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C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 © 2001Académiedessciences/ÉditionsscientifiquesetmédicalesElsevierSAS.Tousdroitsréservés S0764446901013774/REV Pointsur/Concisereview Is there chaos in the brain? I. Concepts of nonlinear dynamics and methods of investigation PhilippeFaure,HenriKorn* Biologiecellulaireetmoléculaireduneurone(InsermV261),InstitutPasteur,25rueDocteurRoux,75724 ParisCedex15,France Received18June2001;accepted2July2001 CommunicatedbyPierreBuser Abstract– In the light of results obtained during the last two decades in a number of laboratories,itappearsthatsomeofthetoolsofnonlineardynamics,firstdevelopedand improved for the physical sciences and engineering, are well-suited for studies of biological phenomena. In particular it has become clear that the different regimes of activities undergone by nerve cells, neural assemblies and behavioural patterns, the linkagebetweenthem,andtheirmodificationsovertime,cannotbefullyunderstoodin the context of even integrative physiology, without using these new techniques. This report,whichisthefirstoftworelatedpapers,isaimedatintroducingthenonexpertto thefundamentalaspectsofnonlineardynamics,themostspectacularaspectofwhichis chaostheory.Afterageneralhistoryanddefinitionofchaostheprinciplesofanalysisof time series in phase space and the general properties of chaotic trajectories will be described as will be the classical measures which allow a process to be classified as chaoticinidealsystemsandmodels.Wewillthenproceedtoshowhowthesemethods needtobeadaptedforhandlingexperimentaltimeseries;thedangersandpitfallsfaced when dealing with non stationary and often noisy data will be stressed, and specific criteriaforsuspectingdeterminisminneuronalcellsand/orassemblieswillbedescribed. Wewillfinallyaddresstwofundamentalquestions,namelyi)whetherandhowcanone distinguish,deterministicpatternsfromstochasticones,and,ii)whatistheadvantageof chaos over randomness: we will explain why and how the former can be controlled whereas,notoriously,thelattercannotbetamed.Inthesecondpaperoftheseries,results obtainedatthelevelofsinglecellsandtheirmembraneconductancesinrealneuronal networksandinthestudyofhigherbrainfunctions,willbecriticallyreviewed.Itwillbe shown that the tools of nonlinear dynamics can be irreplaceable for revealing hidden mechanisms subserving, for example, neuronal synchronization and periodic oscillations. The benefits for the brain of adopting chaotic regimes with their wide range of potentialbehavioursandtheiraptitudetoquicklyreacttochangingconditionswillalso be considered. © 2001 Académie des sciences/Éditions scientifiques et médicales ElsevierSAS nonlineardynamics/determinism/unpredictability/phasespace/entropy/randomness/control 1. Introduction grounds for this conviction: reductionism has been tre- mendously successful in recent decades in all fields of Manybiologists,includingneuroscientists,believethat scienceparticularlyfordissectingvariouspartsofphysical living systems such as the brain, can be understood by orbiologicalsystems,includingatthemolecularlevel.But application of a reductionist approach. There are strong despite the identification of ionic channels and the char *Correspondenceandreprints. E-mailaddress:[email protected](H.Korn). 