fatiguing. The children at Kids in Distress Residential and Day Care Center in Southeast Florida, called KIDS, tended to be small for their ages. As a psychiatric consultant to the Center, | often summarized an evaluation of both their physical and intellectual development as “delayed.” Looking like almost completely formed adult-like personalities, however, they were developmentally “advanced.” | heard in a child analytic seminar at the Psychoanalytic Institute of Southern California that traumatized children often hurry through the dangerous developmental ambiguity of openness and flexibility to the predictable, fixed attitudes and behavior of adults. It was common to find prematurely wise young children serving as parents in chaotically dysfunctional families. In residence at the Center, set free from their pathogenic homes by social workers and family law judges, these premature caregivers lost sleep worrying about who was taking up their obligations to the sisters and brothers left behind. Trauma-induced possibility pruning was often obvious in the young refugees at Kids in Distress. Having been soaked in alcohol containing, nutritionally deficient, crack-laced amniotic fluid, young babies were then left in dirty cribs behind locked doors to cry themselves into exhausted despair. Their mothers were working the streets for drugs. The children that survived often demonstrate personality styles that are reduced in variety. They came to use a few, individualized, and stereotyped techniques for survival. Some children’s insulated detachment was_hollowly disguised as interpersonal caring. Others used driven and rigid compulsion to maintain the appearance of conscientious good citizenship. For some children, paranoid thoughts were realistic expectations. . Arriving at the Center | heard “Dr. Arnold! Dr. Arnold!’ in high-pitched screams. Several children ran up to me at once, demanding to be held. Some leaped into my arms for a hug. Trying to get and hold their visual gaze was another 32 HOUSE_OVERSIGHT_013532

matter. Their eyes darted back and forth across my face, not stopping at my eyes, as though checking for danger. It felt like a strange mix of physical clinging and interpersonal distantiation. Many articles in the International University Press’s Psychoanalytic Studies of the Child book series, described these prematurely formed child personality types: the paranoid scouts, the detached as /f children pretending to feel, the desperate to please obsessionals, the charismatically seductive hysterics and the unconscionable psychopaths. Experiments simulating trauma and neglect in young animals also demonstrate acceleration in biobehavioral development. Possibilities, the number of available states, ©, brain entropies as S = log Q, become casualties of traumatic and neglected early life. Like one trick ponies, these abused and abandoned children take up singular patterns of behavior that seem to work and stick to them. One doesn’t anticipate seeing such narrowly fixated personality patterns until late adolescence or adulthood. They appear at ages too young to qualify for the character pathology coding of the Diagnostic and Statistical Manual IV. Yet the labels of adult personality disorder seem inescapable when one sees a four-year- old child trapped in a compulsive hand washing ritual or a panty flashing five-year- old girl with a seductive gait. Four-year-old Alicia rubbed the lumps in my right hip pocket containing caramel candies. Her blue eyes twinkled. Her long blonde hair was in bangs and her lips in a pout. She kept a hand on her hip and tilted her pelvis as she spoke. Listening to children’s stories, she straddled the reader's thigh and rocked. Alicia had a history of sexual abuse in a home that was a hang out for drug dealers. There were rumors that she talked to strange men late at night on the phone. On admission to the Center, she was found to have genital herpes. Both of her parents had been in and out of prison for drug-related crimes. The Center’s staff spoke of Alicia’s seductive smiles, incessant demands, irritable complaints and tantrums. With the back of her hand held against her forehead, she said that it was too hot to pick up the toys she had scattered around the fenced yard. Ordered to comply, Alicia took three steps into Florida’s summer heat and fainted. Each morning, she spent the better part of an hour in front of the mirror, trying on all four of her dresses 33 HOUSE_OVERSIGHT_013533

and their scarf and belt accessories before choosing one for her appearance at the breakfast table. Five-year-old Grace was a suspicious and dictatorial presence in the Center’s kindergarten class. Articulate and righteous, she confronted children and staff alike with evidence for the unfairness she found everywhere. In legalistic defense of her rights and sometimes those of her peers, she used her strong wide face, penetrating look and quick and observant mind aggressively. Her somewhat intimidated childcare worker maintained Grace's cornrowed hair with care. Sensitive to criticism and quick to anger, she competed with her teacher for control of the class. Her drug abusing young mother had escaped from her own mother’s authoritarian house, leaving six-month-old Grace in the care of her commanding grandmother, a matronly church elder. Recent studies by David Reiss and associates at George Washington University assessed psychosocial dynamics in genetically varied families. They found that genetic similarities amplified the expression of individual characteristics of interpersonal relating through what might be called personality resonance. Relatives often commented that Grace and her grandmother, being alike, deserved one another. Shortly after her fourth birthday Grace was removed from her grandmother's home while the circumstances surrounding the accidental scalding of the bottom half of her body in an overheated bath were being investigated. She began her first conversation with me, “Hey doctor baldy, why are your bottom teeth so crooked?” Damon was darkly handsome, with teasing eyes and a gleaming smile. Talking to his legal guardian on the pay phone in the afternoon of his second day at KIDS, he was heard to be making charges of mistreatment by the staff. He asked his guardian, loud enough to be heard throughout the day room, “What does it take to get someone fired around here?” Six years old and abandoned by his mother at the age of three, Damon came to KIDS with a history of provoking administrative conflicts at several children’s shelters. His record showed that once he successfully used accusations of beatings to get a staff member fired employing charges that were later shown to have been fabricated. He argued persuasively, manufacturing events and quoting imaginary conversations with smooth confidence. He could 34 HOUSE_OVERSIGHT_013534

change stories midstream without apparent loss of continuity or confidence. He learned the power of a claim of abuse, and used the threat of it to control his environment. Damon talked other children out of their candy allotments, cheated at games and stole clothes from other children’s lockers. Debbie, age eight, was the eldest of four children. Her mother was a street prostitute with an expensive drug habit. Debbie was thin, restless and worried. A self-appointed mother from the age of four, Debbie felt responsible for the care and feeding of her brother and two sisters. With a history of physical and sexual abuse by a series of her mother’s boyfriend-pimps, Debbie spent most of her time cleaning and recleaning their small apartment and worrying about obtaining enough food for her brothers and sisters. Her mother was often gone for one or two days at a time, and food supplies were not dependable. On several occasions, Debbie was caught stealing food from all night grocers. The investigative social worker reported that Debbie had learned to sell oral sex to the men who loitered behind a neighborhood bar. She used the money to buy food. For several days after admission to the crisis home, Debbie was anxious and sleepless. She worried endlessly about the welfare of her sisters and brother despite reassurances that they were in caring foster homes. She checked on them as frequently as allowed by phone. In a playroom therapy session, wielding a rubber knife, she pointed to a scar on her left forearm and told a story about the time that she cut herself with a kitchen knife and fed her blood to her infant sister when there wasn’t any food in the house. Debbie kept her room very tidy, did all her chores and sometimes those of other children. Even after several months in residence, always-busy Debbie didn’t have even one close relationship with any of the other children or members of the staff. Despite the superficial differences, there are subtle and pervasive similarities among the personality styles of Alicia, Grace, Damon and Debbie. Like overgrown and tasteless cabbages, pale and four feet across, growing from seeds over-treated with gibberellin or auxin plant hormones, the inner lives of these prematurely big little people are relatively empty of stable interpersonal objects. The pantheon of indwelling companions are either malignant, absent or both. There is a deficiency of internalized significant others with qualities we more healthy neurotics paste onto 35 HOUSE_OVERSIGHT_013535

new faces which we then love and hate. Instead, every interpersonal arrangement is new, suspect and run on a cash-and-carry basis. We are made to feel like there are no seats for us inside of them. Even Debbie, with her history of selfless motherly devotion to her “children,” felt like an empty husk, encased in the exoskeletal armor of compulsive correctness. With their inner life unpeopled, the best we on the outside can hope for is to be valuable to them as tools, like forks and chairs. In new and potentially therapeutic settings, for example a genuinely loving foster family, these children manipulate, testing for the feared loss and abuse that first generated their detachment. They provoke the very mistrust they fear. The sexually exploited child is seductive. The physically abused child provokes attack. Personality constellations which can be adaptive, when narrowed and fixated, become impediments to new and reparative experience. It is in this way that personality disorders are self-maintaining. An irony is that these interpersonally empty and rigid patterns in personality tend to occur in the most constitutionally robust of the abused and neglected children. They are those who have escaped early death from failure to thrive, severe neuropsychological impairment, chronic depression, severe social withdrawal or the pediatric psychotic disorders. The children with sufficient flexibility to adapt quickly and survive often settle into empty-centered rigid caricatures of adult personality styles. Of course, well-defined and characteristic personality patterns do not require abandonment and abuse or the pathological simplification of traumatic deforestation of neuronal connectivities in order to emerge. Demanding social selection of particular personality proclivities that are competitively advantageous for highly sought positions also results in the appearance of well-defined personality styles. no tt Common examples are the technical types, “techies,” “nerds,” whose work require long hours alone to master and execute, as in doing mathematical proofs, solving problems in theoretical physics, unraveling computer programming problems or writing highly technical tracks. These activities can be aided by the personality inclinations of shyness and distantiation, the experience of discomfort in social occasions along with a rich private fantasy life. Diagnostically oriented mental health 36 HOUSE_OVERSIGHT_013536

professionals (and lonely mates) may label these interpersonally distant, engineering rocket science people, “high functioning” sufferers of Asperger's autistic spectrum disorder. Things going on inside get most of the attention, having more impelling importance than those on the outside involving other people. A recent study by Cambridge University’s Autism Research Center compares the empathizing (E) versus systemizing (S) ability of normal controls and adults with Asperger Syndrome and find the quasi-autistic adults are deficient in E and superior in S. They call it the E-S theory of autistic spectrum diseases. Psychotherapists of these autistic spectrum personality types, patients who characteristically do not seek therapy but are forced into the office by marital or family conflict, speak of their long, patient and mighty struggles to make intimate contact with these clients. A more philosophical question involves issues of what are acceptable individual differences and why it is that these high functioning, highly paid and successful professionals have any diagnosis at all. It is not surprising that the highest paid members of corporations producing technical products and services such as IBM and Oracle are those rare individuals in technical sales that are able to combine the skills and insights of introverted scientists and technicians with those of the gregariously successful salespersons. In business schools such a blend is seen in people who combine talents in both marketing and finance. In architecture this combination might take the form of a graphic-design artist with computational mechanical engineering skills. Recruiters know that it is difficult to find people for what is called engineering sales. From all over the United States, professional instrumental musicians that began to experience severe technical difficulties that defied their teachers as well as more extended practice time came to see Chicago’s music guru, Carl Boardstadt. He was a nationally known consultant to classical and jazz professionals in the 1920’s and 30’s. His particular specialty involved those who had “hit the wall,” those whose progress toward advanced musical mastery and accession into the higher echelons of the business had been truncated. His recommendations were often eccentric indeed. For the wind musician with breadth control problems, it might be blowing uniform bubbles through a long tube held at increasing depths of a filled 37 HOUSE_OVERSIGHT_013537

bathtub or feeling the seductively diaphragmatically oscillating belly of a taxi dancer. Pianists with speed problems worked at specially constructed up-side-down keyboards with the rationale being that finger lifting was more rate limiting than finger placing. He said that his most hopeless cases were those whose personalities didn’t fit their choices of instrument, too often made by what position remained open in the high school band rather than following a personal interview. He claimed that trombonists should be sensually languorous; clarinetists, nervously impatient; double reed instrument players, obsessional and withdrawn; brass players, athletic and exhibitionistic. As one of the team physicians of the San Diego Chargers in the years 1971- 1975, | spent several days a week in their summer training camps, on the team plane to and from games, in the locker room and on the sidelines during games. | was involved particularly in player drafts. Unbeknown to candidate players and other teams, we used a system of what social scientists call unobtrusive measures of their personalities as part of their evaluations. College football players are sent questionnaires each year by professional teams asking about a variety of life events and attitudes including their goals for the future. Filled out by hand, they served as repeated measure, handwriting samples. Twenty years of them were available in the Charger’s record room. Using 30 standard signs from the French graphology literature and three trained raters, we evaluated the hand writing characteristics of players, National Football League wide, who obtained and retained playing, not reserve, positions in the League for at least three years. After studying handwriting profiles from close to a thousand established NFL players, and hundreds of hours of individual interviews of members of many teams, it became clear that, athletic abilities being equal, success was more likely when the player's personality type fit his football position. What amounts to a series of selective filters are operated by coaches, scouts and managers throughout the playing careers of these players in grammar schools, high schools, universities and, 38 HOUSE_OVERSIGHT_013538

ultimately, the NFL draft. Choices obviously involved more than height, weight, time in the 40-yard dash and performance in motor coordination tasks. The players behavior, carefully studied on the field, in multiple camera angle game films, direct and collateral interviews and observations under game conditions constituted a high level of selective pressure that brought with it the emergence of characteristic personality types. Tens to hundreds of thousands of candidates are winnowed down to several hundred highly paid players in this selective process. Distinctive personality patterns accompany success at a particular position. Structure loving, politically more conservative, choreographed in detail and repeatedly rehearsed, offensive players keep their lockers more organized and tidy. More rebellious, resentful of structure, politically more libertarian, thematically instructed but principally opportunistic, defensive players, particularly linemen and linebacker’s lockers had messy lockers. Defensive team players were most often in trouble with the law. Offensive lineman including centers, guards, tackles and some tight ends tend to be patiently enduring and tenacious, their aggression taking the form of stubbornness. This contrasts with the temperamental explosiveness of the defensive line and linebackers. We could speak of the volubility of centers, the loyal and caring kindness of offensive tackles, the narcissistic exhibitionism of wide receivers, the murderous rage of the defensive end, the sullen and paranoid depressiveness of the defensive back, the joyfully impulsive unpredictability of broken field running backs and the good citizenship egolessness of the blocking fullback. Some quarterbacks lead and play fearlessly in a religious state of grace, some are members of the Fellowship of Christian Athletes. Others lead as fearlessly, but in the style of an unconscionably calm psychopathic bank robbing professional. Influenced by our findings, the San Diego Chargers drafted the Hall of Fame quarterback and one time ABC Monday Night Football commentator, Dan Fouts. Skinny and hurt several times during his college years as a quarterback in Oregon, he was passed over in the NFL draft until the third round. The scouts “knock” on him was that they thought that he lacked psychological and physical toughness; the ability to get up after a hit and to ignore the on coming tons of defensive linemen 39 HOUSE_OVERSIGHT_013539

while calmly and quickly surveying the routes of several potential receivers. The pattern found in his handwriting features, however, resembled those Johnnie Unitas, the Hall of Fame quarterback of the Baltimore (then) Colts who, in spite of his small size, famously played with great courage and physical toughness. In chronic and severe back pain, he played regularly until retirement in his early 40’s. Fouts drafted in the third round with a small five-figure bonus, proved to be a great bargain for the Charger franchise. Given the theoretically infinite number of ways that a personality can be, it is remarkable that the latest Diagnostic and Statistical Manual of the American Psychiatric Association, DMS-IV, describes only eight types, which form three subsets of exaggerated expressions of stable personality styles called personality disorders. All eight personality disorders can be grouped into: (1) Cluster A - Odd and eccentric types, whose anxiety is related to the felt threat of disintegration and annihilation of the self and whose style is dominated by mistrustful paranoia, a schizoid, detached and emotionally flat pattern or the isolated strange eccentricism of schizotypal characters, (2) Cluster B - Unstable and impulsive types whose anxiety is related to loss of the stable self and whose style is dominated by irresponsible antisocial behavior, chronic instability with high amplitude fluctuations in behavior called borderline, or patterns of excessive emotionality and dramatic display associated with histrionic characters; and (3) Cluster C - Fearful types whose anxiety is related to hypersensitivity to criticism, guilt and feelings of inadequacy or loss of control, and whose style is dominated by interpersonal avoidance, clinging dependency, or rigid lock up into obsessive-compulsive efforts to do the right thing and avoid disapproval. This remarkably small array of stylistically consistent global behaviors selected from a practically infinite number of imaginable possibilities establishes a small set of invariants of some, perhaps abstract, property. These characteristic patterns inspire our search for the implied brain and behavioral conservation laws that may underlie them. 40 HOUSE_OVERSIGHT_013540

Further Readings for Doesn’t Everybody The Evangelicals, David F. Wells and John D. Woodbridge, Abingdon Press, Nashville, 1975. Godtalk, Travels in Spiritual America, Brad Gooch, Knopf, N.Y. 2002 The Value of Science, Essential Writings of Henri Poincare’ Stephen Jay Gould, Modern Library, Random House, N.Y. 2001 From Being to Becoming, /lya Prigogine, Freeman, San Francisco, 1980 The Development of Mathematics, E. 7. Bell, McGraw Hill, N.Y. 1945 Deterministic Chaos, An Introduction, Heinz, George Schuster, VCH, Weinheim, 1989 Lectures on Dynamical Systems, Structural Stability and their Applications, Kotic K. Lee, World Scientific, Hongkong, 1992 The Psychobiology of Behavioral Development, Ronald Gandelman, Oxford, N.Y. 1992 Handbook of Character Studies, Psychoanalytic Explorations, Manfred Kets de Vries and Sidney Perzow, International Universities Press, Madison, 1991 Cognitive Style, Five Approaches and Relevant Research, Kenneth M. Goldstein and Sheldon Blackman, Wiley, N.Y. 1975 41 HOUSE_OVERSIGHT_013541

CHAPTER 3: TRANSMOGRIFICATIONS OF ENERGIES After several of months of running, 12 miles most days, | felt an energetically calm, self-containment and a growing loss of interest in things sexual. My increasingly impoverished fantasy life led my training psychoanalyst to suggest that | was running away from the critical, females issues of my psychoanalysis. He said | was becoming more out of reach as | became more socially pleasant. This was decades before Prozac, Paxil and other serotonin reuptake inhibitors were inducing similar hyposexual, withdrawn states of cordiality in millions of Americans. Recall that Norman Geschwind, the Harvard Professor of Neurology, reported similar conditions of high energy sexual disinterest and abstract metaphysical preoccupation in patients with right temporal lobe epilepsy. For reasons other than the loss of church property rights and the spread of syphilis to the clergy, it felt like | was being readied for Pope Gregory VII’s Eleventh Century celibacy reforms for abbots and clerics of the Catholic Church. It was true that my feelings of dependence on my analyst for understanding and approval were being reduced as | ran into less emotional involvement. | was becoming a more rationally objective observer of others and myself. It wasn’t the first time that my over-ardent practice led to this warning. Baba Muktanada, my Hindi guru, told me to reduce my daily sitting time of meditation. He said my spacey 42 HOUSE_OVERSIGHT_013542

social smile belied a growing disinterest in the welfare of others. | was getting hooked on the hard training high of not really being there for other people. Several articles in Runner’s World said that many runners become addicted after even a few months of running over six miles per day. It’s true that over fifteen years | missed less than 10 days of running per year. | ran in driving rain, sweltering heat and dangerous places. In New York’s Central Park after dark, | followed a freshly strewn trail of torn woman’s garments that ended in shredded panties and a bra on the Park’s bridle path. In Oklahoma City at 104,° | was chased and bitten by a terrier. In Munich at 4:30 AM, before delivering a morning lecture, the black uniformed police stopped me for a shakedown. In Ann Arbor, | shuffled along in two feet of snow. By the Seine, at 14°, paranoid barge hounds barked in big dog baritones. | ran on the Hebrew University track a block away from a loud Palestinian bomb left in a refrigerator near a busy street corner. Breathless at nine thousand feet in Aspen, gagging on the strong manure smell of Sacramento Valley farms, in the hot wetness of Houston and dry heat of Palm Springs. | wore out three to four pairs of Nike running shoes per year. What | did not tell my training analyst was that this felt like a chase after God. As in most spiritual transformations, His messages and music could emerge quite suddenly. Even after stretching, it was painful to begin and that was my daily sacrifice. | was readying myself to follow the God of the Hebrews and make the “three days journey into the desert” as in Exodus and Paul’s recommended presentation of my body “as a living sacrifice, holy and well pleasing to God.” After three miles of running, the hip pain, back stiffness and leg heaviness lifted, difficult breathing became easier. A burst of new energy appeared suddenly. The first pop usually took the form of assertive feelings fueled by new personal power, an undoing of the lethargy and depression of a helpless sinner. New and big, | felt like | could fix almost anything. Up bubbled an aggressive speech to the Dean about his refusal of our recent request for an increase in departmental research space. As for the National Institute of Health’s recent return of one of our grant proposals, it was now clear that the reviewers were wrong. | would resubmit but this time ask for twice the amount of money. | rehearsed a new list of necessary and routine laboratory chores 43 HOUSE_OVERSIGHT_013543

for my most rebellious post-doctoral student. | would tell my teen-age son that he must wait another year for his own car. | felt generally intolerant. In an article in Runners World, | labeled my run’s first global brain state transition, the first second wind. |t energized me with the cool firmness but ready-to- be angry righteousness of modern religious orthodoxy: Orthodox Jews gunning down Hamas terrorists as retribution for bus bombing children which was itself a retribution; Muslim suicide bombing as vengeance for cultural contamination; Catholic Bishops refusing the Eucharist to pro-choice politicians; Charismatic Christians gay bashing defense of the sanctity of marriage; Mohammed’s early Sufi- like poetry of love turning into territorial aggression and Jew killing in his later years. Once in while, unpredictably, past the first hour of running and after the first second wind, a fatigue easing second burst of energy followed the second stage of exhaustion. | called this running-induced, second global brain state transition to a softer loving energy, the second second wind. Colors became intense, clouds breathed and my body lightened. Running once again became easy. | was flooded with empathic and generous thoughts. | understood that the Dean was faced with too many space demands to satisfy; the grant reviewers’ criticisms of the budget were meant to be constructive. | recalled that strong minded, rebellious post- doctoral students often made the most creative contributions to science. | realized that my son’s urgent desire for his own car was a proposal in the direction of the independence that would be required of him the following year when he was going to be hundreds of miles away at a university. Filled with benign optimism, | felt the compassionate perspective afforded those with energy but without envy, anger or fear. William James, in Varieties of Religious Experience, A.C. Underwood’s book, Conversion, Christian and Non-Christian and Gobi Krishna’s The Awakening of the Kundalini, among many others before and since, describe the sudden appearance of long lasting states of optimistic energy and loving empathy that can emerge after long episodes of suffering, especially following periods of privation of spiritual meaning and the loss of a previously strong faith. These episodes are painfully chronicled by St. John of the Cross in his Dark Night of the Soul. 44 HOUSE_OVERSIGHT_013544

In the long distance running model of spiritual transformation, the first energy appears suddenly in the middle of painful fatigue and feels like a vigorous implementation of Halachic commands or Canon Law. The second burst of energy emerges from readiness for resignation and ends in humane comprehension and empathy. In some Christian monastic practice, a similar transition is represented in the ritual of Tenebrae (or Darkness). Fifteen lit, unbleached candles are extinguished, one by one over the night, while reading the Psalms. The practice is said to represent the desertion of Christ by his disciples, as the church grows darker over the night. After the singing of the Benedictus, the one remaining light is quenched, plunging the church into total darkness. In Myth and Ritual in Christianity, Alan Watts suggests that the loss of the last light of Tenebrae induces the realization that “| am nothing.” This reduction in egocentrism, along with a dark- piercing alertness is said to facilitate an invasion by a loving God that precipitates the fasting, sleep deprived and praying petitioners into long lasting ecstatic states. These uses of energy and its attendant characteristics are not physically specifiable but rather hermeneutic of a force. It is both a potential and a realization, observed and inferred. It is the “energy stuff’ of Freud’s /ibido, Wilhelm Reich’s orgone energy, Pavlov’s drive, Rudolph Steiner etheric formative force, the arousal and attention of brain wave and consciousness research, the Ch’ of Chinese medicine, the Hindu divine energy of Shakti, the Hebraic ruach, the Cabalist’s Yesod, the Sufis Baraka, the Christian Holy Spirit, the Yogic breath energy, prana, Mesmer’s animal magnetism, Galvani’s life force, Goethe’s Gestaltung, Madam Blavatsky’s astral light, Georg Groddeck’s it, Henri Bergson’s elan vitale, Schroedingers entropy, Abraham Maslow, Ruth Benedict and Buckminister Fuller's synergy, Bertalanffy’s anamorphosis, Colin Wilson’s x factor and George De la Warr’s biomagnetism. Of course, by nationality, culture and field of study, there are many more examples, each locally defined by its particular context and haunting with its promise of universality. Energy in the context of mathematical physics is intuitional, abstract and relational. It is not created or destroyed, but rather transformed. Consistent with his deceptively simple style of physical intuition training of the young, Feynman’s 45 HOUSE_OVERSIGHT_013545

discussion of thermodynamic energy and its conservation in Lectures in Physics begins with the premise that it is a numerical quantity that does not change when one or many alterations in the system occurs. His heurism for energy and its conservation involves the premise that Dennis the Menace has 28 indivisible blocks, a number which his parents find constant at the end of every day of play. If one day a count yielded 27, an investigation would reveal that a block could be found elsewhere, say under the rug. If at the end of the day, the count was 29, the extra one had to come from somewhere else, perhaps Dennis’s playmate Bruce. If Dennis locked some of his blocks in the toy box and threw some into a bathtub of dirty water and (1) A block weighed three ounces; (2) The box alone weighed 16 ounces; and (3) Each block raised the water level of 6 inches by one fourth of an inch, then this metaphoric energy relation can be expressed: \4 (weight of box)-16 ounces (height of water)-six inches (blocks seen = constant (28) 3 ounces 1/4 inch Feynman notes that this representation of an energy relation, computed as a number of blocks, will always remain the same. If there were no blocks in sight, and one used this energy conservation relation with blocks as units of energy, we find no blocks as such in the expression at all. The abstract and formal idea of energy in physics first arose in mechanics and was generalized to electrostatics and electrodynamics. If one idealizes these systems, eliminating real world factors such as friction, temperature gradients, temperature dependence of the properties of materials, viscosity, hysteresis and other nonlinear behavior, then the energy conservation law says that in an isolated and interacting set of systems, the sum of the energies of the several systems remains constant. If, on the other hand, a system interacts with its surroundings, not isolated and interacting, then the increase in the energy of the index system is equal to the work done on the system by its surrounds. Like pre- Enron bookkeeping of corporate cash flow and balancing ones personal checking account, energy, like money, does not disappear; it is only changed in expression. As in the context of currency equivalent value, energy can represent a very general quantity applicable to a wide array of specific objects and activities. The results of 46 HOUSE_OVERSIGHT_013546

the early studies by Professor Seymore Kety of Harvard and Dr. Harold Himwich of the Thudicum Laboratory in Galesberg, Illinois, using measures of whole brain oxygen and glucose utilization as indices of energy generation and utilization by the brain, surprised many of us. They indicated that energy use by the whole brain was relatively constant when states of relaxed awakeness, mathematical cognition and deep sleep were compared. Of course, modern studies have indicated that relative regional brain energy utilization is state dependent and may vary quite widely. More spiritual aspects of energies and their transformations were made clearer during several month visits to Baba Muktananda’s, now Gurumayi Chidvilasananda’s, Sidha Yoga Ashrams. Baba Muktananda loved and worshipped his Hindu Guru, Bhagawan Nityananda. Baba had been a restlessly wandering, guru-hunting, young man. Nityananda said he had “wheels for feet.” After many years of devoted meditation, chanting and service, sadhanna, all the while being prohibited from eating mangos, his favorite food, his passive, taciturn, ecstatic guru, Nityananda, presented voluble, energetic, joyful Baba with the guru’s rather aromatic and worn sandals. This symbolically acknowledged Baba’s successful absorption of the guru’s transforming spiritual energy, shaktipat, the power of his enlightenment. At Nityananda death, Baba, using world tours, spiritual fellowship meetings, satsangs (public conversations) and spiritual training sessions called “intensives’”, organized Ashrams in West Coast sites such as Oakland and Venice, and on the East Coast, in South Fallsburg, New York, buying several old residence hotels in the Borscht Belt. Baba was introduced to America by one of his first advance men, Be Here Now Baba Ram Das, Timothy Leary’s co-investigator in the Harvard Student LSD project when his name was Richard Alpert. ESTs Werner Erhard was another of Baba’s advance men. Baba discipled and disciplined a sister and brother who, when 18 and 11 respectively, were sent to live in his Ashram in Ganeshpuri India by their parents. The girl was known as Malti when she served as a translator for Baba and Gurumayi Chidvilasananda after receiving the energy of her enlightenment. The younger brother was given the name of Baba’s guru, Nityananda. When Baba took 47 HOUSE_OVERSIGHT_013547

a guru’s ecstatic death, Samadhi, both Gurumayi and young Nityananda became co-gurus. Following three years of the usual covert power struggles of succession in organizations, Gurumayi took over the guru lineage of Siddha Yoga. Her lively brother’s worldly preoccupations with jazz drumming and confessions of promiscuity led to his giving up of the orange robe of the denunciate, sanyasi, for the blue robe of worldliness, exchanging one kind of energy for another. Brad Gooch who visited Gurumayi’s Ashram in Ganeshpuri, India, wrote in his recent book, Godta/k, that she looks like a “synthesis of Indira Gandhi and Bianca Jagger.” In what reads like a Hunter Thompson episode in an unwritten book called Fear and Loathing Along the Guru Trail, Godtalk’s explication of Siddha Yoga was dominated by yellow journalistic rumors such as the one about Baba’s use of a gynecologist’s table with stirrups for non-ejaculatory Tantric practice with some female followers. This unconfirmed claim remains, as Gooch says, in the realm of “...he said, she said.” Gooch’s exploration almost ignores the deeper meanings of Kashmir Shavism, Buddhism and Kundalini Yoga that compose the philosophical foundations of Siddha Yoga. The importance of knowing, loving and becoming one with the God within trivializes all but ungenerous or hurtful interpersonal behavior. Even the tougher version of the Ten Commandments in Leviticus 19 would not necessarily disagree. When a Los Angeles Times reporter tried to chide Baba about being driven about in his “worldly” Mercedes sedan, he explained that a very wealthy Indian merchant had given it to him and “...| have to put my behind somewhere.” Similarly, why would Gooch’s account of Baba’s Tantric practice, even if true, ruin the imago of him in my mind unless | had already surrendered to the pantheon of good and evil absolutes of Judeo-Christian taboo? My knowledge of these non-materialistic meanings of apparent materialism began with one of the favorite finds of Baba’s youthful days of guru hunting: Zipruanna, who, wearing only a loincloth spent all day, every day, on a stool in the middle of a garbage dump. Remarkable changes occurred in people who spent time there in his presence. Baba said the identity of guru was established by the results experienced by those that spent time in his presence. It could not be defined by the physical features or ritual conduct of the 48 HOUSE_OVERSIGHT_013548

interaction. People become spiritually energized and change in Zipruanna’s smelly, garbage-filled presence. | keep a picture of him on my desk. Gooch, in his implicitly and superficially righteous preoccupation with what he considered disenfranchising human vulnerability, recalls how the medieval church used the difficult to impossible vow of chastity for political control of their priesthood. He seemed to have missed Baba’s lessons about the remarkably simple sounding practices for mobilizing the energy of the God-receptive state. Once in this new state, the rest of the metaphysical work almost takes care of itself. |, like many others, adopted Baba’s mantra, Om Namah Shivaya, “| worship the God within me (and you)” that he was given by his guru. The inner chant of this mantra brings me to an internal quiet in which things become clearer. Meditation, chanting and service to the guru was motivated by his promise that my egoistic concerns ranging from the number of publications on my curriculum vitae, to the size and adroitness of my penis, would disappear autonomously in the Baba state of bliss. This sounds very much like the role of the transition to an “active intellect.” in Abraham Abulafia’s 13" Century Commentary on the Secrets. Arduous study of the spiritually dense writings of Sri Aurobindo during the days with Professor Spiegelberg at Stanford gave me a peak into the simple but difficult to execute idea of “simply” becoming the transcendently comprehending state of existence-consciousness-bliss. Whereas Baba would occasionally lapse into terse Sanskrit verse and its multiplicity of potential meanings, Gurumayi keeps things simple. Sitting silently and immobile at satsang for hours, she radiates transformational energy, shakti, that makes ruminations about human affairs seem unimportant. The work is about getting the self concerned head noise of ones preoccupations sufficiently out of the way to allow the discovery of the God who has been waiting patiently within. A fellow ashramite gave me a photograph of my first audience with Gurymayi. It showed me on my knees in front of her. She appears to be dismissing me with a baleful, almost disdainful look as my introducer, gesturing broadly, was, unasked, reciting a list of my professional bona fides. The picture caught her waving me off with a long, peacock-feathered stick. Obviously unimpressed, she is sending me back to my all night, every night, tent cleaning labors at the Ashram. Rich Indian 49 HOUSE_OVERSIGHT_013549

businessmen, whose large donations were a major source of support of the Ashrams, faired little better. They seldom received a personal audience or favorable seating at Darshan, the evening public time of question and answers with the guru. In contrast with the relatively easy public availability, mischievous play, provocative humor and worldly sophistication of Baba, the ambience of Gurumayi is more private, simple, serious and subtle. It is as powerful, but in another way. In response to Gurumayi’s ascension to Siddha Yoga’s singular guru, | imagined hearing Baba saying that God energy was at least androgynous, if the dimension of sexual identity was relevant at all. Baba taught that divine energy, by necessity, is expressed through a wide variety of particular personalities and cultures and should not be confused with the details of its manifestations. This included the sexual identity of the chosen Vehicle. Guramayi’s central theme, as | understand it, concerns the simple, quiet and pervasive powers of love and faith. Some say Baba took the path, marga, of selfless action, karma-marga, whereas Gurumayi took the bhakti-marga, the road of loving devotion and faith. The third marga is jnana-marga, my inclination, is the road of intellectual study and knowledge. Aldous Huxley related the choice among these three categories of yoga practice, to the physical and personality types of William Sheldon’s 1954 Atlas of Man. Karma yoga corresponded to the mesomorphic body type and the assertive boldness, high energy, and interpersonal callousness of the somatotonic personality. Bhakti Yoga was the characteristic choice of endomorphic body types with the viscerotonic personality traits of sociability, good will, tolerance and love. Huxley associated Jnana Yoga with ectomorphic body type and the cerebrotonic characteristic of shyness, sensitivity and intellectuality. My summers with Baba at his temporary Ashram in Venice, California and the permanent American Ashram in South Fallsburg, New York, were spent in daily, very early morning, chanting of the gurugita after most of the night spent taking down, cleaning and putting up large tarpaulin meeting tents. | was assigned this simple, arduously manual, all night work after being interviewed and found out to be a professor and chairperson of a medical school department. Baba instructed his assignment committee that many if not all professorial egos would benefit from what 50 HOUSE_OVERSIGHT_013550

Andrew Carnegie famously called the dignity of real work. Spicy one dish vegetarian meals, twice a day meditation and brief stolen naps consumed the rest of the day. | found myself meditating for longer and longer times, chasing the promised Blue Pearl that Baba said appeared behind the eyes near the supreme meditative end point. Beside care with the _ titration of meditation-induced interpersonal disconnection, detachment with love is the desired end point of most Hindu and Buddhist meditative practice, another set of “side effects” of the energy arising early in the course of too much meditation is called kriyas, spontaneous episodes of involuntary behaviors and postures of the body such as unprovoked chanting and writhing and stereotyped hand positions called mudras. Baba told us one of his kriyas took the form of spontaneous erections that occurred during his first experiences with deep meditative states. | recall a woman physician and fellow ashramite in Los Angeles telling me that her panties often got so soaked during meditation that she worried about being stuck to her cushion. Beyond these initial somatic overflows of Divine Energy, shakti, emerges a vision of the Blue Pear, bindu, Baba’s “gift from the Goddess Kundalini.” As he entered this stage, he said that his mind filled with “joyous contentment.” Jewish mysticism of the 1300’s acknowledged the neighborhood relations of Eros and the Sacred. More formal and scientific uses of the word, energy, like all objects of thought embeddable in a mathematical context, are abstract and relational. In his book, Mathematics-The Music of Reason, Jean Dieudonne’ treats mathematical objects as objects of thought. Dieudonne”s book documents the 19" Century transition from concrete, visualizable, classical mathematics to abstract, nonvisualizable relational ideas. This conceptual transition to abstract, relational thought objects that are no longer representable by pictures or accessible to our senses of mathematics and physics is yet to reach the concrete DNA-causal religionists of modern molecular biology. In 20" Century mathematics, Dieudonne’ observes that “...the primary role in theory is played by the relations between mathematical objects concerned rather than the nature of the objects themselves...these relations are often the same for objects which appear to be very different and therefore they must 51 HOUSE_OVERSIGHT_013551

be expressed in ways which do not take these appearances into account...and can be specialized at will... DNA sequences are, as MIT molecular biologist, Eric Lander observed, nothing more than an elementary “...list of parts...” In fact, since about 1% of the nucleotides are relevant to functional genes, one might say that the important members of this list of parts are distributed very thinly among many more apparently unimportant ones. The next frontier will certainly involve § an understanding of the dynamics of the interactions among elemental parts and in more abstract laws about molecular biological relations; a focus on the dynamics, not the structural parts, that regulate and control their expression. * * * | made a pilgrimage to spend eighteen months within Rene’ Thom’s penumbra, living among mathematicians in his “ashram” in Bures sur Y’vette, France. Thom was one of the founders of the /nstitute des Hautes D’Etudes, IHES, Institute for Advanced Scientific Studies, created to stanch the flow of high-level scientific talent away from France after the Second World War. It is in Bures sur Y’vette, deep in a green forested valley, 50 or so miles South of Paris, in a building packed with small, thin walled, big windows-on-the-woods offices. Each office contained a single hard chair, an old office desk, two walls of blackboards and a box of white only chalk. The use of colored chalk was felt to be without mathematical rigor because its use substitutes colors as dimensional descriptors for more demanding abstract and formal representations. Color was cheating. Meditation in this ashram was practiced by staring, pacing, scribbling, and humming, mumbling, belching and farting through the Institute’s thin office walls. The building, though almost completely occupied, was otherwise silent. The Institute was populated by such world-class mathematicians and theoretical physicists that once inside that building, | felt so intimidated that | almost never spoke above a whisper. Listening to excellent William Thurston’s casual use of a tiled bathroom floor to motivate a unique partition of a topological space, | was attacked by the awe of an early morning visit to an almost empty Notre Dame Cathedral in Paris or standing in front of Michelangelo’s radiant marble statue of Mary and Jesus the Infant in the Vatican. 52 HOUSE_OVERSIGHT_013552

Though the environment was one of tranquil academic scholarship, | lived charged with anticipated performance anxiety about the seminars on the brain as a dynamical system | was scheduled to present to these (| feared) ready-to-be- disdainful, prize-winning, pure mathematicians and theoretical physicists. My dorm-style sleeping room at /HES was, in winter, painfully cold and drafty; the narrow iron bed’s thin mattress contained lumps of persistently disturbing dreams, the small scratched table for work shim-irreparably wobbled. A faded poster of Van Gogh’s garden was tacked crookedly on the door facing the toilet in the dank, dimly lit small bathroom. A dwelling for distracted young mathematicians. A retired but still famous Parisian chef cooked many course, elegant meals every afternoon. The food was accompanied by so many liters of unlabeled red wine and peer pressure to be French and socially drink it that it became a choice between dulled, blunted,. sleepy post-prandial afternoons or living on bread, many cheeses, apples and Perrier water, alone in my room. | chose the latter. Thom’s gifts to us _ theoretically oriented non-mathematicians were diagrammatic, easy-to-visualize pictures that allow the intuitive capture of counter- intuitive discontinuities in functions. How we might imagine that a smooth and continuous change in a cause of something can lead to a big, discontinuous change in the results. His system of topological (shape not size) diagrams was useful when considering up to four causal variables and one to two dependent variables that described how things behaved. For an important real life example, in modern clinical pharmacology, the smooth dose-response curve consistent with the physician’s intuition that if a little drug didn’t work, a little more may do so, should become an up and down search for the dose-region for the desired effect which may involve a lower amount than a previously ineffective drug dose. The therapeutic effect may occur in the middle of a narrow dose range with too much or no effect occurring out of this span. In many physical systems, sudden and global transitions in state, from incoherent light rays to coherent lasing and from laminar flow of fluids to turbulence, emerge unexpectedly when causal parameter are moved into what some call the critical region of the values of control parameters. Outside this region, cause and result 53 HOUSE_OVERSIGHT_013553

were behaving linearly and smoothly whereas within this region we observe global and dramatic changes via a forced discontinuity in what Thom called a catastrophe and others use related words such as bifurcation or phase transition. The transitions from painful fatigue to running rage and then to ecstatic transcendence feels like the gifts from two kinds of Gods, the first, bearing the righteous lawfulness of the Old Testament, the second bringing the empathic forgiveness of the New Testament Jesus. Catastrophe and bifurcation theories predict and keep track of these transitions using mathematically describable changes in global characteristics of the “motion” using technical descriptors such as eigenvalues, germs and jets. Thom taught me my first catastrophe, called the cusp, in words during our late afternoon walks along a shadowed green wooded path on the grounds of the Institute des Hautes Etudes, outside of Paris. My homework consisted of trying to visualize his verbal descriptions. It was not until weeks later that he drew the geometric object being discussed on the blackboard. With eyes twinkling and in his provocatively playful style, he said, “Imagine an empty rectangular box with the front edge of its roof buckled into an *S’ and the back edge, an unfolded, left-to-right gradually rising simple smooth curve. If one moves the causal force from low to high, from left to right along the back of the box, the changing effect (represented by height) would be smooth; moving from left to right in the front encounters a sudden drop off at the S shaped buckling, a discontinuity in roof height indicating a discontinuity in effect. The energy equivalent height of the roof graphically indicates the amount of result. The roof is the manifold upon which the result of causal change is portrayed. The two dimensional floor of the box represents a graph of the two causal parameters, the increasing amount of normal factor going left to right along the °x’ dimension, the increasing amount of splitting factor (taking one from the back to the front to the region of the buckling) going back to front along the ‘y’ dimension.” He gave me some examples of systems that showed cataclysmic changes in effect from smooth changes of normal and splitting factors. About the onset of a war: “At the back of the top surface of the box, the manifold, the normal factor increasing from left to right is the amount of the perceived threat. The splitting factor 54 HOUSE_OVERSIGHT_013554

decreasing from front to back is the cost (and ability to pay) for war. Without the financial capacity to make war, threat goes from left to right smoothly at the back of the box as tension gradually increases without the onset of armed conflict. When effective fighting capacity is cheap and/or already well funded, the country well armed, the increases in threat go from left to right at the front edge of the box and encounter the cliff of catastrophe and war is declared. Cost of, or ability to wage war varies from the front to back, and serves as the splitting factor. Considering prison riots, social tension is the normal factor and alienation (degree of identification with prison authority) is the splitting factor.” Using factial expressions of dogs sketched by the Konrad Lorenz, Christopher Zeeman then of Warwick Mathematics Institute in England, considered countenances reflecting increasing rage as the normal factor, the amount of fear was the splitting factor. Increasing rage at high fear increased smoothly at the back of the box; at low fear, increasing rage falls off the cliff to an animal attack at the front of the box.” He paced as he talked, occasionally looking up to see if | was following him. He continued, “A light above the box casts a shadow from the roof to the floor, outlining the gradually widening fold created by the transition from the smoothly rising back of the roof to its ‘S-shaped’ front. This triangle on the x-y causal floor is the region in which the discontinuity in the result surface roof results and is called the bifurcation set. An increasing amount of the causal ‘normal factor’ is represented from left to right along the *x’ dimension, the results of which change smoothly at the back of the roof but encounter a discontinuous jump up or fall down crossing the inaccessible crevice in the °S’ fold at the front of the roof. Again, the triangular shadow on the floor made by the fold indicates the parameter region in which discontinuous changes in the result surface occur. The reason the parameter that determines the front to back location of the left to right movement of the ‘normal factor’ is called the ‘splitting factor’ becomes obvious. Its value determines whether the results induced by increasing amounts of ‘normal factor’ will be smoothly changing or generate a discontinuous jump. The entire visualizable object is called a cusp catastrophe and it along with higher dimensional parameter region-inspired shapes such as the 55 HOUSE_OVERSIGHT_013555

swallowtail and butterfly buy back the smooth DE deterministic intuition lost with discontinuous changes in results.” He grinned mischievously as he asked, “Can you see it?” Thom’s catastrophes serve as accessible and powerful theoretical settings for the use of energy as a generalizable, one dimensional, dependent, resulting effect, influenced by one or several, sometimes conflicting, independent, causal, variables. For more examples: the weight of a ship (smaller to greater, left to right, along the x, normal dimension) and the position of center of gravity (smaller to greater, front to back, along the y splitting dimension) are causal with a jump in roof- height energy from stability to capsizing, a discontinuity emerging from initially smooth changes in stability. As above, gradually increasing tension (the left to right normal factor) and alienation (the back to front (splitting factor) in inmates generate a sudden increment in energy, from subtlety increasing tension in relative quiet to the sudden outbreak in a riot in the prison population. Embryological notochord somitogenesis, (that which become the vertebrate of the spinal column) has a smooth (left to right) causal influence that Chris Zeeman named a normal factor. It is the smooth growth of the material wave of mesodermal (to become muscle, connective tissue and bone) tissue. Zeeman called the front to back dimensional gradient of influence, the secondary wave of adhesiveness, the splitting factor. The value of this secondary wave co-determined a critical-valued interaction between these causal parameters leading to a discontinuous change in the “energy” equivalent continuity of developmental growth and vertebral column segmentation. A little more technically: Thom’s basic mathematical contributions were in differential topology and analysis with particular emphasis on what is called structural stability of surfaces representing and supporting actions called manifolds. For example, in a graph of a function, say F(x), such that a change in cause x determines what happens to the result y= F(x), the stability question involves what happens when one perturbs F(x) with a littles, i.e. 6 + F(x). Do the topological properties of the surface representing the potential range of actions of the system (such as nearness of an originally close point set, continuity and connectedness of the surface, its dimensionality, its compactness as a generalization of finiteness) 56 HOUSE_OVERSIGHT_013556

remain the same after perturbation? Note that the inter-data point metric distances are not considered. If they do, the two dynamical objects being compared are topologically equivalent. The test of this equivalence requires the mapping one set onto the other with, at most, smooth distortions of either or both surfaces. In the context of catastrophe-related bifurcation theory, if a 5 converts a steady valued fixed point to an oscillating cycle on a manifold of potential actions, also called a state space, then the fixed point system was not structurally stable. In phase space, this is seen as a change-in-causal-parameter induced transformation of a dot to a circle. If the one frequency circle is perturbed to a manifold of the system’s actions consisting of two independent frequencies, the circle takes the topological form of the crust of a doughnut, one frequency graphed spiral winding around the doughnut, the other winding along the doughnut around its orifice, the circle is not structurally stable. If 5 distorts the frequency-amplitude relations on a surface such that the manifold of possible actions is distorted from a doughnut to a tea cup, both topological manifolds being one holed surfaces and therefore topologically equivalent, the system is structurally stable. Perturbed systems that maintain the sequence of points in time in sequential order (though the distances between the points may be different), are generally structurally stable. The seductive possibility, one which Thom realized so successfully, was that in the language of distance-independent differential topological forms, there would exist a small, finite set of shapes categorically describing the causes and result parameter spaces from which, even without specific quantities, universal qualitative (including discontinuous) behavior could be described and sometimes predicted. A formal yet general categorical system within which a small set of universal discontinuous changes in global qualities could be rationalized seemed seductively applicable to the enlightenment transitions, spiritual transformations, appearing suddenly after months and years of disciplined spiritual practice. The Platonic view is that the universal forms of discontinuous change existed before they could be about anything specific, before the universe was born. In this era of nonlinear dynamics and dynamical system, common dynamical scenarios give accounts of smooth changes in causes leading to discontinuous 57 HOUSE_OVERSIGHT_013557

changes in results. The Nobel Prize winning solid-state physicist, Phillip Anderson, in a short but memorable piece in Science in the 1970’s said it tersely, “More is different.” This general, qualitative mathematical theory of discontinuous change models nicely the sudden delivery of the first and second second winds from gradually and continuously increasing running distances as well as the abrupt transmission of the guru’s “energy”, shaktipat, from smoothly increasing amounts of chanting, meditation, guru service and Baba love. Gradually changing forces leading to sudden changes in an energy-equivalent result are found in most rigorous form in Rene’ Thom’s singularity-bifurcation-catastrophe theory applied to rational mechanics and geometric optics. Here the existence of already solvable computational formalisms makes this more qualitative approach superfluous. On the other hand, the power of this both basic and applied mathematical orientation and method lies in its approach to the qualitative understanding of variously induced global and sudden changes in an energy-equivalent observable in biological, psychological, spiritual and social systems, fields of study in which little abstract and formal lawfulness presently exists. Oxford’s Chris Zeeman’s more accessible applications of Thom’s deeper, more generally ramifying, almost mystical (due to their apparent wide generality) results, include approaches to real world problems such those above as well as the sudden change in excitable membrane potential accompanying the generation of the heart beat and neuronal discharge; mechanisms of opinion change, stock market crashes and, as noted above, the social science of riots. Whereas Thom’s On Structural Stability and Morphogenesis can be said to be scriptural, Zeeman’s Selected Papers, 1972-1977 constitute the Book of Common Prayer of this church. To review and place catastrophe and bifurcation theories in the context of the differential equations of mathematical physics and biology, causal determinism implied by differential equations conventionally requires continuity and smoothness in behavior to be credible. Our intuitions as well as the formal conditions for the generic differential equations of mathematics and physics imply that smoothly increasing amounts of cause lead to smoothly increasing results and yield at least local predictability: a little more leads to a little more, a little less leads to a little less. 58 HOUSE_OVERSIGHT_013558

This smoothness-dependent intuition of determinism breaks down in nonlinear equations as well as in a wide variety of the machines of experimental physics, from the sudden coherent lasing of previously incoherent light to the vortices and turbulence in suitably bounded rotating or flowing fluid. It took me a while for these topological still shots and movies of the head to become real. Nevertheless, the enrichment of intuition was well worth it. Of course one could smoothly increase the normal factor weight of a ship until it gradually sank, but if one moved the center of gravity splitting factor to an eccentric position in the ship in the parameter region of the bifurcation set, a sudden global capsize before weight-induced gradual sinking made sense. | could see it. Indeed, increasing normal factor tension in a prison population that was identified, not alienated, from the officials and mores of the penal institution, would increase social symptomotology gradually. However, increasing the splitting factor of social and institutional alienation results in the cataclysmic change of a riot with increasing tension. | could see it. Do we need to know the causal equations to anticipate instability and discontinuity in our lives? Zeeman making Thom’s thoughts accessible to us plain mortals said no. He suggested that we could use several diagnostic phenomenological signs to make a good guess about whether we are near or within the bifurcation set. Depending upon the route that the causal variables take through the shadow of the bifurcation set, we may see very large fluctuations in our observable. The Dow or S&P stock indices in the neighborhood of a sudden large change is often presaged, sometimes for weeks, by a marked increase in volatility, fluctuations between extreme values. Theorists call the statistical properties of a time series of values behaving this way anomalous variance. For several months, | did psychotherapy with a genuinely spiritual Catholic priest who only some Sundays served the Eucharist, the corporal presence of our Lord at Communion, wearing no trousers or underpants beneath his robes. A sudden change in a stock index in response to the “shock” of a terrorist attack takes much /onger to settle down if a cataclysmically bigger change is in the neighborhood. This extension of the system’s usual relaxation time is sometimes called critical slowing. In the bifurcation 59 HOUSE_OVERSIGHT_013559

regime of a schizophrenic break down, critical slowing can be both global and literal as the patient freezes in catatonic postures. In the neighborhood of the bifurcation set, big jumps in the stock index, up or down, are possible under almost the same surrounding conditions. This stock analyst-humbling phenomenon is called bimodality. Jimmy Swaggert’s Saturdays were often spent watching the show at naked dance parlors and buying videos at the pornography shops of Metairie Highway near Schwegmann’s Grocery outside New Orleans. Sundays found him on national television engaged with infectiously real, transcendent experiences in the public arena of the pulpit. The ecstatic congregation was deeply moved by his eloquent and tearful sermons about sin and salvation. Counter to most suspicions, this is less conscious fakery than the genuinely felt alternating states intrinsic to the bimodality in neighborhoods of spiritually unstable, born again transitions. Similarly, beginning with nearly the same initial values near the boundary of the bifurcation set, very similar motions lead to dramatically different results. This counter-intuitive behavior has been called divergence. At UCLA’s Neuropsychiatric Institute, | interviewed a pair of lively teenage, genetically identical male twins raised by a loving family in Los Angeles’s Valley. One was president of his high school class, a Sunday school nursery school volunteer and a Saturday soup server to the poor. The other twin sold pot and cocaine to support his habit. Deep and potentially dark mysteries live in these spiritual bifurcation sets. They leave us pondering child sexual abuse by deeply religious clergy and the massacre by mass suicide of a New Christian congregation by James Jones. We wonder why it is that fundamentalists (Jewish, Christian and Muslim) have the most ecstatic and direct validating experiences of God and do the most shooting and bombing of other people. In Burt Lancaster's portrayal of bifurcation set dweller, Elmer Gantry, charismatic believer and exploitative psychopath, were simultaneous and both credibly real. Another feature of the occupancy of this bifurcation region in control space is that the values producing a sudden jump that occur passing through going one way along the “normal” dimension usually jump back much further along when moving 60 HOUSE_OVERSIGHT_013560

the other way. Theorists call this characteristic sign of bifurcation land, hysteresis. It is generally known that sudden healing changes of the first born again experience can arrive magically fast whereas a run at it a second time, another born again state after the loss of the first one, comes, if at all, with much more effort and difficulty. Members of Alcoholic’s Anonymous know that getting on the AA wagon the first time may be quick, joyful and easy. Getting back on this wagon after a fall is much more painfully slow and demanding, analogous to the Carmalite monk; St. John’s lost faith engendered suffering of the Dark Night of the Soul. Viewing the instabilities and extremes near the boundary of a bifurcation brings inquiries and advice about why a rational compromise, some form of disciplined moderation, would not be more desirable. It turns out that in this parameter regime, the in-between state is intrinsically inaccessible. The pocket in the S shaped fold of the upper manifold cannot be attained, at least for very long, by varying the values of the two parameters. However, if one increases the number of controls, it might be possible to stabilize a small island in a parametric sea of instabilities. In an application of this strategy, Smith College and Harvard Professors James Callahan and Jerome Sashin used a geometric representation of the difficult to stabilize region of normal weight on a double cusp manifold representing the behaviors of patients with eating disorders with both anorexia nervosa and bulimia. They varied five controls to stabilize a very small result area representing normal eating by varying the control values for ability to verbalize feelings, to imagine solutions, to defend against anxiety with unconscious forgetting called repression, to make contact with realistic rationality and to modulate feelings with say exercise, meditative practice or psychopharmaceuticals. My experiences with the so-called borderline personality, with the tendency toward sudden and global personality change, from Sunday school teacher to Harlot in the space of a breath, has been both sexually exciting and personally ruinous for me in my life. | could feel the instabilities in these dwellers of the bifurcation pockets and my heart raced at the promise of mutually unconsidered impulses, the blurring of orificial identities, the experiments with sexual roles and modes and the incipiency of collapse into regressive mud play. Most of all, | anticipated that their 61 HOUSE_OVERSIGHT_013561

screaming orgasms, potentiated by a natural inclination to bifurcate, would be so messianic as to carry me along to a transcendentally erotic new place. Unfortunately, paranoid rages, bursts of promiscuity and hopeless inconsistency of goals and efforts dominated the remainder of our living days. Further Readings for TRANSMOGRIFICATIONS OF ENERGIES Religions in Four Dimensions; Existential, Aesthetic, Historical, Comparative, Walter Kaufman, Reader’s Digest Press, 1976 Religious and Spiritual Groups in Modern America, Robert S. Ellwood, Prentice- Hall, Englewood Cliffs, 1973. The Evangelicals, What They Believe, Who They Are, Where They are Changing, David F. Wells and John D. Woodbridge, Abington Press, Nashville, 1975 A Nation of Believers, Martin Marty, Univ. Chicago Press, Chicago, 1976 Conversion: Christian and Non-Christian, Alfred C. Underwood, George Allen, Unwin Ltd., London, 1925 Eros and the Sacred, Paul Avis, SPCK, London, 1989 Mukteshwari, The Way of Muktananda, SYDA Foundation, Ganeshpuri, India, 1972 Godtalk, Travels in Spiritual America, Brad Gooch, Knopf, N.Y. 2002 The Beat of a Different Drum; The Life and Science of Richard Feynman, Jadish Mehra, Clarendon Press, Oxford, 1994 62 HOUSE_OVERSIGHT_013562

The Shape of Space, Jeffrey Weeks, Dekker, NY, 1985 The Topological Picture Book, George K. Francis, Springer-Verlag, NY 1988 Mathematical Models of Morphogenesis, Rene Thom, Wiley, NY 1983 Catastrophe Theory, Selected Papers, 1972-1977, Christopher Zeeman, Addison- Wesley Reading, MA 1977 63 HOUSE_OVERSIGHT_013563

