the brain as a dynamic physical system

Pergamon 0306-4522(94)E0008-R

Neuroscience Vol. 60, No. 3, 587-605, pp. 1994 Elsevier Science Ltd

IBRO Printed in Great Britain

0306-4522/94 $7.00 + 0.00

COMMENTARY

THE BRAIN AS A DYNAMIC PHYSICAL SYSTEM

T. M. MCKENNA,*~ T. A. MCMULLEN? and M. F. SHLESINGER~

TDivision of Cognitive and Neural Sciences and IDivision of Physics, Office of Naval Research, Arlington, VA 22217, U.S.A.

Abstract-The brain is a dynamic system that is non-linear at multiple levels of analysis. Characterization of its non-linear dynamics is fundamental to our understanding of brain function. Identifying families of attractors in phase space analysis, an approach which has proven valuable in describing non-linear mechanical and electrical systems, can prove valuable in describing a range of behaviors and associated neural activity including sensory and motor repertoires. Additionally, transitions between attractors may serve as useful descriptors for analysing state changes in neurons and neural ensembles.

Recent observations of synchronous neural activity, and the emerging capability to record the spatiotemporal dynamics of neural activity by voltage-sensitive dyes and electrode arrays, provide opportunities for observing the population dynamics of neural ensembles within a dynamic systems context. New developments in the experimental physics of complex systems, such as the control of chaotic systems, selection of attractors, attractor switching and transient states, can be a source of powerful new analytical tools and insights into the dynamics of neural systems.

CONTENTS

1. Introduction 2. Single neuron dynamics 3. Transitions between dynamic states: driving parameters 4. Transitions produced by graded inputs 5. Perturbation-induced transitions 6. Noise-induced transitions 7. Neural ensemble dynamics at the mesoscale 8. Intermittent synchronized oscillations 9. Spatiotemporal modes of activity in the brain

IO. Coupled non-linear oscillators in motor control: chains of oscillators 11. Mammalian olivocerebellar system as a two-dimensional isochronous sheet of

coupled oscillators 12. Conclusions

References

587 588 590 591 592 592 594 594 598 600

601 602

1. INTRODUCTION

Until recently, physics lacked the tools to contribute to the analysis of complex biological systems. Physics has traditionally focused on the very large, the very small and, in general, the very simple physical systems. However, the recent emergence of a physics of complex systems which explicitly considers non- linear, non-equilibrium, non-stationary, open and strongly coupled systems has provided a new oppor- tunity for neuroscience to pursue the analysis of the brain as a dynamic physical system. This is a timely development, since current neuroscience has provided abundant evidence that, at all levels, the nervous

*To whom correspondence should be addressed at: ONR Code 342 CN, 800 N. Quincy Street, Arlington, VA 222 17-5660, U.S.A.

Abbreviations: EEG, electroencephalogram: IO, inferior olive; MEG, magnetoencephalogram; RF, receptor field; SR, stochastic resonance.

system is rich with complex, non-linear dynamic behavior. From single neuron burst patterns to global electroencephalogram (EEG) measures, phase space descriptors have revealed a wide range of non-linear dynamic phenomena. While earlier neural models incorporated non-linear equations and attempts were made to account for non-linear behavior in experimental data (cf. Refs 10, 22, 27, 37, 72 and 97), in the last decade there have been a number of advances in the understanding of the rich behaviors of non- linear systems, including many complex behaviors which only became apparent with the widespread use of high performance computers. These newer developments may give important insights into the behavior of complex living systems, including neural systems.

We will review the evidence on non-linear dynamic system behavior in neural systems, including the single neuron level, neural ensembles in sensory and motor systems, and spatiotemporal modes of activity

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in large neural ensembles. In addition to recounting these significant new results, we hope to provide the neuroscience reader with a glimpse of the tremendous potential for dynamic systems theory in under-

standing the sequential changes in system dynamics underlying sensorimotor behavior. Just as physical

systems can be characterized as evolving across different regimes (periodic, quasi-periodic, chaotic, etc.) under the control of one or more system parameters, neurons and neural ensembles also pass through a repertoire of regimes under the control of driving inputs such as precisely timed synaptic

inputs. An important physical property playing a critical role in such state transitions is noise, and we will review two new concepts on the role of noise in neural systems. Most man-made systems, and engineering paradigms that have spawned many of the analogies that guide neural models (batteries, electric power grid, telephone switchboard, servo- control, electronic circuits, digital computers), are designed to be stable. By contrast, we advance the hypothesis that real neural systems operate near instability, and this confers the ability to respond rapidly and with a large flexible repertoire of sensory and motor patterns.

It is helpful to define the terms used in characterizing dynamic systems. The “phase space” of a dynamic system is a mathematical space with independent co-ordinates representing the dynamic variables needed to specify the instantaneous state of the system.4.65 A common example in mechanical systems would be a plot of position vs velocity. In the case of neural activity, one might plot two or more measurable variables. A “phase space embedding” is a type of multidimensional phase plot which represents the relations among F(t), F(t + At), F(t + 281) where F(t) is a measurable variable (e.g. voltage) at time t and A\t is an appropriate lag. This is particularly useful with experimental neural data, where it is much easier to obtain measures of a single variable at many times, but it may be difficult to directly measure other system parameters which reflect independent degrees of freedom. An “attractor” is a trajectory or point in phase space to which the system will converge from a set of initial co-ordinates. A non-linear system may have more than one attractor. One important example of an attractor is the limit cycle. We will discuss limit cycles in relation to central pattern generators for motor control.

One can measure the average rate at which neighboring trajectories on an attractor diverge in a phase plot via a rate called the “Lyapunov exponent”. If it is negative, orbits of the trajectory tend to converge and the orbit is stable. If the Lyapunov exponent is positive for at least one degree of freedom, adjacent orbits diverge exponentially in time, and the orbit is “chaotic”. If the attractor for the orbit is fractal, then the attractor is called a “strange attractor”. Strange attractors without chaotic orbits also exist. Several examples exist where

a periodically driven non-linear system possesses a strange chaotic attractor, and the same system when driven quasi-periodically possesses a strange non-chaotic attractor (quasi-periodic means driven by two incommensurate frequencies). When a dynamic system exhibits a strange attractor, it also exhibits high sensitivity to initial conditions. Identify- ing attractors in phase space is not the end goal of

neural dynamics; however, they provide the building blocks for describing much more complex system behavior. For example, in the domain of the physics of complex systems (see Vohra et LZ~.~~), small changes in the conditions in systems operating in a very non-linear fashion, near instabilities, can change

attractors to repellers, change fixed points from stable to unstable, make attractors collide, and create, destroy, modify or reverse the sequence of changes in phase space. These changes can be used to detect, control and synchronize signals in physical

systems, and they provide a rich repertoire for describing the dynamics of neural systems. Another advantage of describing neural activity in terms of attractors in phase space is that it may permit the use of the same descriptors at different levels of organization of the nervous system, e.g. single neurons and neural ensemble activity, and permits comparison across sensory and motor systems as well as across species. Phase space descriptors help to identify dynamic patterns in data that are not obvious in traditional physiological measurements. The examples we will present are meant to be illustrative, rather than definitive, for these pioneering studies are at the beginning of a long series of investigations that will likely produce far more sophisticated dynamic descriptors of neural systems as the full range of neural phenomena are considered.

2. SINGLE NEURON DYNAMICS

Currently, the best characterized neurons, in terms of dynamic analysis, are the invertebrate bursting neurons. In both experimental and modeling analyses, evidence has emerged that these neurons exhibit chaotic attractors in phase space.‘6~‘7~20@~70 Mpistos et a1.6*.69 analysed the bursting pattern in motorneu- rons in the sea slug Pleurobranchea. The time series derived from the envelopes of the high frequency bursts of these neurons exhibited the sign of strange chaotic attractors: a positive Lyapunov exponent. In addition, low correlation dimensions were calcu- lated, which is consistent with clear structure in the Poincare plots. [For a time series one can calculate how many points N(E) are within E in value of a chosen point. The manner in which this varies with E determines the “correlation dimension” d, as N(E) E Ed, as E tends to zero.] “Poincare plots” or “sections” are slices cut through the phase space embedding plots. An example is shown in Fig. 1.68 At the top is the low-pass filtered burst pattern of a single neuron. Below it is a 2-D phase space

Non-linear dynamics of neural systems

Bursting Pattern from Invertebrate Neuron

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l!.,., ‘I I 8.1, c ‘1 0 25 50 75 100 125 150 175

Time Ls)

Phase Space Embedding PoincarCt Section

f.0 0.8 w3

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Fig. I. Phase space analysis of the bursting pattern of a motor neuron from the mollusc P~e~~o~r~chea during a bout of the bite-swallow motor pattern. Top: frequency of discharge of the neuron (reciprocal of interspike intervaIs). Bottom left: 2-D phase portrait of band-pass filtered version of frequency plot at top. A Poincare section was taken at the horizontal line. Bottom right: Poincare section or map of return times as a function of preceding return time, constructed by plotting the successive times at which trajectories pass in the same direction through the horizontal line. The numbers next to each point indicate

the sequence of ordered pairs. After Mpistos et a1.68.69

embedding, and a Poincare section of the embedding, taken from the indicated slice through the embedding. Each 360” turn of the trajectory in the phase embedding corresponds to a burst of the neuron. In the Poincare section, points show the return time where the trajectories cross the 1-D section, and the time ordered sequence of numbered points characterize the system dynamics. In a later section we will show how physicists use real-time Poincare plots to track and control the dynamics of physical systems and isolated hearts.

