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, σ 2 , the variance; the third moment, σ 3 , the distribution’s
asymmetry, the skew; and the fourth moment, σ”, its relative peakedness with
respect to the probability mass in the tail, called the kurtosis. In these moment
computations of an observable
q
x i
’s deviation from the mean, |x − x| , the value for
q accentuate particular regions of the density distribution. Similarly, the q’s of the
i
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“generalized dimensions,” D q , emphasize different aspects of the relative point
density that are assumed to be uniform in the computation of D 0 . We recall from
above that the power law slope constituting D
ln M( ε)
lim
ε ln( ε)
0 = →o
. If we emphasize the
component of the probability (measure, µ) or, equivalently, time spent by the orbit in
cube i, µ( C i
) instead of simply the number of cubes occupied by any points, M(ε),
along with the different length scales of the cube as ε→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 ( ε )
Iq
D = 1 lim ln ( , ε)
q
, where Iq ( , ε ) = ∑[ µ ( C i
)]
q −1ε
→ 0 ln( ε)
i=
1
the dominance of the higher probability cubes,
q
. The higher the q, the greater
µ( C i
). 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 D q , can vary continuously, functions are created that shows how D q 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
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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 D 2 ,
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
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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, D 2 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 D 2 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 λ( + )
(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 D 2 of the attactor was reduced significantly (Yamamoto et al,
1993). The latencies and amplitudes of the visual evoked potential failed to
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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 D 2 (Schmeisser et al, 1993). Marginal
qualitative differences in optokinetic nystagmus were quantitatively significant when
studied as the D 2 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 D 2 (and a decrease in λ( + ) correlated with
age in an adult population (Keuner et al, 1996; Vliegen et al, 1996). D 2 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 D 0 in 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 D 0 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 D 0 in the latter (Guillemin et al, 1983; Mandell,
1986).
The number of examples of the use of D 2 on orbital point geometries in
explorations of physiological and pharmacological regulation are increasing. The D 2
of respiratory rhythms is higher with intact vagal afferents than without (Sammon
and Bruce, 1991). Histamine induced an increase in D 2 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
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(Edwards and Griffith, 1997). The role of central and autonomic innervation in
cardiac interval dynamics has been explored using D 2 in various ways. For
examples, the transplanted heart rhythm in man has a lower D 2 than that of the
normal heart (Guzzetti et al, 1996) and general anesthesia and cholinergic (but not
β-adrenergic) blockade decreased multisystem D 2 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 D 2 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 D 2 as it did in the kindled in vitro hippocampal
slice (Koch et al, 1992). D 2 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 D 2
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).
D 2 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 D 2 than REM. D 2 of the EEG record was selectively reduced in
Stage II 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
higher EEG D 2 values were frontal in schizophrenic patients and more central in
controls (Elbert et al, 1992). The D 2 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 1 in their EEG attractors than the children of a normal
control group (Ehlers et al, 1995). Higher I.Q. correlated with EEG D 2 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 D 2 in the EEG of specific brain regions (Heffernan, 1996). Memory for
but not induced pain increased EEG D 2 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 D 2 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 D 2 computations (Stam et al,
1998). In equally problematic “atypical seizure” syndromes in children, D 2 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 λ( + ) 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 λ( + ) → λ( 0)
may auger a nearby bifurcation), the presence of a fractional scaling
exponent,
D i
, does not in and of itself implicate a chaotic dynamical state. A nice
example of a nonchaotic dynamic with
λ = 0 that has a fractional scaling exponent,
D = 0.538, is the “Feigenbaum” point where the above noted “infinite” series of
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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 λ = 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,
λ = 0 , D 0 and D 1 ≠ 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, θ to α, 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
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:
λ = 0 , D 0 and D 1 ≠ Integer, and a signatory power spectral distribution in which the
number of peaks, N, with amplitudes greater than ϖ, N(ϖ ), went as ϖ -α , 1 < α < 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 (D 0
and D 1 ≠ Integer) from picosecond fluctuations of neural membrane proteins to the
decades of bipolar phenomena and since λ = 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
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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 λ( + )) 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
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,
−∑ p i
log p i
(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 ln
(1/3)) = 1.098 > -2(1/2 ln (1/2) = 0.6931 loge and in computer relevant log 2 , 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 Rössler and Lorenz
systems, one can visualize the joint actions of λ( + ) and λ( − ) 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.
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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, h T
and h M , proven to be the upper and lower bounds of any estimate of the entropy in a
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, h T . They record new
happenings, the growth rate of the diversity of orbits, and not how likely with respect
241
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, h T , and log λ( + ), 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, h T
= log λ( + ) whereas the largest value that h M 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 h T and h M (Adler et al, 1977; Adler and Marcus, 1979).
If the sum of the densities in each j box were normalized so as to sum to 1.0,
such that each is a probability, p j , then - Σ p j log p j represents the metric entropy,
h M . h M 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. h M is maximal for the equidistributed, uniformly
expansive, C or Axiom A systems (see above). As noted above, generally h T = the
maximum estimate of the entropy and h M the minimum estimate (Adler and Weiss,
1965). h T = h M in uniformly hyperbolic systems (Bowen, 1975) and the difference,
|h T – h M | 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 h T and h M
below which might seem forbidding to those “not up on their linear algebra,” we note
242
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 h T 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 h T . 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 h T 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 h M , requires repeatedly exponentiating a Markoff matrix constructed
from relatively short samples and generates the probabilistic (eigenvector) “dual” of
h T. h M computed in this way serves as a useful quantity, h M called by some the
Kolmogorov entropy in comparisons of control and experimental conditions of the
same sample lengths. Systematic decreases in h M (“Kolmogorov entropy”) have
been shown to accompany increasing “depth” of sleep using standard sleep staging
techniques (Gallez and Babloyantz, 1991) and increases in h M were associated with
both positive and negative emotional states induced by movies (Aftanas et al,
1997).
243
h T and λ( + ) 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 therapistdirected
“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
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 h T , h M and λ( + ), resemble very closely those
245
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, ε, 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 h T and h M. 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 λ( + ) -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
(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
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 , λ( + ), is compensated by a decrease in D 0 leaving h T 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
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
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 nonintrusive
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 (Iasemidis 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
References
Aasen,T.,Kugiumtzis,D.,Nordahl,G. (1997) Procedure for estimating the correlation
dimension of optokinetic nystagmus signals. Comput Biomed Res. 30:95-116
Accardo, A., Mumolo, E.(1998): An algorithm for the automatic differentiation
between the speech of normals and patients with Friedreich's ataxia based on the
short-time fractal dimension. Comput Biol. Med. 28:75-89
Adler, R.J., Feldman, R.E., Taqqu, M. (1998) A Practical Guide to Heavy Tails;
Statistical Techniques and Applications. Birkhauser. Boston
Adler, R.L., Goodwyn, L.W., Weiss, B. (1977) Equivalence of topological Markov
shifts. Isreal J. Math. 27:49-63
Adler, R.L., Konheim, A.G., McAndrew, M.H. (1964) Topological entropy. Trans.
Amer. Math. Soc. 114:309-319
Adler, R.L., Marcus, B. (1979) Topological entropy and equivalence of dynamical
systems. Mem. Amer. Math. Soc. 219:
Adler, R.L., Weiss, B. (1967) Entropy, a complete metric invariant for
authomorphisms of the torus. Proc. Natl. Acad. Sci. 57:1573-1576
Aftanas, L.I.,Lotova, N.V., Koshkarov, V.I., Pokrosvskaja, V.L., Popov, S.A.,
Makhnev, V.P (1997) Non-linear analysis of emotion EEG: calculation of
Kolmogorov entropy and the principal Lyapunov exponent. Neurosci Lett 226:13-16
251
Aihara, K., Matsumoto, G., Ikegaya, Y. (1984): Periodic and non-periodic responses
of a periodically forced Hodgkin-Huxley oscillator. J. Theor. Biol. 140:27-38.
Aihara, K., Numajiri, T., Matsumoto, G., Kotani, M. (1986): Structures of attractors in
periodically forced neural oscillators. Phys. Lett. A116:313-317
Akay, M., Mulder, E.J. (1998) Effects of maternal alcohol intake on fractal properties
in human fetal breathing dynamics. IEEE Trans Biomed Eng 45:1097-1103
Alexeev, V.M., Jacobson, M.V. (1981) Symbolic dynamics and hyperbolic dynamical
systems. Phys. Rep. 75:287-325.
Alonso, A., Faure, M.-P., Beaudet, A. (1994): Neurotensin promotes oscillatory
bursting behavior and is internalized in basal forebrain cholinergic neurons. J.
Neurosci. 14:5778-5792.