CHAPTER 4: SENSUAL IN-BETWEEN ENTROPIES Since the early teens, I’ve been beguiled by girls and women that have what might be regarded as exquisite sensibility, perhaps more precisely, exquisite self sensibility. These inhabitants of the near transformational neighborhoods of bifurcation sets, are grandly responsive receivers of emotionally significant information arising from their insides and the world. They are the canaries in the deep mines of human experience. Not the usual one lively-eye, one sober-eye, binocular difference of most of us, both their eyes sparkle, their feeling antennae await a happening and each is regarded as new. | spot these brains in a crowd within minutes and am compulsively drawn to know them better, to become part of them, to vicariously experience and serve them. They seem to have little inhibitory control of even weak sensory information on its way to their strong, global feelings. Near ecstasy and excruciating pain await. They feel their anticipations with their body, down to their painted toes. Their receptivity brings me lower abdominal warmth in remembrance. At sixteen in my Dad-purchased second hand Ford convertible, | was parked with my new girl friend on Sarasota’s Lido Beach, hearing and seeing dark shadows of the Gulf of Mexico’s waves hit white sand against the night sky. | took her flat party shoes off to message her feet. When | kissed her left foot and sucked gently on her toes, she gasped and became faint. She told me that a strong electric shock 64 HOUSE_OVERSIGHT_013564

had run up her back. The passionate licking and sucking of her musky, moist, pink labial lips brought what she said were explosions of pink and blue lights. She had several ecstatic multicolored crises in a row, sometimes without pause. She begged me to stop. | was as pleased as a sexually inexperienced young man in love could have possibly been. Bowled over by what seemed to be the uniquely sensual properties of her brain, | began to wonder if her sensitivity was more general when she asked me to keep the windows open or top down, even in the cool of a Florida January, because the exhaust smell in my car was suffocating, though | couldn’t smell it. The car had been checked and registered negative for abnormal fumes and leaks by Anderson Ford. She asked me never to wear any kind of after-shave lotion because it choked her. Jazz music on the car radio had to be played quietly. On-coming headlights gave her headaches. Her mother, sometimes desperate, called me for help during her daughters episodes of premenstrual emotionality and early menstrual discomfort. During these times, we would drive together for hours as she explained the many different colors of lower abdominal pain and how this particular kind yawned darkly before it cramped. It was more purple then any of the others. | tried to explain what | intuited but didn’t understand to her mother about the her gift of unfiltered information coming through her nerve endings, her ever readiness for surprise and her brain’s unwillingness or inability dampen or ignore what it didn’t like. She saw things in art, heard things in music that | only saw, and heard after her telling. She had tearful smiles listening to Debussy’s Afternoon of a Fawn. The flatted fifths of Charley Parker and the laconic riffs of Miles Davis made her anxious. Since then and for all these many years, the same sensually susceptible brains showed up in my life carrying a variety of woman’s names and | never lost my fascination for them. | learned that their heightened awareness extended to the spiritual realm with unusually strong metaphysical inclinations and readiness for transcendent experience. They seemed to live closer to the direct experience of God. Attending Assembly of God and other Pentecostal midweek service, | found that praying in tongues and dying in the Lord came as easily and dramatically to them as their orgasmic experiences. At the same time, distant bad news could 65 HOUSE_OVERSIGHT_013565

suddenly become immediate and loud in a litany of threatening thoughts that hooked and persisted through sleepless nights. They taught me to see genuinely the delicate beauty of flowers and to know in my stomach that some forms of sadness felt hollow like homesickness. In medical school | found that that many of them were the clinic patients, women and men, with unusual sensitivity to chemical odors, think Gulf War Syndrome, and fibromyalgia, which | heard as unusually sensitive awareness of normal sensory information about posture and position coming in from the bones and muscles of the body but experienced as pain. This background of odorific and somatic information is usually repressed from consciousness by the rest of us. Their medical histories contained detailed accounts about how each of their organs was feeling at the time, sensations that the textbooks say we are incapable of consciously knowing. Internists and psychiatrists often dismissed their accounts as signs of somatoform disorder, psychological conflicts expressed in the language of body feelings. In the psychophysiological laboratory, | learned these brains tended not to habituate. Each of a series of noises continued to elicit startle responses that could be picked up in brain wave recordings or in the running record of a psychophysiological, lie detector, machine. In psychoanalytic training, | learned that these brains remembered their dreams more richly than the rest of us and that treatment with over twice a week analytic sessions was potentially dangerous. The psychoanalytical situation-engendered fantasies and feelings could get too strong and exaggerated, too real. Professor Iris Bell of University of Arizona’s Alternative Medicine Research Program has, studying these brains, found slower reaction times, defects in divided attention psychological tasks, longer latencies to the first dream, and unusual patterns of odor reception called cacosmia or dysosmia. Using brain wave and cardiac interbeat interval data as markers, Bell reports the increase in the amount of alpha awake brain waves and decreases in cardiac interbeat interval variation associated with increasing sensitivity, rather than habituation, with repeated exposure to a variety of smells over time. In spite of these brains usually requiring what is known as high maintenance 66 HOUSE_OVERSIGHT_013566

in relationships, | continue to be erotically spellbound, in love with them in all their forms. Questions about how to think about these exquisitely sensitive women, Bell’s Syndrome exists but is rarer in men, continue to drive aspects of my scientific research. It has been variegated quest, which began with trying to find a general conceptual framework that would help my understanding of this unique capacity to be aware and process large amounts of internal and external information that escape the awareness of most of us. As one might guess, this search led to fundamental ideas about information and its inverse, the entropy indicating the amount of information transport capacity, with respect to their characterization, quantification and measurement. To get to the end from close to the beginning, we recall that it was Claude Shannon and his followers who both mathematically proved and experimentally verified that a receiver must have more entropy, less already fixed knowledge and more wondering, than the sending source, in order for the message to be sensitively and reliably received and encoded. Sensibility seems to have something to do with the readiness for information transmission afforded by the brain’s high entropy, minimal fixed information states, in its resting dynamics. Their remarkable receptivity derives from a baseline brain state like the formless emptiness of the t bodhisattva’s “...no form, no sound, no_ feelings, no _ perceptions, no n consciousness...” of transcendent Tibetan Buddhism as described in the Heart Sutra of The Dalai Lama. In Chinese Medicine, xu, meaning emptiness, contrasts with shi, the word for fullness, both of these complementary opposites having multiple specific meanings. Most metaphysically relevant is the characterization of xu as the emptiness of the deepest reality of being and the highest state of human spirituality. Like that aspect of Lao-Tsu’s ineffable Dao, The Way that is empty, xu indicates a mind devoid of desire, being lucid and serene. In the context of dynamical form, xu shares the structureless, non-imagery of maximal entropy systems and shi the lower dynamical entropy of fixations on form, desires and beliefs. Shigehisa Kuriyama’s The Expressiveness of the Body, elucidating historical and conceptual divergences of Greek and Chinese Medicine, notes that xu was the supreme end of self-cultivation 67 HOUSE_OVERSIGHT_013567

and the secret to vigor and longevity. “...to achieve fullness of life one had to abide in empty nothingness, xuwu.” In Lao-Tsu’s Tao-Te-Ching, “...the Way is gained by daily loss, loss upon loss until...by letting go, it all gets done...” William James, in The Principles of Psychology, tried to capture the subjective dynamics of the brain as an on-going preconscious stream of statistical wave processes. He envisioned autonomously increasing and decreasing coherence emerging spontaneously and from sensorial evoked thoughts via the confluence tt and disaggregation of statistical wave processes, “...wave crests and hollows...” that achieved temporary statistical stability by “...feelings of relation, consubstantial with our feelings or thoughts of the terms between which they (only temporarily) obtain.” In the more receptive, higher entropy brain systems, fleeting forms change without continuity, Jumping from one to another with “magical rapidity,” but being not already engaged, are available for use for self-organized structure evoked by new information. Without ordered, low entropy, preconceived ideational defects in the resting random brain field, the full attentional statistical machine is available to sensitively respond in self-organized, quasi-stable states of cognitive, conative and affective integration. They then disappear; this brain relaxes quickly, ready for new experience. This contrasts with those brains that are dominated by islands of order composed of personality fixations and rigid belief systems, low entropy defects, which interfere with sensorially responsive sel/f-organization. 68 HOUSE_OVERSIGHT_013568

As in most systems of authoritarian premises, precise definitions and what appears to be strict logical continuity, as in discussions of Torah among Orthodox Jews and Canon Law by Catholic bishops, classical equilibrium thermodynamic ideas that are borrowed for use out of the context of their origins, risk the calumny of their physicist practitioners. We have probably already earned more than a little distain from those quarters with our use of none-minimal or none-maximal but in- between entropies. This phrase cannot be found in the literature of physics or, as such, in the writings of communication and information theory. In the modern theory of nonlinear motion called dynamical systems, in-between entropies can be generated by chaotic systems that are non-uniform in their rates of separation of near by points and convergence of far-away points in dynamics that have been previously described as nonuniformly hyperbolic. The energies and their transformations that fuel and support karmic escape from the personality fixations of samsara and accession to unmanifest Divine Life can occur without the loss of the richness and multiplicity of apparent reality. Big internal changes without external sign can occur in the arrangements of the ineffable and mysterious formless silence within which we have associated with states of high, but not maximal, in-between entropy. For examples, the Indian Saint, Sri Aurobindo, in the early 20" Century, the Catholic metaphysical anthropologist, Teilhard de Chardin and currently American pandits (spiritual seekers with intellectual and academic inclinations) such as Ken Wilber, among many others over the millennia, direct us toward the goal of Nirvanically changeless emptiness without the properties of space or time. At the same time, we maintain an astute and effective yet distantiated appreciation for existential realities. The non-dual enlightenment of Integral Being or Yoga involves realizing emptiness through the world of form. There is a way of thinking about and even computing that “nothing within” and its changes. As John R. Pierce suggested in the 1981 revision of his book that made the theorems of the father of communication theory, Claude Shannon, so accessible, “_..If we want to understand information-related entropies, it is perhaps best to clear our minds of any (physical) ideas associated with the entropy of physics.” 69 HOUSE_OVERSIGHT_013569

Nonetheless, historical comments about what the classical thermodynamic term, entropy, is and is not about are in order. We recall that Richard Feynmann, in his well-known 1962 class notes, Lectures on Physics, said that the subject of thermodynamics is the study of relationships among the heat, energetic and organizational properties of materials, without knowing their internal structure. Historically, the relational formalisms of equilibrium thermodynamics emerged before our knowledge of the internal structure of matter. For examples, the pressure in an insulated container of gas is due to molecular bombardment of the container walls, which increases with heat or compression of its volume. Compression of its volume increases its temperature and expansion of its volume leads to cooling. Note that these relationships hold without specifying the constituents and the specifics of a particular gas or solid. In his lectures, Feynman’s intuitively accessible examples of reversible thermodynamic properties are reminiscent of his on camera performance at the Senatorial hearings about the Challenger disaster. Recall that he dropped an O-ring in a glass of iced water demonstrating cold-induced rigidification of the rubber ring, which he postulated to be the cause of the fuel leak and resulting explosion. In his Lectures, he said that if one holds a rubber band between ones lips as a crude thermometer, stretching a rubber band heats up the lips and relaxing it cools them. Working the same system in reverse, and equilibrium thermodynamic systems are classically reversible, we find that heating a rubber band makes it contract. These changes involve complicated alterations in the internal arrangements of the polymeric strands of rubber, their structural properties, the details of which, for the purpose of global thermodynamic characterization, need not be known. The relationships between physical state, energy and temperature in this material were predictable from thermodynamic laws even without specific knowledge of the complex internal structure and physical dynamics of rubber. Thermodynamic theory, which makes deep conceptual connections between quantitatively measurable primitives such as heat, hotness and work and the invisible in the form of derived ideas such as energy and entropy, yielded an 70 HOUSE_OVERSIGHT_013570

enormously rich and logically consistent intellectual framework from within which to characterize macroscopic behavior composed of unknown molecular mechanisms. Ideas about entropy grew out of William Thomson's (a.k.a Lord Kelvin) thermodynamic laws about energy conservation and its allowable transformations. Later Clausius decomposed the energy into that which was available for mechanical work, called work-content, and that which was not, called transformation content. He referred to the transformation content, a reflection of what changes in the internal order properties of the system that occurred as a concomitant of changes in energy and heat, as the entropy. Rudolph Clausius added the word entropy as a thermodynamic property to the conceptual armamentarium of theoretical physics in about 1865. This followed the earlier work of the French engineer, Nicolas Leonard Sadi Carnot, who was trying to develop a theoretical framework within which efficiencies in heat- generating engines might be understood. It implicated positive, > 0, changes, d, in entropy, S, with changes in time, ¢, .e. “ > 0, entropy is increasing in time, as a concomitant of the inevitable mechanical inefficiencies in an energy driven system. The resulting losses in the form of wasted energy show up as increases in molecular motion, which could be estimated from the increases in heat. Wasted energy dissipated as heat increases the amount of random motion and volume occupied by the surrounding molecules in physical processes involving heat, pressure, vaporization, condensation and work; all elements of that era’s dominant physical metaphor, the steam engine. The highly developed, multifaceted, often quite abstract formal characteristics of the inferred property, entropy, prevent glib definitions and generalizations. In the context of Kelvin-Clausius theory, the entropy of a closed system will remain the same if it is isolated from any matter or energy exchanges with the environment. If heating a system such that the change, d, in heat, Q, is positive, i.e. dQ > 0, it experiences a rearrangement in its microstructural motions, but the temperature is left unchanged. The (inferred) entropy, S, increases (i.e., dS > Q) as the ratio of change in added heat, dQ, over the unchanging, absolute 71 HOUSE_OVERSIGHT_013571

temperature, T. Thus, one definition of entropy change is dS = dQ/T. In classical contexts, dS is expressed in units of heat called Joules per degree of absolute temperature in units Kelvin, the temperature in Centigrade plus 273.16°. The best- known physical image involves the heat-energy transfer to and from heat baths called reservoirs as intermediate actions of the work of the heat driven engine executing what has come to be known as the Carnot Cycle. The same formulation emerges in this more concrete context: the heat, Q, transfer, dQ, at a particular absolute temperature, T, dQ/T, has been used to define an entropy change, dS = dQ/T related to some not-need-to-know-about specific alteration(s) in a system’s internal physical properties. lf one allows some loose thinking about heat-induced increases in the Statistical randomness of molecular motion in the above reservoir that is associated with the loss of useable energy, the positive entropy change, dS > 0, is vaguely relatable to the kinds of information entropies to be discussed below. If a gas trapped in an insulated, physically isolated, closed cylinder is allowed to expand infinitely slowly, reversibly, called adiabatically, pushing up the piston that closed off its end, the gas will become cooler, energy having been expended doing the work of lifting the piston. Defined as an isolated system (of course no where in the real, non- laboratory, world can this condition of absent exchanges of energy or matter with the environment be found), it is a reversible process, because returning the energy of the work by, again, infinitely slowly pushing down on the piston and compressing the gas to its original volume, returns it to its former temperature-defined energy state. In this historically prominent thought-toy of physics, there has been a reversible change in energy but no changes in the entropy, dS = 0. The gas’s heat, temperature (and energy and volume) can be completely restored in this metaphysically mythic classical thermodynmical tale of an entropy-conserving, reversible process. While fixed entropy and independence of the specific path is the case for the above noted abstract reversible cycle, in the real, irreversible orbits of most physical and all biological systems, entropy increases, dS > 0. Walter Nernst’s 1907 heat theorem yields a zero point from which to determine a difference measure in the 72 HOUSE_OVERSIGHT_013572

postulated, real physical world of ever-increasing entropy. He showed that at an absolute temperature of zero, entropy is zero. We can illustrate an approach to this singular state by placing a heated metal rod in ice water which would result in a decrease in the entropy of the rod’s molecular motions by dQ/ T; < 0, the cooling reducing the complexity of molecular motion in the metal bar and an increase in the entropy of the water by dQ/T2 > 0 indicating an increase in the amount and complexity of the surrounding water’s molecular motions. Of course the heat moves from metal rod to the water as 7; +72 making dQ > 0 positive and the entropy change, dS = dQ/ Tz - dQ/ T;, also positive. In another simple example, producing friction by rubbing a surface generates heat, dQ > O, at a temperature 7. This induces a positive change in entropy, dQ/ T > 0, in the form of increasing amount and complexity of the patterns of molecular motion in the air surrounding the rubbed surface. Using another related and well-known thermodynamic thought toy, the original isolated, insulated body of gas in the cylinder is partitioned by a membrane into two chambers, one containing all the gas with its temperature, pressure and ability to do mechanical work and the other a vacuum without these properties. This equilibrium state is changed into another equilibrium state by suddenly removing the membrane, filling both chambers with gas and, while increasing its entropy irreversibly, dS > 0, removes at least some of the gas’s ability to do piston raising work. In the context of classical thermodynamics, it is in this way that irreversibility can be defined by its associated increase in entropy. Though there has been no change in total energy in this insulated closed system, an increase in entropy means a decrease of the energy available for work. The increased disorder in the gas is associated with the loss of ability to convert heat, thermal energy, into mechanical energy. Historically important and still available elementary texts by Enrico Fermi (1936), Mark Zemansky (1957) and Herbert Callen (1985), among many others, explicate clearly the formal, but far from biologically relevant, classical theory of the physical entropy of closed equilibrium thermodynamic systems. Growing in part out of the formal thermodynamics of physics, statistical mechanics offers yet another set of intuitions about the not-necessarily-known 73 HOUSE_OVERSIGHT_013573

molecular details associated with changes in entropy. These ideas are closer to applicability in problems of making measures on the behavior of biological systems. Very generally, in the statistical mechanical context, an increase in entropy means a decrease in the order, which can be a quantitative observable reflecting a decrease in predictability and/or knowledge about the system. For example, we can locate the molecules of the gas more accurately when they are all on one side of the membrane-partitioned cylinder compared with the situation when the membrane is suddenly removed. This accompanying increase in ambiguity and decrease in knowledge in locating a set of gas particles reflects a statistical mechanical view of increases in entropy. Can anything general be said about the bounds on an increase in entropy? The statistical developments of the Yale mathematical physicist, Josiah Willard Gibbs (about 1875), consonant with the logical arguments of the Greek mathematician, Constantin Caratheodory (about 1910), conclude that the entropy increase goes to the maximum allowed by the constraints imposed by or upon the system. A change in likelihood as a probability is a characteristic way to quantify the entropy change, reflecting an alteration in knowledge or its reciprocal complement, uncertainty. The system’s entropic uncertainty said more colloquially, and relevant to the Bell Syndrome’s women of my life, is its capacity for surprise. A statistical mechanical approach to the total entropy of a bounded set of molecules in motion involves summing this property across all the participating molecules. We let N be the number of particles involved. As a problem in Newtonian mechanics, each of the N particles is represented in 6N dimensional phase space. That means that each point represents one of the N molecules in the three dimensions of location space plus three dimensions of motion space as its velocity, more specifically, the product of mass times velocity called momentum. This adds up to 6 dimensions of measurement. This so called phase space reconstruction of the molecules of a gas as individual particles are a daunting task, though fast computers and new algorithms are making computations from first principles more generally attainable. Those based on the first principles of short-range repulsion and long-range weak attraction among particles and the bumper-car collision 74 HOUSE_OVERSIGHT_013574

dynamics between them can now be implemented if the system of particles being simulated is sufficiently small and the computer simulation is for very short times. To transform the entropy into something more statistical and global, we return to the theoretical work of Ludwig Boltzmann whose formalism was used previously to quantitate pathological developmental simplification. He assumed that given a set of constraints, say the closed volume, V, of a box, B, of a fixed size, V (B), the orbit of each particle would eventually explore all the space in the box that was available to it. Boltzmann’s entropy became a constraint dependent, n- dimensional volume measure, with the assumption that the entropy, S, equals the logarithm of this volume measure, S = /n V (B). To calculate a value for the entropy, compute the volume of the molecular motion as determined by the invariant constraints of the system, such as the volume, temperature, pressure and/or its total number of molecules. We may partition, discretize, the volume up to some limit of resolution such that it is divided into © small boxes, each containing the representation of a particular state. Making the same assumptions of closed system, equilibrium thermodynamics, such a system is completely isolated from outside sources of matter and energy, it spends equal time in each of its © available states. In such a case, the characteristic occupancy time of any state is inverse to the number of States available, e.g. 1/0, and the system’s entropy is maximal for that set of states. Under these conditions, S = k In(Q), where the k term is the Boltzmann constant that contributes to the numeric units of entropy, as above, in Joules of heat /degrees Kelvin of the temperature. If the system is in contact with a heat bath, but cannot exchange matter with its environment, it is called diathermally isolated. The distribution of times spent in the available states of a classical diathermally isolated system of gas molecules can be represented by what is called a Boltzmann distribution of probabilities of state occupancies, p (as a function of their energy level, more measurably, their responsiveness, susceptibility, to heat). Here the characteristic time of the system spent in each state varies as the particular state’s probability. 75 HOUSE_OVERSIGHT_013575

Leaving the framework of physical thermodynamic entropies entirely, the entropy of information was introduced in the context of communication engineering in electrical and electronic devices. The metaphorical machine for the current age of entropy, analogous to the role of heat and steam engines in_ classical thermodynamics, is the computer. Energy in this context is a relatively trivial property. Ammeters and other monitors of load are unable to discriminate between a computer actively engaged in encoding and computation or one simply maintaining its dynamic memory while resting in computational readiness. This situation is very analogous to the results of early work discussed previously on the metabolic rates and sources of the whole brain’s energy, oxygen and glucose metabolism, by National Institutes of Mental Heath’s Seymore Kety and Louis Sokoloff and the State of Illinois Thudicum Laboratory’s Harold Himwich. Using whole head arterial-venous, energy-in, energy-out, differences, they could not demonstrate differences in rates of whole brain metabolism between states in which the human subjects were engaged in solving mathematical problems or deeply sleep. In today’s brain imaging research, using a variety of physical reflections of the brain’s metabolic activity, it is the differences in regional distributions of metabolic activity that are relatable to subjective and behavioral states, not differences in total amount of energy expended. In _ graphically coded representations of the regional metabolism of the brain in action, one or another or many areas “light up” and others “grow dark” in correlation with changes in thinking, feeling and action. The entropy first developed by Claude Shannon was formalized for use in 1948 in what was then called communication theory and now information theory. It represented a measure of the ambiguity and uncertainty that had the potential for being resolved by new knowledge. In this context, entropy and information were obviously complementary descriptors. A message that informs us about which of ten possibilities should be chosen contains less information than one that informs us about the proper choice to be made from among a thousand possibilities. The entropy of communication theory is a measure that is computed on uncertainty. The information reception capacity of a system is dependent upon the amount of 76 HOUSE_OVERSIGHT_013576

uncertainty in the receiver that pre-existed the receipt of the message. |n the binary coding scheme of digital electronic operations, the unit of information is the bit, a choice made between 0 or 1 in the resolution of a two state ambiguity at each place of some power of two number of places. Our relatively common computers these days have 32 or 64 bit processors. If these 0,1 choices are made in a random sequence in which each step is independent of the previous one, the sequential probabilities, _, are multiplicative: e.g. the probability of getting two 1’s (heads in a fair coin) in a row are the product of each 0.5 probability: p,;=0.5 x p2=0.5 = P1 P2 = 0.25. Using the common base ten system of logarithms to demonstrate the algebraic fact that multiplicative probabilities are logarithmically additive (and ignoring the minus sign that comes with making logarithms of the decimal fractions of probability), we notice that /og70(0.5) = 0.693147 and /og70(0.25) = 1.386294 and that 0.693147 + 0.693147 = 1.386294. The dot-dash choices of Morse code machines, the go, no-go gates of transistors, the open versus closed ion channel-mediated neuronal membrane discharge and the left, right spins of the single electrons of today’s quantum computers lead naturally to an information encoding of multiplicative sequences as the sum of logarithms in base (equal to the number of available states) two, each p= 0.5 choice called, /og2(0.5) = 1, a bit. Shannon’s 1938 master’s thesis mapped George Boole’s algebraic scheme for doing yes-no, either-or computation onto current switching devices such that circuit closed was “true” and circuit open was “false.” Using Boole’s laws such as “Not(A and B)” always equals “(Not A) or (Not B)” led to schemes for circuit routing through electronic gates which also serve for information storage in gadgets ranging from cell phone directories to computer hard disks. Following Claude Shannon, each logarithmically additive entropy term is expressed as the sums, ~%, of its probability, p,, times the probability’s logarithm, =.(p.x /ogz) (p,in base two. A logarithm is an exponent of its relevant base such that, for example, the logarithm, base two, of 2 x 2 x 2, 2° = 3 and 3 bits can encode eight binary (0,1) numbers: (000, 001, 010,011,100,101,110, and 111). Shannon used a hill-like, called convex, entropy function S (p)= -X(p In (p)). The amount of 77 HOUSE_OVERSIGHT_013577

information required to gain knowledge of an event is dependent upon the probability of its occurrence. log2(0.5) = 1 is the maximal entropy when modeling the equilibrium entropy of an independent random 0,1, (heads or tails) series of informational states as might result from flipping a fair coin a large number of times. This value would be maximal when the coin was fair, p(heads, tails) = 0.5, and the entropy would be 2(number of allowed states)x0.5(probability of occupying each state)x/ogio (0.5) = 0.693147...or in bits, log2(0.5) = 1. More generally, if system’s behavior is distributed equally among its possible states, the Shannon entropy is maximal and equal to the logarithm of the number of defined states, for example, log2 (2) = 1. Shannon’s classical equation about information content says the amount of information, / = -p /og2 p, measured in bits. The minus sign in this reciprocal relation indicates that the information content of data, /, goes up as the probability of occurrence of the observed data, p, goes down. Since soon we will be talking about brains and their various styles of information encoded content as well as its transmission, we note the other famous Shannon theorem dealing with limits on the channel capacity, C, for information transport is C = Wlog2(1+S/N) where W is bandwidth, the range of frequencies available for information transport, S is the strength of the signal and WN is the strength of the noise. Recall that the /og2(7) = 0 so only the signal-to-noise ratio, S/N contributes to the value of the product of the multiplication by bandwidth, W. Transparent clinical examples come from studies of the perceptual and cognitive decline in normal geriatric patients in which the range of aural frequencies (W) heard without augmentation decreases with age as does the frequency range (W) observed in their resting brain waves. The inattentiveness of the obsessively worried ruminator can be used as an example of brain channel capacity being reduced by the amount of on going head noise, an increase N, which, of course, reduces the value of S/N and therefore C. Measures of the informational complexity of systems in motion, in contrast with the information content of a static equilibrium state, are of dynamical entropy. Dynamical entropy is often called H, in contrast with thermodynamic and/or informational entropy, S. One can begin with a representational image of the 78 HOUSE_OVERSIGHT_013578

location, velocity and directional tendency of every point generated by a dynamical system by an arrow on the surface of action, the manifold, of a dynamical system. This field of arrows indicating directional and strength of motional tendencies is called a vector field. A vector represents its location at the base of the arrow, its velocity by the length of the arrow (called the modulus) and the direction of the motion by the direction of the arrow. If we regard all moduli as equal to one, every vector on the surface has the same length. The resulting graphs are called direction fields. Looking at a stop-action photograph of any point on this surface, its associated vector informs about where the system would take it over the next unit of time. The whole surface can be marked by initial points, which the dynamical systems move as they generate patterns of orbits of moving arrows in time. The following two brain and behavioral experimental circumstances make this depiction and its relevance to dynamical entropy more concrete. We review in more detail the concrete and visualizable findings from experiments requiring the quantification of characteristic patterns of motion in animals and man. They can be embedded into a similar surface-like setting, which might be called a behavioral manifold. For examples, my students from the past, Martin Paulus and Mark Geyer, now Professors at the Medical School of the La Jolla branch of the University of California studied the effects of psychotropic drugs on the patterns made on the floor by rats of various genetic strains while they wandered about, in exploratory behavior in a bounded space. Monitored by a video camera placed above the ceiling less cages, the patterns made by the paths taken by the rats over time were reconstructed as vectorial orbits on a behavioral manifold. This manifold was then repeatedly partitioned, covered with, from just a few large, in graded progression, to many smaller boxes, each partition composed of rectangular lattices of a particular size. Units of time were also partitioned into range of units from larger to smaller durations of observation. Differences in the rat’s genetic strain as well as injections of stimulants, antidepressants or antipsychotic drugs resulted in characteristic and discriminable path geometries mapped onto the behavioral manifold as orbital patterns. Each path was encoded as a sequence of size-dependent numbered boxes that were entered and occupied 79 HOUSE_OVERSIGHT_013579

or left. The new information being generated by the pattern of spatial orbits took the form of sequences of numbers or symbols representing the sequence of labeled boxes. The complexity of these numeric or symbol sequences was then quantified in a variety of ways including the use of two fundamental measures of dynamical entropy. One measure reflects how many new, previously unexplored boxes were entered by the rat per unit of time. This rate represents a percent of the possible. The second measure reflects how much of the time did the rat in each box visited as a distribution of the probable. The rate of expansion of the possible and the relative time in occupancy of these possibles, the probables, form the bases for the computation of these two kinds of entropies. For example, the work of Paulus and Geyer showed that the administration of a very small amount of stimulant drug, compared with a salt water control, led to an increase in the first measure of the number of new, previously unexplored, boxes entered per unit time. With respect to the second measure, the stimulant drug augmented exploratory activity was also more uniformly distributed over the possible boxes, making for more uniform probability. Administration of higher doses of stimulant drugs, at a critical dose, led suddenly to more spatially and temporally restricted and stereotyped patterns of motion of the rats, compulsive circling alternating with frozen sniffing. Both contributed to a decrease in the possible and nonuniformity in the distribution of the probabilities. In man, low doses of amphetamine tend to increase the rate and creativity of thought streams and high doses generate fixed ideas and paranoid delusions. In the statistical approach to nonlinear dynamical systems, time- dependent generation of new possibilities is called topological entropy, H; and the entropy associated with the distribution of probabilities is called the metric entropy, Hy. These kinds of entropies have also been used to quantitate characteristic patterns of in human behavior as well. We have previously mentioned these measures as used in human experiments by Karen Selz, a Research Professor of Psychiatry at Emory University in Atlanta. Recall that she devised a set of experiments leading to unobtrusive measures made on human subjects by asking them to remove, as many as they 80 HOUSE_OVERSIGHT_013580

could, the dots in a lattice, one by one, from the computer screen, by clicking on each point with a mouse. In some experiments, after removal, the dot reappeared in fifty milliseconds, in the “fast return condition”, or after one-second delay in the “slow return condition.” Unbeknown to the subject, the path made by the motions of their mouse on the computer screen over time while removing dots were reconstructed as a path on a fine to coarse grained box-partitioned behavioral manifold. Entropic indices of the rate of expansion of the possible, number of new boxes entered, reflecting H; , and the relative occupancy of the partition of the possible, reflecting Hy, the distribution of probabilities with respect to the boxes, could then be computed. For examples, Selz found that the spatial and temporal patterns of computer mouse motions made in this dot search and destroy task correlated highly with the subjects’ age, sex and personality types as defined by profiles from the Minnesota Multiphasic Personality Inventory, MMPI, and the Structured Clinical Interview, SCI, associated with the standard Diagnostic and Statistical Manual, DSM IV. She found that subjects whose personalities were like my high self-sensibility girlfriends demonstrated high indices of both Hrand Hy. The actions of nonintegrable nonlinear differential equations, not solvable by the usual techniques of integration, can be transformed into graphical images by plotting their orbits in abstract phase spaces with the three physically measurable coordinates of location x (or some other temporarily fixed value), velocity y (the rate of change in the location or measured value) and z acceleration (the rate of change of the rate of change in location or value) in x, y, z space. Graphical representations of the system in action in phase space can serve in place of analytic solutions to the equations. This idea was one of Henri Poincare’s major contributions to mathematics and physics, and has come to be the centerpiece of the qualitative theory of differential equations. The often point-to-point unpredictable but globally and qualitatively characteristic geometric shapes of the orbital patterns in abstract phase space are the objects of interest. There are visualizable representations such as cycles as circles and statistical measures made on these objects such as the Hy and Hy entropies and the in-betweenness (neither maximal nor minimal) of their difference. 81 HOUSE_OVERSIGHT_013581

A global statistical context for these qualitative differential systems was inspired by the Russian mathematician, Andrei Nikolaevic Kolmogorov. In his now famous foundational talk about the stability of classical mechanical systems in the final session of the 1954 International Congress of Mathematics, he gave public birth to, among other ideas, what has come to be called the ergodic or statistical, measure theory of dynamical systems. Here, ergodic means the existence of an invariant statistical measure on the phase space attractor of the system that can be obtained using a variety of equivalent methods and beginning the count at any of its points. Two phase space objects generated by a dynamical system may look different in phase space but their statistical measures may all be the same, L.e. invariant. These qualitative orbits in a box-partitioned space can be visualized as Paulus and Geyer’s rats exploring a space and Selz’s path sequences of computer screen dot quenches produced by clicking on them with a computer mouse. A precursor of Kolmogorov’s ergodicity was the earlier ergodicity of Ludwig Boltzmann. This describes a suitably partitioned system such that equivalent values come from quantitating the behavior of one single orbit exploring the space of the lattice of boxes over very long times time as those obtained from a single aggregate photograph of a// orbits run from all possible starting places simultaneously. The ergodicity of gas-like molecular randomness implicates systems being in one of only two possible equilibrium statistical states: measure zero (at most occupying a single point, zero, minimal entropy) or its “complement,” full measure one (occupying all available space in a state of maximal entropy). Joseph Goldstein, a well known teacher of meditation, giving advice recorded in Daniel Goleman’s 1977 book on the subject said that all methods of nirvana directed meditation amounted to “...simple mathematics ...all systems aiming for One or Zero—union with God or emptiness.” In place of the maximal or minimal values for the H; and Hy entropies of these states of transcendence, we in the world of samsara are stuck in states of in- between entropy which invariant statistical measures of on phase space shapes help quantify. To generalize measures made on rat and computer mouse paths to more general and idealized systems, after plotting an orbital path in a phase space, we 82 HOUSE_OVERSIGHT_013582

may partition the space of values taken by the journey of the orbital action generated by the equation over time with rectangular grids of increasing fineness. The result is an equipartition of phase space such that there is at most one orbital point in each rectangle of the grid, with, of course, many rectangles in the finer grids being empty. This final grid partition is called a generating partition. The proportion of the available boxes of the partition occupied by points is called its area or volume measure. This measure has been given a variety of names including Liouville, Haar and Lesbegue measures. lf every box is occupied, it has measure one. If at most one box, it has measure zero. If we allow partitions to be non-uniform and/or not fine enough to be generating and apply probability weightings for how many points fall into each particular box of the grid, the method is called the Sinai-Ruelle-Bowen or SRB measure after Kolmogorov’s students and followers, the Russian, Ya Sinai, the Belgian Frenchmen, David Ruelle and the American, Rufus Bowen. Similar to the SRB measure, the distribution of box occupancy probabilities multiplied by their logarithms and summed over all cells of the partition yields a statistical measure that is close to the informational entropy of Claude Shannon as described above. It is called the metric entropy ( Hy = -X(p; In(pi)), where H means entropy and gj; is the proportion of the total observations that occupy cell i of the phase space or state space partition. It was the above noted Russian father of modern dynamical systems, Kolmogorov, who in 1956 proved that the Shannon metric entropy is a quantifiable invariant of systems even in very complicated motion. Stanford University's Donald Ornstein won a Field’s Medal (the under forty year old mathematician’s Nobel Prize) for his late 1960’s work proving that the Shannon metric entropy, Hy, was the only invariant for a large class of appropriately defined, expansive (near by points separating in time) dynamical systems. Recall that we refer to metric entropy reflecting the relative occupancy as probability among the possible boxes (or states) as Hy. Hy is maximal when the percentage occupancy of all occupied boxes is uniform. IBM’s Roy Adler in New York and Brian Marcus in California, Hebrew University’s Benjamin Weiss, Warwick University’s English mathematicians, William Parry, Peter Walters, Mark Pollicott and others developed and proved the relevance 83 HOUSE_OVERSIGHT_013583

of a related measure of the rapidity of dynamical expansion, the generation of new information seen as the rate of entering new boxes of the partition, a logarithmic rate of expansion of the possible. Counting the number of previously unoccupied squares entered by the dynamical systems orbit per unit time over the generating partition, for instance, yields an estimate of entropy that, as in the rat and computer mouse examples above, is called the topological entropy, Hr. Hr, is about how much new information is being generated by the system per unit time. Theorems have been proven that Hr is a maximal estimate of the global dynamical entropy with Hy proven to be a minimum estimate. Monitoring single or aggregate molecular motion in a system with the maximum randomness of a space filling gas, we find that, on the average, every box is entered and occupied uniformly such that H; = Hy or said another way, H7 — Hy = 0. As evidenced by the above described experiments in rats and people, the same entropic relations (but usually not with maximal or minimal measure) can be found in biological systems. We have previously described the manifold geometry of a generic (typical, idealized) nonlinear dynamical systems as hyperbolic defined by the presence of simultaneous but decomposable components of the motion including the straight ahead and round and round actions on the center manifold, the new possibility generating, expansive, away from the center manifold motions along unstable manifolds and the back to the center manifold, contracting motions, along the stable manifolds. Uniform expansive and contractive influences in the flow leads to mixing of the order of the initial sequence of the values inscribed by the orbits. This results in maximization of the entropies and satisfaction of a concomitant of the uniformly hyperbolic condition, H7 — Hy = 0. These clean and mathematically proven findings do not hold for the quasi- mess that is human neuropsychobiology. Enmeshed as most of us are in only intermittently random or nonuniformly hyperbolic systems with the in-between entropies of the only apparently real world of maya, Hr — Hy # 0. How the H;— Hy = 0 of uniform hyperbolicity fails, H7 — Hy # 0, and along with it the dispassionate detachment of entropic emptiness and fullness, becomes a problem not unrelated to the existence and quantitative qualities of personality styles and their dissolution 84 HOUSE_OVERSIGHT_013584

with return toward but not reaching the maximally entropic openness, flexibility and naive credulousness of the in Jesus and Holy Ghost occupying transcendent dynamical states. We are all stuck somewhere in the range of measures indicating in-between entropies. Further Readings for Sensual In-Between Entropies Ecstasy in Secular and Religious Experience, Marghanita Laski, Tarcher, Los Angeles, 1961. The Role of Neural Plasticity in Chemical Intolerance, Barbara A. Sorg and Iris R. Bell, Ann. N.Y. Acad. Sci. Vol. 933, 2001 The neuropsychiatric and somatic characteristics of young adults with and without self-reported chemical odor intolerance and chemical sensitivity, |.D. Bell, C.S. Miller, G.E. Schwartz, Arch. Environ. Health 51:9-21, 1996. Application of entropy measures derived from the ergodic theory of dynamical systems to rat locomotor behavior, M. Paulus, M. Geyer, L. Gold, A. Mandell, Proc. Natl. Acad. Sci. (USA) 87:723-727, 1990. Long-range interactions in sequences of human behavior, Martin Paulus, Phys. Rev. E. 55:3249-3256, 1997. Mixing properties in human behavioral style and time dependencies in behavior identification: The modeling and application of a universal dynamical law. Karen A. Selz, UMI, Ann Arbor, 1992. A family of autocorrelation graph equivalence classes on symbolic dynamics as models of individual differences in human behavioral style, Karen A. Selz and 85 HOUSE_OVERSIGHT_013585

Arnold J. Mandell, In (ed. R.R. Vallacher and A.J. Nowak), Dynamical Systems in Social Psychology, Academic Press, San Diego, 1994. Toward a neuropsychopharmacologicy of habituation: a vertical integration. Arnold J. Mandell, Math. Modeling 7:809-888, 1986. Thermodynamics, Enrico Fermi, Dover, N.Y. 1956. Thermodynamics and Statistical Mechanics, Peter T. Landsberg, Dover, N.Y. 1978. Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane and C. Series, Oxford, Oxford, 1991. The Mathematical Theory of Communication, Claude E. Shannon and Warren Weaver, U. of Illinois Press, Urbana, 1963. Science and Information Theory, Leon Brillouin, Academic Press, N.Y. 1962. Brain Metabolism and Cerebral Disorders, Harold E. Himwich, Williams and Wilkins, Baltimore, 1951. 86 HOUSE_OVERSIGHT_013586

CHAPTER 5: SOME ENTHEOGENIC ENTROPIES In the spring of 1968, members of my laboratory team were looking for new brain metabolic pathways of the essential amino acid tryptophan, the dietary precursor of the human mood, sleep and libidinal neurotransmitter, serotonin. After struggling for several months to identify an apparently new compound, which turned out not to be new but only new in the brain, we collected evidence for a human brain enzyme that could catalyze the production of an LSD-like hallucinogen, dimethyltryptamine, DMT. Tracing its metabolic origins, we found that DMT was derived from tryptamine, a common metabolite of the essential and omnipresent amino acid, tryptophan. This enzyme and its metabolic product were located in highest concentrations in brain stem systems that influence the neural regulation of the heart, blood pressure, temperature, breathing, vomiting and primitive approach- avoidance behavior. It was also found in limbic brain nuclei thought to modulate the emotional coloring of perception and thought. Richard Wyatt, working at the National Institutes of Mental Health found DMT in the urine of schizophrenic humans. He also showed that DMT increased significantly if tryptamine’s normal pathway for degradation was blocked by monoamine oxidase inhibitors, such as 87 HOUSE_OVERSIGHT_013587

Nardil, Marplan, Eutony, Parnate and others of a then common family of antidepressant drugs. The presence of a DMT-generating enzyme in human brain was particularly exciting because we knew from the work of Harvard botanist, Richard Shultes and others, that DMT and the monoamine oxidase inhibitor, beta carboline, are combined in a mixture of the leaves of a shrub and the bark of a vine, both Amazonian plants, used together by the shaman of Peru, Colombia and Ecuador for thousands of years to evoke mystical experiences in themselves. In their state of chemically-facilitated, spiritual transformation, they were better able to engage in healing and divination of others. More recently this and other similarly acting biochemicals have been called entheogenic, “connecting to the sacred within.” Consistent with our neurochemical findings in human brain, the shamanic concoction, called by many names including ayahuasca and yage, combined the DMT containing plant, Psychotria viridis, with an extract of a vine with the powerful monoamine oxidase inhibitor properties of the beta carbolines found in Banisteriospsis caapi. |n 1975, working with a graduate student, Louise Hsu, we found that the mammalian brain could also synthesize beta carbolines. This family of compounds from the vine protects the tryptamine substrate as well as DMT from metabolic degradation such that it could circulate in the blood long enough after oral ingestion for enough to cross the blood brain barrier to induced prolonged and dramatic alterations in perceptions, feelings and thoughts. In addition, the carbolines of the Benisteriospsis component extended the time of action of DMT beyond the 15-30 minutes of effect of DMT when injected alone in human subjects. We found it fascinating that the human brain made combinations of DMT and beta carbolines similar to the blend that indigenous shamamic chemists discovered as an entheogenic from plant sources. Ralph Metzner, in the introduction to his 1999 collection of papers called Ayahuasca concluded that “...it is widely recognized by anthropologists as being...the most powerful and most widespread of the shamanic hallucinogens.” William Burrough in a 1953 City Lights published book written with Allen Ginsberg, tt The Yage Letters, said that yage “...gave entrance to a city where all human 88 HOUSE_OVERSIGHT_013588

potential is spread out in a silent market...” It was generally believed that with adequate spiritual preparation, ayahuasca could generate transcendent states that allowed access to ones inner being and the beings of other worlds that could serve as sources of mystical knowledge and healing. The Shams dervish of the 13" Century, wandering the Turkish portion of the Silk Road, used the word sohbet to describe the inner land of mystical conversations about mystical subjects that their turning meditation, whirling, and the shaman’s entheogenic compounds such as DMT give entrance. The question was whether our finding of DMT and its human brain enzyme had been an artifact, an accidental laboratory fluke. Members of my neurochemical research teams at the University of California Medical Schools in Irvine and La Jolla, notably Dr. Lee Poth, now a professor of pediatric endocrinology at the Uniform Services Medical School in Washington D.C., demonstrated that the DMT synthesizing enzyme existed in the brains of recent accident victims that as far as we were able to learn from their family and social histories, had been completely psychologically normal. More than a little bit startled by this finding and worried about making a sensational scientific mistake, we repeated the experiments with a variety of controls with the same findings. Though our original estimates of the human brain enzyme concentration were on the high side, we confirmed the general finding and published them in Science in 1969 and Nature in 1970. Our carboline work was published in the Journal of Neurochemistry in 1975. A year or so after our Nature paper was published, the Nobel Prize winning neuropharmacologist at the National Institutes of Mental Health, Julius Axelrod, confirmed the presence of the DMT biosynthetic enzyme that converted the tryptophan product, tryptamine, to DMT in mammalian brain tissue. We were both delighted and relieved. We speculate, perhaps too grandly, that this finding, along with the beta carboline human brain synthesizing capacity, supplies one of many possible neurobiological and neurochemical mechanisms for the claims of the cross-cultural universality of mystical experience. We all had human brains with these enzymes. The idea that the phenomena accompanying primary religious experience were common to all cultures was a major theme of the life’s work of the philosopher- 89 HOUSE_OVERSIGHT_013589

psychologist, William James, and was studied using fieldwork by anthropologists such as Bronislaw Malinolowki as described in his classic book, Magic, Science and Religion. Wa hopeless, without materialistic solutions and trapped in a belief system of spiritual nihilism? Was this a brain chemical transcendence escape and spiritual delivery system for the suprapsychological survival of those in dire need? As the 13" Century Islamic mystic, Jelaluddin Rumi, has written, “lf a tree could fly off, it n wouldn’t suffer the saw...” and more concretely, “...if you can’t go somewhere, move into the passageways of the self...,” a spiritual escape via a neurobiological road to the God-space within. What followed were a few years of occasional exploration of an “inside out” understanding of the mystical states evoked by the entheogenic family of chemicals. There were varieties of settings for these personal experiments. | found myself LSD-lost, circling endlessly in the tall silence of a Northern California redwood forest. | tried on Hunter Thompson’s mescaline lenses for the experience of Las Vegas unfiltered. | was expertly mentored in these quests by a distinguished collection of guides: Cultural anthropologist Michael Harner who taught me about the yage and datura use among the shaman of the Jivaro; Social anthropologist, Barbara Meyerhoff introduced me to the personal renewal rituals of the peyote cactus-using Huichol Indians of the Southwestern Sonora Desert; Neurochemically sophisticated Sidney Cohen, founding director of the National Institutes of Health’s Institute on Drug Abuse, told me stories of his involvement with Aldous Huxley and Barbara Brown in the Los Angeles covey of early American LSD explorers; organic 90 HOUSE_OVERSIGHT_013590

did some work with the dissociated anesthesias (producing wide awake but not there states) having consulted with John Lilly, a brain scientist who used these agents as a courageous self-medicating explorer of sensory isolation tanks; | met several native shamanic practitioners including the Huichol Indian that was the model for Don Juan in Carlos Castanada’s five volumes of pseudoethnography written up in my essay “Is Don Juan Alive and Well?” in The Pushcart Prize of 1977. Issues of culture and brain chemistry came together in several accounts about entheogenic, mescaline-containing peyote use among the Huichol Indians in a book edited by Kathleen Berrin and Thomas Seligman of the San Francisco Art Museum called Art of the Huichol Indians. Over these years | collected many nauseating, upper and lower bowel wrenching and ecstatically transcendent and exhausting day-long episodes of the angular geometries of visual pattern-generating DMT, the animistic breathing of bush and flower breathing peyote cactus, the darkly forbidding shadows of the psylocybin-containing mushrooms, the irreversible rocket launches into the electrically buzzing, kaleidoscopic circus of LSD-containing vials from Sandoz and the optimistic, trust engendering, expansively warm rush of six of Sacha Shulgin’s gregarious, rave dancing, chlorinated, methoxylated and _ ethoxylated phenylethylamines which he had, years before, synthesized for “an undisclosed purpose” for the Dow Chemical Corporation under contract with the U.S. Army Chemical Corps. The best known of the latter group remains part of the rave culture as Ecstasy. These agent’s peaks are flooded with exaggerated, caricaturizing images of people’s faces and a belief in the mindedness of animals and even the embodiment of inanimate things. Evoked are simultaneous and diametrically conflicting interpretations of the same social context, heteromodal sensory fusion called synesthesia so that sound bespoke color and smells induced music, habitual thoughts rearranged as new ideas in what is experienced as exciting new insights, and, most of all, that which Louis Lewin, Berlin’s early 20" Century Freud of psychotropic drugs in his book Fantastica, called gladness of the soul. Timothy Leary wrote of entheogenic escape from the habitual human brain’s mental- 91 HOUSE_OVERSIGHT_013591

manipulative and socio-sexual circuits gaining access to the rapture and ecstasy brain pathways on the way to the new planet within. What is seldom written about is the aftermath of chemical entheogenic agents. After the several hours of fireworks, all of these entheogenic agents, some more than others, gifted me with weeks to months of more self-sufficient, emotional fullness and ease in the conduct of living that was less contaminated by narcissistic preoccupation or defensive distantiation. | was left with increased interpersonal sensitivity and a noticeable repair of my deficiencies in aesthetic sensibility, particularly for the visual arts and landscapes. What were once two dimensional, trivial, beside-the-point, scattered copses of trees and apparently casual arrays of plant life in the Boboli Gardens behind the Palazzo Pitti in Florence, became the grandly structured, botanical wonder of increased dimension, communicating awe filed new perceptions of its previously unseen beauty. For the first time, | found myself walking slowly and stopping for several minutes, wordless, spellbound, in front of the modern art pieces of New York’s Guggenheim Museum. Lost in the experience, | found myself exclaiming to no one in particular, “| can see!” The delicacy and deliciousness of post-entheogenic agent’s new and beautiful everything made me tiptoe watchfully so as not to injure an ant. Feelings of omnipersonal kindness and generous compassion were without prideful self- reflection. This state of grace felt like an invasion of a shimmering presence that made contact with my other, generally unknown to me, life. It brought new perceptions, feelings and ideas for which | was moved to give thanks. | began to think | understood a little bit about what was meant by living in the Spirit and merging with God. Mircea Eliade, the French, University of Chicag In the state that this requires, “...all nature is capable of revealing itself as cosmic sacrality....” The entire world can become a hierophany with what Abraham Abulafia called an activated mind, the Jewish soul of emergent properties called the Nefesh. This entirely new world, Rudolf Otto in his 1917 Das Helige (The Sacred) called it ganz andere, (wholly other, something else), seemed to emerge 92 HOUSE_OVERSIGHT_013592

spontaneously along with an instantaneous knowing-how-it-is-with-you-and-|l-and- all-of-us that made even vicious killers appear sympathetic. Is this what the Charismatic New Testament Book Churches mean by redemption through forgiveness of others, requiring the genuine sincerity of this thought before qualifying for Communion? Is this Christ’s undemanding gift of grace as in Romans 4: where Paul observed that all of us fall short of the full glory of God unless justified freely by His grace. Was this the New Testament’s spiritual technological advance from the Old Testament’s and Koran’s eye-for-an-eye? Did this chemically triggered transcendent experience differ significantly from the supernatural transformation of individuals by the Holy Spirit of Christian revivalist teachings? Martin Marty, University of Chicago’s Professor of Modern Church History, dates the institutionalization of this personal transformation in the United States to the post- Civil War period. Did this mean that the mysteriously selfless love of Christian agape and the altruism of E.O. Wilson’s sociobiology lay waiting in the brain and could appear spontaneously, by grace, without lawful directive, repetitive recitation or the discipline of catechism? As one might have suspected, the urgency of my inner and outer search for a new spiritual ecology of mind was driven by more personal needs. My spiritual hunger was made acute a couple of years before our laboratory’s DMT discovery when as a 30 year old Assistant Professor of Psychiatry and Neuroscience at UCLA in West Los Angeles, | was living in a small, heavily mortgaged house in Brentwood with my graduate student wife and two young sons. A testicular lump was an accidental discovery made while showering. After surgical biopsy and radical lymph node dissection, the professor of urology gave me a diagnosis of right testicular choriocarcinoma. All by itself, my testicle had given birth to a mass containing all the embryological tissues of a fetus, and had thrown in some maternal placental cells Unlike now, when the group of testicular neoplasms are treated successfully with a high survival rate (think Lance Armstrong), at that time, follow up research of this young man’s disease by the Army Medical Corps promised a five- year survival rate of only 5% to 10%. The news filled me with fear and the ensuing hopeless resignation detached me from life with a dread broken up only by 93 HOUSE_OVERSIGHT_013593

episodes of rageful envy of everyone else in the world that had been spared. My wife escaped into an alcoholic flirtation with her major professor; my sons grew increasingly ensconced in the generous and kind neighborhood homes of their playmates. | metered as many hours as possible in equity growing, long lonely days in a small, dark, couch filled, university office, listening to Beverly Hills, Brentwood and West Los Angles citizens as they psychoanalyzed their mysterious lack of emotional fulfillment from materialistic fulfillment. Legend has it that Gautama’s sudden insight about the universality of this sated, bored condition occurred in 528 B.C. after 49 days of sitting in the lotus position under the bodhi tree, now called ficus religiosa. In contrast with Buddha’s illumination, my psychoanalytic training- induced, Freudian-Darwinian instinctual conflict, driven by fears of starvation and castration, drew me tighter into the world of meaningless, coin flip probabilities. Our house was a block away from a West Los Angeles synagogue and we knew the Rabbi and his family well. Our sons played together frequently. The Rabbi tried to bring comfort to me on my death watch, with hours of discussions about trans-individual, ethnic belonging and a deeper foray into philosophical humanism. Both felt completely irrelevant to my condition. As an intern tending to those dying at night in Ochsner Foundation Hospital in New Orleans, it seemed to me that Jews tended to die more noisily than Catholics. For my personal escape from low-lying dread, | needed the metrically linear time of chronos to become the metric-free, topological, continuous surface of the twisted circular ribbon of a Mobius loop, with the view from each moment a kairos, a stretchable infinity of each moment’s internal multiplicity of times. The ruthlessly reasonable Hebraic historicity, configured by the tooth-for-a- tooth, Mosaic and Roman falion law, the reciprocal, economic, exchange-calculating brains of Barkow, Cosmide and Tooby’s The Adapted Mind (1992) and the terrifying stories of the Five Books of Moses, made the hopelessness of this sinner’s plight inevitable. It felt like my dichotomous choice of God-type was between One of merciless fairness and the He and She of unconditionally forgiving generosity. The mind set of logical problem solving applied to the question about which of these two represented the true character of God lead to a momentarily distracting, metaphoric 94 HOUSE_OVERSIGHT_013594

ecclesial exercise: what were the minimal number of four magical cards need we turn over with preconditions or results on the upsides and downsides if what was showing was: (1) Beatifically good; (2) Cursed with extraordinarily bad luck; (3) Not dependent upon personal virtue; (4) Inordinately fortunate in all of life’s trials. The pay-as-you-go God people would need to pick up (1) and find fortunate life and (2) to find the fate of the non-believer to establish that God was coldheartedly true and fair with the results of flipping (3) and (4) being none contributory. The grace-to-all- sinners God people need to turn over card (3) to find good life and (4) to find sometime sinners nonetheless fortunate to confirm their belief in the unconditionally of the loving generosity of God and making finding out about the underside of cards (1) and (2) unnecessary. This liturgical discussion and gamble with God’s cards, perhaps a caricature of the Talmudic, rational discussions with the rabbi, felt irrelevant to my spiritual needs. Missing was mysticism’s promise of the disappearance of | into a union with the divine, the Heart Sutra’s eternal emptiness of form and the eternal form of emptiness that gifts with spiritual perspective and not-necessarily-logical intuition about unseen Absolute Reality. Forced either-or, binary, card-turning cognition in the search for God’s logic is unrewarding. As the Dalai Lama, in his Heart of Wisdom Teaching, says, “...all phenomena are emptiness, without defining characteristics, they are not born, they do not cease..." In trying to penetrate the mystery and promise of this emptiness, it was difficult to surrender my internal parody of what sounded like that day’s Southern California New Age stuff about global nonaggression, sexual politics, Beadles music, distressed jeans and pot. In the synagogue of my neighborhood, experience with a deeply felt, never-you-mind- about-anything God of detachment with love, was not on the menus of Friday night or Saturday morning services. All | could feel was a faithless and nonnegotiable fear. In the work of many mysticism-positive scholars, a classic being Evelyn Underhill’s Mysticism, 1961, it has been speculated that this ineffable state as a union with a powerful unknown, transcending description in language, becomes more socially prominent during times of cultural efflorescence. She pointed to the 95 HOUSE_OVERSIGHT_013595