What is the significance of chaotic-like bursting to the organism? In this case, Mpistos et a1.69,70 speculate that the chaotic and variable activity of neurons could provide a route for pattern switching in the neural ensemble. The sensitivity to initial conditions leads to the blending of several behavioral responses in the presence of disparate cues which alone elicit different responses, via a self-organizing co- operativity within an ensemble, as well as providing a history and context sensitivity to the ensemble.

Recently, Byrne and his associates’6 demonstrated seven different attractors in the bursting pattern of model ApZysiu R15 neurons in realistic simulations. These model neurons show seven different modes of bursting. In phase plane plots of the two most slowfy changing parameters during the bursts of model neuron (internal Ca2+ and activation of a slow inward current), seven separate attractors are revealed (Fig, 2). Some of these attractors are limit cycles, and some are strange chaotic attractors. In the simulations, the neuron will remain in one attractor (for many thousands of orbits) until a synaptic input is applied at a particular phase of the burst, then the neuron shifts to another attractor. This result raises a number of issues. First, under what conditions and by what means does the organism use this attractor switching? Is this used for multiplexing in neuron-sparse invertebrate circuits? Is there an equivalent m~hanism in vertebrates? The ability of perturbations to switch dynamic modes in a phase- sensitive manner brings us to the general issue of how

590 T. M. MCKENNA rt al.

.6 -

I I I I 250 300 350 400 4!

Internal Free Calcium Concentration (nM)

Fig. 2. Phase plane projection of dynamic activity of a model of the R15 neuron from the mollusc Aplysiu. The values of two model quantities, activation of a slow inward current and internal calcium concentration, were plotted for each attractor for 200 s of simulation time, after 6 h of simulated time had elapsed to assure steady state activity. The different trajectories correspond to distinct discharge patterns produced by different initial values of the state variables. I, Bursting limit cycle; II, outer chaotic attractor;

III-V, limit cycle; VI, chaotic attractor; VII, beating limit cycle. After Canavier er al.”

complex systems move through multiple dynamic states.

3. TRANSITIONS BETWEEN DYNAMIC STATES: DRIVING PARAMETERS

One way to track the qualitative changes in the dynamics of a system is the bifurcation diagram. The original use of this diagram was to plot changes in the solution to a differential equation as a parameter is varied. However, it can also be used to plot experimental data, and can reveal transitions to different dynamic states. Figure 3 shows a bifurcation diagam for a physical system simulation. A horizontal line in this figure corresponds to periodic steady-state solution or harmonic component; at bifurcation points two new periodic solutions arise. Following some of these bifurcations, a chaotic regime with a broad complex spectrum can arise. This sequence of behavioral changes is referred to as the period doubling route to deterministic chaos. Byrne has previously provided evidence that the RI5 burster can proceed through this period doubling route to chaos. However, physicists have identified additional routes to chaos in complex systems: intermittency, in which chaotic bursts become progressively longer as a system parameter is varied, and the quasi-periodic route to

chaos in which several simultaneous independent frequencies appear before the chaotic regime is entered. These routes have not been demonstrated in neural systems, nor adequately investigated. Additionally, chaotic crises and transient chaos are active areas of investigation in physical systems, with strong implications for physiological systems. Chaotic crises occur where a small change in some parameter produces a large change in attractor structure.24,h5

The current diagnostic tools for deterministic chaos require that the process studied is stationary. Typically, in a physical system the first 100 or so periods are eliminated from the analysis, and thousands of periods are used for definitive evidence of deterministic chaos. In physiological systems, non- stationarity is common. and transient signals are likely to be of paramount importance. Fortunately, the new emphasis on characterizing transient dynamic states is very timely.‘” Our current limitations on identifying pure deterministic chaos in the brain from physical measurements has become some- thing of a distraction. The real issues are identifying sequences and transitions among dynamic states or attractor structures in brain systems, identifying what controls the transitions between these states and determining the extent to which dynamic system analysis will permit us to find new candidate codes

Non-linear dynamics of neural systems 591

Period 3 Multiple Windows

Increasing Nonlinearity /

Saddle Node Bifurcation

Driving Parameter

Fig 3. Evolution of the solutions for equations describing a non-linear system (periodically forced pendulum) as the driving parameter is increased. The system proceeds through a sequence of state changes, through period doublings or bifurcations and into chaotic states. At some stages, small changes in the driving parameter can produce dramatic changes in state. We propose that some neural systems may reside near points of instability in order to effect rapid transitions in state. This plot was produced by simulations

run by M. Shlesinger. A similar mechanical example is developed in detail in Baker and Gollub.4

and representations for behavioral and perceptional sequences. It is also apparent that analysis techniques for identifying non-stationary non-linear dynamic systems are needed.

One can categorize neural dynamic phenomena either by the nature of the neural activity and its sequential changes as represented by measurements or states or, alternatively, according to the kinds of controlling inputs or system parameters which regulate transitions between dynamic states. The latter approach is similar in spirit to neuroscience experiments in which stimuli or chemicals are presented and the effects described. For the phenomena present at the level of single neurons, we can divide the inputs which promote transitions into three broad classes: graded or smoothly varying, sharp pertubations and noise.

4. TRANSITIONS PRODUCED BY GRADED INPUTS

One class of neural state transitions that has been described experimentally and analytically for single neurons is bistability. Bistability has been demonstrated in both vertebrate and invertebrate neurons, including neurons in thalamus,40,64 basal ganglia,lo3 inferior olive,57 cerebellar Purkinje cellis and stomatogastric ganglion.42.82 Neuro- physiologists use the term bistable to indicate that neurons have two nearly stable states. To physicists, bistability may mean that a system has two states

with the same parameter values, but experimental neuroscientists usually have no way of determining if a governing parameter is identical. Wang and Rinzellw have performed a phase plane analysis of bistable behavior for a pair of inhibitory neurons, and demonstrated two mechanisms. One mechanism involves the decay rate of synaptic potentials and the other slowly inactivating currents, which produce synchronous or alternating oscillations in membrane potential. Bistable neuronal discharge patterns can be controlled by application of neuromodulators. In vertebrates, bistability of discharge in thalamic neurons and motor neurons can be induced by neuromodulators such as norepinephrine, serotonin and acetycholine, which are known to be released by diffusely projecting neurons whose activity level varies with behavioral state.44.M In invertebrates, neuromodulators induce bistability in neurons by altering intrinsic neuronal properties. In ensembles, neuromodulators can trigger motor patterns or co-ordinate multiple motor patterns.42.82

Another graded parameter that potentially controls neuronal and ensemble dynamic state is the average level of activity (discharge rate) in the entire network. (The average rate is likely to be some combination of extrinsic and recurrent input activity, the transfer function of the neurons and the “spontaneous” neuron discharge. The latter two can be regulated by neuromodulators.) Three separate


model analyses have demonstrated that the electrotonic structure and time constants of cerebral cortical pyramidal neurons,’ basal ganglia neurons”’ and Purkinje cells” can be strongly modulated by the level of “background” synaptic input on the dendrites. The dynamic consequences of such background activity have not been explored. Its relevance for neuron dynamics is all the more likely given that some extrinsic inputs are often strongly modulated during behavior. Graded changes in synaptic efficacy or weight may also control transitions to different dynamic regions. In simple artificiat neural networks with as few as three neurons, complex dynamics have been observed which produce bifurcation plots similar to Fig. 3, including bifurcations and chaotic regimes.” The control parameter in this model is synaptic weight. While the neurons in such simulations are not realistic, the role of synaptic weight, which can be modified quickly via long-term potentiation, in ensemble dynamics remains largely unexplored in terms of dynamic system analysis.

5. PERTURBATION-INDUCED TRANSITIONS

Perturbations are very effective in shifting dynamic systems from one regime to another (see, for example, Moo@). In the neural context these perturbations can be precisely timed synaptic inputs (e.g. the shift between attractors in the work of Byrne et u/.,‘~

volleys or synchronized inputs,’ or even transient sensory inputs”‘). Physicists have devised means for controlling magnetoelastic chaotic systems, and selecting out stable periodic motions by moving the stable point and repeller and attractor axes by tracking the Poincare map, but not necessarily knowing the underlying equations of motion. This technique has been extended to electrical and laser systems, and successfully applied to control of the heart.” This experiment is pa~icularly instructive in indicating the potential of dynamic systems techniques in characterizing the dynamics of neural systems. A cardiac arrhythmia was induced in an isolated heart by pharmacological means. At successive stages during arrhythmias, the spontaneous period beating passed through period 2 and higher order periodicities into an aperiodic regime. A Poincare plot of successive interbeat intervals was computed in real time. Based on the theory of Ott et al., stable (attractor) and unstable (repeller) manifolds (axes) (Fig. 4) were computed. By following the successive points on this return map, these investigators could compute precisely when to deliver an electrical perturbation or impulse that would bring the next point on the Poincare plot close to a stable manifold or axis. When this control was applied. the chaotic aperiodic beating was replaced by periodic beating. The precise timing of the perturbations required to move the system to a new regime could be computed by tracking the empirical system dymunics in a Poincare section of the phase diagram. Such empirical Poin-

care maps provide a significant new tool for the neurophysiologist concerned with the dynamics of single neurons and ensembles.

6. NOISE-INDUCED TRANSITIONS

Noise is also an important cause of phase transitions in dynamic systems. Noise in neural systems

can arise from many sources, including channel noise, multiple types of synaptic noise and noise in the sensory inputs and transduction processes. Recently, two new theories arising from non-linear dynamic systems have been advanced on the role of noise. In the first theory, noise facilitates state transitions and

the resulting phenomenon, called stochastic resonance (SR). In the second theory, noise can stabilize systems in certain regions of their attractors.