Anderson, C.A., Mandell, A.J., Selz, K.A., Terry, L.M., Smotherman, W.P., Wong,
C.H., Robinson, S.R., Robertson, S.S., Nathanielsz, P.W. (1998) The development
of nuchal atonia associated with active (REM) sleep in fetal sheep: presence of
recurrent fractal organization. Brain Res. 787:351-357
Arnold, V.I. (1984) Catastrophe Theory. Springer-Verlag. N.Y.
Arnold, V.I., Avez, A. (1968) Ergodic Problems in Classical Mechanics. Addison-
Wesley. Redwood City, CA
Babloyantz, A. (1986) Evidence of chaotic dynamics of brain activity during the
sleep cycle. In (ed. Mayer-Kress, G) Dimensions and Entropies in Chaotic Systems.
Springer-Verlag, 1986
252
Babloyantz, A., Destexhe, A. (1986) Low dimensional chaos in an instance of
epilepsy. Proc. Natl. Acad. Sci. 83:3513-3517.
Baker, G.L., Gollub, J.P. (1991) Chaotic Dynamics; An Introduction. Cambridge
University Press. Cambridge
Bassingthwaighte, J., Liebovitch, L., West, B. (1994) Fractal Physiology. Oxford
University Press. Oxford.
Bayne,W.F., Hwang, S.S. (1985) Effect of nonlinear protein binding on equilibration
times for different initial conditions. J Pharm Sci 74:120-123
Bazhenov, M., Timofeev, I., Steriade, M., Sejnowski, T.J. (1998) Computational
models of thalamocortical augmenting responses. J. Neurosci. 18:6444-6465
Bennett, C.H. (1982) Thermodynamics of computation; A review. Intl. J. Theor.
Phys. 21:905-946
Bennett, C.H. (1986) On the nature and origin of complexity in discrete,
homogeneous, locally-interacting systems. Found. Phys. 16:585-604
Beard, D.A., Bassingthwaighte, J.B. (1998) Power-law kinetics of tracer washout
from physiological systems. Ann Biomed Eng 26:775-779
Berge′,P., Pomeau, Y., Vidal, C. (1984) Order Within Chaos. Wiley-Interscience.
N.Y.
Berlin, Yu. A., Miller, J.R., Plonka, A. (1996) Rate Processes with Kinetic
Parameters Distributed over Time and Space. Chem. Physics 212: Special Issue
253
Berns, G.S., Sejnowski, T.J. (1998) A computational model of how the basal ganglia
produce sequences. J. Cogn. Neurosci. 10:108-121
Birkhoff, G.D. (1922): Surface transformations and their dynamical applications.
Acta Math. 43:1-119
Boiteux, A., Goldbeter, A., Hess, B. (1975): Control of oscillating glycolysis of yeast
by stochastic, periodic, and steady source of substrate: a model and experimental
study. Proc Natl Acad Sci. 72:3829-33
Boltzmann, L. (1909) Wissenschaftliche Abhandlungen (ed. Hassenohrl, F.) Barth,
Liepzig
Bores, P.A., Bertelli, G.M., Gilli, G. (1988) Temperature dependence of the binding
of mu, delta, and kappa agonists to the opirate receptors in guinea-pig brain. Eur. J.
Pharmacol. 146:247-252
Bowen, R. (1970) Markoff partitions and minimal sets of Axiom A diffeomorphisms
Amer. J. Math. 92:907-918
Bowen, R. (1975): Equilibrium States and the Ergodic Theory of Anosov
Diffeomorphisms. Lect. Notes. Math. 35
Bowen, R. (1978) On Axiom A Diffeomorphisms. CBMS Regional Conference
Series in Mathematics. Vol 35 A.M.S. Providence, RI
Bressler, S.L. (1995): Large-scale cortical networks and cognition. Brain. Res. Rev.
20:288-304
254
Briggs, K. (1990) An improved method for estimating Lyapounov exponents of
chaotic time series. Phys. Lett. A151:27-32
Brodskii,V.(1998) The nature of the circahoralian (ultradian) intracellular rhythms.
Similarity to fractals Izv Akad Nauk Ser Biol 3:316-29
Broomhead, D.S., King, G.P. (1986) Extracting qualitative dynamics from
expeirmental data. Physica D20:217-236.
Brown, R., Bryant, P., Abarbanel, H.D. (1991) Computing the Lyapounov spectrum
of a dynamical system from an observed time series. Phys. Rev. A43:2787-2806
Bryant, P., Brown, R., Abarbanel, H.D. I. (1990) Lyapounov exponents from
observed time series. Phys. Rev. Lett. 65:1523-1526
Bullard, W.P., Guthrie, P.B., Russo, P.V., Mandell, A.J. (1978) Regional and
subcellular distribution and some factors in the regulation of reduced pterins in rat
brain. J. Pharmacol. Exp. Therap. 206:4-20
Buzug, Th., Reimers, T., Pfister, G. (1990) Optimal reconstruction of strangeattractors
from purely geometrical arguments. Europhys. Lett. 13:605-610
Callahan, J., Sashin, J.I. (1987) Models of affect-response and anorexia nervosa.
Ann. N.Y. Acad. Sci. 504:241-259
Canavier, C.C., Clark, J.W., Byrne, J.H. (1990): Routes to chaos in a bursting
neuron. Biophys J. 57:1245-1251
Carpenter, G.A. (1979): Bursting phenomena in excitable membranes. SIAM J.
Appl. Math. 36:334-352
255
Casdagli, M. (1991) Chaos and deterministic versus stochastic nonlinear modeling.
J. Royal. Stat. Soc. 54B:303-328
Casdagli, M.C., Iasemidis, L.D., Savit, R.S., Gilmore, R.L., Roper, S.N.,
Sacklellares, J.C. (1997) Non-linearity in invasive EEG recordings from patients with
temporal lobe epilepsy. Electroenceph. Clin. Neurophys. 102:98-105
Cartwright, M.I., Littlewood, J. (1945): On nonlinear differential equations of the
second order. J. London. Math. Soc. 20: 180-189
Celletti, A.,Villa, A.E.(1996) Low-dimensional chaotic attractors in the rat brain. Biol
Cybern 74:387-393
Cerf, R., Sefrioui, M., Toussaint, M., Luthringer, R., Macher, J.P. (1996) Lowdimensional
dynamic self-organization in delta-sleep: effect of partial sleep
deprivation Biol Cybern 74:395-403
Chaitin, G.J. (1974) Information theoretical computational complexity. IEEE Trans.
Inform. Theory IT20:10-15.
Chay, T.R., Rinzel, J. (1985) Bursting, beating and chaos in an excitable membrane
model. Biophys. J. 47:357-366
Chen, Y-Q., Ku, Y-H. (1992) Properties of favored patterns in spontaneous spike
trains and responses of favored patterns to electroacupuncture in evoked trains.
Brain Res. 578:297-304
Clausius, R. (1897) The Mechanical Theory of Heat. MacMillan London
256
Coffman, K., McCormick, W.D., Swinney, H.L. (1986) Multiplicity in a chemical
reactions with one-dimensional dynamics. Phys. Rev. Lett. 56:999-1002
Contreras, D., Dextexhe, A., Sejnowski, T.J., Steriade, M. (1997) Spatiotemporal
patterns of spindle oscillations in cortex and thalamus. J. Neurosci. 17:1179-1196
Cooper, N.G. (1987): From Cardinals to Chaos, Cambridge University Press,
Cambridge.
Cornfield, I.P., Fomin, S.V., Sinai, Ya.G. (1982) Ergodic Theory. Springer-Verlag.
Berlin
Crevier,D.W., Meister,M (1998): Synchronous period-doubling in flicker vision of
salamander and man. J Neurophysiol. 79:1869-1878
Crutchfield, J.P., Young, K. (1989a) Inferring statistical complexity. Phys. Rev. Lett.
63:105-108
Crutchfield, J.P., Young, K. (1989b) Computation at the onset of chaos. In (ed.
Zurek, W.H.) Complexity, Entropy and the Physics of Information. Addison-Wesley.
Redwood City, CA.pp 223-269.
Cvitanovic′, P. (1989): Universality in Chaos. Adam-Hilger, Bristol.
Dayhoff, J.E. (1984) Distinguished words in data seqauences; analysis and
applications to neural coding and other fields. Bull. Math. Biol. 46:529-543
Dayhoff, J.E., Gerstein, G.L. (1983a) Favored patterns in spike trains. I. Detection.
J. Neurophysiol. 49: 1334-1346
257
Dayhoff, J.E., Gerstein, G.L. (1983b) Favored patterns in spike trains. II.
Applications. J. Neurophysiol. 49:1347-1363
DeBrouwer, S. Edwards, D.H.,Griffith, T.M (1998): Simplification of the
quasiperiodic route to chaos in agonist-induced vasomotion by iterative circle
maps.Am J Physiol. 274:H1315-26
de Gennes, P-G. (1979) Scaling Concepts in Polymer Physics. Cornell University
Press, Ithica, N.Y.