flowering of mysticism in epochs of the high cultural achievements at the close of the Classical Period in the Third Century, the Medieval Period in the Fourteenth Century, the Renaissance in the Seventeenth Century and, now, as we know, in the Western World toward the end of the Twentieth Century. An increase in general acceptance of talk, writing and practice focused on mystical experience is said by many to accompany historical high points in intellectual, literary and political achievement. One might include as a component of our growing cultural richness, the new science about chemical dialogues with the brain. Although no central nervous system agents were ever allowed in the ashrams of Baba Muktananda, it was common during some evening sessions of questioning, called satsangs, for him to acknowledge that one or a few experiences with entheogenic agents can open many recalcitrant folks to the existence of the God within. This, in turn, led them to the drug free spiritual exercises, sadhana, of love, se/f-truth, and spontaneity (each according to their nature) as well as abstinent discipline, meditation, chanting and yoga to maintain the knowledge. We might speak of participating in the creation and maintenance of the spiritual ecology of ones inner and outer being. Underhill said that the cultural richness of an efflorescent epoch is taken inward and accompanies personal and societal mutations into states and institutions involving higher spiritual consciousness. In addition to an increase in the common outward manifestations of having had a mystical experience, such as an increase in compassion, forgiveness and more respectful and reverential attitudes toward the Earth and all its creatures (currently taking the forms of deep ecology, ecofeminism, herbal medicine, organic farming and the like), these times bring more public consideration of the nature of reality itself, apart from its material manifestations. The theme of the life’s work of the Dominican priest, Thomas Aquinas, made master of theology by papal dispensation in 1259, involved the existential recognition of this dichotomy of existence, esse, and essence, nature and grace, the material world and God. William James wrote famously about mystical experience penetrating the thin veil between these two worlds. Those with a mystical orientation attribute reality to inner experience in relationship to a transcendental, supernatural world. Whereas 96 HOUSE_OVERSIGHT_013596

everyday events are subject to perceptual ambiguity and its attendant variety of interpretations, mystical union is claimed to bring the existence and meaning of Absolute Reality into direct experience. This kind of knowing is more akin to the Platonic view of mathematics, that theorems have been everlastingly existent, from before our physical world, then it is to the here and now, physically based, finite computations involving the experimental machines of physics. The philosopher-mathematician father of phenomenology, Edmund Husserl, criticized the physics-want-to-be orientation of the 1860 empirical, objective measure psychologies of Fechner and Wundt. He understood the best of their findings as simply correlations between subjective and observable events. Using mathematical discoveries as examples, Husserl spent his life arguing for the possibility of abstract truths relevant to mind being more reliable and valid if grasped via direct experience. Knowing by what the popular mid-twentieth century writer of science fiction, Robert Heinlein, called grocking it. This is antithetical to the attitudes of today’s human cognitive and brain sciences which disallow such knowing as deeply suspect unless accompanied by objectively definable observables such as changes in electrical or imaging indices of brain activity in one neural region or other. The modern psycholinguistics of brain mechanics can be _ called neolocationism. Using modern technology to measure regional blood flow, energy metabolism and/or electrovoltage or magnetic field activity, stories of function are spun that closely resemble those imagined more than a century ago by the first locationists, such as Ramon Cajal. These neuroanatomists spent thousands of hours looking at cell clusters and their connections in stained slides of human brain tissue using microscopes and imagined their singular and integrated function. Today, Lewis Judd, long time chairperson of the Department of Psychiatry at UCSD in La Jolla, carries a full sized, polymeric, three-dimensional model of the human brain when teaching his students about human subjective experience and interpersonal behavior. In his weekly grand rounds, he explains that day’s psychiatric patient’s problems pointing here and there at regions in this plastic surrogate for our electrical jellied brain. Few, if any, of the psychiatry students in his class was inclined to ask the foundational question: how it is that a finger point and 97 HOUSE_OVERSIGHT_013597

a name of a brain place can describe, much less explain in the language of physical or physiological mechanism, a patient’s illogical thoughts, feelings of hopelessness, irrational rage or prayerful gratitude. There remains a wide gap between ideas about the mechanisms of human symbolic processing and those involving the structures and functions of neuronal components and their connectivities in the brain, particularly when perceived as regionally segmented meat. Yet this report of Professor Judd’s finger-pointing plastic brain ritual should not elicit surprise since iconic manipulation is certainly not new to the practices of priesthood. In contrast with neuropsychiatry’s behavioral attributions to brain parts as an explanatory pantheon of mysterious doers, absent of mechanical specifics, the fields of physics turn to more abstract and general mathematical and statistical, so- called phenomenological laws, such as those of thermodynamics and statistical mechanics. The accounts of Feynman’s abstract and general thermodynamic development of conservation of energy as well as equilibrium thermodynamics discussed previously serve as relevant examples. These abstract models have been found to capture the behavior common to diverse physical systems involving (often still unknown) differing physical mechanisms. Consistency of description, reliability, weighs in before predictive validity, which, with maturation of the research area, gradually becomes detailed mechanistic understanding with the eventual goal being derivation from the first principles of physics. The painful truth is that that in spite of evocative claims made to the contrary in the 1990-2000 Decade of the Brain, this level of understanding at the interface of neurobiological hardware and software remains unbreached. Some recent attempts are interesting. One of the current research themes about real single neurons in real brains (in contrast with the silicon chip modules used in neural network computer simulations), involve widely distributed neurons that discharge in temporal synchrony. These phenomena have been described by Max Planck’s Wolf Singer, Christoff Koch of California Institute of Technology and Florida Atlantic University’s Steven Bressler and others with words such as synchronization, phase locking, coherence and binding. Binding is an intuitively seductive word that premises that two, even widely spatially separated, brain regions that manifest neuronal signals of 98 HOUSE_OVERSIGHT_013598

activation locked together in time are assumed to be functionally integrated. Another time-dependent neuronal characteristic of current interest involve neurons or neuronal clusters that beat with almost strict periodicity, the oscillatory pacemakers. For example, the program of research by Professor Al Selverson at University of California at San Diego, among others, has elucidated the role of these rhythmic pattern generators, both autonomous and those emerging from particular patterns of network connections. A wide variety of functional links involving neuronal pacemakers has been demonstrated. They range from the oscillatory transport of calcium through membrane channels in neurons and heart muscle, smooth muscle oscillations of the pylorus muscle of the stomach, the neuronal ganglion driven chewing motions of the jaws of invertebrates and the retina-to-brain hypothalamic cells gating human circadian rhythms coupling our body’s hormonal clocks to light cycles. Though regular rhythmicity in neuronal discharges is an intuitively attractive idea and relatively easy to quantitate using simple sine wave trigonometric transformations, in the real brain it is statistically rare. The commonest neuronal discharge pattern observed is that of intermittent bursting, clusters of neuronal discharges in time in which the inter-discharge intervals irregularly stretch and contract like the bellow pleats of a syncopated accordion. Bursts of repeated firing of some unpredictable length followed by silences of equally mysterious durations. Their behavior can be represented as statistical measures using non-normal, /ong tailed distributions and in-between entropies described previously. For a whole human example, although the rhythm of manic depression is commonly thought to involve periodic cycles, careful study using motility patterns of the timing through life of these episodes of extreme mood states by Professor Allan Gottschalk at the University of Pennsylvania and others have demonstrated an irregularly intermittent bursting pattern in manic-depressive episodes, getting more frequent with age. Neuronal inter-discharge intervals seldom demonstrate what is called a regression to the mean like the normal distribution of heights, as one increases the number of people measured, the tighter the distribution around the mean. Neurons, much like our own irregular pattern of doing things (in spite of our plans), the statistical 99 HOUSE_OVERSIGHT_013599

distributions of neuronal interspike intervals have increasingly /ong tails. Contrary to the behavior of a normally distributed observable, the larger the series of neuronal spike observed, the more likely that a longer interspike interval than had been seen before will occur. Counter-intuitively, long intervals tend to be followed by more long intervals as more shorts follow short intervals. Manic attacks cluster in time as does a number of other brain and body diseases. Maybe it is intuitively obvious that bad stuff tends to cause more bad stuff and good stuff is self-propagating. Having suffered recently does not mean fate owes you one. The brain’s syncopated segmentations of time can be translated into a creatively arrhythmic dance. What makes neurologizing conversations like these about subtle human experience possible are the human subjective scenarios we have agreed to short hand with names of brain parts and neurochemicals. The how is where conceptual connection is filled with post 19'° Century Spanish microscopic neuroanatomist, Santiago Ramon y Cajal-like, intuitions about the functional role of brain structures: we think motor automaticity and pacing when hearing the brain place names such as caudate, putamen and cerebellum; we think limbic lobe when musing about sexuality, rage and depression; we short hand /eft versus right hemispheric places for verbal and sequential versus intuitive and geometric shape cognition; we point to the frontal lobe for the future work of executive control, anticipation and paranoia; the hypothalamus for primitively expressed appetites and to the brain stem for our vital functions such as breathing and blood pressure. With respect to the brain juices, we say dopamine for aggressive activity, norepinephrine for attention and sensory discrimination and serotonin for hunger, mood and sexual inclination. No matter how avant guarde our experimental techniques such as monitoring local functional blood supply by fMRI, regional brain glucose utilization maps, time- dependent changes in_ skull surface voltage using a cap studded with electroencephalographic, EEG, leads, monitoring these voltage field via their transverse magnetic fields by the frozen helmets of magnetoencephalography, MEG, we conclude our work by calling forth named but still enigmatic brain parts and their juices as mysteriously powerful little men and women executing remarkably complex and subtle tasks, sometimes even when called upon. 100 HOUSE_OVERSIGHT_013600

Current neurochemical research using molecular biological tools such as mice knockouts (the ablation of specific proteins though interference with their nucleotide-mediated protein biosynthesis), for example, the production of animals missing a subunit of their hippocampal glutamate receptors associated with the loss of some memory functions, conclude the memorial mechanism to be a specific cellular region, such as hippocampal CA3 cells. Technology advances but continues to support a primitive philosophic animism of named brain parts which pop science icons like the late Francis Crick called “The Amazing Hypothesis.” He and his fellow brain philosophers implicate brain mechanisms such as the amygdaloidal nucleus man who can emotionally color even affectually neutral information that is transported through him. Imaging data showing amygdala man lighting up is used to tell us that circulating sensory information through the differentially behaving amygdaloid nucleus is used for fight or flight interpretive significance. Emotionally expressive human faces light up inferior parietal cortex. The lowa University Professors, the husband and wife Damasios, have located even the criminal psychopath man in specific locations in the brain. As we have argued, perhaps ad nausem, using multimillion-dollar imaging and molecular biological technology and no new thoughts that weren’t around during the era of the 19th Century’s neuroanatomists, specific brain regions continue to gain implicative properties like the task-specialized gods of the Roman and Greek pantheons. Crick implied that God is a brain part. At the same time, those of us that have been in the brain business for a while, recall skyscraper window washers, standing steady, high up on rope lashed planks, suffering from congenital absence of the cerebellum, the supposed sine qua non brain part supporting motor coordination and balance in humans. More generally, there is much evidence that if young enough and willing to work, many of the functions of missing parts of the brain can be taken on remarkably well by other brain parts thought not to be involved in these functions at all. In addition, since evidence of neuronal responding to loud noise or bright light perturbation can be found almost everywhere in the hyper-connected human brain, because anticipation and brain time inversions make before and after indicate little about human 101 HOUSE_OVERSIGHT_013601

neuropsychological causality, and inhibitory on or off and activating on or off are a priori functionally equivalent with respect to the logic gates of information encoding, transport or storage, the modern study of brain mechanisms in emotion, cognition and behavior remains almost as mysterious as ever. The only human mind-brain observations that are doubted consistently, and treated as unpublishable by the editors of the journals of science, are those that result from direct human experience using subjective reports from within. They are called unscientific. Often ignored are logically consistent mathematical and computational contexts, which, as abstract and general tools of thinking and imagining, have the capacity to frame, rigorously define and describe thinking about both the subjective and objective aspects of brain-generated phenomena. These mathematically configured metaphors can lead to consistencies in description, this is behaving like that, in what are called equivalence relations expressed both as intuitive imagery; for a concrete example, a one holed bagel and one handled tea cup are topologically equivalent because, sculpting in clay, they can be smoothly transformed into each other. We have seen that invariant measures in computable statistical flows can come out of a mess of data. Professor Paul Rapp of the University of Pennsylvania has been able to mathematically encode the verbal content of the patient’s free associations and the therapist's responses, using tape recordings of hours of psychoanalytical treatment. Examples of quantifiable qualities found useful in this regard involve a variety of characteristic statistical patterns in what are called entropies and information as well as various measures of what with a wide range of definitions is called complexity. These quantifiable properties, measures, can help in the struggle with the intrinsic tension of Absolute Reality between the “eternal emptiness of form and the eternal form of emptiness.” We resort to measures of entropy, information and complexity when confronted with our ignorance, “emptiness,” great or little, with respect to either cause or result, about what exactly is going on. Entropy in its forms relevant to information quantifies our 102 HOUSE_OVERSIGHT_013602

ignorance, the emptiness and its mystery. Computations of the entropy of systems in motion convert questions and answers concerning the detailed workings of the leg’s neuromuscular machinery to global statistical descriptions of more abstract thematic motifs, forms, expressed in the dance. Patterns of behavior of these properties can suggest intuitive ideas and imagery about global mechanisms, approach/avoid, smooth/discrete, wildtame, as well as correlated and objective physical observables. To learn more about this abstract, topology tinged (none numeric) style of model building, we can go to school on a long studied physical example. It connects a simple and well understood rea/ world observable with abstract statistical patterns resulting from motions using the one-to-one correspondence (the equivalence relation called isomorphism) between their entropies. As we have discussed, the Stanford mathematician and Field’s Medal Winner, Donald Ornstein, proved that in statistical studies of even point-to-point unpredictable, chaotic systems, entropy is the only isomorphism. The hardware of this physical example is what the statistical physicists call a dilute gas of some fixed number, n, of uniform hard spheres, moving scatterers, that, absent of dissipative friction, wander continuously around, changing their directions when bumping into each other. In a two dimensional bounded arena of randomly rolling balls, this game has been called Sinai’s billiards. It was named for previously mentioned Ya Sinai, an eminent Russian mathematician He is now at Princeton and was previously a student of Andrei Nikolaevic Kolmogorov, the Russian guru of many of the Twentieth Century’s world- class Russian mathematicians. Kolmogorov axiomatized the field of probability and, more relevantly, initiated the theory of statistical descriptions, the ergodic theory, of nonlinear dynamical systems. |In the language of statistical physics, we will see that the same system produced by high number of elements executing Newton’s deterministic laws can be generated by a so-called random system such as that resulting from flipping a suitably biased coin. Our example can also serve as a metaphor, used extensively in the mathematical biology of the late Professor Art Winfree, for the temporal features of life on a topological circle: the natural irregularities of the recurrent beat of the heart, the in and out breathing of lungs, the 103 HOUSE_OVERSIGHT_013603

up and down voltage of brain waves, the pendulum swings of our blood hormone levels, the cyclic procession of our days, months and years and at large scale, our body’s journey from dust to dust. The angular deviation theta, 8 from the initial reference direction of a single moving sphere, gets rotated to a new angle theta, 9 8 by a collision with another sphere. It has been shown that the new angle 0 is the previous angle, _ times twice the average distance traveled between collisions called the mean free path, here symbolized by delta, 6, divided by the diameter, D, of the sphere. Algebraically, 0-—0 the new angle is equal to twice the mean free path divided by the diameter of the spheres times the original directional angle of the sphere’s motion. If we symbolize the time between collisions with tau, 1, after an elapsed time of experimental observation, tf, we can say that the deviations from the initial direction of the sphere changes like (a The exponent, t/, represents the time of the experimental observation divided by the average time between collisions of the spheres, i.e. the time we’ve been watching, ft, is expressed as units of inter-collision interval, t. Of course, the circular deviation in the angle from the initial direction rotates repeatedly around a circle as the number of collisions increase. If a point on a circle marks the angular change resulting from each collision and the system runs long enough, it has been shown that the circle will eventually be completely covered by points. An estimate of the entropy, S, being generated by each sphere labeled with some index i, Si, is positive because the recurrent motion is deviating continuously from the initial direction. It can be computed for each sphere as the /ogarithm of the intercollision time-averaged deviation from the initial direction, S; = “loe(=>) and the T entropy of the whole n hard sphere system is the sum of the n entropies, which can be expressed as nxS;. If we keep books by registering the points when each sphere’s makes a stop on the top half of the sphere’s circle as 1 and the bottom half as O (and we must arbitrarily decide between 0 and 1 if it falls exactly on the 104 HOUSE_OVERSIGHT_013604

division between top and bottom and do so in a consistent way), then we can keep score with a random looking binary series such as 11001001010.... that describes the sequence of rotations. The advantage that accrues by doing so is that this coin flip counting eliminates details in favor of a computable over all measure and supports several forms of entropy calculations for its use in deciding if this system is behaving like that system, an equivalence relation. One can imagine a series of coin flips with 1 being heads and 0 being tails such that the statistics of a characteristic series is determined by the fairness of the coin. As noted above, Donald Ornstein’s famous theorem says that the entropy of these kinds of hardware and software systems is the only general basis for finding correspondence between characterizations of two such irregularly behaving systems. The important idea here is that a series of 1’s and 0’s may not be identical but the two systems can be isomorphically equivalent with respect to their entropy. Notice again that the physical process of hard spheres bouncing off each other on a flat surface has been captured by an abstract representation in binary numbers that, like a series of coin flips, can be quantified as entropies (which would be maximal for an ideal, fair coin). After describing the process of real number representation by the binary code, we will show how entropies can be computed for these binary series. We remind ourselves that we are struggling to obtain some kind of knowing in a representative system manifesting the tension and mystery between emptiness and form. We can translate all finite real numbers into this language, making them accessible to standard entropy computations. The following discussion of the process of transforming numbers into binary series is in the spirit of the famous number theory theorem that every natural number (the positive integers such as 1, 2, 3, 4...) can be expressed as the sum of at most four squared numbers. Encoding any number by a series of 0’s or 1’s in what is called a binary transformation, begins with its separation, called partition, into a sum of powers of 2, for example, 100 = 64 (2°) + 32 (2°) + 4 (2). A short hand description of this sum begins with a form indicating the presence or absence of each successive power by a 1 or 0 coming before the relevant power of two; i.e. 100 = 1 x 2°+1 x 2°+0 x 24+0 x 2°+1 x 105 HOUSE_OVERSIGHT_013605

2? +0 x 2'+0 x 2°(in which the last term, arbitrarily, is 2° = 1, since anything to the power O = 1). This can be written even more simply as a series of 0’s or 1’s, their presence indicating whether the power represented by each place in the left to right descending sequence of powers of two participates in the sum of the partition. It is in this way that in binary numbers, 100 = 110010. As another example, if we similarly partition the decimal number 729 = 512 (2°) + 128 (2”) + 64 (2°) + 16 (2°) + 8 (2°) + 1(2°), we find that its binary transformation results in 729 = 1011011001, the 0’s representing the descending powers of two that are absent in the powers of two partition. One can compute the binary representations of lower valued numbers immediately; for example, 4 = 1 x 27 +0 x 2'+0 x 2°s0 that there is a 1 in the multiply-the-power-of- two column and 0 the power 1 and power O columns so in binary representation, 4 = 100. Similarly, 6 = 1 x 22+ 1 x 2'+0 x 2° making the binary transformation of 6 = 110. It was the co-inventor (with Isaac Newton) of the calculus, Gottfried Wilhelm Leibniz, in about 1665, who fully developed the binary representation of all decimal numbers. |n a state of wonderment about the simplicity, power and completeness of this 1 and 0 encoding, he is said to have the beliefs that 0 symbolized the emptiness of the universe’s beginnings, 1 represented the complete fullness of God and that this transformation served as metaphoric evidence consistent with God’s creation of the universe out of nothing. The simplicity of binary expressions as in the dynamics of hard spheres or rotations on the circle as well as the transformations such as 729 = 1011011001 make them propitious for exemplifying the methods for computing the entropies of the growth rate of the possible, called the topological entropy, Hr, and the probable, the metric entropy, Hy, which was introduced in a previous chapter called “Sensual In-Between Entropies.” The following exemplify the computations of measures of topological and metric entropies, Hr and Hy, another computable idea called algorithmic complexity, AC and finally, the well known (to statisticians) standard run score, src. Their descriptions have as their purpose a demonstration for the reader that these apparently abstract, perhaps nebulous sounding, words can be transformed into well-defined, concrete, quantitative and computable form of reality. 106 HOUSE_OVERSIGHT_013606

lf acceptance of this idea does not constitute a problem for the reader (and you do not find it fun to follow along with a computer math program and/or a pencil), then the following several paragraphs can be quickly scanned or skipped entirely. The computations of H; and Hy begins with keeping track of how many 0 > 1 and 1 > 0 transitions are found going from left to right in the binary series. For example, in the binary expression of 729, 1011011001, one starts counting with a 1— 0 transition followed by a 0 + 1 transition and then a 1 — 1 transition and so on. A useful way to record the count is via entries into a 2 x 2 matrix for score keeping in which the horizontal rows are labeled 0 on top and 1 below and the vertical columns are labeled O on the left and 1 on the right. The number of each kind of transitions (from the vertical label to the horizontal label) are counted and summed in the appropriate box of the two box by two box matrix; for examples: for a 0 > O transition, a tally mark is entered in the upper left corner of the matrix; for a 0 —>1 transition, a tally mark is entered in the upper right corner; a 1— 0 tally goes in the left lower corner and a 1-> 1 Is tallied in the right lower corner. The resulting transition incidence counting matrix, M; for the 729 binary transformation series 1 3 looks like M; = 32 indicating one 0 — O, three 0 > 1, three 1 > 0, and two 1 > 1 transitions have been tallied. Although this series alone is too short for computing reliable statistical measures, if we assume that the pattern of transitions observed in this short series is stationary, that is its transition behavior will remain the same if the binary series continued on to be infinite in length, the assumption being that the dynamics of now will be the same as always, 729 will stay 729, then we can use two forms of this transition matrix in the computation of the topological entropy reflecting the growth rate of the possible, H;, and the metric entropy from the statistical weights of allowed choices among them, the probable, Hy. To obtain the entropy representing the growth rate over time of the new possibles, the computation of H7, the topological entropy, involves first transforming M; into an transition incidence matrix, Mz; a 0 or 1 matrix indicating whether each box has been entered at all (or not). Since in the binary representation of 729, all 107 HOUSE_OVERSIGHT_013607

1 1 four boxes of M; are occupied, the Mz; = rf indicates that all four kinds of transitions are possible. Since we remain in the context of a 0,1, two state system, the growth rate of the possible equals the logarithm, base two, of the sum of the entries in the boxes of the left-top-row to right-bottom-row diagonal called the trace and Hr = log (1 + 1) = log2(2) = 1. Consistent with intuition, since every transition is possible, the topological entropy of M; as indicated in its M;; is maximal (= 1). Another expression equal to the sum of the trace (the sum of the upper left to lower right diagonal) in a square matrix, is its leading eigenva/ue, most often symbolized with a lambda, 41. The logarithm of the leading eigenvalue of the transition incidence matrix is equal to its topological entropy. Symbolically, Hz (Mz) = logz (A1 ) = log2 (2) = 1. Standard elementary linear algebra texts describe how to compute eigenvalues, these relations and related operations as well as their foundational theorems. Before computing the entropy of the distribution of probabilities among the possibles as the metric entropy, Hy, let us notice again that the occupancies in the 1 3 four entry boxes of the transition matrix M; are not uniform, M; = 3 9° This leads naturally to the intuition that for this series of binary transitions, Hy, in contrast with Hy, will not be maximal, i.e. not equal to 1 and the nonuniformity of H; and Hy is a computational expression of what we mean by a state of in-between entropy. These entropies are identical and their difference = O for transitions reflecting maximal entropy, as might be realized in a very long series of fair coin flips in which the entropies = 1. Entropy will be minimal when flipping a two headed coin, here the entropies = 0. More compactly, the non-uniform probabilistic, metric entropy, differing from the maximal topological entropy indicates that the system is in a dynamical state of in-between entropy, written as H7 - Hy + 0. In the computation of the metric entropy, Hy, the M; is transformed into a transition probability matrix, M;p, called a Markov matrix named for one of the two great Russian mathematicians, both students of Pafnuti Lvovich Chebyshev, the Markov brothers. The entries of each row in the MM; are transformed into transition 108 HOUSE_OVERSIGHT_013608

probabilities, so that the sum of the decimal fraction parts of all the boxes in each horizontal row add up to 100%, or as a real number, 1.00. Recall that in the example we’ve been using, the binary expansion of the natural number 729, the 1 3 transition incidence matrix is M; = 44 and its Markov matrix is top row, 1/4, 3/4 0.25 0.75 and bottom row 3/5 , 2/5, i.e. Mrp = Ane Od . Matrix multiplication of Mz» by itself repeatedly is equivalent to tracking the temporal evolution of the transition matrix’s probabilities until the resulting matrices move toward, converge onto, a steady state; each self matrix multiplication step represents what results from the passage of one unit of time. The convergence to equilibrium values is continuous and gradual. When the steady state is reached, both rows become identical. For this example, 0.5125 0.4875. 4 _ 0.4527 0.5472 g _ 0.4445 0.5554_ Mi x Mip or Min? = > Mip” = = te Sue" 9.3900 0.6100” ~——«0.4377-0.5622" 0.4443. 0.5556 te _ 0.4444 0.5555 = which for the first four decimal places remain the same for 0.4444 0.5555 tp additional times of self multiplication. Note the convergence of the top and bottom rows to the same asymptotic values. Books discussing the multiplicative and other behavior of these nonnegative matrices are numerous and frequently appear in matrix algebra texts under the rubric of the Frobenius-Perron theorems. Using the entropy formalism of Claude Shannon as developed previously, Hy is computed as the sum across either of the identical rows of each probability times its logarithm, px /og(p.p2x/og(p2)) remembering from above that we are working in base 2 logarithms and to change the minus sign (resulting from taking the logarithms of decimal fractions) to plus: Hy (Mip) = .4444 x log(.4444) + 5555 x log(.5555) = .9911 The nonuniformity of the box occupancy probabilities is reflected in the difference between the topological (maximal estimate) and metric (minimal estimate) entropies and is therefore quantifiable and computable: H7 - Hy # 0 = 1.00 - 0.9911 = 0. 0089. If the maximal and minimal estimates of the entropy were equal and all the probabilities boxes in each row asymptotically contained the same 109 HOUSE_OVERSIGHT_013609

probabilities as in M; = as es" it would retain these values across an infinite number of self multiplications such that Hy = .5 x log(.5) + .5 x log(.5) = 1 and Hr - Hy = 1.00 - 1.00 = 0.0. Complexity is a more general and variously defined descriptive expression than that of the topological and metric entropies and as such brings with it many kinds of definitions and computational approaches. One choice that’s intuitively appealing assumes that the relative complexity of an expression representing, say an outcome of an observation or experiment, is reflected in the minimum length of the most compressed program (algorithm) from which, given a suitable dictionary of symbolic equivalencies, one can reconstitute the original expression. Increases in what some have called algorithmic complexity, AC, are reflected in the growth of this minimally descriptive symbol series length. Karen Selz’s approach to compression and AC, similar to one proposed by Paul Rapp, involves the identification and symbolic representation of repeated blocks of symbols called words. For example, given an_ arbitrary, exemplifying binary — series: 011011101010001010101001001010011, we first find the longest repeated word [1010100] and represent it with the symbol, a, yielding a shortening in the original series, 011011a010a1010011. The next longest repeated word is [011] is replaced with b, yielding a further compression, 66a010a1010b. The next remaining binary word is of length equal to the previous one, [010], which, when replaced by c results in the series bbacat1cb. This can be further compressed to the final representation with four symbols and for the sequentially repeated b, one exponent of degree two, b*aca1cb. From this representation and a dictionary of letter equivalent words, the original binary expression can be recovered. For a quantitative index of the algorithmic complexity, AC, of the compression, Selz computes the sum of the number of distinct symbols plus the sum of the natural logarithms of the exponents: 4 + log (2) = 4.6931. The binary representation of 729, 1011011001, discussed above, is compressed by making two [101]’s = a and two 0’s = b resulting in a71b*1. Having three distinct symbols, a,o, and 1, and two exponents of two, its algorithmic complexity is equal to, AC = 3 + 2 x log(2) = 4.38. 110 HOUSE_OVERSIGHT_013610

In addition to H7, Hy and AC, if computable in a meaningful way, the deviation of the binary series under study from the idealized random behavior of a fair coin could serve as another index of complexity. Common descriptions of the amount of randomness in a series are indices of run length If a run length is defined by number of elements in a series of the same symbol before it stops, counting the number of run boundaries by reading along the binary series and counting the number of switches from 0 —+1 or 1 — 0, then the binary expression of 729, 1011011001, has six runs. The great analytic probabilist, William Feller, among many others, including the distinguished 18 Century Swiss family of mathematicians, the Bernoulli's, proved that computing a standard run score, srs, involves three terms, the theoretical expectation, E, of the number of runs, r, that is E(r), the number of runs actually observed, Obs(r) and the variance of the expectation of the number of runs, Var( E(r). If the srs is less than zero, then the binary series is more random than that resulting from the flipping of a fair coin. Interestingly, when a normal group of subjects are instructed to simulate what they think of as a random coin flip determined series of 0’s and 1’s, their srs tends to be lower than zero, over-estimating the degree of irregularity that randomness represents. Long runs occur by chance far more often than intuition would dictate. If srs is more than zero, than the binary, coin-flip series is more ordered than random. If srs equal to zero, the binary series is not discriminable from fair coin flipping randomness. The expected number of runs, E(r), can be estimated by a fraction formed by twice the product of the number of heads times tails divided by the sum of the 2x6x4 heads and tails to which is added one. That is, E(r) = ZA + +1=5.8. The average variation around this expectation called the variance, Var, of the expectation, Var(E(r)), is estimated by a fraction formed by (take a breath) twice the product of the number of heads times tails x twice the product of the number of heads times tails minus the number of heads and minus the number of tails, all over the product of the sum of the heads and tails squared, times the sum of the number of heads 111 HOUSE_OVERSIGHT_013611

(2x6x4)(2x6x4-6-4) (6+4) x(6+4-1) E(r)—Obs(r) _ 5.8—6,0 \Var(E(r) V2.03 the standard run score of the binary series is less than zero and therefore more and tails minus one. That is, Var(E(r)) = = 2.03. From these =-0.140. We conclude that three terms, we compute srs = random than the expected random behavior of a fair coin. Recall from the last chapter that Karen Selz, Martin Paulus and others have shown that various personality types and psychiatric diagnoses are associated with characteristic deviations of srs from zero. When the winners of the 2002 Annual World Rock, Paper Scissors Championships held in Montreal Canada were interviewed, they said that sensing their opponent’s characteristic style of deviations from randomness in what we would call the continuum from maximal to minimal entropy determined their successes. We characteristically use all of these measures to estimate quantitate the deviation from randomness standard run score, Srs, algorithmic complexity, AC, as well as H; and Hy,, the topological and metric entropies. The encounter with mystical Absolute Reality, though sought by arduous contemplative and other practice, emerges spontaneously, most often during times of apparent mental emptiness, detachment, a state in which rationally instructive thought and the choral background of brain voiced, emotion-ladened, commentary have disappeared into the entropic soup of formless silence. It is this indescribable, ineffable, stillness that we think serves as the psychophysiological anlage of mystical experience. The mathematical systems yielding quantitative metaphors, descriptive ideas about dynamical entropic statistical emptiness and form inspire the use of mathematical structures in place of localized lumps in brain meat as personalized icons of doing. Our wedding of well-defined mathematical objects to metaphoric elements of more general nonverbal intuition has a long tradition. Rene Thom’s 1990 book, 112 HOUSE_OVERSIGHT_013612

Semiophysics, discusses mathematical mechanisms and their representations in mind and the real world, analogizing mathematical objects and the intuitions they generate to mechanical tools. Similar ideas are found among the four liberal arts of the ancients: Number, Geometry, Music and Cosmology. The epistemologies of all four require, then and now, the intuitive use of mathematical objects, conscious or unconscious. Examples can be found in conceptual issues of Geometry and Number with implications for relationships between man’s physical and psychological worlds. One set of articulations were attributed to the shapes of Platonic solids found among the Neolithic stone circles in Aberdeenshire, Scotland, 2000 years before Plato. Each symbolized particular physical and psychological themes. All manifested equal edges and every face of each solid was the same perfect polygon. The solid with four equilateral triangles manifesting four vertices and faces, the tetrahedron, represented the physical element, Fire, and the personal psychological climate of a choleric, fiery nature. A Platonic solid composed of eight equilateral triangular faces, two tetrahedrons annealed, the octahedron, signified Air in physical composition and optimistic hopefulness in psychological disposition. Six square faces together making a cube, evoked the elemental physical component, Earth, and its human expression as a phlegmatic, apathetic personal style. Twenty faces, all equilateral triangles, constitute an icosahedron indicating Water and a dominant feeling state of melancholic sadness. Like onomatopoeic words and pictorial script, the three dimensional geometry of these Platonic solids feel like what they came to symbolize. The personality styles symbolized and evoked by the Platonic solids continue to be used to this day. For example, they compose the basic elements of the constitutional categories of remedy in homeopathic medicine as introduced over 200 years ago by Dr. Samuel Hahnemann in his classical Organon of the Medical Art. The assignment of clinical remedy in homeopathic treatment combines consideration of the presenting physical symptoms and signs, the what, with intuitive discernment of the patient’s constitutional type, the who. To the homeopathic physician, tetrahedral fire is suggested by the traits of personal magnetism, courage and inspiration as well as egotism, strong desire and rage. 113 HOUSE_OVERSIGHT_013613

Octahedral Air people intellectualize objectively in confident and insensitive aloofness. Those symbolized by cubic Earth are realistic and practical, a what-you- see-is-what-there-is belief along with rigid, materialistic ways. /cosahedral Water types experience emotions strongly and are sensitive, intuitive, nurturing and can be overly sensitive and dependent. What intuitions and observations relevant to self, subjective and objective, emanate from the stylistic properties of feelings as derived from a time series of observations of their associated actions suggested by their statistical measures, Hr, Hu, AC and srs? The yield is rich and unexpected. We find an enjoining of values of these measures, characteristic and invariant for each person, with the brain and behavioral actions of entheogenic agents and Zen meditation in contrast with worldlier focused attitude adjusting experiences and drugs. The range of their potential values helps rationalize a person’s inclinations along the continuum of attachment and detachment. This quantifiable dimension augers positively and negatively with respect to the requirements for mystical experience as poetically described by the ambivalent warrior prince, Ardjuna, in conversations with Lord Krishna in the Bhagavad-Gita, the most famous and influential component of the Mahabharata of Hindu scripture, A similar theme relevant to the occupancy of a propitious range of values for the measures, H7, Hy, AC and srs, is found in what is often called the Second Nobe/ Truth as explained by the Buddha, Siddhartha Gautama, in lectures recorded in a deer park near Benares. We begin with the results of some drug experiments conducted by behavioral neurophysiologists and end with suggestions about the intuitive relevance of the conceptual content of these measures to the universals of mystical experience and perhaps to elements of spiritual transformation. As described previously, the brain and behavioral process of habituation is characterized by a decrease in the strength of an observable response to the repetition of an evocative stimulus. Imagine the decrease in our startle responding when a once unexpected loud noise continues to occur. Sir Charles Sherrington, the early Twentieth Century British pioneer in neurophysiology showed that animals and humans gradually stopped the withdrawal of their limbs with stimulation of its 114 HOUSE_OVERSIGHT_013614

skin when it was repeated several times. Columbia University’s Nobelist in the brain sciences, Eric Kandel studied the neural mechanisms of habituation as a primitive, accessible and fundamental example of learning, the association of a nonresponse to a usually evocative stimulus, in Aplysia californica. The sea snails learned not to respond to a local irritation with a gill-withdrawal response when exposed to it many times. They learned to stop paying attention to the perturbation. The background noise appears to disappear after a little time in the Mall. Though his exploration of its synaptic mechanisms involved the neural circuit of the gill-withdrawal reflex in the marine snail, its generality and human relevance is well established. Hundreds of papers can be found reporting the results of studies of habituation in normal humans under all kinds of circumstances as well as in psychopathological conditions. That it samples something both fundamental and persistent is suggested by studies in children by one of Kandel’s students, Michael Lewis. He found that the rate of habituation of a startle response to a bright light in one-year-old human infants predicted success in many kinds of learning and other cognitive functions when the children were tested again at the age of four. Pavlov’s experiments studied habituation of the classically conditioned salivary response to meat powder- coupled bell sounds in dogs in which the bell was followed by nothing, not only led to inhibition of the salivary response with unreinforced trial repetition but generalization of the inhibitory state such that dogs were observed to freeze in motionless catatonic states for hours. In the language of our statistical measures, the fixation of the dog’s behavior would manifest minimal entropy in the form of H7 = Hu = 0 and the lowest complexity values for AC and srs. Entheogenic agents like LSD or mescaline inhibit the process of habituation and fixation, maximizing the entropy of behavioral measures, H7, Hy > 1 and high complexity values for AC and srs. Mark Geyer and David Braff, Professors at the University of California in La Jolla and Michael Davis, a Professor at Yale’s School of Medicine, found that entheogenic agents, such as mescaline and LSD, as well as naturally occurring indoleamines, such as DMT, which occurs naturally in human brain, prevented habituation of startle responding in mammals. Each sound repetition was treated as 115 HOUSE_OVERSIGHT_013615

though it were new. The baby is Buddha is an Eastern philosophical aphorism that captures the fresh spiritual state of each moment’s openness and readiness, the in- between entropies for new information surprise. Geyer and Martin Paulus found that entheogenic agents such as Ecstasy also increased the complexity of the patterns of spontaneous motor movement made by rats exploring a bounded space. Recall that they partitioned the floor to document the exploratory motion in the context of a sequence of location transitions, readying the data for the computation of some of the measures previously described. Following the administration of entheogenic agents, the partitioning of the space that the animals were exploring, into a lattice of discrete boxes and the encoding of each square with a symbol, the computable entropic and complexity measures such as H;, Hy, AC and srs were increased. In contrast, the administration of amphetamine-like stimulants led to a different kind of behavioral activation than that induced by entheogenic agents. The measures of Hr, Hu, AC and srs reflected decreases in entropy and complexity. As University of California’s David Segal and others documented in the 1960’s, high doses of amphetamine led to animals into in a minimal entropic state, they were frozen in stereotyped rocking, nodding and circling motions. High dose amphetamine-treated humans develop rigid fixation of ideas, low H7, Hv, AC and srs, in man this is seen as inescapable obsession and paranoid delusion. There is considerable medical evidence that Hitler took large doses of amphetamine (Benzedrine) daily for the last 20 years of his life. The entheogenic drug-induced phenomena of naive openness and absence of fixation, states of high entropy and complexity, behavior generating higher than control measures tending toward maximal values of H7, Hy, AC and srs , are subjectively reflected in the results of personal experiments of University of Chicago’s Heinrich Kluver as described in his Mescal and Mechanisms of Hallucinations (1966). Observing himself after the self administration of a crude preparation of peyote cactus, he said that it led to glad feelings of unfamiliarity and a marked reduction in his tendency for boredom (habituation), a detachment from old ways of thinking and a new openness to a rush of seen again for the first time experiences. Everything in his personal world, no matter how mundane, became a 116 HOUSE_OVERSIGHT_013616

source of new interest and fascination. New thoughts replaced old ideas in a continuing process of new formulation. All of these things feel like they emerge spontaneously, making ideas about being born again and personal renewal concrete. We remember that Timothy Leary and his wife in their privately circulated pamphlet, Neurologic, described their entheogenic drug-induced escape from the habitual order as supported by the learned and established “...mental-manipulative and socio-sexual brain circuits...,” an escape to a fresh new planet of possibilities. Louis Lewin, the early Twentieth Century German pioneering ethnopharmacologist described his subjective responses to peyote as a flood of lively, numerous, random fantastic creations of perception and thought, all demanding his fresh attention. To complement these subjective reports, experimental tasks involving habituation, such as the disappearance of a brain wave sign of arousal to sound or light stimulation, called alpha blocking, the eyes-closed resting pattern of 8-14 cycles per second, hz, waves perturbed into the arousal pattern of >20 hz, did not habituate when the subjects were pretreated with entheogenic drugs. This finding was also true for the results of years of meditative practice. In his 1974 Psychophysiology of Zen, Hirai reported that Soto Zen monks, after many years of practice in mindful, one pointed, be here now meditation, unlike normal controls, continued to show alpha blocking surprise, brain wave arousal patterns, throughout the course of repeated stimulation with auditory clicks. James Austin in his monumental book, Zen and the Brain (2000) summarizes other studies of habituation in TM practitioners and other mediators in which eyes open versus eyes closed, the set and setting and variations in other experimental variables blurred these results to some degree. He develops the case that years of meditation-induced brain states of emptiness, we would say of maximal entropy and minimal form, set the stage for the ecstatically insightful flood accompanying the sudden insight into a Zen koan’s solution or the transcendent startle induced by a roshi’s shout. A meditative struggle concerns how one can think about not thinking. That is, thinking of nothing. This is generally thought to be the most important part of Zen meditation, called zazen. Achieving high values for brain and behavioral Hr, Hm, AC and srs supply the formless infrastructure for ecstatic transformation. 117 HOUSE_OVERSIGHT_013617

In healthy people, an awareness of self is not lost during this time of invasion by and fusion with what feels like an independent agency. At full force, the mystical experience is transfixing, tending to paralyze movement and speech, and at the same time bringing with it the capacity for clear sensory and sensory-integrative lucidity. This new seeing brings previously unnoticed things to attention and makes old things new. Perhaps most striking is the passive (unsought) experience of the unification of erstwhile disparate, apparently unrelated thoughts and feelings. The yield can be the sudden emergence of deep relationships between apparently very different constructs, beliefs and formalisms leading to unanticipated and unsought integrative connections. In mathematics, this experience can lead to entirely new kinds of theorems and proofs; in the physical and biological sciences, a previously unseen organization of the data generating new global relationships and potential scientific laws. In our spiritual life, the ineffable richness of the direct experience of God. Mysticism-negative interpretations of these experiences have always been attendant. To the extent that the mystic’s inward turn is seen as a detachment and implicit derogation of the external, consensually real world, it is often seen as alienating from established institutions of religion and government. Psychoanalytic practitioners may label it a regression to primary narcissism. Most churches tend to discourage its practice as counter to the dominant social hierarchy and _ its governance. Governments pass laws against its practice and manifestations, a current example being modern Chinese governmental reactions to the Tibetan Buddhism of the Dalai Lama and the yogic practices of the Falon Gong. Agencies of established society such as the institutions of licensed medical practice make the dominance of the inner world of mysticism subject to diagnoses ranging from the narcissistic character disorders to interpretations of the reported extraordinary experiences as manifestations of schizophrenia, manic-depressive disorder or temporal lobe epilepsy. Rejection and fear of the transcendent states lead to uninformed and politicized anti-narcotic laws, grouping heroine and cocaine with the entheogenic (recall: engendering connection with the sacred within) agents such as the Huichol Indian’s peyote and the Amazonian Indian’s yage, obstruct and socially 118 HOUSE_OVERSIGHT_013618

taint the personal use of plants and practices that facilitate access to the mystic way. Rational, socially responsible and otherwise kind and tolerant Presbyterians, Unitarians and Reformed Jews can be suspicious and rejecting of what appears to them as the politically tinged mass hysteria of praying in tongues and other rituals of Charismatic Christian rebirth and renewal or the ecstatic states of Orthodox Jewish chant-dancing. Modern brain and behavioral scientists, remaining under the philosophical spell of logical positivism and its requirement for operational definitions and (external) experimental disconfirmability, operate from the position of strong doubt when mystical experience is addressed. What is striking and strange about how science plays the game of mysticism research is exemplified by the publishable increment in credibility concerning a meditation-induced change in state of consciousness when Boston’ University’s William Benson reported the accompanying relaxation response, a sudden decrease in heart rate---much like the dive reflex of a seal or what the heart rate does when you duck your head suddenly forward into a sink full of water. Decades are spent getting professorial tenure for research yielding things we have already experienced and know directly and for ourselves. Recall that the existence of visual imagery in the human, doubted by an experimental psychology of the time in which William James _ self-exploratory observations were viewed as revolutionary, was made more credible by evidence for the existence of a subjective spatial metric: verbally reporting subjects, when timed, took longer in their minds to go from one room to another one that was down the hall then going to the room that was immediately next door. We use brain chemical, pharmacological, neurophysiological and neuroanatomical localization and computation of characteristic statistical patterns in time dependent brain and behavioral observations to the same end. Further Readings for SOME ENTHEOGENIC ENTROPIES 119 HOUSE_OVERSIGHT_013619

Phantastica: A Classic Survey on the Use and Abuse of Mind-Altering Plants. Louis Lewin, Park Street Press, Rochester, Vermont, 1998 (First published in 1924) Indole(ethyl)amine N-methyltransferase in the human brain. M. Morgan (Poth) and A. J. Mandell, Science 165:492-493, 1969 Enzymatic formation of tetrahydro-beta-barboline from tryptamine and 5- methitetrahydrofolic acid in rat brain fractions. L.L Hus and A.J. Mandell, J. Neurochemistry 24:631-636 The Sacred and Profane, The Nature of Relgion. Mircea Eliade, Harvest Books, Harcourt, San Diego, 1957 Hashish and Mental Iliness. J. J. Moreau, Raven Press, N.Y. 1973 (First published in 1848) The Neurochemistry of Religious Insight and Ecstacy, A.J. Mandell in Art of the Huichol Indians, Fine Arts Museum of San Francisco, Abrams, N.Y. 1978 Altered States of Consciousness: A Book of Readings. Charles T. Tart, John Wiley, N.Y. 1969 Soul; God, Self and the New Cosmology. A. Tilby, Doubleday, N.Y. 1992 Pihkal: A Chemical Love Story. Alexander Shulgin and Ann Shulgin, Transform Press, Berkeley, CA 1991 Psychochemial Research Strategies in Man, A. J. Mandell and M.P. Mandell, Academic Press, N.Y. 1969 120 HOUSE_OVERSIGHT_013620

The Biology of Transcendence, J. C. Pierce, Park Street Press, Rochester, Vermont, 2002 Psychiatry and Mysticism, S.R. Dean, Nelson-Hall, Chicago, 1975 Zen and the Brain, James H. Austin, MIT Press, Boston, 2000 Perspectives in Biological Dynamics and Theoretical Medicine. Eds. S.H. Koslow, A.J. Mandell and M.F. Shlesinger, Ann. N. Y. Acad of Sci. Volume 504, 1987 Consciousness and the binding problem, W. Singer, Ann. N.Y. Acad. Sci. 929:123- 146. Mixing properties in Human Behavioral Style, Karen A. Selz, U.M.I., Ann Arbor, MI. 1992 Dynamical Systems and Ergodic Theory, M. Pollicott and M. Yuri, London Mathematical Society, 1998. Introduction to the Modern Theory of Dynamical Systems, A. Katok and B. Hasselblatt, Cambridge University Press, Cambridge, 1995 121 HOUSE_OVERSIGHT_013621

CHAPTER 6: PENTECOSTAL PHASE TRANSITIONS By their late teens, my two offspring, sons of an Alcohol Anonymous, born again, originally Christian Science mother and a spiritually struggling and mostly secular Jewish psychiatrist father, had been unfulfilled in their hungry search for the experience of a personally meaningful God. After years of perhaps too academic conversations with their parents, visits to a variety of houses of worship, talks with University of California religion professors and evenings with a Ph.D. psychologist- rabbi and friends at the neighborhood synagogue, they turned somewhere else. Some of their high school friends who were Evangelical Christians took them to their Assembly of God, Pentecostal and other Christian, direct experience of God, churches. They came to love what they sometimes called their Wednesday night and Sunday morning “rock and roll,” services. Struggling with the post-Vietnam cynical mistrust of authority and the Marijuana apathetic nihilism of the 60’s and 70’s, and clearly not enticed by what they regarded as their father's vacuous mélange of New Age Eastern Religions and secular brain science, they spoke about their sudden and life-changing experiences. They studied, memorized and quoted the Scriptures as part of their commitment to their word churches. As erstwhile cynical teenagers, now positive and brimming with faith, | secretly called it denial, they described what was happening to them as New 122 HOUSE_OVERSIGHT_013622

Birth. They told me that, paraphrasing Paul in Romans, they had been saved and were living New Life, not earned by good works as in Hebraic Law, but by faith through God’s Grace. Jesus had “paid their bills’ through His sacrifice at Gethsemane. They both tried to explain inexplicable feelings of new energy, the unseen hand of spiritual guidance and peace. One told me that the wind of the Holy Ghost had taken him to the front of the pulpit, tearfully, thankfully, on his knees, to accept Jesus as his personal Savior. They described how they had opened their lives to the spiritual strength of /iving in Jesus. Many things about them changed: their tastes in food, from hamburgers to vegetables and fruit; from the jazz of John Coltrane and McCoy Tyner and the cynicism of Frank Zappa’s “...only fourteen and knows how to nasty...,” to playing strum guitar and singing the hymns of Wednesday night healing services; from t- shirts hanging out of raggedy, Southern California, boutique store purchased, stressed jeans, to polished dark shoes, starched white shirts and gray or tan khaki slacks, sometimes with ties. They became cool, respectful, rational and more distant with me. They repeated often the scriptural story about young Jesus, accidentally separated from his parents on a visit to Jerusalem. When by standers asked Him about where His parents were, He answered, “| have no mother and father.” They told me that they, like God’s son Jesus, were filled to completeness with the Father and the Holy Ghost. On one hand, their experiences sounded like those of the activated mind state of Abraham Abulafia, a suddenly emergent Nevesh and my father’s metaphysical talks about personal transformation. My personal secular- computational brain God spoke to me of the mechanisms of sudden personality change, a phase transition in complex systems, in the context of the nonlinear dynamics of brain and behavior. On the other hand, their global changes in mind felt both alien and threatening. When | came to learn their churches’ full list of expectations, rules, requirements and sociopolitical policies, | found that | could not identify with this system of spiritual knowing at all. It felt rigid, righteous, unforgiving, even angry, and it frightened me. | never anticipated that my culturally enriched, intellectually sophisticated sons would be quoting Pat Robertson and Jerry Falwell. 123 HOUSE_OVERSIGHT_013623

The Freudian psychoanalyst of my younger days tried to write off these (to me) cataclysmic changes as manifestations of male sons’ unconscious oedipal strivings to father kill and thus become. After some mulling, my theory did not wash. They spent time accompanying themselves on guitars, singing hymns and shouted Corinthian Paulisms to small curious crowds gathered in beach parking lots, city parks and inner city street corners of Southern California. They passed out pamphlets containing New Testament tracts and formulaic aphorisms promising the post-repentance blessings of Jesus. The eldest, articulate, bright and prematurely worldly, had been an ardent memorizer and appreciator of Shakespeare, especially the mystical Tempest, the music of Aaron Copeland and Igor Stravinsky, the improvisations of Charlie Parker and Cannon Ball Adderley and the provocative literature of the time including Jack Kerouc’s On the Road and Hunter Thompson’s Fear and Loathing in Las Vegas. They loved riffing with the Voltairean pungency of Frank Zappa’s lyrics. Now, nihilistic humor had become an anathema. Several weeks after my eldest son’s transformation, | found him in the garage using a hammer and an empty barrel for disposal as he destroyed his modern jazz and early rock record collection. He ridded himself of all of his fiction and most of the nonfiction books in his young but relatively large personal library. His new energy and high purpose emerged as a clearly defined set of rules of behavior, a strong stand against abortion, frequent talk about the need to escape from the contaminating influence of MTV culture, as well as our years of talk about the biological and physical sciences. Both boys were particularly critical of my Darwinian flavored attempts at scientific explanation of man’s inner life using the selective and adaptive neurobiology of brain mechanisms and behavior. They spent increasing amounts of time with Church friends, seldom seeing their old ones. The eldest’s college goals turned from plans for a U.C. Berkeley equipped career in literature and creative writing to a none spiritually challenging, objective and practical, Christian free market finance and accounting degree from U.C.’s Business School. Gone were shared magical hours of intellectually stimulating, humorous, even scholarly discussions. In place of evidential talk in areas of philosophy, 124 HOUSE_OVERSIGHT_013624

literature and science, their opinions and claims derived exclusively from biblical quotation. Their particularly favorites were Paul’s letters and some of the later prophets, particularly Jesus-auguring Isaiah. “In the beginning was the word...” became the real reality. The meaning of life was Scripture as explicated by their book church pastors. They scribbled notes in the margins of their Bible pages during sermons They were displeased when | interpreted the wild imagery and 666 symbolism of Revelations from the point of view of the historicity of encoded political messages, meanings hidden for the safety of the early Jews in their world of Greco- Roman governance. Twenty-five years, before the glut of books by Tim LaHaye, my well-educated sons claimed that Revelations was literal and foretold the coming tribulation that augured the end of the world and ascension to heaven of the believers. My youngest, since childhood a well-read history buff, now viewed New Testament scripture as sui generisly, divinely and literally true. They said the conduct of their lives their meaning had been clarified by the biblical truths revealed to them by The Book. What | did not say was that much of the talk seemed to me to be an intellectually and spiritually impoverished miasma of cant and righteousness. At the same time, their remarkable transformation appeared to be the expression of a powerful and mystical force, the scientific understanding of which has been the ostensible focus my life’s work. Why did their alterations appear so alien, strange and forbidding? Born to a home of psychoanalytically and scientifically oriented political liberals, these precociously bright and worldly sophisticated young men were suddenly transformed into, unrecognizable to me, radical Christian Fundamentalists. They are now in their late thirties and remain just as ardent, Christian patriotic, Right Wing voters to this day. The eldest is now an executive in Morris Cerullo’s San Diego based, worldwide missionary movement, raising money for revival and media ministries. He travels to and is involved with hundreds of Fundamentalist Christian churches in countries ranging from Argentina and Africa to the Middle East and Russia. He hasn’t allow me to contact his children, my grandson and granddaughter, because, in vague talk and mostly silent implication, | and people like me are seen as sources of potentially satanic, worldly 125 HOUSE_OVERSIGHT_013625

contamination. He feels wronged by the way | am. He once chided me about what he saw as my futile spiritual search in what he called the “health food” Eastern and brain religions. My youngest, only a little less ardent and critical, visits occasionally, and, hands in the air and speaking in tongues, prays to the Lord for my salvation. Of course, this sudden and long lasting personal transformation in the direction of Fundamentalism is well known and almost commonplace in modern American and European Jewish, Christian and Moslem college educated middle class families. The Saudi Arabian World Trade Center bombers were, mostly, well supported children of the educated middle class We recall the famously tragic American radical Moslem, Richard Reid, the would be airplane shoe bomber. My stomach clenched as | heard Richard’s sophisticated and obviously caring father share his confusion and struggle to rationalize what had happened to his son. The commonality of this kind of spiritual and life transformations in the educated young makes each event no less painful. On the other hand, we know that healing transformations in the name and spirit of the Christian God can lead to quite positive realities. They are effective in even quasi-secular disguise as in Alcoholics and Narcotics Anonymous, Synanon and in the rehabilitation of the Charismatic Christian, ex-alcoholic, Southern Methodist politician, George W. Bush. Paul Holmer, Professor of Theology at Yale Divinity School gives thanks to the evangelicals who “...keep alive the radical breach that the gospel is from the nous of this world...they (Fundamentalists, Evangelicals) look marginal if you are churchy...intolerant if you are ecumenical...anti-intellectual if you are trying to systematize... in their roughness and ...abrasiveness.” | bring personal and painful witness to these claims. To get to the personal meaning and mechanisms of these transformations, | had to start from somewhere. | am wedded to the belief of the Jewish ecstatic, Abraham Abulafia, and not those of Moses Maimonides, that the human mind in an altered state of activated intellect, man’s Nevesh, can understand such mystical happenings. | would continue to work at it. One of the early personal church experiences with my sons’ religious path came after accepting an invitation to go with them to a Sunday service at their current charismatic church. By then, the eldest was married with children, the 126 HOUSE_OVERSIGHT_013626

youngest, unmarried, was teaching bilingual mathematics in high school. | had waited several years for this occasion.. The meeting took place in a large, gray, unmarked warehouse building that was crowded in back with high stacks of storage cartons. The large, cement floored, open space in front of the storage boxes was occupied by rows of metal folding chairs. They faced an unadorned, elevated wooden platform upon which was a lectern and microphone. Behind the lectern stood a casual array of a dozen or so young people, singing hymns and playing a variety of instruments. These included piano, two or three guitars and upright bass, tenor saxophone, trombone, trumpet, mouth organ and two snare drums. Sounding a bit like a Salivation Army Band, they played and sang, “They cast their nets in Galilee just off the hills of brown; such happy simple fisher folk, before the Lord came down...the peace of God, it is no peace, but strife closed in the sod. Yet let us pray for but one thing, the marvelous peace of God.” The building, used for commercial storage, packaging and mass mailings during the week and a Charismatic Christian word church on Sunday, was located at the rear of an unfinished strip mall. A new and well-polished yellow Cadillac Deville was the only vehicle parked in the no parking zone immediately in front of the entrance to the warehouse. My youngest explained that the car belonged to Carl Austin, the self-discovered and declared pastor, who spontaneously rose up to lead without academic religious training or a conventional ordination. The bright yellow car was explained as evidence of the power of God. Paraphrasing Mark, my son told me “...he who does not doubt in his heart and believes that those things he says will come to pass, he will have whatever he says...whatever things you ask for when you pray, believe that you receive them, and you will have them.” The car served as a glorious instantiation of the church’s major promise of the rewards of faith. Pastor Carl Austin, a tall, blonde, portly man in his early thirties with a resonant tenor voice, was the youngest of several children of a poor Midwest farm family. He had been a state college drop out and without a career or a job. His sermons contained stories about how he had caught spiritual fire at a revival meeting conducted by Kenneth Hagin of Kenneth Hagin Ministries, aka Rhema 127 HOUSE_OVERSIGHT_013627

