Is There Chaos in the Brain? II - Experimental Evidence and Related Models

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1631-069doi:10.10C. R. Biologies 326 (2003) 787840

Neurosciences

Is there chaos in the brain? II. Experimental evidenceand related models

Henri Korn , Philippe FaureRcepteurs et Cognition, CNRS 2182, Institut Pasteur, 25, rue du Docteur-Roux, 75724 Paris cedex 15, France

Received 16 September 2003; accepted 17 September 2003

Presented by Pierre Buser

t

earch for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science.sults and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the firstions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysisrimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/orindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have

d over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions.although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single

pled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributedg to chaotic patterns at all levels of its various forms of hierarchy.

chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn,cad. Sci. Paris, Ser. III 324 (2001) 773793] that was focused on the basic concepts of nonlinear dynamics and methodstigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioningntrol using external perturbations. Here we present the data and main arguments that support the existence of chaos ats from the simplest to the most complex forms of organization of the nervous system.rst provide a short mathematical description of the models of excitable cells and of the different modes of firing ofneurons (Section 1). The deterministic behavior reported in giant axons (principally squid), in pacemaker cells, inor in paired neurons of Invertebrates acting as coupled oscillators is then described (Section 2). We also consider

processes exhibited by coupled Vertebrate neurons and of several components of Central Pattern Generators (Section 3).n shown that as indicated by studies of synaptic noise, deterministic patterns of firing in presynaptic interneurons aretransmitted, to their postsynaptic targets, via probabilistic synapses (Section 4). This raises the more general issue of

s a possible neuronal code and of the emerging concept of stochastic resonance Considerations on cortical dynamicsEGs are divided in two parts. The first concerns the early attempts by several pioneer authors to demonstrate chaos inental material such as the olfactory system or in human recordings during various forms of epilepsies, and the beliefmical diseases (Section 5). The second part explores the more recent period during which surrogate-testing, definitionble periodic orbits and period-doubling bifurcations have been used to establish more firmly the nonlinear features

al and cortical activities and to define predictors of epileptic seizures (Section 6). Finally studies of multidimensional

responding author.ail addresses: [email protected] (H. Korn), [email protected] (P. Faure).

1/$ see front matter 2003 Acadmie des sciences. Published by Elsevier SAS. All rights reserved.

16/j.crvi.2003.09.011

788 H. Korn, P. Faure / C. R. Biologies 326 (2003) 787840

systems have founded radical hypothesis on the role of neuronal attractors in information processing, perception and memoryand two innemodific e toand des logicunderly ith(Section ory ostrongly needin the b 6 (2 2003 rese

Keyword erless

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l contifiThechaothe pspaces and the general properties of dynamicries as well as the coarse-grained measures,permit a process to be classified as chaotic inystems and models. We insisted on how theses need to be adapted for handling biologicalries and on the pitfalls faced when dealing withtionary and most often noisy data. Special at-was paid to two fundamental issues.first was whether, and how, one can distinguishinistic patterns from stochastic ones. This ques-

such a control can be achieved by taking advantage ofthe sensitivity of chaotic trajectories to initial condi-tions and to redirect them, with a small perturbation,along a selected unstable periodic orbit, toward a de-sired state. A related and almost philosophical prob-lem, which we will not consider further, is whetherthe output of a given organism can be under its owncontrol as opposed to being fully determined by in-principle-knowable causal factors [6]; The metaphys-ical counterpart of this query consists in speculating,elaborate models of the internal states of the brain (i.e. wations during cognitive functions are given special attention dupite the difficulties that still exist in the practical use of topoing correlates. The reality of neurochaos and its relations w

8) where are also emphasized the similarities between the thechallenge computationalism and suggest that new models are

rain. To cite this article: H. Korn, P. Faure, C. R. Biologies 32Acadmie des sciences. Published by Elsevier SAS. All rights

s: neuronal dynamics; neurochaos; networks; chaotic itinerancy; winn

oduction

re is growing evidence that future research onsystems and higher brain functions will be aation of classical (sometimes called reduction-roscience with the more recent nonlinear sci-his conclusion will remain valid despite the dif-s in applying the tools and concepts developedribe low dimensional and noise-free mathemat-dels of deterministic chaos to the brain and tocal systems. Indeed, it has become obvious inber of laboratories over the last two decadese different regimes of activities generated byells, neural assemblies and behavioral patterns,kage and their modifications over time cannotunderstood in the context of any integrative

logy without using the tools and models that es-a connection between the microscopic and thecopic levels of the investigated processes.I of this review [1] was focused on briefly pre-the fundamental aspects of nonlinear dynam-most publicized aspect of which is chaos the-

ore fundamental text books can also be con-by mathematically oriented reader [25]. Af-eneral history and definition of this theory we

tiowhtiopofordacluindesur

calmidieteman

orddisbethevan

icaide

ofofrless competition and chaotic itinerancy). Theirtheir functional and adaptive capabilities (Section 7)al profiles in a state space to identify the physical

information theory are discussed in the conclusionf chaos and that of dynamical systems. Both theoriesed to describe how the external world is represented003).rved.

competition; representation; neuronal code

particularly important in the nervous systemvariability is the rule at all levels of organiza-and where for example time series of synaptic

als or trains of spikes are often qualified as con-to Poisson distributions on the basis of stan-

ter event histograms (see also [7]). Yet this con-can be ruled out if the same data are analyzedwith nonlinear tools such as first or second or-

rn maps and using the above-mentioned mea-onfronted with those of randomly shuffled datasurrogates. The critical issue is here to deter-intrinsic variability, which is an essential ingre-successful behavior and survival in living sys-

eflects true randomness or if it is produced byrently stochastic underlying determinism and

n other words how can the effects of noise beished from those resulting from a small num-

nteracting nonlinear elements. In the latter caseo appear as highly unpredictable but their ad-is that they can be dissected out and the phys-

rrelates of their interacting parameters can beed physiologically.second issue concerned the possible benefitstic systems over stochastic processes, namelyossibility to control the former. Theoretically

H. Korn, P. Faure / C. R. Biologies 326 (2003) 787840 789

as did number of authors, about the existence and na-ture of

In tcriticallevel ofin realfunctiofor judconstraparticu(which[1])). Tdemonweak esufficiestory. Idynamhiddensynchrstudiesas dynone ca

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th thd atmpgle itermsingicalfree-will [8,9]...he present part II of this review, we willly examine most of the results obtained at thesingle cells and their membrane conductances,networks and during studies of higher brain

ns, in the light of the most recent criteriaging the validity of claims for chaos. Theseints have become progressively more rigorouslarly with the advent of the surrogate strategyhowever can also be misleading (references inhus experts can easily argue that some earlystrations of deterministic chaos founded onxperimental evidence were accepted withoutnt analysis [9]. But this is only one side of thendeed we will see that the tools of nonlinearics have become irreplaceable for revealingmechanisms subserving, for example, neuronalonization, periodic oscillations and also for

of cognitive functions and behavior viewedamic phenomena rather than processes thatn study in isolation from their environmentalt.history of the search for chaos in the nervous

, of its successes and its errors, and of the ad-what has become neurodynamics is truly fasci-It starts in the 1980s (see [10]) with the obser-

that when rabbits inhale an odorant, their EEGsoscillations in the high-frequency range of 20hat Bressler and Freeman [11] named gammaogy to the high end of the X-ray spectrum! Odoration was then shown to exist as a pattern ofactivity that could be discriminated wheneveras a change in the odor environment or after. Furthermore the carrier wave of this infor-was aperiodic. Further dissection of the exper-

l data led to the conclusion that the activity ofactory bulb is chaotic and may switch to any) perceptual state (or attractor) at any time. Tonsate for experimental limitations the olfactorys then simulated by constructing arrays of localors interconnected by excitatory synapses thatted a common waveform. The inclusion of in-y cells and synapses facilitated the emergencelitude fluctuations in the waveform. Learningtrengthen the synapses between oscillators and

the formation of Hebbian nerve cell assem-a self-regulatory manner which opened new

sto

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clasindeclotistrepresentations of the outside world.complementary, experimental and theoreticalh of Freeman and his collaborators was similarf other authors searching for chaos, during the

eriod, in the temporal structure of the firing pat-f squid axons, of invertebrate pacemaker cellstemporal patterns of human epileptic EEGs.l show that regardless of todays judgment onsty conclusions and naive enthusiasm that re-ill-adapted measures for multidimensional andystems these precursors had amazingly sharp. Not only were their conclusions often vindi-ith more sophisticated methods but they blos-more recently, in the form of the dynamicalh of brain operations and cognition.

have certainly omitted several important issuesis general overview which is largely a chrono-description of the successes, and occasionalantments, of this still evolving field. One can

n the problem of the stabilization of chaos byf the phylogeny and evolution of neural chaotic

s, whether or not coupled chaotic systems be-one and the nature of their feedbacks, to name

These issues will most likely be addressed inn the context of research on the complex sys-

which the brain obviously belongs.