SR is a special example of the result of combining periodic and stochastic forcing in a multistable non- linear system. hh In the 1-D case, an overdamped

particle in a double well potential is subject to random forces from thermal fluctuations, and a periodic external modulation which is weak and additive (effectively raising and lowering the potential wells). SR arises because of the interplay between the modulating frequency and the mean switching rate (Kramers’ rate) between wells. This interaction accounts for the coherence between the response and modulation frequency. There is an optimum noise intensity that maximizes coherence (resonance), and in real physical systems the power spectrum exhibits a sequence of sharp peaks with decreasing amplitude at odd or all integer multiples of the modulating frequency. SR has been examined in electronic circuits, ring lasers, superconducting quantum interference devices, electron paramagnetic resonance. magnetoelastic beams and in afferent neurons in crayfish mechanoreceptors and monkey auditory nerve.“‘~~4~*x 60.66.67 In the electronic systems, the goal is to use SR to detect very weak periodic signals in noise by using the noise-enhanced switching between system states. SR can account for the exact form of the multimodal interspike interval histograms of monkey auditory nerve with tone driving. The data can bc matched by adjusting a single parameter in electronic simulations (either noise intensity or stimulus intensity).“” *” Hence this very simple physical mechanism is sufficient to account for the neural

data. More recently, Moss and co-workers have tested

the SR mechanism in crayfish me~hanoreceptors.‘~ Sharp peaks in the power spectral density of the compound action potentials were observed at multiples of the driving input riding on Lorentzian noise, as predicted by the theory (Fig. 5). Addition- ally, the interspike intervals also exhibit peaks at integer multiples of the mechanical driving frequency, and the Poincare plots of return times show a non-random structure. Many neurons in the CNS also receive both stochastic and weak


Time (5)

1

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800 1000 1200 n

Fig. 4. Control of heart beat by analysis of chaotic behavior in real-time phase plane analysis. Top right: recordings of monophasic action potentials during arrythmia induced by ouabain-epinephrine in typical rabbit septa. Left: Poincare map of interbeat intervals. The parabolic pattern of the intervals is diagnostic of chaos for an appropriate parameter range. By tracking the sequence of data points in real time, a stable and unstable manifold can be identified. When interval points occur near the stable manifold, the next interval point will occur closer to the fixed or stable point. By contrast, in cases where interval points occur close to the unstable manifold, the next point will occur farther from the fixed point. Tracking the actual interspike intervals on this Poincare plot permits one to specify the instant at which an electrical stimulus should be delivered to move the next interval closer to the stable manifold. Bottom right: when this control technique was applied at every third beat, and maintained on an arrhythmic heart beating chaotically, period 3 beating was observed, which reverted to chaotic beating when the control was removed. After

Garfinkel er a/.r’

periodic inputs (e.g. 10 Hz in cerebellum and cerebral cortex and 25-50 Hz in cerebra1 cortex; see sections on ensemble dynamics).

A new type of phenomenon was recently described in non-linear electronic circuits which is also referred to as SR. In classical SR, there are two potential wells or point attractors; in this new form the “particles” jump between two chaotic attractors which are generated by a non-linear electronic circuit developed by Chua called the double scroll circuit.’ For both kinds of SR, there is now strong interest in using arrays

of stochastic resonators for tasks such as adaptive image processing or designing more effective auditory prostheses. This new type of SR in Chua’s circuit has not been examined explicitly as a neural model, but it may be formally equivalent to a simplified version of Freeman’s model of the olfactory system.”

Noise can also stabilize dynamic systems. In recent neuron simulations based on Van der Pol equations for a highly non-linear relaxation oscillator, increasing the noise level added many loops to the attractor orbits, and increased the density of long dwell times

594 T. M. MCKENNA rt ul.

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xE 0

Fig. 5. Experimental evidence for a stochastic resonance mechanism in crayfish mechanoreceptors. Bottom: a power spectrum obtained from the compound action potential from crayfish mechanoreceptor afferents with the associated hair stimulated at 35 Hz. Note the delta-function-like signal at the stimulus frequency and second harmonic, and the Lorentzian-like noise background. Top left: interspike interval histogram of mechanoreceptor afferents, exhibiting multi-peaked interspike intervals with amplitude distribution predictable from stochastic resonance theory. Top right: signal-to-noise ratio (SNR) of afferent discharge as a function of temperature. The non-monotonic function is predicted by stochastic resonance theory when temperature is equated with internal noise of the mechanoreceptor transduction

process. After Moss. unpublished; Bulsara cr (11.”

in a saddle-sink region of the attractor.“’ In neural terms this was interpreted as increasing the influence of slower membrane events such as adaptation, and long duration afterhyperpolarization that would promote burst modes of firing. When applied to cortical EEGs, the interpretation given was that increasing noise amplitude input by non-specific brainstem inputs would serve to select and stabilize the EEG in the single spectral mode (one dominant peak in the EEG spectrum), multiple mode and then spindle-burst patterns.

7. NEURAL ENSEMBLE DYNAMICS AT THE MESOSCALE

Evidence of non-linear dynamics is also emerging at the level of neural ensembles. We review some examples in this section. Unfortunately, there has not yet emerged a systematic account which relates the single neuron dynamics to the ensemble dynamics, nor uniquely characterizes the dynamics of these levels, nor identifies the specific neural substrates which dominate the dynamics at different levels. An important exception is provided by Freeman’s” earlier work, which analysed the olfactory bulb from cellular to ensemble levels using classical systems theory.

8. INTERMITTENT SYNCHRONIZED OSCILLATIONS

An intriguing example of non-linear ensemble dynamics is provided by the intermittent synchronized oscillations observed in cortical ensembles. These oscillations may appear over a narrow range of the spectrum (25550 Hz), are synchronous between separate brain sites, last for a few hundred milliseconds or less, and correlated oscillations are observed at multiple sites in sensory regions in response to particular sensory configurations’4.26” or between sites in sensory and motor regions at particular phases of a sensorimotor task.y.71.79 Cross- correlogams for neural activity recorded from two separate electrodes show peaks at phase lags between 0 and a few milliseconds. This phenomenon and other examples of “anomalous dispersion” (Ref. 29 and see below) appear to have too short a time lag to be accounted for by direct axonal conduction velocity, reflecting either common driving or non-linear system properties.

In the case of cat visual cortex areas 17, 18 and 19, these synchronized oscillations or neuronal bursts can be recorded at nearby sites or at sites separated by 7 mm.” Cross-correlograms reveal synchronized 25-50 Hz oscillations during activity driven by effective visual stimuli in the respective receptive fields


(RFs) of the two sites. The degree of synchrony depends on the stimulus configuration, e.g. single

bars across both RFs produce stronger cross-corre-

lation peaks than do separate bars in the same direction crossing the RFs, while two bars in opposed directions to the RFs give zero cross-correlation. Although these oscillations are elicited by the visual stimulus, they are not locked to the stimulus.

Cluster recordings from multiple neurons, and some- times single neurons, exhibit the rhythmic bursting. Neurons at a given site can shift from one synchronous ensemble to another depending on the stimulus conditions.25 For example, for four cortical sites with overlapping RFs, cells can be synchronized in all possible pair combinations in response to single light bars with the appropriate orientation, but simultaneous presentation of two different orientations can selectively synchronize a small subset of pairs of sites depending on their orientations. Transient synchronized oscillations are also observed in non-anesthetized monkey visual area MT49 and anesthetized area Vl in squirrel monkey.50

Transient coherent activity has also been observed between visual and sensorimotor cortical sites in behaving monkeys, although the spectral content is much broader than reported for cat sensory cortex. Bressler and Nakamura9 recorded evoked potentials at multiple sites in visual, parietal and sensorimotor cortex in monkeys performing a go/no-go discrimi-

nation task. During the response in the go condition, a subset of paired sites in visual and sensorimotor cortex showed a strong peak in coherence (cross- power). The coherence peak. was spectrally broad over the range of O-100 Hz, with amplitude falling off in a l/fmanner. However, with band-pass filtering in the gamma region, oscillations are evident in individual records from visual and motor cortex, which are in phase for four to five cycles just preceding the peak of coherence in the unfiltered records (Fig. 6).9