Bible Church, Tulsa, Oklahoma. The pastor's witness of the Holy Ghost acting through his life was his personal cure, by transformative Grace, of a triad of self- destructively sinful addictions: alcohol, gambling and promiscuity. Self-chosen and self-declared, he now served this two and a half year old growing congregation of over 200, mostly young, working families. The young men in attendance at the warehouse church were in shirts and ties, very unlike the more casual garments of even dressy occasions in Southern California at that time. Women were dressed simply and modestly. Most of the children were in Sunday school in a small neighboring store in the strip mall during the adult service. The few that accompanied their parents were remarkably well behaved | was told that most t families tithed 10% of their income. They quoted Hebrews, “...king of the righteous...to whom also Abraham gave a tenth part of all....” They believed that their tithe would be returned manifold and the yellow Cadillac Deville served as Pastor Carl Austin’s personal evidence. From these funds, the congregation supported the pastor, his car, the rental expenses of the Sunday warehouse church and an orphanage in a small Mexican border town. Some of these children, several neurologically disabled, were bussed to the Sunday service for healing. They sat together in a section in the front of the congregation and were the beneficiaries of the second Sunday collection plate, passed around after the first one that was designated for the church and its pastor. The first Sunday sermon | heard in the warehouse followed several awkward minutes of Pastor-directed warm up hugs of neighboring strangers while the choir sang hymns. The songs were accompanied by instruments playing the melody in unison sans harmony, and accented by the beats of two loud drums. As the volume and pace of singing increased, | saw several episodes of ecstatic looks and fainting, dying in the Lord and shouts of praise with upraised hands. The intermittent elevation of the hands during prayer and song appeared to be spontaneous. | was told that the arms were up as antennae, feeling the energy of Lord all around us. tt The pastor’s topic was forgiveness. From Ephesians, “...let all bitterness, wrath, anger, clamor, and evil speaking be put away from you, along with all malice... be kind to one another, tender hearted, forgiving one another, just as God in Christ 128 HOUSE_OVERSIGHT_013628

also forgave you.” In the middle of his sermon, which built slowly in tension and volume, the pastor introduced a forty-ish, sparkly eyed, somewhat overweight, dark haired, slightly made up woman who the Pastor said was a witness for the ultimate in Christian forgiveness. She was someone from whom all of us could learn. She was the mother of the 7-year-old boy that he, the Pastor, had, four years before, accidentally killed during a drunken driving episode in his “other life.” That was the one he was living before he was saved. | was told that he presented her in a service at least once a year. The woman said that her successful struggle for forgiveness led to her being saved. She quoted Ephesians, "And you who were dead in trespasses and sins hath he quickened.” She looked radiant and hugged the pastor. When my sons introduced me to him as we filed out at the end of the service, the pastor told me that my visit was important to the congregation. He told me that Jews were special in Charismatic Christianity since we would play an important role in the return. He said he hoped he would see more of me. My boys seemed pleased to have invited me. | accompanied them to their church most Sundays, and often for what they called the “rock and role healing services” on Wednesday night, for over two years. Within three or four months | found myself, the first time while awakening out of a deep sleep, mumbling sounds that | was told sounded like some unknown language, | was praying in tongues. At some services it happened spontaneously accompanied by an almost ecstatic feeling accompanying the surrender of willful control. This was usually accompanied by the release of new energy. | recall thinking that the spontaneous, nonsensical linguistics shorted out my verbal and obsessionally logical left brain allowing the unbridled expression of my hysterical right brain. Sometimes in agreement with an insight offered in a sermon or when particularly moved by a hymn, | found my hands lifting skyward, right hand and arm higher than left, with a high feeling of trust and delicious surrender of conscious cognitive control. Reading the New Testament’s Acts, | learned that we were re-enacting the scene of the Apostles in the upper room. Those gathered there were the ones chosen by the risen Jesus to be able to see Him, the list including Peter, James, 129 HOUSE_OVERSIGHT_013629

John, Andrew, Philip, Thomas, Bartholomew, Matthew, James, Simon and Judas. “_..they were all filled with the Holy Spirit and began to speak with other tongues, as the Spirit gave them utterance...” The secular psychoanalyst in me tried to make an analogy with the joyful jazz lyrics of Ella Fitzgerald’s scat singing, I’d done a little of that during my small jazz group pianistics as an a adolescent. | thought about how verbally paralyzed stutterers could be articulate when singing what they mean when they could not talk it. | wondered about the relevance of the spontaneous poetry of slams and Hip Hop rapping. We attended what my sons called charismatic black Baptist churches in South Los Angeles and Long Beach. These often four hour services usually featured two wonderfully harmonic echoing choirs with organ and drum punctuation of the speech-singing, sermonizing Reverend. Large and beautifully dressed black women sang operatically and danced gracefully down the aisles. | joined my sons in this joyful noise for these long services and, exhausted, | was forced to go home for a Sunday afternoon nap. In spite of what could be regarded as validating experiences with the real life Holy Spirit, | continued to be generally confused and even more deeply estranged. An inner voice kept recalling my spiritual failure as a parent and being traitorous to my Jewish ethnic identity by Christian church attendance. | tried to understand how my sons had traveled from where | thought we were living together to this entirely new world. How did it happen? Could the path going there and back be meaningfully reconstructed and then reversed? This idea is consistent with the medical dictum that knowing the cause, the treatment logical follows. My education had shown me such assumptions of reversibility need not be true. Contrary to the beliefs of early physical mechanics, medical psychiatric history takers and psychoanalysts reconstructing childhood events, the modern physics and mathematics of complex systems says phase transitions in complex systems are probably not reversible, at least not simply so. One of the features of global changes in complex systems, often called bifurcations or phase transitions (think heated water going suddenly to a boil), is their dramatic discontinuities in behavior. Knowing only the initial and end state, phase transitions in complex systems do not allow for point-to-point backtracking or specific linear-causal 130 HOUSE_OVERSIGHT_013630

understanding. These discontinuous and global transformations are the stuff of miracles, especially for physicists. Even with respect to initial and end-states, rather than using straight forward phenomenological observation, the mathematical and physical theories of phase transitions are usually dependent on not necessarily intuitive, derivative physical quantities. Their verbal representations are often not concrete but metaphoric. This retreat to derived and abstract, far from the primary data computables, may be more evidence of man’s many insufficiencies in understanding of the mysteries that are often placed in the spiritual realm. Driven by an effect that contributes to cause, like the faith-driven abandonment to God that generates more faith, a drop of water hanging from a faucet is pulled down by its own gravitational field as the thinning neck of the drop facilitates its own further thinning. A gobbet connected by a thick neck to the main drop begins to separate. The neck between them thins and breaks, and one becomes suddenly and irreversibly two. A continuous structure has suddenly become discontinuous in finite time at what is called a singularity. Since the single measurable feature that dominates the water’s behavior around this singularity is the diameter of the thinning neck, a derivative physical, one-dimensional observable, neither the details about where it all began (called the initial conditions) nor the path it followed to get to the moment of fracture, are predictively relevant with respect to the sudden transition. Considering this kind of phenomenon going on in our brains, choosing between theories of behavior that involve changes in brain cell groups and/or brain chemicals versus those that involve behavioral quantities, may be neither possible nor necessary. The challenge is to place the problems of cataclysmic change in brain and behavior in sufficiently abstract and universal terms that can be represented in some low dimensional, computationally accessible space of variables. The simplification and stereotypy of behavior around singularities reduce the number of features that are required to discuss the dynamics of change in what 131 HOUSE_OVERSIGHT_013631

would otherwise be a complicated beyond reach situation. One of the properties found around singularities, is the Joss of absoluteness in contextual characteristics such as the scale of the observation. We no longer can say that what we are studying happens in inches or miles, in seconds or days, now or in the past. In the place of a single unit of relevant measurement, we have a distribution of spatial and temporal feature sizes that stretch toward both the infinitely small and the infinitely large. We can illustrate a dynamical transition involving the passage of the system through a singularity by using the metaphor of another kind of water experiment. If we pour a small amount of water through a filter full of coffee grounds, or watch our coffee maker do it, the first spurt of water makes an incomplete path of wet grounds in the bed of dry ones. The next bit of water soaks this path more thoroughly and may form additional and multiple, new and branching, incompletely penetrating paths. Eventually, on just one more of these pourings, a connection in the paths occur, such that the water snakes all the way through the coffee grounds and the first brown drop of coffee falls into the pot. At this flow singularity and opposite to the dynamic of a faucet water drop, a discontinuous system of pathways becomes continuous in finite time in a process called percolation, Trying to set up a predictive model, we can count the number of water deliveries that occur before the first drop finds its way through. Repeating the experiment many times yields a span of the number of pours required to reach the singular point of percolation. If we do the experiment enough times, the distribution of the number of pours required to reach percolation will range from one toward infinite. In the neighborhood of the transition, time as recorded as the number of small pouring events may stretch. In aa comparable system, as elegantly described by Detrich Stauffer in his Springer-Verlag book on percolation, multiple hot spots in the woods can suddenly fuse into a forest fire. Isaiah said, “...glorify the Lord in the growing fires of dawn...” Faith fires spreading through a faithless dense forest, its hot irregular front damped by the disbelief of water-filled leaves, or disillusionment gaps of already burned out trees, can, under the right motivating conditions of dryness, wind velocity, tree 132 HOUSE_OVERSIGHT_013632

density, kindling temperature and desperation-induced willing of faith, sweep through the entire woods in a sudden blaze. This is the spirit of percolation. Computer simulations of percolating blazes generate a multiplicity of life times of forest fires near the singularity that represents the transition to a global conflagration. Mentioned previously is Rudolf Otto’s 1917 book about the characteristics of religious experience, Das Heilige, The Sacred, which described phases in the discontinuous transition from everyday life to the wholly other (ganz andere) reality of the world of the sacred. They include intense rofane. In his 1958 book, Patterns in Comparative Religions, this well-known historian of religion called the revelatory occurrence of sacred reality an hierophany. Eliade’s classic work, The Sacred and the Profane, contrasts the homogenous, spiritually formless and relative world of the profane with the results of passage through spatial and temporal singularities to a place and time that are not of this world. Poincaré said that the brain did not know of absolute space, but rather established a model of it through internal reconstructions of sequential sensory experiences that_accompanied our exploratory movements. A world. It was Poincare’s habit to topologize the dynamics of motion in mathematical problems that lacked analytic solutions. In this way, simple algebraic operations replace some of the insoluble problems of the calculus. Eliade’s sacred space defining singularity in the plane that breaks profane homogeneousness, a center point that is no longer a circle, can be viewed also as Poincaré’s topological center. real physical movements around such singular fixed points. The operational object called groups defines this kind of algebraic, mathematical structure and motion. 133 HOUSE_OVERSIGHT_013633

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associated with the loss of habitual temporal-spatial contextual moorings. A mind at time one and the same mind at time two are unconnected. They are wholly other. In much the same sense, for Eliade, sacred time, like space, is neither homogenous nor linearly continuous. Sacred time is circular, recoverable and reversible. Past, primordial, mythical time can exist in the present. Religious festivals are recurrently ontological, allowing the recovery of the sacred time such that their past and present expressions are the same. Rebirth is new birth. |In the language of the North American Indian Tribe, the Yokuts, the term for world (cosmos) and year are the same. A year and the world has gone by, only to start again. The Dakota Tribe says that the Year goes around the World. As Elaide has said, “...at each New Year...the world (is) recreated and to do this is also to create time...the sick man becomes well because he begins life again with its sum of the energy intact.” Healing by becoming The quality of separateness, discontinuity in states, as occurs in the same- different inside world, is much like that found in the stages of anesthesia. Each stage of anesthesia is ganz andere from the others. In Stage | anesthesia, fast frequency, low voltage brain waves are observed and accompanied by a two Martini-like, mildly activated, sedated but exhilarated high. Stage //, the next deeper stage of anesthesia, is marked by the sudden emergence of intermittent bursts of high amplitude brain waves, and animals and man demonstrate bizarre postures, hallucinatory phenomena, fixed staring, and sometimes movements that look like acting out some symbolic drama. This stage marks the beginnings of the loss of responsiveness to painful stimuli. In the sudden drop into Stage ///, a low voltage mix of mostly slow and some fast brain waves can be seen associated with depressed consciousness, complete insensitivity to pain, slow regular respiration and an unexcitable cardiovascular system. Stage /V is the deepest stage of anesthesia. This state is characterized by very low voltage, almost flat brain waves, a loss of spontaneous breathing, the collapse of blood pressure and, finally, cardiac irregularities and death in cardiac arrest. These are both discontinuous and global brain state phase transitions. 135 HOUSE_OVERSIGHT_013635

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primary process by Freud and his followers. This forgotten language of the unconscious, an archaic needs and fear-driven tongue lurking beneath our supposedly objective discourse, comes to dominate themes of communication in the middle of these unfinished spiritual transitions. The Rorschach Test of master meditaters and LSD users overflow with conflictual primary process images, as does the talk of patients on the verge of schizophrenic decompensation. The primitive symbolism of primary process provides the major current in the overwritten prose of the hyper-religious temporal lobe limbic epileptics described previously and called the Geschwind Syndrome and in the regressed and iconic transference concerns of patients with tendencies for global and sudden phase transitions, prostitute to saint, righteous obsessional to conscienceless psychopath, called borderline personality disorder. Primary process represents a dynamical brain state, one unburdened by linearly predictive connections with reality. It is a state without even a transient single defining physical time or other fixed measure of order. It is without the causal logic or knowledge of an outside reality that a brain implies in supposing to know. Its primitively instinctual style and goals contrast with more physically time-locked, reality oriented thinking which Freud called secondary process and Penn-Lewis referred to as ordinary and religiously lawful “reasoning faculties.” An absence of absolute time and space scales with which the executive ego orders internal and external time and events, and therefore their relations, results in primary process thinking characterized by condensations of several, often incompatible, representations into one. Dueling, conflictual and simultaneous feelings and thoughts float from their relevant objects to others. In the transitional transcendent state, there may be confusion of self with others, of objects with their labels, of parts with the whole and of symbols with the things that they symbolize. This facilitates living in the spirits of the Father, the Son and the Holy Ghost at the same time. Mixed inextricably with saintly awareness and charisma, there are signatures of instinctually driven and configured primary process. Freud’s classical work on s/lips of the tongue concerned the intrusion of these instinctual thought stream condensations from the world of the ganz andere and displacements into 138 HOUSE_OVERSIGHT_013638

everyday life. In this intense and quasi-fluid state, saintly priests slip seamlessly into sexual predation; an ecstatic Jewish Orthodox fundamentalist shoots 29 praying Moslems in a cave near Abraham’s burial plot for Sarah in Hebron; what were lovingly mystical, Jelaluddin Rumi’s Afghanistan (Balkh) descendents become people bashing and women stoning morality police; committed and mesmerizing Christian televangelists attend peep shows and seek child pornography; devoted Islamists crash airplanes into tall New York buildings. In the physics of condensed matter, two common forms of multi-molecular or polyatomic cooperative arrangements are the crystalline condition and in some ways its opposite, the amorphous glassy state that results from rapid cooling through a melting temperature. The microscopic atomic arrangement in glasses, in contrast with the crystalline state, exhibits no spatial periodicity or long-range order. In contrast with fluids, the friction of passage of molecular elements of glasses past each other, their shear viscosity, is large enough such that their macroscopic shapes are maintained in the very slow flow for very long times. In-between the crystalline and glassy states their exists a multiplicity of possible unstable arrangements which result from what physicists call frustration, the inability of a system to find a unique, lowest energy, ground state. The generic example of a ferromagnetic crystal has two types of ordering principles: (1) The mutual alignment of the atomic magnetic moments, visualizable as the lining up of dipole, positive to negative, magnetic arrows; (2) The geometric crystalline low energy ground state described above. When the symmetry of these two ordering principles are incompatible, imagine an arrangement of neighboring atoms that prefer anti-alignment of the magnetic moments which are placed on a geometrically triangular rather than a square lattice, there is no single arrangement that can satisfy both magnetic and geometric principles. What emerges in this state of frustration is the potential for a multiplicity of nearly equal energy states. Water has the potential for both geometric ice crystal symmetry as well as arrangements of hydrogen proton (+) to oxygen electron (-) magnetic moments (with well-ordered oxygen lattices but disorder among the hydrogen positions). It is therefore not surprising that a multiplicity of 139 HOUSE_OVERSIGHT_013639

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indirectly by my sons and church elders about joining a study group for personal conversion. | was surprised to learn that discussions of current political topics were a regular part of these discussions as well as the Sunday and Wednesday night services. We received a weekly political action committee report. Their issues involved abortion, school vouchers, sex education in schools, family planning, school prayer and carefully chosen Christian elected officials for school boards and the Congress. As a congregation, we frequently held hands in small circles and prayed for the electoral success of our issues and candidates. Twenty years later, this movement has evolved into the public political morality play of the Republican base of George W. Bush. Laying on of hands, dying in the Lord, speaking in tongues, dancing in the aisles and praying with up stretched arms were routine in the hymn dense services. The goal for all was the spiritual transformation of mind as in Romans, “...be not fashioned according to this world, but be ye transformed by the renewing of your mind that ye may prove what is the good and well-pleasing and perfect will of God...” The pastor told us that the world ruled mind could not grasp spiritual things as in Corinthians “...they are foolishness unto him and he cannot know them, because they are spiritually understood.” My research took me to a collaborative project at a European mathematics institute for three months. | returned to our town very late on a Saturday night. | planned to surprise my sons by appearing at their usual choice of the middle service the next day. | drove up to the warehouse church fifteen minutes before the service was scheduled and found that the parking lot of the strip mall was nearly empty. There was no Cadillac parked at the front door. | banged on the double door when | found it locked. More then a little surprised, | called my eldest. He told me that four weeks before, the pastor disappeared, | later found that his disappearance accompanied that of the congregation’s bank account, and no one knew where he had gone. He had not warned or informed anyone in the congregation about his plans. Calmly and without apparent awareness of my surprise and distress, my 141 HOUSE_OVERSIGHT_013641

eldest asked me if | would like to attend the late Sunday morning service at their newly chosen Charismatic Christian church. He gave me its address and told me that the service started at 11:00 AM. There still was enough time for us to meet there. | wondered how the Pastor Carl Austin would use this incident in sermons about sin and redemption to his next congregation. Further Readings for Pentecostal Phase Transitions Religious and Spiritual Groups in Modern America. Robert S. Elliwood, Prentice- Hall, Englewood, N.J. 1973. The Name of Jesus. Kenneth E. Hagin, Rhema Bible Church. Tulsa, Oklahoma. 1979. War on the Saints. Jessie Penn-Lewis, Robert Lowe, N.Y. 1973. Discipleship, David Watson, Hodder and Stoughton, London, 1981. Mysticism. Evelyn Underwood, Dutton, N.Y. 1911. A Nation of Believers Martin Marty, Univ. Chicago Press, Chicago. 1976. Introduction to Percolation Theory. Dietrich Stauffer. Taylor and Francis. London. 1985. Modern Theory of Critical Phenomena. Shang-Keng Ma, Benjamin/Cummings. Reading, MA. 1976. A Modern Course in Statistical Physics. Linda E. Reichl, Univ. Texas Press, Austin, 1980. 142 HOUSE_OVERSIGHT_013642

Manic-Depressive Illness. Fred K. Goodwin and Kay R. Jamison, Oxford Univ. Press, N.Y. 1990. The Pharmacological Basis of Therapeutics. Louis S. Goodman and Alfred Gilman, MacMillan, N.Y. 1975. Statistical Mechanics of Phase Transitions. J.M. Yeomans, Clarendon Press, Oxford. 1992. 143 HOUSE_OVERSIGHT_013643

CHAPTER 7: AMPHETAMINE ROLL-UP AND SPLITTING We try to understand the metaphysics and inner dynamical life of the committed, judgmental, fundamentalist believer. In these sacerdotaly rigid and faithful, disenfranchisement and righteous intolerance toward other denominations are simultaneous with spiritual compassion, mercy and forgiveness for the members of their own. This splitting between the good people and latent evil doers is seen by psychoanalysts and dynamically oriented brain scientists as an all too common, sometimes psychopathological, solution to the inevitable ambiguities of living. | am certainly not alone in being fearful of Fundamentalists: Jewish, Christian, Moslem and Hindu. From the overpass above the freeway, bearded Jewish Orthodox men rained rocks onto the roof of my rented car because | was driving on Sabbath. A research project had taken me to Jerusalem Mental Health Center’s neurochemical laboratories for collaborative work with mostly secular Jewish scientists. Halachic considerations, those of Jewish lawfulness, comparable to the constraints of Muslim shirah, forbids working, even driving, on the Sabbath. Orthodox Jews live walking distance from synagogues or benefit from a rabbinicaly blessed, network of symbolically covered walkways for going longer distances on the Sabbath. This 144 HOUSE_OVERSIGHT_013644

Sabbarian grid of permission obviously did not cover driving on the free way to the mental health center. It is the splitting of us from them that leads to the breakdown in empathy and compassionate identification with others. Studies of the dominance of direction of rotation within a closed space in small mammals have shown that amphetamine- induced intensification makes the choice of right versus left (or left versus right) rotation, broken symmetry, more statistically significant. In contrast, the Hefner Foundation of Switzerland has shown that entheogenic drugs such as psilocybin in man facilitate seeing both of the conflicting, simultaneously presented, right eye and left eye images in place of the usual dominance of just one of the two representations. A precondition of compassion might be that a person’s brain be able to see and comprehend both or several sides of apparently conflicting points of view at the same time. The Fundamentalists do not see things that way. In the Koran, Mohammed says, “...give sustenance to the poor man, the orphan, the captive...and for the unbelievers We have prepared fetters and chains and a blazing fire....” In the New Testament’s Mark we find the final words of the risen tt Jesus, “...whoever believes and is baptized will be saved but whoever does not believe will be damned.” The Crusaders’ claimed scriptural support for their murderous marches to reclaim Jerusalem. Carl Jung wrote about the New Testament’s Revelations in his Answer to Job: “...a terrifying picture that blatantly contradicts all ideas of Christian humility, tolerance, love of your neighbor and your enemies and makes nonsense of a loving father in heaven and rescuer of mankind. A veritable orgy of hatred, wrath, vindictiveness and blind destructive fury that revels in fantastic images of terror breaks out...overwhelming a world which Christ endeavored to restore to the original state of innocence and loving communion with God...” As Princeton University philosopher, Walter Kaufman, has noted in his Religion in Four t Dimensions “...compassion for unbelievers is implicitly condemned and proscribed...Augustine argued expressly against compassion for the damned and Luther used invectives against his (religious) enemies...” How can this be God’s 145 HOUSE_OVERSIGHT_013645

tt setting for the spiritual work toward that promised in John “...that you love one another; even as | have loved you, that you also love one another.” In contrast with what has been described in previous chapters as the entheogenic drug-induced transitions to a spiritual mind, one is tempted to describe these Fundamentalists’ states as the amphetamine religions. The Los Angeles Ram’s Hall of Fame defensive end, on very high doses of amphetamine (125 milligrams compared with the diet dose of 5 milligrams) taken four hours before the Sunday games, the Baptist minister, Deacon Jones, used his famous and consciousness annihilating head slap to daze the opposing offensive tackle in order to gain access to and injure the other team’s quarterback. Before taking the handful of Dexedrine spansuls, he would tell me, “See you on Tuesday.” Along with the Deacon’s destructive aggression was the other invariant feature of the actions of high doses of amphetamine, compulsive stereotypy, the fixity and driven repetition of over simplified actions and thoughts along with the loss of breadth of vision and adaptive flexibility. Deacon consistently rushed inside, took the inside lane, in spite of offensive linemen, who having studied previous game films, being set up to expect his route. They used this knowledge to take him out of the play. In modern theological parlance, judgmental rigidity and thinly veiled disapproval take the place of the more flexibly curious and lovingly humane feelings of the participants in the evolution in spiritual understanding of today’s /iberal Protestant process religions. These are the ones that believe that the properties of God evolve along with our biology, our brains and our growing scientific understanding of ourselves and the world. Angry splitting is not just a stimulant drug effect. Recall my experience of the sudden emergence of a first second wind after a mile or so of my daily ten miles of running. It was frequently accompanied by inner bursts of obsessive, paranoid thoughts. Taking five milligrams of amphetamine felt much like the first second wind. | am full of energy with arrogant feelings of power, mind fixated in grand and simple ideas that | believe to be absolute and correct. | feel irritably intolerant about anyone or anything different. It is my virtuous duty to set everyone straight. 146 HOUSE_OVERSIGHT_013646

In the 1980’s, Moishe Zar, a desert castle dwelling, settlement organizing, ardent Orthodox Jewish Zionist, now 65 years old, was the leading vigilante of the West Bank He planted bombs in the cars of Arab mayors and plotted to blow up the Dome of the Rock. Buying up farmland from the Palestinians beginning in 1979, many of whom were then killed by their own because they were seen as collaborators, Zar and his group of young volunteer settlers took over harvesting the Palestinian’s olive trees and shooting rifles over the heads of those that would take them back. Fundamentalist Christians share his vision that the coming of the Messiah, the second for Christians, the first for the Jews, is dependent upon the complete return of all of the land of Israel to the Jews. | recall that in the middle 1940’s, my father took me to a fund raising dinner for the local chapter of the Jewish Antidefamation League. The whispered talk was about blowing up a warehouse in which anti-Semitic pamphlets were stored, planned for the middle of the night when it was unoccupied. Even at the age of 10, | could tell that their quiet anger and firm commitment made these threatened men feel less vulnerable. | understood a little more about the motivation for this proposed nighttime property destruction when, the following year, my father explained the reason for our being refused overnight rooms at several motels as we drove along I- 95 in Southeast Florida. It took us until late night to find a place to sleep. This was America’s muted version of what Hitler and his legions were doing to Jews that, at that time, was not generally known, except for Walter Winchell, in America. Resonant with our chemical-cultural theme are the many reports that Hitler was taking an amphetamine drug, Benzedrine, daily and in high doses for the last 20 or more years of his life. One can hear the characteristic, amphetamine-induced, higher pitched, rants in his recorded radio tirades. Compare the pitch and strained voice quality of the singing of Bob Dylan in his early records made while he was on speed with the gravely, much lower pitched voice, now that he is not. In our behavioral neuropharmacology laboratory at the Brain Research Institute at UCLA, Professor Charles Spooner and | used an audiographic oscilloscope to monitor the sounds of baby chicks whose peeps became higher in pitch and rate following injections of amphetamine. The earliest members of the methadrine-amphetamine 147 HOUSE_OVERSIGHT_013647

chemical family were synthesized by the great organic chemists of the German pharmaceutical industry in the early 1930’s. The sequence of parallel streets in the neighborhood of my home and first grammar school in Kansas City, Missouri were my street, Virginia, then Tracy, Forrest and Troost. My school, Bancroft Elementary, was on Tracy and one block down that street was the Lutheran Day School established by German immigrants under the aegis of the Missouri Synod. Starting in the third grade in 1943, | was intermittently and unpredictably chased by rock throwing, “damn Jew” and “Christ killer’ shouting boys from the Lutheran Day School. | had my choice of running for safety directly from Tracy to my family’s half duplex at 4232 Virginia Street, or moving away from school via Troost and then down several blocks and around to sneak back to my home on Virginia without being spotted. One run-for-it afternoon, my parents took me to the emergency room of the Menorah Hospital to have my scalp sewed up where a sharp rock had landed. When | asked my synagogue’s young people’s spiritual counselor, Rabbi Kleigfeld, to explain the feelings and actions of these children of Martin Luther’s Post-Reformation Christian Church, he answered that | already knew about similarly difficult places and times of our Twelve Tribes’ like Rome, Medieval Europe, the Spanish Inquisition, Persia (Iran) and, it was rumored, in Germany as we spoke. “Conversion or death” was its most benign form, in places like Spain and Iran, many Jews faked it, staying alive and practicing Judaism secretly. Kleigfeld told me that the causes of this historical theme of persecution of Jews were complex. Among the frequently unmentioned events recorded in the later part of the worldly life of Mohammed, who lived from 570 to 632 AD was, ”...in the name of Allah, the Compassionate, the Merciful...” his participation in the crushing of the Jewish tribe of al-Nadhir in 626 A.D., the beheading of 800 Jewish men of the tribe of Qurayza who refused to accept Allah as their God in 627 A.D. and putting to the sword the Jews of Khaybar in 629 A.D. As in the section of the Koran called The Cow, Mohammed proposed to “...fight against them (the infidels) until idolatry is no more and Allah’s religions reigns supreme...” In contrast, the more entheogenic spiritual orientation of the ecstatic followers of Mohammed in his earlier years 148 HOUSE_OVERSIGHT_013648

speaks of the multiplicity of valid Ways to Deep Truth. The acceptability of many ways is supported in the tales from the millennial oral tradition of the Sufi Masters in their Teaching Stories. One of them, What Befell the Three, is attributed to the early 18" Century Sufi teacher, the Dervish Murad Shami. In it, an apparition is mobilized by the concentrated Truth seeking efforts of three Sufi Dervishes named Yak, one, Do, two and Se, three. When this “...white smoke head of the very old man...” was asked what he was, he answered “...| am what you think me to be...have you never heard the saying ‘There are as many ways to the Deep Truth as there hearts of man.” In the narratives about the lives of the Mevievi Islam dervishes called Munaquib el-Arafin (1353), Jalaludin Rumi, the Sufi saint, instructs his ill and troubled petitioner to ask forgiveness from the Christian he recently spat on saying “...whether a ruby or a pebble, there is a place on His hill, there is a place for all...” Cole Barks and Michael Green’s The Illuminated Prayer (2000) notes that the Rumi follower, Bawa Muhaiyaddeen, a modern Sufi guru, was said to be keenly aware how quickly spiritual entheogenic systems can become amphetamine-like and “...develop rigid marching orders ...which turn into a dumb obsession with other people’s behavior...” It appears that entheogenic and amphetamine spiritualities can coexist contemporaneously, in Islam as well as in all the other of the world’s great religions. One day, sneaking home from school, taking the long way around via Troost, | was spotted and chased up some stairs into an apartment building’s dark hall. Terrified, | swung hard and hit the leading angry and noisy head with a propitiously found snow shovel that had been left near the apartment’s entrance. An ambulance was called to tend to the twelve-year-old, transiently unconscious, Lutheran boy. He recovered completely within a day and the chases after school and my desperate escapes stopped suddenly, never to reappear. After several months, our family crossed the socioeconomic divide in Kansas City to a more tolerant, upper middle class, Southside neighborhood near Rockhill Road, to a suburban home, one block from Missouri’s border with Kansas. There, persecution for my Jewishness took more subtle forms such as not being permitted to play teen-age golf with my friends, though invited, on their Blue Hills and Kansas City Country Club’s golf courses. It 149 HOUSE_OVERSIGHT_013649

was decades later that the first Jewish member of the KCCC was the founder of H and R Block. Unable to afford membership in the single all Jewish country club of the region, | practiced for my high school golf team on Armour Hills Public Golf Course, where, at the time, mostly white working class golfers played. How can it be that spiritual states include both personal humbleness and loving mercy toward some of mankind and judgmentalness, nonacceptance and commitment to seduction, threat and even violence in the service of invoking changes in the beliefs of others. How can the high energy calm of being home at last in the born again condition with its new freedom from self assaults about sin, most importantly that of disbelief, but also peccadilloes such as drunkenness, promiscuity and familial abuse, be associated with readiness to judge, harass even persecute others. Psychoanalysts would say that it is a riddance mechanism, the projection of unwanted personal traits onto others. From the standpoint of rational thought, this seems more like non-Aristotelian cognition, two, not either-or, countervailing orientations toward mankind held simultaneously. The newborn parishioners of these charismatic amphetamine churches express their fealty to God with strongly held beliefs that diagram logically as contradictions. The perception of the world’s peoples into believers and infidels, good and evil, our people and your people, ourselves and the others. It is generally believed among social psychologists that it is the perceived nonpersonness of others, which allows the cruelty that empathic identification with them would never permit. Splitting feels like resolution, its stereotypy reducing the complexity of spiritual thought as well as true to life perception. A concrete laboratory example of amphetamine conversion, the sudden transition to a high energy, fixated, and delusional state called amphetamine psychosis, is supplied by experiments in humans conducted by Professor John Griffith at Vanderbilt University in the 1960’s. These experiments would not be allowed by today’s human research committees or medical ethicists. Each one of a group of psychologically screened-as-normal graduate student volunteers, at an individually unique amphetamine dose, developed suddenly a personally unique and peculiar system of new beliefs, obsessionally held as rational thoughts. Ten 150 HOUSE_OVERSIGHT_013650

milligrams of amphetamine were administered to volunteer subjects every hour until every subject crossed their particular threshold for personality change. The graduate students underwent a global mind-brain-person transition at differing total doses of amphetamine. The subject’s world was suddenly transformed into one of enemies and friends. The syndrome dissipated over several hours when the drug was stopped and the plasma levels of amphetamine and its metabolites declined. As amphetamine makes memory formation and recall stronger, the subjects were embarrassed when remembering what strange and forbidding yet uneatable things they so strongly believed. These included such things as: they as good people were caught in a network of bad person Russian spies; some threatening others arranged for poison gas to be seeping out of the water faucet; the white coated scientists were CIA undercover intelligence officers hoping to get information about their small pornography collection. The subject's world had become divided in, for each person, a stereotyped way. After a couple of weeks of return to normal living, the experiment was repeated. Each subject again developed his or her individually unique set of good- guy, bad-guy delusional beliefs and at the same dose of amphetamine as before. Like those of strong faith, their ideas once again resisted the logical arguments made by the professional staff: that the new realities were neuropsychological and had an obvious pharmacological origin. While on the drug, all stuck to their story, even while being shown the movie record of their first drug-induced episode. There is reliable scientific literature describing kamikaze pilots on high doses of amphetamine in an ecstatic state of Shinto nationalism. With their planes loaded with explosives, they deliberately crashed their planes onto American aircraft carriers in the Pacific Theater of World War II. One wonders if these drug-induced states occur in the drug-free condition in today’s abstemious Muslim suicide bombers. A more abstract and general way of thinking about the sudden emergence of fixation, repetitiousness and splitting in feelings and thoughts involves the emergence of regular /imit cycle oscillations in a complex system that was behaving previously in a stable but flexible way. Locking up into a fixed, closed loop, is a 151 HOUSE_OVERSIGHT_013651

common way for electrical circuits, computer programs, brain mechanisms and other complicated systems, even cultural or spiritual movements, to behave when one or more important control parameters crosses a threshold. Doyne Farmer of the Los Alamos’s Prediction Company once said about this vulnerability in complex system, “Those things can hardly wait to roll up.” The /imit cycle lock-up occurs most often as a sudden, discontinuous change, called a_ bifurcation, into autonomous Sself-oscillations from an equilibrium state around which there was some random variation. A bifurcation, a discontinuous change in outcome from a smooth changes cause, characteristically occurs when the amount of an important influence, a metabolic state, a drug, a psychodynamic conflict or level of emotional stimulation crosses some critical value. The switch from one type of dynamical behavior to another looks like the system has suddenly changed into something else with an entirely new kind of life of its own. In the new life of rolled up, locked-up repetitious motion, almost all new starting conditions follow pathways that lead into the same limit cycle pattern. Evangelical Christians talk about a// born again life being in Jesus, fixed in a complete set of moral, social and political beliefs, ideas and judgments. The limit cycle gets its name because the end state of the orbits of almost all starting points of the dynamics winds up being drawn into the same fixed, repetitious pattern of a stable cycle. Visualizing the simulation of one kind of bifurcation to a limit cycle on a computer screen, we see a slightly jiggling point explode suddenly into an orbit of ceaseless rotations around a circle. Ralph Abraham, the University of California at Santa Cruz pioneer in graphical approaches to nonlinear systems, describes, cinemagraphically, the emergence of limit cycles from a single point. He starts with a picture of an attractor of water flow in the shape of a basin. All water that enters the basin, rolls down its sides to the bottom, to what physicists say represents a potential energy minimum. A little more technically, this attractor basin is composed of the set of all points such that the orbits that flow from them tend to end up inside the basin as time goes toward infinity, no matter where they start. Changing the value of a control parameter of the system changes the shape of this basin-like landscape, of the surface of the systems dynamical actions called a manifold, which can intuitively 152 HOUSE_OVERSIGHT_013652

predict how the fluid will flow upon it. If we start with a simple bowl, a parabolic basin, then the attractor itself is a point at the bowl’s very bottom. Changing the value of some influential parameter may induce the sudden formation of a small hill, growing at the center of the basin’s bottom. Now fluid flow in the attractor bowl runs down to a path around the hill at its bottom. The autonomous motion of the fluid flows takes place now in a circular orbit. The basin of the new attractor is the original bowl minus the point at the top of the central hill. The fluid flow around the hill at the bottom of the basin is circular and is called a limit cycle. Note that the direction of the rotation of the limit cycle can circle in one direction or the other. |n some computational simulations, motion alternates between directions. This suggests the aspect of the born again amphetamine religions, splitting. There is an unstable and intermixed probability of right versus left turning directions and their alternation. This vulnerability to directional splitting and often unpredictable alterations in action themes can represent what seem to be _ paradoxical combinations of both good and evil in the same strongly faithful, for example, the apparent bidirectional morality of generous and loving, pederast priests. These mathematically flavored images of the sudden emergence of a limit cycle in complex systems was made biologically concrete to me by research conducted by one of my first graduate students, David Segal. He is now a professor of psychiatry at the University of California in San Diego. His program of work involved the administration of very gradually increasing doses of amphetamine to rats while their behavior was being monitored and recorded by a continuously running video camera. He documented the behavior of rats in a walled rectangular space within which, without drugs, they first wandered about randomly and then settled down to rest in an individually selected, favorite home corner. Segal called all of these phenomena, patterns of exploratory behavior. At doses of amphetamine below 2.5 milligrams (mg) per kilogram weight (kg), the exploration of the entire bounded space proceeded faster than was the case with their salt-water treated controls, their paths being more uniformly distributed throughout the box. They spent less time resting in their home corner. At almost precisely 2.5 mg/kg, the rat’s behavior changed dramatically into an entirely new pattern of continuous circling. As 153 HOUSE_OVERSIGHT_013653

was the case in the abstract manifold picture of bifurcations to limit cycles, some rats tended to circle their chamber to the left and some to the right and switching between them was often seen. The influence of amphetamine and other brain dopamine neurotransmitter- mediated drug manipulations on directional turning tendencies in rats, mice and cats were the focus of brain and behavioral research of Professor Stanley Glick of the University of Massachusetts. The asymmetry of dopamine concentrations in the two sides of the brain, particularly in the medial prefrontal cortex and the brain stem’s nucleus accumbens, predicted both the paw preference for pellet reaching and direction of turning in several studies in rats. These findings were statistically true over a population of rats, but not necessarily predictive for any single one. Reminiscent of the conflict between good and evil in our human spiritual analogy, naturally right turning male rats and left turning female rats, when compared with the opposite paired group, were greater voluntary ingesters of alcohol placed in their water bottles. Splitting as a part of the phenomenology of /imit cycle bifurcations, with directional implications for good and evil, has neurological support in humans as well. In the context of contrasting right versus left hemispheric temporal lobe syndromes, recall that temporal lobe seizures with a right side excitatory focus leads to the development of the Geshwind Syndrome, a high, softly energetic and saintly state of spiritual preoccupation and voluminous writings, loving and generous kindness toward all and the complete disappearance of sexual interest but not sexual potency. A left temporal lobe excitatory focus leads to the development of the Kluver-Bucy Syndrome of indiscriminate aggressiveness and hypersexuality. Experimental simulations of this syndrome in cats lead to them mounting and attacking living and nonliving things, even chairs. A variety of manipulations of the symmetry of brain dopamine concentration and dynamics by its characteristic drug, amphetamine, interact with lateral brain lesions such that we conclude that the stimulant-induced limit cycle lockup remains a phenomena influenced by drugs, sex, genetic predisposition and several other experimental conditions. This situation is 154 HOUSE_OVERSIGHT_013654

perhaps not so different in variety and complexity from the range of representations in art and literature of the /eft hand of evil and the right hand of grace. Oscillations that appear spontaneously in nonlinear systems without external periodic input were known to Henri Poincaré in 1882, and were systematically studied and made accessible to non-mathematicians by early 20" Century Russian mathematicians and physicists, well represented by a 1949 book, Theory of Oscillations by the Russian engineer-mathematicians, A. A. Andropov and C.E. Chaikin. Another relatively early classic is Nonlinear Oscillations by Nicholas Minorsky. The most common form of transition from a fixed point to a limit cycle was pictured as changes in the surface of the action, the bow/-hillock manifold in the paragraphs above, and is called a Hopf bifurcation. Recall that bifurcation means a discontinuous change in an observable over a continuous change in what is known as a control parameter, such as dose of amphetamine or intensity of an experience. The mathematical mechanism resulting in circular directional motion represented by the (eigen)vectorial states, was named for the German mathematician, Eduard Hopf. His 1942 paper was a mathematical proof of its existence and was discussed in the context of fluid flows that role up such that circling vortices arise from smooth, called jaminar, water flow, at a critical value of the flow rate. Hurricanes are another example of these kinds of dynamics. The Hopf bifurcation to limit cycles has been found in several, many dimensional, physical, chemical and biological systems. The latter include calcium conductance oscillations in the excitable membranes of muscle, heart and the brain, cardiac arrhythmias such as ventricular flutter as well as oscillations in population numbers in foxes and rabbits, predator-prey systems. California Institute of Technology’s Professor, James Old and Johns Hopkins Professor, Joseph Brady made experimentally obvious the potential for the rigid irrationality implied by the brain’s inclination to be locked up into limit cycle behavior. They demonstrated that animals, from rats to monkeys, could get locked up in apparent self torture, repeatedly and endlessly pushing a bar to deliver current to pain systems in the their brains. These pushes induced almost unremitting screams in monkeys and 155 HOUSE_OVERSIGHT_013655

what appeared to be rageful biting and then immobilized resignation in behaviorally depressed rats. Freud’s last paper, Analysis, Terminable and Interminable (1939), featured examples of what he perceived to be the unsolvable mystery of helpless psychological entrapment in repetitious patterns of self-destructive behavior. He blamed the lliad’s and Odyssey’s villainous immortal, Thanatos, the ever- threatening spirit of death and destruction to contrast with the good, life giving Eros. The Yiddish word for a personified Thanatos is Moloch ha-Moves. A range of fixations in self-excitatory, repetitious, self-mutilating behaviors is documented in domesticated animals. Dogs, particularly German Shepherds and Labrador Retrievers, can lock up in compulsive grooming cycles of what is called acral lick in which endless licking of paws or flanks lead to the break down of skin into seeping- sore dermatitis, which, in turn, stimulates more licking. Mark Twain wrote a story about his getting stuck in ceaseless mental repetitions of a catchy, clangy poem. He could not stop reciting it to himself even after days of sleep loss and anorexia. He was finally cured by relating his problem and the poem to his pastor who he then unwittingly heard creating a community epidemic by including the rhyme in his following Sunday’s sermon. Psychologists, who study this form of human mental limit cycle attacks, call this state of internal, repetitiously recited, poetic stuckness, earworms. There are additional invariants of sudden transformations into spiritual-mind- brain bifurcations into a limit cycle lockups and, as discussed, one of them is psychological splitting. |n psychoanalytic theory, as first suggested by Freud in his 1937 written and posthumously published paper, Splitting of the Ego in the Process of Defense (1940), splitting implies two simultaneous and contrary psychological reactions, one can be conscious and the other unconscious. They can both emerge in conflictual situations involving adaptive efforts of the personality to deal with the opposition between some form of powerful instinctual pressure and attendant perceived or imagined danger. Otto Fenichel’s Psychoanalytic Theory of Neurosis (1950) elucidates multiple manifestations of splitting of the | (more technically, the ego) into a conscious part that knows reality versus an unconscious part that denies 156 HOUSE_OVERSIGHT_013656

it. In some situations, a logical view contends with a more irrational, magical one. Today, the morning group praying, evening hymn singing, Christian Republican Right Wing feed their feelings of being on the side of God by dividing people into those that are like them and good and those that President Bush and Attorney General John Ashcroft calls the evil doing “bad guys.” As noted previously, psychoanalytic theory posits that the evil doing others may represent the projected repository of our own unacceptable impulses and inclinations. It became quite clear in my own psychoanalysis and psychoanalytic training that it is in healing our split and knowledge of our own unacceptable things that will lead to our understanding and forgiveness of others. As we dig deeper into global brain-mind dynamics of emergent high-energy fixation, stuck repetitiousness and splitting, we encounter their universality in the structures of mathematical thought. Did we just make them fit? Do these thought forms map onto internal and external physical reality? Are these abstract concepts and operations simply products of our biological brains manifested as psychological mechanics and used to explain to ourselves what we perceive and think? Does a square have external reality or is it a universally imagined something, and, as such, represented only in our minds and the pictures of it we draw? Is mathematical understanding simply inborn perceptual skills combined with developed and practiced logical cognition? Or, do we take the Platonic view of mathematical relations: these abstractions are the ultimate realities, antedating and persisting through the past, present and future of the universe and omnipresent. Where can the conceptual boundary be drawn between the physical reality of the Babylonian surveyors use of the Pythagorean theorem to calculate distances, that the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of its hypotenuse, and its abstract, pencil-marks-on- paper, algebraic development as in the definition of Pythagorean numbers, a,b, and c such that a*+ b*= c*. The dichotomy between the abstract and concrete, consistently blurred in our work, is between a natural science with ideas that can be disconfirmed, directly or indirectly, by experimental observation and the thinking of mathematics as an a priori field in the sense of Kant. The modern Platonic view 157 HOUSE_OVERSIGHT_013657

such as that held by Rene Thom is that once accepting a set of natural givens, called the axioms, the rest of the knowledge of this reality grows in the form of theorems that relate to the axioms and each other through their logical consistency. Knowledge of reality is moved by the ever-forward mathematical refinement of a priori conditions to do away with the theorems’ exceptions, called counter examples. The Hebraic Bible’s view of signifiers such as words and symbols Is close to, but not identical with, the Platonic view of mathematical formalism. According to the Torah, God made the word with words. God spoke and the world became real. The Aramaic for “Il create in speaking” is avara k’davara , or as the magician says, as he waves his wand over an apparently empty black high hat, abracadabra. The Hebrew word for word, davar, also signifies thing. This view contrasts with the mathematical formalists, among them Hilbert, who considered the signifiers of abstract mathematics simply symbols used in a game, the rules of which being arbitrary, must include proofs of consistencies among them. Consistency from the point of view of physics was addressed by Hertz, in Die Prinzipien der Mechanik,(1894), where he expressed the formalist theoretical physicist’s work as “...within our own minds we create images or symbols of the external objects, and we construct them in such a way that the logically necessary consequences of the images are again the images of the physically necessary consequences of the objects.” In another set of related contrasts, the constructionist mathematician will argue that mathematical assertions are only true if they can be demonstrated, found or constructed. In contrast, the classical school of mathematics can develop the case for the truth of mathematical statements if they are consistent with field’s network of theorems and proofs, even if, up to the current time, no specific example of this truth can be demonstrated. The former can be thought of as a builder, the latter as a discoverer. For example, suppose we try to make a proposition about perfect numbers where a perfect number is defined as being equal to half the sum of its divisors. Using the perfect number 6, we find that its non-identity divisors are 1, 2, and 3 and half of their sum = 6. Our proposition: either there exists an odd perfect number, or else there exists no odd perfect number. An expression of this 158 HOUSE_OVERSIGHT_013658

forced decision between yes and no is called the excluded middle. The constructionist mathematician, an orientation without the excluded middle, asserts that “an odd perfect number exists” would only be meaningful if one could show that such a number had been found or constructed. The classical mathematician would find the phrase “no odd perfect number exists” meaningful without a concrete example, if the assumption of its existence would lead to a no (versus yes) contradiction encountered in the proof-relevant network of established theorems and their relations. The symbolic operations of these formal schools of mathematics and their relationship to the objective and ideational realities of brain-mind-spiritual life have been viewed by some as Western cultural products rather than expressions of secular or spiritual Absolutes. Still others have argued that cultural relativism is not relevant here because mathematicians worldwide constitute a monoculture. With respect to the real world existence of abstract mathematical structure, our Platonic bias must be obvious. The thrilling experience of a new reality | get to know from finally understanding how a theorem works and the rush of peering into the grandeur of the Grand Canyon feel like the same kind of full-of-wonder high to me. | blend them here without reservation. Perhaps this world of spiritual abstraction is closer to the orientation of the school of intuitionist mathematics. |ts founder, L.E.J. Brouwer, required that every mathematical construction be so immediately apparent to the human mind that no formal proof was necessary. This became my form of spiritual transcendence, which led naturally to a mathematical, mystical faith. We carry the explication of this kind of reality further. Reflections of the good and evil, right and left, moral directional biases and their relative weightings in born again bifurcations to invariant circles called limit cycles, can be symbolically represented in what are called the complex eigenvalues of matrices describing the system’s set of orthogonal motions with changes in their control parameters. The behavior of these complex eigenvalues underlies and characterizes the mathematical mechanism of the Hopf bifurcation. 159 HOUSE_OVERSIGHT_013659

The subject of complex eigenvalues brings up in me the emotionally disturbing subject of imaginary and complex numbers. | can still feel a little of my earlier anxiety. The episode started benignly enough. Our high school’s freshman algebra class was studying how to solve quadratic equations, equations in which the highest power of an expression was two. Told to work at the blackboard in front of the class, | was given the problem of finding the two values of x that were the roots of the equation, 5x? + 3x + 4 = 0. | had been taught to use the memorized _ + af 2 quadratic formula, rs in which a = 5, b = 3 and c = 4. | always a calculated the square root part first and wound up with the expression, /9—80 =J/-71.| can still feel the sinking feeling in my stomach as | looked at the result. | anticipated the usual snide remarks and embarrassment as | contemplated doing what | did not know how to do, take the square root of a negative number. Mr. Kirby, the retired mechanical engineer who was my high school freshman algebra teacher tried to help, but | did not trust him. It seemed to me that he had already humiliated me in front of the class, several times. He asked, “... what number when squared, multiplied by itself, would equal —7.” He then asked it another way: solve the following equation for x: x°+1 = 0. Seeing something | could do, | wrote the next line quickly x° = -7 and then, taking the square root of both sides, | wrote x = V-1. He then asked me what that meant. | answered by writing quickly, glibly and blindly that that meant that J—1xV/—1=-1. He asked me to explain what that meant by giving him an example from the real world. Not yet knowing about imaginary and complex (combine real and imaginary numbers), | stood head down, ashamed and silent, thinking that my smart friend Jerry Blau would get the answer immediately. Mr. Kirby said he would go on with the class while | continued to stand in front of the blackboard and thought about it. He told me to interrupt him when | was ready to answer. Some classmates were smirking, others giggled aloud. They had seen him do this to me before. Mr. Kirby, a short, muscular man, an ex-marine with a military haircut and a brusque manner, lectured that mathematical competence and obedience to authority and class discipline were all of a piece. | asked him about mathematical 160 HOUSE_OVERSIGHT_013660

creativity and he said that this class was certainly not about that. | disliked and feared him. He seemed to feel (and wrote a note to my parents to the effect) that, being “too arrogant” | needed to be “brought down a peg or two.” | had gotten the best grades in the first two exams and was enjoying the role of after school tutor for some of my friends. | suspect | was getting pretty egotistical. In class, | found myself eagerly shouting out answers without holding up my hand, behavior that Mr. Kirby met with his characteristic look of fatigued disgust. Twice | was thrown out of class for my introjections. He then began to give me problems that | could not do, for which | was not prepared. This left me standing at the blackboard until the end of the hour, after all the rest of the students had solved theirs and sat down. On parent’s night, Mr. Kirby told my father that | needed more “social and intellectual discipline.” Inspired and personally directed hard work and socially defined correct behavior were not synonymous to this arrogant 13 year old who had already brought chagrin to his mother, the conservatory classical piano instructor, with his satirical pianistic jazzy composition called “How High the Moonlight Sonata.” | was also a secret reader of the book on the top back shelf in my father’s library by Jack Hanley called “How to Make Mary; A Gentlemen’s Guide to Seduction.” In Mr. Kirby’s class, inspired by the book, | sometimes reached behind me, through the crack in my desk seat, to caress the inside part of the long smooth legs and sometimes moist panties of a well developed, tall and beautiful brunette girl behind me. | was never caught and she pretended that nothing was happening. In fact, she never talked to me outside of class. | felt then, vaguely, and now, more specifically, that a content enriched, instinctually titillated and excited unconscious could lead me to the solutions of intellectual challenges if it were both sufficiently indulged and untrammeled, left alone in its work of being itself. Mr. Kirby did not see things that way. Since then, among my graduate and post-doctoral students in the neurosciences, | have learned that the Mr. Kirby’s of modern American educational practice have ruined generations of potential mathematicians and physical scientists. Worse, they have created generations of very bright math phobics who 161 HOUSE_OVERSIGHT_013661

run to other graduate fields such as biology and medicine and come to resist the potentially humiliating incursions of new and potentially helpful abstract ideas and operations from mathematics and physics into their fields. They do not want their persecutory versions of Mr. Kirby to take up residence once again in their heads. | can still feel his negative presence during long hours of struggle with the ego deflating feelings of dumbness that an understanding of almost any new mathematical concept requires of me. Holding Mr. Kirby’s voice off as long as | can until, sometimes, the wonderful “aha!” experience arrives. | have tried to forgive him since but forgetting him has not been possible. It turns out that in the world of elementary, physically representative, real numbers, the square root of a negative number has no meaning. Such a number has understandably come to be called imaginary. Was this the answer Mr. Kirby wanted? There was some conflict among mathematicians in the 17" and 18" Century about the arbitrary definition of V-las an imaginary number. |t was symbolized by a letter, /, that is J-1 =i. The existence of i extended the range of algebraic definitions so that a solution of the quadratic formula as above could be found for the square root of a negative number. A further expansion of this idea was to that of a complex number that can have both a real and an imaginary part. For example, letting letters be generalized representations of numbers, a complex number might be written, a + bi, real number a + real number b times /, the letters such as a, b, c, d... symbolized real numbers. Consistent with membership in an algebraic system, a + bi and c + di can be added and multiplied. This extension of the real numbers into the imaginary realm permitted d’Alembert’s and Gauss’s proofs (and many, more complete ones since) of the powerful Fundamental Theorem of Algebra from which the faith derives about always being able to find at least one solution to an algebraic equation. It was proven that any n” degree algebraic equation (e.g. x"+x"'+...=0) with real or complex coefficients always has at least one real or complex root. Closer to an image that helps make intuitive connections with human born again bifurcations, limit cycles and directional splitting is the geometric interpretation of a complex number, let us now call it z. As above, algebraically, z is the sum of a 162 HOUSE_OVERSIGHT_013662

real part a, plus b times the imaginary part, b/’; that is, z = a + bi. We can then set up a geometric space to represent z by imagining a two dimensional plane with the horizontal real axis extending from left to right, the usual x axis, and the vertical dimension, called the imaginary axis, extending from bottom to top like the standard y axis. These two axes, going from negative values to positive ones, left to right and bottom to top, cross at the shared value of 0. Thus a and 6 can be visualized as the rectangular coordinates of a point in the plane and the point locates the complex number, z = a + bi. Since real parts and imaginary parts are like apples and pears and for addition, like must be added to like, if two complex numbers, a+ bi and c + di are equal, then a = c and b = d and their sum Is written (a+ c)+(b+d)i. Now that we’ve set up a point z on the plane, located with a complex number at z = a + bi, we can then draw an arrow, called a vector, from the intersection of the imaginary and real axis at 0 to this point z. Its length from 0 to z, 0z, we'll call that length p, is the size or amplitude-like modulus of the complex number, z = a +i. The angle this 0z vector makes with the real, Oa-axis, lets call this angle ¢, is called the argument of complex number z = a +bi. p is a length that can grow or shrink, ¢ is an angle that can rotate. We imagine vectorial movement like that of a variable length hand of a clock. This geometric explication of complex numbers prepares us to visualize complex numbered eigenvalue solutions to matrices representing the relevant equations that bifurcate to limit cycles and directional good and evil splitting. represents the dilatable clock’s radial amplitude of circular motion and ¢, the angle of vectorial turning from the 0a-axis. The complex conjugate of the complex number, a + bi is the complex number a — bi in which the sign of the imaginary part is reversed. Geometrically, this means that a pair of complex conjugate numbers with the »’s of both having below zero values relative to the Oa-axis, that is negative real parts, could be imagined as the points indicated by two same sized, mirror image, clock hands pointing at 8:00 and 10:00 o’clock. Note that the ¢, the angle of vectorial deviation of the arrow pair from the Oa-axis, turn in opposite directions in these mirror image moving clock hand vectors. Without going deeper into the representation of the actions of the system in 163 HOUSE_OVERSIGHT_013663

question (its differential equation) in the form of what is called its Jacobian matrix of partial derivatives (a matrix representation of the differential equation indicating orthogonal directional velocities of change of locations of the components of the motion with respect to changing values of the control parameter), we know that when the p of the matrix’s set of two complex conjugate eigenvalues is less than zero, p <0, the orbit representing the system, spirals into a stable fixed point. This is analogous to going to the bottom of the parabolic attractor basin as described above. Values of the invisible eigenvalues and their changes constitute the abstract mathematical mechanisms underlying the observable dynamics of the system observable physically. The mathematical mechanism underlying the Hopf bifurcation of fixed points into limit cycles (associated with bi-directional splitting that accompanies the amphetamine transformation into limit cycle stereotypy of rigid ideas and equally likely mirror image motions in the directions of good versus evil) is the crossing of the systems real valued parts, p’s, of its complex conjugate eigenvalues into positive territory, o > 0. The mirror image of clock arrows is transformed from 8:00 and 10:00 o'clock to the clock locations of 4:00 and 2:00. At a Hopf bifurcation, a pair of complex conjugate eigenvalues crosses the imaginary (vertical) axis such that is real parts have positive value. In the orbit representing the motions of the system itself, the fixed point disappears to be replaced by the action spiraling out to an invariant circle. This is analogous to our manifold image of the disappearance of the central attractive point and the sudden appearance of a small hill at the bottom of a parabolic basic of attraction.. The new attractor is an invariant circular path around the hill, with the spiraling out to the invariant circle being a two dimensional picture of the disappearance of the bowl-bottom and appearance of a missing point, hill top fixed point and a spiral flow to the path circling the hill. Underlying the transition from a fixed point to a limit cycling, invariant circle, are a pair of mirror image complex conjugate eigenvalues that turn in mirror image, we could say, good versus evil, opposite directions. The Hopf bifurcating system inevitably has both. The implications of this very abstract metaphor for the emergent limit cycle- splitting style of spiritual transformation can be made deeper by considering the 164 HOUSE_OVERSIGHT_013664

common practice of Rumi’s Mevlevi (and other) orders of Islamic Dervishes that facilitate the onset and maintenance of their ecstatic states by an improvisational dance which goes from rocking to irregular whirling. The Dervish teaching tales place a symbolic emphasis on the power of the rotating wheel, the circling of the heavenly bodies, the mill wheel and the millstone. As Rumi said, “The mountain of the sun I'll fashion to a mill. And as my waters run, I'll turn thee at my will.” Note that their work toward spiritual transformation results in neither the emergence of the involuntary and rigid limit cycles of invariant circles or the associated divisive internal eigensplitting of good self from evil other. The Sufi compass points to an integrated field of divine consciousness, which contains the appearance of the world’s multiplicity. In this profound unity, all humankind is perceived as one family. The singular direction of all prayer, Salat, five times a day, at dawn, high noon, afternoon, sunset and an hour after sunset, turns the entire world into a unified directional field of prayer. At its center, the Islamic pilgrims wander round and round the black cube of the ancient shrine of Kaaba, This leaves one with the speculation that we started with: that the simple, authoritarian rules of the amphetamine, roll-up and splitting religions may be intrinsically more vulnerable to unpredictable breakouts into morally inconsistent actions and that the righteously rigid limit cyclists are invariantly split into ambivalence. In contrast, the more free form, chaotic turns of the entheogenic dervish define us all as belonging to one unified ecstatic field. Further Readings for Amphetamine Roll-Up And Splitting Psychology and Religion. Carl G Jung, Princeton Univ. Press, N.J. 1938. The Faith of a Heretic, Walter Kaufmann, Meridian, N.Y. 1959. Nightmare Season. Arnold J. Mandell, Random House, N.Y. 1976. 165 HOUSE_OVERSIGHT_013665