cellular and cellular levels

fully controlled experiments during which itsible to collect large amounts of stationary dataambiguously demonstrated chaotic dynamics

evel of neurons systems This conclusion wasusing classical intracellular electrophysiolog-

ordings of action potentials in single neurons,e additional help of macroscopic models. Thesedescribe the dynamical modes of neuronal fir-enable a comparison of results of simulations

ose obtained in living cells. On the other handa lower level of analysis, the advent of patchtechniques to study directly the properties ofon channels did not make it necessary to invokeinistic equations to describe the opening andof these channels which show the same sta-

features as random Markov processes [12,13],


although deterministic chaotic models may be consis-tent wi

It isin the boften oessentitrains.their rand clSincemodifieionic mfor thisHuxleysimplifimodel

Brierametefor whimode otions, fsive bifthe sigposed bthors is

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ance-bunderlytentialsto thein whicheld tofull accfound tterminewhichteries Vbrane cvoltagenels pewhichstimulathe medirectio

. 1. IoEquiactiv

ductaer iontery aexplathe ipagatvatede theiulsesequilapted

ls, ao phn grapolarcedolarth channel dynamics [7,14,15].generally believed that information is securedrain by trains of impulses, or action potentials,rganized in sequences of bursts. It is thereforeal to determine the temporal patterns of suchThe generation of action potentials and of

hythmic behavior are linked to the openingosing of selected classes of ionic channels.the membrane potential of neurons can bed by acting on a combination of differentechanisms, the most common models usedapproach take advantage of the Hodgkin andequations (see [1618] as first pioneered anded by FitzHugh [19] in the FitzHughNagumo

[20]).fly, knowing the physical counterpart of the pa-rs of these models, it becomes easy to determinech of these terms, and for what values the firingf the simulated neurons undergoes transforma-rom rest to different attractors, through succes-urcations. A quick reminder of the history andnificance of the mathematical formalism pro-y Hodgkin and Huxley and later by other au-necessary for clarifying this paradigm.

odels of excitable cells and of neuronal firing

The Hodgkin and Huxley modelthe paving-stone upon which most conduct-

ased models are built. The ionic mechanismsing the initiation and propagation of action po-have been beautifully elucidated by applying

squid giant axon the voltage clamp technique,h the membrane potential can be displaced anda new value by an electronic feedback (for aount see [2123]). As shown in Fig. 1A, it washat the membrane potential of the axon is de-d by three conductances, i.e. gNa, gK and gL,

are placed in series with their associated bat-Na, VK and VL and in parallel with the mem-apacitance C. Before activation the membrane, V , is at rest and the voltage-dependent chan-rmeable to sodium (Na+) and potassium (K+),can be viewed as closed. Under the effect of ation, the capacitor is led to discharge so thatmbrane potential is shifted in the depolarizingn and due to the subsequent opening of chan-

Fig(A)twocon

othbatforofproactinotimpthe(Ad

ne

twtiodeplarepnic currents involved in the generation of action potentials.valent circuit of a patch of excitable membrane. There aree conductances gNa and gK, and a third passive leaknce gL which is relatively unimportant and which carriess, including chloride. Each of them is associated to and is placed in parallel with the capacitance C (see textnations). Vertical arrows (labeled I ) point the direction

ndicated ionic currents. (B) Theoretical solution for aed action potential (V , broken line) and its underlying

conductances (gNa and gK), as a function of time;r good agreement with those of experimentally recorded. The upper and lower horizontal dashed lines designateibrium potential of sodium and potassium, respectively.from [21], with permission of the Journal of Physiology.)

current is generated. This current consists inases. First sodium moves down its concentra-dient thus giving rise to an inward current and aization. Second, this transient component is re-by an outward potassium current and the axonizes (Fig. 1B).


To describe the changes in potassium conductancesHodgkstates,constanformal

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cerebellar Purkinje cells (for details about authorsd equuireeac

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.2. TalysiA sijustimbr

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= V

=ich,ns w

b an

e [20Anmaliin and Huxley assumed that a channel has twoopen and closed, with voltage-dependent ratets for transition between them. That relation is

ly expressed as,

(1)n(V )(1 n) n(V )nn is the probability that a single particle is inht place, V is the voltage, n and n are ratets.ng the experimental data to this relationship re-gK gK n4 where gK is the maximal conduc-hus it was postulated that four particles or sen-

ed to undergo transitions for a channel to open.ly, for the sodium channel, it was postulatedee events, each with a probability m, open thed that a single event, with a probability (1 h)it. Then

(2)m(V )(1 n) m(V )m

(3)h(V )(1 h) h(V )hportant point here is that this formalism restslinear differential equations which, in addition,pled by the membrane potential, V . It follows,total membrane current density is:

V

dt+ gK n4(V VK)

(4)gNa m3 h(V VNa)+ gL(V VK)) to (4), which underly the generation of actions in a limited patch of membrane, can beted to account for the propagation of actionals along the core of axons by including to thisism equations pertaining to their specific cableies (see [24,25]).Hodgkin and Huxley model has been, and

s extremely fruitful for the studies of neuronsproduces with great accuracy the behavior ofle cells such as their firing threshold, steadyctivation and inactivation, bursting properties,lity, to name a few of their characteristics. Forle it has been successfully used, with somed adjustments of the rate constants of specifictances, to reproduce the action potentials ofcells (whether nodal or myocardial) and of

an

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whtioa,

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forations, see [26]). However, its implementations an exact and prior knowledge of the kineticsh of the numerous conductances acting in

set of cells. Furthermore the diversity ofurrents in various cell types coupled withplexity of their distribution over the cell,that number of parameters are involved in

erent neuronal compartments, for example, ines (see [23,27]). This diversity can precludeanalytic solutions and further understanding ofarameter is critical for a particular function.void these drawbacks and to reduce the num-parameters, global macroscopic models havenstructed by taking advantage of the theory ofcal systems. One can then highlight the qual-features of the dynamics shared by numerousof neurons and/or of ensemble of cells suchbistability, their responses to applied currentsptic inputs, their repetitive firing and oscilla-

ocesses. This topological approach yields geo-l solutions expressed in term of limit cycles,of attraction and strange attractors, as definedFor more details, one can consult several otherhensive books and articles written for physiol-18,28,29].

he FitzHughNagumo model: space phases

mplification of the Hodgkin and Huxley modelfied by the observation that, changes in theane potential related to (i) sodium activation,sodium inactivation and potassium activation,

during a spike on a fast and slow time course,ively. Thus the reduction consists of taking twoes into account instead of four, a fast (V ) and a

) one, according to:

(5) V3

3W + 1

(6)(V + a bW)again, are nonlinear coupled differential equa-here Eq. (5) is polynomial and where the termsd in Eq. (6) are dimensionless and posi-,29].important aspect of the FitzHughNagumosm is that since it is a two-variable model it


is well suited for phase plane studies in which thevariabl(howevon HodsinglecannotprovidequalitaequatioVan deoscillatpacemamodelof neuHuxleyby Moof thewhen athe simspike ta transcycle vunstablFig. 2Bnodes athe posone sid

2.1.3.A fe

the phaborrowchangeparamevalue.evolutisystemmissedseveralthe trajnodes (dle poipel theinitionthat kevalue ocycle ifurcatio

. 2. TrLeft.absccase

mbranximalte thady sttableuron

er tre of areced

a) duht. Cing thg (f

cissa)es V and W can be shown as functions of timeer, it can be noted that although models basedgkin and Huxley equations can generate chaos,two dimensional FitzHughNagumo neurons). These plots called phase plane portraits

a geometrical representation, which illustratestive features of the solution of differentialns. The basic relationships were derived byr Pol [30] who was interested in nonlinearors and they were first applied to the cardiacker [31]. It is therefore not surprising that this

was used later on to study the bursting behaviorrons, sometimes linked with the Hodgkin andequations in the form of a mosaic, as proposed

rris and Lecar [32] to describe the excitabilitybarnacle muscle fiber (see [18]). Specifically,n appropriate family of currents is injected intoulated neurons the behavior of the evoked

rains appears in the phase space to undergoition from a steady state to a repetitive limitia Hopf bifurcations which can be smooth ande (supercritical, Fig. 2A) or abrupt (subcritical,), or via homoclinic bifurcations, i.e. at saddlend regular saddles (not shown, see [33]), withsible hysteresis when the current I varies frome to the other of its optimal values (Fig. 2C).