773 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 acterizationoftheirresponsestovoltage,aphenomenon like,forinstance,theactionpotentialonlymakessensein terms of an ‘integrated’ point of view, thus the need of Hodgkin–Huxley model to understand its generation. Indeed,complexsystemscangiverisetocollectivebehav- iours, which are not simply the sum of their individual componentsandinvolvehugeconglomerationsofrelated units constantly interacting with their environment: the wayinwhichthishappensisstillamystery.Understand- ingtheemergenceoforderedbehaviourofspatio-temporal patterns and adaptive functions appears to require additional,andmoreglobal,conceptsandtools. Asomewhatrelatedandcommonlyacceptedviewpoint is that the strength of science lies in its ability to trace causalrelationsandsotopredictfutureevents.Thegoalof scientificendeavorwouldbetoattainlong-termpredict- abilityandthisisperhaps“thefoundingmythofclassical science” [1]. This credo is rooted in Newtonian physics: oncethelawsofgravitywereknown,itbecamepossible to anticipate accurately eclipses thousand years in advance.Otherwisestated,theLaplaciandogmaaccord- ingtowhichrandomnessisonlyameasureofour“igno- ranceofthedifferentcausesinvolvedintheproductionof events....”[2]dominatestheimplicitphilosophyoftoday’s neuroscience. Conflicting with this view is the evidence that,forexample,somebasicmechanismsofthetransmis- sionofinformationbetweenneuronsappeartobelargely governedbychance(referencesin[3,4]). For a long time it was thought that the fate of a deter- ministicsystemispredictableandthesedesignationswere twonamesforthesamething.Thisequivalencearosefrom a mathematical truth: deterministic systems are specified bydifferentialequationsthatmakenoreferencetochance and follow a unique trajectory. Poincaré was the first to showthelimitsofthisfaith:withafewwordshebecame Figure1. Noiseversusorderedtimeseries.(A)Computergenerated the forerunner of a complete epistemological revolution trainofactionpotentialsproducedbytheHindmarshandRosemodel “... it may happen that small differences in the initial (1984). At first sight this sequence looks random. (B) Probability conditions produce very great ones in the final phenom- density function of time intervals between spikes with an almost ena.Asmallerrorintheformerwillproduceanenormous exponentialdecaysuggestingindependencebetweenthesuccessive errorinthelatter.Predictionbecomesimpossible,andwe spikes.(C)EachintervalIn(axis)isplottedagainstthenextoneIn+1 havethefortuitousphenomenon.”[5]. (ordinates),indicatingastrictrelationshipbetweenthem.Thispattern reveals that the sequence in (A) is produced by a deterministic Systems behaving in this manner are now called ‘cha- process(FaureandKorn,unpublished). otic’. They are essentially nonlinear meaning that initial errors in measurements do not remain constant, rather they grow and decay nonlinearly (in this case exponen- describedbyHindmarshandRose[7]thispatternwould tially) with time. Since prediction becomes impossible, beinterpretedasrandomonthebasisofclassicalstatisti- thesesystemscanatfirstglanceappeartobestochasticbut cal methods analysing interval distributions suggesting this randomness is only apparent because the origin of exponential probability densities (figure 1B); however, a theirirregularitiesisdifferent:theyareintrinsic,ratherthan different representation of the interspike intervals (figure duetoexternalinfluences.Thus,asstatedbyVidal,chaos 1C)revealsawellorderedunderlyinggeneratingmecha- theory“isthechallengetothemeaningandtothescopeof nism. More generally, observation of exponential prob- the ideas of determinism and chance, as we are accus- ability density functions is not sufficient to identify a tomed to practice them today” and a revision of our processasconformingtoaPoissondistribution[8]andthe definitionsisnowimperative[6]. sameremarkappliestootherformsofdistributions. Therelevanceoftheseconsiderationstobrainfunctions The essentials of the discovery of chaos can be traced and neurosciences may not at first be clear. To take an back to the turn of the last century in the mathematical example, a train of action potentials was simulated workofthreeFrenchmathematicians(see[9]).Hadamard (figure1A), using a system of differential equations. First andDuhemwereinterestedinthemovementofaballon 774 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 anegativelycurvedsurfaceandonthefailuretopredictits