The Rabbinic Mind. Max Kadushin, Bloch , N.Y. 1972. Coming of (Middle) Age. Arnold J. Mandell, Simon and Schuster, N.Y. 1978. Introduction to Islamic Theology and Law. |gnaz Goldziher, Princeton Univ. Press, N.J. 1981. Tales of the Dervishes. |dries Shah, Dutton, N.Y. 1970. Open Secret; Versions of Rumi. J. Moyne and C. Barks, Threshold Books, Putney, Vermont. 1984. Amphetamine Psychosis, P.H. Connell, Oxford University Press, Oxford, 1958. Amphetamine Use, Misuse and Abuse. David Smith, Hall, Boston. 1979. Long-term Administration of D-Amphetamine. David S. Segal and Arnold J. Mandell, Pharmacology, Biochemistry and Behavior. 2:249-255. 1974. Amphetamine Enhancement of Reward Asymmetry. S.D. Glick, L.M. Weaver and R.C. Meibach, Psychopharmacology 73:323-327, 1981. Hopf Bifurcation and Its Applications, Appl. Math. Sci. Vol. 19,. Springer-Verlag, N.Y., N.Y. 1976. Dynamics, The Geometry of Behavior, I-IV, Aerial Press, P.O. Box Office 1360, Santa Cruz, CA 1982. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. John Guckenheimer and Phillip Holmes, Springer-Verlag, N.Y. 1983. 166 HOUSE_OVERSIGHT_013666

Psychiatric Aspects of Neurologic Disease. D. Frank Benson and Dietrich Blumer, Grune and Stratton, N.Y. 1975. Drives and Reinforcements. James Olds. Raven, N.Y. 1977 Neurobiology of Stereotyped Behavior. S.J. Cooper and C.T. Dourish. Clarendon, Oxford, 1990. Mathematics Unlimited—2001 and Beyond. B. Engquist and W. Schmid, Springer, N.Y. 2000. 167 HOUSE_OVERSIGHT_013667

CHAPTER 8: FAITH AND RATIONALITY It was my belief that, without subjective evidence of Holy Spirit Energy, the rush of reconfiguring transcendent experience, some glimmering of grace no matter how fleeting, an experience of intoxication with God, Martin Buber’s self authenticating /-Thou encounter, the many good citizens of this world, without these moments of illumination, must be attending church or temple to negotiate a better now and hereafter. Attending synagogue or church without the promise of a mystical high seemed like a superstitious rabbit foot rubbing for personal health and safety and a sharing of propitious contacts for social and economic advantage. Why else? | have had the good feel of what Jews call 7zedakah, the sharing of supplies by the haves for the betterment of the have nots. | have known the quiet calm of human right action as in the Unitarian Universalist’s serving the needy, open and flexible, intimate mindfulness of others and their needs. Considering E.O. Wilson’s brand of brain herd biology of altruism gives me a warm feeling about the potentially intrinsic goodness of man. But compared with the Jamesian brands of ecstatic transcendence, minds blown in Sufi twirling, Orthodox Jewish chanting, rocking and dancing, hands-in-the-air praying and hands-on-the-head healings of Wednesday night Pentecostal services, the soberly serious social engagement and 168 HOUSE_OVERSIGHT_013668

responsibility sermons of Reformed Judaism and the Unitarians as well as the 19" Century hymns and high |.Q. apologetics of some Presbyterian and Methodist clergy, are like near beer. Formally equivalent but without the rush and the delicious risk and promise of life long addiction. National opinion polls have found my preference for churchly fireworks in religious experience quite common. My Charismatic Christian sons are among the many with a preference for and loving labeling of these kinds of houses of worship as rock and roll churches. In a recent survey of Americans, 46% of respondents claim to be twice born, Evangelical Christians. Perhaps unfortunate with respect to their children’s academic and professional ambitions, 48% do not accept a Darwinian view of biology. Fifty million American readers are now buying books with plots taken from the Babylonian prophecies and anticipate the Rapture of Return with weekly, joyful, mini-rehearsals. They include praying in tongues as the Spirit moves them like Peter, John, James and the rest of the one hundred and twenty in the upper room on the day of Pentecost. Those of us with two or more available cable religious networks can, on any given Sunday morning, choose a smiling, kind, Proverbs quoting, rational Presbyterian liturgical stylist. In his seventies, standing tall with a full head of white hair and in a quietly resonant voice, he delivers a sermon about seven ways to avoid growing old. His list includes learning new things and continuing to work. His spiritual proposal was about personal faith, always leaning on the Lord. On another network, the three hundred pound, restlessly pacing preacher of the Cornerstone Assembly of God Church of San Antonio, Texas, stood in front of large maps of Iraq and the Middle East. He preached from Ezekiel about the refleshing of dry bones and a return of all Jews to Israel. He said that contributions to his church over the past year helped finance the return of 4000 Russian Jews to Israel. He reiterated the promise that, when the return was completed, there would be a massive Islamic attack on Jerusalem and “we will all rise up to Heaven” in an_ ecstatic disappearance. Jews, as long as they accepted Jesus as their Savior, were welcomed along on the ride. More then two thousand parishioners erupted into loud applause along with shouts of “praise Jesus.” 169 HOUSE_OVERSIGHT_013669

An inkling of something entirely different, neither human psychology nor frenzy, was an unanticipated benefit of being at England’s Warwick University in sabbatical residence in Math House #2. This large, round, many windows and black boards, study with a small upstairs bedroom was one of the apartments for visiting professors behind the Warwick Mathematics Institute in the English Midlands. | attended a variety of churches and synagogues on the weekends. The perspective that emerged for me at Warwick was that rabbinic Haggadah, inferences to be drawn from imaginatively spawned narrative, isn’t the same thing as Halakhah, the law dictated by Jewish legal tradition; that geometric insight and other intuitions aren’t the same as mathematical proofs; that the mystical visions of the English romantic poet and illustrator, William Blake, were not necessarily consistent with the scientific observations and logical arguments of the contemporary Scottish philosopher, David Hume. Paul Tillich wrote that the wisdom attendant to primary spiritual experience that was without the unconditional character of sensible moral obligation was not to be trusted without critical analyses. | learned that among High Episcopal and Reformed Jewish English academics, God is not a hallucinogen, but more like a spiritually based, social contract. In his 1929 essay, Mysticism and Logic, Bertrand Russell noted mysticism’s preference for: (a) Insight over discursive analytic knowledge; (b) Belief in the unity of all things over oppositions or divisions in representational thought; (c) The denial of the reality of time, even in the divisions of past, present and future; (d) Belief that evil is unreal, manufactured by the innate divisiveness in some analytic intellects. In modern brain hemispheric and other neuropsychological philosophies, these countervailing descriptions of external observables can grow naturally out of the brain’s abilities to maintain logically incompatible perspectives simultaneously. Right-brain aesthetic holism in contrast with /eft-brain categorical analytics recalls a popular example. Would one chose Blake or Hume to better explain how the time dimensions of memory disappear with the scent of a past lover or the hearing of his favorite music for lovemaking. In the inevitable mix of primitive instinct with high purpose, the visiting professors’ Math House #2 had an aura of infamy. It was the one in which, by the 170 HOUSE_OVERSIGHT_013670

accidental intrusion of a campus security officer, the brilliantly eccentric Northern California mathematician, Ralph Abraham, was famously arrested for pot smoking. The campus officer told me that, late one night, thinking he had smelled fire, he used his master key to make an unwelcome entrance. The incident became part of the record in House of Commons hearings about the intellectual and moral decay of English Universities. Apparently, even among English intellectuals, there were trivial and politicized definitions of virtue. Christopher Zeeman, the head of the Mathematics Institute was a world- class topologist who, among other things, demonstrated biological and social- psychological applications of Rene Thom’s Catastrophe Theory. | was invited as a brain person and amateur mathematician, to see what might result from mixing me with members of his fine mathematics faculty. In addition to learning some bifurcation and lots of ergodic (statistical) theory, my chats with Christian and Jewish mathematicians on Saturday and Sunday morning visits to the synagogues and chapels of Oxford and Cambridge introduced me to an English intellectual’s religious tradition. The spirit of C.S. Lewis was still very much alive. Surprising, however, was that more than a few of these scholars had the elements of Christian faith in full menu: virgin birth, incarnation, crucifixion, resurrection, original sin and the promise of salvation. | was disabused of my belief that these elements of Christian belief were incompatible with high mental capacity and _ intellectual sophistication. Yet, the spiritual climate of these English intellectual Christians were different from today’s post, post Vietnam return of the religious themes of the turn of the Twentieth Century, big tent revivalism and Billy Sunday’s brand of Christian patriotic America. Today’s religious patriotism infuses George W. Bush’s Republican base, National Security Adviser Condoleezza Rice’s after dinner hymns and Attorney General John Ashcroft’s early morning bible study groups for his Assistant Attorney Generals. Even the most religious of my English math buddies are without what seems like adventitious baggage of today’s faith based Republicans: the belief in the immorality and godlessness of teaching evolution in schools, what has been called the massacre of the innocents in stem cell research and abortion clinics, the 171 HOUSE_OVERSIGHT_013671

right to bear machine guns and the intrinsically venal sinfulness of a man’s commitment in love of another man. Was the clustering of these apparently diverse concerns the accidental result of a sociopolitical-religious short circuit, a class- resentment-driven spiritual split in geographic, socioeconomic and educational class? Tim LeHay is selling millions of books, whole tables full at Wal-Mart’s, which come packaged with these assumptions. Surely higher-level theists would make today’s evil more subtle, abstract and pervasive, perhaps involving inner life themes of envy, vengeance and aggression; goodness implicating empathically made moral choices involving interpersonal kindness and evidence of caring about the well being of others. My contact with some English academicians taught me that even the mathematics of hard science can be viewed as a gift of grace and belief in the possibility of a continually emerging, Christ-centered, evolutionary process. Protestant philosopher mathematician Alfred North Whitehead in his 1926 Religion in the Making, Catholic anthropologist priest, Pierre Teilhard de Chardin in his The Phenomenon of Man and the more modern process theologists of New York’s Union Theological Seminary do not exclude Christ's involvement in evolving science and other new knowledge. They see Him participating in a spiritual evolutionary progress which does not gather the barnacles of irrational ideas about the murder of less than hundred-cell blastula or the psychoneurohormonally determined sexual partner preference. They know about the ever-changing cultural and political appearances of faux and real evil. Nonetheless, what | learned from my Christian and Jewish friends at the mathematics institute was that, though the definitions of evil may change, evil as a construct and spiritual mechanism is an apparently essential component of the Christian experience. On Rosh Hashanah, even the reformed Jews commit themselves to Teshuvah, making up for past evil deeds. The good versus evil dichotomous view of man’s existence is true in the lives of Assembly of God Fundamentalists of Georgia as well as the sophisticated Readers, Professors and Dons of the high Episcopal churches and university chapels of Oxford and Cambridge. 172 HOUSE_OVERSIGHT_013672

Finding high-level mathematical thinkers at home in metaphysical surrounds and metaphysicians diligently practicing mathematics are certainly not new. Some instructive examples include, famously, the Pythagoreans, the 15" Century Catholic Cardinal Nicholas Von Cusa, who used geometric symbols to record his spiritual philosophy, and the Talmudic-Cartesian style of argumentation of Nicholas de Spinoza. This approach to an examination of metaphysical systems, sometimes called mathematicism, exploits the machinery of the mathematical mind to evaluate the consistency and completeness of thoughts, to create representative axiomatic structures and to operate within them using syntactic calculus. The practice of the rational dialectic of mathematicism, working for moral purity of heart, develops a brain-somatic discipline much like the exercises of Yoga. This approach flies in the face of the major premise of these essays, my belief in the necessity of what William James and others have called the primary religious experience in order to know God. Recall that my father’s favorite Jewish mystic, Abraham Abulafia, said this experience gives birth to an activated mind that can then immediately and completely inform the Spirit. Among the religious English mathematicians, | learned that it doesn’t have to happen this way. One can apparently think oneself to It. A well known example of a modern theistic Oxford type, the Magdalene College English tutor and Don, C.S. Lewis, in his introduction tt to St. Athanasius’s The Incarnation of the Word of God, wrote, “...| believe that many who find that nothing happens when they sit down or kneel down with a book of devotion, might find that their heart would sing unbidden while they are working their way through a tough bit of theology with a pipe in their teeth and a pencil in their hand...” In contrast, without my personal experiences with joyful transcendence, the direct feeling of His presence, | would not have known about the goals of his more analytic efforts. It was a struggle for me to use a rational mind to share the meanings of the poetic ruminations in his BBC lectures, Mere Christianity. This Reader from Oxford with two firsts in Latin and Greek followed by another first tt in English Literature, described the world as “...enemy occupied territory...” the omnipresence of the Good Power turned Dark Power of the Prince of Darkness and the Christian as “...a man who is enabled to repent and pick himself up...” 173 HOUSE_OVERSIGHT_013673

For C.S. Lewis, religious faith came from intellectual hard work. He was put off by spirituality that arrived by thoughtless fiat. He rejected the idea of living in simple and loving direct conversation with the God within, as described by Brother Lawrence. Lawrence was described as the simple “great awkward fellow who broke everything.” Lewis had little faith in what he perceived as the mindless spiritual methodology of this selfless, silent, hard working Parisian monastery cook for a hundred fellow monks who was also their dedicated smelly sandal repairer. Perhaps reflecting his place in the British intellectual class system, Lewis wrote that Lawrence’s conversations and letters in the brief pamphlet, Practice of the Presence of God, “...full of truth... but unctuous and repulsive.” At the same time, Lewis spoke of his own experiential evidence for God in Surprised by Joy in which he admits, “| am an empirical theist. | have arrived at God by induction.” It is likely that Brother Lawrence did not know and did not need to know the difference between an inductive and deductive argument. For most of my years, | have been a subject of Jamesian transcendent experience, LSD expansive visions, Sufi moving meditation, long distance running, Black Baptist shouting, Tantric orgasmic withholding, Yiddish Labovicher dancing, Charismatic Christian Church rock and rolling, Hindi meditative rising Kundalini, almost any ecstatic crisis inducing, God type. Recall that | am from a generation that a Donovan song inspired to smoke bananas. | did not personally access Brother Lawrence’s calm, work-a-day, devotional, quietly persistent, perspective yielding, inner conversations with God until my sixth decade. The opportunity came from my growingly severe, unfixably chronic, pain. The counter-intuitive insight and helpful identification was gained from reading about Joseph de Beaufort’s conversations with Brother Lawrence. Beaufort said Lawrence was born with the name Nicholas Herman in 1611 and renamed Lawrence in honor of his parish priest. As young soldier in the Thirty Years War of the 17" Century, he was severely injured. He was left with both sciatic nerves trapped between bone spurs and tissue scarring from his early twenties. These injuries, involving the two biggest pain- conducting nerves in the body, left him crippled in gait and in chronically severe lower back and leg pain from which he would never be free. It was after this time 174 HOUSE_OVERSIGHT_013674

and a few years of looking for God in what he called “wondering in the wilderness” that he began his 40 years of monastery service as cook and sandal maker. He was tt described as amazingly selfless and a “...gentle man of joyful spirit...” who “...continually walked with God...not from the head but from the heart...” Doing long hours of selfless work with such painful disabilities, how was it that he maintained his joyful, loving and calm contact with God and his fellow man? How did he do it? | found that, as with all miracles of God contact for me, it happened by itself. | suffered my first testicular cancer in my thirties. | felt the little hard rock by accident while scratching. It was on the left side. Surgical removal was followed by a five-hour radical abdominal lymph node dissection that left me with incidental abdominal sympathetic nerve damage, urinary hesitancy and ejaculating backwards into my bladder. The tissue diagnosis was of embryonic cell carcinoma with chorionic elements. The U.S. Armed Service Pathology Department’s statistical book gave me 5% chance of living beyond two years. My second testicular cancer occurred in my fifties and on the right, two little joined lumps found by my wife. It was a seminoma with cure rate of 85% but requiring four weeks of almost daily x- ray treatment. The combination of radiation induced blood vessel scarring (they had to blast widely since my earlier lymph node dissection confused the usual radiological anatomy), a pre-existing laterally curved spinal column and the arthritic changes resulting from fifteen years of running over 10 miles per day with this kind of back led eventually to the degeneration and collapse of several of the bodies of my vertebrate pinching several leg nerves between bone spurs and radiation- induced scarring. | have been in increasingly severe back and leg pain for fifteen years. It was in this way that | fell heir to both Brother Lawrence pain syndrome and what | now think was his strong inclination to live in the Spirit, as far as possible outside the concerns with his own mental and physical body. In my experience, this led naturally to a decreased in my life long narcissistic preoccupations, diminished my ego-driven achievement desperation, setting up a more comfortable inner seating for conversations about and with God. The choice was between fully embracing a God-oriented place for most of my daily existence or the chronic use of 175 HOUSE_OVERSIGHT_013675

enough narcotics to eliminate complexity of thought, real interpersonal feelings and hope for meaningfully creative work. The remarkable thing to me was that people began to talk about my “improved disposition,” an increase in out-of-my- psychiatrist's office personal empathy and kindness as well as a_ significant decrement in my overweening, ego-stoking ambitious and competitive urges. Any return to the earth body of tense readiness to competitively succeed, protect with ego defensive anger, fantasies of assertive sexuality, stand tall grandiose notions of intellectual superiority, even getting up for scientific combat, was accompanied by the return to this world of pain. Only lovingly detached, unpretentious, other directed, quietly calm inner dialogue with Him was a place that | could live. This was an inner land of still another kind of God than | had previously known. | could even read and struggle with theological ideas thoughtfully, without referencing personal mystical, psychopharmacological, Holy Ghost-mimetic, experiences. | could enjoy the rational, social responsibility valuing, spiritual peace of a white Protestant Sunday morning service. | could attend Reformed Jewish Friday night services about man’s responsibility to man without restless boredom. No longer seeking the feeling of God’s thrill, | could think about it, even without being in the state of my father’s and Abulafia’s activated mind. If | had been benefited with a classical language education beyond the high school and early college Latin of Julius Caeser and Cicero or matriculated in an academic theological seminary, | would have already studied, maybe even worn out, the deeper aspects of what seemed like a paradox of the consonance of faith and reason. | would have been familiar with the rhetorical argumentation in the patristic Latin commentary on sacred texts by Tertullian and other Fathers of the early Christian Church, the Talmudic discussions (the Mishna in Hebrew and Gemora in Aramaic) of the oral Torah by the Rabbinate, the Muslim explication of Koranic Islam in the oral tradition of the Hadith. Robert Wilken in his recent The Spirit of Early Christian Thought was in no doubt about the harmonic relationship between rationality and faith: “...by putting itself in the service of truth, faith enables reason to exercise its power in realms to which it would otherwise have no 176 HOUSE_OVERSIGHT_013676

access...” It is perhaps strange to come to this common knowledge so late, but | came to my life with my forbearers and father’s magical, mystical biases. My father had parodied what he thought was the “wasteful time” spent in rational, Talmudic discussion. He said that is what Jewish men spent their time doing to avoid physical work while sitting near the city gates. It was the women who raised the crops and cared for the cattle and children. He had a favorite conundrum satirizing the village gate discussions. Jewish males, after the age of thirteen, accompany their morning prayer of commitment to loving and serving God with the ritual of wrapping scripture embedded animal skin, tefillin, and winding them seven times around the left arm, near the heart, and around the head, symbolizing the mind. This contextualizes how my father made fun of a typical topic of these all male Talmudic seminars: “If one had seven arms, would one wrap the fefillin once around each appendage or seven times about one of them. If the latter is the case, how would one chose which one.” In fact, there remains an on-going debate about the order with which the embedded four passages from Exodus and Deuteronomy should be arranged and inserted in the ftefillin such that some compromising orthodox Jews wear two types of fefi/lin, each representing one of the theoretically justified orderings. | know now that there is an implicitly positive confirmation of a jointly held faith and feeling of ethnic belonging achieved by such apparently abstract discourse and argumentation. In truth, | had not come to Warwick to explore the relationships between faith and rationality using the cognitive style of mathematicism, but rather to be saved by the mathematical miracles of the Brain God. Not unrelated to what C.S. Lewis saw as a prominent characteristic of spiritual experience, “wonder,” and what Philip Davis and Reuben Hersh in their 1981 book, The Mathematical Experience, spoke of as “beauty” and “surprise.” | know about the attack of excitement that comes with the sudden emergence of counterintuitive conceptual connections while exploring new mathematical ideas. In energetic high, | start skip reading, underlining the book frantically, jotting commentary on the margins, copying the relevant equations into my notebook. Was this the same break through to a glimmering of grace, everything beautifully in order and precious, that | experienced on LSD while sitting for hours 177 HOUSE_OVERSIGHT_013677

inside Paris’s towering, echoing, Notre Dame Cathedral, hearing Latin chants in the dank sweet smell of old church and chained, swinging canisters of smoking incense as the pipe organ roared? Those realities that George Berkeley, the 1721 author of Treatise Concerning the Principles of Human Knowledge, the theist whose name was given to a mostly agnostic Northern California city, saw as grounded in the spirituality of God’s infinite mind and broadcast as universal ideas through our derivative, finite minds. Rational religion and mystical religion joined in faith by the presence of implicit and universal mathematical structure | spent about two years at a mathematics institute in France, /nstitute des Hautes Etudes, IHES, sitting at the guru feet of the mathematical great and metaphysician, Rene Thom. His mathematical pallet was breathtakingly broad, a taste of what in past centuries was called natural philosophy and what seemed to me to be about the unapologetic geometrization of the Intuitive God of the Mind. Natalie Angiers, erstwhile mathematician, now reporter and atheistic hard ass, writing in the New York Times, called Thom’s ideas the talk of “...an Emperor without clothes...” The Kantian theme of the personal a priori status of an intuitive geometry, an already in us representation of all that’s out there, was implicit in his Catastrophe Theory research program and was published first in his classical Structural Stability of Morphogenesis (1977) and made more overt in his later (1990) Semiophysics. To get a feeling for the rational-logical versus mystical-intuitive spiritual issue in a mathematical context, consider the following: most of us remember the struggle to unify the strange and difficult cognitive duality of the high school geometry experience. On one hand, shapes and their relations and rearrangements could be intuitively grasped, even manipulated; on the other hand, we were taught that these mental images and the results of their intuitive transformations were not to be trusted. In mathematics, as in my belief in the fireworks of primary religious experience, seeing is not necessarily believing. In my high school geometry class, what was to be believed was what followed from the proper practice of the tightly organized, Euclidean system of axioms, postulates and the derivative logical 178 HOUSE_OVERSIGHT_013678

operations resulting in the surety of proofs. The unresolved tension about what | believed from intuitive experience and what | was allowed to believe from the logic of theorem and proof, perhaps not unlike my belief in the transcendent experience over logical theological argument as Reality, continued throughout my life. For example, many decades later at HES, | saw the world class dynamical systems theorist and differential geometer-topologist, Dennis Sullivan, use a projector to display a computer-generated, intricate and beautiful, mathematical object, the well known, computer screen saver, Mandelbrot set. It represents the control parameter plane of the well studied complex analytic map, z > z* + c. Sullivan, pointing to a small, discrete complicated little part of it that looked like a little version of the whole of it, from a distance looking like a point, said, “An important Ph.D. dissertation is waiting to be done on the question: is this (pointing to the little object) really there?” In the audience of about a hundred professional mathematicians and one amateur, | was the only one that laughed. Historians of mathematics point to the successful generalization of Euclidian geometry via its abstract axioms, postulates and logical operations to a new, not naturally intuitable, almost nonvisualizable, non-Euclidean geometry (with the new geometric axiom, parallel lines do meet at infinity), as evidence against the Kantian idea of the intuitively accessible, a priori status of geometry. This served as an example of where mathematics naturally resides, and argues in favor of the thought control imposed by the modern set theoretic and logical rituals of mathematical theorem and proof. Thom, in a_hereditary-evolutionary biological argument developed in Semiophysics, said “Objections raised to the Kantian apriority of Euclidean geometry after the discovery of non-Euclidean geometries, and the theories of twentieth century physics (restricted and general relativity, quantum mechanics) appear to me to be irrelevant...they deal with ...the infinitely small and infinitely large...which lies outside the usual cognitive activity of ancient man.” In my discussions with him, Thom found equivalence relations between mental and real world objects and their behaviors. He described what he called an abstract physicalist truth that describes a psychic universe, which, in turn, simulates outside things and processes. Much like the transcendent experiential God | have 179 HOUSE_OVERSIGHT_013679

experienced, seek and think | know about, Thom was not after the logical proofs of geometry but rather viewed mathematical theorem and proof work as activity derived from intuitive experience with geometric relations as the thought forms that represented rea/ Reality. Though a Field’s Medal winner in mathematics (recall that it is the Nobel Prize in mathematics awarded every four years at the International Congress of Mathematics) and for his life time, one of the most brilliant and fecund mathematicians in the world, so many mathematicians admit that they got the seeds of their life work from his throw away remarks, Thom, with a little smile and his eyes twinkling, admitted to me with apparent pleasure that “| have never proven any theorem in my life.” All his discoveries came from insightful moments of grace and the courage to pursue them. Riding back from Paris late one night on a train that didn’t stop at /HES’s town of Bures sur Yvette, | watched him use the red emergency phone to call the train’s engineer to stop the train suddenly for our exit. | loved him, in part, because he had the courage to believe in and act on my kind of intuitively realizable, experiential God. In keeping with his characteristic style of generalizing mathematical systems beyond their carefully defined specifics, Thom defined the concept of singularity very broadly, speaking of them as distinctive and noteworthy things, points where the usual or expected properties, laws and definitions fail, where smooth and continuous processes become discontinuous. For Thom, these were the settings for the unexpected and miraculous. He believed that his work and that of many others, now and in the future, would indicate that the set of miraculous singularities were finite, systematic, universal and describable. Most importantly for our purposes, Thom believed them to be archetypal. It was through the structure of archetypal singularities that he regarded inside and outside realities as mutually reflective. | was blessed by hours of discussion with him during his car travels to lecture around France. Thom often asked me to accompany him as he drove from IHES to various branches of the University of Paris. He used these times to exercise my geometrically flavored, mathematical intuitions. He used words to create visualizable structures without the diagrammatic aid of a blackboard. He used mental topological structures created by the properties of imagined motions, 180 HOUSE_OVERSIGHT_013680

flows, which led to examples of some of his universal singularities that he claimed could be found in all real physical, biological and psychological systems. For some examples: One of his archetypal singularities was a boundary at x = 0 such that the flow couldn’t spread from where it was in x>0O into x< 0 and was therefore like the border, the membrane, between the inside and outside of a cell as well as the hoped for sociopolitical functions of the Great Wall of China and the Maginot Line. If we were to blow up the boundary line from two to three dimensions, R’—R®, the straight boundary line becomes a cylinder for directionally organizing and connecting flows as in blood vessels, oil pipes, cables and wires. Since production and delivery need not occur at similar rates, temporary storage is required and may take the form of a spherical blow-up in the vertical segment of R® leading to an open bottle which may serve as a dead end storage branch of a network of connected cylinders. In the conceptual reductionism of Semiophysics, Thom said, “...life is essentially a question of embankment, canalization and the struggle to stem dispersion.” These structures of mind and world are built and maintained. Coagulation of blood is an example of a canalized fluid repairing gaps like a tubeless tire. Thom considered apparent the problem of making something from nothing, birth, that of finding the hidden sources: the bubbling spring emerges from an unseen, underground network of canalized fluid flow converging on the apparent source, birth being the invisible becoming visible. In contrast, a canalized flow emptying into lake can represent disappearance as a flow. Mathematicians from all over the world attended Thom’s 65" birthday celebration at HES. His Field’s Medal winning work on the topology of differentiable (smooth) manifolds, cobordism and related ideas, was mentioned frequently, and great homage paid to him with respect to these areas of his work. However, in two days of lectures of personal and professional tribute by the world’s great mathematicians, his work relevant to Catastrophe Theory and Semiophysics was not mentioned, even once. The form taken by mathematicians’ most severe judgments is silence. As the New York Times’ Natalie Angier’s comments indicated, this is not the time for the intuitive conduct of applied mathematics. 181 HOUSE_OVERSIGHT_013681

It was upon Thom’s recommendation, that | spent the year in the Mathematics Institute in Warwick, England. Using the Math House #2 s home base, | made many trips to Oxford University and a few to Cambridge. It was in these places that | learned first hand that belief in the Resurrection was not simply a matter of socioeconomic class. | tried to schedule my trips to Oxford or Cambridge to coincide with the weekend so | could hear the remarkably literate sermons at the Universities college chapels. In these places, for hundreds of years, just because one was a top-notch practitioner of mathematics or linguistics did not mean that the Don did not have within him the full panoply of beliefs attendant to the Christian God. Maybe this easy combination of logic and Spirit derives from the character of English mathematics. There are graduates with professorially enfranchising Masters of Art Degrees in Mathematics from Cambridge University where the subject is considered by many to be part of the culture of the humanities, closely akin to philosophy and linguistics.. In the universities of United States, for example the Massachusetts Institute of Technology, an academic degree of Ph.D. in mathematics is seen by most faculty as an indication of the intellectual equipment required for a life of scientific work in which disconfirmable experiments are the ultimate criteria for knowledge. The field of pure mathematics (not ostensibly relevant to the real world outside the mind) has itself evolved in this direction. Recently, a physical scientist, a theoretical physicist, Edward Witten, was given the mathematician’s ultimate award, the Field’s Medal. In American universities in general, very few mathematics departments are in schools of the humanities. Most are in the schools of science. This variation in bureaucratic, metaphysical, sorting reflects our continuing struggle with the true nature of reality and the role of mathematics in its knowing. The now emergent field of computer science removes mathematics even further from intuition and Spirit. Difficult problems such as proofs of theorems can be systematically examined for all possibilities quickly by trying them out in what is now known as a computational proof. On the other hand, pointing at this computation’s graphics, the theorem and proof, real mathematicians can ask, is this really there? 182 HOUSE_OVERSIGHT_013682

Mentioned briefly above was one of humankind’s beacons, Pythagoras, the intellectual and spiritual progenitor of Plato. He taught the disciples of the Pythagorean Brotherhood in Crotona, Italy, that reality at its deepest level was mathematical thought. Studies there included philosophy, geometry, music and astronomy, all at the service of achieving closer union with the Divine. Pythagoras and his school, only his student’s writings remain, was said to be working at unifying elements of the ancient tribal mystery cults with the observables of worldly events through meditative, mathematical, philosophical mysticism. Knowledge was gained through spiritual intuition made harmonious with formal systems of thought. As Plato later said and as quoted by Thomas Heath in his 1921 History of Greek Mathematics, about the study of the motion of stars, “...leave the heavens alone...” because what one sees is only an approximation of the real and more perfect mathematical structures involving points, lines and circles. To which Newton added an elongated circle, the ellipse, and Nineteenth and Twentieth Century mathematicians and physicists, the geometries of positively and negatively curved space. It is perhaps not an accident that debates about evidence for the existence and location of God and where the ideas and structures of mathematics live and breath generally involves a stand off between those that believe that both are out there and can be seen, like thoughtful, humanistic actions and caring service for needful others, versus those that feel the phenomenology of both are projections of the psychobiologically intuitive Brain God and can be felt like an ecstatic rush of insightful illumination. Further Reading for Faith And Rationality Introduction of Comparative Mysticism. Jacques De Marquette, Philosophical Library, N.Y. 1949, 183 HOUSE_OVERSIGHT_013683

Mysticism and Logic. Bertrand Russell. Norton. N.Y. 1929. Sefer shel Devarium (The Book of Words). Lawrence Kushner, Jewish Lights, Woodstock, Vermont. 1998. Semiophysics: A Sketch, Aristotelian Physics and Catastrophe Theory. Rene Thom. Addison-Wesley, Reading, MA. 1990. Mere Christianity. C-S. Lewis. MacMillan, N.Y. 1952. Sacred Geometry. Miranda Lundy. Walker, N.Y. 1998. Fractals, Form, Chance and Dimension, B.B. Mandelbrot, Freeman, San Francisco, 1977. Non-Euclidean Geometry, H.S.M. Coxeter, University of Toronto Press, Toronto, 1957. Fundamentals of Mathematics, Vol. |. H. Behnke, F. Bachmann, K. Fladt and W. Suss, MIT Press, Cambridge. 1983. Religion Explained, The Evolutionary Origins of Religious Thought. Pascal Boyer. Basic Books, N.Y. 2001. Catastrophe Theory. Alexander Woodcock and Monte Davis, Dutton, N.Y. 1978. lt Must Be Beautiful; Great Equations of Modern Science. Graham Farmelo, Granta, London, 2002. Neurobiological Barriers to Euphoria. Arnold J. Mandell, American Scientist 61: 565- 573, 1973. 184 HOUSE_OVERSIGHT_013684

Brain Physics and the Respiritualization of Healing. Arnold J. Mandell, Bulletin of the National Guild of Catholic Psychiatrists. 28:19-24, 1983. Toward a Psychobiology of Transcendence, God in the Brain. Arnold J. Mandell, |In Psychobiology of Consciousness (eds. J.M. Davidson and R.J. Davidson). Plenum, N.Y. 1980. 185 HOUSE_OVERSIGHT_013685

APPENDIX AN INTUITIVE GUIDE TO THE IDEAS AND METHODS OF DYNAMICAL SYSTEMS FOR THE LIFE SCIENCES Arnold J. Mandell and Karen A. Selz Biological Scientists Can Understand and Use Ideas and Methods of Nonlinear Science A yield of advances in computer hardware and software is that even quite difficult applied nonlinear mathematics can become accessible to experimentally oriented biological scientists. Before this time, the development and analysis of a particular set of nonlinear differential equations, describing the actions of a neurobiological system in motion, involved decades of specialty training, rare insight and many hours of highly skilled, trial and error computations by hand. Since the idiosyncrasies of each nonlinear system were considered unique, the results of their analyses were thought to concern only the particular nonlinear system being studied. Often a shift in hypothetical mechanism meant starting the long and painful process all over again. In addition, these findings were usually communicated only to a small and arcane mathematical community in the form of dense theorems and difficult to follow proofs, insurmountable language barriers to biological researchers wishing to use them to better describe and understand their experimental observations. For today’s neuroscientist with a desktop computer, an inclination to program and access to computer algebra and numerical software such as Maple, Mathematica or MatLab, operational definitions and computational empiricism can replace the theorem and proof continuity required to do old style applied mathematics. For those of us without sufficient facility in algebraic manipulation to easily follow the arguments of professional mathematicians, a computer algebra program such as Maple serves as a delightfully accessible consultant with which to “check out what the guy is saying". Those motivated enough to write their own data generating or analytic programs in C, Fortran, Pascal or Basic (though not 186 HOUSE_OVERSIGHT_013686

essential) can find easy-to-use algorithmic help in Cambridge University Press’s Numerical Recipes series (Press et al, 1991). The conceptual and communication gaps between applied mathematicians and physicists and the bench practitioners of the neurosciences, that inevitably lead one or the other, most often both, to surrender their deepest intuitions to jointly shared images that are inevitably more simplistic, are no longer inevitable. With her own hands on both the quantitative conjectural and experimental machinery, the motivated practicing neuroscientist can honor her own insights, read about and construct symbolic representations from her intuitions and do her own quantitative theory. Computerized numerical techniques have become so powerful and accessible that, even in academic settings, there is debate about whether fundamental analytic tools, such as series expansions, should be taught in undergraduate courses about differential equations. The practice of “try it and see what happens", with the current name of experimental, computational mathematics, is accessible to all. In addition to the powerful general mathematical programs noted above, there exist several sets of more specifically targeted software with the capacity to generate, portray and quantify the behavior of nonlinear continuous and discrete abstract and real dynamical systems. These often also include algorithmic modules that are useful in tailoring new models and measures (see for examples, Parker and Chua, 1989; Baker and Gollub, 1991; Nusse and Yorke, 1991; Sprott and Rowlands, 1991; Sprott, 1993; Korsch and Jodl, 1994; Enns and McGuire, 1997). Learning from and using this software, along with only a little programming in the high level languages and computer algebra programs listed above, permit the non-mathematician neuroscientist, willing to read in the literature such as that described below, to do independent, cutting edge research in applied dynamical systems. Described below will be the computational discoveries in abstract systems and real neuroscientific data that have led to multiple contexts of quantitative description. These include those that are: (1) Geometric and conserve metric distances; (2) Topological and conserve relative positions but not distances; (3) Single or multiple global quantitative descriptors such as scaling numbers or scaling 187 HOUSE_OVERSIGHT_013687

number spectra; (4) Non-Gaussian distributions with heavy tails and correlations reflected in their Hurst, Fano, Allan and Levy exponents; (5) Statistical dynamical descriptions of trajectories of the system in their embedding space such as Lyapounov exponents, Hausdorff-Mandelbrot dimensions, Sinai-Ruelle-Bowen measures, and Adler-Weiss-Ornstein topological and metric entropies. Characteristics which discriminate between experimental versus control conditions in parametric computational and real physiological and pharmacological experiments serve to generate and test ideas and imagery arising out of behavior observed in both biological and abstract dynamical realms. New experiments can be suggested by the implicative structure of dynamical systems theory as well as neurobiological findings and intuitions. As examples, the sudden “switch” of manic- depressive bipolarity syndromes may be a “bifurcation” in nonlinear dynamical systems; the “noise” of the statistical physicist may be the “arousal” of the brain stem-thalamic biogenic amine and reticular formation neurophysiologist; aspects of “thought disorder” in the pathophysiology of schizophrenic patients may be an entropic sequencing idiosyncrasy in the “symbolic dynamics” of a particular brain system attractor; neuronal “bursting” may be the “intermittency” of a neurodynamical system; a multiplicity of “discrete ion channel conductances” may be a single “global scaling hierarchy” of conductances times. The number of published examples of this fusion of ideas and methodology in the biological-relevant literature is already in the several hundreds and Medline counts indicate is growing exponentially. Representative samples of these are described below. In addition to the technological advances in computational hardware and software, the major scientific surprise making this new era possible is the discovery of universalities, the finite set of behaviors characteristic of most, if not all nonlinear systems, across most if not all of the specific equations or neural systems being explored. This makes the emergence of semi-quantitative equivalence relations between model and data not only possible but likely, even though we don’t now and perhaps never will know enough to either write or solve completely the specific and detailed equations for the biological system of interest. We neuroscientists need not be apologetic for using these ideas and tools qualitatively and empirically. In fact, 188 HOUSE_OVERSIGHT_013688

unanticipated results of analog and digital computer experiments were responsible for most if not all of the discoveries underlying the current era’s revolution in applied nonlinear mathematics Modern Applied Dynamical Systems Emerged from Accidental Computational Discoveries A medical student named Herr, in his thesis research with the “radio engineer’, Van der Pol (1926), was simulating cardiac electrophysiology with an analog device which permitted real time, exploration of a full range of parameter values long before there were fast enough digital processors to do so. Studying the behavior of equations of a periodically, pace maker, driven, nonlinear triode oscillator, Herr found orbital points that appeared to belong to two different periods simultaneously thus violating the uniqueness of solutions of differential equation theory. The Van der Pol relaxation oscillator equations, with their slow buildup and sudden discharge of membrane potential are good models for the slow-fast processes of repolarization and depolarization of Hodgkin-Huxley type equations (Rinzel, 1985). Periodically driven, nonlinear differential equations of the Van der Pol type are generally applicable to the multiplicity of dynamical regimes of neuronal dynamics (Carpenter, 1979; Aihara et al, 1984; Chay and Rinzel, 1985) and, with periodic and aperiodic driving and noise, can be made relevant to particular mammalian neuronal subsystems in the context of clinically relevant global electrophysiological phenomena such as Magoun’s (1954) brain stem evoked EEG and behavioral arousal (Nicolis, 1986; Selz and Mandell, 1992; Mandell and Selz, 1993). In the early 1940’s, using the pre-publication results of similar analog computer studies (Levinson, 1949), the Cambridge mathematicians, Mary Cartwright and Joe Littlewood (1945; McMurran, S., Tattersal, J.,1999) used geometric methods to prove that the highly nonlinear, periodically driven Van der Pol equations, depending upon one or two changing parameters, generated fixed point (“homeostatic”), periodic (“cyclic”), subharmonic (“period doubling”), quasiperiodic (“multiply periodic”), intermittent (“bursting”) and “deterministically 189 HOUSE_OVERSIGHT_013689

random” patterns. We now know such phenomena to be universal characteristics of bifurcation scenarios in nonlinear dynamical systems where bifurcation means discontinuous changes in patterns of behavior (dependent variables) resulting from smooth changes in parameters (independent variables). Alerted to their presence in computer experiments with biologically relevant nonlinear differential equations, these phenomena have since been found in time series from patch clamped membrane channels, single neurons, neuronal networks, neuroendocrine systems, brain waves and patterns of behavior in animals and man (see below). Cartwright- Littlewood found that the inner and outer edges of the domains of attraction (all the initial values that eventually wind up in the attractor—the limit set of all bounded solutions) of two different sets of subharmonic periods for the same parameter settings were interlaced at many scales in what is today called a fractal basin boundary. It was in this way that the specific values of the end state are understood to be indeterminate since the starting values in the fractal basin boundary are impossible to isolate and specify with adequate experimental precision. Similar biologically-relevant analog computer discoveries about the Van der Pol and comparable periodically forced, dissipative (energy utilizing) Duffing equations (Zeeman, 1976) were made in the early 1960’s by electrical engineer, Yoshi Ueda (1992), but his thesis director, Chihiro Hayashi of Japan’s Kyoto University, was sufficiently disturbed by this evidence for the existence of bounded solutions (attractors) that were neither fixed points (equilibria) nor periodic orbits (cycles), the only ones known at the time and therefore “strange,” that he refused to let Ueda publish his findings until he did so as an independent investigator in the 1970's. In the early 1960’s, Edward Lorenz (1963), a meteorologist and student of the Harvard mathematician and dynamical systems pioneer, George Birkoff (1922), was computing the output of a very reduced subset of Saltzman’s differential equations for predicting the weather (1962). Lorenz found that numerically integrated trajectories manifested unpredictable times and directions of motion between the two spiral orbits of what has come to be known as the Lorenz attractor. Very small differences in starting values led to widely diverse final values, and, just 190 HOUSE_OVERSIGHT_013690

as importantly, far apart initial values could be found close together in the limit set. This behavior was called “sensitivity to initial conditions” by David Ruelle (1978; Ruelle and Takens, 19771). It is noteworthy, however, that over a range of values of the parameters, the overall pattern of the orbits of the Lorenz attractor results in characteristic geometric pictures as well as invariant statistical descriptors. Qualitative and quantitative global similarities were gained while specific solutions were lost in these “strange attractors” of nonlinear systems. Analog computer simulation of a simpler set of equations inspired by nonlinear chemical reaction kinetics led to the discovery by Rdéssler (1976) of another early and generic strange attractor combining sensitivity to initial conditions and characteristic geometries and measures. It was the Russian mathematicians, A.N. Kolmogorov (1957), Sinai (1959) and V.I. Arnold (Arnold and Avez, 1968), the French mathematicians, Rene Thom (1972) and David Ruelle (1978) and the U.C. Berkeley mathematicians, Steve Smale (1967) and his student, Rufus Bowen (1975), and their associates who gathered together these and other related computational discoveries and embedded them in a qualitative theory of nonlinear differential equations, using a variety of formalisms, including point set and differential topology, geometry, analysis and ergodic (having an invariant statistical description) measure theory that formally established the fundamentals for research in nonlinear dynamical systems. Here a dynamical system refers generally and simply to the components and nonlinear processes (transformations) that move points (values) in discrete (“map”) or continuous (“differential equation”) time around in an appropriately defined space. The phrase, “nonlinear transformation” in this context does not imply easily solvable curved functions, such as those representing the sigmoid kinetic or threshold functions of enzymes and neuronal networks or those that smoothly log transform the amplitudes of auditory or other sensory modalities in man, but rather allude to expressions containing products, powers and functions of the computational and/or experimental variables x,, such as x,x,, (x,)° Orsin(x) . 191 HOUSE_OVERSIGHT_013691

As noted above, the cross-disciplinary cohesiveness of such a vaguely defined field occurred as the result of the unanticipated discovery of a relatively small set of nonlinear phenomena, universalities, that implicated many fields of mathematics, from differential geometry to number theory, and were found in a broad range of physical and biological realizations, from turbulent plasmas and chemical and enzymatic reactions to neuroendocrine hormone release patterns. It is perhaps counter-intuitive but, whereas linear systems can generate an infinite number of solutions locating points anywhere the person writing the equations wants them to go, nonlinear systems are generally restricted to a finite set of global dynamics and these emerge on their own from the intrinsic dynamics of the system. Trying to make these systems follow orders, not unlike finding the most clinically effective dosage range of a psychopharmacological agent, require the empiricism of trial and error experiments. A second class of computational accidents involving nonlinear systems resulted in unanticipated coherence rather than unpredictable disorder. Using one of the early “high speed” digital computers at Los Alamos, MANIC I, Enrico Fermi with Pasta and Ulam (1955) attempted to obtain a many-body statistical thermodynamic equilibrium analogous to heat generated noise by coupling 64 particles together with nonlinear springs. They found only a few low period modes that oscillated indefinitely. Instead of equidistribution of the energy into 128 degrees of freedom (64 locations x 64 velocities in 128 dimensional phase space), they found it gathered up into only few coherent modes. Although the relevance to biological science of nonlinear multifrequency coherence is a bit off from our focus, it is worthwhile noting that a recent (Karhunen-Loeve) decomposition of the alpha band of the resting alert human EEG revealed only three dominant temporal-spatial modes: anterior-posterior, rotational and standing (Friedrich et al, 1991) and “few frequency coherence’ is a frontier of inquiry in brain wave research. A heterogeneous collection of coupled nonlinear elements in the form of widely distributed, multi-location, multifrequency systems such as cross-cortical, brain stem-thalamic-cortical and interconnected spinal motor neurons can generate 192 HOUSE_OVERSIGHT_013692

coherent activity. This temporal and phase coherence plays an important role in current theory of sensory-associative-motor integrative function, how distributed attributions come together in the brain representation of a single object or process, in the context of the so-called “binding problem” (Singer, 1993; Bressler, 1995; Nicolelis, 1995; Schiff et al, 1997). Diffusely distributed neurochemical variables have been invoked. For example, the role of metabotropic glutamate receptors in driving the synchronization of interneuronal networks has been suggested as a mechanistic model (Whittington et al, 1995). The objects of relevance to the discovery of Fermi-Pasta-Ulam are studied as the nonlinear physics of nondissapative wave processes and are called solitons (Zabusky and Kruskal, 1965). They have been invoked to model nerve conduction and information transport in brain (Scott, 1990). A third counter-intuitive set of accidental computational findings is in an area of research called symbolic dynamics which involves the universal parameter- dependent coding language and capacity of nonlinear systems. In the early 1960’s, a group around Stan Ulam at Los Alamos (Cooper, 1987) used one of the early “high powered” computers, MANIAC Il, to iterate (letting the output of the action of a discrete time function serve as its input the next time around) simple equations they called “maps.” These reduced dimensional objects shaped like tents, sine functions and parabolas can be extracted from and represent the behavior of higher dimensional, nonlinear differential equations (see Devaney, 1989; Schuster, 1989 or Moon, 1992 for intuitive descriptions). They varied a parameter, such as the height of the tent or parabola, to systematically change the period and/or phase (order) of the symbol sequence (Metropolis et al, 1973). Normalizing the range of values of the output to [0,1] and transforming the series of values into a binary code, L < 0.5 and R > 0.5, they found an invariant, one parameter dependent, progression of ordered periods, R, RLR, RLRR...RLLRL..., in all such single maximum maps. This “U (universal)sequence” has also been found as singly or multiply present in a variety of real systems, including complicated chemical reactions (Simoyi et al, 1982; Coffman et al, 1986). This means one can “dial” the parameter to generate “words” of sufficient computational complexity to serve as a language. These 193 HOUSE_OVERSIGHT_013693

computer experimental findings had already been anticipated in a remarkable mathematical proof by Sharkovskii (1964). The dynamical richness of these simple, single maximum, one dimensional maps was computationally explored in the context of ecological and epidemiological issues in the classical studies of Robert May (1976). It has been possible to relate the individually characteristic L, R sequence behavior of human subjects on a computer task to a unique parameter of a tent map generating those sequences which predicted age and discriminated subclinical obsessive compulsive from borderline syndromes (Selz and Mandell, 1993). The dynamical entropy of unstructured L,R behavior also discriminated a population of schizophrenic patients from normals (Paulus et al, 1996). More generally, parameter dependent dynamical coding, built into the universal behavior of its constitutive equations, is a mechanism with which a nonlinear dynamical system, such as nerve membrane equations as above, or in the aggregate, the middle layer of a completely connected neural network, can encode, Morse code-like, messages (Paulus et al, 1989). Bifurcations in Biologically Relevant Dynamical Systems Bifurcations, “splitting into (two) branches,” are observed over a smooth change in control parameter(s) (independent variables), as a discontinuous and qualitative change in the dynamical (time-dependent) pattern of the observable (Guckenheimer and Holmes, 1993; Wiggens, 1990; see Strogatz, 1994, for a particularly intuitive description). Qualitative here means how the dynamics of the trajectory appear as a geometric-topological (relative shaped not necessarily sized) pattern in phase space. In such a space, the orbital points are located along the x- axis by their value, x at time f, and along the y-axis by their time rate of change at that ¢, = To visualize a representative phase portrait in the plane, start by imagining the pattern made by mass hanging on a linear spring at rest as dx represented by a point centered at x=0,y= a 0. When perturbed from rest, the 194 HOUSE_OVERSIGHT_013694

phase portrait of the motion of this “harmonic oscillator,” is composed of a (continuous) series of points representing its location, graphed along x, its rate of motion graphed along , y= = x and = co-localize the circular orbit as it speeds up and slows down while it bobs up and down. The transition from a fixed point (the mass at rest) to a circle (the bobbing mass), a bifurcation in phase space, results in the loss of topological equivalence. That is, the phase space geometries before and after the bifurcation cannot be smoothly distorted into each other. Continuity and connectedness of the space is lost. For topological equivalence, stretching, bending and warping are allowed but not tearing apart and/or gluing together. Following the bifurcation of a fixed point into a circle, even limitless shrinking of the ring leaves a hole. The appearance or disappearance of an equilibrium fixed point (called a “saddle-node” bifurcation), splitting into two (“period doubling” bifurcation), its exploding into a circle (“Hopf’ bifurcation to a limit cycle), a circle splitting into two or more incommensurate cycles (“secondary Hopf’ bifurcation) and these multiperiodic (“quasiperiodic”) dynamics breaking down into a recursive spirals (“homoclinic bifurcation to chaos”) are among the common bifurcations in nonlinear dynamical systems, and all of them have been observed in many neurobiological settings. In the forced-dissipative (energetically driven and energy consuming) dynamical systems relevant to the neurosciences---this characteristic contrasts with the dissipation free momentum of the classical mechanics of astrophysical bodies--- there are four “most generic” bifurcation scenarios as a parameter changes that may, but need not, lead to chaos (see below for definition) (for early and physically oriented treatments see Eckmann, 1981; Ott, 1981; Berge’ et al, 1984, Kaneko, 1983). These scenarios are: (1) Fixed point or cycle splittings into twice-as-long period lengths 1>2—>4-—8->16...called the subharmonic or “period doubling route”; (2) The transformation of fixed points to one and then more periodic orbits, multiple independent (nonharmonic, incommensurate) frequency oscillations, their mode lockings and then breakdown called the “quasiperiodic route”; (3) Fixed point or cyclic equilibria metamorphosed into irregular bursting patterns called the “intermittency route”; and (4) In the context of quasiperiodic dynamics, adjacent 195 HOUSE_OVERSIGHT_013695

nonharmonic frequency encoding parameter spaces fusing, resulting in new periods that are the sums of their adjacent ones: period 2 + period 3 = period 5, in what is called the “period adding route”. Technically precise classification of bifurcations involve much more careful definitions and well studied technical constraints involving such issues as the symmetries and dimensionality of the system of observables, how many control parameters (“codimensions”) are required to reasonably realize the bifurcation and the particular way the fixed points of the system become unstable, all of which are directly explorable when the equations are known or can be hypothetically inferred from the qualitative behavior of real data. We note a few examples from the wide variety of bifurcating systems that can be found in the biomedical literature of interest for the biological sciences. With substrate input rate as the bifurcation parameter, the phosphofructokinase regulated glycolytic cycle in yeast extract was found to change among steady state, periodic and period doubling (subharmonic) regimes (Boiteux et al, 1975). Transitions between steady state, oscillatory and chaotic patterns have been reported in variety of physiological measures in man including respiratory rhythms and circulating blood cell concentrations over time (Mackey and Glass, 1977; Glass and Mackey, 1988 ) and models of dopamine cell dynamics (King et al, 1984). Flow rate parameter sensitive periodic, bursting and chaotic behavior has been found in a peroxidase-oxidase system (Olsen and Degn, 1977). A brain enzyme, substantia nigral dopaminergic tyrosine hydroxylase, manifested different saturation and fluctuation patterns, including bursting and periodicity, in experiments in which low (physiological) levels of tetrahydrobiopterin cofactor were the bifurcation parameters and adrenergic drugs were used as modulators (Mandell and Russo, 1981). All four of the generic bifurcation routes to chaos, period doubling, changing multifrequency (quasiperiodicity), period adding and bursting (called “intermittency”) were observed in self-sustained oscillations induced in the neural membranes of space clamped, giant squid axons that were immersed in a 550mM NaCl, and electrically stimulated over changing amplitudes and frequencies (Aihara et al, 1986; Takahashi et al, 1990). With external stimulus current level as the control 196 HOUSE_OVERSIGHT_013696

parameter, the R15 cell of the abdominal ganglion of the Aplysia demonstrates transitions between bursting and periodic modes as well as period doubling, a signatory period 3 and the Lyapounov characteristic exponent evidence (see below) for the discontinuous onset of chaos (Canavier et al, 1990). Manipulating feed back delay, the human pupillary light reflex will bifurcate into regular oscillations (Milton and Longtin, 1990). A transition between a regime of irregular discharging to oscillatory bursting behavior was induced in basal forebrain cholinergic neurons by neurotensin (Alonso et al, 1994). Sympathetic nerve discharge in decerebrate, ventilated cats demonstrated transitions between periodic, multiple periodic (quasiperiodic with changing ratios to the ventilation frequency) and subharmonic behavior in response to inferior vena cava occlusion, vagotomy, aortic constriction and spinalization (Porta et al, 1996). Period adding bifurcations were induced by changing calcium concentrations or the addition of a potassium channel blocker in the “pacemaker” formed when (rat) sciatic nerve is chronically injured (Ren et al, 1997). Changing levels of the L-type calcium channel antagonist, verapramil, alter the pattern of vasomotion of rabbit ear arteries among sets of multiple independent periods, “quasi-periodicity,” mode locking and chaos (De Brower, 1998). At critical intensity and frequency, flicker visual stimulation of the salamander generates a pharmacologically modifiable period doubling bifurcation in their ganglion cells (one spike for every two flickers) which is also seen subjectively and in occipital lobe evoked potentials at critical frequencies in bright, full-field flickered humans (Crevier and Meister, 1998). Qualitative and Quantitative Universality in Nonlinear Dynamical Systems “Universality” (see above) entered the parlance of physics in the context of the statistical mechanics of phase transitions near their critical points (Stanley, 1971; Stauffer, 1985; Yeomans, 1993) and has come to refer to the finite set of transitions and quantities common to nonlinear systems arising in_ their neighborhoods. A common physical example is the triple point of water-ice-steam on the temperature-pressure phase plane where a small change in temperature or 197 HOUSE_OVERSIGHT_013697

pressure leads to a global qualitative change in physical state. Analogously, the loss of topological equivalence occurs at the fixed point that, for examples, splits into two or explodes into a cyclic orbit in phase space. The same critical point behaviors and quantities occur in a wide variety of specific processes and their equations, and they are independent of the way the trajectories first arrived in the fixed point neighborhood. Once the system enters the regime of critical behavior, the predictive significance of its dynamical history is lost. This may also be the case for emergent psychiatric disorder (Mandell et al, 1985; Mandell and Selz, 1992; Ehlers, 1995; Paulus et al, 1996; Huber et al, 1999). There are diagnostic patterns of behavior when a nonlinear system is in a neighborhood of a potential bifurcation. They include sudden and/or large jumps resulting from a small change in experimental conditions, the appearance of big baseline fluctuations (anomalously large variance), the lengthening of the time on required to relax following evoked or spontaneous perturbation (“critical slowing”), the same global change in state occurring at different values of the parameter when increasing versus decreasing a parameter’s value (“hysteresis”), the existence of some range of values of the observable that cannot be attained by manipulation of the parameter (“inaccessibility”) and the availability of two or more distinct states in the same parameter neighborhood (“modality”) (Thom, 1972; Arnold, 1984; Gilmore, 1981). It is perhaps relevant to polydrug psychopharmacology and clinical management that the higher the co-dimension (the greater number of effective parameters being manipulated), the greater the accessibility and control of selected state stability becomes with respect to difficult to obtain behaviors. Examples of the potential advantages of simultaneous manipulation of multiple influences have been developed for affect disorder and anorexia nervosa (Callahan and Sashin, 1987). As evidence for the independence of critical behavior from specific history, the qualitatively universal bifurcations along the four canonical routes to chaos manifest dimensionless ratios of parameter and phase space geometries between bifurcations. These ratios are quantitatively universal. The formalisms that rescale the distances from fixed points in parameter and observable spaces result in the same picture across scale, a dilatational symmetry (also called self similarity or 198 HOUSE_OVERSIGHT_013698

affinity). They are called renormalization group equations, and, with respect to prediction, they replace any or all of the original specific predictive equations for the particular system under study (Cvitanovic’, 1989). Whereas the U sequence and critical point behaviors are manifestations of qualitative universality, these scaling numbers are manifestations of quantitative universality. We discuss them here because their omnipresence in computationally realized differential equations as well as physical and chemical experiments along with their quantitative specificity (with values in all systems as “constant” as x) constitute a most persuasive argument for the substantiality of modern dynamical systems approaches to brain and other biological research. The physical and physiological requirements for manifestations of these universal bifurcation scenarios can appear to be remarkably minimal. In physics, for example a full panoply can be observed in a “dripping faucet” (Shaw,1984). Similarly, a small piece of extirpated and perfused myenteric or femoral artery will demonstrate these transitions in vasomotion spontaneously and almost independent of flow rate (Stergiopulos et al, 1998). Feigenbaum discovered that in dynamical systems manifesting a series of period doubling bifurcations, the ratio of the parameter value at which the next period doubling bifurcation occurred relative to the last one ~ ... and the ratio of the magnitude of the spawning point to the one spawned ~ 2.5. (Feigenbaum, 1979). By “rescaling” distances along a parameter value (see below) using what is called a “universal renormalization operator” the geometric situation around each bifurcation point (though of different absolute “size) remains relatively the same. In intermittent systems, burst length varies as the inverse square root of the distance of the value of the parameter from that value that elicited the fixed point (Manneville and Pomeau, 1980). The universal characteristic of the third common parametric route to chaos, quasiperiodicity, is that the ratio of independent frequencies found most resistant to mode locking and breakdown into chaos is, “i = 1618... the Ost number to which the ratio of adjacent Fibonocci numbers converge (1, 2, 3, 5, 8, 13, 199 HOUSE_OVERSIGHT_013699