Definitionsw definitions of some of the events observed inse space become necessary. Their description ised from Hilborn [34]. A bifurcation is a suddenin the dynamics of the system; it occurs when ater used for describing it takes a characteristic

At bifurcation points the solutions of the time-on equations are unstable and in many reals (other than mathematical) these points can bebecause they are perturbed by noise. There aretypes of fixed points (that is of points at whichectory of a system tends to stay). Among themor sinks) attract close by trajectories while sad-nts attract them on one side of the space but re-m on the other (see also Section 6.1 for the def-of a saddle). There are also repellors (sources)ep away nearby trajectories. When for a givenf the parameter, a point gives birth to a limitt is called a Hopf bifurcation, a common bi-n, which can be supercritical if the limit cycle

Fig(A)Thethisme

ma

Nosteauns

a ne

(lowcas

is pare

Rigdurfirinabsansitions from a steady state to an oscillatory firing mode.Diagram bifurcation of a supercritical Hopf bifurcation.

issa represents the intensity of the control parameter, inan intracellularly applied current, I . The ordinate is thee potential. The repetitive firing state is indicated by the(max) and minimal (min) amplitudes of the oscillations.

t for a critical value of I (arrow) the system shifts from aate to an oscillatory mode (solid curve) on either side of anpoint (dashed line). Right. Corresponding firing pattern of(upper trace) produced by a current, I , of constant intensityace). (B) Left. Same presentation as above of events in thesubcritical Hopf bifurcation. The stable oscillatory branched by an unstable phase (vertical dashed line in shaded

ring which the steady state and the oscillations coexist.urrent pulses of low amplitude can reset the oscillationsis unstable state (bistability). (C) Plot of the frequency of, ordinate) versus the intensity of the applied current (I ,. (Adapted from [33], with permission of the MIT Press.)


takes its origin at the point itself (Fig. 2A) or subcriti-cal if th(Fig. 2nonlinemoclinhas toble (ouare forstrict mdle poiSectionappearchangesect; thmanifomanifocur thehomocto chao

2.1.4.neuron

Thisneuros

modelsame a

the Fittant prospike inmaticaof oscironal ctwo vaone fortion, W

dVdt

=dWdt

=whereconstanlinear factual dobservdescribdegreefit to th

. 3. DFor ito bo

aotic)uenceibitory

tabilrrenttableaddetails

= y

= g

= rere

rrentnamis in

ustaiDesy ofe solution of the equation is at a finite distanceB) due to amplification of instabilities by thearities [35]. To get a feeling for what are ho-

ic and heteroclinic bifurcations and orbits onerefer to the invariant stable (insets) and unsta-tsets) manifolds and to the saddle cycles whichmed by trajectories as they head, according toathematical rules, toward and away from sad-nts, respectively (for more details see [1] and

6.1). Specifically, a homoclinic intersections on Poincar maps when a control parameter isd and insets and outsets of a saddle point inter-ere is a heteroclinic intersection when the stableld of one saddle point intersects with the stableld of another one. Once these intersections oc-y repeat infinitely and connected points formlinic or heteroclinic orbits that eventually leadtic behavior.

The Hindmarsh and Rose model of burstings

algorithm becomes increasingly popular incience. It is derived from the two-variableof the action potential presented earlier by theuthors [36], which was a modified version ofzHughNagumo model and it has the impor-perty of generating oscillations with long inter-tervals [37,38]. It is one of the simplest mathe-

l representation of the widespread phenomenonllatory burst discharges that occur in real neu-ells. The initial Hindmarsh and Rose model hasriables, one for the membrane potential, V , andthe the ionic channels subserving accommoda-. The basic equations are:

(7)(r f (v)+ I)

(8) (g(v) r)

I is the applied current, , , and are ratets, and where f (V ) is cubic and g(V ) is not aunction. This model allows to take into accountata: its right-hand side (vector field) can fit the

ed current/voltage relationship for the cells ites. Hence, it is possible to determine how manys of freedom are needed to make polynomiale I/V characteristics. These equation generate

Fig(A)top(chseqinh

biscu

bisis(dedxdtdydtdzdtwhcu

dyfirea s

ertifferent firing patterns of Rose and Hindmarsh neurons.ncreased values of an injected current, I (as indicated, fromttom), the model cell produced short, long and irregularbursts of action potentials. (B) Example of out of phase

s of bursts generated by two strongly reciprocally coupledneurons (after Faure and Korn, unpublished).

ity and to produce bursting, a slow adaptation, z, which moves the voltage in and out of theregime and which terminates spike discharges

d. Changing variables V and W into x and yin [37]), one obtains the three-variable model:

(9) f (x) z+ I

(10)(x) y

(11)(h1(x) z)

r is the time scale of the slow adaptationand h1 is the scale of the influence of the slowcs [39], which determines whether the neurona tonic or in a burst mode when it is exposed toned current input [38].pite some limitations in describing every prop-spike-bursting neurons, for example the rela-


tion between bursting frequency and amplitude of thereboundata [4advantaual celthat arare cou

Firsthe valneuron

periodior highexampa HindinjectePart I o

Secolinkedchemichibitorcal moences i(Fig. 3or to acouplin

2.2. Ex

2.2.1.The

for detwell dosive thThe reactionlarly conal conduce aterizeditive fiity of aoscillatcally erium pan unstSimulaley equternal i

. 4. PPerior exp

ernall, re/3.5liminducedillatioural oe 136el, th

the[42

oretic

imesta

,43]Exte

d hisne r

d Hudalthe ss. Thscopd potential versus current observed in some real0], the Hindmarsh and Rose model has majorges for studies of: (i) spikes trains in individ-

ls, and (ii) the cooperative behavior of neuronsises when cells belonging to large assembliespled with each other [40,41].t, as shown in Fig. 3A, and depending onues of parameters in the equations above, thes can be in a steady state or they can generate ac low-frequency repetitive firing, chaotic bursts-frequency discharges of action potentials (an

le of period-doubling of spike discharges ofmarsh and Rose neuron, as a function of thed current is illustrated in Fig. 14 displayed inf this review [1]).nd, Rose and Hindmarsh neurons can be easily

using equations accounting for electrical and/oral junctions (the latter can be excitatory or in-y) which underlie synchronization in theoreti-dels as they do in experimental material (refer-n [39]). Such a linkage can lead to out of phaseB) or to in phase bursting in neighboring cellschaotic behavior, depending on the degree ofg between the investigated neurons.

perimental data from single cells

Isolated axonsnonlinear behavior of axons and the potential

erministic chaos of excitable cells have beencumented both experimentally and with exten-eoretical models of the investigated systems.sults obtained with intracellular recordings ofpotentials in the squid giant axon are particu-nvincing. Specifically, by changing the exter-centration of sodium (Na), it is possible to pro-switch from the resting state to a state charac-by (i) self sustained oscillations and (ii) repet-

ring of action potentials that mimic the activ-pacemaker neuron (Fig. 4A). The resting and

ory states were found to be thermodynami-quivalent to an asymptotically stable equilib-oint and a stable limit cycle, respectively, withable equilibrium point between them (Fig. 4B).tions based upon modified Hodgkin and Hux-ations successfully predicted the range of ex-onic concentrations accounting for the bistable

Fig(A)afteextNaCa 1subproosc

natwer

panandfromThe

regthe[42

an

braan

soiofterboeriodic and non-periodic behavior of a squid giant axon.dic oscillations (left) and membrane potential at rest (right)osure of the preparation for 0.25 and 6.25 min to ansolution containing the equivalent of 530 and 550 mMspectively. (B) Bistable behavior of an axon placed inmixture of NSW 550 mM NaCl. Note the switch fromal (left) to supraliminal (right) self-sustained oscillations,by a stimulating pulse of increasing intensity. (C) Chaotic

ns in response to sinusoidal currents. The values of thescillating frequency and the stimulating frequency (fn)and 328 Hz (left) and 228 and 303 Hz (right). In each

e upper and lower traces represent the membrane potentialactivating current, respectively. (A and B are modified], C is from [44], with permission of the Journal ofal Biology.)

and the transition between the unstable andble periodic behavior via a Hopf bifurcation.nding their work on the squid giant axon, Ahiracollaborators [43,44] have studied the mem-

esponse of both this preparation and a Hodgkinxley oscillator to an externally applied sinu-

current with the amplitude and the frequencytimulating current taken as bifurcation parame-e forced oscillations were analyzed with stro-

ic and Poincar plots. The results showed that,


in agreement with the experimental results, the forcedoscillatperiodilationsthe appchaos wdoublindefined

WithIshizukrent apto studmembrlusk, OHuxleybifurcaas theThe dithose o(Fig. 5

Ano[46] sh

can be encoded chaotically, needs to be mentioned.