trajectory due to the lack of knowledge of its initial con- dition. Poincaré tried to solve the so-called three body problem. He found that even though the movement of three celestial bodies is governed by the laws of motion andisthereforedeterministic,theirbehaviour,influenced by gravity, could be so complex as to defy complete understanding. The nonlinearity is brought about in this casebytheinverselawofgravitationalattraction. Present part I of this series of reviews describes chaos and the main methods and theoretical concepts now availableforstudyingnonlineartimeserieswithafocuson themostrelevanttobiologicaldata.InpartII(inprepara- tion), we will show that despite the failure of earlier attemptstodemonstratechaosinthebrainconvincingly, data are now available which are compatible with the notionthatnonlineardynamicsiscommoninthecentral nervoussystem. Thisleadstothequestion:thenwhat?Wewillshowthat such studies can bring new insights into brain functions, andfurthermorethatnonlineardynamicsmayallowneu- ral networks to be controlled, using very small perturba- tions,fortherapeuticpurposes. Figure2. Sensitivitytoinitialconditions.Twoinitiallyclosetrajecto- riesofabilliardball(thickanddashedlines,respectively)quickly 2. Introduction to chaos theory diverge,althoughtheyhitthesameconvexobstaclesalongtheirway (seealsofigure1of[9]). 2.1. Whatischaos? Contrarytoitscommonusage,themathematicalsense This account is the Western version, but it is far from ofthewordchaosdoesnotmeandisorderorconfusion.It beingcomplete.Fairnesspromptsothernamestobeadded designatesaspecificclassofdynamicalbehaviour.Inthe to the above, those of Russian scientists who exploited past two types of dynamics were considered, growth or Poincaré’slegacylongbeforeothers.Theirschoollaidthe decay towards a fixed point and periodic oscillations. foundations of the modern theory of dynamical systems Chaotic behaviour is more complex and it was first andof‘nonlinearmechanics’,themostpublicizedaspect observed in abstract mathematical models. Despite its of which is chaos. A detailed description of their efforts ‘banality’ [6] it was not discovered until the advent of andsuccesses,forexamplethoseinthefieldsofnonlinear modern digital computing: nonlinear differential equa- physicsandvibrations,ofselfmaintainedoscillations,of tions for which there are no analytical solutions and, as bifurcationtheory,andoftherelationsbetweenstatistical importantly,noeasywaytodrawcomprehensivepictures mechanicsanddynamicalsystemscanbefoundin[17]. oftheirtrajectories,couldthenbesolved. TheauthordescribesthecontributionsofA.Kolmogorov, Amongmanyinvestigators,pioneersthatpavedtheway Y.G.Sinaiandtheircollaboratorsinthecharacterizationof of modern theory of chaos were the meteorologist E. chaos and of its relations with probabilistic laws and Lorenz [10] who modeled atmospheric convection in informationtheory. terms of three differential equations and described their Thereisnosimplepowerfulandcomprehensivetheory extreme sensitivity to the starting values used for their ofchaoticphenomena,butratheraclusteroftheoretical calculations, and the ethologist R. May [11, 12] who models,mathematicaltoolsandexperimentaltechniques. showedthatevensimplesystems(inthiscaseinteracting AccordingtoKellert[18],chaostheoryis“thequalitative populations) could display very “complicated and disor- study of unstable aperiodic behaviour in deterministic dered” behaviour. Others were D. Ruelle and F. Takens dynamical systems”. Rapp [19], who also acknowledges [13, 14] who related the still mysterious turbulence of thelackofageneraldefinitionofchaoticsystemsconsid- fluidstochaosandwerethefirsttousethename‘strange ers, however, that they share three essential properties. attractors’. Soon thereafter M. Feigenbaum [15] revealed First, they are dramatically sensitive to initial conditions, patterns in chaotic behaviour by showing how the qua- as shown in figure 2. Second, they can display a highly draticmapswitchesfromonestatetoanotherviaperiod disordered behaviour; and third, despite this last feature, doubling(seedefinitioninsection2.3).Theterm‘chaos’ they are deterministic, that is they obey some laws that hadbeenalreadyintroducedbyT.