Devaney, R.L. (1989): An Introduction to Chaotic Dynamical Systems, 2 nd ed.
Addison-Wesley, Redwood City, CA.
Dewey, T.G. (1997) Fractals in Molecular Biophysics Oxford University Press,
Oxford
Ding, M., Grebogi, C., Ott, E. (1989) Dimensions of strange nonchaotic attractors.
Phys. Lett. A137:167- 172
Ding, M., Grebogi, C., Ott, E., Sauer, T., Yorke, J. (1993) Plateau onset for
correlation dimension: When does it occur. Phys. Rev. Lett. 70:3872-3875.
Duke, D.W. and Pritchard, W.S. (eds) (1991)Measuring Chaos in the Human Brain.
World Scientific. Singapore.
Dvorak, I. and Holden, A. (eds) (1991) Mathematical Approaches to Brain
Functioning Diagnostics. Manchester University Press. Manchester
Ebling, W., Jimenez-Montano, M.A. (1980) On grammars, complexity and
information measures of biological molecules. Math. Biosci 52:53-71
258
Eckmann, J.-P. (1981): Roads to turbulence in dissipative dynamical systems. Rev.
Mod. Phys. 53:643-654
Eckmann, J., Ruelle, D. (1985) Ergodic theory of chaos and strange attractors. Rev.
Mod. Phys. 57:617-656
Eckmann, J.P., Ruelle, D. (1992) Fundamental limitations for estimating dimensions
and Lyapounov exponents in dynamical systems. Physica 56D:185-187
Eckmann, J.-P., Kamphorst, S.O., Ruelle, D., Cilberto, S. (1986) Lyapounov
exponents from time series. Phys. Rev. A34:4971-4979
Edwards, D.H., Griffith, T.M. (1997) Entrained ion transport systems generate the
membrane component of chaotic agonist-induced vasomotion. Am J Physiol
273:H909-920
Efremova, T.M., Kulikov, M.A.(1997) The chaotic component of the high-frequency
EEG of the rabbit cerebral cortex in the formation of a defensive conditioned reflex.
Zh Vyssh Nerv Deiat Im I P Pavlova 47:858-869
Elger,C., Lehnertz, K. (1998) Seizure prediction by non-linear time series analysis of
brain electrical activity. Eur J Neurosci.10:786-789
Enns, R.H., McGuire, G.C. (1997) Nonlinear Physics with Maple for Scientists and
Engineers. Birkhauser. Boston
Ehlers, C.L. (1995) Chaos and complexity. Arch. Gen. Psychiat. 52:960-964
Ehlers, C.L., Havstad, J.W., Garfinkel, A., Kupfer, D.J. (1991) Nonlinear analysis of
EEG sleep states. Neuropsychopharmacology 5:167-176
259
Ehlers, C.L., Havstad, J., Prichard, D., Theiler, J. (1998) Low doses of ethanol
reduce evidence for nonlinear structure in brain activity. J Neurosci 18:7474-86
Ehlers, C., Havstad, J.W., Schuckit, M.A. (1995) EEG dimension in sons of
alcoholics. Alcohol Clin. Exp Res 19:992-998
Elbert, T., Lutzenberger, W., Rockstroh, B., Berg, P., Cohen, R.(1992) Physical
aspects of the EEG in schizophrenics. Biol Psychiatry 32:595-606
Erdos, P., Renyi, A. (1970). On a new law of large numbers. J. Anal. Math. 22:103-
111
Farmer, D., Crutchfield, J., Froehling, H., Packard, N., Shaw, R. (1980) Power
spectra and mixing properties of strange attractors. Ann. N.Y. Acad. Sci. 357:453-
472
Farmer, J.D., Ott, E., Yorke, J.A. (1983) The dimension of chaotic attractors.
Physica D7: 153-180
Feder, J. (1988) Fractals. Plenum. N.Y.
Feigenbaum, M.J. (1979) The universal metric properties of nonlinear
transformations, J. Stat. Phys. 21:669-702
Fell, J.,Roschke, J., Beckmann, P. (1993) Deterministic chaos and the first positive
Lyapunov exponent: a nonlinear analysis of the human electroencephalogram
during sleep. Biol Cybern 69:139-46
Fell, J., Roschke, J., Schaffner, C. (1996) Surrogate data analysis of sleep
electroencephalograms reveals evidence for nonlinearity. Biol Cybern 75:85-92
260
Feller, W. (1968) An Introduction to Probability Theory and Its Applications. Wiley.
NY. Vol. I., Chapter 3
Fermi, E., Pasta, J., Ulam, S. (1955): Studies of nonlinear problems. Los Alamos
Scientific Laboratory Report # 1940
Fessard, A., Gerard, R.,W. , Konorski, J. (1961) (eds) Brain Mechanisms and
Learning. Blackwell, Oxford.
Flory, P. (1971) Principles of Polymer Physics. Cornell University Press, Ithica, N.Y.
Frank, G.W., Lookman, T., Nerenberg, M.A.H., Essex, C., Lemieux, J., Blume, W.
(1990) Chaotic time series of epileptic seizures. Physica D46:427-438
Friedrich, R., Fuchs, A., Haken, H. (1991): Spatio-temporal EEG patterns In (ed.
H.Haken and H.P. Kopchen) Synergetics of Rhythms. Springer. Berlin
Gallez, D.,Babloyantz, A. (1991) Predictability of human EEG: a dynamical
approach. Biol Cybern 64:381-391
Ganz,R.E., Faustman, P.M. (1996) Functional coupling between the centralautonomic
and the cognitive systems is enhanced in states after optic neuritis and
early multiple sclerosis. Int J Neurosci, 86:87-93
Geist, K., Parlitz, U., Lauterborn, W. (1990) Comparison of different methods for
computing Lyapunov exponents. Prog. Theor. Physics 83:875-893
Gershenfeld, N.A. (1992) Dimension measurement on high-dimensional systems
Physica D55: 135-154
261
Gerstein, G.L., Mandelbrot, B.B. (1964) Random walk models for the spike activity
of a single neuron. Biophys. J. 4:41-68.
Gibbs, J.W. (1902) Elementery Principles in Statistical Mechanics, Yale University
Press, New Haven.
Gilmore, R. (1981): Catastrophe Theory for Scientists and Engineers. Wiley. N.Y.
Glass, L., Mackey, M. (1988) From Clocks to Chaos; The Rhythms of Life.
Princeton University Press. Princeton.
Goldberger, A.L., Rigney, D.R., West, B.J. (1990) Chaos and fractals in human
physiology. Scientific American 262:42-49
Goldberger, A.L. (1996) Non-linear dynamics for clinicians: chaos theory, fractals
and complexity at the bedside. Lancet 347:1312-1314
Gong, Y., Xu, J., Ren, W., Hu, S., Wang, F.(1998) Determining the degree of chaos
from analysis of ISI time series in the nervous system: a comparison between
correlation dimension and nonlinear forecasting methods. Biol Cybern 78:159-65
Gottschalk, A., Bauer, M.S., Whybrow, P.C. (1995) Evidence of chaotic mood
variation in bipolar disorder. Arch. Gen. Psychiat. 52:947-959
Grebogi, C., Ott, E., Pellikan, S., Yorke, J.A. (1984) Strange attractors that are not
chaotic. Physica 13D:261-268
Grassberger, P. (1981) On the Hausdorf dimension of fractal attractors. J. Stat.
Phys. 26:173-179
262
Grassberger, P. (1983) Generalized dimensions of strange attractors. Phys. Lett.
A97:227-230.
Grassberger, P, Procaccia, I. (1983) Measuring the strangeness of strange
attractors. Physica 9D:189-208
Gregson, R.A.,Britton, L.A., Campbell, E.A., Gates, G.R. (1990) Comparisons of the
nonlinear dynamics of electroencephalograms under various task loading
conditions: a preliminary report. Biol Psychol 31:173-191
Grimmett, G. (1989) Percolation. Springer-Verlag.
Guckenheimer, J., Holmes, P. (1983): Nonlinear Oscillations, Dynamical Systems
and Bifurcations of Vector Fields. Springer. N.Y.
Guillemin, R.C., Brazeau, P., Briskin, A., Mandell, A.J. (1983) Evidence for
synergetic dynamics in a mammalian pituitary cell perfusion system in (ed. E.Basar,
H. Flohr, H. Haken, A.J. Mandell) Synergetics of Brain. Springer-Verlag. pp 155-162
Gupta, V., Suryanarayanan, S., Reddy, N.P. (1997) Fractal analysis of surface EMG
signals from the biceps. Int J Med Inf 45:185-192
Guzzetti, S.,Signorini,M.G., Cogliati, C., Messetti,S., Porta, A., Cerutti, S., Malliani,
A. (1996) Non-linear dynamics and chaotic indices in heart rate variability of normal
subjects and heart-transplanted patients. Cardiovasc Res 31:441-446
Hadamard, J. (1898) Les surfaces a courbures opposees et leur lignes geodesic. J.