Fig. 5. D A3) r(B1B3) kes co(A1B1) ith tw(Adaptedischarge patterns of a pacemaker neuron caused by a dc current (A1One-dimensional Poincar maps of the corresponding sequence of spiRegular discharges of action potentials. (A2B2) Periodic firing wfrom [45], with permission of the Journal of Theoretical Biology.)epresentative samples of the recorded membrane potential.nstructed using the delay method (see [1] for explanations).o spikes per burst. (A3B3) Chaotic bursting discharges.or exhibited not only periodic but also non-c motions (i.e. quasi-periodic or chaotic oscil-) depending on the amplitude and frequency oflied current (Fig. 4C). Further, several routes toere distinguished, such as successive period-

g bifurcations or intermittent chaotic waves (asin Part I, [1]).a somewhat similar rationale, Hayashi and

a [45] used as a control parameter a dc cur-plied intracellularly through a single electrodey the dynamical properties of discharges of theane of the pacemaker neuron of a marine mol-nchidium verraculatum. Again, a Hodgkin andmodel did show a sequence of period-doubling

tions from a beating mode to a chaotic stateintensity of the inward current was modified.fferent patterns shared a close similarity withbserved experimentally in the same conditionsA1A3).ther and interesting report by Jianxue et al.owing that action potentials along a nerve fiber

Spontaneous spikes produced by injured fibers ofthe sciatic nerve of anaesthetized rats were recordedand studied with different methods. Spectral analy-sis and calculations of correlation dimensions wereimplemented first, but with limited success due tothe influence of spurious noise. However other ap-proaches turned out to be more reliable and fruitful.Based on a study of interspike intervals (ISI), theyincluded return (or Poincar) maps (ISI(n + 1) ver-sus ISI(n); Fig. 5B1B3) and a nonlinear forecastingmethod combined with gaussian scaled surrogate data.Conclusions that the time series were chaotic foundadditional support in the calculations of Lyapunov ex-ponents after adjusting the parameters in the programof Wolf et al. [47], which is believed to be relativelyinsensitive to noise.

2.2.2. Chaos in axonal membranes: commentsGeneral self criticism by Aihiara et al. [44] as to

which chaos with dimensions of the strange attrac-tors between 2 and 3 in their experiments was ob-


served under rather artificial conditions is important.This crior replationsphysioproduccillatora sinusplies thronal osynapstrains i

2.3. Si

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otheriticism applies to all forms of nonlinear behav-orted previously: in every instance the stimu-, whether electrical or chemical, were far fromlogical. However chaotic oscillations can beed by both the forced Hodgkin and Huxley os-and the giant axon when a pulse train [44] oroidal current [43] are used. This already im-at, as will be confirmed below, nonlinear neu-scillators connected by chemical or electricales can supply macroscopic fluctuations of spiken the brain.

ngle neurons

familiar to electrophysiologists that neuronalossess a large repertoire of firing patterns.le cell can behave in different modes i.e. suchnerator of single or repetitive pulses, bursts ofpotentials, or as a beating oscillator, to nameThis richness of behavioral states, which isled by external inputs, such as variations in thenvironment caused by the effects of synapticnd by neuromodulators, has prompted number

obiologists to investigate if, in addition to theses, chaotic spike trains can also be producedvidual neurons. If so, such spike trains woulde serious candidate neural codes as postulatedsly for other forms of signals thought to playr role as information carriers in the nervous[48,49]. Analytical proof that this hypothesiswell grounded has been presented for the

loch and Pitts neuron model [50].zled by the variability of activities in the buccal-urons of a sea slug Pleurobranchae californica,s et al. [51] recorded from individual cells withd techniques and analyzed the responses gen-in deafferented preparations in order to studyporal patterns of signals produced by the cen-vous system itself. The recorded cells, calledfor buccal-cerebral neurons) were particularlyting since they can act as either an autonomousor as part of a network that produces coordi-hythmic movements of all buccal-oral behav-everal criteria of chaos were apparently satis-

the analysis of the spike trains. These testsd the organization of the phase portraits andr maps which revealed attractors with clear ex-

pograreltherietio

calCadirR1dotonjecis btatifieimtherithimbuacttieproapbliwe

inglenattco

wa

ityionsynsw

pa

FohatheAPa bbetoanLyapunov exponents (assessed with the pro-f Wolf et al. [47]) and relatively constant cor-dimensions. The authors recognized however

itations of these conclusions since their time se-re quite short and often non-stationary. In addi-rogates were not used in their study.otic regimes were described with mathemati-els of neuron R15 of another mollusk, Aplysia

nica, but their reality has only been confirmedwith recordings from the actual cell. Neuron

d been known for long to fire in a normal, en-ous, bursting mode [52] and in a beating (i.e.

ode if a constant depolarizing current is in-nto the cell or if the sodium potassium pumped. These activities were first mimicked quali-by Plant and Kim [53] with the help of a mod-rsion of the Hodgkin and Huxley model. Whenented further for additional conductances andnamics by Canavier et al. [5456], the algo-

predicted different modes of activity and, morently, that a chaotic regime exists between the

g and beating modes of firing. That is, chaoticcould well be the result of intrinsic proper-

ndividual neurons and need not be an emergenty of neural assemblies. Furthermore the modelhed chaos from both regimes via period dou-

ifurcations. It was also suggested that these asother modes of firing, such as periodic burst-

rsts of spikes separated by regular periods of si-orrespond, in a phase space, to stable multiplers (Fig. 6A1A3 and B1B3). These attractorsed at given sets of parameters for which thereore than one mathematical solution (bistabil-ally, it was predicted that variations in external

oncentration (of sodium or calcium), transientc inputs and modulatory agents (serotonin) canthe activity of the cell from one stable firingto the other.eriments confirmed these prophecies in part.mple, transitions between bursting and beatingeady been observed in R15 in response tolication of the blocker 4-aminopyridine (4-ggesting that potassium channels may act as

cation parameter [57]. Also transitions fromto doublet and triplet spiking and finally

rsting regime were described in response toK+ channel blocker, tetraethyl ammonium


Fig. 6. Sinputs (aduring ainitial coaccordinBistabili(bottom)originalare in the

which,was cr

identifi[58,59]ensitivity of bursts to external stimuli. (A1A3 and B1B3) Control of model responses. (A1A3) A short (1 s, 4 Hz) train of synapticrrow) delivered immediately after a burst (A1) induces after a brief initial transient a transition to a beating mode (A2) which persistsn hour. In the phase plane projection (A3), the original pattern is shown in cyan and the final attractor is in red. (B1B3) Identicalnditions as in A1, but the stimulus that is delivered earlier (B1) induces a prolonged shift into a new mode of firing (apparently chaoticg to the authors) (B2). In the corresponding phase plane (B3), the initial attractor is cyan and the final one is magenta. (C1C2)ty in a intracellularly recorded R15 neuron. Shift of the cell from a bursting to a beating mode of activity. (C1) A brief current pulse

delivered during an interburst hyperpolarization is followed by a sustained beating after which the spiking activity returns to thebursting pattern. (C2) Successive transitions between identical bursting and beating episodes (above), whether current pulses (bottom)

depolarizing or the depolarizing direction. (A1B3 from [56]; C1C2 from [60], with permission of the Journal of Neurophysiology.)

in addition to this pharmacological property,edited to induce chaotic-like discharges ined neurons of the mollusc Lymnae Stagnalis. More critically, recordings from R15 were

performed by Lechner et al. [60] to determine whethermultistability is indeed an intrinsic property of the celland if it could be regulated by serotonin. It was foundthat R15 cells could exhibit two modes of oscillatory


Fig. 7. Dpresencedelay methe slow

activityperturband inswhich(Fig. 6of serotransitiperiods

Thepropertimportron ofster, Pative bpulsesynamic changes of the membrane potential of a LP neuron. Left column: intracellularly monitored slow oscillations and spikes in theof the indicated values of directly applied currents. Right column: corresponding state phase reconstructions obtained with the timethod (see text for explanations). The original coordinates are rotated so that the fast spiking motion takes place in the xy plane andbursting motion moves along the z-axis. (Modified from [40], with permission of the Journal of Neurophysiology.)

(instead of eight in models) and that briefations such as current pulses induced abrupttantaneous transitions from bursting to beatinglasted from several seconds to tens of minutesC1 and C2). In presence of low concentrationstonin the probability of occurrence of theseons and the duration of the resulting beatingwere gradually increased.contribution of ionic channels in the dynamicies of isolated cells has been demonstrated byant studies of the anterior burster (AB) neu-the stomatogastric ganglion of the spiny lob-ncibirus Interruptus. In contrast to constitu-

ursters, which continue to fire rhythmic im-when completely isolated from all synaptic in-

put, this neuron is a conditional burster, meaning thatthe ionic mechanisms that generate its rhythmic firingmust be activated by some modulatory input. It is theprimary pacemaker neuron in the central pattern gen-erator (see Section 3.2) for the pyloric rhythm in thelobster stomach. With the help of intracellular record-ings, HarrisWarrick and Flamm [61] have shown thatthe monoamines dopamine, serotonin and octopamineconvert silent AB neurons into bursting ones, the firsttwo amines acting primarily upon Na+ entry and thelatter on the calcium currents, although each cell canburst via more than one ionic channel (see also [61,62]). These experimental results were exploited on byGuckenheimer et al. [63] who characterized the ba-sic properties of the involved channels in a model


combining the formulations of Hodgkin and Huxley,and ofthe intmodelof Hopsionalrhythmservedtal protof dyning theauthorsvantageof thesitive pment rehavior.the nertor wheganizin

Theclusionnals prstomatcellulajectedAs thetern ofodic (Fstructutinguistion ofchaos,tors intion anto distichaos (be morreliable