-Y.LiandJ.Yorke[16] completely describe their motion. A more complete duringtheiranalysisofthesamemap(theseconceptsare description,althoughinlargepartssimilar,hasbeengiven furtherdescribedbelow). byKaplanandGlass[20]whodefinechaosas“aperiodic 775 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 Figure3. Severalpredictableattractorsintheirphasespaces.(A)Dynamicalsystemwiththreeinterconnectedvariables.(B)Representationofthe trajectoryofthesystemshowninA,inathreedimensionalphasespacedefinedbythevariablesx,yandz.(C1-C3)Twodimensionalphasespaces. (C1)Thesimplestattractorisafixedpoint;afterafewoscillations(dashedlines)apendulumsubjectedtofrictionalwayssettlesinthesameposition ofrest,indicatedbyadot.(C2)Alimitcyclethatdescribesstableoscillationsformsacloseloopinthestatespace.Thesamestateisreached whateverthedeparturepointofthetrajectory.(C3)Quasiperiodicbehaviourresultingfromthemotionoftwosystemsthatoscillateatfrequency f andf ,respectively,andconformingtoatorus. 1 2 bounded dynamics in a deterministic system with sensi- opedfromstudiesofmathematicalobjects(models)before tive dependance on initial conditions”. The additional consideringindetailhowtheyapplytobiologicaldata. wordaperiodicreinforcesthepointthatthesamestateis neverrepeatedtwice. 2.2. Phasespace,strangeattractorsandPoincarésections More generally, chaos theory is a specialized applica- Thephasespace(asynonymoustermisstatespace)isa tion of dynamical system theory. Nonlinear terms in the mathematicalandabstractconstruct,withorthogonalcoor- equationsofthesesystemscaninvolvealgebraicorother dinatedirectionsrepresentingeachofthevariablesneeded morecomplicatedfunctionsandvariablesandtheseterms to specify the instantaneous state of a system such as mayhaveaphysicalcounterpart,suchasforcesofinertia velocity and position (figure 3A). Plotting the numerical thatdamposcillationsofapendulum,viscosityofafluid, values of all the variables at a given time provides a nonlinear electronic circuits or the limits of growth of description of the state of the system at that time. Its biologicalpopulations,tonameafew.Sincethisnonlin- dynamics,orevolution,isindicatedbytracingapath,or earityrendersaclosed-formoftheequationsimpossible, trajectory, in that same space (figure 3B). A remarkable investigations of chaotic phenomena seek qualitative feature of the phase space is its ability to represent a ratherthanquantitativeaccountsofthebehaviourofnon- complexbehaviourinageometricandthereforecompre- lineardifferentiabledynamicalsystems.Theydonottryto hensibleform. findaformulathatwillmakeexactnumericalpredictions A classical example, and the simplest, is that of the of a future state from the present state. Instead, they use pendulum. Its motion is determined by two variables, other techniques to “provide some idea about the long- position and velocity. In this case the phase space is a termbehaviourofthesolutions”[21]. plane,andthestateisapointwhosecoordinatesarethese Constructing a ‘state space’ is commonly the first and variablesatagiventime,t.Asthependulumswingsback obligatorystepforcharacterizingthebehaviourofsystems and forth the state moves along a path, or orbit. If the andtheirvariationsintime.Thisapproachbeganwiththe pendulum moves with friction (as does a dissipative sys- workofPoincaré.Currentresearchinthisfieldgoesbythe tem), it is damped and finally comes to a halt. That is, it name ‘dynamical systems theory’, and it typically asks approaches a fixed point, that attracts the closest orbits suchquestionsaswhatcharacteristicswillallthesolutions (figure 3C1). This point is an attractor (the term attractor of this system ultimately exhibit? We will first give the refers to a geometrical configuration in a phase space to generalprinciplesofthistheoryastheyhavebeendevel- whichallnearbytrajectoriestendtoconvergeovertime). 