Math. Pures. Appl. 4:27-73
263
Hagerman, I.,Berglund, M., Lorin, M., Nowak, J., Sylven, C.(1996) Chaos-related
deterministic regulation of heart rate variability in time- and frequency domains:
effects of autonomic blockade and exercise. Cardiovasc 31:410-418
Haken, H. (1997) Visions of synergetics. Int. J. Bifurcat. Chaos. 7:1927-1951
Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procacccia, I., Shraiman, B.I. (1986)
Fractal measures and their singularities: the generalization of strange sets. Phys.
Rev A33:1141-1151
Hartman, M., Pincus, S., Johnson, M., Matthews, D., Faunt, L., Vance, M., Thorner,
M., Veldhuis, J. (1994): Enhanced basal and disorderly growth hormone secretion
distinguish acromegalic from normal pulsatile growth hormone release. J. Clin.
Invest. 94:1277-1288.
Hausdorff, J.M., Cudkowicz, M.E., Firtion, R., Wei, J.Y., Goldberger, A.L. (1998)
Gait variability and basal gangia disorders: stride-to-stride variations in gait cycle
timing in Parkinson’s and Huntington’s disease. Mov. Disord. 13:428-437
Hausdorff, J.M., Peng, C.-K., Ladin, Z., Wei, J.Y., Goldberger, A.L. (1995) Is
walking a random walk? Evidence for long range correlations in the stride interval of
human gait. J. Appl. Physiol. 78:349-358
Hausdorff. J.M., Purdon, P.L., Peng, C.-K., Ladin, Z., Wei, J.Y., Goldberger, A.L.
(1996) Fractal dynamics of human gait stability of long-range correlations in stride
interval fluctuations. J. Appl. Physiol. 80:1448-1457
Heffernan, M.S.(1996) Comparative effects of microcurrent stimulation on EEG
spectrum and correlation dimension. Integr Physiol Behav Sci 31:202-209
264
Hentschel, H.G.E., Procaccia, I. (1983) The infinite number of generalized
dimensions of fractals and strange attractors. Physica D8:435-444
Herman, M.L., Pincus, S.M., Johnson, M.L.,Matthews, D.L.., Faunt, , L.M.., Vance,
M.L.., Thorner, M.O., Veldhuis, J.D. (1994) Enhanced basal and disorderly growth
hormone secretion distinguish acromegalic from normal pulsatile growth hormone
release. J. Clin. Invest. 94:1277-1288.
Hitzemann, R., Murphy, M., Curell, J. (1985) Opiate receptor thermodynamics:
agonist and antogonist binding. Eur. J. Pharmacol. 108:171-177
Hoyer, D.,Pompe, B., Herzel, H., Zwiener, U. (1998) Nonlinear coordination of
cardiovascular autonomic control. IEEE Eng Med Biol Mag 17:17-21
Huber, M.T., Braun, H.A., Krief, J.C. (1999) Effects of noise on different states of
recurrent affective disorder. Biol. Psychiat. In press.
Huberman, B.A. (1987) A model for dysfunctions in smooth pursuit eye movement.
Ann. N.Y. Acad. Sci. 504:260-273
Hughes, B.D. (1995) Random Walks and Random Environments. Clarendon.
Oxford.
Hurewicz, W., Wallman, H. (1948) Dimension Theory. Princeton University Press,
Princeton
Iasemidis, L.D., Sackellares, J.C. (1996) Chaos theory and epilepsy The
Neuroscientist. 2:118-128.
Iasemidis, J., Sacklellares, H., Zaveri,H., Williams, W.J. (1990): Phase space
topolography and the Laypounov exponent of electrocorticograms in partial
seizures. Brain Topolography 2:187-201.
265
Iasemidis, LD., Sveri, H.P. Sackellares, J.C., William, W.J., Hood, T.W. (1988)
Nonlinear dynamics of electrocorticographic data. J. Clin. Neurophysiol. 5:339
Inouye, T., Shinosaki, K., Sakamoto, H., Toi, S., Ukai, S., Iyama, A., Katsuda, Y.,
Hirano, M. (1991) Quantification of EEG irregularity by use of the entropy of the
power spectrum. Electroencephalogr Clin Neurophysiol 79:204-210
Inouye, T., Shinosaki, K., Toi, S., Ukai, S., Iyama, A., Katsuda, Y., Hirano, M. (1992)
Abnormality of background EEG determined by the entropy of power spectra in
epileptic patients. Electroencephalogr Clin Neurophysiol 82:203-207
Jansen, B.H. (1996) Nonlinear dynamics and quantitative EEG analysis.
Electroencephalogr Clin Neurophysiol Suppl 45:39-56
Jansen, B.H. and Brandt, M.E. (eds) 1993 Nonlinear Dynamical Analysis of the
EEG. World Scientific. Singapore
Jeong, J., Joung, M.K., Kim, S.Y. (1998) Quantification of emotion by nonlinear
analysis of the chaotic dynamics of electroencephalograms during perception of 1/f
music. Biol Cybern 78:217-25
Jeong,J., Kim, S.Y., Han, S.H. (1998) Non-linear dynamical analysis of the EEG in
Alzheimer's disease with optimal embedding dimension. Electroencephalogr Clin
Neurophysiol 106:220-228
Kaneko, K. (1983): Similarity structure and scaling property of the period adding
phenomena. Prog. Theor. Phys. 69:403-414
Kaplan, D.T. (1994) Exceptional events as evidence of determinism. Physica
D73:38-48
266
Kaplan, D.T., Glass, L. (1992) Direct test for determinism in a time series. Phys.
Rev. Lett. 68:427-430
Katz, B. (1966) Nerve, Muscle and Synapse. McGraw-Hill. N.Y.
Kelly,O.E., Johnson, D.H., Delgutte, B., Cariani, P. (1996)Fractal noise strength in
auditory-nerve fiber recordings. J Acoust Soc Am 99:2210-20
Keunen, R.W.,Vliegen, J.H., Stam,C.J., Tavy, D.L.(1996) Nonlinear transcranial
Doppler analysis demonstrates age-related changes of cerebral hemodynamics.
Ultrasound Med Biol 22:383-390
Khinchin, A.I. (1957) Mathematical Foundations of Information Theory. Dover. N.Y.
King, R., Barchus, J.D., Huberman, B.A. (1984) Chaotic behavior in dopamine
neurodynamics. Proc. Natl. Acad. Sci. 81:1244-1247
Klafter, J., Shlesinger, M.F., Zumofen, G. (1996) Beyond brownian motion. Physics
Today Feb. pp 33-39
Klemm, W.R. (1990) Historical and introductory perspectives on brain-stem
mediated behaviors In (ed. Klemm, W.R., Vertes, R.P.) Wiley-Interscience, N.Y.
Klemm, W.R., Sherry, C.J. (1981) Serial ordering in spike trains: what's it "trying to
tell us"? Int J Neurosci 14:15-33
Knapp, S., Mandell, A.J. (1983) Scattering kinetics in a complex tryptophan
hydroxylase preparation from rat brainstem raphe nuclei: statistical evidence that
267
the lithium-induced sigmoid velocity function reflects two states of available catalytic
potential. J. Neural Trans. 58:169-182
Knapp, S., Mandell, A.J. (1984) TRH influences the pterin-cofactor and timedependent
instabilities of rate raphe trypophan hydroxylase activity assessed under
far-from-equilibrium conditions. Neurochem. Internat. 6:801-812
Knapp, S., Mandell, A.J., Russo, P.V., Vitto, A., Stewart, K. (1981) Strain
differences in kinetic and themal stability of two mouse strain tryptophan
hydroxylase activities. Brain Res. 230:317-336
Koch, C.D.,Palovcik, R.A., Uthman, B.M., Principe, J.C. (1992) Chaotic activity
during iron-induced "epileptiform" discharge in rat hippocampal slices. IEEE Trans
Biomed Eng 39:1152-1160
Koch,H.P, Zajcek, H. (1991) Fractals also in pharmacokinetics? Pharmazie 46:870-
871
Kolmogorov, A.N. (1950) Foundations of Probability Theory. Chelsea, N.Y.
Kolmogorov, A.N. (1957): General theory of dynamical systems and classical
mechanics. In Proc. 1954 International Congress of Mathematics, North-Holland.