Recthat dychaoticinto a swere into a strsake ofof phasthe res

. 8. Nizonron. Aasse

lysis:r (NP

responhe Ph

tputoritht atoch

thorsnor

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aoticenco

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re, aalysithemize cthe cyfish. Thee seRinzel and Lee [64]. Specifically, changes inrinsic firing and oscillatory properties of theAB neuron were correlated with the boundariesf and saddle-node bifurcations on two dimen-maps for specific ion conductances. Complexic patterns, including chaotic ones, were ob-in conditions matching those of the experimen-ocols. In addition to demonstrating the efficacyamical systems theory as a means for describ-

various oscillatory behaviors of neurons, theproposed that there may be evolutionary ad-s for a nerve cell to operate in such regions

parameter space: bifurcations then locate sen-oints at which small alterations in the environ-sult in qualitative changes in the systems be-Thus, using a notion introduced by Thom [65]ve cell can function as a sensitive signal detec-n operating at a point corresponding to an or-g center.above mentioned studies met a rewarding con-when Abarbanel et al. [40] analyzed the sig-

oduced in the isolated LP cells from the lobsterogastric ganglion. The data consisted of intra-rly recorded voltage traces from neurons sub-to an applied current of different amplitudes.intensity of the current was varied, the pat-firing shifted via bifurcations, from a peri-

ig. 7A and B) to a chaotic like (Fig. 7CE)re. The authors could not mathematically dis-h chaotic behavior from a nonlinear amplifica-noise. Yet, several arguments strongly favoredsuch as the robust substructure of the attrac-Fig. 7C and D. The average mutual informa-d the test of false nearest neighbors allowednguish between noise (high-dimensional) andlow-dimensional). This procedure was found toe adequate than the Wolf method which is onlyfor the largest exponents.

ent investigations on isolated cells have shownnamical information can be preserved when ainput, such as a Rssler signal, is convertedpike train [66]. Specifically, the recorded cellsvitro sensory neurons of rats skin subjected

etch provided by a Rssler system, and, for thecomparison, to a stochastic signal consisting

e randomized surrogates. The determinism ofulting inter spike intervals (monitored in the

Fighorneu

was

ana

erro

cor

of t

ou

algthaa s

au

thestoas

chtostim

rec

hean

ma

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ulitimormalized prediction error as a function of the predictedfor chaotic and random (surrogate) signals, in a sensory

n embedding dimension of three was used, significancessed with two-tailed tests. Inset: results from the statisticals is the standard deviation of the normalized predictionE) for the surrogate trials; dashed line: significance levelding to the indicated p value. (From [66], with permission

ysical Review Letters.)

nerve) was tested with a nonlinear predictionm, as described in [1]. The results indicatedchaotic signal could be distinguished fromastic one (Fig. 8). That is, and quoting the, for prediction horizons up to 36 steps,malized prediction error (NPE) value for thetically evoked ISI series were all near 1.0,osed to significantly smaller values for theally driven ones. Thus sensory neurons are ablede the structure of high-dimensional externalinto distinct spike trains.ough based on studies of non isolated cellsd in vitro, another report can be mentioned

t least, as a reminder of the pitfalls facing thes of large neuronal networks with nonlinearatical tools. It represents an attempt to charac-

haos in the dynamics of spike trains producedaudal photoreceptor in the sixth ganglion of theProcambarus clarkii subjected to visual stim-authors [67] rely on the sole presence in their

ries of first order unstable periodic orbits sta-


tistically confirmed with gaussian surrogates, despiteevidencing [68

3. Pair

A fathat enactionperimethat for(closedtic paroperatilow dimthis nochaos,neuralal. [69]tions ofactorssince thneuron

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sube that this criterion alone is far from convinc-].

s of neurons and small neuronal networks

miliar observation to most neurobiologists issembles of cells often produce synchronizedpotentials and/or rhythmical oscillations. Ex-ntal data and realistic models have indicatedsome geometrical connectivity of the networktopologies) and for given values of the synap-

ameters linking the involved neurons, the co-ve dynamics of cells can take the form of a

ensional chaos. Yet a direct confirmation oftion, validated by unambiguous measures forhas only been obtained in a limited sample ofcircuits. In principle, as noted by Selverston et, network operations depend upon the interac-f numerous geometrical synaptic and cellular, many of which are inherently nonlinear. Butese properties vary among different classes of

s, it follows that although often taken as an end-y itself a reductionist determination of theirentation can be useful for a complete descrip-networks global activity patterns. So far, suched analysis has only been achieved successfullyfew invertebrate and lower vertebrate prepara-

inciples of network organization

n extensive review of the factors that governk operations, Getting [70] remarked that indi-conductances are not as important as the prop-hat they impart. Instead, he insists on two mainof elements. The first defines the functionaltivity. It includes the sign (excitatory or in-y) and the strength of the synaptic connections,lative placement on the postsynaptic cell (somaritic tree) and the temporal properties of these

ns. The second, i.e. the anatomical connectivi-rmines the constraints on the network and whowhom. Despite the complexity and the vast

r of possible pathways between large groupsrons, several elementary anatomical building

Figtatiinhjuntriaspe

blowo

ver

havproor

citren

intcan

theco

or

tonan

thethrthevo

chare

lobsivcanmple building blocks of connectivity. (A) Recurrent exci-) Mutual inhibition. (C) Recurrent inhibition. (D) Cyclic. (E) Coupling by way of directly opposed electrotonic. (F) Electrical coupling via presynaptic fibers. Symbols:and dots indicate excitatory and inhibitory synapses, re-

y; resistors correspond to electrical junctions.

which contribute to the nonlinearity of the net-can be encountered in both invertebrate andate nervous system. Such simple configurationsutual (or recurrent) excitation (Fig. 9A) whiches synchrony in firing, and reciprocal (Fig. 9B)rrent (Fig. 9C) inhibitions which regulate ex-ty and can produce patterned outputs. Recur-clic inhibition corresponds to a group of cellsnnected by inhibitory synapses (Fig. 9D), and iterate oscillatory bursts with as many phases ase cells in the ring [71]. In addition cells can beby electrical junctions either directly (Fig. 9E)

ay of presynaptic fibers (Fig. 9F). Such electro-upling favors synchrony between neighboring

synergistic neurons [72].mber of systems can be simplified according tostricted schemes [73], which remain conservedout phylogeny. As described below, such is

e in the Central Pattern Generators (CPGs) in-in specific behaviors that include rhythmic dis-

of neurons acting in concert when animalsing, swimming or flying. One prototype is thestomatogastric ganglion [74], in which exten-dies have indicated that (i) a single networkserve several different functions and partici-


Fig. 10. Daction popresentatand the m

pate inganizatby modgiven apotentinew co

and areterns [classic

3.2. Co

Whetronic date nonetermistic behavior of coupled formal neurons. (A) Two excitable cells modeled according to Rose and Hindmarsh generate periodictentials at the rate of 26 and 33 Hz, respectively (left) and each of these frequencies is visualized on a return map (right). (B) Sameion as above showing that when the neurons are coupled, for example by an electrotonic junction, their respective frequency is modifiedap exhibits a chaotic-like pattern. Note the driving effect of the faster cell on the less active one. (After Faure and Korn, unpublished.)

more than one behavior, (ii) the functional or-ion of a network can be substantially modifiedulatory mechanisms within the constraints of anatomy, and (iii) neural networks acquire theiral by combining sets of building blocks intonfigurations which however, remain nonlinearstill able to generate oscillatory antiphasic pat-

75]. These three features run contrary to theal view of neural networks.

upled neurons

n they are coupled, oscillators, such as elec-evices, pendula, chemical reactions, can gener-linear deterministic behavior (Refs. [7,76]) and

this property extends to oscillating neurons, as shownby models (Fig. 10) and by some experimental data.

Makarenko and Llinas [77] provided one of themost compelling demonstration of chaos in the cen-tral nervous system. The experimental material, i.e.guinea-pig inferior olivary neurons was particularlyfavorable for such a study. These cells give rise to theclimbing fibers that mediate a complex activation ofthe distant Purkinje cells of the cerebellum. They arecoupled by way of electrotonic junctions, and slices ofthe brainstem which contain their somata can be main-tained in vitro for intracellular recordings. Subthresh-old oscillations resembling sinusoidal waveforms witha frequency of 46 Hz and an amplitude of 510 mVwere found to occur spontaneously in the tested cellsand to be the main determinant of spike generation and


Fig. 11. Pon the twand 2 anbursts dy

collectNonlinments oin pairteria bculatiothe Lyachaos wsynchrpresumses neu

Rathal. [79]ually ghase portraits of the slow oscillations in two coupled PD neurons as a function of the indicated external conductance ga. The projectionso planes of variables VF1(t), VF2(t) in the left column, that is of the low-pass filtered (5 Hz) of the membrane potential V of cells 1d of VF1(t), VF1(t + td) in the right column characterize the level of synchrony of bursts in the neurons, and the complexity of thenamics, respectively. (From [79], with permission of the Physical Review Letters.)