21...) (Shenker and Kadanoff, 1982). Similar quantitative scaling properties were also discovered in the parametric period adding route (Kaneko, 1983). All of these scaling numbers have been found in experiments and in remarkable agreement with theory. Examples have been discovered in electronic circuits, hydrodynamic and mercury flows, acoustic systems, laser dynamics and oscillating chemical reactions (see Cvitanovic’, 1989, for representative list of references). Whereas qualitative evidence for all of these bifurcation scenarios have been found in brain relevant experiments, there is yet to be a bifurcating experimental biological system with adequate precision across a sufficient range of magnitudes such that quantitative universality could be demonstrated across a sufficient range of values to be convincing. We remind ourselves that in order to establish a Fiegenbaum number, each period doubling bifurcation of the several required necessitates about a five-fold improvement in the experimenter’s ability to specify the control parameter. Using Invariant Measures of Dynamical Neurobiological Systems Before the modern era of dissipatively forced (energy utilizing) dynamical systems research, the known attactors of an experiment’s initial values resulted from their convergence onto either a fixed point or a limit cycle. An attractor can be regarded as a set which remains in bounded space and to which all orbits in this neighborhood converge (Milnor, 1985). Since by the rules of differential equations, orbits are required to be both smooth (graphable without lifting the pencil) and unique (different trajectories don’t intersect since the point of intersection would no longer be unique), the foundational Poincare-Bendixon theorem says that any such orbit confined to a two dimensional phase space that doesn’t converge to a fixed point must, no matter how long it irregularly wanders, must, eventually intersect with itself and then go around the same route again in a (perhaps very long) cycle. In most neuroscience research as well, we have generally regarded our data as manifesting either tolerable (or intolerable) fluctuations around mean values (fixed 200 HOUSE_OVERSIGHT_013700

points) or more or less regular cycles. We analyze our “fixed point” data using quantities such as the mean and variance of distributional statistics and the cycle data using the amplitude, frequency, cycle length and phase of trigonometric functions. In central tendency-oriented research, rare, very high amplitude events have usually been considered aberrations and tossed, and imperfect periodic behavior is treated by “cosiner analysis” as regular cycles contaminated by measurement or system noise. Whereas technically, chaotic dynamics must live in dimension greater than two (for orbits to be more than a fixed point or limit cycle, able to snake around without necessarily intersecting ), the Lorenz attractor has dimension just a little over two, our difficulties with establishing the “true” physiological dimension of real biological observables (see below) makes such a consideration more theoretical than practical. The orbits of a forced-dissipative dynamical system in a parameter regime engendering chaos, converge onto an attractor which is neither a fixed point nor a limit cycle, thus the origin of the name “strange attractor” (Ruelle and Takens, 1971). It was James Yorke that first named these dynamics “chaos” (Li and Yorke, 1975). The necessarily statistical properties of the chaotic orbits on strange attractors follow from the generic characteristics of their motions (see Shaw, 1981 for a still conceptually current, non-mathematical treatment). These kinds of statistics are studied in a research context called the “ergodic theory of dynamical systems” (Ruelle, 1979; Eckmann and Ruelle, 1985). Ergodic is a word used to characterize a system with (or without) a particular condition placed on its statistical measures: the existence of an invariant measure which is undecomposabile into two invariant measures and, equivalently (though not obviously) one in which the time average equals its average in the geometric space into which it is embedded. One may arrive at the same ergodic measure from studying a single very long orbit or from summing across many individual but shorter orbits. This ergodic equivalence is made possible due to the definitional existence of at least one invariant statistical measure and the dynamics of the system which ideally include a uniformly, sequence disordering process called “mixing” (see below). 201 HOUSE_OVERSIGHT_013701

Of course, most real biological dynamics are not uniformly mixing and so are non-ergodic, but we shall see that the ways they fail to be ergodic (and thus remain in the conceptual context of ergodic measures) are descriptively useful (Mandell and Selz, 1997a). The emergence of many statistical approaches to characterizing these motions have been accompanied by the expected controversies about which is best or correct (see below) and have been applied to the problem of diagnosis and clinical discrimination in a variety of neuroscience settings. In ideal abstract chaotic dynamical systems called Axiom A (Russians called them “C systems’), where most mathematical theorems are proven (Smale, 1967), all these measures, if properly computed, are equivalent. In real life, as in the related case of ergodicity, they are not, and since no single one is complete, the more (incomplete) measures we use in our studies along with interest in the way that they differ, supplies more useful information about the system. Though researching and elucidating the most reliable and valid ways of computing these measures are a valuable goal, the current debates focused on the superiority of a single particular measure, constructed in a particular way in relationship to issues of insoluble absolutes like “randomness” versus “deterministic chaos may not be particularly valuable for uncovering new characteristics and potential mechanisms underlying a specific set of real neurobiological observables. Emphasizing diversity and relevance to the clinical biological sciences, we note that quantifying patterns in ergodic (non-ergodic) measures have aided: the discrimination between normal and abnormal opticokinetic nystagmus in neurology patients (Aeson et al, 1997); localizing a two year old subcortical stroke in an EEG of a patient with no other signs or neurological findings (Molnar et al, 1997); the diagnosis of early (not late) multiple sclerosis, as a nonspecific long tract disorder, in patients with mild optical neuritis using cardiac rate dynamics (Ganz and Faustman, 1996); seizure prediction from minutes to hours before the event in which subthreshold, pre-phase transition spatial diffusion and oscillations in characteristic changes in these measures can be found (Martinerie et al, 1998; Elger and Lehnertz, 1998; Pign et al, 1997; lasemidis et al, 1990); using these measures on the EEG to differentially predict hereditary predisposition to alcoholism 202 HOUSE_OVERSIGHT_013702

(Ehlers et al, 1995); indicating the presence or absence of septic encephalopathy (Straver et al, 1998); using time series from jejunal manometry to discriminate objectifiable somatic from psychological conversion related irritable bowel syndrome (Wackerbauer et al, 1998); analyzing time-dependent patterns in plasma hormone levels to discriminate between the presence or absence of a functioning tumor (Hartman et al, 1994, Mandell and Selz, 1997a); automated differentiation of ataxic from normal speech (Accardo and Menulo, 1998); and discrimination of temporomandibular joint dysfunction from normal patterns of chewing motions (Morinushi et al, 1998). Styles of Orbital Motions in Chaotic Dynamical Systems In chaotic dynamics, in various specific ways, an initial hypothetical handful of points lined up along the trajectory and acted on over time by the nonlinear differential equation (“operator”), get out of order in an unpredictable way. Here the hypothetical handful can come from a statistical aggregate of initial conditions or from a single recursive orbit studied over long times. As noted above, ergodic theorists call this getting out of order “mixing” and how and to what degree this happens consumes many mathematical theorems but for purposes of brain research, it can be best described using a variety of statistical measures. For example, visualizing the Lorenz attractor (see above) as a butterfly in phase space, the points get out of order because as they spiral out (“stretching”) to the edge of one wing and return (“folding”) to the unstable fixed point on the butterfly’s body whence they either jump to some place on the other wing to spiral to its edge or return to the same wing to spiral out again. Which one of these is chosen is exquisitely sensitive to very small changes in where the trajectory started and very small fluctuations in where it returned to the unstable fixed point on the butterfly’s body. In fact, specification of these locations is beyond the precision of any real, thermodynamically vulnerable system. Chaotic trajectories on the Réssler attractor (see above) wind out (“stretch”) to the edge along the inside of a conch shell in phase space and then are mapped 203 HOUSE_OVERSIGHT_013703

back (“fold”) into the spiral unpredictably somewhere in a mixing mechanism that has been called “displaced reinjection.” In the slow-fast oscillations of the forced van der Pol in the chaotic regime, points in the slow phase (“repolarization”) jitter around and step on each other’s heels, getting out of order while waiting on the ledge before jumping (“depolarization”) to the next slow phase (“repolarization”) at some unpredictable time, thus generating a variably irregular series of interspike intervals. Stretching and folding are also responsible for getting points get out of order in the single maximum map of the unit interval (studied for universal qualitative and quantitative properties by May and Feigenbaum and others as described above). With increases in parameter values, the parabolic hill function onto which the unit line has been stretched gets steeper, more stretched. Mapping points on the hill back onto the straight line of the unit interval results in what amounts to the line folding back on itself. This stretching and folding eventually fills the line with points, but their sequence, from end to end, gets shuffled like a deck of cards. As described more generally above, points that start as neighbors may get separated (“divergence along the attractor’) and those that start at a distance from each other may be thrown together (“compression back onto the attractor’). These expanding and folding motions that characterize the chaotic behavior on strange attractors have been likened to the actions of a taffy puller (Réssler, 1976). It is in this way that nearby points can separate without leaving the attractor. It is also the case that once indistinguishably close but then separated points may be compressed together again generating new, temporary (unstable) cycles of all possible period lengths. These unstable fixed points may be the most important feature of chaotic systems from the standpoint of new ideas about brain mechanisms (Pei and Moss, 1996; So et al, 1997). This aggregation of unstable loops can occur from points fluctuating away and back to the attractor as well as during the crowding of points at the turns after their stretching out on more linear parts of the flow. Under the mixing flow of a chaotic dynamics, it is also true that a single point eventually explores the entire attractor, no attractor location is inaccessible to it. 204 HOUSE_OVERSIGHT_013704

Although counter-intuitive when expressed in words, the trajectories that one sees in the graphics of chaotic attractors result from the actions along the “unstable” directions of the stretching distortion; the actions in the otherwise invisible stable directions “iron down” the points onto this unstable manifold (n dimensional abstract surface). As might be expected from this set of characteristic motions, the diagnostic triad of chaotic dynamical systems are: (1) Sensitivity to initial conditions—tiny distances between starting points are magnified and large distances between starting points are reduced under the stretching and folding actions of the system; (2) The presence of a theoretically infinite but countable number of unstable periodic orbits of theoretically all period lengths—points in phase space can be viewed an attractive-repellers, visited and left by the orbits recursively as the dynamics proceed; and (3) Indecomposability—the attractor is not separable into isolated regions and no points escape (see Devaney, 1989, for one of the clearest definitions). Of particular relevance to information encoding and transport by brain mechanisms, it is important to visualize that new information in the form of unstable periodic orbits is being created as well as destroyed by the dynamics. The logarithmic rate of formation of these new orbits is computed as the system’s topological entropy (see below). Assuming the real neurobiological system under study is behaving in these ways (and often much has to be done to help justify such a claim), the observables take the form of an irregular and/or episodic time series of amplitudes, as in repeated sample, neuroendocrine studies of plasma hormone levels (Veldhuis and Johnson, 1992) or a sequence of times between events as in neuronal interspike intervals (Katz, 1966; Perkel et al, 1967). These time or time-sequence series are generally studied from three relatively distinct yet complementary quantitative perspectives: (1) As stochastic (“random”) processes with various amounts of sequential dependency (autocorrelations) and scale (sample length) dependencies; (2) As “deterministic” smooth or discrete, vectorial geometries in phase space following reconstruction and/or embedding of the series as phase portraits or return maps; (3) As information generating and transporting, topological (about relative 205 HOUSE_OVERSIGHT_013705

nearness and sequential order not absolute distances), symbolic dynamical processes which as either (1) or (2) can be analyzed with respect to its various entropies, algorithmic complexities and word content and syntax. A variety of techniques aimed at deciding between the relevance of one or another of these underlying assumptions (such as series and Fourier phase shuffling to destroy statistical autocorrelations and vectorial continuities but leaving the probability density distributions intact ) may at times help emphasize one or another of these orientations in the analyses (see Ott et al, 1994 for a collection of articles on this topic). Nonconvergent Distributions and Power Law Scaling in Biologically Relevant Time Series The statistical distribution with which most of us are familiar is the Gaussian which can be generated by summing and averaging a series of independent random events. The average behavior head/tails probabilities observed by one person flipping a fair coin for a very long time or by many people flipping similar coins for shorter times converges upon the invariant measure of 0.5. The variance, “second moment” in the distribution of a population of coin flipping sequences will be finite and computable. In a graph of this distribution, the tails will converge to the xX axis in a Gaussian exponential manner. The longer or the more numerous the “sample” series of observations, the closer they will approximate the “ergodic” invariant measures representing the true “central moments” of the behavior of this “population” of fair flipping coins. Since the coins are not changing their relevant characteristics over the time of observation, we say that the series is not time dependent but instead is “stationary.” Computation of correlations over increasing lags to determine how much and for how many flips the sequences continue to resemble themselves yield an exponential decay with a single characteristic correlation length. This reflects the existence of a finite variance from which its amplitude is derived and serves as the single characteristic temporal scale of the random process. 206 HOUSE_OVERSIGHT_013706

Before describing the relatively new set of measures of biological processes designed to find and quantitate what are assumed to be relatively sample size insensitive, distributionally nonconvergent and multiply correlated processes that are without a single time or space scale, we should remind ourselves that there is already much more apparent “order” in a generically random situation than our intuitions would lead us to believe. For example, if we keep cumulative scores in a competition between heads and tails and determine the distribution of trials between those in which the number of heads and tails are even, we will get periods between zero crossings of many lengths with very short ones and very (very) long ones being most statistically prominent. The distribution of these wavelengths is shaped like a symmetrically fat-tailed, bowl (Feller, 1968). As another illustration, expected runs of heads or tails in this Gaussian random task are longer and more frequent than we might suspect. It has been proven that the expected run length grows with n coin flips (as an order of magnitude estimate) like the logarithm (for a fair coin, base 1/p = 1/0.5 = 2) of n. For example, in 512 ( e.g. 2°) tosses, we cannot report a run of 9 heads as a evidence for a biased coin or the sign of some deterministic coin tossing mechanism (Erdos and Renyi, 1970). If we had a 0.6 head biased coin, then the observation of a run of 13 heads couldn’t dissuade us from a random mechanism! Unlike our random coin task, the variances of many, perhaps most, time series of biologically-relevant events, do not tend to converge onto a limiting value as sample size, n, grows, but rather continue to increase (or decrease) with n ina scale invariant manner. Instead of “regressing to the mean” with increasing sample length or number, the likelinood of a larger deviation than previously observed increases with n. Analyses of inter-event intervals reveals a multiplicity of characteristic times. One interpretation of these finding might be that this represents evidence for the inherent “nonstationarity” of biological mechanisms as reflected in, for examples, the frequency of saccades concomitant with ceaselessly shifting foci of visual attention (Steriade and McCarley, 1990), or our inability to not think of “white bear’ when so instructed (Wegner, 1994). Hermann Haken, the father of laser-inspired “synergetics,” has said that biological mechanisms are not in a steady 207 HOUSE_OVERSIGHT_013707

state for very long, spontaneously and irregularly jumping from one unstable dynamical state to another (1997). This suggests that meaningful tension between experimental sample lengths long enough to minimize statistical error and short enough to be stationary may be, for the biological sciences, more apparent than relevant. The studies reviewed below exploit measures arising from the view that the noisy statistics of nonstationarity in biological processes are not a sign of measurement error, but rather evidence consonant with the statistical physics of nonequilibrium states and phase transitions (Stanley, 1971; Stauffer, 1985; Yeomans, 1993). Very high amplitude fluctuations and multiple, up to infinite, correlation lengths are characteristic of the normal, on-going biological dynamical behaviors, which are apparently without characteristic amplitude and time scales. From this point of view, if most or all information is widely distributed in the brain (e.g., serial order of visual tasks involving motor cortical neurons, Carpenter et al, 1999) ) then the “binding problem” (see above) could also be solved by multiple, up to infinite spatial and temporal correlation lengths in place of the current theories of monofrequency resonances (Singer, 1993). Hierarchical neurodynamical mechanisms communicating across many mechanistic temporal and spatial scales, brain information transport analogous to the energy cascade of hydrodynamic turbulent velocities (Tennekes and Lumley, 1972), would be likely in the parametric vicinity of incipient bifurcations and phase transitions. Three closely related techniques for quantifying the systematic changes in average fluctuation amplitudes with n (scale, sample length) involve a “power law,” linear slope relationship between the logarithm of an index of variability and the logarithm of sample segment sizes. These easy, yet powerful methods were brought to experimentalists’ attention by Benoit Mandelbrot (Montroll and Badger, 1974; Mandelbrot, 1983; Fedor, 1988; Bassingthwaighte et al, 1994; Liebovitch, 1998). To estimate the exponent in Hurst rescaled range analysis, we compute the standard deviation and the range of the deviation of the running sum from the mean on sequential subsamples of increasing size. The Hurst power law exponent is the slope of the straight line formed by graphing the logarithm of the subsample length 208 HOUSE_OVERSIGHT_013708

along the x axis and the logarithm of the ratio of the range to the standard deviation on the y axis. An independent random system has a Hurst of 0.5. If a sequential increase or decrease in an amplitude or inter-event time tends to be followed by a change in the same direction, the Hurst > 0.5. If an increase in the measure tends to be followed by a decrease, then Hurst < 0.5. Computation of the Fano factor (power law exponent) exploits the same general strategy using the variance/mean in place of the range/variance and counting the number of events (such as single neuron discharges or heartbeats) in time windows of increasing length, generating a similar log-log graph. There is a relatively long history of the use of spike-number variance-to mean ratio in studies of response variability in visual cortical neurons (see Teich et al, 1996 for a review). The Allen factor (power law exponent) tends to reduce the influence of local trends by a computation of the variance of the difference between the number of events in two successive time windows divided by twice the mean number of events in the window. Each system’s invariant logarithmic slope across sample segment sizes takes the place of its missing finite variance in characterizing experimental data in which the distributional tails do not converge (or do so very slowly) to the x axis. Recent approaches to these measures in the context of stochastic analysis of DNA sequences, but also applied to normal and pathological cardiac inter-beat intervals and gait interval sequences, have dealt with the influence of non-stationarity due to apparent trends in the data on a-equivalent indices by local mean-normalization of the fluctuations at each window size (Peng et al, 1993; Peng et al, 1995; Hausdorff et al, 1995). The rate of decay of the densities in the tails of the probability distribution as they approach extreme values along the x axis, called the Levy exponent when represented in Fourier space (technically, as a “characteristic function” of the probability distribution) (Shlesinger, 1988; Shlesinger et al, 1995), can also be computed directly on the distribution by fitting the tails with a two parameter curve quantifying their “fatness” and rates of decay (Mantegna, 1991). We can speak of a Gaussian tail as having an exponential decay rate representable by a = 2 implying 209 HOUSE_OVERSIGHT_013709

finite variance. A tail with a nonconvergent decay rate of 1 < a < 2 indicates non- finite variance in the data such that the usual “normal curve” derived, standard deviation dependent tests of statistical significance are without meaning. a < 1 indicates the data is without a consequential mean and will require the use of interquartile measures to locate the center of the distribution (Adler et al, 1998). Recalling that the Hurst, Fano and Allan indices are invariant across sample segment size, we remind ourselves that, as is the case in the finite mean and variance, a = 2, Gaussian, any of the other “a tails” also retain their value (“shape”) across all partitions that might be used to sort and sum the observable. This property is called convolutional, a, stability. In passing it should be noted that the last outpost of convergence of a probability density distribution with a = 2 is called “log-normal,” in which the tails along the x axis are “pulled in” by the variable being plotted as its logarithm. A Hurst exponent of > 0.5 in the data is associated with a Levy exponent of < 2.0, and both would be indicative of a process in which the characteristic style of change, rather than decay with some finite correlation length, would persist across all time. Using a bursting neuron as a generic example, a short interspike interval would, on the average, be followed by another short one and a long one by another long one, and this behavior, unlike our fair coin flipping sequence of observables, would not become uncorrelated with itself even over infinite time. Another way to represent this infinite, innumerably lengthed, correlation property is via its implicate frequency (inverse wavelength) content by computing its best fit assortment (along with their densities) of a range of short to long sine waves forming the Fourier transformation of the correlation function. The condition of correlated fluctuations across many measured temporal scales yields yet another power law slope when graphed as the logarithm of its range of frequencies, f, plotted along the x axis, versus their corresponding amplitudes squared, powers, plotted along the y axis. Naming this spectral power law exponent B, the system’s characteristic scaling law is usually expressed as Gr (Fedor, 1988; Hughes, 1995; Shlesinger, 1996; 210 HOUSE_OVERSIGHT_013710

Liebovitch, 1998). We see that the Hurst exponent, Fano and Allen factors, Levy exponent and power spectral scaling exponent are kindred statistical descriptors. They are most usefully applicable to systems with distributions that fail to be Gaussian or asymmetrically Poisson, the latter from random data sequences with only positive x values, thus backed up toward zero by a minimum inter-event interval or amplitude. These time series are sequentially dependent, not conventionally stationary, without finite central moments and with self-correlations that don’t demonstrate Gaussian exponential decay with sample length or time. The following are some examples of the use of these measures in studies of biological dynamics. . Examples of Biological Data with Divergent Distributions and Power Law Scaling A paradigm challenging group of experiments involved models and measures of the distribution of characteristic open and closed times of membrane ion conductance channels. The usual approach to this problem assumed the existence of a small set of distinguishable channel types that were reflected in discrete conductance events with a small set of characteristic open and closed times. The distributions of each of could be fitted with its own, Markov process derived, exponential. With technical advances and improved temporal resolution, more characteristic times and their associated a = 2 exponentials were reported with as many as three not being unusual. Liebovitch (and Sullivan,1987; 1989) used analogue to digital transformation of current recordings from the unselective corneal epithelial channels and voltage dependent potassium channels in cultured mouse hippocampal cells at temporal resolutions ranging from 170 to 5000 Hz and found similarly shaped, a < 2, nonconvergent distributions across temporal scales. This led these investigators to suggest that, related to the >16 recorded magnitudes of characteristic times, from picoseconds to months, in autonomous protein motion (Careri et al, 1975; Gurd and Rothgeb, 1979), that there was an “a stable” hierarchy 211 HOUSE_OVERSIGHT_013711

of lifetimes of states, observable at almost any temporal resolution that methods would allow. Early and representative studies comparing the fit of the data with hierarchical scaling functions versus a sum of a small number of Markovian exponentials included studies of a calcium activated potassium channel in human fibroblasts (Stockbridge and French, 1989) which yielded evidence to support both models, as did studies of membrane conductances in corneal epithelial cells by another group (Korn and Horn, 1988). In a systematic comparison of scaling and Markov exponential modes of the gating kinetics of GABA activated chloride channels, acetylcholine activated end plate potentials, calcium activated potassium channels and fast chloride channels (McManus et al, 1988), it was found that the latter fit the data best in most experiments. Similar results were reported in studies of the glutamate and delayed rectifier potassium channel with respect to distributions of open and closed times (Sansom et al, 1989). Space does not permit a systematic account of the continuing debate and conflicting studies about these representations and the implicit biophysics of discrete, finite versus continuous, hierarchical channel event heterogeneity. It is interesting that recent experiments making use of Hurst rescaled range analyses of time series of whole cell membrane voltage fluctuations (without the assumptions and current renormalizing procedures associated with patch clamping) have yielded additional evidence for multiply correlated, Hurst > 0.5, a < 2 power law behavior of what some might regard more generally as a protein relaxation time mediated hierarchical array of ion conductance behaviors (Liebovitch and Todorov, 1996). Following the discovery of (very) subsaturating (“far from equilibrium”) rat brain levels of the common cofactor for tyrosine and tryptophan hydroxylases, tetrahydrobiopterin (Bullard et al, 1978), studies of amino acid substrate saturation functions and time courses determined at these low, physiological co-reactant concentrations manifested patterns of hierarchical multiplicity and discontinuities suggestive of bifurcations and time-dependent fluctuations with fractional (hierarchical) time scaling exponents that were sensitive to psychotropic drugs 212 HOUSE_OVERSIGHT_013712

(Mandell and Russo, 1981; Knapp and Mandell, 1983; Russo and Mandell, 1984a; Russo and Mandell, 1986). Similar bifurcating and power law kinetics were found in receptor-ligand binding systems (Mandell, 1984) which were confirmed by more recent studies of diffusion-limited binding kinetics with receptors immobilized on a biosensor surface (Sadana, 1998). Hierarchical kinetics have also been reported in time courses of drug and metabolite levels (Koch and Zajcek, 1991), tissue tracer washout studies (Beard and Bassingthwaighte, 1998), carrier mediated transport processes (Ogihara et al, 1998), general pharmacokinetic functions (Macheras et al, 1996) and biochemical networks (Yates, 1992). It is likely that bifurcating and hierarchical, power law kinetic functions will be studied more commonly in the chemical literature in general (Shlesinger and Zaslavsky, 1996; Berlin et al, 1996) as well as applied to a variety of protein-mediated biological functions (Dewey, 1997). The first demonstration of and stochastic model for nonconvergent distributions of interspike intervals of a single neuron was by Gerstein and Mandelbrot (1964). Though rich with possibilities, it has been only very recently that additional work from this point of view has been published. This is likely due to the fact that most neuroscience oriented statistical packages, with rare exceptions, are without techniques for computing descriptive parameters for these divergent probability density distributions. This has not been the case for economic time series, download STABLE from http:///www.cas.american.edu/~jpnolan. Recently, applications of the Fano and Allan factor as well as power spectral scaling exponents to observed and shuffled series of spike counts and interspike intervals in the auditory and visual systems (including spatial and/or time resolved single unit recordings in retinal ganglion, lateral geniculate and lateral superior olivary cells as well a auditory nerve fibers) demonstrate the characteristic behavior of nonconvergent, hierarchical stochastic systems (Teich, 1989; Teich et al, 1990; Lowen and Teich, 1992; Kumar and Johnson, 1993; Kelly et al, 1996; Teich et al, 1997). These statistical techniques are well suited to the characterization of the irregularly intermittent bursting patterns generic for activity in single neurons as well 213 HOUSE_OVERSIGHT_013713

as in nonlinear equations representing them and other brain processes (Mandell, 1983). An early study of power spectral scaling in the EEG reported alpha band fluctuations that extended a <P 6B ~ 1 pattern to 0.02 Hz (Musha, 1981), as did other applications of the log-log power spectrum to the EEG in man (Hu and Hu, 1988; Prichard, 1992). This power law scaling led naturally to the suggestion that the range of frequencies available in the electromagnetic signal from the calivarial surface extends far beyond those currently appreciated and may be available for study using relatively noise free recording techniques such as_ the magnetoelectroencephalogram (Mandell and Selz, 1991). A not surprising range of intrinsic correlation lengths reflected in Hurst > 0.5 and/or Levy exponents < 2 have been reported in lamb fetal breathing patterns (Szeto et al, 1992). The exponent has been shown to be sensitive to maternal alcohol intake in humans (Akay and Mulder, 1998), rat neonatal motoric activity (Selz et al, 1995), and nuchal atonia duration sequences (associated with putative intra-uterine REM sleep) in fetal sheep (Anderson et al, 1998). Sequential amplitudes in 1 Hz stimulated soleus spinal cord H-reflex demonstrated a $F 6B ~ 0.83 in control subjects and, reflecting the decrement in correlations, by 0.31 in patients with losses in supraspinal input from spinal cord injury (Nozaki et al, 1996). Whereas the sequences of fixation times in eye movements of normal control subjects reading difficult material demonstrated an exponentially decaying distribution, those of schizophrenic patients demonstrated a power law tail, consistent with more sequential correlations (Yokoyama et al, 1996). This finding may be related to the appearance of velocity arrests, runs of sticky fixed points, in a spatially oscillating target task, called “smooth pursuit eye movement dysfunction” in schizophrenic patients which has been modeled as a parametric disorder in a periodically driven nonlinear dynamical system (Huberman, 1987). The “short time fractal dimension” has been used to discriminate acoustic signal transformations from the speech of normal subjects and ataxic patients (Accardo 214 HOUSE_OVERSIGHT_013714

and Mumolo, 1998). Spontaneous changes in the apparent syllabic sound made by regularly presented, word-like auditory stimuli emerge irregularly, the duration of perceived sameness demonstrating a power law distribution of “dwell” times (Tuller et al, 1998). The same kind of power law distribution of characteristic “brain times” can be found in studies of gait cycle durations in normal walking (Hausdorff et al, 1996) with a decrease in this locally detrended, a-like index compared with controls (0.91+0.05) in patients with the basal ganglia disorders of Parkinson’s (0.82+0.06) and Huntington’s (0.60+0.04) Diseases (Hausdorff et al, 1998). Hurst > 0.5 has been speculated to more accurately quantitate the fundamental time structure of cells that was previously called circahoralian (ultradian) intracellular rhythms (Brodski, 1998). Reconstructions of Time Series as Orbital Geometries Rene Thom (1972), extending the ideas of Poincaré and D’Arcy Thompson (1942), argued that experimentally useful, intuitive connections between the qualities of biological processes and the quantities of an explicit (equations known) or implicit (equations unknown) dynamical system could be best achieved through the use of graphic representations of their geometric and topological forms. Notably successful examples can be found in the work of Thom, Arnold (1984) and Zeeman (1977), who were inspired by “caustics” (the shapes made on surfaces by the coincidence of reflected or refracted light rays) and Whitney’s representation of parametric manifolds (surfaces) by the shadows they would make on a plane when back lit (Whitney, 1955). This led to a small number of qualitatively predictive, no tt number-of-independent-parameters dependent shapes, such as “folds” “cusps” and “wavefronts.” Experimentally crossing the values of these independent variable forms at their singular boundaries successfully predicted discontinuities in the otherwise smooth alterations in the dependent variable; i.e. bifurcations (“catastrophes”) in the behavior of the observable. This approach was best suited to the study of systems with many independent variables and one dependent variable that could be mapped on the axis of the latter to represent a continuum of 215 HOUSE_OVERSIGHT_013715

operationally defined “energy states.” Smooth changes along the path of the nonlinear parameter manifold generated discontinuous changes in energy levels indicating states of the observable. Crossing a wrinkle in an “independent variable” (some call it “order parameter” to indicate its emergence rather than availability for predictable manipulation) such as the nonlinear parameter surface of the countervailing influences of survival fear and financial cost, may lead to a bifurcation in behavior from peace (“low energy”) to war (“high energy”) (Zeeman, 1977). In a similar geometric spirit but dealing with nonequilibrium systems. in motion, the conditions such that one could “smoothly” embed a trajectory like a continuously recorded EEG record, a complicatedly coiled snake into a three or higher dimensional box without loss of its essential dynamical or statistically measureable properties, was settled by Whitney in what is now referred to as the “embedding theorem” (Whitney, 1936). Starting with a tangled knot of overlapping vectorial orbits with apparent “non-invertable points” (given a point, one cannot chose among or between the more than one point that it apparently came from), it can always be unwrapped into a non-crossing trajectory satisfying uniqueness when reconstructed in a box of a little more than twice the parameter-determined dimension of the original space of observables. A common technique for the spatial reconstruction of the output of a dynamical system is called a “time delay embedding.” This approach, first suggested by Ruelle (1987, pg. 28) replaced the value, x, versus the time derivative, = phase portrait plot described for a continuously perturbed bob on a spring above. A sequence of observables over time, in, for example, three dimensional “phase space” (Packard et al, 1980; Takens, 1981; Sauer et al, 1991), is depicted by a curve representing the system’s trajectory at times 4, f, fs, by sliding one-by-one down the series and plotting each pj, Po, ps, location with respect to each other along the x, y and z axes respectively. The choice of time interval between the points, the delay, can be delicate and usually some standard fraction of the decay time of the sequence’s autocorrelation length, “the decay time of mutual information” is chosen. There are many technical considerations, 216 HOUSE_OVERSIGHT_013716

including those involving the choice of the embedding space vis a vis the “true” dimension of the attractor. This becomes an issue when, for example, the attractor shrinks over time to some subspace of the initial embedding (Liebert et al, 1991 and references therein). lf we imagine the process of time series reconstruction to inscribe an attractor’s untidy ball-of-string of recurrent trajectories in three dimensions, we can then, by making the z-dimension a constant value, cut the ball with a two dimensional plane, a “Poincaré surface of section.” This could yield a roundish cloud of discrete points on the x,y plane and t,-1 — tp would be the time between two piercings of this surface. It has been proven that almost any cut, as long as it is made transverse to the direction of the orbital trajectories, is equally valid and useful for further analyses (Oseledec, 1968). If the original embedding and subsequence section was in high enough dimension to allow invertability, we might have enough (trial and error) knowledge to be able to write a discrete equation, a “return map,” f, that would move one point to the next on the plane as (x,y)/, , <> (x,y)¢,. What can sometimes be case with real systems (Coffman et al, 1986), is that reducing the geometric reconstruction still one dimension further, accepting non- invertability, ironing down the points in the plane onto the x axis line (normalized to [0,1]), and plotting the values at x; against x1 (“mapping the unit interval to itself’), can generate points in the general shape of a parabola with dynamics representable by the same family of one parameter, single maximum discrete equations that generated May's sequence of bifurcations, Feigenbaum’s scaling and Metropolis, Stein and Stein’s (and Sharkovskii’s) U sequence (see discussions of qualitative and quantitative universalities above). Although sometimes a significant change in brain system physiology, such as penicillin-induced epileptic neuron spiking activity is revealed simply by a change in the graphic appearance of suitably embedded time series data (Zimmerman and Rapp, 1991), more often statistical measures made on the geometric dynamics of the points on the attractor are required. 217 HOUSE_OVERSIGHT_013717

Orbital Divergence Characterizes Expansive Dynamics on_ Biological Attractors In the dynamical world of equilibria (fixed points in phase space) and periodic cycles (fixed points of a return map), a common concern involves their stability. What happens if an adventitious jiggle moves the orbit a little distance away from the fixed point? Would the wind wiggled suspension bridge start to flap with increasing amplitude or would it damp back down quickly to its stable state. A “Lyapounov functional,” L, is constructed which can be visualized like a smooth potential bowl around the fixed point such that any L stable solution that starts at its bottom tends to stay there or is asymptotically L stable if the solution converges to the fixed point at the bowl’s bottom as +». If the point is not L stable, it is L unstable. The modern study of nonlinear systems have produced another kind of stability issue with a similar appellation yielding other direction specific indices, the Lyapounov characteristic exponents, 42 (Oseledec, 1968; Eckmann and Ruelle, 1985; Ruelle, 1990; Ott et al, 1994). In this context, the instability is not one of perturbative escape from a fixed point, but of the average rate with which the (theoretically infinitesimal) distances among a handful of points representing a set of initial conditions (each a precision limited, hypothetical repetition of the same experiment), are being stretched apart by the expansive action of a strange attractor system. In three dimensions, one can envision a ball of initial conditions being elongated along the unstable direction and ironed down from both sides along the stable direction over time, transforming the ball into an ellipsoid and then into a (recurrent) curve. In simplest terms and thinking about a one dimensional scalar time series, the Lyapounov exponent reflects the multiplicative average (logarithmic addition) of the sequence of slopes of the series of straight lines connecting the points. An average slope of > 45 is expansive such that a linear distance on the x- axis is increased when mapped onto the y axis. A slope of < 45 is a contraction mapping reducing the linear distance of the x-axis when mapped to the y axis. 218 HOUSE_OVERSIGHT_013718

Réssler’s generic chaotic system (see above) moving recurrently in a three dimensional box can be orthogonally decomposed into three directional motions in a moving frame, each with a signatory sign of 2. The unstable direction of expansive stretching is characterized by some number > 0, A(+), the stable direction of contractive folding, some number, < 0, A(-), and the neutrally stable direction of recurrence, (0). For The “Lyapounov spectrum” of the Réssler attractor is [A(4), A(-),4(0)] (Shaw, 1981). An n-dimensional dynamical systems has n one- dimensional Lyapounov exponents, and it is sometimes the case in relatively noise free, finite semi-stationary data lengths of the neurosciences, that a 2>0 can be shown to exist for a second one, in a dynamical situation called “hyperchaos” by (Rossler, 1979). For example, two and sometimes three 4(+) have been reported in the flows on the EEG attractor of normal alert subjects (Gallez and Babloyantz, 1991). The presence of measurement noise, the finiteness of neurophysiological sample lengths as well as the relatively small expansive actions in some directions in the chaotic attractors of brain dynamics lead to the finding that most often, only one “leading Lyapounov exponent,” 4(+), is reliably computable (Sano and Sawada, 1885; Wolf et al, 1985; Eckmann et al, 1986). A counter-intuitive fact about the stability of a dynamical system when a decrease in the value of 4(+)is observed such that 2(+)—> A(0), is that this more neutral stability augers a global bifurcation (Guckenheimer and Holmes, 1983). A small perturbation does not change the global dynamics of an already expanding and contracting (called “hyperbolic”) dynamical system, it will maintain the style of its motions. However, when 4(+)—> 1(0), a velocity changing perturbation evokes a bifurcation to a new dynamic in what is called “loss of hyperbolic stability.” The best examples come from the observations of this kind of change in the EEG predicting the onset of epileptic seizures in patients with focal or temporal lobe epilepsy (lasemidis et al, 1988,1990; lasemidis and Sackellares, 1996 ). 219 HOUSE_OVERSIGHT_013719

The number and variety of algorithmic strategies for computing Lyapounov exponents that are applicable to real data divide naturally into those that compute directly the average rate of separation of neighboring points from the “fiduciary” orbit, as observed on the reconstructed attractor, from which only the largest 2can be obtained (Wolf et al, 1985), and a variety of techniques based on assumed model maps of the unknown flow along which the sequential products of the local derivatives are computed. The logarithms of the straight line slopes of the sequence of directionally decomposed local tangent vectors multiplied, yield as many Lyapounov exponents as directions (Sano and Sawaka, 1985; Eckmann et al, 1986; Geist et al, 1990). The techniques of regularization by which these model processes approximate the unknown flow include those with least squares, linear fit assumptions (Eckmann et al, 1986; Sato et al, 1987; Buzug et al, 1990), more detailed fits involving polynomial expressions in higher powers (Briggs, 1990; Brown et al, 1991; Bryant et al, 1991) and techniques such as “singular value decomposition” which decomposes the flow into orthogonal components before computing the logarithmic rate of divergence of nearby points on each of them (Stoop and Parisi, 1991). A clever check on the Lyapounov number obtained is to study the flow backwards so that, for example, some rate of separation of points in the forward direction would approximate the rate of convergence in the time reversed data (Parlitz, 1992). Among the sources of spurious Lyapounov exponents are sample lengths that are too short and/or too measurement-noisy to compute a statistically stable average, embedding dimensions that are too high or low and attractors (many of physiological relevance) that have geometric features such as sharp corners or tight folds as in the Réssler (where points gather) or delicate boundary points such as those on the body on the Lorenz butterfly (see above) where very small distances determine whether the orbit makes big jumps to the right or left wing leading to uncharacteristically large separations. This “nonuniformity” in the rates of expansion and contraction in the dynamics over the attractor, a source of error in computations of statistical indices of the average behavior, becomes a _ useful tool in characterizing individual differences in sets of neurobiological data ranging from 220 HOUSE_OVERSIGHT_013720

brain enzyme kinetics (Mandell, 1984) and single neuron firing patterns (Selz and Mandell, 1991) to human psychomotor and cognitive behavior (Selz, 1992; Selz and Mandell, 1993). The Leading 4(+) of Some Biologically Relevant Time Series An early application of a simplified form of leading Lyapounov exponent to brain data involved the computation of the one dimensional averaged slope of in vitro studies of psychopharmacological drug and peptide effects on time series of catecholamine and indoleamine biosynthetic enzyme _ activities studied at physiological, far-from-equilibrium reactant concentrations (Russo and Mandell, 1984b; Knapp and Mandell, 1984). A contemporaneous study also suggested the influence of differences in initial conditions for pharmacokinetic equilibrium times in drug binding kinetics by proteins (Bayne and Hwang, 1985). The most extensive applications to the clincial neurosciences of the Lyapounov measure of the exponential divergence of orbital points has involved reconstructed brain wave attractors from the intracranial or scalp recordings of the EEG (Duke and Pritchard, 1991; Dvorak and Holden, 1991; Jansen and Brandt, 1993). Space prevents us from surveying more than a small representative set of the studies (Jansen, 1996). It should be noted, however, that this is an area in which “state of the art” research has grown quite complicated and somewhat controversial with respect to technical issues. The choices of the digitizing frequency of the smooth record, the dimension of the embedding space and time delays continue to be debated in the context of numerical computations of 2 and dimension measures (Mayer-Kress, 1986; Ott et al, 1994). Controls for the implicitly required statistical discrimination between “randomness” and “deterministic chaos” consist of sequence and (Fourier) phase randomization generating “surrogate data” which conserve the probability distributions and destroy the correlation properties and attractor geometries (Sauer et al, 1991; Ott et al, 1994). Since neither bring with them any connections with 221 HOUSE_OVERSIGHT_013721

known or explorable brain mechanisms, one might argue that at this early stage of the work it would be more desirable to simply report the quantitative findings, leaving unanswerable questions about ultimate causality for later discussion (see below). The first EEG 4(+) was reported in a patient with epilepsy (Babloyantz and Destexhe, 1986) which was confirmed by others (lasemedia et al, 1988; Frank et all, 1990). An important study of simultaneous time series from 16 subdural electrodes placed in the right temporal cortex of a patient with a right medial temporal lobe epileptogenic focus demonstrated that a decrease in a single lead’s A(+) reliably anteceded and localized the first signs of the incipient seizure. The rest of the leads followed with similarly decreased positivity in their leading Lyapounov exponents associated with spatially coherent patterns of behavior. In addition, the averaged value of the leading Lyapounov exponents in the 16 leads increased post-ictally over the averaged values of 4(+)in the pre-ictal state (lasemidis et al, 1988,1990). These findings, including seizure anticipation for 25 minutes, were confirmed using intracranial recordings in 16 patients with temporal lobe epilepsy (Elger and Lehnertz, 1998). The para-ictal decrease and post-ictal increase in 4(+) found in patients with focal temporal lobe seizures was confirmed more generally in left and right pre-frontal-to-mastoid EEG recordings made before, during and after electroconvulsive shock treatment of psychiatric patients (Krystal and Weiner, 1991). Pre-ictal changes were also found six minutes before seizure onset from scalp EEG recordings in 17/19 patients with chronic focal epilepsy (Martinerie et al, 1998). The most exciting potential application of this approach is its use, in real time, for the prediction and prophylactic treatment of incipient seizures, minutes to hours before the event, in place of or augmenting long term drug management (lasemidis and Sackellares, 1996). There is a growing literature about leading Lyapounov exponent(s) in the reconstructed attractor of the EEG associated with a variety of normal and pathological human behavioral states. For examples, two and sometimes three 222 HOUSE_OVERSIGHT_013722

A(+) were reported in awake relaxed subjects and were lost in deep sleep (Stage IV) and coma (advanced Jakob-Creutzfeld disease), suggesting that level of consciousness correlated positively with amount of orbital divergence (Gallez and Bablioyantz, 1991). A “pathologically low’ leading 4(+)was also found to be characteristic of the EEG of patients with Alzheimer’s syndromes (Jeong et al, 1998). Technically defined sleep stages (I, Il, Ill, IV, REM) were found to correlate well with the values of the leading 4(+) of the EEG in normal subjects (Fell et al, 1993; 1996; Pradhan and Sadasivan, 1996). EEG recordings during problem solving sometimes, but not always, demonstrated a relationship between values of A(+) and the kind or amount of load of the task (Micheloyannis et al, 1998; Popivanovov et al, 1998; Meyer-Lindenberg et al, 1998). Both emotionally positive and negative videos increased the value of the leading 4(+) (Aftanos et al, 1997) as did computer generated music with sounds that exploited a “pleasing” hierarchical, 1/f but not an “unpleasant” 1/f 7 frequency spectrum (see previous section about power law scaling) (Jeong et al, 1998). The EEG theta rhythm of “day dreaming” manifested a lower 4(+) than the “relaxed alert awake” alpha rhythm (Roschke et al, 1997). Relationships between the Lyapounov spectra demonstrated both regional independence and task-related dependence in the magnetoencephalography record in man (Kowalik and Elbert, 1995). These and other studies suggest that divergence rate of orbits on a geometrically reconstructed attractor is a subtle measure, which can be quantified as a continuous variable and which has been found to be useful in a variety of neuroscience-related, experimental contexts. The range’ includes’ the characterization of the discharge pattern of a single somatic or renal sympathetic nerve fiber (Gong et al, 1998;Zhang and Johns, 1998); quantifying the results of perturbing autonomic nervous system activity, for examples, exercise, atropine and propranolol decrease 4(+)in the cardiac interbeat interval attractor (Hagerman et al, 1996) and interference with the function of the baroreflex or clonidine alters the A(+)in the blood pressure attractor in man and animals (Wagner et al, 1996; 223 HOUSE_OVERSIGHT_013723