Kolmogorov, A.N. (1958) A new metric invariant of transient dynamical systems and
automorphisms in Lebesgue space. Dokl. Acad. Nauk. SSSR 119:861-864
Kolmogorov, A.N. (1959) Entropy per unit time as a metric invariant of
automorphisms. Dokl. Acad. Nauk. SSSR 124:754-758
268
Korn, S.J., Horn, R. (1988) Statistical discrimination of fractal and Markov models of
single-channel gating. Biophys J. 54:871-877
Korsch, H.J., Jodl, H. (1993) Chaos; A Program Collection for the PC. Springer-
Verlag.
Krebs-Thomson, K., Lehmann-Masten, V., Naiem, S., Paulus, M.P., Geyer, M.A.
(1998a) Modulation of phencyclidine-induced changes in locomotor activity and
patterns in rats by serotonin. Eur. J. Pharmacol. 343:135-143
Krebs-Thomson, K., Paulus, M.P., Geyer, M.A. (1998b) Effects of hallucinogens on
locomotor and investigatory activity and patterns: influence of 5-HT2A and 5-HT2C
receptors. Neuropsychophamracology. 18:339-351
Kowalik, Z.J., Elbert, T. (1995) A practical method for the measurement of the
chaoticity of electric and magnetic brain activity. Int. J. Bifurcation. Chaos 5:475-490
Kumar, A.R., Johnson, D.H. (1993) Analyzing and modeling fractal intensity point
processes. J Acoust Soc Am 93:3365-3373
Krystal, A.D., Weiner, R.D. (1991) The largest Lypounov exponent of the EEG in
ECT seizures. In (ed. D.Duke and W. Pritchard) Measureing Chaos in the Human
Brain. World Scientific. Singapore
Lasota, A., Yorke, J. (1973) On the existence of invariant measures for piecewise
monotonic transformations. Trans. Amer. Math. Soc. 186:481-488
Lathrop, D.P., Kostelich, E.J. (1989) Characterization of an experimental strange
attractor by periodic orbirts. Phys. Rev. A40:4028-4031
269
Levinson, N. (1949): A second order differential equation with singular solutions..
Ann. Math. 50:127-153
Li, T.Y., Yorke, J.A. (1975): Period three implies chaos. Amer. Math. Mon. 82:985-
992
Liebovitch, L.S. (1989) Analysis of fractal ion channel gating kinetics: kinetic rates,
energy levels, and activation energies. Math Biosci. 93:97-115
Liebovitch, L.S. (1998) Fractals and Chaos; Simplified for the Life Sciences. Oxford
University Press, Oxford.
Liebovitch,L.S., Sullivan, J.M. (1987) Fractal analysis of a voltage-dependent
potassium channel from cultured mouse hippocampal neurons. Biophys J. 52:979-
88
Liebovitch, L.S., Todorov, A.T. Using fractals and nonlinear dynamics to determine
the physical properties of ion channel proteins. (1996) Crit Rev Neurobiol 10:169-87
Liebovitch, L.S., Todorov, A.T. (1996) Invited editorial. J. Appl. Physiol. 80:1446-
1447
Liebert, W., Pawelzik, K., Schuster, H.G. (1991) Optimal embeddings of chaotic
attractors from topological considerations. Europhys. Lett. 14:521-526
Lorenz, E.N. (1963): Deterministic nonperiodic flow. J. Atmos. Sci. 20:130-141
Lowen, S.B.,Teich, M.C.(1992) Auditory-nerve action potentials form a nonrenewal
point process over short as well as long time scales. J Acoust Soc Am 92:803-6
270
Lutzenberger, W., Flor, H., Birbaumer, N. (1997) Enhanced dimensional complexity
of the EEG during memory for personal pain in chronic pain patients. Neurosci Lett
226:167-170
Lutzenberger, W., Birbaumer,N., Flor, H., Rockstroh, B., Elbert, T. (1992)
Dimensional analysis of the human EEG and intelligence Neurosci Lett 143:10-14
Macheras, P.,Argyrakis, P., Polymilis, C. (1996) Fractal geometry, fractal kinetics
and chaos en route to biopharmaceutical sciences. Eur J Drug Metab
Pharmacokinet 21:77-86
Mackey, M. (1992) Time’s Arrow: The Origins of Thermodynamic Behavior.
Springer-Verlag. NY
Mackey, M., Glass, L. (1977) Oscillation and chaos in physiological control systems.
Science 197:287-289
Magoun, H.W. (1954) The ascending reticular formation and wakefulness. In (ed.
E.D. Adrian, F. Bremer, H.H. Jasper. Brain Mechanisms and Consciousness.
Blackwell. Oxford pp 1-20
Makarenko, V., Llinas, R.(1998) Experimentally determined chaotic phase
synchronization in a neuronal system. Proc Natl Acad Sci 95:15747-15752
Mandelbrot, B.B. (1967) How long is the coast of Britain? Statistical self-similarity
and fractional dimension. Science 155:636-638
Mandelbrot, B.B. (1974) Intermittent turbulence in self-similar cascades: divergence
of high moments and dimension of the carrier. J. Fluid Mech. 62:331-358
271
Mandelbrot, B.B. (1975) Les Objets Fractals:Forme, Hasard et Dimension.
Flammarion. Paris
Mandelbrot, B.B. (1977) Fractals; Form, Chance and Dimension. Freeman. N.Y.
Mandelbrot, B.B. (1977) Fractals: Form, Chance and Dimension, Freeman. San
Francisco
Mandell, A.J. (1983) From intermittency to transitivity in neuropsychobiological
flows. Am. J. Physiol. 245: R484-R494
Mandell, A.J. (1984) Non-equilibrium behavior of some brain enzyme and receptor
systems. Ann. Rev. Pharmacol. Toxicol. 24:237-274
Mandell, A.J. (1986) The hyperbolic helix hypothesis in (ed. L Pietronero, E. Tosatti)
Fractals in Physics. Elsevier. N.Y.
Mandell, A.J. (1987) Dynamical complexity and pathological order in the cardiac
monitoring problem. Physica D27:235-242
Mandell, A.J., Kelso, J.A.S. (1991) Dissipative and statistical mechanics of amine
neuron activity. In (eds. Ellison, J.A., Uberall, H.) Essays on Classical and Quantum
Dynamics. Gordon Beach. Philadelphia pp 203-236
Mandell, A.J., Knapp, S., Ehlers, C., Russo, P.V. (1985) The stability of constrained
randomness: lithium prophylaxis at several neurobiological levels. In (ed. R.M. Post,
J. Ballenger). Neurobiology of Mood Disorders. Williams and Wilkins. Baltimore.
pp744-776
272
Mandell, A.J., Russo, P.V. (1981): Striatal tyrosine hydroxylase:multiple
conformational kinetic oscillators and product concentration frequencies. J.
Neuroscience 1:380-389
Mandell, A.J., Russo, P.V., Knapp, S. (1982) Strange stability in hierarchically
coupled neuropsychobiological systems. In (ed. Haken, H.) Evolution of Order and
Chaos. Springer-Verlag. Berlin. pp 270-286
Mandell, A.J., Selz, K.A. (1991) Period adding, hierarchical protein modes and
electroencephalographically defined states of consciousness. In (eds. Vohra, S.,
Spano, M., Shlesinger, M.F., Pecora, L., Ditto, W. ) Proc. 1 st Experimental Chaos
Conference. World Scientific. Hong Kong
Mandell, A.J., Selz, K.A. (1992) Dynamical systems in psychiatry: now what? Biol.
Psychiat. 32:299-301.
Mandell, A.J., Selz, K.A. (1993): Brain stem neuronal noise and neocortical
“resonance.” J. Stat. Phys. 70:355-371
Mandell, A.J., Selz, K.A. (1994) Resonance, synchroniation and lexical redundancy
in the expanding dynamics of brain stem neurons. SPIE Proc 2036:86-99
Mandell, A.J. and Selz, K.A. (1994) Resonance, synchronization and lexical
redundancy in the expanding dynamics of brain stem neurons. Chaos in Medicine
and Biology SPIE Proc: 2036:86-99
Mandell, A.J., Selz, K.A. (1995) Nonlinear dynamical patterns as personality theory
for neurobiology and psychiatry. Psychiatry 58:371-390
273
Mandell, A.J., Selz, K.A. (1997a) Entropy conservation as h
neurobiological dynamical systems. Chaos 7:67-81
≈ + d in
t µ
λ µ µ
Mandell, A.J. and Selz, K.A. (1997b) Is the EEG a strange attractor? Brainstem
neuronal discharge patterns and electroencehphalographic rhytms in (ed. Grebogi,
C., Yorke, J) The Impact of Chaos on Science and Society United Nations
University Press, Tokyo, pp 64-96
Mandell, A.J., Selz, K.A., Shlesinger, M.F. (1991) Some comments on the weaving
of contemporaneous minds: Resonant neuronal quasiperiodicity, period adding and
O(2) symmetry in the EEG In (ed. Pritchard W., Duke, D.) Chaos in the Brain World
Scientific Singapore pp 136-155
Manneville, P., Pomeau, Y. (1980): Different ways to turbulence in dissipative
systems. Physica 1D:219-226
Mantegna, R.N. (1991) Levy walks and enhanced diffusion in Milan stock exchange.
