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

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