Murthy and Fetz” observed synchronous 25-35 Hz oscillations and zero phase lag in sites 20 mm distant in somatosensory and motor cortex of behaving monkeys. The transient nature of the synchronous activity in sensorimotor areas is further evident in the observations of Sanes and Donoghue.79 Local field potentials were recorded simultaneously at up to 12 sites in the motor and premotor cortex of monkeys performing visually guided, instructed delay tasks using wrist or finger movements. Oscillations at 25-50 Hz were observed most frequently prior to the signal to start movements. Significantly, the synchronization was observed at long distances and at individual sites the periodic waves could be synchronized for many cycles, phase-shifted or go in-and-out of phase within a few cycles. Neither the relevant sensory, motor nor behavioral variables, and the full topography of these phase differences between neural ensembles, have been characterized in these experiments. However, striking new results from Simmons on auditory system in

bats, described below, show a systematic relation between phase shifting of oscillations and stimulus

parameters.23,39

The biosonar of the bat enables it to discriminate fine surface features of targets as well as minute differences in the distance of different target surfaces. In psychophysical tasks, bats are capable of echo resolution in the 1 ps range.87 This raises a problem

for auditory neurophysiology. How can single neurons which operate in the millisecond range reliably encode microsecond differences in acoustic signals? Simmons’ data suggest that the solution is found at the neural ensemble level, and involves the systematic phase shift of population oscillations in response to acoustic stimuIi.23,39 Multi-unit recordings in inferior colliculus of the big brown bat revealed that oscillations are evoked in response to FM sweep pairs corresponding to emission and echoes, presented binaurally via earphones. As seen in Fig. 7, the average population waves last four to seven cycles, at approx. 20 Hz. As the interaural delay is systematically varied over a range of O-SO ps, the oscillatory responses to the emission-echo pair are phase advanced by several milliseconds. The phase-shifted oscillations produce a time expansion of the echo delay by a factor of 20-50. Moreover, the cycle by cycle structure of the local averaged response (at best frequency sites) may be a model of the echo waveform; a 360” echo shift produces a 360” response phase shift. Similar oscillations have also been observed in bat auditory cortex (Simmons, unpublished observations). Phase plane analysis of the cortical oscillations reveal that the waves are non-periodic and have a deterministic structure (Solinsky and Simmons, unpublished observations). Further analysis of the waves and the possible non-linear oscillators that could generate such waves are under way. Although the mechanism by which an interaural delay results in a particular phase shift of the inferior colliculus oscillation is unknown, these results indicate that phase differences in ensemble oscillatory activity are capable of encoding stimulus dimensions. Simmons has further interpreted the results to indicate that multiple time scales can be encoded in the neural population oscillation responses (i.e. millisecond scale for emission-echo delay, microsecond scale for interaural delay and echo waveform).

Llinas and associates55~78 have observed in human magnetoencephalogram (MEG) recordings a 40 Hz oscillation which sweeps from rostra1 to caudal in cerebral hemispheres. Acoustic stimuli reset this oscillator, and a synchronized, higher amplitude 40 Hz oscillation is produced for 100-200 ms following stimulus onset. This synchronous activity is phase-shifted rostra1 to caudal, so that a phase shift of 46 ms is seen from frontal to occipital-temporal cortex. These results indicate that the use of resettable and synchronizable oscillators may be a broadly used neural phenomenon. Llinas et al. have also obtained direct evidence for thalamocortical involvement in


Unfiltered

Normalized amplitude

Correlation’

Time (ms)

y-Filtered .-

Normalized amplitude

f Striate

Correlation*

.5

Time (ms)

Fig. 6. Time series of coherence (normalized cross-power) between electrode sites in motor and striate cortex in monkeys performing a go/no-go task. (A) Unfiltered waveforms of striate and motor cortex for a single go trial, with squared cross-correlation plotted below. (B) The same waveform with digital band-pass filtering in the gamma-frequency range (3 dB down at 30 and 80 Hz). After Bressler and

Nakamura;’ Bressler, Coppola and Nakamura, unpublished.

the 40 Hz oscillations. Layer 4 neurons in neocortex exhibit subthresho~d oscillations in this range” and thalamic projection neurons can produce rebound oscillations at 40Hz.‘06 They have pointed out that thalamocortical resonance could be a factor in the widespead synchronizations.

Previous interpretations of the synchronous oscillations (eg. Koch45) have emphasized a distinction between the synchronous nature of the activity and the oscillations, which may or may not be evident in individual records. Evidence for synchronous activity between neural ensembles has been more readily accepted than for oscillatory activity, perhaps in part because prior theoretical analyses of cortical activity have emphasized the significance of coinci- dent inputs. Examples of this are the “synfire” chains

of neural assemblies which propagate synchronous activity along diverging and converging pathways,’ and the theory that temporal synchrony could underlie visual segmentation and that correlated activity within a group of cells can reflect their dynamic grouping. 9y In fact, the original observation of synchronous oscillations in visual cortex by Gray et a1.35 led Crick and Koch2’ to hypothesize that this activity constituted a neural substrate of sensory binding, in this case binding elements of a sensory stimulus into a coherent entity. Recently, Lhnas and Pares4 have hypothesized that the temporal correlation between sensory inputs and widespread endogenous thalamocortica1 rhythmic activity (40 Hz) is an essential component of the perception of that sensory input.

Non-linear dynamics of neural systems

12.5

lime (ms)

Fig. 7. Neural ensemble oscillating responses in bat inferior colliculus with phase shift dependent on interaural delay. Top left: spectrogram of emitted call of bat Eptesicus. Top right: diagram of stimulus delivery system. The bat’s sonar emissions were picked at microphones (m), digitally delayed and then returned to the bat from loudspeakers (s) as echoes. Bottom: responses to emission and echo (6 ms between emission and echo) recorded from small groups of neurons in the inferior colliculus with glass micropipettes with impedance of 5-10 Ma. Each trace represents an average response (n = x) to the emission-echo stimulus, at a particular interaural difference in presentation of the echo, with the delay indicated in microseconds. Note the systematic phase shift, in milliseconds, as the interaural delay is

increased, in microseconds. After Haresign ef af.‘9

The notion that synchronous activation can be used as a substrate for cognitive binding has been exploited by Shastris3 in his novel approach to hybrid neural architectures. Shastri and Ajjanagadde@ have developed a spatiotemporal neural net in which the strength of activation and firing time of a node depend explicitly on a spatiotemporal integration of input signals, and the representational state of the net depends on the relative firing times of nodes. They have used this connectionist net to encode complex conjunctive rules and perform inferences. The model achieves these by representing dynamic bindings as the synchronous firing of appropriate nodes, and representing rules as interconnection patterns which direct the propagation of rhythmic activity.84 Naturally, this system goes well beyond current neuroscience and the nodes are not realistic neurons, but it is indicative of how much computational potential resides in the simple notion of synchronous neural activity.

The neural ensemble oscillations, per se, have been somewhat sceptically received, perhaps in part because reductionist neurophysiologists are more comfortable with bursting in individual neurons than with waves in neural ensembles, particularly in the sensory domain. It may also result from the influence of linear sytems-based engineering, in which instabilities are designed out of systems, but device physics lead to imperfectly implemented designs. In fact, it has been suggested that the oscillations are “epiphenomena”.~ Additionally, there are currently conflictirig experimental results.34.‘07 Indeed, aperiodic activity with a broad spectrum is a common observation in many, if not most, neural regions. For those who wish to characterize the non-linear dynamics of neural activity, this observation is more challenging than periodic oscillations, but also more interesting.

Moreover, from the dynamic systems view, we wish to draw attention to the significance of the

598 T. M. MCKENNA PI al.

intermittent nature of the activity, the possible physiological control of the phase shift and the nature of the waveform of the periodic waves, when they are present.

Intermittency is characteristic of dynamic systems near instability and could indicate the transition from one attractor regime to another. The control of the phase shift, as indicated in Simmons’ experiments,

needs to be explored in terms of biophysical mechanisms and the analysis extended to other sensory and sensorimotor systems. As we will see later in the section on motor controt by coupled non-linear oscitlators, phase shifting between coupled non-linear oscillators may be a key factor in producing co- ordinated movement. Finally, the nature of the waveform will help to identify the dynamic state of the generator and how it could serve as a signal to another ensemble. Of relevance to the latter issue is the recent work on synchronization of chaotic systems, an achievement previously thought imposs- ible given the sensitivity to initial conditions of chaotic systems. Recently, two physicists, Pecora and Carro1,‘8.74.75 designed a system for synchronizing two chaotic systems within a few milliseconds by sending a control signal from one system to another. A parent chaotic system generates a complex signal which causes synchronization with duplicate subsystems. The criterion for this synchronization is that the sign of the Lyapunov exponents of the subsystems is negative, which means the subsystems are stable in the absence of a driving signal. This scheme has been demonstrated in electronic circuits. This type of synchronization is important when one cannot use periodic forcing of timing signals because the responding subsystem is operating in the multiperiod domain (e.g. period doubling). Neural ensemble activity generally exhibits multiple spectral peaks: moreover, the modular or~nization of neural architectures (identical subsystems?} is ubiquito~ls. Extending the analogy further, one can imagine master signal generators which may be aggregates of modules or subsystems which have acquired an additional degree of freedom, generating pseudo- periodic signals to co-ordinate additional modules in a motor or sensory sequence. Alternatively, special- ized global systems, such as non-specific systems,54,b’ may serve this role by driving the subsystems into regimes where they co-ordinate their activity in qualitatively different manners.