Physica A 179:232-242
Martinerie, J.,Adam, C., Le Van Quyen, M., Baulac, M., Clemenceau, S., Renault,
B., Varela, F.J. (1998) Epileptic seizures can be anticipated by non-linear analysis
Nat Med 4:1173-1176
May, R.M. (1976): Simple mathematical models with very complicated dynamics.
Nature 261:459-467
Mayer-Kress, G. (1986) Dimensions and Entropies in Chaotic Systems. Springer-
Verlag. N.Y.
274
McManus, O.B., Weiss, D.S., Blatz, A.L., Magleby, K.L. (1988) Fractal models are
inadequate for the kinetics of four different ion channels. Biophys J 54:859-70
McMullen, C. T. (1994) Complex Dynamics and Renormalization. Princeton
University Press. Princeton.
McMurran, S., Tattersal, J. (1999): Mary Cartwright. Notices, AMS 46:214-220
Mestivier, D., Dabire, H., Safar, M., Chau, N.P. (1998) Use of nonlinear methods to
assess effects of clonidine on blood pressure in spontaneously hypertensive rats. J
Appl Physiol 84:1795-800
Metropolis, N., Stein, M.L., Stein, P.R. (1973): On limit sets for transfomrations on
the unit interval . J. Combinatorial Theory 15:25-44
Meyer-Lindenberg, A., Bauer, U., Krieger, S., Lis, S., Vehmeyer, K., Schuler, G.,
Gallhofer, B. (1998) The topography of non-linear cortical dynamics at rest, in
mental calculation and moving shape perception. Brain Topogr 10:291-9
Micheloyannis, S., Flitzanis, N., Papanikolaou, E., Bourkas, M., Terzakis, D.,
Arvanitis, S., Stam, C.J. (1998) Usefulness of non-linear EEG analysis Acta Neurol
Scand 97:13-9
Milnor, J. (1985): On the concept of attractor. Commun. Math. Phys. 99:177-195
Milton, J.G., Longtin, A. (1990): Evaluation of pupil constriction and dilation from
cycling measurements. Vision Res. 30:515-525
275
Misiurewicz, M. (1995) Thirty years after Sharkovskii’s theorem. Int. J. Bifurcation
Chaos 5:1275-1281
Mogilevskii, A.Ya., Derzhiruk, L.P., Panchekha, A.P., Dershiruk, E.A.(1998)
Adaptive regulation of the nonlinear dynamics of electrical activity of the brain.
Neurosci Behav Physiol 28:366-375
Molnar, M., Gacs,G., Ujvari, G., Skinner, J.E., Karmos, G. (1997) Dimensional
complexity of the EEG in subcortical stroke--a case study. Int J Psychophysiol.
25:193-199
Montroll, E.W., Badger, W.W. (1974) Introduction to Quantitative Aspects of Social
Phenomena. Gordon & Breach. N.Y.
Montroll, E.W., Shlesinger, M.F. (1984) On the wonderful world of random walks. In
(ed. Liebowitz, J.L, Montroll, E.W.) Nonequilibrium Phenomena II: From Stohastics
to Hydrodynamics. Elsevier. North-Holland. Amsterdam
Moon, F. (1992): Chaotic and Fractal Dynamics; An Introduction for Applied
Scientists and Engineers. Second Edition, Wiley, N.Y.
Morinushi, T., Kawasaki, H., Masumoto, Y., Shigeta, K., Ogura, T., Takigawa, M.
Examination of the diagnostic value and estimation of the chaos phenomenon in
masticatory movement using fractal dimension in patients with temporomandibular
dysfunction syndrome. (1998): J. Oral Rehabil.25:386-94
Morse, M. (1921) A one-to-one representation of geodesics on a surface of negative
curvature. Amer. J. Math. 43:33-51
Morse, M. Hedlund, G.A. (1938) Symbolic dynamics. Amer. J. Math. 60:815-866.
276
Moruzzi, G. (1960) Synchronizing influences of the brain stem and the inhibitory
mechanisms underlying the production of sleep by sensory stimulation.
Electroencephal. Clin. Neurophysiol. Suppl. 13:231-256
Moruzzi, G., Magoun, R.W. (1949) Brain stem reticular formation and activation of
the EEG. Electroencephal. Clin. Neurophysiol. 1:455-473
Musha, T. (1981) 1/f fluctuations in biological systems. In Sixth International
Conference on Noise in Physical Systems. National Bureau of Standards #614 :
143-146
Nicolelis, M.A.L., Baccala, L.A., Lin, R.C.S., Chapin, J.K. (1995): Sensorimotor
encoding by synchronous neural ensemble activity at multiple levels of the
somatosensory system. Science 268:1353-1358
Nicolis, J.S. (1986) Chaotic dynamics applied to information processing. Rep. Prog.
Phys. 49:1109-1196
Nieminen, H., Takala, E.P.(1996) Evidence of deterministic chaos in the myoelectric
signal. Electromyogr Clin Neurophysiol 36:49-58
Nozaki,D., Nakazawa, K., Yamamoto, Y.(1996) Supraspinal effects on the fractal
correlation in human H-reflex. Exp Brain Res 112:112-118
Nusse, H.E., Yorke, J.A. (1991). Dynamics: Numerical Explorations. Springer-
Verlag. N.Y.
277
Ogihara,T., Tamai, I., Tsuji, A. (1998) Application of fractal kinetics for carriermediated
transport of drugs across intestinal epithelial membrane. Pharm Res
15:620-625
Olsen, L.F., Degn, H. (1977): Chaos in an enzymel system. Nature 267:177-178
Ornstein, D.S. (1989) Ergodic theory, randomness and chaos. Science 243:182-
198.
Ott, E. (1981): Strange attractors and chaotic motions of dynamical systems. Rev.
Mod. Phys. 53:655-671
Ott, E., Sauer, T., Yorke, J.A. (1994) Coping with Chaos. Wiley-Interscience. N.Y.
Packard, N., Crutchfield, J., Farmer, D., Shaw, R. (1980) Geometry from a time
series. Phys. Rev. Lett. 45:712-716
Parker, T.S., Chua, L.O. (1989) Practical Numerical Algorithms for Chaotic
Systems. Springer-Verlag. N.Y.
Parlitz, U. (1992) Identification of true and spurious Lyapounov exponents from time
series. Int. J. Bifurcation and Chaos 2:155-165
Parry, W. (1964) Intrinsic Markoff chains . Trans. Amer. Math. Soc. 112:55-66
Paulus, M.P., Bakshi, V.P., Geyer, M.A. (1998) Isolation rearing affects sequential
organization of motor behavior in post-pubertal but not pre-pubertal Lister and
Sprague-Dawley rats. Behavior Brain Res. 94:271-280
278
Paulus, M.P., Gass, S.F., Mandell, A.J. (1989): A realistic , minimal “middle layer”
for neural networks. Physica D40:135-155
Paulus, M.P., Geyer, M.A. (1992) The effects of MDMA and other methylenedioxysubstituted
phenylalkylamines on the structure of rat locomotor activity.
Neuropsychopharmacology 7:15-31
Paulus, M.P, Geyer,M.A., Braff, D.L. (1994) The assessesment of sequential
response organization in schizophrenic and control subjects. Prog.
Neuropsychopharm. & Biol. Psychiat. 18:1169-1185
Paulus, M.P., Geyer, M.A., Braff, D.L. (1996): The use of methods from chaos
theory to quantify a fundamental dysfunction in the behavioral organization of
schizophrenic patients. Am. J. Psychiat. 153:714-717
Paulus, M.P., Geyer, M.A., Gold, L.H., Mandell, A.J. (1990) Application of entropy
measures derived from the ergodic theory of dynamical systems to rat locomotor
behavior. Proc. Natl. Acad. Sci. 87:723-727
Paulus, M.P., Geyer, M.A., Mandell, A.J. (1991) Statistical mechanics of a
neurobiological dynamical system: The spectrum of local entropies, s(α) applied to
cocaine-perturbed behavior. Physica A174:567-577
Pei, X., Moss, F. (1996) Characterization of low-dimensional dynamics in the
crayfish caudal photoreceptor. Nature 379:618-621
Peng, C.-K., Buldyrev, S.V., Goldberger, A.L., Havlin, S., Simons, M., Stanley, H.E.