Mestivier et al, 1998); and predicting defects in visual learning functions from decreases in the A(+)of the cardiac interbeat interval attractor in patients with multiple sclerosis (Ganz and Faustman, 1996). We recall that on theoretical grounds (Guckenheimer and Holmes, 1983), a decrease in the positivity of 4(+)—>/(0)in a delay coordinate, geometric reconstruction of a time series of observables may auger an incipient global bifurcation in the system’s dynamics. As reviewed above, this has turned out to be the case in several studies of the EEG and electrocorticogram in epileptic patients. Futher research will be required to see if this idea has substance more generally for predicting “catastrophic” changes in other brain-related systems. Power Law Scaling of Orbital Geometries in Time Series Reconstructions Benoit Mandelbrot’s book in its first incarnation was derived from his lectures at College de France in 1973 and 1974 and was called Les Objets Fractals: Forme, Hasard et Dimension (Mandelbrot, 1975). This essay was translated into English as Fractals, Form Chance and Dimension (Mandelbrot, 1977). Later expanded and reworked editions displayed another title, The Fractal Geometry of Nature (Mandelbrot, 1982) but the deep conceptual, sometimes poetic fusion and confusion generated by the apparent identity among the objects of his first title remains. “Fractal,” along with “chaos” and “strange attractor” are among the most widely familiar new words in modern dynamical systems research. Fractal is the most difficult to rigorously define and is commonly misunderstood due to the evocative yet dream-like cognitive condensations provoked by the first title and its reflections in Mandelbrot’s prose. A common conceptual confusion is exemplified by the assumed relation between “fractal time event distributions” of the cardiac interbeat interval and the “fractal like” anatomy of the purkinje network of the cardiac conduction system. Data from both contexts are often shown juxtaposed in the same illustration as though their relationships were obvious (Goldberger et al, 1990; Goldberger, 1996; Liebovitch and Todorov, 1996). “Fractal times” and “fractal 224 HOUSE_OVERSIGHT_013724

geometries” are not related to each other essentially, either in the mathematical or physiological domain, but are often made vaguely equivalent on the basis of their lexical similarity. An experimentally meaningful relationship between fractal statistics (hazard), dynamical fractals (dimension) and fractal geometries (form), has to be proven ona case by case basis and not assumed from their common designation. Among the informal attempts to do this have been those that involve the branching pattern of nerves and the associated reductions in their diameter-dependent characteristic conduction velocities yielding a multiplicity of “arrival times.” There is, however, a more central idea common to these concatenated meanings of fractal: the statistical, dynamical and geometric expressions of “scaling,” a word which is not mentioned in Mandelbrot’s book titles. The cluster of theories, theorems and methods associated with the idea of scaling (and renormalization) have led to Nobel Prizes for Flory (1971), Wilson (1975) and de Gennes (1979) and the (equivalent mathematical) Field’s Medal for McMullen (1994). There is speculation that the last two awards were supported by the inspiration and interest given their research by Mandelbrot’s intuitions and books. Scaling laws take the place of (unknown causal) physical laws by indicating the proportion by which observables of a system can be changed in relationship to each other such that some statement about them, “this varies with that,” still holds. In a cross species comparison, as the average weight of a mammalian body, called lb, increases, the skeletal weight, called w, increases at an exponentially greater rate: w goes like Ib'°® where Ib'° would indicate that they grew across species at the same rate. Plotting log (Ib) on the x axis and log (w) on the y axis in a log-log plot results in a straignt line with a slope that indicates the power law scaling relationship between body weight and skeletal weight across mammals. The slope of the scaling exponent of 1.08 is a little over 45 = 1. In contrast, the metabolic rate, 0.75 r, goes like (Ib)°”, r ~ (Ib)°”°. Larger animals (relative to their weight) have lower basal metabolic rates (Schmidt-Nielsen, 1984). We don’t completely know the chain of intervening mechanisms that relate these variables to each other but we do know 225 HOUSE_OVERSIGHT_013725

invariant scaling laws that describe their relationships within some limits on the range of values. In describing the functional size, radius of gyration, Rg, of a polymer such as a polypeptide, composed of N monomers, assume each of the amino acids to be the same and that they are in a “good” hydrophobic solvent that didn’t stick the polymer together in a fold. Flory (1971) found a scaling law for certain broad classes of polymers and solvents, Rg ~ aN” where the exponent, v = 3/5, was universal, N indicated the number of monomers in the chain and the value of “pre-factor” a depended upon the particular monomer and solvent chosen. Log Rg plotted against log N has a “power law” slope of 0.60. For an equally static but less physical example, there is the well known Zipf law of “vocabulary balance’(Zipf, 1949). First reported for the 260,450 words of James Joyce’s Ulysses, the slope of the log of the rank of the words found (ordered from most to least along x) plotted against the log of their frequency (along y) results in a power law that is (generally) true for other collections of words and in other languages. An accessible example of a dynamical scaling law arises in a two dimensional lattice model of a forest which is to be set on fire with probability p independent random single tree ignitions. At some critical p, pc, the fire sweeps through the entire forest (“percolates”) and the correlation length of the connected clusters grows as |p-p,|’ with a universal scaling exponent, y = 4/3, for all Monte Carlo, two dimensional percolation problems (Stauffer, 1985; Grimmett, 1989). Mandelbrot’s scheme for the power laws that compose his fractal geometry of dynamical objects is a measure made on the pattern of occupancy in the embedding space by the reconstructed orbits of an attractor. It is, generally, mass = length” in which Do (the subscript that of the “capacity dimension”) is not the whole number of Euclidian dimensions, d, of the space in which the orbits are embedded. After Hausdorffs “convergence of external and internal measures” (Hurewicz and Wallman, 1948), the (capacity) fractal dimension Do is also defined as being larger than its topological dimension and smaller than its Euclidian embedding dimension. Graphing a time series on a plane one can think of its 226 HOUSE_OVERSIGHT_013726

topological dimension as that of a line equal to one. If each time step had the largest up or down amplitude as possible, its fractal dimension would approach (but not reach) that of the embedding plane, Euclidean d = 2. The Do of the one dimensional Richardson technique (Mandelbrot, 1967) can be computed by covering the one dimensional surface of a time series with a number, #, of line segments of several orders of magnitude range of lengths, / -Graphing log(l) along the x-axis and log #(I) along the y-axis yields a negative linear slope, -s. As defined, 1- s = Do noting that (-(-s)+s) such that 1 < Do = 1+s < 2. Strain differences and peptide and psychotropic drug-induced changes in Do computed in this way were found in time series of fluctuations in rat brainstem tyrosine and tryptophan hydroxylase activities under far-from-equilibrium co- reactant concentrations (Mandell and Russo, 1981; Knapp et al, 1981; Knapp and Mandell, 1983; 1984). Systematic influences of stimulant drug dose on Do were found as well in these systems (Mandell et al, 1982). This simple measure, made directly on the “roughness” of the graph of a one dimensional time series rather than on its orbital reconstruction, has been used to discriminate the pattern of fluctuations in daily mood scales in normal subjects and mood disordered patients (Woyshville et al, 1999). These findings confirmed dimensional scaling exponents on higher dimensional embeddings of similar time series in mood disordered patients (Gottschalk et al, 1995; Pezard et al, 1996). Due to the ease and rapidity of its computation, techniques involving Do on one dimensional time series are currently in development as possible real time epilepsy predictors when analyzing the output of a large number of EEG leads simultaneously. If M(e) is the minimum number of d-dimensional cubes of side ¢ required to cover the d-dimensionally embedded attractor, plotting a logarithmic range of rulers of length ¢ (as e—0) along the x axis and a logarithmic range of number of cubes, M(e), each of corresponding «-edge size, along the y axis, results in a negative (more smaller M(e) ‘s and fewer bigger M(e) ‘s) power law slope Do. Here the numbered covering cubes, M(e), are those in which the probability of containing at least one point (its “probability density measure,” often called uw) is not zero. We 227 HOUSE_OVERSIGHT_013727

note that changing the ratios of the numbers of cubes that are dense in point probability to those that are sparse would not influence the value of Do. This helps differentiate Do from other dimensions and, as noted above, Do as a maximal estimate of the fractal dimension, is called the capacity dimension and by convention the scaling law is written M(<)~ « ”. More specifically, Do is calculated by repeatedly dividing the d-dimensionally embedded phase space into equal d- dimensional hypercubes and plotting the log of the fraction of the hypercubes containing data points versus the log of the (normalized) linear dimension (“length scale”) of the hypercubes. The slope fitted to the most linear part of the slope (usually the middle 50%) indicates the capacity dimension. Do is computed for increasing embedding (and cube) dimension, d, until it achieves an asymptotic plateau, it “saturates”. This is but one of a range of geometric scaling exponents, “dimensions,” that are currently being computed (Farmer et al, 1983; Grassberger and Procaccia, 1983; Meyer-Kress, 1986; Theiler, J. (1990); Gershenfeld, 1992; Ott et al,, 1994). Although still subject to debate, convention has it that the sample length required to determine this most primitive of dimension computations goes like 10” (e.g. a dimension of 2.45 requires a sample length of at least 282 points). Assuming robust findings using Do as indicated by non-parametric tests of significance in test-retest, before and after, drug treatment designs, this arbitrary criteria sounds more like ritual than meaningful help for the clinical neuroscientist with (say) 100 spinal fluid hormone and metabolite samples painfully and laboriously collected from a patient’s indwelling catheter over 48 hours. In the context of real data (and not numerical studies of differential equations), we are dealing with empirical findings that must find their meaning (or lack of) in the context of questions about issues in the neurosciences, not in abstract questions such as those about the number of dimensions that an unknown differential equation would require to represent the data (Broomhead and King, 1986). In a similar arbitrary spirit, a system manifesting a Do > 5 is considered not discriminable from a random process; e.g. the difference between Do = 5 versus Do = 7 (though perhaps statistically significant) is thought to be without meaning. Since in neurobiological 228 HOUSE_OVERSIGHT_013728

research, “random” (if it doesn’t mean measurement error) indicates unknown degrees of freedom, this Do> 5 rule is also without relevance for brain research. D, is called the “information dimension” and is computed by counting the number of e-cubes, M(g), it takes to cover the points constituting some fixed fraction of all of the points of the set of orbital points on the attractor and can be regarded as the “core dimension” (without the outliers) of the set. The counterintuitive finding is that D, is nearly constant across a range of fixed fractions that are less than the whole measure (Farmer et al, 1983). The invariance of D; can even be taken to the extreme by computing the D, = lim in (4) around (typical, not all) single points. In this context, D; is called the “pointwise dimension” or “singularity exponent” and, as might be anticipated, its value is usually less than that of Do. The scaling exponent that is both sensitive to point densities and easiest to compute from real data is the “correlation dimension,” D2 Here, analogous to the relationship between the amplitudes of the variance and the correlation function in conventional statistics, the measure squared is of interest for the computation of Dz, M(e) e.g. /(2,¢€)= SLAC, )F (see below for this use of measure u on sum = of cubes C)). i=l The selection of Dz as the fractal measure dominates the studies that invoke scaling exponents to quantify the distributions of points on the attractor as reconstructed from time series in the neurosciences (Grassberger and Procaccia, 1983; Mayer- Kress, 1986; Ott et al, 1994). Several sets of programs are available for its computation (for example, Sprott and Rowlands, 1991). Generally, a correlation sum (“integral’, R(e) ) is computed from a starting point by counting all subsequent point pairs with distances between them less than ¢ as e—>0 and plotting __ lim In(R(e) 逗>0 Ine) , . D2 is computed for increasing embedding (and therefore hypercube) dimension, d, until Dz achieves an asymptotic plateau, it “saturates” (Ding et al, 1993). It is generally the case that Do > D;> Dz (Farmer et al, 1983). 229 HOUSE_OVERSIGHT_013729

In his statistical explorations of experimental results in hydrodynamic turbulence, Mandelbrot (1974) called attention to the need for a multiplicity of characteristic scaling exponents, a range of values for each exponent and their sensitivity to orbital point density distributions (the latter called the Sinai-Ruelle- Bowen or natural measure (Eckmann and Ruelle, 1985)). These needs grew out of the intrinsic heterogeneity in the time dynamics and the nonuniform point distributions in phase space of orbitally divergent, real physical systems. Even with relatively uniform orbital point distributions, it is intuitively obvious that as e > 0, the smaller e- cubes are over-represented and larger e- cubes are under-represented in the M(e) computation (Farmer et al, 1983). For a concrete example, the fraction of the total number of cubes containing say 75% of the points would obviously decrease as the e-lengths studied gets smaller. Normalizing the Dj measures with respect to point densities would correct for this systematic distortion. In addition, the non-systematic influence of real system heterogeneity and non-uniformity in both time and reconstruction space distributions makes the need for relating the Dj measures to the natural measure even more pressing. The derivation of many separate scaling exponents, as well as global generalized exponents and the incorporation of point densities in their computation, has been approached by a kind of method of moments (Renyi, 1970; Grassberger, 1983; Hentschel and Procaccia, 1983; Halsey et al, 1986; Mayer-Kress, 1986; Ott et al, 1994). We outline the general arguments here so that the reader will be generally familiar with the ideas and terms, not to serve as a definitive summary. It is a complicated area and the reader will find the required detailed descriptions in the references. . We recall that with respect to a statistical distribution, the first moment is the mean; the second moment, o”, the variance; the third moment, o°, the distribution’s asymmetry, the skew; and the fourth moment, o”, its relative peakedness with respect to the probability mass in the tail, called the kurtosis. In these moment computations of an observable x,’s deviation from the mean, |x, — x|*, the value for q accentuate particular regions of the density distribution. Similarly, the q’s of the 230 HOUSE_OVERSIGHT_013730

“generalized dimensions,” Dg, emphasize different aspects of the relative point density that are assumed to be uniform in the computation of Do. We recall from above that the power law slope constituting D, = lim a 7 é>0 sal é . If we emphasize the component of the probability (measure, uw) or, equivalently, time spent by the orbit in cube i, “(C,) instead of simply the number of cubes occupied by any points, M(e), along with the different length scales of the cube as e—0 we have a generalized dimension. A common expression for the generalized dimension includes the fractional pre-factor in q written so as to make things come out right: : M(e) D, = ,__finn sel where /(qg,¢) = SLAC I. The higher the q, the greater q-le—>0 In(e) = the dominance of the higher probability cubes, ~(C,). To see how this q-induced separation in emphasis might work, if the ratio for q = 2 between the probability containing cubes 0.25 and 0.05 is 25, their ratio for q = 3 is 125. For q = 0, the scaling exponent is the capacity dimension. This result of the actions of a changing q has been analogized to the way changing temperature in a thermodynamic system evokes different aspects of its behavior. The “multifractal formalism” generally begins by determining the statistical densities over a range of scale lengths by one means or another including wavelet transformations across wavelength scale (Arneodo et al, 1988). These densities by scale are then systematically raised to a range of q exponents. Since q, and therefore Dg, can vary continuously, functions are created that shows how D, varies with q. These are then further transformed, resulting in a single maximum parabolic curve whose shape and size is sensitive to the conditions of the experiment (Halsey et al, 1986). Generalized dimensions decrease as q increases. A unique neuropsychopharmacological application of the multifractal technique to a study of the behavioral influence of increasing amounts of cocaine on the time-dependent patterns of spatial exploration, temporal-spatial fluctuations, in rats, demonstrated a global splitting in the parabolic distribution suggestive of a cocaine-induced global phase transition, not unlike the well-known, dose-dependent, amphetamine-induced 231 HOUSE_OVERSIGHT_013731

shift from hyperactivity to motor stereotypy (Paulus et al, 1991). Studies that followed demonstrated that “q-moment” distributions of heterogeneous scaling exponents and their relative statistical weightings were useful in making subtle discriminations between effects of psychopharmacological agents and behavioral (isolation) influences on animal behavior as well as patterns of simple psychomotor behavior in normal subjects and schizophrenic patients (Paulus et al, 1994; 1996; 1998; Krebs-Thomson et al, 1998a; 1998b). Fractal Scaling Measures on Reconstructed Time Series from Biological Dynamics Publications involving the applications of various D measures, particularly Dz, to brain-relevant times series number in the hundreds and are growing exponentially. The following constitutes a brief review of a representative set of empirical findings. In doing so, for the reasons discussed below, we ignore what some might consider the rather abstract and philosophical issue of “determinism” versus “randomness” or “error” (Sugihara and May, 1990; Casdagli, 1991; Wayland et al, 1993; Kaplan and Glass, 1992; Kaplan, 1994) since this question is relatively unproductive with respect to generating new neurobiological insights, novel experiments or new quantitative approaches to brain dynamics. In addition, as noted in the final section, this discrimination may not even have definitive theoretical meaning in that the conduct of much of the rigorous mathematics about “deterministic dynamical systems” involve Markoff partitions and matrices which are also the generic operators of formal probability theory (Sullivan, 1979; Kolmogorov, 1950). For example, N-dimensional non-linear Markoff processes can be shown to capture the dynamics of multidimensional neurobiological processes such as the EEG (Silipo et al, 1998). We have also ignored the related issue of the presence or absence of “low dimensional structure” (Theiler and Rapp, 1996; Rapp, 1995) which, from the authors’ point of view, resulted from an unfortunately concrete interpretation of the word “dimensions.” With respect to experimental brain data, dimensions are defined 232 HOUSE_OVERSIGHT_013732

most relevantly by their computational procedures and what are computed are empirical scaling exponents describing real observables as limited by the precision of the observations, their resolution and series lengths (Smith, 1988; Eckmann and Ruelle, 1992). The “correlation integral,” the probability that two vectors chosen at random from the phase space reconstruction lie within “r” distance of each other, not unrelated to the phase randomization controlled, Dz measure, yields statements about amount of “nonlinearity” (not accountable by the linear regressively capturable component of the power spectrum), which are also difficult to translate into experimentally or theoretically useful concepts (Casdagli et al, 1997). These efforts contrast with a more direct attempt to establish a spiking neuron system’s dynamical “dimension” using trial and error prediction in which “dimension” was defined as the number of potentially physiologically relevant variables required to make the predictive equations fit (Segundo et al, 1998). Computations of scaling exponent descriptors of orbital point distributions on reconstructed attractors of the brain sciences have proven to be most useful as atheoretical, empirical techniques discriminating experimental, clinical and/or treatment conditions with various approaches to statistical significance. In this regard, one can say that D2 is often found to be superior to central tendency oriented statistics in making these discriminations. Dimension and _ correlation integral descriptors appear least useful when dealing with global issues such as chaos, randomness, linearity and the “underlying dimensions” of (unknown) differential equations. We discuss below the possibility that the failure to find chaos in the more recent EEG studies (Theiler and Rapp, 1996; Prichard et al, 1996) may be because the EEG attractor is better characterized as a “strange nonchaotic atttractor” with orbital patterns manifesting fractional scaling exponents but no 2(+) (Grebogi et al, 1984; Mandell and Selz, 1993). The relatively subtle influence of high altitude (Mt. Everest) oxygen concentrations was not seen in the central moments of the cardiac interbeat intervals, but the D2 of the attactor was reduced significantly (Yamamoto et al, 1993). The latencies and amplitudes of the visual evoked potential failed to 233 HOUSE_OVERSIGHT_013733

discriminate normal subjects from those with early glaucoma, but the reconstructed attractor of the steady state visual cortical response to full field flicker demonstrated a statistically significant decrease in D2 (Schmeisser et al, 1993). Marginal qualitative differences in optokinetic nystagmus were quantitatively significant when studied as the Dz of the attractor’s points in patients with vertigo compared with controls (Aasen et al, 1997). Reconstructions of maximum velocity waves from Doppler studies of middle cerebral artery hemodynamics (using phase random “controls”) demonstrated an increase in D2 (and a decrease in 4(+) correlated with age in an adult population (Keuner et al, 1996; Vliegen et al, 1996). D2 served as a sensitive descriptor of functional changes in the EMG from the surface of the biceps muscle, increasing with muscle load and rate of flexion and extension and decreasing with muscle fatigue (Rapp et al, 1993; Nieminen and Takala, 1996; Gupta et al, 1997), suggesting its use in suspected early myotonic dystrophies and myasthenias. Reconstructed time series of stomatognathic motions in high school students with temporomandipular joint syndromes compared with those with malocclusion revealed a specific decrease in Doin the plane of horizontal motion in the former (Morinushi et al, 1998). Time series of plasma growth hormone levels in acromegalic patients with functioning pituitary adenomas manifested a statistically significant increase in Do when compared with age-matched controls (Mandell and Selz, 1997) which corresponded nicely to the reduction in “approximate entropy” (Pincus, 1991a) computed on this same data set (Hartman et al, 1994). On the other hand, comparative in vitro studies of growth hormone release patterns in normal rat pituitary cells and their neoplastically transformed relatives, the GH3 strain, demonstrate a decrease in Do in the latter (Guillemin et al, 1983; Mandell, 1986). The number of examples of the use of D2 on orbital point geometries in explorations of physiological and pharmacological regulation are increasing. The D2 of respiratory rhythms is higher with intact vagal afferents than without (Sammon and Bruce, 1991). Histamine induced an increase in Dz in the attractor point distribution of rabbit ear artery vasomotion, attributed to calcium-activated membrane potassium channels in that TEA prevented and reversed the change 234 HOUSE_OVERSIGHT_013734

(Edwards and Griffith, 1997). The role of central and autonomic innervation in cardiac interval dynamics has been explored using D2 in various ways. For examples, the transplanted heart rhythm in man has a lower Dz than that of the normal heart (Guzzetti et al, 1996) and general anesthesia and cholinergic (but not B-adrenergic) blockade decreased multisystem Dz in a series of multiparameter (respiration, mean blood pressure and heart rate) studies in piglets (Zwiener et al, 1996; Hoyer et al, 1998). The activities of single and aggregates of neurons are being described and differentiated by the D2 of their interevent interval attractors. Early and important studies related to both neuronal and field electrical activity indicated their promise (Rapp et al, 1985; Zimmerman and Rapp, 1991). The olefactory bulb demonstrated spatially uniform scaling dimensions that changed with event-related perturbation (Skinner et al, 1990). An iron-induced spiking focus in the rat hippocampus in vivo manifested the same decrease in D2 as it did in the kindled in vitro hippocampal slice (Koch et al, 1992). D2 also differentiated among characteristic single unit time series in norepinephrine, dopamine and serotonin neurons (Selz and Mandell, 1991) and among A8, A9 and A10 dopamine neurons (Selz and Mandell, 1992). Attractors reconstructed from single unit interspike intervals in the substantia nigra pars compacta and the auditory thalamus manifested discriminatable values for D2 in neurons recorded by the same electrode (Celletti and Villa, 1996) and changes in state manifested in patterns of subthreshold oscillations in single neurons in the inferioir olivary nucleus could be characterized using this index (Makarenko and Llinas, 1998). Dz reliably discriminated between states of arousal and between the multiparameter (eye movements, neck muscle tone, EEG stage) defined EEG stages of sleep (Bablyoyantz, 1986; Rapp et al, 1989; Ehlers et al, 1991) with non- REM having a lower Dz than REM. Dz of the EEG record was selectively reduced in Stage Il and REM in schizophrenic patients compared with controls (Roschke and Aldenhoff, 1993), this difference was made more prominent by treatment with the aminodiazopoxide, lorazepam (Roschke and Aldenhoff, 1992). In the waking state, 235 HOUSE_OVERSIGHT_013735

higher EEG Dz values were frontal in schizophrenic patients and more central in controls (Elbert et al, 1992). The Dz computed on the EEG during Stage IV (“delta”) sleep was sensitive to acute sleep deprivation and recovery, but demonstrated compensation (Cerf et al, 1996). Non-alcholic children of alcoholic parents manifested lower values for D; in their EEG attractors than the children of a normal control group (Ehlers et al, 1995). Higher |.Q. correlated with EEG Dz in most leads in the resting state but not during a visual imagery task (Lutzenberger et al, 1992). These differences also correlated with individual differences in task performance in a perceptual pattern predictive task (Gregson et al, 1990) and with a working memory task load with regional differences most marked in the right fronto-temporal cortex (Sammer, 1996). Peripheral nerve stimulation in the earlobe and trapezius muscle induced increments in Dz in the EEG of specific brain regions (Heffernan, 1996). Memory for but not induced pain increased EEG D2 in chronic pain patients but not in normal controls (Lutzenberger et al, 1997). Using contingent reinforcement of brain wave modes by hypothalamic, but not cerebral hemispheric, stimulation reduced Dz in the EEG (Mogilevskii et al, 1998) resembling the changes accompanying defensive reflex conditioning in the rabbit between the early and late stages of the process (Efremova and Kulikov, 1997). Difficult to diagnose “periodic lateralized epileptiform discharge” syndromes have apparently yielded to D2 computations (Stam et al, 1998). In equally problematic “atypical seizure” syndromes in children, D2 computed on the autocovariance functions of 200 Hz digitized EEG records from multiple channels demonstrated characteristic changes (Yaylali et al, 1996). Unlike computing a reliable leading 4(+)on a point set of a time series reconstruction denoting the “sensitivity to initial conditions” requirement for the diagnosis of chaos (and a potential for change such that a decrease in the positivity of A(+) > 4(0) may auger a nearby bifurcation), the presence of a fractional scaling exponent, D,, does not in and of itself implicate a chaotic dynamical state. A nice example of a nonchaotic dynamic with 4 = 0 that has a fractional scaling exponent, D = 0.538, is the “Feigenbaum” point where the above noted “infinite” series of 236 HOUSE_OVERSIGHT_013736

period doubling bifurcations accumulate (Grassberger, 1981). This is a dust-like region, which when endlessly dilated looks like the same dust. Some mathematicians call these objects “Lebesgue points” because even though at low magnifications when they look rather solid, they are not. Composed of points, they have topological measure zero (a line has measure one) and non-integer fractal dimension. These 4=0, D = Integer, period doubling accumulation points can be found in a wide variety of attractors, though in each case the parameter space in which they are located is so small (in point set topology also called “Lebesgue measure zero”) that they are very difficult to locate and therefore have little chance of being physiologically significant. This constrasts with a relatively new category of dynamical systems which promises to be important in studies of the nervous system. These are ones that are driven by two or more independent frequencies (called quasiperiodic driving). We found them to be relevant to brain stem, thalamocortical neurophysiology of perceptual processes and states of consciousness. They have the properties, A=0, Do and D; # integer and a characteristic scaling “spectral distribution function” (see below). They have been named “strange nonchaotic attractors” (Grebogi et al, 1984; Romeiras et al, 1987; Ding et al, 1989). In addition, the strange nonchaotic behavior of these quasiperiodically-driven, nonlinear oscillators has positive (>0) measure in parameter space and thus is of potential physiological significance. A good demonstration of a multiple frequency driven strange nonchaotic attractor can be found and manipulated in the software package of Nusse and Yorke (1991). The neurobiological substrate for this system is the brain stem neuronal modulatory driving of on- going thalamocortcal oscillatory brain waves (once called “recruitment waves” in the 7-14 Hz, 6 to a, day dreaming to quiet alert range) and as perturbed by multifrequency driving in what was once called “reticular formation arousal” are realized as dominant EEG modes and associated states of perceptual acuity and consciousness (Moruzzi and Magoun 1949; Moruzzi, 1960; Klemm, 1990; Steriade and McCarley, 1990; Contreras et al, 1997). In addition to intrinsic 237 HOUSE_OVERSIGHT_013737

multiply periodic and aperiodic oscillations of thalamic and cortical cells and their recursive, feedback coupling, the brain stem manifests more than two orders of magnitude of “independent” neuronal driving frequencies ranging from serotonin discharges at 1 Hz, cortically direct dopamine and norepinephrine neurons in the 10-50Hz range and mesencephalic reticular neurons discharging as fast as 100 to 200 Hz. The “thalamocortical brain wave oscillator” as their target has been a fixture in global state neurophysiology since the 1940’s and 1950’s and is of great current interest (Fessard et al, 1961; Bazhenov et al, 1998). We have explored the relationships between strange nonchaotic dynamics and brain-stem neuronal and thalamocortical physiology from the standpoint of neuronal coding and the properties of the EEG attractor. (Mandell et al, 1991; Mandell and Kelso, 1991; Mandell and Selz, 1992; 1993;1994:1997a). We found that the EEG attractor could be characterized by the diagnostic triad identifying strange nonchaotic attractors: A=0, Do and D, # Integer, and a signatory power spectral distribution in which the number of peaks, N, with amplitudes greater than ow, N(w ), went as wo", 1<a <2 (Romeiras et al, 1987; Mandell et al, 1991). In addition to being consistent with known multifrequency, brain stem driving of thalamocortical oscillations, the EEG as a strange, nonchaotic attractor is intuitively appealing in that it has the necessary mechanisms for the power law scaling of a wide range of characteristic times (Do and D, =~ Integer) from picosecond fluctuations of neural membrane proteins to the decades of bipolar phenomena and since 2=0, the orbital points don’t tend to “mix’(get out of order) on the attractor, thus protecting the fidelity of sequence dependent brain information transport (Berns and Sejnowski, 1998). Entropies, Unstable Periodic Orbits and Shadowing; Short Time Series Can Discriminate Experimental Conditions in Studies of Biological Dynamics We avoid the temptation to deal with the deep analogy between thermodynamic entropy (Clausius, 1897) and information theoretic entropy (Shannon and Weaver, 1949), constraining our discussion to the context of an operational equivalence (in healthy systems) between gain of information and 238 HOUSE_OVERSIGHT_013738

decrease in entropy in brain-relevant dynamical systems. As we shall see, certain pathophysiological processes appear to manifest themselves as reductions in background or “resting” state entropy which then limits its supply with respect to information gain and/or transport. Relationships between “physical” thermodynamic observables, such as changes in heat capacity or temperature dependence of kinetic constants, and information-transport driven, neurotransmitter evoked conformational changes in neural membrane proteins may someday come together in an experimentally productive way (Hitzemann et al, 1985; Zeman et al, 1987; Borea et al, 1988), but they are beyond the scope of this paper. The idea of taming the orbit of an expanding flow (with at least one 2(+) ) by partitioning the geometric space supporting its actions, its “manifold,” and then labeling each box so that its trajectory is representable by a symbol string of box indices is the way “symbolic dynamics” are applied to dynamical systems. Symbolic dynamics arose in pure mathematics in the context of obtaining a one-to-one, topological (sequence not distance preserving ) representation of a difficult to characterize system of “geodesics on surfaces of negative curvature” (Hadamard, 1898; Morse, 1917; Morse and Hedlund, 1938). Geodesics here are the shortest lines in this curved, non-Euclidean space in which nearby lines spread apart and far away ones came together with (in Euclidian space) parallel lines meeting at infinity. Remarkably, symbolic dynamic encoding of the motions on this abstract manifold of negative curvature also capture how uniformly divergent (and convergent), “hyperbolic” chaotic systems, such as brain systems, behave in Euclidean space, an intuitive similarity about which Poincare experienced his famous vacation bus trip epiphany (Stillwell, 1985). It should also be noted that encoding neural spike trains in one dimension for symbolic dynamical comparisons of sequence structure and recurrances, “favored patterns” has been developed independently of orbital dynamics on manifolds (Dayhoff, 1984; Dayhoff and Gerstein, 1983a; 1983b). A similar approach has been used to characterize firing patterns and their response to acupuncture in dopamine neurons in the substantia nigra and hypothalamic neurons (Chen and Ku, 1992). 239 HOUSE_OVERSIGHT_013739

For real neurobiological data, a time series and its n time delays are first reconstructed as a trajectory in an n+1 dimensional geometric embedding space and, following partition of that geometric space into n+1 dimensional lettered boxes (the choice of partition being a sensitive step), what was once an orbit has become a sequence of symbols. Dynamical systems in geometric space become symbolic dynamics in sequence space. It was Kolmogoroff (1958) who first applied Shannon’s ideas of entropy and information (Shannon and Weaver, 1949; Khinchin, 1957) to the quantification of these dynamical system’s telegraphic messages as discrete, “stochastic” (random, probabilistic) output. Kolmogoroff turned to Shannon entropy, -S'p, log p, (where p = 1/n and n = number of possibilities) to decide the question whether a dynamical system that naturally partitioned into a two or three box system per unit time had the same entropy. His answer was no, that —3(1/3 In (1/3)) = 1.098 > -2(1/2 In (1/2) = 0.6931 loge and in computer relevant logz, 1.5850 > 1.0 (Kolmogorov, 1959). Entropy increases with possibility. Nonlinear differential equations representing brain-relevant expanding dynamical systems replace Shannon’s linguistically weighted and serially ordered, Markoff-dependent random number generator of probabilistic language. As noted above, in the case of the Sharkovskii sequences (Sharkovskii, 1964; Metropolis et al, 1973; Misiurewicz, 1995), a small change in the single parameter of an entire class of single maximum maps generating motions that are coded from their position at the left or right of center of the unit interval, alters and determines precisely the periodic output such as {1,0,0,1,0,1,1,0,0,1,0,1...) of its binary message. In higher dimensional examples such as the Rossler and Lorenz systems, one can visualize the joint actions of 24(+) and A(-) moving the trajectory so as to both enter, “create,” new boxes and generate new letters as well as visit old ones, unstable fixed points, thus forming unstable periodic orbits. The latter, one of three diagnostic features of chaotic attractors (see above), can also be seen as resulting from the “coarse-grained” imprecision of real world neurobiological measurement such that two points that are brought close to attractive-repelling points are, within measurement error, recorded as having the same value. 240 HOUSE_OVERSIGHT_013740

Problems of measurement precision, amplified by the expansive actions of systems that are sensitive to initial conditions, yield parameter sensitive entropies of two (mathematically) fundamental kinds called topological and metric entropies, hr and hw, proven to be the upper and lower bounds of any estimate of the entropy ina uniformly expanding and/or equidistributed system (Adler and Weiss, 1965). Measures of entropy, as “missing information related to the number of alternatives which remain possible to a physical system” (Boltzmann, 1909), “index of probability” (Gibbs, 1902) or the “amount of uncertainty associated with a finite scheme” (Khinchin,1957) are obviously sensitive to the partition rules and its fineness of the grain. The most theoretically defensible partition is called the “generating partition” in which no box contains more than one point. Comparisons of control and experimental data can be differentially sensitive to partition construction, so that if a generating partition is not practicable due to sample length or dense curdling in the point distribution, some arbitrary choices have to be made. These have included naturally renormalized variational partitions, such that in one dimension the boxes are defined by +1, +2, +3,...standard deviations, or quartiles or quintile, above and below the mean and in n dimensions. Partitions have also been constructed and used to described drug effects on rat exploratory behavior by sequential partitioning along the dimension of the highest remaining variation (after the previous partition) called the “KD” partition (Paulus et al, 1991). Partition strategies to capture entropic measures on serial ordering (Klemm and Sherry, 1981; Strong et al, 1998) can grow from knowledge or hypotheses about the physiological sources of temporal irregularities and discontinuities in brain dynamics including characteristic interval(s) of refractoriness, relaxation times of the inhibitory surround, correlation time in dendritic tree summation, the time course of reciprocal inhibition and its decay and chemical influences such as the synaptic half-life and time of action of inhibitory influences such as GABA on cell firing. The logarithmic growth rates of occupancy of new symbolically indexed boxes or, equivalently, the growth rates of visitations to old ones generating unstable periodic orbits, are called topological entropies, hy . They record new happenings, the growth rate of the diversity of orbits, and not how likely with respect 241 HOUSE_OVERSIGHT_013741

to box occupancy densities they are likely to occur ( Adler et al, 1964; Alexeev and Jacobson, 1981; Cornfield et al, 1982; Ornstein, 1989; Ruelle, 1990). The close relationships in real brain observables between the appearance rate of new symbols or new unstable periodic orbits, hr , and log 4(+), reflecting the rate of divergence from the next expected value generating a new, unexpected value, is not surprising. In fact, a maximal estimate of the entropy of a dynamical system, hr = log 4(+) whereas the largest value that hy can attain is log(#of states). A great deal of substantial mathematics has gone into proofs that similarities (“equivalence relations”) and differences between dynamical patterns are robustly indicated by differences in hy and hy (Adler et al, 1977; Adler and Marcus, 1979). lf the sum of the densities in each | box were normalized so as to sum to 1.0, such that each is a probability, pj , then - X pj log p; represents the metric entropy, hu. hy was first described in the dynamical context by Kolmogorov (1958;1959). The sum having a —1 prefactor converts the negative log of < 1 to a meaningful positive value in the expression. hy is maximal for the equidistributed, uniformly expansive, C or Axiom A systems (see above). As noted above, generally hy = the maximum estimate of the entropy and hy the minimum estimate (Adler and Weiss, 1965). ht = hw in uniformly hyperbolic systems (Bowen, 1975) and the difference, [hz — hy] is an index of non-uniformity found useful in discriminating among classes of single neurons from their discharge patterns (Mandell, 1987; Selz and Mandell, 1992: Mandell and Selz, 1993; Mandell and Selz, 1997a). These measures applied to temporal and spatial patterns of rat exploratory behavior have been used to discriminate among stimulant drug effects (Paulus et al, 1990; Paulus and Geyer, 1992). Similar computations involving the symbolic dynamics and disallowed transitions have been used to study the complexity of the the EEG (Xu, 1994) in which both extremely low (fixed point, periodic) and high (Gaussian random) entropies are seen as manifesting low “complexity as a function of the diversity of the available patterns of behavior (Crutchfield and Young, 1989a). Before describing the simple but definitional matrix operations for ht and hy below which might seem forbidding to those “not up on their linear algebra,” we note 242 HOUSE_OVERSIGHT_013742

that procedures such exponentiation of a matrix can be carried out automatically using computer algebra programs such as Maple or for data processing available as computational modules in MatLab. One of the techniques for the computation of hy involves determining the logarithm of the asymptotic growth rate of the major diagonal (“trace”) in the transition matrix symbolically encoding the trajectory which would therefore count the “self visitations” of each indexed boxes as the dynamics proceed. This involves setting up a transition incidence matrix, each box scored for a disallowed, 0, or allowed, 1, transitions and the matrix is exponentiated t times with the logarithm of the asymptotic growth rate of the sum of the diagonal values serving as a (leading eigenvalue) estimate of hr. More technical considerations involving the Frobenius- Perron theorem guaranteeing the existence of such an logarithmic index of new information generation rates, even in random matrices (Seneta, 1981), will not be discussed here. We have found that computing hz in this way is empirically useful for difficult to obtain or only transiently stationary brain data series. Even with relatively short samples lengths, if one is willing to make the pragmatic assumption of “temporary stationarity” or “things as they are right now will, for the sake of argument, go on forever’ (perhaps the best we can do with intrinsically transient brain phenomena) then this “freeze framed” representation of reality yields an asymptotic measure on relatively short sample lengths since they are computationally infinite. A similar approach to hy, requires repeatedly exponentiating a Markoff matrix constructed from relatively short samples and generates the probabilistic (eigenvector) “dual” of hr. hy computed in this way serves as a useful quantity, hy called by some the Kolmogorov entropy in comparisons of control and experimental conditions of the same sample lengths. Systematic decreases in hw (“Kolmogorov entropy”) have been shown to accompany increasing “depth” of sleep using standard sleep staging techniques (Gallez and Babloyantz, 1991) and increases in hy were associated with both positive and negative emotional states induced by movies (Aftanas et al, 1997). 243 HOUSE_OVERSIGHT_013743

ht and 4(+) have been analogized to what is called algorithmic complexity, which quantifies a computer algorithm’s minimal representation of a symbol sequence as it grows longer (Chaitin, 1974; Bennett, C.H., 1982; Nicolis, 1986; Rissanen, 1982; Crutchfield and Young, 1989b). Examples of applications of a pseudocomputational compression scheme have quantified differences among protein sequences (Ebling and Jimenez-Montano, 1980), discriminated therapist- directed “transference” manifestations in verbally encoded processes in psychotherapy (Rapp et al, 1991), characterized neural spike train patterns in a penicillin kindled spike focus (Rapp et al, 1994), differentiated among spike sequence patterns of biogenic amine families of brain stem neurons (Mandell and Selz, 1994) and as a sample length-dependent rate, in content-free, mouse driven computer tasks differentiated borderline from obsessive-compulsive personality patterns (Selz and Mandell, 1997). Computation of lexical complexity is a good example of this approach. This procedure recursively surveys the sequence of symbols for the longest word, where “words” are subsequences that appear at least three times if they contain two letters or at least twice if they contain more than two letters. Upon finding a longest repeated word, the compression algorithm replaces all occurances of this word with a single distinct (new) symbol and looks again for the longest repeated word in the modified sequence. When the sequence cannot be further recursively compressed, there may remain identical adjacent symbols in the sequence. These are coded as the symbol raised to the power of the number of its adjacent occurances. This exponent cannot exceed five because six adjacent identical symbols would be two occurances of a three letter word. The numerical value of the lexical complexity is simply the sum of the number of distinct symbols and the (sum of the) logarithm of the exponents of the symbol sequences (Ebling and Jimenez-Montano, 1980). A clear account of algorithmic and lexical complexity in relationship to other measures of “complexity” in the context of brain relevant research data can be found in Rapp and Schmah (1996). The relationship between thermodynamic and ergodic, measure theories in relationship to forced-dissipative dynamics and the 244 HOUSE_OVERSIGHT_013744

role of self-intersection on manifolds in this new source of irreversibility (with a resulting “arrow of time”) is developed in Mackey (1992). As noted, the skeleton which configures attractors is composed of unstable, “saddle” fixed points, each of which attract (iron down) the trajectory along one dimension and repel or spread it out along another. Systems fulfilling the criteria for a chaotic dynamical system have the property of a countably infinite number of unstable periodic orbits composed of these unstable fixed points. Depending upon parameters, the orbital points can pull up their tails to be discrete with respect to each other or spread along the unstable direction to connect smoothly with others along a curve such as a saddle cycle. Parametric control of the strengths and structures of the saddle point skeleton of typical attractors can be used to change both the rate of generation of novel symbols as well as recurrances to old ones in the symbolic dynamics generating a brain dynamical system’s lexagraphic products (Bowen, 1978; Alexeev and Jacobson, 1981)). Using a variety of techniques to algorithmically register “return times,” experimental condition-sensitive “saddle orbits” composing unstable periodic orbits have been demonstrated in geometric reconstructions of real data series generated by a 40+ component chemical reaction (Lathrop and Kostelich, 1989), in response to natural stimuli in the time dependent behavior of the crayfish caudal photoreceptor (Pei and Moss, 1996) and in the interburst interval sequences recorded in hippocampal slices of the rat (So et al, 1997; So et al, 1998). If the reader uses the software listed above to simulate the time evolution of one of these attractors of abstract or real systems , she will learn that a remarkably small number of points, a very short time sample, will outline, “shadow” (Bowen, 1978), the complete array of unstable fixed points before filling in the attractor. It is tempting to speculate about the potential nervous system relevance of this dynamical anticipation of the attractor’s recognizable geometry, as well as a precis of what the symbolic dynamics are going to say occurs many time steps before filling in the attractor and its asymptotic message. Values of the measures made on the early unstable periodic orbit arrays such ht, hu and 4(+), resemble very closely those 245 HOUSE_OVERSIGHT_013745

made on their attractors when they were much more densely filled (Lathrop and Kostelich , 1989). Bowen’s “shadow lemma” in support of a thin film of points over the skelton of unstable fixed points of attractors is the fundamental reason that short sample length time series can often discriminate between control and experimental conditions in brain research studies. Another recently implemented entropy, called “approximate entropy,” is exploiting the underlying unstable fixed point skeletal shadowing principle in expansive dynamical systems to find statistically significant differences between control and experimental results in reasonably short, physiologically realistic, sample lengths (Pincus, 1991; Pincus et al, 1991). This algorithm is somewhat derivative of those involved in the computation of the correlation dimension (see above). Instead of computing across a range (and taking the limits) of embedding dimensions, d, and sequential paired-vectorial distances, ¢«, it empirically tailors and fixes them to compute a “logarithmic likelihood” that points remains close through incremental change in the time series. The “approximate entropy” is not easily relatable to either hy and hy. One is tempted to predict that this geometrically oriented algorithm might be fooled into a postive entropy diagnosis if applied to strange, nonchaotic dynamical systems with fractal dimension but no 4(+) -related mixing. Since sequence position is conserved in this computation, two simultaneously studied (“multiparameter”) systems can be examined for their mutual coherence as the “cross approximate entropy.” Among the interesting findings from applications of this index to neuroendocrine studies are an increase in approximate entropy in LH and FSH secretory patterns with age in both sexes, perhaps quantitatively heralding menopause (Pincus and Minkin, 1998) and decreased cross approximate entropy, a decrease in regulatory coupling between ACTH and cortisol secretion patterns in patients with Cushing’s syndrome (Roelfsema et al, 1998). Among the many of other empirically derived entropies, one is called “power spectral entropy,” which is equivalent to the normalized variance of the distribution of frequencies in a power spectral transformation of a time series (Farmer et al, 1980). This has been successfully applied to brain enzyme and receptor fluctuations 246 HOUSE_OVERSIGHT_013746

(Russo and Mandell, 1984a; Mandell, 1984), and, more recently, to multiple simultaneously EEG leads which demonstrated focal increases in epileptic patients (Inouye et al, 1991; 1992). An entropy derived from the quantification of the failures in temporal forecasting of EEG signals increased in the fronto-temporal region with drug treatment in patients with Alzheimer’s syndrome (Pezard et al, 1998). With respect to their implications for the clinical neurosciences, changes in dynamical entropy in behavior of brain dynamical systems has been regarded in two general ways: (1) Since representation of information requires the resolution of relevant ambiguity, a nonrelevant and global reduction in the dynamical entropy of a brain system (Stage IV sleep EEG slow waves, neuronal fixed point or regularly periodic activity, extrapyramidal motor tremor, fixed paranoid or obsessional mentation, the actions of some anxiolytics and antipsychotics ) reduces its potential for information encoding and transport. In contrast, “arousal” induced increases in the measures of entropy in brain wave and neuronal discharge patterns (pre-task warning signals, motivating conditions, stimulant drugs) are associated with improved psychophysical receptive and discrimination functions, learning rates and memory. (2) Regarding as potentially pathophysiological both of the two extremes of entropy generation, fixed point and periodic behavior as the lowest and fair coin flipping, “Bernoulli” randomness as the highest, another descriptor, “complexity” is defined as maximal (optimal) midway through the entropy range, making a new kind of parabolic entropy curve (Bennett, 1986; Crutchfield and Young, 1989a). In analogy with an optimal amalgam of periodic rotations and coin flips, in higher dimension, the most meaningful maximum complexity of real, nonuniformly expansive processes may derive from a multiplicity of measure invariants, symmetries, of the system such as the growth rate of unstable periodic orbits, divergence of the tail of a density distribution and specifiable linguistic variables such as word length and redundancy. The more symmetries, the more potential for complicated information encoding and transport with the maximum complexity located midrange in each one. We have pursued the hypothesis that entropy is a conserved property in the healthy brain and that complementarity in other statistical measure mechanisms make that possible. For example, in uniformly expansive, 247 HOUSE_OVERSIGHT_013747

idealized systems, topological entropy has been proven be equivalent to the product of an index of expansion and the dimension of the support such that an increase in expansiveness , 4(+), is compensated by a decrease in Do leaving hr invariant (Manning,1981). This relationship has also been found in the behavior of some nonuniformly expansive neuroendocrine, neuronal and human behavioral systems (Mandell and Selz, 1995; Smotherman et al, 1996; Mandell and Selz, 1997a;). Is Randomness Versus Determinism a Productive Question for the Biological Sciences? Are There Better Ones? Measures made on realistically nonuniformly expansive behavior of dynamical systems emerging from nonlinear differential equations and that arising from a variety of non-classical random walk models overlap such that making what may be more a metaphysical discrimination at this point is labor intensive, contentious and unproductive for generating new experimental work in the neurosciences. It is important to note that random walk theory and computation has matured to such an extent that almost any “nonlinear dynamical behavior” can, with respect to statistical measure, be modeled using one of many varieties. For examples, power law distributions in continuous time random walks (times of movement are also randomly chosen) , random walks with traps (temporarily immobilizing the trajectory like unstable fixed points), random walks in random environments, time of passage of ants in a labyrinth and Levy leaps and local diffusive exploration (looking for a wallet) among many others can represent much of the irregular behavior we observe in the brain (Shlesinger et al, 1982; Montroll and Shlesinger, 1984; Hughes, 1995; Klafter et al, 1996). On the other hand, (Markoff) partition of the sequence and a probabilistic style of analysis of nonlinear dynamical systems has been a major strategy for description and quantification from the field’s beginnings (Parry, 1964; Adler and Weiss, 1967; Bowen, 1970; Lasota and Yorke, 1973). The issue of randomness versus determinism remains current although many if not most properties of deterministic dynamical systems can 248 HOUSE_OVERSIGHT_013748

be simulated with a suitably constructed random process and all of our random number generators are deterministic. This theoretical blind alley is reminiscent of the decades lost partialing out causal attributes of nature versus nurture before knowledge of dynamical influences on nucleotide dynamics was available. It is perhaps unfortunate that for finite length real data, “house keeping requirements” (Ruelle, 1990; Rapp, 1993;1994) and “warnings on the label” with various random sequence, random phase controls (“surrogate data”) have become so intimidating to those of us in the early stages of exploring the use of these theories and methods in the brain sciences. Currently the “controls” are more relevant to abstract statistical processes and what can be said about them rather than generating and addressing new claims and the controls for them related to quantitatively oriented, experimental brain physiology. Statistical caveats have arisen to retard the emergence of potentially important and robust neurophysiologically-relevant phenomena. For example, a recent well conducted and analyzed study of the influence of low doses of ethanol in 32 normal male subjects, which honored almost all of the current analytic rituals including sequence and phase randomized surrogate data and searches for the continuity features of deterministic dynamical systems such as time asymmetry, concluded that the drug “reduced the evidence for nonlinear dynamical structure” in the brain (Ehlers et al, 1998). Though honoring the currently popular statistical rituals, what appears to be missing here are suggestions for new neurobiological or mathematical intuitions that will lead to the design of the next experiment. We now see that it is now possible to use these new ideas and methods to ask and at least partially answer more specific questions relevant to the clinically oriented neurosciences such as: whether increases in lithium-induced expansiveness and mixing in the dynamics of brain enzymes, neurons and behavior help explicate a mechanism of de-coherence in bipolar disease (Mandell et al, 1985); do these approaches to membrane conductance fluctuations suggest a new way to think about ion channel dynamics (Liebovitch, 1990); can alcohol-induced changes in statistical dynamics of the EEG predict genetic predilection in males to 249 HOUSE_OVERSIGHT_013749

alcoholism (Ehlers et al, 1995); do these approaches suggest a new neural dynamical mechanism for the actions of anticonvulsant drugs (Zimmerman et al, 1991); can these measures made on non-verbal, psychomotor tasks yield a non- intrusive measure of personality and character (Selz, 1992); can these approaches to deviant patterns of psychomotor sequencing in schizophrenics give us some insight into potential (cerebeller-basal ganglia?) mechanisms of the thought disorder in schizophrenia (Paulus et al, 1994); does cocaine induce new patterns of behavior that conserve pre-treatment entropy in developing animals (Smotherman et al, 1996); will these quantities applied to objective gait observables supply early diagnoses and quantification of clinical course in patients with extra-pyramidal disorders or taking anti-psychotic medication (Hausdorff et al, 1998); can these transformations of time series on the EEG give us an early diagnostic approach to Alzheimer’s disease (Jeong et al, 1998) or a new acute preventive pharmacological approach to patients with psychomotor and partial seizures (lasemidis et al, 1990). To end where we began: We think that if neuroscientists “did their own” nonlinear dynamical theory and analysis, shaped and tailored by intuitions growing out of their own experimental work and thinking, abstract and philosophical questions about what is determinism and what is random would retreat in favor of new specific ideas and experiments about brain dynamical mechanisms and their pathophysiology. From the studies reviewed here, it appears that a robust move in this direction in the brain sciences is well underway. 250 HOUSE_OVERSIGHT_013750

References Aasen,T.,Kugiumtzis,D.,Nordahl,G. (1997) Procedure for estimating the correlation dimension of optokinetic nystagmus signals. Comput Biomed Res. 30:95-116 Accardo, A., Mumolo, E.(1998): An algorithm for the automatic differentiation between the speech of normals and patients with Friedreich's ataxia based on the short-time fractal dimension. Comput Biol. Med. 28:75-89 Adler, R.J., Feldman, R.E., Taqqu, M. (1998) A Practical Guide to Heavy Tails; Statistical Techniques and Applications. Birkhauser. Boston Adler, R.L., Goodwyn, L.W., Weiss, B. (1977) Equivalence of topological Markov shifts. Isreal J. Math. 27:49-63 Adler, R.L., Konheim, A.G., McAndrew, M.H. (1964) Topological entropy. Trans. Amer. Math. Soc. 114:309-319 Adler, R.L., Marcus, B. (1979) Topological entropy and equivalence of dynamical systems. Mem. Amer. Math. Soc. 219: Adler, R.L., Weiss, B. (1967) Entropy, a complete metric invariant for authomorphisms of the torus. Proc. Natl. Acad. Sci. 57:1573-1576 Aftanas, L.|.,Lotova, N.V., Koshkarov, V.I., Pokrosvskaja, V.L., Popov, S.A., Makhnev, V.P (1997) Non-linear analysis of emotion EEG: calculation of Kolmogorov entropy and the principal Lyapunov exponent. Neurosci Lett 226:13-16 251 HOUSE_OVERSIGHT_013751

Aihara, K., Matsumoto, G., Ikegaya, Y. (1984): Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator. J. Theor. Biol. 140:27-38. Aihara, K., Numajiri, T., Matsumoto, G., Kotani, M. (1986): Structures of attractors in periodically forced neural oscillators. Phys. Lett. A116:313-317 Akay, M., Mulder, E.J. (1998) Effects of maternal alcohol intake on fractal properties in human fetal breathing dynamics. IEEE Trans Biomed Eng 45:1097-1103 Alexeev, V.M., Jacobson, M.V. (1981) Symbolic dynamics and hyperbolic dynamical systems. Phys. Rep. 75:287-325. Alonso, A., Faure, M.-P., Beaudet, A. (1994): Neurotensin promotes oscillatory bursting behavior and is internalized in basal forebrain cholinergic neurons. J. Neurosci. 14:5778-5792. Anderson, C.A., Mandell, A.J., Selz, K.A., Terry, L.M., Smotherman, W.P., Wong, C.H., Robinson, S.R., Robertson, S.S., Nathanielsz, P.W. (1998) The development of nuchal atonia associated with active (REM) sleep in fetal sheep: presence of recurrent fractal organization. Brain Res. 787:351-357 Arnold, V.|. (1984) Catastrophe Theory. Springer-Verlag. N.Y. Arnold, V.|., Avez, A. (1968) Ergodic Problems in Classical Mechanics. Addison- Wesley. Redwood City, CA Babloyantz, A. (1986) Evidence of chaotic dynamics of brain activity during the sleep cycle. In (ed. Mayer-Kress, G) Dimensions and Entropies in Chaotic Systems. Springer-Verlag, 1986 252 HOUSE_OVERSIGHT_013752

Babloyantz, A., Destexhe, A. (1986) Low dimensional chaos in an instance of epilepsy. Proc. Natl. Acad. Sci. 83:3513-3517. Baker, G.L., Gollub, J.P. (1991) Chaotic Dynamics; An Introduction. Cambridge University Press. Cambridge Bassingthwaighte, J., Liebovitch, L., West, B. (1994) Fractal Physiology. Oxford University Press. Oxford. Bayne,W.F., Hwang, S.S. (1985) Effect of nonlinear protein binding on equilibration times for different initial conditions. J Pharm Sci 74:120-123 Bazhenov, M., Timofeev, |., Steriade, M., Sejnowski, T.J. (1998) Computational models of thalamocortical augmenting responses. J. Neurosci. 18:6444-6465 Bennett, C.H. (1982) Thermodynamics of computation; A review. Intl. J. Theor. Phys. 21:905-946 Bennett, C.H. (1986) On the nature and origin of complexity in discrete, homogeneous, locally-interacting systems. Found. Phys. 16:585-604 Beard, D.A., Bassingthwaighte, J.B. (1998) Power-law kinetics of tracer washout from physiological systems. Ann Biomed Eng 26:775-779 Berge’,P., Pomeau, Y., Vidal, C. (1984) Order Within Chaos. Wiley-Interscience. N.Y. Berlin, Yu. A., Miller, J.R., Plonka, A. (1996) Rate Processes with Kinetic Parameters Distributed over Time and Space. Chem. Physics 212: Special Issue 253 HOUSE_OVERSIGHT_013753

Berns, G.S., Sejnowski, T.J. (1998) A computational model of how the basal ganglia produce sequences. J. Cogn. Neurosci. 10:108-121 Birkhoff, G.D. (1922): Surface transformations and their dynamical applications. Acta Math. 43:1-119 Boiteux, A., Goldbeter, A., Hess, B. (1975): Control of oscillating glycolysis of yeast by stochastic, periodic, and steady source of substrate: a model and experimental study. Proc Natl Acad Sci. 72:3829-33 Boltzmann, L. (1909) Wissenschaftliche Abhandlungen (ed. Hassenohrl, F.) Barth, Liepzig Bores, P.A., Bertelli, G.M., Gilli, G. (1988) Temperature dependence of the binding of mu, delta, and kappa agonists to the opirate receptors in guinea-pig brain. Eur. J. Pharmacol. 146:247-252 Bowen, R. (1970) Markoff partitions and minimal sets of Axiom A diffeomorphisms Amer. J. Math. 92:907-918 Bowen, R. (1975): Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lect. Notes. Math. 35 Bowen, R. (1978) On Axiom A Diffeomorphisms. CBMS Regional Conference Series in Mathematics. Vol 35 A.M.S. Providence, RI Bressler, S.L. (1995): Large-scale cortical networks and cognition. Brain. Res. Rev. 20:288-304 254 HOUSE_OVERSIGHT_013754

Briggs, K. (1990) An improved method for estimating Lyapounov exponents of chaotic time series. Phys. Lett. A151:27-32 Brodskii,V.(1998) The nature of the circahoralian (ultradian) intracellular rhythms. Similarity to fractals Izv Akad Nauk Ser Biol 3:316-29 Broomhead, D.S., King, G.P. (1986) Extracting qualitative dynamics from expeirmental data. Physica D20:217-236. Brown, R., Bryant, P., Abarbanel, H.D. (1991) Computing the Lyapounov spectrum of a dynamical system from an observed time series. Phys. Rev. A43:2787-2806 Bryant, P., Brown, R., Abarbanel, H.D. |. (1990) Lyapounov exponents from observed time series. Phys. Rev. Lett. 65:1523-1526 Bullard, W.P., Guthrie, P.B., Russo, P.V., Mandell, A.J. (1978) Regional and subcellular distribution and some factors in the regulation of reduced pterins in rat brain. J. Pharmacol. Exp. Therap. 206:4-20 Buzug, Th., Reimers, T., Pfister, G. (1990) Optimal reconstruction of strange- attractors from purely geometrical arguments. Europhys. Lett. 13:605-610 Callahan, J., Sashin, J.|. (1987) Models of affect-response and anorexia nervosa. Ann. N.Y. Acad. Sci. 504:241-259 Canavier, C.C., Clark, J.W., Byrne, J.H. (1990): Routes to chaos in a bursting neuron. Biophys J. 57:1245-1251 Carpenter, G.A. (1979): Bursting phenomena in excitable membranes. SIAM J. Appl. Math. 36:334-352 255 HOUSE_OVERSIGHT_013755

Casdagli, M. (1991) Chaos and deterministic versus stochastic nonlinear modeling. J. Royal. Stat. Soc. 54B:303-328 Casdagli, M.C., lasemidis, L.D., Savit, R.S., Gilmore, R.L., Roper, S.N., Sacklellares, J.C. (1997) Non-linearity in invasive EEG recordings from patients with temporal lobe epilepsy. Electroenceph. Clin. Neurophys. 102:98-105 Cartwright, M.I., Littlewood, J. (1945): On nonlinear differential equations of the second order. J. London. Math. Soc. 20: 180-189 Celletti, A., Villa, A.E.(1996) Low-dimensional chaotic attractors in the rat brain. Biol Cybern 74:387-393 Cerf, R., Sefrioui, M., Toussaint, M., Luthringer, R., Macher, J.P. (1996) Low- dimensional dynamic self-organization in delta-sleep: effect of partial sleep deprivation Biol Cybern 74:395-403 Chaitin, G.J. (1974) Information theoretical computational complexity. IEEE Trans. Inform. Theory 1T20:10-15. Chay, T.R., Rinzel, J. (1985) Bursting, beating and chaos in an excitable membrane model. Biophys. J. 47:357-366 Chen, Y-Q., Ku, Y-H. (1992) Properties of favored patterns in spontaneous spike trains and responses of favored patterns to electroacupuncture in evoked trains. Brain Res. 578:297-304 Clausius, R. (1897) The Mechanical Theory of Heat. MacMillan London 256 HOUSE_OVERSIGHT_013756

Coffman, K., McCormick, W.D., Swinney, H.L. (1986) Multiplicity in a chemical reactions with one-dimensional dynamics. Phys. Rev. Lett. 56:999-1002 Contreras, D., Dextexhe, A., Sejnowski, T.J., Steriade, M. (1997) Spatiotemporal patterns of spindle oscillations in cortex and thalamus. J. Neurosci. 17:1179-1196 Cooper, N.G. (1987): From Cardinals to Chaos, Cambridge University Press, Cambridge. Cornfield, |.P., Fomin, S.V., Sinai, Ya.G. (1982) Ergodic Theory. Springer-Verlag. Berlin Crevier,D.W., Meister,M (1998): Synchronous period-doubling in flicker vision of salamander and man. J Neurophysiol. 79:1869-1878 Crutchfield, J.P., Young, K. (1989a) Inferring statistical complexity. Phys. Rev. Lett. 63:105-108 Crutchfield, J.P., Young, K. (1989b) Computation at the onset of chaos. In (ed. Zurek, W.H.) Complexity, Entropy and the Physics of Information. Addison-Wesley. Redwood City, CA.pp 223-269. Cvitanovic’, P. (1989): Universality in Chaos. Adam-Hilger, Bristol. Dayhoff, J.E. (1984) Distinguished words in data seqauences; analysis and applications to neural coding and other fields. Bull. Math. Biol. 46:529-543 Dayhoff, J.E., Gerstein, G.L. (1983a) Favored patterns in spike trains. |. Detection. J. Neurophysiol. 49: 1334-1346 257 HOUSE_OVERSIGHT_013757