776 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 Figure4. Lackoflong-termpredictabilityoftheLorentzattractor.(A)Thetwocurves(thickandthinlines)startatinitiallocationsthatdifferbyonly 0.0001.Notetheirrandomappearanceandrapiddivergence.(B)Effectsofinaccuracyofmeasurements:10000trajectoriesstartinthesameregion (arrow,totheleft).Attheindicatedtime,theycanbefoundanywhereontheattractor(thindottedlines)andinregionsprogressivelymoredistant fromeachother.Predictionquicklybecomesimpossible. Inabsenceoffriction,orifthelatteriscompensatedbya Anotherandamorecomplicatedtypeofattractoristhe weight or another force, the pendulum behaves like a two dimensional torus which resembles the surface of a clockthatrepeatsthesamemotioncontinuously.Thistype tire.Itdescribesthemotionoftwoindependentoscillators ofmotioncorrespondstoacycle,orperiodicorbit,andthe with frequencies that are not related by integers correspondingattractoriscalledalimitcycle(figure3C2). (figure3C3). In some conditions the motion of such a Manybiologicalsystems,forinstancetheheartandnumer- systemissaidtobequasiperiodicbecauseitneverrepeats ousneuronalcells,behaveassuchoscillators. exactly itself. Rhythms that can be considered as quasi 777 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 periodic are found, for example in cardiac arrhythmias, becausepacemakerswhicharenolongercoupledtoeach otherindependentlymaintaintheirownrhythm[20]. Fixedpoints,limitcyclesandtoriweretheonlyknown (and predictable on a long-term) attractors until Lorenz [10] discovered a new system that displays completely differentdynamics.Inhisattemptstoforecasttheweather, hedevelopedasimplemodeltodescribetheinterrelations oftemperaturevariationandconvectormotion.Themodel involvesonlythreedifferentialequations: (cid:1)dx/dt=−σx+σy dy/dt=−xz+rx−y (1) dz/dt=xy−bz where σ, r, and b are parameters that characterize the Figure5. Poincarésection.Thesectionofathreedimensionalphase spaceisatwodimensionalsurface(whichisplacedhereinthex,z properties of a fluid and of the thermal and geometric plane).Crossingpoints1,2,3arerecordedeverytimethetrajectory configurationofthesystem.Thevariablexisrelatedtothe hitsthissection,resultingina“strobscopic”portraitoftheattractor. fluid’sflow,yisproportionaltothetemperaturedifference between the upward and downward moving parts of a convection roll, and z describes the nonlinearity in tem- by taking a simple slice of the attractor (figure 5), thus peraturedifferencealongtheroll.Thenumericalsolution resultinginareturnmap.Reducingthephasespacedimen- of these equations with parameter values σ=10, r=28, sion in this manner corresponds to sampling the system andb=8/3leadstoanattractorwhichcanbevisualized everytimethetrajectoryhitstheplaneofthesection.This in a three-dimensional space with coordinates (x, y, z) procedure, which simplifies the flow without altering its sincethesystemhasthreedegreesoffreedom.Becauseof essentialproperties,isparticularlyusefulforthestudiesof its complexity, this geometric pattern, which looks like a biologicaldata,especiallyoftimeseriesobtainedduring butterfly,isthemostpopular‘strangeattractor’(figure4). studiesofthenervoussystem(seebelow).Itisalsovalu- Numeroussetsofequationsleadingtostrangeattractors, ablefordealingwithhighdimensionalsystems(i.e.more alsoshowntobechaotic,havenowbeendescribedinthe thanabout5). literature (Rossler, Hénon, Ikeda attractors, etc). In all cases, the systems are deterministic: the corresponding 2.3. Fromordertochaos:perioddoubling trajectories are confined to a region of the phase