ive behavior in the olivo-cerebellar system [78].ear analysis of prolonged and stationary seg-f those oscillations, monitored in single and/or

s of IO neurons was achieved with strict cri-ased on the average mutual information, cal-n of the global embedding dimensions and ofpunov exponent. It unambiguously indicated aith a dimension of 2.85 and a chaotic phase

onization between coupled adjacent cells whichably accounts for the functional binding of the-rons when they activate their cerebellar targets.er than concentrating on chaos per se, Elson etclarified how two neurons which can individ-

enerate slow oscillations underlying bursts of

spikes (that is spiking bursting and seemingly chaoticactivities) may or may not synchronize their dis-charges when they are coupled. For this purpose theyinvestigated two electrically connected neurons (thepyloric dilatators, PDs) from the pyloric CPG of thelobster stomatogastric ganglion (STG). In parallel tothe natural coupling linking these cells, they estab-lished an artificial coupling using a dynamic clampdevice that enabled direct injections of, equal and op-posite currents in the recorded neurons, different fromto the procedure described in [80], in that they usedan active analog device which allowed the change inconductivity, including sign, and thus varied the totalconductivity between neurons. The neurons had been


Fig. 12.pyloric prhythm,a state ofpattern iof phaseal. [87],

isolatedThe aucillatiodespitemationpling wsentingbifurcaAddingthe neutiphasethe fastthreshocludedthe synConnecting an electronic neuron to isolated neurons via artificial synapses restores regular bursting. Above: experimental setup. Theacemaker group of the lobster consists in four electronically coupled neurons. These are the anterior burster (AB), which organizes thetwo coupled pyloric dilatator (PD) and the ventral dilatator (VD). Here AB is replaced by an electrotonic neuron (EN), set to behave inchaotic oscillations. (A1A2) Neurons disconnected exhibit chaotic discharges of action potentials. (B1B2) Generation of a bursting

n the mixed network after coupling the cells. IPD is the current flowing into PD from EN. Note that the bursts are in phase, or out, in EN and PD depending whether the coupling conductance is positive (B1) or negative (B2), respectively. (Adapted from Szucs etwith permission of NeuroReport.)

from their input as described in Bal et al. [81].thors found that with natural coupling, slow os-ns and fast spikes are synchronized in both cellscomplex dynamics (Fig. 11A). But in confir-of earlier predictions from models [40], uncou-ith additional negative current (taken as repre-an inhibitory synaptic conductance) produced

tions and desynchronized the cells (Fig. 11B).further negative coupling conductance caused

rons to become synchronized again, but in an-(Fig. 11C). Similar bifurcations occurred forspikes and slow oscillations, but at a different

ld for both types of signals. The authors con-from these observations that the mechanism forchronization of the slow oscillations resembled

that seen in dissipatively coupled chaotic circuits [82]whereas the synchronization of the faster occurringspikes was comparable to the so-called threshold syn-chronization in the same circuits [83]. The same ex-perimental material and protocols were later exploitedby Varona et al. [87,91] who suggested, after using amodel developed by Falke et al. [84], that slow sub-cellular processes such as the release of endoplasmiccalcium could also be involved in the synchronizationand regularization of otherwise individual chaotic ac-tivities. It can be noted here that the role of synapticplasticity in the establishment and enhancement of ro-bust neural synchronization has been recently exploredin details [85] with Hodgkin and Huxley models ofcoupled neurons showing that synchronization is more


Fig. 13.of 165)phase prrecordingthe expo

rapid atimingnection

Conrons ca

connec

stored.used anfiring pThis ENthat waand Roof thevated awere birregulindicatever, s

that seeControl of bursting by inhibitory inputs in a LP neuron. Left column: superimposed traces, with individual bursts (n 30% of a totalsweeps are aligned at time 0 ms which corresponds to the point of minimum variance. Negative times indicate the hyperpolarizingeceding each bursts onset. Right column: variance as a function of time between the voltage traces calculated fom the entire sample ofs in each condition. (AC) See text for explanations. Open circles mark the mean time of burst termination. The arrow in C signals

nential tail of the plot (from Elson et al. [90], with permission of the Journal of Neurophysiology).

nd more robust against noise in case of spikeplasticity of the Hebbian type [86] than for con-s with constant strength.versely, isolated, non regular and chaotic neu-n produce regular rhythms again once theirtions with their original networks are fully re-This was demonstrated by Szucs et al. [87] whoanalog electronic neuron (EN) that mimickedatterns observed in the lobster pyloric CPG.

was a three degree of freedom analog devices built according to the model of Hindmarshse. When the anterior burster (AB) which is onemain pacemakers of the STG was photoinacti-nd when synaptic connections between the cellslocked pharmacologically, the PD neurons firedarly (Fig. 12A1 and A2) and nonlinear analysised high-dimensional chaotic dynamics. How-ynchronized bursting, at a frequency close ton in physiological conditions, appeared imme-

diately after bidirectional coupling was established (aswith an electrotonic junction) between the pyloric cellsand the EN, previously set to behave as a replacementpacemaker neuron (Fig. 12B1). Furthermore switch-ing the sign of coupling to produce a negative conduc-tance that mimicked inhibitory chemical connectionsresulted in an even more regular and robust antiphasicbursting which was indistinguishable from that seen inthe intact pyloric network (Fig. 12B2). These data con-firmed earlier predictions obtained with models sug-gesting the regulatory role of inhibitory coupling oncechaotic cells become members of larger neuronal as-semblies [88,89].

The LP neuron receives strong inhibitory inputsfrom three electrically coupled pacemaker neurons ofthe STG. These are the anterior burster (AB) and twopyloric dilator (PD) cells. As shown above, this set-ting had already been exploited by Elson et al. [79]to strengthen the notion that the intrinsic instabilities


of circuit neurons may be regulated by inhibitory af-ferentsspontanregularFig. 13strongvated aperiodifectedtial in sulationnamicsimportlated fractivitysystemtive nonential

3.3. Le

Thetweensectedthe resmodel.parameterestinvalue oidentichas bee(Fig. 1formatgressivof exciphase swhichone. Thtion [8to modcally an

Senhave bescribedtwo disequationeuron

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. 14.dmarntiphductaametehase0.5).

ctedC) nede etse n

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uationg [. Furthermore, in control conditions [90], theeous bursts generated by the LP neuron are ir-

, as illustrated by the superimposed traces ofA. However forcing inhibitory inputs had astabilizing effect. When the latter were acti-t 65 Hz the bursts were relatively stable andc and their timing and duration were both af-(Fig. 13B). This means that inhibition is essen-mall assemblies of cells for producing the reg-of the chaotic oscillations prevalent in the dy-of the isolated neurons (see also [91]). Equally

ant is that in confirmation, when cells were iso-om all their synaptic inputs their free-runningresembled that of a typical nonlinear dynamicshowing chaotic oscillations with some addi-

ise, a property that could account for the expo-tail of their computed variance (Fig. 13C).

ssons from modeling minimal circuits (CPGs)

role of the different forms of coupling be-two chaotic neurons has been carefully dis-by Abarbanel et al. [40] in studies based onults obtained with the Hindmarsh and RoseAlthough the values of some of the couplingters may be out of physiological ranges, in-g insights emerged from this work: for a highf the coupling coefficient , synchronization ofal chaotic motions can occur. This propositionn verified for coupling via electrical synapses

4A1A3) with measurements of the mutual in-ion and of Lyapunov exponents. Similarly, pro-ely higher values of symmetrical inhibition, ortatory coupling, lead to in phase and out ofynchronization of the bursts of two generatorscan then exhibit the same chaotic behavior asis phenomenon is called chaotic synchroniza-2,92]. The authors extended these conclusionserately noisy neurons and to non symmetri-d non identical coupled chaotic neurons.

sory dependent dynamics of neural ensemblesen explored by Rabinovich et al. [93] who de-the behavior of individual neurons present intinct circuits, modeled by conductance basedns of the Hodgkin Huxley type. These formals belonged to an already mentioned CPG, theection 3.2) and to coupled pairs of intercon-

FigHinin acon

parin p >

ne

(Triathema

ne

Thofspowirepsysne

var

in-(Fifewa n

are

eqpliSynchronization of two electrically coupled Rose andsh neurons. The membrane potentials x1(t) and x2(t) arease for a low value of the coupling parameter which is ance of the wire connecting them (A1, = 0.02). As thisr is increased, the synchronization is incomplete and nearly(A2, = 0.4) and it is finally complete and in phase (A3,(From [40], with permission of Neural Computation.)

thalamic reticular (RE) and thalamo corticalurons that were previously investigated by Ste-al. [94]. Although the functional role played by

etworks is very different (the latter passes infor-to the cerebral cortex), both of them are con-by antagonistic coupling (Fig. 15A1 and A2).hibit bistability and hysteresis in a wide rangeling strengths. The authors investigated the re-of both circuits to trains of excitatory spikesrying interspike intervals, Tp, taken as simplentations of inputs generated in external sensorys. They found different responses in the con-cells, depending upon the value of Tp. That is,ns in interspike intervals led to changes frome to out-of-phase oscillations, and vice-versa5B1 and B2). These shifts happened within akes and were maintained in the reset state untilnput signal was received.e bistability occurs in the CPG when there

distinct solutions to the conductance-basedns within a given range of electrical cou-93], the authors further investigated the range