The technique of Pecora and Carrel is related to Haken’s’” concept of synchronization of systems near singuiarities like bifur~tjons. The point is that if two neural ensembles are near critical points in their dynamics, an appropriate periodic or more complex signal with the correct waveform can synchronize their activities within a few cycles. The results arc analogous to those of Freeman in his olfactory model; when a sensory cue has been learned. the particular spatiotemporal pattern of oscillations resulting from the non-linear processing of the sen-

sory input can drive the olfactory cortex from a basal attractor observed during spontaneous activity into attractor wings that correspond to the learned signal class.“,*04

9. SPATIOTEMPORAL MODES OF ACTIVITY IN THE BRAIN

New tools for dynamic imaging of brain activity, such as voltage-sensitive dyes and coherence analysis of MEG or EEG recording arrays, have been used to explore the spatiotemporal domains of activity correlated with ~havioral states and sensorimotor processing. The spatiotemporal pattern of activity in brain is a fertile area for dynamic system analysis.“’

In one recent series of experiments, Kelso and co- workers analysed a phase transition in both human behavior and neuromagnetic field patterns.“,4” MEG recordings were made with an array of 37 superconducting quantum interference devices in human subjects performing a task in which they were instructed to syncopate a manual response between tones delivered at one per second and higher rates. As the rate of tone presentation rose, a stage was reached ~‘transition”) where subjects suddenly lost syncopation and shifted to a predictive mode, where the response became synchronous with the next tone, This is evident at a jump in response latency, measured in terms of relative phase of the tone delivery cycle. This transition in sensorimotor behavior is accompanied by a change in the power spectrum of the MEG and in the spatial pattern of activity. Spatial analysis of the coherence (cross- power) revealed that, at the behavioral transition, the spatial locus of maximum cross-power expands considerably, then shrinks back to the pretransition locus at higher presentation rates (see also the discussion of transient correlation structures in cerebellar cortex in a later section). The recorded activity also showed a change in phase of the response and topographic distribution of phase relation to tone stimulus when the transition point was reached. Eigenfunction decomposition of the spatial pattern of MEG activity reveals a large drop in the most dominant mode at the transition. They are currently using this spatial mode information to infer the dynamics of the neuromagnetic field generators. in terms of coupled non-linear oscillators. Kelso concludes that the results support his view of the brain as a self-or~njzjng pattern-forming system that operates close to instability points, which allows it to switch llexibly and spontaneously from one coherent state to another.

The development of voltage-sensitive dye recording has produced significant mappings of the spatial organization of the visual cortex. Static features such as orientation domains, ocular dominance bands. etc., have been revealed in great detail8 Recently, this technique has been extended to dynamic mapping of


brain activity. 85,9o In response to moving bar stimuli,

Kaplan and colleagues recorded the entire spatiotemporal pattern of neural ensemble activity within a 3 mm x 3 mm window of visual cortex using voltage- sensitive dyes. The complex sequence of spatiotemporal patterns observed were analysed by a “snapshot” technique of Karhunen-Loeve decomposition, which has also been used recently for the analysis of patterns of turbulence. The spatial eigenfunctions capture the salient features of the pattern of activity. By averaging over the spatial array, the time course and power spectrum of the individual eigenfunctions were computed. Peaks at 9, 21 and 36 Hz were evident in the first eigenfunction from cat visual cortex. This preliminary study has shown that it is technically feasible to analyse the tremendous amount of multidimensional data generated by voltage-sensitive dyes for dynamic stimuli. Among the many questions that remain to be addressed are the relationship between the eigenfunctions and known cortical structures, such as ocular dominance columns, orientation columns and color blobs. Another issue is the relationship between the stimulus parameters and the eigenfunctions. More impor- tantly, can a model be synthesized to account for the spatiotemporal pattern, and can a new dynamic theory emerge from this analysis. One of the issues raised in an earlier section, the topography of synchronization, might also be addressed by performing local averages of the eigenfunctions to observe local time series of eigenfunctions or cross-power analysis to search for coherence patterns. The use of eigenfunction decompositions is part of a larger search for the natural basic functions of spatiotemporal brain activity. In the case of turbulent channel flow, the use of empirical eigenfunctions led to the discovery of hitherto unknown phenomena (propagating turbulent wavesssm90), and the eigenfunctions themselves can be used in specifying mathematical models that produce the main features of the attractors underlying the physical system. Hence, this approach appears very promising for analysis of patterns of neural activity revealed by voltage-sensitive dyes, and the synthesis of dynamic models of the cortex.

Perhaps the most extensively investigated and modeled neural system with regard to spatiotemporal patterns of activity is the olfactory system.29.30,63 Sim- ultaneous recordings from as many as 64 sites in the olfactory bulb reveal a complex spatial pattern of oscillations riding on the respiratory rhythm when a rabbit inhales a learned odor. The spatial pattern of the phase of the burst is consistent with a self- organizing, co-operative dynamic rather than an external pacemaker. This is based on the observation that the phase pattern in the bulb is a cone in circular co-ordinates, whose apex varies spatially at random and whose sign can shift from lead to lag, observations which are inconsistent with pacemaker activity. The spatial configuration of the amplitude modulation of the oscillations encodes odors in a

context and history-dependent manner.29.30,9’ The

phase patterns can also be regarded as a case of anomalous dispersion. The evidence for this is based on the observations that the speed of the phase gradient of the coherent oscillations is much greater than either calculations of synaptic and axon delays or the speed of the spread of evoked potentials elicited by focal electrical stimulation of the bulb. Because the olfactory bulb has a repeating micro- circuitry, Freeman has produced a model based on interconnected modules which are composed of lumped parameter models which represent distinct neuron types. Due to realistic feedback loops within the olfactory bulb, anterior olfactory nucleus and

prepyriform cortex, each are capable of oscillation with a different characteristic frequency. The bulb oscillator output can exhibit a positive Lyapunov exponent, and the distributed delay lines and positive and negative feedback connecting these three brain areas lead to a large dynamic repertoire, including aperiodic chaotic waveforms. The entire system can

simulate the real EEG. The basal attractor appears in phase space as a collapsed hypertorus. A dynamic non-linearity within the bulb, which arises because the gain depends on the input amplitude, produces a spectrally rich burst of activity on each inhalation. In the animal experiments, discriminant analysis of the spatial configuration of the amplitude modulaton of the 64-electrode array permitted correct classification of learned odors. Based on these results, a version of the model was implemented with 64 modules within the olfactory bulb (KIII model’“). This model was tested on a difficult classification task involving the identification of industrial parts from the pattern of ultrasound scatter. The scattered ultrasound was collected at eight sensors, and these signals provided 64 vector inputs to the KIII model. The model outperformed a statistical classifier, backpropagation neural networks and Hopfield network in classification accuracy.lo5 It was observed that when the network learned a new signal class, a unique set of attractors was observed in the phase plots of co-varying activity in the different neuron types.‘“,io5 This raises the issue of the conditions under which neural system or module attractors can represent signal class. Based on experimental results, in the simulations, learning produces Hebbian synaptic strength changes in the excitatory feedback connections. These synaptic changes produce a bifurcation in the system. Learning adds a new attractor, with a lower dimension, to the attractor of the basal state.

Freeman’s analysis of the olfactory system provides a number of observations which are highly relevant to the discussion of the significance of the sychronized oscillations seen between sites in visual and sensorimotor cortices. One is that multistage neural systems may take advantage of the spatiotemporal pattern of synchronized activity to extract weak signals from noise. For example, the olfactory

600 T. M. MCKENNA et uf.

bulb exhibits spatiaIly distributed synchronized carrier waves in response to specific peripheral events. The bulb, in turn, projects to the prepyriform cortex, where bulb input is widespread and diffuse, and where individual neurons receive widely convergent input. Because of this divergent-convergent pattern of connectivity, the spatially coherent oscillations of the bulb become reflected in the neural activity of the

cortex with broad spatial distribution, even though the input coherent signals are embedded in large uncorrelated noise activity. Second, the form of the oscillations is significant. The information resides in the spatial distribution of the amplitude moduIation of the carrier. Third, the target regions, by virtue of their own dynamics, temporal dispersion, feedback connections and possible plasticity of intrinsic connections, particpate in and modify the global

dynamics of the system. 3o Fourth, this type of analysis points out how a hierarchy of neural levels might take advantage of non-linear dynamics. Neural ensembles capable of chaotic behavior are sensitive to initial conditions; hence, in principle, small patterns of activity in a few neurons could induce a global pattern shift in the ensemble.

10. COUPLED NON-LINEAR OSCILLATORS fN MOTOR CONTROL: CHAINS OF OSCILLATORS

In earlier sections, we described observations of synchronous oscillations in sensory and motor systems. While there have been some attempts to produce models which can account for the zero small phase lag in these oscillations,‘5~4’.45 47 these models have not predicted the generation of spatiotemporal patterns that could specify sensory or motor behavior. There have also been preliminary attempts to identify the generator of spatiotemporal patterns of neural activity in terms of coupled oscillators? Aside from the work of Freeman, these analyses have not proceeded very far.

However, in the case of neuronal generation of motor control, there has been success in accounting for the adaptive rhythmic generation of activity for locomotion.5~‘~~36~7”~*6~‘o* During swimming in the lamprey, a traveling lateral mechanical wave propa- gates down the body, producing forward propulsion. Recordings from spinal ventral roots or segmental muscles show a rhythmic output at each segment (the lamprey has up to 100 segments).‘” While the time lag between activity bursts in different segments is a function of swimnling speed or cycle duration, the phase lag per segment is a constant proportion of the cycle period, independent of speed. This leads to approximately one wavelength of this traveling wave being on the body at all times, and this minimizes lateral forces. Hence, the constant proportion phase lag is a significant constraint on co-ordinated locomotion in the lamprey. How can neural circuitry produce co-ordinated motor output that fullils this constraint over a range of swimming speeds. for

both forward and backward locomotion? The neural circuitry underlying motion co-ordination has been successfully modeled as a chain of coupled non-linear oscillators.4R~‘0’*‘02 In the model, each segment has a pair of oscillators, each with an intrinsic frequency and a limit cycle attractor in phase space. When coupled to its neighboring oscillators (rostrocaudal), each oscillator will increase or decrease the frequency of the receiving oscillator in proportion to the phase difference between the sending and receiving oscillators. A simple set of linear ascending and descending coupling functions (frequency change as a function of intersegmental phase lag) can be defined which produces stable backward or forward swimming. This simple system satisfies the constraint that phase lag between segments is independent of speed. In the lamprey the specific neuron types and connectivities of these segmental oscillators have been characterized. This general model can be mapped into a model with specific neural circuits.” These simulations also satisfy the locomotion constraints and the critical connections can be identified. Unfor- tunately, for many motor systems, we lack this cellular detail, but dynamic models can provide significant insights into essential properties which can focus the search for neural substrates. There are a number of advantages in the use of coupled non- linear oscillators for motor control and robotics. Where the individual oscillators generate limit cycles, they are stable in the face of perturbations. Addition- ally, in this model, the coupling of the oscillators confers additional adaptiveness in response to perturbations or drift: an increase in the phase difference produces an increase in coupling, with an attendant increase in ensemble frequency and hence a decrease in phase differences between segments. (Williams has suggested that this mechanism may also apply to synchronized cortical oscillations.)‘“’ This approach simplifies the motor control problem. lnstead of tracking movement trajectories, which may be difficult to adaptively compute in real time for limbs and may be unreliable in the face of outside forces, the system tracks and adjusts limit cycles of the oscillators which control components of the movement. This produces a rapid adaptation lo internal and external perturbations and drifts. This approach also implies a broader strategy for motor control, in which neural systems embody a library of non-linear oscillators that can be coupled in different ways to enable flexible, real-time movement control. We will explore this idea again in relation to the cerebellum with its 2-D sheet of neural elements.