(1993) Finite size effects on long range correlations: implications for analyzing DNA
sequences. Phys. Rev. E47:3730-3733
279
Peng, C.-K., Havlin, S., Stanley, H.E., Goldberger, A.L. (1995) Quantification of
scaling exponents and crossover phenomena in nonstationary heartbeat time
series. Chaos 6:82-87
Perkel, D.H., Gerstein, G.L., Moore, G.P. (1967) Neuronal spike trains and
stochastic point processes. Biophys. J. 7:419-440
Pezard, L., Martinerie, J., Varela, F.J., Bouchet, F., Guez, D., Derouesne, C.,
Renault, B. (1998) Entropy maps characterize drug effects on brain dynamics in
Alzheimer's disease Neurosci Lett 253:5-8
Pezard, L., Nandrino, J.L., Renault, B., Massioui, F.E., Allilaire, F., Muller, J. (1996)
Depression as a dynamic disease. Biol. Psychiat. 39:991-999
Pijn, J., Velis, D., van der Heyden, M., DeGoede, J., van Veelen, C.W., Lopes da
Silva, F. (1997): Nonlinear dynamics of epileptic seizures on basis of intracranial
EEG recordings. Brain Topogr. 9:249-70
Pincus, S.M. (1991) Approximate entropy as a measure of a system’s complexity.
Proc. Natl. Acad. Sci. 88:2297-2301
Pincus, S.M., Gladstone, I.M., Ehrenkranz, R.A. (1991) A regularity statistic for
medical data analysis. J Clin Monit 7:335-245
Pincus, S.M.,Minkin, M.J. (1998) Assessing sequential irregularity of both endocrine
and heart rate rhythms. Curr Opin Obstet Gynecol 10:281-291
Popivanov, D., Mineva, A., Dushanova, J. (1998) Tracking EEG signal dynamics
during mental tasks. A combined linear/nonlinear approach. IEEE Eng Med Biol
17:89-95
280
Porta, A., Baselli, G., Montano, N., Gnecchi-Ruscone, T., Lombardi, F., Malliani, A.,
Cerutti, S. (1996): Classification of coupling patterns among spontaneous rhythms
and ventilation in the sympathetic discharge of decerebrate cats. Biol Cybern.
75:163-172
Pradhan, N. Sadasivan, P.K. (1996) The nature of dominant Lyapunov exponent
and attractor dimension curves of EEG in sleep. Comput Biol Med 26:419-428
Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. (1991) Numerical
Recipies; The Art of Scientific Computing. Cambridge University Press, Cambridge.
Pritchard, W.S. (1992) The brain in fractal time: 1/f-like power spectrum scaling of
the human electroencephalogram. Int J Neurosci 66:119-129
Pritchard, W.S., Krieble, K.K., Duke, D.W. (1996) Application of dimension
estimation and surrogate data to the time evolution of EEG topographic variables.
Int. J. Psychophysiol. 24:189-195
Rapp, P. E. (1993) Chaos in the neurosciences: Cautionary tales from the frontier.
The Biologist (London) 40:89-94.
Rapp, P.E. (1994) A guide to dynamical analysis. Integrat. Physiol. Behavioral. Sci.
29:311-327
Rapp, P.E. (1995) Is there evidence for chaos in the human central nervous system.
In (ed. R. Robertson and A. Combs) Chaos Theory in Psychology and the Life
Sciences. Larence Erlbaum. Mahway, N.J. pp 89-100
Rapp, P. E., Bashore, T.R., Martinerie, J.M., Albano, A.M., Mees, A.I. (1989)
Dynamics of brain electrical activity. Brain Topolography 2:99-118
281
Rapp, P.E., Goldberg, G., Albano, A.M., Janicki, M.B., Nurphy, D., Niemeyer, E.,
Jimenez-Montano, M.A. (1993) Using coarse-grained measures to characterize
electromyographic signals. Int. J. Bifurcat. Chaos. 3:525-542.
Rapp, P.E., Jimenez-Montano, M.A., Langs, R.J., Thomson, L. (1991) Quantitative
characteristization of patient-therapist communication. Math. Biosci. 105:207-227
Rapp, P.E., Schmah, T. (1996) Complexity measures in molecular psychiatry.
1:408-416
Rapp, P.E., Zimmerman, I.D., Vining, E.P., Cohen, N., Albano, A.M., Jimenez-
Montano, M.A. (1994) The algorithmic complexity of neural spike trains increases
during focal seizures. J. Neurosci. 14:4731-4739
Ren, W., Hu. S.J., Zhang, B.J., Wang, F.Z. (1997): Period-adding bifurcation with
chaos in the interspike intervals generated by an experimental neural pacemaker.
Int. J. Bifurc. Chaos 7:1867-1872
Renyi, A. (1970) Probability Theory. North-Holland. Amsterdam
Rinzel, J. (1985) Excitation dynamics: insights from simplified membrane models.
Fed. Proc. 44:2944-2946
Rissanen, J. (1982) Estimation of structure by minimum desciption length. Circuit
Systems Signal Processing 1:395-396
Roelfsema, F., Pincus, S.M., Veldhuis, J.D. Patients with Cushing's disease secrete
adrenocorticotropin and cortisol jointly more asynchronously than healthy subjects.
J Clin Endocrinol Metab 83:688-692
282
Romeiras, F.J., Bondeson, A., Ott, E., Antonsen, T.M., Grebogi, C. (19987)
Quasioperiodically forced dyanmical systems with strange, nonchaotic attractors.
Physica 26D:277-294.
Roschke, J. (1992) Strange attractors, chaotic behavior and informational aspects
of sleep EEG data. Neuropsychobiology 25:172-176
Roschke, J., Aldenhoff, J.B.(1992) A nonlinear approach to brain function:
deterministic chaos and sleep EEG. Sleep 15:95-101
Roschke, J., Aldenhoff, J.B. (1993) Estimation of the dimensionality of sleep-EEG
data in schizophrenics Eur Arch Psychiatry Clin Neurosci 242:191-196
Roschke, J., Fell, J., Mann, K. (1997) Non-linear dynamics of alpha and theta
rhythm: correlation dimensions and Lyapunov exponents from healthy subject's
spontaneous EEG Int J Psychophysiol 26:251-61
Rössler, O. (1976): An equation for continuous chaos. Phys. Lett. A57:397-398
Rössler, O.E. (1979) An equation for hyperchaos. Phys. Lett. 71A:155-157
Ruelle, D. (1978) What are measures describing turbulence. Prog. Theor. Phys.
Supp. 64:339-345
Ruelle, D.(1979) Ergodic theory of differentiable dynamical systmes. Publ. Math.
IHES 50:27-58
Ruelle, D. (1987) Chaotic Evolution and Strange Attractors. Cambridge University
Press. Cambridge
283
Ruelle, D. (1990) Deterministic chaos: The science and fiction. Proc. Roy. Soc.
Lond. 427A:241-248.
Ruelle, D., Takens, F. (1971): On the nature of turbulence. Commun. Math. Phys.
20:167-192; 23:343-344
Russo, P.V., Mandell, A.J. (1984a) A kinetic scattering approach to the nonlinear
instabilities of rat brain tyrosine hydroxylase preparations at several levels of
tetrahydrobiopterin cofactor demonstrates evolutionary behavior characteristic of
global dynamical systems. Brain Res. 299:313-322
Russo, P.V., Mandell, A.J. (1984b) Metrics from nonlinear dynamics adapted for
characterizing the behavior of non-equilibrium enzymatic rate functions. Anal.
Biochem. 139:91-99
Russo, P.V., Mandell, A.J. (1986) Sonication, calcium and peptides have
systermatic effects on the nonlinear dynamics of tyrosine hydroxylase activity.
Neurochem. Int. 9:171-176
Sadana,A.,(1998) Analyte-receptor binding kinetics for biosensor applications. An
analysis of the influence of the fractal dimension on the binding rate coefficient. Appl
Biochem Biotechnol 73:89-112
Saltzman, B. (1962): Finite amplitude free convection as an initial value problem. J.
Atmos. Sci. 19:329-341
Sammer, G.(1996) Working-memory load and dimensional complexity of the EEG.
Int J Psychophysiol 24:173-182
284
Sammon, M.P., Bruce, E.H. (1991) Vagal afferent activity increases dynamical
dimension of respiration in rats. J Appl Physiol 70:1748-1762
Sano, M., Sawada, Y. (1985) Measurement of the Lyapounov spectrum from
chaotic time series. Phys. Rev. Lett. 55:1082-1085
Sansom, M.S., Ball, F.G., Kerry, C.J., McGee, R., Ramsey, R.L., Usherwood, P.N.