Fig. 15.networkjunctionthe effecRE-TC ([93], wit

of thethe REthat thespace,illustradependtwo baportingily prodspondstractors

Largtical nResponses of formal circuits to train of excitatory inputs. (A1A2) Diagrams of the STG circuits (A1) and of the thalamo-cortical(A2). Solid and empty dots indicate inhibitory and excitatory coupling connections. The resistor symbol in the CPG denotes a gapbetween the two neurons. External signals were introduced at loci indicated by arrows. (B1B2) Time series (upper traces) showingt of external forcing by 1-s period trains of spikes (lower traces) at the indicated interspike intervals (Tp), in the CPG (B1) and in theB2) circuits. Action potentials from each of the two cells are indicated by solid and dashed vertical lines, respectively. (Adapted fromh permission of Physical Review E.)

strength of the inhibitory coupling over whichTC cells act in the same fashion. It turned outre were two distinct phase portraits in the stateeach one for a solution set (Fig. 16). Here theyte two distinct attractors, and the one that winss on the initial conditions of the system. Thesins of attraction are close to each other, sup-the fact that a switch between them can be eas-uced by new spike trains. This behavior corre-to what the authors call calculation with at- [91].er cortical assemblies aimed at mimicking cor-

etworks were also modeled in order to char-

acterize the irregularities of spike patterns in a tar-get neuron subjected to balanced excitatory and in-hibitory inputs [95]. The model of neurons was asimple one, involving two state units sparsely con-nected by strong synapses. They were either activeor inactive if the value of their inputs exceeded afixed threshold. Despite the absence of noise in thesystem, the resulting state was highly irregular, witha disorderly appearance strongly suggesting a deter-ministic chaos. This feature was in a good agree-ment with experimentally obtained histograms of fir-ing rates of neurons in the monkey prefrontal cor-tex.


Fig. 16.RETCspace ofin the sathe twothat spikbetweenactivatiopermissi

3.4. Conetwor

Moswhichmembeas oppocells reones, s

This cal. [96as a cr

given bfoundexhibita task,when tThis isclosedthe chCPGsevolvedof dyna

Whain a reg

corporated in the nervous system? Rather than con-trateneur

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Neu

In alutsiabils var

isecess

tputs firnt octiore istoc

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h-didetaState space portrait of two coexisting attractors of thesystem. The solid line is the orbit in [V (t), IT(t), Ih(t)]the in-phase oscillations. The dotted line is the path takenme state space by the out of phase oscillations. Note thatattractors are close to each other, supporting the notione trains with appropriate intervals can induce transitionsthem. Abbreviations: V membrane potential, IT and Ih:

n and inactivation of ionic channels. (From [93], withon of Physical Review E.)

mments on the role of chaos in neuralks

t of the above reported data pertain to CPGs inevery neuron is reciprocally connected to otherrs of the network. This is a closed topology,sed to an open geometry where one or severalceive inputs but do not send output to othero that there are some cells without feedback.ase was examined theoretically by Huerta et] using a Hindmarsh and Rose model. Takingiterion the ability of a network to perform aiological function such as that of a CPG, they

that although open topologies of neurons thatregular voltage oscillations can achieve suchthis functional criterion selects a closed one

he model cells are replaced by chaotic neurons.consistent with previous claims that (i) a fullyset of interconnections are well fit to regularizeaotic behavior of individual components of[41] and (ii) real networks, even if open, have

to exploit mechanisms revealed by the theorymical systems [97].t is the fate of chaotic neurons which oscillateular and predictable fashion once they are in-

cen

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higinon the difficulties of capturing the dynamicsons in three or four degrees of freedom Rabi-et al. [89] addressed a broader and more quali-sue in a somewhat opinionated fashion. Thatasked how is chaos employed by natural sys-accomplish biologically important goals, or,

ise stated, why evolution has selected chaosical pattern of behavior in isolated cells. Theyhat the benefit of the instability inherent tomotions facilitates the ability of neural systemsly adapt and to make transitions from one pat-another when the environment is altered. Ac-to this viewpoint, chaos is required to main-robustness of the CPGs while they are con-

to each other, and it is most likely suppressedollective action of a larger assembly, generallynhibition alone.

ral assemblies: studies of synaptic noise

l central neurons the summation of intermittentfrom presynaptic cells, combined with the un-ity of synaptic transmission produces continu-iations of membrane potential called synaptic[98]. Little is known about this disconcerting, except that it contributes to shape the inputrelation of neurons (references in [99,100]). Itst attributed to a random synaptic bombard-f the neurons and the view that it degrades theirn has remained prevalent over the years [101].mportant, it has been commonly assumed tohastic [102104] and is most often modeled[95,105,106]. Therefore the most popularizedon synaptic noise have mostly concentrated onr or not, and under which conditions, such a

process contributes to the variability of neu-ring [107109]. Yet recent data which are sum-

below suggest that synaptic noise can be de-stic and reflect the chaotic behavior of inputst to the recorded cells. These somewhat uncon-al studies were motivated by a notion whichn and remains too often overlooked by physi-i.e. that at first glance, deterministic processes

e the appearance of stochasticity, particularly inmensional systems. This question is addressedils in [1]. As will be shown in the remaining


sections of this review, this notion brings about funda-mentalnisms u

4.1. Ch

Conaratinging synmeasur

stochasear dynstructuin the Mron wh

Sevdifferenries (Ftime denon ran

were a

accord(Fig. 1next onthe matook thdicatinpost-sy(Fig. 1the inipopulatudes hriodicitso-callvertebrible wicorrelafrequentweenstable pand 3 oreturn mstrengtthe %entropywhichnoise (

A model of coupled Hindmarsh and Rose neu-s, ge fr

g. 18r to tr, thcon

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mplerks [changes to our most common views of mecha-nderlying brain functions.

aos in synaptic noise

ventional histograms of the time intervals sep-synaptic potentials and/or currents compris-

aptic noise suggest random distributions of thise. However since a chaotic process can appeartic at first glance (see [1]), the tools of nonlin-amics have been used to reassess the temporal

re of inhibitory synaptic noise recorded, in vivo,authner (M-)cell of teleosts, the central neu-

ich triggers the animals vital escape reaction.eral features of chaos were extracted from thetiated representation of the original time se-

ig. 17A). Recurrence plots obtained with thelay method already suggested the existence ofdom motion [110]. Return (or Poincar) maps

lso constructed with subsets of events selecteding to their amplitude by varying a threshold 7B) and plotting each interval (n) against thee (n + 1). As was progressively lowered,

ps first disclosed a striking configuration whiche form of a triangular motif, with its apex in-g a dominant frequency, fp, of the inhibitorynaptic potentials that build up synaptic noise7C1). Subtracting events associated with fp intial time series further revealed at least threetions of IPSPs of progressively smaller ampli-aving in consecutive return maps, distinct pe-ies p,s,t (Fig. 17C1 and C2), all in theed gamma range commonly observed in higherates. Two series of observations were compat-th chaotic patterns, (i) mutual interactions andtions between the events associated with thesecies were consistent with a weak coupling be-

underlying generating oscillators and, (ii) un-eriodic orbits (Fig. 17D) as well as period 1, 2rbits (see also Section 6.1) were detected in theaps [39]. The notion of a possible chaos was

hened by the results of measures such as that ofof determinism and of the KolmogorovSinai

[111] combined with the use of surrogates,confirmed the nonlinear properties of synapticFig. 17E).

ron

sam

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4.2

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woenerating low-frequency periodic spikes at theequencies as those detected in synaptic noiseA) produced return maps having features sim-hose of the actual time series providing, how-at their terminal synapses had different quan-tents (Fig. 18C1 and B1 versus C1 and B2).e simulations the quantal content varied in theetermined experimentally for a representativeion of the presynaptic inhibitory interneuronsenerate synaptic noise in the M-cell [112]. Thement of synaptic efficacies in the transmissionmical patterns from the pre- to the postsynaptics verified experimentally, taking advantage ofing that the strength of the M-cells inhibitory

ns are modified, in vivo, by long-term tetanication (LTP), a classical paradigm of learningn be induced in teleosts by repeated auditory. It was found (not illustrated here) that this in-of synaptic strength enhances measures of de-sm in synaptic noise without affecting the peri-of the presynaptic oscillators [39].

haos as a neural code

nature of the neural code has been the subjecttless speculations (for reviews, see [113115])spite innumerable schemes, it remains an al-tractable notion (for a definition of the termhistory, see [116]). For example, it has beend [48,117119] that the coding of informationentral Nervous System (CNS) emerges fromt firing patterns. As noted by Perkel [49] nonl codes involve several aspects of the temporale of impulse trains (including burst character-nd some cells are measurably sensitive to vari-f such characteristics, implying that the latterread by neurons (review in [120]). Also, a richire of discharge forms, including chaotic ones,een disclosed by applying nonlinear analysissionality, predictability) to different forms ofains (references in [121]). Putative codes maythe rate of action potentials [104,122], wellsynchronous activities of the gamma type