Paired chains of oscillators have also formed the basis of models of legged locomotion. Following the suggestion of Pearson, models of six coupled oscillators can produce sequences of gaits in hexa- pods that confer stability during locomotion at different speeds.’ These models have been recast in terms of dynamic systems.h.h2 The advantages of this formulation are terms of analysis, particularly in the


absence of complete knowledge of the specific neural circuits, and in the ability to explicitly regard the organism and the physical environments as a coupled dynamic system.6 System dynamics are also essential in the biomechanics of legged locomotion, as evident in the analysis of dynamic stability of cockroach running.” Under conditions of static stability (slow locomotion), the center of gravity is maintained within a tripod of legs in contact with the substrate. However, during rapid running, the tripod gait is replaced by an “effective leg” (sum of leg force vectors), with a trajectory which orbits the center of gravity during the stepping cycle.96

11. MAMMALIAN OLIVOCEREBELLAR SYSTEM AS A TWO-DIMENSIONAL ISOCHRONOUS SHEET OF

COUPLED OSCILLATORS

The cerebellar lobules contain detailed but complex maps of the body via tactile and proprioceptive inputs, receive relayed input from the other motor regions such as motor cortex and map systematically onto motor output (premotor) systems. The cerebellar cortex has a very regular cellular architecture, which includes a monolayer of large Purkinje neurons, each receiving about 100,000 synapses from a beam of parallel fibers and one large climbing fiber originating in the inferior olive (IO), which synapses repeatedly on the Purkinje cell dendrites. The IO also receives somatosensory input and contains neurons which are locally connected by gap junctions. By virtue of intrinsic neuronal properties and the coupling it generates a 5-15 Hz rhythm.52 The coup-

ling of these neurons is controlled by feedback from the Purkinje neurons via the deep cerebellar nuclei (including dentate nucleus). The inhibitory synapses of the deep cerebellar nuclei (e.g. dentate nucleus) synapse directly on the gap junctions between the IO neurons.92 Hence, cerebellar input is capable of fractionating the IO oscillator into local domains. The conduction velocity of the IO axons to cerebellar cortex is biologically regulated so that the time of arrival of all IO fibers to the cerebellar sheet is identical, i.e. the cerebellar cortex is an isochronous sheet in terms of olivocerebellar transactions.93 Llinas and co-workersS’,80*8’ have recorded simultaneously from 30 to 96 single Purkinje neurons in the cerebellum of the rat. Cross-correlation analysis of recordings reveal that rostral-caudal beams of Purkinje neurons are correlated with a 10 Hz rhythm which depends on IO input, and whose pattern is consistent with the anatomy of IO-cerebellar projections. Under the same conditions, activity of neurons in the mediolateral dimension of the Purkinje cell array is not correlated. This basic pattern requires the active involvement of the cerebellar inhibitory feedback to IO via dentate.5’ By contrast, preceding and during movements, a complex pattern of mediolateral corre- lations is seen transiently over long distances in the cerebellar array (Ref. 80; Llinas, unpublished observations) (Fig. 8).

This leads to a new view of the coding of movements in cerebellum. In this view, the 10 Hz oscillation and motor tremor generated by the olivocerebellar system play a role in co-ordinating movements by providing a clock for synchronization.”

A. NON-MOVEMENT PERIODS 8. MOVEMENT PERIODS

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Fig. 8. Cross-correlation patterns among simultaneously recorded Purkinje cells in rat cerebellum. Top: magnitude of cross-correlation coefficient of discharge of Purkinje neurons relative to neuron at site M, for non-movement periods (A) and movement periods (B). Note ephemeral correlation pattern during movement. Bottom: topography of electrode array in rat cerebellum. After Llina$’ Llinas,

unpublished.

602 T. M. M~KENNA et al

Moreover, it is hypothesized that these transient correlation structures in the Purkinje cell array are coding the movement. A large number of motor programs can be stored by means of the large number of correlation structures possible in the large 2-D Purkinje cell array. Compared to the chain of oscillators described in the lamprey signal cord and invertebrate ganglia, the olivocerebellar

system provides much greater flexibility and capacity for movement programs. In a chain of oscillators local connectivity among local oscillators domi- nates and the repertoire is limited. In the 2-D Purkinje cell array, a large number of combinations of Purkinje neurons can be linked in a correlation structure by the IO-cerebellar loops. Hence, the mammalian motor system has placed the oscillators in a separate neural structure which can be frac- tionated into local oscillators under control of the cerebellum. This strategy may lead to greater flexibility while retaining the speed advantages of motor control by coupled oscillators. Obviously, further work is needed to specify the relation of the correlation structures to movements, but this approach could potentially provide exciting new insights into the dynamic population coding of motor control. This approach should also be extended to motor cortex in order to specify the unique roles that these two neuronal arrays play in motor control and planning. This new approach requires state-of- the-art recording techniques combined with the appropriate analytical tools for dynamic system characterization. Fortunately, the repetitive nature of the cerebellar architecture provides significant experiment advantages.

12. CONCLUSIONS

In summary, the brain can be regarded as a dynamic system that is non-linear at multiple levels of analysis. A characterization of its non-linear dynamics is fundamental to our understanding of brain operation. An extensive array of recently developed mathematical tools for analysing non-linear systems in physics can be used to great advantage by the neuroscience community. For example, phase space analysis can be applied to describe neural

activity at different levels of the nervous system. The familes of attractors in phase space that characterize

complex non-linear physical systems can prove valuable in describing a range of behaviors and associated neural activity, including sensory and motor repertoires, in neural systems. These techniques can identify functionally significant states of single neurons and neural ensembles not apparent by

more traditional methods of analysis. Transitions between attractors and transit times within attractors may serve as useful descriptors for analysing state changes in neurons and neural ensembles, including

state changes induced by sensory inputs. Among the likely neural parameters that regulate dynamic state transitions are neuromodulators, synaptic weights and the non-specific systems that determine background level of neural activity in ensembles. Pertur- bation-induced state transitions may be produced by precise timing of synaptic inputs or synchronous inputs relative to the phase of endogenous cycles of activity. Noise-induced transitions may enhance the responses to weak periodic inputs of neurons via simple non-linear physical mechanisms.

The recent observations of synchronous cortical oscillations in the 25-50 Hz range may provide insight into neural ensemble dynamics. From the view

of non-linear dynamic systems, neural subsystems arc being co-ordinated or synchronized. Key factors from the dynamic systems viewpoint are the phase shift and its systematic control, the intermittent nature of the synchronization and the identification of the waveform and its modulation across the spatial dimensions of the neural array. New techniques for the visualization of spatiotemporal dynamics in neural ensembles (including voltage-sensitive dyes, arrays of recording electrodes for single neurons. MEG and EEG) provide opportunities for observing coherent spatial structures, and the natural basis functions of neural population activity involved in the control of movement and perception. New

developments in the experimental physics of complex

systems, such as the control of chaotic systems, selection of attractors, attractor switching and transient states, can be a source of powerful new analytical tools and insights into the dynamics of neural systems.

REFERENCES

Abeles M. (I 982) Studies of‘ Bruin Function, Vol. 6, Locul Corticctl Circuits: An Elrctrophysiological Study. Springer, Berlin. Anishchenko V. S., Safonova M. A. and Chua L. 0. (1992) Stochastic resonance in Chud’s circuit. Int. J. B$irc. Chaos 2, 397401. Bak P., Tang C. and Wiesenfeld K. (1988) Self-organized criticality. Phys. Reu. A38, 364 374. Baker G. L. and Gollub J. P. (1990) Chaotic Dynumicst An Imroduction. Cambridge University Press, Cambridge. Beer R. (1990) Intelligencr as Adaptive Behavior: An Experiment in Computational Neuroethology. Academic Press, Boston. Beer R. (1993) A dynamical systems perspective on autonomous agents. AI J. (in press). Bernander 0.. Douglas R., Martin K. and Koch C. (1991) Synaptic background activity influences spatiotemporal integration in single pyramidal cells. Proc. natn. Acad. Sci. U.S.A. 88, 11569-11573. Blasdel G. (1992) Differential imaging of ocular dominance and orientation selectivity in monkey striate cortex. J. Neurosci. 12, 3115-3138.


9. Bressler S. and Nakamura R. (1993) Inter-area synchronization in macaque neocortex during a visual pattern discrimination task. Comput. neural Syst. (in press).