(1989) Markov, fractal, diffusion and related models of ion channel gating. A
comparison with experimental data from two ion channels. Biophys J 56:1229-43
Sato, S., Sano, M., Sawada, Y. (1987) Practical methods of meaureing the
generalized dimension and largest Lyapounov exponent in high dimensional chaotic
systems. Prog. Theor. Phys. 77:1-5
Sauer, T., Yorke, J.A., Casdagli, M. (1991) Embedology. J. Stat. Phys. 65:579-616
Schiff, S.J., So, P., Chang, T., Burke, R.E., Sauer, T. (1997): Detecting generalized
synchrony through mutual prediction in a spinal neural ensemble. Preprint
Schierwagen, A.K. (1990) Growth, structure and dynamics of real neurons: model
studies and experimental results Biomed Biochim Acta 49:709-722
Schmeisser, E.T.(1993) Fractal analysis of steady-state-flicker visual evoked
potentials: feasibility. J Opt Soc Am 10:1637-1641
Schuster, H.G. (1989): Deterministic Chaos. Second Edition, VCH, Weinheim,
Germany.
Scott, A. (1990) The solitary wave: An enduring pulse first seen in water may
convey energy in the cell. Science 30:28-35
285
Segundo, J.P., Sugihara, G., Dixon, P., Stiber, M.,Bersier, L.F.(1998) The spike
trains of inhibited pacemaker neurons seen through the magnifying glass of
nonlinear analyses. Neuroscience 87:741-66
Selz, K.A. (1992) Mixing Properties in Human Behavioral Style and Time
Dependencies in Behavior Identification; The Modeling of a Universal Dynamical
Law. Doctoral Disseration. UMI. N.Y.
Selz, K.A., Mandell, A.J. (1991) Bernoulli partition-equivalence of intermittent
neuronal discharge patterns. Int. J. Bifurcation Chaos 1:717-722.
Selz, K.A., Mandell, A.J. (1992) Critical coherence and characteristic times in brain
stem neuronal discharge patterns. In (ed. T. McKenna, J. Davis, S.F. Zornetzer)
Single Neuron Computation, Academic. N.Y. pp 525-560.
Selz, K.A., Mandell, A.J. (1993): Style as mechanism: from man to a map of the
interval and back. In (ed. W. Ditto) Chaos in Medicine and Biology, Proc. SPIE
2036:174-182
Selz, K.A., Mandell, A.J. (1997) The power of style: differential operator scaling in
the lexical compression of sequences generated by psychological, content-free
computer tasks. Complexity 2:50-55
Selz, K.A., Mandell, A.J., Anderson, C.A., Smotherman, W., Teicher, M. (1995)
Distribution of local Mandelbrot-Hurst exponents: motor activity in cocaine treated
fetal rats and manic depressive patients. Fractals 3:893-904
Seneta, E. (1981) Non-negative Matrices and Markoff Chains. Springer-Verlag, New
York.
286
Shannon, C.D., Weaver, W. (1949) The Mathematical Theory of Communication.
University of Illinois Press, Urbana
Sharkovskii, A.N. (1964): Coexistence of the cycles of a continuous mapping of the
line to itself. Ukranian Mat. Z. 16:61-71
Shaw, R. (1981): Strange attractors, chaotic behavior and information flow. Z.
Naturfursch. 36a:80-112
Shenker, S.J., Kadanoff, L.P. (1982): Critical behavior of a KAM surface. J. Stat.
Phys. 27:631-645
Shlesinger, M.F. (1988) Fractal time in condensed matter. Ann. Rev. Phys. Chem.
39:269-290
Shlesinger, M.F. (1996) Random Processes. Encyclopedia of Applied Physics.
16:45-70
Shlesinger, M.F., Klafter, J., Wong, Y.M. (1982) Random walks with infinite spatial
and temporal moments. J. Stat. Phys. 27:499-512
Shlesinger, M.F., Zaslavsky, G.M. (1996) Strange kinetics. Physics Today. Feb. pg.
33-39.
Shlesinger, M.F., Zaslavsky, G.M., Frisch, U. (1995) Levy Flights and Related
Topics in Physics. Springer-Verlag
287
Silipo, R., Deco, G., Vergassola, R., Bartsch, H.(1998)Dynamics extraction in
multivariate biomedical time series. Biol Cybern 79:15-27
Simoyi, R.H., Wolf, A., Swinnery, H.L (1982): One dimensional dynamics in a
multicomponent chemical reaction. Phys. Rev. Lett. 49:245-248
Sinai, Ya G. (1959) On the concept of entropy of a dynamical system. Dokl. Akad.
Nauk SSSR 124:768
Singer, W. (1993): Synchronization of cortical activity and its putative role in
information processing and learning. Ann. Rev. Physiol. 55:349-374
Skinner, J.E., Martin, J.L., Landisman, C.E., Mommer, M.M., Fulton, K., Mitra, M.,
Burton, W.D., Saltzberg,B. (1990) Chaotic attractors in a model of neocortex:
dimensionalities of olfactory bulb surface potentaisl are spatially uniform and eventrelated.
In (eds. Basar, E., Bullock, T.H.) Brain Dynamics, Progress and
Perspectives. Springer-Verlag, Berlin. pp 119-134
Smale, S. (1967): Differentiable dynanmical systems. Bull. AMS 13:714-817
Smith, L.A. (1988) Intrinsic limits on dimension calculations Phys. Lett. A133:283-
288
Smith, L.A. (1992) Identification and prediction of low-dimensional dynamics.
Physica D58:50-76
288
Smotherman, W.P., Selz, K.A., Mandell, A.J. (1996) Dynamical entropy is
conserved during cocaine-induced changes in fetal rat motor patterns.
Psychoendocrinology 21:173-187
So, P., Francis, J.T., Netoff, T.I., Gluckman, J., Schiff, S.J. (1998) Periodic orbits: a
new language for neuronal dynamics. Biophys. J. 74:2776-2785
So, P., Ott, E., Sauer, T., Gluckman, B.J., Grebogi, C., Schiff, S.J. (1997) Extracting
unstable periodic orbits from chaotic time series. Phys. Rev. E55:5398-5417
Sprott, J.C. (1993) Strange Attractors; Creating Patterns in Chaos. M&T Books.
N.Y.
Sprott, J.C., Rowlands, G. (1991) Chaotic Data Analyzer. Am. Institute Physics.
N.Y.
Stam, C.J., Nicolai, J., Keunen, R.W. (1998) Nonlinear dynamical analysis of
periodic lateralized epileptiform discharges Clin Electroencephalogr 29:101-105
Stanley, H.E. (1971): Introduction to Phase Transitions and Critical Phenomena.
Oxford University Press, Oxford
Stauffer, C. (1985) Introduction to Percolation Theory. Taylor and Francis. London
Stergiopulos, N., Porret, C.A., De Brower, S., Mesiter, J.J. (1998) Arterial
vasomotion: effect of flow and evidence of nonlinear dynamics. Am J Physiol
274:H1858-1864
Steriade, M., McCarley, R.W. (1990) Brainstem Control of Wakefulness and Sleep.
Plenum. N.Y.
289
Stillwell, J. (1985) Poincare’s Papers on Fuchsian Groups. Springer-Verlag, N.Y. pp
1-50
Stockbridge,L.L., French,A.S. (1989) Characterization of a calcium-activated
potassium channel in human fibroblasts Can J Physiol Pharmacol 67:1300-1307
gating. Biophys J 54:871-877
Stoop, R., Parisi, J. (1991) Calculations of Lyapounov exponents avoiding spurious
elements. Physica D50:89-94
Straver,J., Keunen, R.W., Stam, C.J., Tavy, D., de Ruiter, G., Smith, S., Thijs, L.,
Schellens, R., Gielen, G.(1998): The EEG diagnosis of septic encephalopathy.
Neurol. Res. 20:100-106
Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos. Addison-Wesley.
Strong, S.P., de Ruyter van Steveninck, R.R., Bialek, W., Koberle, R. (1998) On the
application of information theory to neural spike trains. Pac Symp Biocomput 621-
632
Sugihara, G., May, R.M. (1990) Nonlinear forcasting as a way of distinguishing
chaos from measurement error in time series. Nature 344:734-741.
Sullivan, D. (1979) Rufus Bowen. Publ. Math. I.H.E.S. 50:255
Szeto, H.H., Cheng, P.Y., Decena, J.A., Cheng, Y., Wu, D., Dwyer, G. (1992)
Fractal properties in fetal breathing dynamics. Am. J. Physiol. 263:R141-R147.
Takahashi, N., Hanyu, Y., Musha, T., Kubo, R., Matsumoto, G. (1990): Global
bifurcation structure in periodically stimulated giant axons of squid. Physica
D43:318-334
290
Takens, F. (1981) Detecting strange attractors in time series. Lect. Notes. Math.
898:366-381
Teich M.C. (1989) Fractal character of the auditory neural spike train. IEEE Trans
Biomed Eng 36:150-60
Teich, M.C., Heneghan, C., Lowen, S.B., Ozaki, T., Kaplan, E. (1997) Fractal
character of the neural spike train in the visual system of the cat. J Opt Soc Am A
14:529-546