), particularly during binding [123] and morex temporal organization of firing in large net-124,125]. The role of chaos as well as the real-


Fig. 17. Evidence for non random patterns in the synaptic noise of a command neuron. (A) Consecutive IPSPs observed as depolarizingpotentials (dots) recorded at a fast sweep speed (V (t), above) and their derivative (dV/dt , below). The dashed line delineates the backgroundinstrumental noise. (B) Derivative of a segment of synaptic noise, recorded at a slow sweep speed: fast events, each corresponding to an IPSPwere selected by a threshold , having different values (from top to bottom 1, 2, 3); intervals between each selected event, are labeled I (n)and I (n+ 1). (C1C2) Return maps constructed with events selected by 2, i.e. above a level corresponding to an intermediate value of thethreshold. The density was calculated by partitioning the space in 5050 square areas (i.e. with a resolution of 0.420.42 ms) and by countingthe number of points in each of these boxes. Areas in blue, green, red and yellow indicate regions containing less than 4 points, between 4 and8, 8 and 12 or more points, respectively. (C1) The principal and secondary periods, P = 16.25 ms and S = 14.4 ms fit the highest densityof points at the lower edge of the triangular pattern. (C2) A third period, T = 13.3 ms is unmasked at the base of another triangle obtainedafter the events used to construct C1 have been excluded. (D) Unstable period orbits (n = 5) with stable and unstable manifolds determinedby sequences of points that converge towards, and then diverge from, the period-1 orbit (labeled 2), in the indicated order. (E) Variations ofthe significance level of two measure of determinism, the %det and (), as a function of (see [1] for definitions). The vertical dashed lineindicates a confidence level at 2e5 after comparison with surrogates (AD) form. (Modified from [39], with permission of the Journal ofNeurophysiology.)


Fig. 18.(labeledpotentialpropertiesame ran

presentat(C1) andUPOs (n

ity of ation 8.3

Relechaos,may cothe valContribution of synaptic properties to the transmission of presynaptic complex patterns. (A) Modeled Hindmarsh and Rose neurons1 to 4), coupled by way of inhibitory junctions and set to fire at 57, 63, 47 and 69 Hz, respectively. (B1B2) Analysis of postsynaptics produced by uniform junctions. (B1) Top. Same neurons as above implemented with terminal synapses having different releasings but the same quantal content, np. Bottom. Superimposed IPSPs generated by each of the presynaptic cells and fluctuating in thege. As a consequence they are equally selected by the threshold, . (B2) The resulting return map appears as random. (C1C2) Sameion as above but, terminal synapses now have distinct quantal contents. Thus detects preferentially IPSPs from oscillators 1 and 2the corresponding map exhibits a triangular pattern centered on the frequency of the larger events (C2). Note also the presence of= 6). (From [39], with permission of the Journal of Neurophysiology.)

code itself will be further questioned in Sec-.

vant to this issue, it has been suggested thatfound in several areas of the CNS [67,126],ntribute to the neuronal code [95,127129]. Butidation of this hypothesis required a demon-

stration that deterministic patterns can be effectivelytransmitted along neuronal chains. Results summa-rized in the preceding section indicate that, surpris-ingly, the fluctuating properties of synapses favorrather than hamper the degree to which complex ac-tivities in presynaptic networks are recapitulated post-


synaptically [39]. Furthermore, they demonstrate thatthe emnaptic cnon-un

chastic

4.3. St

Thetic resodom flcooperan intetion of[130])ences ivelopetwo staquencefluctuaand it a(referethis pro

Dataconfirmsory symechanpaddlecan als[140] a[142] athe levechroniz

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Ear

Anthe lergence of deterministic structures in a postsy-ell with multiple inputs is made possible by theiform values of synaptic weights and the sto-release of quanta.

ochastic resonance and noise

emerging concept in neurosciences of stochas-nance (SR), which assigns a useful role to ran-uctuations, must be mentioned. It refers to aative phenomenon in nonlinear systems, wherermediate level of activity improves the detec-subthreshold signals (and their time reliabilityby maximizing the signal-to-noise ratio (refer-n [131]). The theory of SR has mostly been de-d with the simplifying assumption of a discretete model [132,133]. It is described as the conse-of interactions between nonlinearity, stochastic

tions and a periodic (i.e. sinusoidal) force [134]pplies to the case of integrate-and-fire models

nces in [135]). The basic concepts underlyingcess are illustrated in Fig. 19A and B.from several experimental preparations have

ed that SR can influence firing rates in sen-stems, such as crayfish [136] and rat [137]oreceptors, the cercal sensory apparatus of

fish [138], and frog cochlear hair cells [139]. Ito play a positive role in rat hippocampal slicesnd in human spindles [141], tactile sensationnd vision [143]. SR is also likely to occur atl of ionic channels [144] and it could favor syn-ation of neuronal oscillators [145].

eral aspects of SR call for deeper investigations,larly since noise, a ubiquitous phenomenon atels of signal transduction [146], may embeddom fluctuations [147]. Enhancement of SRn demonstrated in a FitzhughNagumo modelon driven by colored (1/f ) noise [148], whilec perturbations of the same cells generate phase, quasiperiodic and chaotic responses [149]. Inn, a Hodgkin and Huxley model of mammalianral cold receptors, which naturally exhibits SR, has revealed that noise smooths the nonlin-of deterministic spike trains, suggesting its in-on the systems encoding characteristics [150].ffect termed chaotic resonance appears in thed Lorentz model in the presence of a periodic

Fig(A)noi(hoto tspewitratishois owit

timlowapimno

timterlieSRpalin

5.

inModulation of a periodic signal by stochastic resonance.eled subthreshold sinusoidal signal with added Gaussianch time the sum of the two voltages crosses the thresholdal line) a spike is initiated (above) and a pulse is addede series, as indicated by vertical bars (below). (B) Powerdensity (ordinates) versus signal frequency (abscissae),

arp peak located at 0.5 kHz (arrow). Inset: signal to noiseR-ordinates) as a function of noise intensity (abscissae)that the ability to detect the frequency of the sine wavezed at intermediate values of noise. (Modified from [131],

ission of Nature.)

riation of the control parameters above and be-threshold for the onset of chaos [151]. It alsoin the KIII model [152] involving a discrete

entation of partial differential equations. Hereot only stabilizes aperiodic orbits, since an op-noise level can also act as a control parame-produces chaotic resonance [153] which is be-

o be a feature of self organization [153]. Finallybeen reported in a simple noise-free model of

inhibitory-excitatory neurons, with piece-wiseunction [154].

ly EEG studies of cortical dynamics

enormous amount of efforts has been directedast three decades towards characterizing corti-


cal signals in term of their dimension in order to as-certaincriteriater becof surroterminitonishiavenue

(only sdated t

5.1. Co

Mosthe ana(EEG)with thage ponent, thmationbetweenon-stasuch mpretatiorefinedled to tnot bependinfilteredchaoticterminiThis ramodera[155],chaos,lyzed Elustratea humcombindancy i[166])sion ofcan besignalsby late

. 20.a 90 sthe sce lag.beddin

of qEEG

undanhly silinears. Nomissio

. Ex

Freerela

sis athisEE

mina. Bothernedtire bs strnsethechaos. However, with time, the mathematicalfor obtaining reliable conclusions on this mat-

ame more stringent, particularly with the adventgates aimed at distinguishing random from de-stic time series [155]. Therefore despite the as-ng insights of their authors, who opened news for research, the majority of the pioneer worksome of which will be alluded to below), are out-oday and far from convincing.

rtical nets

t of the initial investigations have relied uponlysis of single channel electroencephalographicsignals, with attempts to estimate dimensione GrassbergerProcaccia algorithm, the aver-intwise dimension [156], the Lyapunov expo-e fractal dimension [157] or the mutual infor-content [158]. But in addition to the conflict

n the requirement of long time series and thetionarity of actual data, serious difficulties ofeasures (such as artefacts or possible misinter-ns) have been pointed out [159,160]. That is,tests comparing measures of segments of EEGshe conclusion [160] that the actual data coulddistinguished from gaussian random processes,g support of the view that EEGs are linearlynoise [161], either because they are not trulyor because they are high dimensional and de-

sm is difficult to detect with current methods.ther strong and negative statement was later onted by evidence that, as pointed out by Theiler

despite the lack of proof for a low-dimensionala nonlinear component is apparent in all ana-EG records [160,162167]. This notion is il-d in Fig. 20, where recordings obtained froman EEG, were analyzed with a method thated the redundancy approach (where the redun-s a function of the KolmogorovSinai entropywith the surrogate data technique. The conclu-this study was that, at least, nonlinear measuresemployed to explore the dynamics of cortical[168]. This view has been strongly vindicated

r investigations ([169], see Section 6.2.2).

Figon

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Is There Chaos in the Brain? II - Experimental Evidence and Related Models

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