10. Broomhead D. S. and King G. (1986) Extracting qualitative dynamics from experimental data. Physica MD, 217-236. Il. Brown R. (1992) Generalizations of the Chua equations. Int. J. Bifurc. Chaos 2(4). 12. Buchanan J. T. (1992) Locomotion in a lower vertebrate: studies of the cellular basis of rhythmogenesis and oscillator

coupling. In Advances in Neural Information Processing Systems 4 (eds Moody J., Hanson S. and Lippmann, R.), pp. 101-108. Kaufmann, San Mateo, CA.

13. Bulsara A., Douglass J. and Moss F. (1993) Nonlinear resonance: noise-assisted information processing in physical and neurophysiological systems. Naval Res. Rev. 45, 23-37.

14. Bulsara A., Jacobs E., Zhou T., Moss F. and Kiss L. (1991) Stochastic resonance in a single neuron model: theory and analog simulation. J. theor. Rio/. 154, 531-555.

15. Bush P. and Douglas R. (1991) Synchronization of bursting action potential discharge in a model network of cortical neurons. Neural Comput. 3, 19-30.

16. Canavier C., Baxter D., Clark J. and Byrne J. (1993) Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity. J. Neurophysiol. 69, 2252-2257.

17. Canavier C., Clark J. and Byrne J. (1990) Routes to chaos in a model of a bursting neuron. Riophys. J. 57, 124551251. 18. Carol T. and Pecora L. (1991) Synchronizing chaotic circuits. I.E.E.E. Trans. Circuits Syst. 38, 453456. 19. Chapeau-Blondeau F. and Chauvet G. (1992) Stable, oscillatory, and chaotic regimes in the dynamics of small neural

networks with delay. Neural Networks 5, 135-743. 20. Chay T. and Rinzel J. (1985) Bursting, beating, and chaos in an excitable membrane model. Biophys. J. 47,357-366. 21. Crick F. and Koch C. (1990) Towards a neurobiological theory of consciousness. Semin. Neurosci. 2, 263-275. 22. Cronin J. (1987) Mathematical Aspects of Hodgkin-Huxley Neural Theory. Cambridge University Press, Cambridge. 23, Dear S. and Simmons J. A. (1993) Temporal organization of range information in the auditory cortex of the big brown

bat: a possible neuronal basis for the representation of acoustic scenes. Meeting of the Assoc. Res. Otoluryngology. Feb. 1993, St Petersburg, FL.

24. Ditto W. L., Spano M. L., Savage H. T., Rauseo S. N., Heagy J. and Ott E. (1990) Experimental observation of strange nonchaotic attractor. Phys. Rev. Lett. 65, 533-536.

25. Engel A., Konig P., Kreiter A., Schillen T. and Singer W. (1992) Temporal coding in the visual cortex: new vistas on integration in the nervous system. Trends Neurosci. 15, 218-226.

26. Engel A., Konig P. and Singer W. (1991) Direct physiological evidence for scene segmentation by temporal coding. Proc. natn. Acad. Sci. U.S.A. 88, 91369140.

27. Fitzhugh R. (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1,4455466. 28. Freeman W. (1975) Muss Action in the Nervous System. Academic Press, New York. 29. Freeman W. J. (1990) Searching for signal and noise in the chaos of brain waves. In The Ubiquity of Chaos (ed. Krasner

S.), pp. 47-55. AAAS, Washington, DC. 30. Freeman W. J. (1992) Tutorial on neurobiology: from single neurons to brain chaos. Int. J. Rifurc. Chaos 2,451482. 31. Friederick R., Fuchs A. and Haken H. (1991) Spatio-temporal EEG patterns. In Rhythms in Physiological Systems

(eds Haken H. and Koepchen H.), pp. 315-338. Springer, Berlin. 32. Fuchs A., Kelso J. and Haken H. (1992) Phase transitions in the human brain: spatial mode dynamics. Int. J. Bifurc.

Chaos 2, 917-939. 33. Garfinkel A., Spano M., Ditto W. L. and Weiss J. (1992) Controlling cardiac chaos. Science 257, 1230-1235. 34. Ghose G. and Freeman R. (1992) Oscillatory discharges in the visual system: does it have a functional role?

J. Neurophysol. 68, 1558-1574. 35. Gray C. M., Koenig P., Engel A. K. and Singer W. (1989) Oscillatory responses in the cat visual cortex exhibit

inter-columnar synchronization which reflects global stimulus properties. Nature 338, 334-337. 36. Grillner S., Wallen P., Brodin L. and Lansner A. (1991) Neuronal network generating locomotor behavior in lamprey:

circuitry, transmitters, membrane properties, and simulation. A. Rev. Neurosci. 14, 1699199. 37. Guttman R. and Barnhill R. (1970) Oscillation and repetitive firing in squid axons. Comparison of experiments with

computations. J. gen. Physiol. 55, 104-l 18. 38. Haken H. (1977) Synergetics. An Introduction, 3rd edn. Springer, Berlin. 39. Haresign T., Ferragamo M. and Simmons J. A. (1993) Averaged multiunit activity in the inferior colhculus of Episicus

jiicus shows sensitivity to phase and microsecond time shifts in echoes. Meeting of the Assoc. Res. Otolaryngology, Feb. 1993, St Petersburg, FL.

40. Jahnsen H. and Lhnas R. (1984) Electrophysiological properties of guinea-pig thalamic neurons: an in vitro study. J. Physiol., Land. 349, 205-247.

41. Kammen D., Holmes P. and Koch C. (1990) Collective oscillations in neuronal networks. In Advances in Neural Information Processing Systems (ed. Touretzky D.), Vol. 2, pp. 7683. Morgan Kaufmann, San Mateo, CA.

42. Katz P. and Harris-Warrick R. (1990) Actions of identified neuromodulatory neurons in a simple motor system. Trends Neurosci. 13, 367-373.

43. Kelso J. A., Bressler S. L., Buchanan S., DeGuzman G. C., Ding M., Fuchs A. and Holroyd T. (1992) A phase transition in human brain and behavior. Phys. Lett. A169, 134144.

44. Kiehn 0. (1991) Plateau potentials and active integration in the “final common pathway” for motor behavior. Trends Neurosci. 14, 68-73.

45. Koch C. (1993) Computational approaches to cognition: the bottom-up view. Curr. Op. Neurobiol. (in press). 46. Koch C. and Schuster H. (1992) A simple network showing burst synchronization without frequency locking. Neural

Comput. 4, 21 l-223. 47. Konig P. and Schillen T. B. (1991) Stimulus-dependent assembly formation of oscillatory responses: I. Synchroniza-

tion. Neural Comput. 3, 155166. 48. Kopcll N. (1988) Toward a theory of modelling central pattern generators. In Neural Control of Rhythmic Movements

in Vertebrates (eds Cohen A., Rossignol S. and Grillner S.), pp. 369413. John Wiley, New York. 49. Kreiter A. K. and Singer W. (1992) Oscillatory neuronal responses in the visual cortex of the awake monkey.

Eur. J. Neurosci. 4, 369-375.

604 T. M. MCKENNA et al

50. Livingston M. S. (1991) Visually-evoked oscillations in monkey striate cortex. Sot. Neurosci. Absw. 17, 73.3. 51. Llinas R. (1991) The noncontinuous nature of movement execution. In Motor Control: Concepts and Issues (eds

Humphrey D. and Freund H.-J.), pp. 223-242. John Wiley, New York. 52. Llinas R., Baker R. and Sotelo C. (1974) Electrotonic coupling between neurons in cat inferior olive. J. Neurophysiol.

37, 560-5-l I. 53. Llinas R., Grace A. and Yarom Y. (1991) In oitro neurons in mammalian cortical layer 4 exhibit intrinsic oscillatory

activity in the lO- to 50-Hz frequency range. Proc. nafn. Acud. Sci. U.S.A. 88, 897901. 54. Llinas R. and Pare D. (1991) Of dreaming and wakefulness. ~~roscje~ce 44, 521-535. 55. Llinas R. and Ribary U. (1992) Rostr~audal scan in human brain: a global characteristic of the #-Hz response during

sensory input. In I~~ce~ Rhyfhms in the Brain (eds Basar E. and Bullock T.), pp. 147-154. Birkhauser, Boston. 56. Llinas R. and Sugimori M. (1980f Electrophysolo~cal properties of in vitro Purkinje cell dendrites in mammalian

cerebellar slices. J. Physiot., Land. 305, 197-213. 57. Llinas R. and Yarom Y. (1981) Properties and distribution of ionic conductances generating electroresponsiveness

of mammalian inferior olivary neurons in vitro. J. Physiol. 315, 5699584. 58. Longtin A. (1993) Stochastic resonance in neuron models. J. stat. Phys. 70, 309-327. 59. Longtin A., Bulsara A. and Moss F. (1991) Time-interval sequences in bistable systems and the noise-induced

transmission of information by sensory systems. Phys. Rev. Ierr. 67, 656659. 60. Longtin A., Bulsara A., Pierson D. and Moss F. (1994) Bistability and the dynamics of periodically forced sensory

neurons. Eiol. Cybern. (in press). 61. Mandell A. and Selz K. (1993) Brain stem neuronal noise and neocortical “resonance”. J. stnf. Phys. 70, 355-373. 62. Marvin D. J. (1992) Coordination of the walking stick insect using a system of nonlinear coupled oscillators. M.Sc.

thesis, University of Maryland. 63. Mayer-Kress G., Barczys C. and Freeman W. (1991) Attractor reconstruction from event related multi-electrode

EEG-data. In Mathema~~ru~ Approaches to Brain Functioning D~agnosfies (ed. Holden A. V.), pp. l-14. World Scientific, Singapore.