space withaspecificshape.Theirtrajectoryrotatesaboutoneor Physicalsystems(aswellasmodeleddata)canundergo twounstablefixedpoints(asdefinedbelow)andeventu- transitions between various dynamics as some of their allyescapestoorbitanotherunstablefixedpointwhichis basicparametersarevaried.Ingeneralasmallchangeina notanattractor.Thisprocessisrepeatedindefinitely,but parameterresultsinaslightmodificationoftheobserved theorbitsofthetrajectoryneverintersecteachother. dynamics, except for priviledged values which produce The solutions of the above model illustrate the main qualitativealterationsinthebehaviour.Forexample,con- featuresofchaos.Inasimpletimeseries,itlooksrandom sideracreekinwhichwaterflowsaroundalargerock,and (figure 4A). The trajectories rapidly diverge even when a device to measure velocity. If the flow is steady, the theyhaveclosestartingpoints.Inthe3Dphasespace,they velocityisconstant,thusafixedpointinthestatespace.As mimicboththeaperiodicityandsensitivedependenceon thespeedofwaterincreases,modulationsofwateraround initialconditions(figure4B).Inapurelylinearsystem,any the rock cause the formation of swirls, with an increase, exponentialgrowthwouldcausethesystemtoheadoffto and then a decrease in the velocity of each swirl. The infinity(toexplode),butthenonlinearityfoldsthegrowth behaviour changes from constant to periodic, having a back.Converselytheexponentialdecayinalinearsystem limit cycle in the same phase space. If speed is further would lead to a steady-state behaviour: the trajectories increased the motion of water may become quasi peri- would converge to a single point. But given that in non- odic,andultimatelyrandomturbulencestaketheformof linear systems there can be exponential growth in some chaos(see[9]).Suchdramaticmaticchangesofbehaviour directions, the effect of the exponential decay in other arecalledbifurcationsandinaphasespace,theyoccurat directionsistoforcethetrajectoriesintoarestrictedregion what are referred to as bifurcation points that serve as ofthephasespace.Sonearbytrajectoriestendtoseparate landmarkswhenstudyingadynamicalsystem. fromoneanotherallthewhilebeingkeptontheattractor. The best studied model of bifurcations is that of the Analysing pictures of strange attractors and their com- logisticequationwhichisafirstorderdifferenceequation plicated paths can prove a complicated matter. Fortu- thattakesthesimpleform nately,threedimensionalphasespacesmaybesimplified usingaPoincarésectionorreturnmap,whichisobtained xn+1=kxn(cid:1)1−xn(cid:2) (2) 778 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 Figure6. Thelogisticmap.(A)Computergenerateddiagramofthebehaviourofthelogisticfunctionx =kx (1–x ),withitsbranchingtree n+1 n n (threebifurcationpointslabelledk ,k andk areindicatedbyverticaldottedlines).Chaos,withdotsapparentlydispersedatrandom,occursat 1 2 3 therighthandsideofthebifurcationpointk=3.569...(arrow).Magnifyingthemapinthisregionwouldrevealfurtherchaoticverticalstripes,and close-upimageswouldlooklikeduplicatesofthewholediagram.(B)Examplesofdifferentasymptoticregimesobtainedfortheindicatedvalues ofk(thickbarsinA,seetextforexplanations). wherekisacontrolparameter.Forexampleconsiderthe tion,whichinturnreproducestheoriginallargenumberof specificcasewherek=0.2andx =3.Thenequation(2) individualsthefollowingyear.Forincreasingvaluesofka 0 yieldsx =0.2(0.3)(1–0.3)=0.63.Usingthisvalueofx , cyclerepeatsevery4years,8years,thenevery16,32,and 1 1 x canbecomputedinthesameway,byaprocesscalled so on, in what is called a ‘period-doubling cascade’, 2 iteration,andsoon. culminatingintoabehaviourthatbecomesfinallychaotic, The logistic ‘map’ (figure 6A) was used by May in his i.e. apparently indistinguishable visually from a random famousinvestigationsofthefluctuationsof‘populations’, process: at this stage “wild fluctuations very effectively which breed and later die off en masse (nonoverlapping maskthesimplicityoftheunderlyingrule”[22]. generations). The size of a generation is bound by envi- Another way to explain the graph of figure 6 is as ronmental conditions (including food supply and living follows.Inafinitedifferenceequationsuchasaquadratic, space), fertility, and interactions with other species. A andonceaninitialconditionx ischosen,thesubsequent number between 0 and 1 indicates this size, which is 0 valuesofxcanbecomputed,i.e.x andfromthereonx , measuredastheactualnumberofindividualsinthespe- 1 2 x ,..., by iteration. This process can be graphical or cies: 0 represents total extinction and 1 is the largest 3 numerical [23, 8]. Successive steps, show for example possible number in a generation. The term k determines [20](figure6B),that: thefateofapopulationofsay,insects:thehigherkis,the – for3.000<k<3.4495,thereisastablecycleofperiod more rapidly a small population grows or a large one 2; becomesextinct.Forlowvaluesofk,theinitialpopulation settlesdowntoastablesizethatwillreproduceitselfeach – for3.4495<k<3.5441,thereisastablecycleofperiod year.Askincreasesthefirstunstablefixedpointappears. 4; Thesuccessivevaluesofxoscillateinatwo-yearcycle:a – for 3.5441< k <3.5644, there is a stable cycle of large number of individuals produces a smaller popula- period8,etc... 779 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 Figure7. Theroadtochaos.(A)Illustrationoftheperioddoublingscenarioshowingthatstablefixedpoints(solidlines),althoughtheycontinue toexist,becomeunstable(dashedline)wheniterationsarepushedawayfrombifurcationpoints.(B1-B2)Graphicaliterationsatindicatedvalues ofthecontrolparameterk(arrowsinA).(B1)Representationofthelogisticfunctionx =2.8x (1–x )(thickline)andofthecorrespondingfixed n+1 n n point(emptycircle)attheintersectionwiththediagonalofthereturnmap.Successiveiteratesconvergetowardthisstablefixedpoint.(B2)Same representationasinB1withk=3.3;thefixedpointisnowunstableanditeratesdivergeawayfromit.(NotethatB1andB2havesamestarting points). (A stable cycle of period 2, 4, 8,... is a cycle that 2.4. Otherroadstochaos alternatesbetween2,4,8,...valuesofx). Among the other roads to chaos two deserve special The emergence on the map of a new stable cycle notice.Oneisviaquasiperiodicity,whenatorusbecomes coincides with the transformation of the previous cycle astrangeattractor.Theotherrouteisthatof‘intermittency’ into an unstable one (for example cycle 2 in figure 7A). whichmeansthataperiodicsignalisinterruptedbyran- This process is illustrated in figure 7B1-B2 which shows dom bursts occurring unpredictably but with increasing howtwodifferentvaluesofthecontrolparameterkmodify frequency as a parameter is modified. Several types of thelocaldynamicsoftheperiod1stablefixedpointintoa intermittencieshasbeenobservedinmodels,andthetype period 1 but unstable, fixed point. As k increases, this depends on whether the system switches back and forth processisrepeateduntilthechaoticstate,whichismade fromperiodictochaoticorquasiperiodicbehaviour[24]; ofaninfinitenumberofunstablecycles,isreached. for theoretical considerations see also [6]. In return (or Theperiod-doublingbifurcationsthatcharacterizelogis- Poincaré)mapsthebeginningofintermittencieshasbeen tic maps obey various beautiful scaling relations. The reported at ‘tangent bifurcations’ of graphical iterations values of the parameter k that which each successive (not shown). Although we have not studied them exten- bifurcation appears grow closer and closer together sively, we have detected similar sequences between the (figure6A,verticaldottedlines).Thatis,successivebifur- intervalsofactionpotentialsrecordedfromburstinginter- cation points are always found at a given value of k and neuronspresynaptictotheMauthnercellsofteleosts. chaosappearsfork=3.569....Furthermore,ifthevalueat A wide variety of self organized systems called ‘com- whichacycleoccursisxthentheratio(k –k )/k is plex’, and best described by a power law [25], have a x x–1 x+1 δ=4.669... This is Feigenbaum’s magic universal con- noisy behaviour. The noise level increases as the power stant[15]. spectrumfrequencydecreaseswitha1/ffrequencydepen 780 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 dence. Although such complex systems are beyond the scope of this review, it is worth noting that intermittency may be one of the mechanisms underlying the 1/f noise foundinnaturalsystems[24,26]. 