Dayhoff, J.E., Gerstein, G.L. (1983b) Favored patterns in spike trains. II. Applications. J. Neurophysiol. 49:1347-1363 DeBrouwer, S. Edwards, D.H.,Griffith, T.M (1998): Simplification of the quasiperiodic route to chaos in agonist-induced vasomotion by iterative circle maps.Am J Physiol. 274:H1315-26 de Gennes, P-G. (1979) Scaling Concepts in Polymer Physics. Cornell University Press, Ithica, N.Y. Devaney, R.L. (1989): An Introduction to Chaotic Dynamical Systems, 2" ed. Addison-Wesley, Redwood City, CA. Dewey, T.G. (1997) Fractals in Molecular Biophysics Oxford University Press, Oxford Ding, M., Grebogi, C., Ott, E. (1989) Dimensions of strange nonchaotic attractors. Phys. Lett. A137:167- 172 Ding, M., Grebogi, C., Ott, E., Sauer, T., Yorke, J. (1993) Plateau onset for correlation dimension: When does it occur. Phys. Rev. Lett. 70:3872-3875. Duke, D.W. and Pritchard, W.S. (eds) (1991)Measuring Chaos in the Human Brain. World Scientific. Singapore. Dvorak, |. and Holden, A. (eds) (1991) Mathematical Approaches to Brain Functioning Diagnostics. Manchester University Press. Manchester Ebling, W., Jimenez-Montano, M.A. (1980) On grammars, complexity and information measures of biological molecules. Math. Biosci 52:53-71 258 HOUSE_OVERSIGHT_013758

Eckmann, J.-P. (1981): Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53:643-654 Eckmann, J., Ruelle, D. (1985) Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57:617-656 Eckmann, J.P., Ruelle, D. (1992) Fundamental limitations for estimating dimensions and Lyapounov exponents in dynamical systems. Physica 56D:185-187 Eckmann, J.-P., Kamphorst, S.O., Ruelle, D., Cilberto, S. (1986) Lyapounov exponents from time series. Phys. Rev. A34:4971-4979 Edwards, D.H., Griffith, T.M. (1997) Entrained ion transport systems generate the membrane component of chaotic agonist-induced vasomotion. Am J Physiol 273:H909-920 Efremova, T.M., Kulikov, M.A.(1997) The chaotic component of the high-frequency EEG of the rabbit cerebral cortex in the formation of a defensive conditioned reflex. Zh Vyssh Nerv Deiat Im | P Pavlova 47:858-869 Elger,C., Lehnertz, K. (1998) Seizure prediction by non-linear time series analysis of brain electrical activity. Eur J Neurosci.10:786-789 Enns, R.H., McGuire, G.C. (1997) Nonlinear Physics with Maple for Scientists and Engineers. Birkhauser. Boston Ehlers, C.L. (1995) Chaos and complexity. Arch. Gen. Psychiat. 52:960-964 Ehlers, C.L., Havstad, J.W., Garfinkel, A., Kupfer, D.J. (1991) Nonlinear analysis of EEG sleep states. Neuropsychopharmacology 5:167-176 259 HOUSE_OVERSIGHT_013759

Ehlers, C.L., Havstad, J., Prichard, D., Theiler, J. (1998) Low doses of ethanol reduce evidence for nonlinear structure in brain activity. J Neurosci 18:7474-86 Ehlers, C., Havstad, J.W., Schuckit, M.A. (1995) EEG dimension in sons of alcoholics. Alcohol Clin. Exp Res 19:992-998 Elbert, T., Lutzenberger, W., Rockstroh, B., Berg, P., Cohen, R.(1992) Physical aspects of the EEG in schizophrenics. Biol Psychiatry 32:595-606 Erdos, P., Renyi, A. (1970). On a new law of large numbers. J. Anal. Math. 22:103- 111 Farmer, D., Crutchfield, J., Froehling, H., Packard, N., Shaw, R. (1980) Power spectra and mixing properties of strange attractors. Ann. N.Y. Acad. Sci. 357:453- 472 Farmer, J.D., Ott, E., Yorke, J.A. (1983) The dimension of chaotic attractors. Physica D7: 153-180 Feder, J. (1988) Fractals. Plenum. N.Y. Feigenbaum, M.J. (1979) The universal metric properties of nonlinear transformations, J. Stat. Phys. 21:669-702 Fell, J.,Roschke, J., Beckmann, P. (1993) Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the human electroencephalogram during sleep. Biol Cybern 69:139-46 Fell, J., Roschke, J., Schaffner, C. (1996) Surrogate data analysis of sleep electroencephalograms reveals evidence for nonlinearity. Biol Cybern 75:85-92 260 HOUSE_OVERSIGHT_013760

Feller, W. (1968) An Introduction to Probability Theory and Its Applications. Wiley. NY. Vol. |., Chapter 3 Fermi, E., Pasta, J., Ulam, S. (1955): Studies of nonlinear problems. Los Alamos Scientific Laboratory Report # 1940 Fessard, A., Gerard, R.,W. , Konorski, J. (1961) (eds) Brain Mechanisms and Learning. Blackwell, Oxford. Flory, P. (1971) Principles of Polymer Physics. Cornell University Press, Ithica, N.Y. Frank, G.W., Lookman, T., Nerenberg, M.A.H., Essex, C., Lemieux, J., Blume, W. (1990) Chaotic time series of epileptic seizures. Physica D46:427-438 Friedrich, R., Fuchs, A., Haken, H. (1991): Spatio-temporal EEG patterns In (ed. H.Haken and H.P. Kopchen) Synergetics of Rhythms. Springer. Berlin Gallez, D.,Babloyantz, A. (1991) Predictability of human EEG: a dynamical approach. Biol Cybern 64:381-391 Ganz,R.E., Faustman, P.M. (1996) Functional coupling between the central- autonomic and the cognitive systems is enhanced in states after optic neuritis and early multiple sclerosis. Int J Neurosci, 86:87-93 Geist, K., Parlitz, U., Lauterborn, W. (1990) Comparison of different methods for computing Lyapunov exponents. Prog. Theor. Physics 83:875-893 Gershenfeld, N.A. (1992) Dimension measurement on high-dimensional systems Physica D55: 135-154 261 HOUSE_OVERSIGHT_013761

Gerstein, G.L., Mandelbrot, B.B. (1964) Random walk models for the spike activity of a single neuron. Biophys. J. 4:41-68. Gibbs, J.W. (1902) Elementery Principles in Statistical Mechanics, Yale University Press, New Haven. Gilmore, R. (1981): Catastrophe Theory for Scientists and Engineers. Wiley. N.Y. Glass, L., Mackey, M. (1988) From Clocks to Chaos; The Rhythms of Life. Princeton University Press. Princeton. Goldberger, A.L., Rigney, D.R., West, B.J. (1990) Chaos and fractals in human physiology. Scientific American 262:42-49 Goldberger, A.L. (1996) Non-linear dynamics for clinicians: chaos theory, fractals and complexity at the bedside. Lancet 347:1312-1314 Gong, Y., Xu, J., Ren, W., Hu, S., Wang, F.(1998) Determining the degree of chaos from analysis of ISI time series in the nervous system: a comparison between correlation dimension and nonlinear forecasting methods. Biol Cybern 78:159-65 Gottschalk, A., Bauer, M.S., Whybrow, P.C. (1995) Evidence of chaotic mood variation in bipolar disorder. Arch. Gen. Psychiat. 52:947-959 Grebogi, C., Ott, E., Pellikan, S., Yorke, J.A. (1984) Strange attractors that are not chaotic. Physica 13D:261-268 Grassberger, P. (1981) On the Hausdorf dimension of fractal attractors. J. Stat. Phys. 26:173-179 262 HOUSE_OVERSIGHT_013762

Grassberger, P. (1983) Generalized dimensions of strange attractors. Phys. Lett. A97:227-230. Grassberger, P, Procaccia, |. (1983) Measuring the strangeness of strange attractors. Physica 9D:189-208 Gregson, R.A.,Britton, L.A., Campbell, E.A., Gates, G.R. (1990) Comparisons of the nonlinear dynamics of electroencephalograms under various task loading conditions: a preliminary report. Biol Psychol 31:173-191 Grimmett, G. (1989) Percolation. Springer-Verlag. Guckenheimer, J., Holmes, P. (1983): Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer. N.Y. Guillemin, R.C., Brazeau, P., Briskin, A., Mandell, A.J. (1983) Evidence for synergetic dynamics in a mammalian pituitary cell perfusion system in (ed. E.Basar, H. Flohr, H. Haken, A.J. Mandell) Synergetics of Brain. Springer-Verlag. pp 155-162 Gupta, V., Suryanarayanan, S., Reddy, N.P. (1997) Fractal analysis of surface EMG signals from the biceps. Int J Med Inf 45:185-192 Guzzetti, S.,Signorini,M.G., Cogliati, C., Messetti,S., Porta, A., Cerutti, S., Malliani, A. (1996) Non-linear dynamics and chaotic indices in heart rate variability of normal subjects and heart-transplanted patients. Cardiovasc Res 31:441-446 Hadamard, J. (1898) Les surfaces a courbures opposees et leur lignes geodesic. J. Math. Pures. Appl. 4:27-73 263 HOUSE_OVERSIGHT_013763

Hagerman, |.,Berglund, M., Lorin, M., Nowak, J., Sylven, C.(1996) Chaos-related deterministic regulation of heart rate variability in time- and frequency domains: effects of autonomic blockade and exercise. Cardiovasc 31:410-418 Haken, H. (1997) Visions of synergetics. Int. J. Bifurcat. Chaos. 7:1927-1951 Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procacccia, |., Shraiman, B.|. (1986) Fractal measures and their singularities: the generalization of strange sets. Phys. Rev A33:1141-1151 Hartman, M., Pincus, S., Johnson, M., Matthews, D., Faunt, L., Vance, M., Thorner, M., Veldhuis, J. (1994): Enhanced basal and disorderly growth hormone secretion distinguish acromegalic from normal pulsatile growth hormone release. J. Clin. Invest. 94:1277-1288. Hausdorff, J.M., Cudkowicz, M.E., Firtion, R., Wei, J.Y., Goldberger, A.L. (1998) Gait variability and basal gangia disorders: stride-to-stride variations in gait cycle timing in Parkinson’s and Huntington’s disease. Mov. Disord. 13:428-437 Hausdorff, J.M., Peng, C.-K., Ladin, Z., Wei, J.Y., Goldberger, A.L. (1995) Is walking a random walk? Evidence for long range correlations in the stride interval of human gait. J. Appl. Physiol. 78:349-358 Hausdorff. J.M., Purdon, P.L., Peng, C.-K., Ladin, Z., Wei, J.Y., Goldberger, A.L. (1996) Fractal dynamics of human gait stability of long-range correlations in stride interval fluctuations. J. Appl. Physiol. 80:1448-1457 Heffernan, M.S.(1996) Comparative effects of microcurrent stimulation on EEG spectrum and correlation dimension. Integr Physiol Behav Sci 31:202-209 264 HOUSE_OVERSIGHT_013764

Hentschel, H.G.E., Procaccia, |. (1983) The infinite number of generalized dimensions of fractals and strange attractors. Physica D8:435-444 Herman, M.L., Pincus, S.M., Johnson, M.L.,Matthews, D.L.., Faunt, , L.M.., Vance, M.L.., Thorner, M.O., Veldhuis, J.D. (1994) Enhanced basal and disorderly growth hormone secretion distinguish acromegalic from normal pulsatile growth hormone release. J. Clin. Invest. 94:1277-1288. Hitzemann, R., Murphy, M., Curell, J. (1985) Opiate receptor thermodynamics: agonist and antogonist binding. Eur. J. Pharmacol. 108:171-177 Hoyer, D.,Pompe, B., Herzel, H., Zwiener, U. (1998) Nonlinear coordination of cardiovascular autonomic control. IEEE Eng Med Biol Mag 17:17-21 Huber, M.T., Braun, H.A., Krief, J.C. (1999) Effects of noise on different states of recurrent affective disorder. Biol. Psychiat. In press. Huberman, B.A. (1987) A model for dysfunctions in smooth pursuit eye movement. Ann. N.Y. Acad. Sci. 504:260-273 Hughes, B.D. (1995) Random Walks and Random Environments. Clarendon. Oxford. Hurewicz, W., Wallman, H. (1948) Dimension Theory. Princeton University Press, Princeton lasemidis, L.D., Sackellares, J.C. (1996) Chaos theory and epilepsy The Neuroscientist. 2:118-128. lasemidis, J., Sacklellares, H., Zaveri,H., Williams, W.J. (1990): Phase space topolography and the Laypounov exponent of electrocorticograms in partial seizures. Brain Topolography 2:187-201. 265 HOUSE_OVERSIGHT_013765

lasemidis, LD., Sveri, H.P. Sackellares, J.C., William, W.J., Hood, T.W. (1988) Nonlinear dynamics of electrocorticographic data. J. Clin. Neurophysiol. 5:339 Inouye, T., Shinosaki, K., Sakamoto, H., Toi, S., Ukai, S., lyama, A., Katsuda, Y., Hirano, M. (1991) Quantification of EEG irregularity by use of the entropy of the power spectrum. Electroencephalogr Clin Neurophysiol 79:204-210 Inouye, T., Shinosaki, K., Toi, S., Ukai, S., lyama, A., Katsuda, Y., Hirano, M. (1992) Abnormality of background EEG determined by the entropy of power spectra in epileptic patients. Electroencephalogr Clin Neurophysiol 82:203-207 Jansen, B.H. (1996) Nonlinear dynamics and quantitative EEG analysis. Electroencephalogr Clin Neurophysiol Suppl 45:39-56 Jansen, B.H. and Brandt, M.E. (eds) 1993 Nonlinear Dynamical Analysis of the EEG. World Scientific. Singapore Jeong, J., Joung, M.K., Kim, S.Y. (1998) Quantification of emotion by nonlinear analysis of the chaotic dynamics of electroencephalograms during perception of 1/f music. Biol Cybern 78:217-25 Jeong,J., Kim, S.Y., Han, S.H. (1998) Non-linear dynamical analysis of the EEG in Alzheimer's disease with optimal embedding dimension. Electroencephalogr Clin Neurophysiol 106:220-228 Kaneko, K. (1983): Similarity structure and scaling property of the period adding phenomena. Prog. Theor. Phys. 69:403-414 Kaplan, D.T. (1994) Exceptional events as evidence of determinism. Physica D73:38-48 266 HOUSE_OVERSIGHT_013766

Kaplan, D.T., Glass, L. (1992) Direct test for determinism in a time series. Phys. Rev. Lett. 68:427-430 Katz, B. (1966) Nerve, Muscle and Synapse. McGraw-Hill. N.Y. Kelly,O.E., Johnson, D.H., Delgutte, B., Cariani, P. (1996)Fractal noise strength in auditory-nerve fiber recordings. J Acoust Soc Am 99:2210-20 Keunen, R.W.,Vliegen, J.H., Stam,C.J., Tavy, D.L.(1996) Nonlinear transcranial Doppler analysis demonstrates age-related changes of cerebral hemodynamics. Ultrasound Med Biol 22:383-390 Khinchin, A.|. (1957) Mathematical Foundations of Information Theory. Dover. N.Y. King, R., Barchus, J.D., Huberman, B.A. (1984) Chaotic behavior in dopamine neurodynamics. Proc. Natl. Acad. Sci. 81:1244-1247 Klafter, J., Shlesinger, M.F., Zumofen, G. (1996) Beyond brownian motion. Physics Today Feb. pp 33-39 Klemm, W.R. (1990) Historical and introductory perspectives on brain-stem mediated behaviors In (ed. Klemm, W.R., Vertes, R.P.) Wiley-Interscience, N.Y. Klemm, W.R., Sherry, C.J. (1981) Serial ordering in spike trains: what's it "trying to tell us"? Int J Neurosci 14:15-33 Knapp, S., Mandell, A.J. (1983) Scattering kinetics in a complex tryptophan hydroxylase preparation from rat brainstem raphe nuclei: statistical evidence that 267 HOUSE_OVERSIGHT_013767

the lithium-induced sigmoid velocity function reflects two states of available catalytic potential. J. Neural Trans. 58:169-182 Knapp, S., Mandell, A.J. (1984) TRH influences the pterin-cofactor and time- dependent instabilities of rate raphe trypophan hydroxylase activity assessed under far-from-equilibrium conditions. Neurochem. Internat. 6:801-812 Knapp, S., Mandell, A.J., Russo, P.V., Vitto, A., Stewart, K. (1981) Strain differences in kinetic and themal stability of two mouse strain tryptophan hydroxylase activities. Brain Res. 230:317-336 Koch, C.D.,Palovcik, R.A., Uthman, B.M., Principe, J.C. (1992) Chaotic activity during iron-induced "epileptiform" discharge in rat hippocampal slices. IEEE Trans Biomed Eng 39:1152-1160 Koch,H.P, Zajcek, H. (1991) Fractals also in pharmacokinetics? Pharmazie 46:870- 871 Kolmogorov, A.N. (1950) Foundations of Probability Theory. Chelsea, N.Y. Kolmogorov, A.N. (1957): General theory of dynamical systems and classical mechanics. In Proc. 1954 International Congress of Mathematics, North-Holland. Kolmogorov, A.N. (1958) A new metric invariant of transient dynamical systems and automorphisms in Lebesgue space. Dokl. Acad. Nauk. SSSR 119:861-864 Kolmogorov, A.N. (1959) Entropy per unit time as a metric invariant of automorphisms. Dokl. Acad. Nauk. SSSR 124:754-758 268 HOUSE_OVERSIGHT_013768

Korn, S.J., Horn, R. (1988) Statistical discrimination of fractal and Markov models of single-channel gating. Biophys J. 54:871-877 Korsch, H.J., Jodl, H. (1993) Chaos; A Program Collection for the PC. Springer- Verlag. Krebs-Thomson, K., Lehmann-Masten, V., Naiem, S., Paulus, M.P., Geyer, M.A. (1998a) Modulation of phencyclidine-induced changes in locomotor activity and patterns in rats by serotonin. Eur. J. Pharmacol. 343:135-143 Krebs-Thomson, K., Paulus, M.P., Geyer, M.A. (1998b) Effects of hallucinogens on locomotor and investigatory activity and patterns: influence of 5-HT2A and 5-HT2C receptors. Neuropsychophamracology. 18:339-351 Kowalik, Z.J., Elbert, T. (1995) A practical method for the measurement of the chaoticity of electric and magnetic brain activity. Int. J. Bifurcation. Chaos 5:475-490 Kumar, A.R., Johnson, D.H. (1993) Analyzing and modeling fractal intensity point processes. J Acoust Soc Am 93:3365-3373 Krystal, A.D., Weiner, R.D. (1991) The largest Lypounov exponent of the EEG in ECT seizures. In (ed. D.Duke and W. Pritchard) Measureing Chaos in the Human Brain. World Scientific. Singapore Lasota, A., Yorke, J. (1973) On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186:481-488 Lathrop, D.P., Kostelich, E.J. (1989) Characterization of an experimental strange attractor by periodic orbirts. Phys. Rev. A40:4028-4031 269 HOUSE_OVERSIGHT_013769

Levinson, N. (1949): A second order differential equation with singular solutions.. Ann. Math. 50:127-153 Li, T.Y., Yorke, J.A. (1975): Period three implies chaos. Amer. Math. Mon. 82:985- 992 Liebovitch, L.S. (1989) Analysis of fractal ion channel gating kinetics: kinetic rates, energy levels, and activation energies. Math Biosci. 93:97-115 Liebovitch, L.S. (1998) Fractals and Chaos; Simplified for the Life Sciences. Oxford University Press, Oxford. Liebovitch,L.S., Sullivan, J.M. (1987) Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons. Biophys J. 52:979- 88 Liebovitch, L.S., Todorov, A.T. Using fractals and nonlinear dynamics to determine the physical properties of ion channel proteins. (1996) Crit Rev Neurobiol 10:169-87 Liebovitch, L.S., Todorov, A.T. (1996) Invited editorial. J. Appl. Physiol. 80:1446- 1447 Liebert, W., Pawelzik, K., Schuster, H.G. (1991) Optimal embeddings of chaotic attractors from topological considerations. Europhys. Lett. 14:521-526 Lorenz, E.N. (1963): Deterministic nonperiodic flow. J. Atmos. Sci. 20:130-141 Lowen, S.B.,Teich, M.C.(1992) Auditory-nerve action potentials form a nonrenewal point process over short as well as long time scales. J Acoust Soc Am 92:803-6 270 HOUSE_OVERSIGHT_013770

Lutzenberger, W., Flor, H., Birbaumer, N. (1997) Enhanced dimensional complexity of the EEG during memory for personal pain in chronic pain patients. Neurosci Lett 226:167-170 Lutzenberger, W., Birbaumer,N., Flor, H., Rockstroh, B., Elbert, T. (1992) Dimensional analysis of the human EEG and intelligence Neurosci Lett 143:10-14 Macheras, P.,Argyrakis, P., Polymilis, C. (1996) Fractal geometry, fractal kinetics and chaos en route to biopharmaceutical sciences. Eur J Drug Metab Pharmacokinet 21:77-86 Mackey, M. (1992) Time’s Arrow: The Origins of Thermodynamic Behavior. Springer-Verlag. NY Mackey, M., Glass, L. (1977) Oscillation and chaos in physiological control systems. Science 197:287-289 Magoun, H.W. (1954) The ascending reticular formation and wakefulness. In (ed. E.D. Adrian, F. Bremer, H.H. Jasper. Brain Mechanisms and Consciousness. Blackwell. Oxford pp 1-20 Makarenko, V., Llinas, R.(1998) Experimentally determined chaotic phase synchronization in a neuronal system. Proc Natl Acad Sci 95:15747-15752 Mandelbrot, B.B. (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 155:636-638 Mandelbrot, B.B. (1974) Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62:331-358 271 HOUSE_OVERSIGHT_013771

Mandelbrot, B.B. (1975) Les Objets Fractals:Forme, Hasard et Dimension. Flammarion. Paris Mandelbrot, B.B. (1977) Fractals; Form, Chance and Dimension. Freeman. N.Y. Mandelbrot, B.B. (1977) Fractals: Form, Chance and Dimension, Freeman. San Francisco Mandell, A.J. (1983) From intermittency to transitivity in neuropsychobiological flows. Am. J. Physiol. 245: R484-R494 Mandell, A.J. (1984) Non-equilibrium behavior of some brain enzyme and receptor systems. Ann. Rev. Pharmacol. Toxicol. 24:237-274 Mandell, A.J. (1986) The hyperbolic helix hypothesis in (ed. L Pietronero, E. Tosatti) Fractals in Physics. Elsevier. N.Y. Mandell, A.J. (1987) Dynamical complexity and pathological order in the cardiac monitoring problem. Physica D27:235-242 Mandell, A.J., Kelso, J.A.S. (1991) Dissipative and statistical mechanics of amine neuron activity. In (eds. Ellison, J.A., Uberall, H.) Essays on Classical and Quantum Dynamics. Gordon Beach. Philadelphia pp 203-236 Mandell, A.J., Knapp, S., Ehlers, C., Russo, P.V. (1985) The stability of constrained randomness: lithium prophylaxis at several neurobiological levels. In (ed. R.M. Post, J. Ballenger). Neurobiology of Mood Disorders. Williams and Wilkins. Baltimore. pp744-776 272 HOUSE_OVERSIGHT_013772

Mandell, A.J., Russo, P.V. (1981): Striatal tyrosine hydroxylase:multiple conformational kinetic oscillators and product concentration frequencies. J. Neuroscience 1:380-389 Mandell, A.J., Russo, P.V., Knapp, S. (1982) Strange stability in hierarchically coupled neuropsychobiological systems. In (ed. Haken, H.) Evolution of Order and Chaos. Springer-Verlag. Berlin. pp 270-286 Mandell, A.J., Selz, K.A. (1991) Period adding, hierarchical protein modes and electroencephalographically defined states of consciousness. In (eds. Vohra, S., Spano, M., Shlesinger, M.F., Pecora, L., Ditto, W. ) Proc. 42 Experimental Chaos Conference. World Scientific. Hong Kong Mandell, A.J., Selz, K.A. (1992) Dynamical systems in psychiatry: now what? Biol. Psychiat. 32:299-301. Mandell, A.J., Selz, K.A. (1993): Brain stem neuronal noise and neocortical “resonance.” J. Stat. Phys. 70:355-371 Mandell, A.J., Selz, K.A. (1994) Resonance, synchroniation and lexical redundancy in the expanding dynamics of brain stem neurons. SPIE Proc 2036:86-99 Mandell, A.J. and Selz, K.A. (1994) Resonance, synchronization and lexical redundancy in the expanding dynamics of brain stem neurons. Chaos in Medicine and Biology SPIE Proc: 2036:86-99 Mandell, A.J., Selz, K.A. (1995) Nonlinear dynamical patterns as personality theory for neurobiology and psychiatry. Psychiatry 58:371-390 273 HOUSE_OVERSIGHT_013773

Mandell, A.J., Selz, K.A. (1997a) Entropy conservation as hy. ~ 1d, in neurobiological dynamical systems. Chaos 7:67-81 Mandell, A.J. and Selz, K.A. (1997b) Is the EEG a strange attractor? Brainstem neuronal discharge patterns and electroencehphalographic rhytms in (ed. Grebogi, C., Yorke, J) The Impact of Chaos on Science and Society United Nations University Press, Tokyo, pp 64-96 Mandell, A.J., Selz, K.A., Shlesinger, M.F. (1991) Some comments on the weaving of contemporaneous minds: Resonant neuronal quasiperiodicity, period adding and O(2) symmetry in the EEG In (ed. Pritchard W., Duke, D.) Chaos in the Brain World Scientific Singapore pp 136-155 Manneville, P., Pomeau, Y. (1980): Different ways to turbulence in dissipative systems. Physica 1D:219-226 Mantegna, R.N. (1991) Levy walks and enhanced diffusion in Milan stock exchange. Physica A 179:232-242 Martinerie, J.,Adam, C., Le Van Quyen, M., Baulac, M., Clemenceau, S., Renault, B., Varela, F.J. (1998) Epileptic seizures can be anticipated by non-linear analysis Nat Med 4:1173-1176 May, R.M. (1976): Simple mathematical models with very complicated dynamics. Nature 261:459-467 Mayer-Kress, G. (1986) Dimensions and Entropies in Chaotic Systems. Springer- Verlag. N.Y. 274 HOUSE_OVERSIGHT_013774

McManus, O.B., Weiss, D.S., Blatz, A.L., Magleby, K.L. (1988) Fractal models are inadequate for the kinetics of four different ion channels. Biophys J 54:859-70 McMullen, C. T. (1994) Complex Dynamics and Renormalization. Princeton University Press. Princeton. McMurran, S., Tattersal, J. (1999): Mary Cartwright. Notices, AMS 46:214-220 Mestivier, D., Dabire, H., Safar, M., Chau, N.P. (1998) Use of nonlinear methods to assess effects of clonidine on blood pressure in spontaneously hypertensive rats. J Appl Physiol 84:1795-800 Metropolis, N., Stein, M.L., Stein, P.R. (1973): On limit sets for transfomrations on the unit interval . J. Combinatorial Theory 15:25-44 Meyer-Lindenberg, A., Bauer, U., Krieger, S., Lis, S., Vehmeyer, K., Schuler, G., Gallhofer, B. (1998) The topography of non-linear cortical dynamics at rest, in mental calculation and moving shape perception. Brain Topogr 10:291-9 Micheloyannis, S., Flitzanis, N., Papanikolaou, E., Bourkas, M., Terzakis, D., Arvanitis, S., Stam, C.J. (1998) Usefulness of non-linear EEG analysis Acta Neurol Scand 97:13-9 Milnor, J. (1985): On the concept of attractor. Commun. Math. Phys. 99:177-195 Milton, J.G., Longtin, A. (1990): Evaluation of pupil constriction and dilation from cycling measurements. Vision Res. 30:515-525 275 HOUSE_OVERSIGHT_013775

Misiurewicz, M. (1995) Thirty years after Sharkovskii’s theorem. Int. J. Bifurcation Chaos 5:1275-1281 Mogilevskii, A.Ya., Derzhiruk, L.P., Panchekha, A.P., Dershiruk, E.A.(1998) Adaptive regulation of the nonlinear dynamics of electrical activity of the brain. Neurosci Behav Physiol 28:366-375 Molnar, M., Gacs,G., Ujvari, G., Skinner, J.E., Karmos, G. (1997) Dimensional complexity of the EEG in subcortical stroke--a case study. Int J Psychophysiol. 25:193-199 Montroll, E.W., Badger, W.W. (1974) Introduction to Quantitative Aspects of Social Phenomena. Gordon & Breach. N.Y. Montroll, E.W., Shlesinger, M.F. (1984) On the wonderful world of random walks. In (ed. Liebowitz, J.L, Montroll, E.W.) Nonequilibrium Phenomena II: From Stohastics to Hydrodynamics. Elsevier. North-Holland. Amsterdam Moon, F. (1992): Chaotic and Fractal Dynamics; An Introduction for Applied Scientists and Engineers. Second Edition, Wiley, N.Y. Morinushi, T., Kawasaki, H., Masumoto, Y., Shigeta, K., Ogura, T., Takigawa, M. Examination of the diagnostic value and estimation of the chaos phenomenon in masticatory movement using fractal dimension in patients with temporomandibular dysfunction syndrome. (1998): J. Oral Rehabil.25:386-94 Morse, M. (1921) A one-to-one representation of geodesics on a surface of negative curvature. Amer. J. Math. 43:33-51 Morse, M. Hedlund, G.A. (1938) Symbolic dynamics. Amer. J. Math. 60:815-866. 276 HOUSE_OVERSIGHT_013776

Moruzzi, G. (1960) Synchronizing influences of the brain stem and the inhibitory mechanisms underlying the production of sleep by sensory stimulation. Electroencephal. Clin. Neurophysiol. Suppl. 13:231-256 Moruzzi, G., Magoun, R.W. (1949) Brain stem reticular formation and activation of the EEG. Electroencephal. Clin. Neurophysiol. 1:455-473 Musha, T. (1981) 1/f fluctuations in biological systems. In Sixth International Conference on Noise in Physical Systems. National Bureau of Standards #614 : 143-146 Nicolelis, M.A.L., Baccala, L.A., Lin, R.C.S., Chapin, J.K. (1995): Sensorimotor encoding by synchronous neural ensemble activity at multiple levels of the somatosensory system. Science 268:1353-1358 Nicolis, J.S. (1986) Chaotic dynamics applied to information processing. Rep. Prog. Phys. 49:1109-1196 Nieminen, H., Takala, E.P.(1996) Evidence of deterministic chaos in the myoelectric signal. Electromyogr Clin Neurophysiol 36:49-58 Nozaki,D., Nakazawa, K., Yamamoto, Y.(1996) Supraspinal effects on the fractal correlation in human H-reflex. Exp Brain Res 112:112-118 Nusse, H.E., Yorke, J.A. (1991). Dynamics: Numerical Explorations. Springer- Verlag. N.Y. 277 HOUSE_OVERSIGHT_013777

Ogihara,T., Tamai, |., Tsuji, A. (1998) Application of fractal kinetics for carrier- mediated transport of drugs across intestinal epithelial membrane. Pharm Res 15:620-625 Olsen, L.F., Degn, H. (1977): Chaos in an enzymel system. Nature 267:177-178 Ornstein, D.S. (1989) Ergodic theory, randomness and chaos. Science 243:182- 198. Ott, E. (1981): Strange attractors and chaotic motions of dynamical systems. Rev. Mod. Phys. 53:655-671 Ott, E., Sauer, T., Yorke, J.A. (1994) Coping with Chaos. Wiley-Interscience. N.Y. Packard, N., Crutchfield, J., Farmer, D., Shaw, R. (1980) Geometry from a time series. Phys. Rev. Lett. 45:712-716 Parker, T.S., Chua, L.O. (1989) Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag. N.Y. Parlitz, U. (1992) Identification of true and spurious Lyapounov exponents from time series. Int. J. Bifurcation and Chaos 2:155-165 Parry, W. (1964) Intrinsic Markoff chains . Trans. Amer. Math. Soc. 112:55-66 Paulus, M.P., Bakshi, V.P., Geyer, M.A. (1998) Isolation rearing affects sequential organization of motor behavior in post-pubertal but not pre-pubertal Lister and Sprague-Dawley rats. Behavior Brain Res. 94:271-280 278 HOUSE_OVERSIGHT_013778

Paulus, M.P., Gass, S.F., Mandell, A.J. (1989): A realistic , minimal “middle layer” for neural networks. Physica D40:135-155 Paulus, M.P., Geyer, M.A. (1992) The effects of MDMA and other methylenedioxy- substituted phenylalkylamines on the structure of rat locomotor activity. Neuropsychopharmacology 7:15-31 Paulus, M.P, Geyer,M.A., Braff, D.L. (1994) The assessesment of sequential response organization in schizophrenic and control subjects. Prog. Neuropsychopharm. & Biol. Psychiat. 18:1169-1185 Paulus, M.P., Geyer, M.A., Braff, D.L. (1996): The use of methods from chaos theory to quantify a fundamental dysfunction in the behavioral organization of schizophrenic patients. Am. J. Psychiat. 153:714-717 Paulus, M.P., Geyer, M.A., Gold, L.H., Mandell, A.J. (1990) Application of entropy measures derived from the ergodic theory of dynamical systems to rat locomotor behavior. Proc. Natl. Acad. Sci. 87:723-727 Paulus, M.P., Geyer, M.A., Mandell, A.J. (1991) Statistical mechanics of a neurobiological dynamical system: The spectrum of local entropies, s(a) applied to cocaine-perturbed behavior. Physica A174:567-577 Pei, X., Moss, F. (1996) Characterization of low-dimensional dynamics in the crayfish caudal photoreceptor. Nature 379:618-621 Peng, C.-K., Buldyrev, S.V., Goldberger, A.L., Havlin, S., Simons, M., Stanley, H.E. (1993) Finite size effects on long range correlations: implications for analyzing DNA sequences. Phys. Rev. E47:3730-3733 279 HOUSE_OVERSIGHT_013779

Peng, C.-K., Havlin, S., Stanley, H.E., Goldberger, A.L. (1995) Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 6:82-87 Perkel, D.H., Gerstein, G.L., Moore, G.P. (1967) Neuronal spike trains and stochastic point processes. Biophys. J. 7:419-440 Pezard, L., Martinerie, J., Varela, F.J., Bouchet, F., Guez, D., Derouesne, C., Renault, B. (1998) Entropy maps characterize drug effects on brain dynamics in Alzheimer’s disease Neurosci Lett 253:5-8 Pezard, L., Nandrino, J.L., Renault, B., Massioui, F.E., Allilaire, F., Muller, J. (1996) Depression as a dynamic disease. Biol. Psychiat. 39:991-999 Pijn, J., Velis, D., van der Heyden, M., DeGoede, J., van Veelen, C.W., Lopes da Silva, F. (1997): Nonlinear dynamics of epileptic seizures on basis of intracranial EEG recordings. Brain Topogr. 9:249-70 Pincus, S.M. (1991) Approximate entropy as a measure of a system’s complexity. Proc. Natl. Acad. Sci. 88:2297-2301 Pincus, S.M., Gladstone, |.M., Ehrenkranz, R.A. (1991) A regularity statistic for medical data analysis. J Clin Monit 7:335-245 Pincus, S.M.,Minkin, M.J. (1998) Assessing sequential irregularity of both endocrine and heart rate rhythms. Curr Opin Obstet Gynecol 10:281-291 Popivanov, D., Mineva, A., Dushanova, J. (1998) Tracking EEG signal dynamics during mental tasks. A combined linear/nonlinear approach. IEEE Eng Med Biol 17:89-95 280 HOUSE_OVERSIGHT_013780

Porta, A., Baselli, G., Montano, N., Gnecchi-Ruscone, T., Lombardi, F., Malliani, A., Cerutti, S. (1996): Classification of coupling patterns among spontaneous rhythms and ventilation in the sympathetic discharge of decerebrate cats. Biol Cybern. 75:163-172 Pradhan, N. Sadasivan, P.K. (1996) The nature of dominant Lyapunov exponent and attractor dimension curves of EEG in sleep. Comput Biol Med 26:419-428 Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. (1991) Numerical Recipies; The Art of Scientific Computing. Cambridge University Press, Cambridge. Pritchard, W.S. (1992) The brain in fractal time: 1/f-like power spectrum scaling of the human electroencephalogram. Int J Neurosci 66:119-129 Pritchard, W.S., Krieble, K.K., Duke, D.W. (1996) Application of dimension estimation and surrogate data to the time evolution of EEG topographic variables. Int. J. Psychophysiol. 24:189-195 Rapp, P. E. (1993) Chaos in the neurosciences: Cautionary tales from the frontier. The Biologist (London) 40:89-94. Rapp, P.E. (1994) A guide to dynamical analysis. Integrat. Physiol. Behavioral. Sci. 29:311-327 Rapp, P.E. (1995) Is there evidence for chaos in the human central nervous system. In (ed. R. Robertson and A. Combs) Chaos Theory in Psychology and the Life Sciences. Larence Erlbaum. Mahway, N.J. pp 89-100 Rapp, P. E., Bashore, T.R., Martinerie, J.M., Albano, A.M., Mees, A.I. (1989) Dynamics of brain electrical activity. Brain Topolography 2:99-118 281 HOUSE_OVERSIGHT_013781

Rapp, P.E., Goldberg, G., Albano, A.M., Janicki, M.B., Nurphy, D., Niemeyer, E., Jimenez-Montano, M.A. (1993) Using coarse-grained measures to characterize electromyographic signals. Int. J. Bifurcat. Chaos. 3:525-542. Rapp, P.E., Jimenez-Montano, M.A., Langs, R.J., Thomson, L. (1991) Quantitative characteristization of patient-therapist communication. Math. Biosci. 105:207-227 Rapp, P.E., Schmah, T. (1996) Complexity measures in molecular psychiatry. 1:408-416 Rapp, P.E., Zimmerman, |.D., Vining, E.P., Cohen, N., Albano, A.M., Jimenez- Montano, M.A. (1994) The algorithmic complexity of neural spike trains increases during focal seizures. J. Neurosci. 14:4731-4739 Ren, W., Hu. $.J., Zhang, B.J., Wang, F.Z. (1997): Period-adding bifurcation with chaos in the interspike intervals generated by an experimental neural pacemaker. Int. J. Bifurc. Chaos 7:1867-1872 Renyi, A. (1970) Probability Theory. North-Holland. Amsterdam Rinzel, J. (1985) Excitation dynamics: insights from simplified membrane models. Fed. Proc. 44:2944-2946 Rissanen, J. (1982) Estimation of structure by minimum desciption length. Circuit Systems Signal Processing 1:395-396 Roelfsema, F., Pincus, S.M., Veldhuis, J.D. Patients with Cushing's disease secrete adrenocorticotropin and cortisol jointly more asynchronously than healthy subjects. J Clin Endocrinol Metab 83:688-692 282 HOUSE_OVERSIGHT_013782

Romeiras, F.J., Bondeson, A., Ott, E., Antonsen, T.M., Grebogi, C. (19987) Quasioperiodically forced dyanmical systems with strange, nonchaotic attractors. Physica 26D:277-294. Roschke, J. (1992) Strange attractors, chaotic behavior and informational aspects of sleep EEG data. Neuropsychobiology 25:172-176 Roschke, J., Aldenhoff, J.B.(1992) A nonlinear approach to brain function: deterministic chaos and sleep EEG. Sleep 15:95-101 Roschke, J., Aldenhoff, J.B. (1993) Estimation of the dimensionality of sleep-EEG data in schizophrenics Eur Arch Psychiatry Clin Neurosci 242:191-196 Roschke, J., Fell, J., Mann, K. (1997) Non-linear dynamics of alpha and theta rhythm: correlation dimensions and Lyapunov exponents from healthy subject's spontaneous EEG Int J Psychophysiol 26:251-61 Réssler, O. (1976): An equation for continuous chaos. Phys. Lett. A57:397-398 Réssler, O.E. (1979) An equation for hyperchaos. Phys. Lett. 71A:155-157 Ruelle, D. (1978) What are measures describing turbulence. Prog. Theor. Phys. Supp. 64:339-345 Ruelle, D.(1979) Ergodic theory of differentiable dynamical systmes. Publ. Math. IHES 50:27-58 Ruelle, D. (1987) Chaotic Evolution and Strange Attractors. Cambridge University Press. Cambridge 283 HOUSE_OVERSIGHT_013783

Ruelle, D. (1990) Deterministic chaos: The science and fiction. Proc. Roy. Soc. Lond. 427A:241-248. Ruelle, D., Takens, F. (1971): On the nature of turbulence. Commun. Math. Phys. 20:167-192; 23:343-344 Russo, P.V., Mandell, A.J. (1984a) A kinetic scattering approach to the nonlinear instabilities of rat brain tyrosine hydroxylase preparations at several levels of tetrahydrobiopterin cofactor demonstrates evolutionary behavior characteristic of global dynamical systems. Brain Res. 299:313-322 Russo, P.V., Mandell, A.J. (1984b) Metrics from nonlinear dynamics adapted for characterizing the behavior of non-equilibrium enzymatic rate functions. Anal. Biochem. 139:91-99 Russo, P.V., Mandell, A.J. (1986) Sonication, calcium and peptides have systermatic effects on the nonlinear dynamics of tyrosine hydroxylase activity. Neurochem. Int. 9:171-176 Sadana,A.,(1998) Analyte-receptor binding kinetics for biosensor applications. An analysis of the influence of the fractal dimension on the binding rate coefficient. Appl Biochem Biotechnol 73:89-112 Saltzman, B. (1962): Finite amplitude free convection as an initial value problem. J. Atmos. Sci. 19:329-341 Sammer, G.(1996) Working-memory load and dimensional complexity of the EEG. Int J Psychophysiol 24:173-182 284 HOUSE_OVERSIGHT_013784

Sammon, M.P., Bruce, E.H. (1991) Vagal afferent activity increases dynamical dimension of respiration in rats. J Appl Physiol 70:1748-1762 Sano, M., Sawada, Y. (1985) Measurement of the Lyapounov spectrum from chaotic time series. Phys. Rev. Lett. 55:1082-1085 Sansom, M.S., Ball, F.G., Kerry, C.J., McGee, R., Ramsey, R.L., Usherwood, P.N. (1989) Markov, fractal, diffusion and related models of ion channel gating. A comparison with experimental data from two ion channels. Biophys J 56:1229-43 Sato, S., Sano, M., Sawada, Y. (1987) Practical methods of meaureing the generalized dimension and largest Lyapounov exponent in high dimensional chaotic systems. Prog. Theor. Phys. 77:1-5 Sauer, T., Yorke, J.A., Casdagli, M. (1991) Embedology. J. Stat. Phys. 65:579-616 Schiff, S.J., So, P., Chang, T., Burke, R.E., Sauer, T. (1997): Detecting generalized synchrony through mutual prediction in a spinal neural ensemble. Preprint Schierwagen, A.K. (1990) Growth, structure and dynamics of real neurons: model studies and experimental results Biomed Biochim Acta 49:709-722 Schmeisser, E.T.(1993) Fractal analysis of steady-state-flicker visual evoked potentials: feasibility. J Opt Soc Am 10:1637-1641 Schuster, H.G. (1989): Deterministic Chaos. Second Edition, VCH, Weinheim, Germany. Scott, A. (1990) The solitary wave: An enduring pulse first seen in water may convey energy in the cell. Science 30:28-35 285 HOUSE_OVERSIGHT_013785

Segundo, J.P., Sugihara, G., Dixon, P., Stiber, M.,Bersier, L.F.(1998) The spike trains of inhibited pacemaker neurons seen through the magnifying glass of nonlinear analyses. Neuroscience 87:741-66 Selz, K.A. (1992) Mixing Properties in Human Behavioral Style and Time Dependencies in Behavior Identification; The Modeling of a Universal Dynamical Law. Doctoral Disseration. UMI. N.Y. Selz, K.A., Mandell, A.J. (1991) Bernoulli partition-equivalence of intermittent neuronal discharge patterns. Int. J. Bifurcation Chaos 1:717-722. Selz, K.A., Mandell, A.J. (1992) Critical coherence and characteristic times in brain stem neuronal discharge patterns. In (ed. T. McKenna, J. Davis, S.F. Zornetzer) Single Neuron Computation, Academic. N.Y. pp 525-560. Selz, K.A., Mandell, A.J. (1993): Style as mechanism: from man to a map of the interval and back. In (ed. W. Ditto) Chaos in Medicine and Biology, Proc. SPIE 2036:174-182 Selz, K.A., Mandell, A.J. (1997) The power of style: differential operator scaling in the lexical compression of sequences generated by psychological, content-free computer tasks. Complexity 2:50-55 Selz, K.A., Mandell, A.J., Anderson, C.A., Smotherman, W., Teicher, M. (1995) Distribution of local Mandelbrot-Hurst exponents: motor activity in cocaine treated fetal rats and manic depressive patients. Fractals 3:893-904 Seneta, E. (1981) Non-negative Matrices and Markoff Chains. Springer-Verlag, New York. 286 HOUSE_OVERSIGHT_013786

Shannon, C.D., Weaver, W. (1949) The Mathematical Theory of Communication. University of Illinois Press, Urbana Sharkovskii, A.N. (1964): Coexistence of the cycles of a continuous mapping of the line to itself. Ukranian Mat. Z. 16:61-71 Shaw, R. (1981): Strange attractors, chaotic behavior and information flow. Z. Naturfursch. 36a:80-112 Shenker, S.J., Kadanoff, L.P. (1982): Critical behavior of a KAM surface. J. Stat. Phys. 27:631-645 Shlesinger, M.F. (1988) Fractal time in condensed matter. Ann. Rev. Phys. Chem. 39:269-290 Shlesinger, M.F. (1996) Random Processes. Encyclopedia of Applied Physics. 16:45-70 Shlesinger, M.F., Klafter, J., Wong, Y.M. (1982) Random walks with infinite spatial and temporal moments. J. Stat. Phys. 27:499-512 Shlesinger, M.F., Zaslavsky, G.M. (1996) Strange kinetics. Physics Today. Feb. pg. 33-39. Shlesinger, M.F., Zaslavsky, G.M., Frisch, U. (1995) Levy Flights and Related Topics in Physics. Springer-Verlag 287 HOUSE_OVERSIGHT_013787

Silipo, R., Deco, G., Vergassola, R., Bartsch, H.(1998)Dynamics extraction in multivariate biomedical time series. Biol Cybern 79:15-27 Simoyi, R.H., Wolf, A., Swinnery, H.L (1982): One dimensional dynamics in a multicomponent chemical reaction. Phys. Rev. Lett. 49:245-248 Sinai, Ya G. (1959) On the concept of entropy of a dynamical system. Dokl. Akad. Nauk SSSR 124:768 Singer, W. (1993): Synchronization of cortical activity and its putative role in information processing and learning. Ann. Rev. Physiol. 55:349-374 Skinner, J.E., Martin, J.L., Landisman, C.E., Mommer, M.M., Fulton, K., Mitra, M., Burton, W.D., Saltzberg,B. (1990) Chaotic attractors in a model of neocortex: dimensionalities of olfactory bulb surface potentaisl are spatially uniform and event- related. In (eds. Basar, E., Bullock, T.H.) Brain Dynamics, Progress and Perspectives. Springer-Verlag, Berlin. pp 119-134 Smale, S. (1967): Differentiable dynanmical systems. Bull. AMS 13:714-817 Smith, L.A. (1988) Intrinsic limits on dimension calculations Phys. Lett. A133:283- 288 Smith, L.A. (1992) Identification and prediction of low-dimensional dynamics. Physica D58:50-76 288 HOUSE_OVERSIGHT_013788

Smotherman, W.P., Selz, K.A., Mandell, A.J. (1996) Dynamical entropy is conserved during cocaine-induced changes in fetal rat motor patterns. Psychoendocrinology 21:173-187 So, P., Francis, J.T., Netoff, T.1., Gluckman, J., Schiff, S.J. (1998) Periodic orbits: a new language for neuronal dynamics. Biophys. J. 74:2776-2785 So, P., Ott, E., Sauer, T., Gluckman, B.J., Grebogi, C., Schiff, S.J. (1997) Extracting unstable periodic orbits from chaotic time series. Phys. Rev. E55:5398-5417 Sprott, J.C. (1993) Strange Attractors; Creating Patterns in Chaos. M&T Books. N.Y. Sprott, J.C., Rowlands, G. (1991) Chaotic Data Analyzer. Am. Institute Physics. N.Y. Stam, C.J., Nicolai, J., Keunen, R.W. (1998) Nonlinear dynamical analysis of periodic lateralized epileptiform discharges Clin Electroencephalogr 29:101-105 Stanley, H.E. (1971): Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, Oxford Stauffer, C. (1985) Introduction to Percolation Theory. Taylor and Francis. London Stergiopulos, N., Porret, C.A., De Brower, S., Mesiter, J.J. (1998) Arterial vasomotion: effect of flow and evidence of nonlinear dynamics. Am J Physiol 274:H1858-1864 Steriade, M., McCarley, R.W. (1990) Brainstem Control of Wakefulness and Sleep. Plenum. N.Y. 289 HOUSE_OVERSIGHT_013789

Stillwell, J. (1985) Poincare’s Papers on Fuchsian Groups. Springer-Verlag, N.Y. pp 1-50 Stockbridge,L.L., French,A.S. (1989) Characterization of a calcium-activated potassium channel in human fibroblasts Can J Physiol Pharmacol 67:1300-1307 gating. Biophys J 54:871-877 Stoop, R., Parisi, J. (1991) Calculations of Lyapounov exponents avoiding spurious elements. Physica D50:89-94 Straver,J., Keunen, R.W., Stam, C.J., Tavy, D., de Ruiter, G., Smith, S., Thijs, L., Schellens, R., Gielen, G.(1998): The EEG diagnosis of septic encephalopathy. Neurol. Res. 20:100-106 Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos. Addison-Wesley. Strong, S.P., de Ruyter van Steveninck, R.R., Bialek, W., Koberle, R. (1998) On the application of information theory to neural spike trains. Pac Symp Biocomput 621- 632 Sugihara, G., May, R.M. (1990) Nonlinear forcasting as a way of distinguishing chaos from measurement error in time series. Nature 344:734-741. Sullivan, D. (1979) Rufus Bowen. Publ. Math. |.H.E.S. 50:255 Szeto, H.H., Cheng, P.Y., Decena, J.A., Cheng, Y., Wu, D., Dwyer, G. (1992) Fractal properties in fetal breathing dynamics. Am. J. Physiol. 263:R141-R147. Takahashi, N., Hanyu, Y., Musha, T., Kubo, R., Matsumoto, G. (1990): Global bifurcation structure in periodically stimulated giant axons of squid. Physica D43:318-334 290 HOUSE_OVERSIGHT_013790

Takens, F. (1981) Detecting strange attractors in time series. Lect. Notes. Math. 898 :366-381 Teich M.C. (1989) Fractal character of the auditory neural spike train. IEEE Trans Biomed Eng 36:150-60 Teich, M.C., Heneghan, C., Lowen, S.B., Ozaki, T., Kaplan, E. (1997) Fractal character of the neural spike train in the visual system of the cat. J Opt Soc Am A 14:529-546 Teich, M.C., Johnson, D.H., Kumar, A.R., Turcott, R.G.(1990) Rate fluctuations and fractional power-law noise recorded from cells in the lower auditory pathway of the cat. Hear Res 46:41-52 Teich, M.C., Turcott, R.G., and Siegel, R.M. (1996) Temporal correlation in cat striate cortex neural spike trains. IEEE Eng. Med. Biol. Sept/Oct pp 79-87 Tennekes, H., Lumley, J.L (1972) A First Course in Turbulence. MIT Press, Cambridge pp 256-267 Theiler, J. (1990) Estimating fractal dimension. J. Opt. Am. A7:1055-1073 Theiler, J.,Rapp, P.E.(1996) Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram. Electroencephalogr Clin Neurophysiol 98:213-222 Thom, R. (1972) Structural Stability and Morphogenesis. Benjamin, Reading. MA Thompson, D’Arcy (1942) On Growth and Form. Dover Mineola, N.Y. 291 HOUSE_OVERSIGHT_013791

Tuller, B., Ding, M., Kelso J.A.(1997) Fractal timing of verbal transforms. Perception 26:913-28 Ueda, Y. (1992): The Road to Chaos, Aerial Press, Santa Cruz, CA Van der Pol, B. (1926): On relaxation oscillations. Philos. Mag. (London) 2:978-980 Veldhuis, J.D., Johnson, M.L. (1992) Deconvolution analysis of hormone data. Methods Enzymol. 210:539-575 Vliegen, J.H.,Stam, C.J., Rombouts, S.A., Keunen, R.W. (1996) Rejection of the filtered noise’ hypothesis to explain the variability of transcranial Doppler signals: a comparison of original TCD data with Gaussian-scaled phase randomized surrogate data sets Neurol Res 18:19-24 Wackerbauer,R., Schmidt,T., Widmer,R., Pfeiffer, A., Morfill, G., Kaess, H.(1998) Discrimination of irritable bowel syndrome by non-linear analysis of 24-h jejunal motility Neurogastroenterol. Motil. 10:331-337 Wagner, C.D., Nafz, B., Persson, P.B. (1996) Chaos in blood pressure control. Cardiovasc Res 31:380-7 Wayland, R., Bromley, D., Pickett, D., Passamante, A. (1993) Recognizing determinism in a time series. Phys. Rev. Lett. 70:580-582 Wegner, D. (1994) White Bears and Other Unwanted Thoughts; Suppression, Obsession and the Psychology of Mental Control. Guilford Press. N.Y. Whitney, H. (1936) Differentiable manifolds. Ann. Math. 37:645-672 292 HOUSE_OVERSIGHT_013792

Whitney, H. (1955) On singularities of mappings of Euclidean spaces. |. Mappings of the plane into the plane. Ann. Math. 62:374-410 Whittington, M.A., Traub, R.D., Jefferys, J.G.R. (1995): Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation. Nature 373:612-615 Wiggins, S. (1990): Introduction to Applied Dynamical Systems and Chaos. Springer. N.Y. Wilson, K.G. (1975) Review of the renormalization group. Rev. Mod. Phys. 47:773- 804 Wolf, A., Swift, J.B., Swinnery, H.L., Vastano, J.A. (1985) Determining Lyapounov exponents from a time series. Physica 16D:285-317 Woyshville, M.J., Lackamp, J.M., Eisengart, J.A., Gilliland, J.A.M. (1999) On the meaning and measurement of affective instability; clues from chaos theory. Bio. Psychiat. 45:261-269 Xu, J. (1994) The measure of sequence complexity for EEG studies. Chaos, Solitons and Fractals 4:2111-2119. Xu, N., Xu, J.H.(1988) The fractal dimension of EEG as a physical measure of conscious human brain activities. Bull Math Biol 50:559-65 Yamamoto, Y., Hughson, R.L., Sutton, J.R., Houston, C.S., Cymerman, A., Fallen, E.L., Kamath, M.V. (1993) Operation Everest II: an indication of deterministic chaos in human heart rate variability at simulated extreme altitude. Biol Cybern 69:205- 212 293 HOUSE_OVERSIGHT_013793

Yates,F.E. (1992) Fractal applications in biology: scaling time in biochemical networks Methods Enzymol 210:636-75 Yaylali, |., Kocak, H., Jayakar, P. (1996) Detection of seizures from small samples using nonlinear dynamic system theory. IEEE Trans Biomed Eng 43:743-751 Yeomans, J.M. (1993) Statistical Mechanics of Phase Transitions. Clarendon Press, Oxford Yokoyama, H., Niwa, S., Itoh K, Mazuka, R. (1996) Fractal property of eye movements in schizophrenia. Biol Cybern 75:137-140 Zabusky, N.J., Kruskal, M.D. (1965): Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15:240-243 Zeeman, C. (1976): Duffing equation in brain modeling. Bull. Inst. Math. Appl. 12:207-214. Zeeman, E.C. (1977) Catastrophy Theory, Selected Papers, 1972-1977. Addison- Wesley, Reading. MA Zeman, P., Toth, G., Kvetnanasky, R. (1987) Thermodynamic analysis of rat brain opioid mu-receptor-ligand interaction. Gen. Physiol. Biophys. 6:237-248 Zimmerman, |.D., Rapp, P.E. (1991) The geometrical characterization of neural activity displays a sensitivity to convulsants. Int. J. Biofurcation Chaos 1:253-259 Zhang, T., Johns, E.J. (1998) Chaotic characteristics of renal nerve peak interval sequence in normotensive and hypertensive rats. Clin Exp Pharmacol Physiol 25:896-903 294 HOUSE_OVERSIGHT_013794

Zipf, G.K. (1949) Human Behavior and the Principal of Least Effort. Addison- Wesley, N.Y. Zwiener, U., Hoyer, D., Bauer, R., Luthke, B., Walter, B., Schmidt, K., Hallmeyer, S.,Kratzsch, B., Eiselt, M.(1996) Deterministic--chaotic and periodic properties of heart rate and arterial pressure fluctuations and their mediation in piglets. Cardiovasc Res 31:455-465 295 HOUSE_OVERSIGHT_013795