64. McCormick D. A., Hugenard J. and Strowbridge B. (1992) Determination of state-dependent processing in thalamus by single neuron properties and neuromodulators. In Singie Neuron Computation (eds McKenna T., Davis J. and Zornetzer S.), pp. 259-290. Academic Press, New York.

65. Moon F. C. (1992) Chaos and Fractal Dynamics: An Introduction ,for Applied Scientiw and Engineers. John Wiley, New York.

66. Moss F. (1994) Stochastic resonance: from the ice ages to the monkey’s ear. In Some Problems in Statistical Physics (ed. Weiss G.), SIAM, Philadelphia, (in press).

67. Moss F., Bulsara A. and Schlesinger M. (1993) Proc. NATO Advanced Research Workshop: Stochastic Resonance in Physics and Biology. Special Issue of J. stat. Phys. 70, l-612.

68. Mpistos G., Burton R., Creech H. and Soinila S. (1988) Evidence for chaos in spike trains of neurons that generate rhythmic motor patterns. Brain Res. Bull. 21, 5299538.

69. Mpistos G., Creech H., Cohan C. and Mendelson M. (1988) Variability and chaos: neurointegrative principles in elf-organi~tion of motor patterns. In Dynamic Patterns in Complex Systems (eds Kelso J., Mandell A. and Shlesinger M.), pp. 162-190. World Scientific, Singapore.

70. Mpistos G. and Soinila S. (1992) In search of a unified theory of biological organizatjon: what does the motor system of a sea slug tell us about human motor integration? In 1991 Lectures in Complex Systems, SFI Studies in the Sciences qf Complexity, Lect. Vol. IV (eds Nadel L. and Stein D.), pp. 677137. Addison-Wesley, Reading, MA.

71. Murthy V. and Fetz E. (1992) Coherent 2% to 50-Hz oscillations in the sensorimotor cortex of awake behaving monkeys. Proc. nutn. Acad. Sci. U.S.A. 89, 5670-5674.

72. Nagumo J. S., Arimoto S. and Yoshizawa S. (1962) An active pulse transmission line simulating nerve axon. Proc. I.R.E. SO, 2061-2070.

73. Pearson K. G. (1985) Are there central pattern generators for walking and flight in insects? In Feedback and Motor Control in Invertebrates and Vertebrates (eds Barnes W. and Gladden M.), pp 307.--315. Croom Helm, London.

74. Pecora L. and Carrol T. (1991) Driving systems with chaotic signals. Phys..Rev. A44, 23742383. 75. Pecora L. and Carrol T. (1992) Drivina nonlinear systems with chaotic sianals. In Proceedings ofthe 1st Exnerimentai

Chaos Conference (eds Vohra S., Span% M., Shlesinger M., Pecora L. and Ditto W.), pp. l>l:l36. World’Scientific, Singapore.

76. Rand R., Cohen A. and Holmes P. (1988) Systems of coupled oscillators as models of central pattern generators. In Neural Conrrol of ~y~hrnjc Movements in Veriebrates (eds Cohen A., Rossignol S. and Grillner S.), pp. 333-367. John Wiley and Sons, New York.

77. Rapp M., Yarom Y. and Segev I. (1992) The impact of parallel fibers background activity on the cable properties of cerebellar Purkinje cells. Neural Compuf. 4, 518-533.

78. Ribary U.. loannides A., Singh K., Bolton J., Lado F., Mogliner A. and Llinas R. (1991) Magnetic field tomography of coherent thalamocortical 40-Hz oscillations in humans. Proc. natn. Acad. Sci. U.S.A. 88. 11.037-I 1.041.

79. Sanes J. and Donoghue J. (1993) Oscillations in local field potentials of the primate motor cortex during voluntary movement. Proc. natn. Acad. Sci. U.S.A. (in press).

80. Sasaki K., Bower J. and Llinas R. (1989) Multiple Purkinje cell recording in rodent cerebellar cortex. Eur. 1. Neurosci. 1, 572-586.

81. Sasaki K. and Llinas R. (1985) Dynamic electronic coupling in mammalian inferior olive as determined by simultaneous multiple Purkinje cell recording. Biophys. J. 47, 53a.

82. Selverston A., Rowat P. and Boyle M. (1992) Modeling a reprogrammable central pattern generating network. In Eioiogical Neural Networks in Invertebrate Neuroethology and Roborics (eds Beer R., Ritzmann R. and McKenna T.), pp. 229-250. Academic Press, New York.

83. Shastri L. (1988) A conn~tionist approach to knowledge representation and limited inference. Cogn. Sci. 12,33 l-392. 84. Shastri L. and A~ana~dde V. (1993) From simple associations to systematic reasoning: a connectionist representation

of rules, variables, and dynamic bindings using temporal synchrony. Behav. Brain Sci. 16, 417494. 85. Shoham D., Ullman S. and Grinvald A. (1991) Characterization of dynamic patterns of cortical activity by a small

number of principal components. Sot. Neurosci. Absfr. 17, 431.8.


86. Sigvardt K. and Williams T. (1992) Models of central pattern generators as oscillators: the lamprey locomotor CPG. Semin. Neurosci. 4, 37-41.

87. Simmons J. A. (1989) A view of the world through the bat’s ear: the formation of acoustic images in echolocation. Cognition 33, U-199.

88. Sirovich L. (1991) Empirical eigenfunctions and low dimensional systems. In New Perspectives in Turbulence (ed. Sirovich L.),.pp. 139-164. Springer, New York.

89. Sirovich L.. Ball K. and Handler R. (1991) Proaaeatine structures in wall-bounded turbulent flows. In Recent ~eve~op~n~s in Turbulence: The Lum&y C~~f~re~~e~(~s~Gatski T., Sarkar S. and Speziate C.), Special Issue of TheoreticaI and Com~taf~ona~ F&id Dynamics 2, 307-317.

90. Sirovich L. and Everson R. (1992) Management and analysis of large scientific datasets. Znr. J. Supercompuf. Appiie. 6, W-68.

91. Skarda C. and Freeman W. (1990) Chaos and the new science of the brain. Concepts Neurosci. 1, 275-285. 92. Sotelo C., Gotow T. and Wassef M. (1986) Localization of glutamic-acid decarboxylase-immunoreactive axon

terminals in the inferior olive of the rat, with special emphasis on anatomical relations among GABAergic synapses and dendrodendritic gap junctions. J. camp. Neural. 252, 32-50.

93. Sugihara I., Lang E. and Llinas R. (1993) Uniform conduction time in the olivocerebellar pathway underlies Purkinje cell complex spike synchronicity in the rat cerebellum. J. Physiol. (Lond) 470, 243-271.

94. Tam D. C. (1992) Vectorial phase-space analysis for detecting dynamical interactions in firing patterns of biological neural networks. Proc. Inf. Joint Conf: Neural Networks, I.E.E.E. III, 97-102.

95. Tam D. C. (1993) Novel cross-interval maps for identifying attractors from multi-unit neural firing patterns. In Nonlinear Dynamical Analysis of the EEG (ed. Jansen B.) pp. 65-77. World Scientific, Singapore, pp. 65-77.

96. Ting L., Blickhan R. and Full R. (1993) Dynamic and static stability in hexapedal runners. J. exp. Viol. (submitted). 97. Tuckwell H. C. (1988) rnfr~ucfion to Theoretical Neurobjology: Vol. 2, Nonlinear and Stochastic Theories. Cambridge

University Press, Cambridge. 98. Vohra S., Spano M., Shlesinger M., Pecora L. and Ditto W. (1992) Proceedings os the fst Ex~r~rne~fa~ Chaos

Conference. World Scientific, Singapore. 99. Von der Malsberg C. (1981) The correlation theory of brain function. Internal Report 81-2, Max-Planck-Institute for

Biophysical Chemistry, Gottingen, Germany. 100. Wang X. J. and Rinzel J. (1992) Alternating and synchronous rhythms in reciprocally inhibitory model neurons.

Neural Comput. 4, M-97. 101. Williams T. (1992) Phase coupling by synaptic spread in chains of coupled neuronal oscillators. Science 258,662-665. 102. Williams T. (1992) Phase coupling in simulated chains of coupled oscillators representing the lamprey spinal cord.

Neural Comput. 4, 546558. 103. Wilson C. J. (1992) Dendritic morphology, inward rectification, and the functional properties of neostriatal neurons.

In Singie Neuron Computation (eds McKetma T., Davis J. and Zornetzer S.), pp. 141-171. Academic Press, New York. 104. Yao Y. and Freeman W. (1990) Model of biological pattern recognition with spatially chaotic dynamics. Neural

Nefworks 3, 153-170. 105. Yao Y., Freeman W., Burke B. and Yang Q. (1991) Pattern recognition by a distributed neural network: an industrial

application. Neural networks 4, 103-121. 106. Yarom Y. and Llinas R. (1990) In~a~llular autostimulation of in vifro guinea-pig thalamic neurons (TH) utilizing

a hardware bio-electric re-entry system. Sot. Neurosci. Abstr. 16, 955. 107. Young M. P., Tanaka K. and Yamane S. (1992) On oscillating neuronal responses in the visual cortex of the monkey.

J. Neurophysiol. 67, 1464-1474.

(Accepted 25 October 1993)

the brain as a dynamic physical system

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