2.5. Quantificationofchaos Several methods and measures are available to recog- nizeandcharacterizechaoticsystems.Despitethesensi- tivity of these systems to initial conditions and the rapid divergenceoftheirtrajectoriesinthephasespace,someof these measures are ‘invariant’, meaning that their results do not depend on the trajectory’s starting point on the attractor,nordotheydependontheunitsusedtodefine the phase space coordinates. These invariants are based ontheassumptionthatstrangeattractorsfulfillthecondi- tionssatisfyingthe‘ergodic’hypothesiswhichpositsthat trajectoriesspendcomparableamountsoftimevisitingthe same regions near the attractor [27]. Three major invari- antswillnowbeconsidered. 2.5.1. TheLyapunovexponent It is a measure of exponential divergence of nearby trajectoriesor,otherwisestated,ofthedifferencebetween agiventrajectoryandthepathitwouldhavefollowedin the absence of perturbation (figure 8A). Assuming two points x and x initially separated from each other by a 1 2 small distance δ , and at time t by distance δ, then the 0 t Lyapunovexponent,λ,isdeterminedbytherelation δ =δ eλt (3) x(cid:1)t(cid:2) x(cid:1)0(cid:2) where λ is positive if the motion is chaotic and equal to zero if the two trajectories are separated by a constant amountasforexampleiftheyareperiodic. 2.5.2. Entropy A chaotic system can be considered as a source of information in the following sense. It makes prediction uncertainduetothesensitivedependenceoninitialcon- ditions.Anyimprecisioninourknowledgeofthestateis Figure8. Principlesunderlyingthemainmeasuresofchaoticinvari- magnifiedastimegoesby.Ameasurementmadeatalater ants.(A)Lyapunovexponents.Asmalldifferenceδ(0)intheinitial x timeprovidesadditionalinformationabouttheinitialstate. pointxofatrajectoryresultsinachangeδ(t)whichisanexponential x Entropy is a thermodynamic quantity describing the functionδ(0)eλt(seeequation4)whereλistheLyapunovexponent. x amount of disorder in a system [28], and it provides an (B)Entropy.ThephasespaceisdividedintoNcellsandthelocation ofallthepointsthatwereinitiallygroupedinonecell(emptycircles) importantapproachtotimeseriesanalysiswhichcanbe isdeterminedatagiventime,t(thickdots).(C)Fractaldimension.A regarded as a source of information [29]. From a micro- Cantor Set is constructed by removing at each successive step the scopicpointofview,thesecondlawofthermodynamics central third of the remaining lines. At the stage m, there are 2M tells us that a system tends to evolve toward the set of segmentsoflength(1/3)Meach. conditionsthathasthelargestnumberofaccessiblestates compatible with the macroscopic conditions [24]. In a tor in N cells and calculate the relative frequency (or phasespace,theentropyofasystemcanthenbewritten probability p) with which the system visits each cell (cid:3)N (figure8B). H=− pi logpi (4) In dynamics, an important form of this measure is the i=1 Kolmogorov–Sinaientropy(K)whichdescribestherateof wherep istheprobabilitythatthesystemisinstatei.In changeoftheentropyasthesystemevolves(fordetailssee i practiceonehastodividetheregioncontainingtheattrac- [29]). K takes into account the entire evolution of the n 781 P.Faure,H.Korn/C.R.Acad.Sci.Paris,Sciencesdelavie/LifeSciences324(2001)773–793 initialsystemafterntimeunitsratherthanconcentrating 3. Detecting chaos in experimental data onafewtrajectories,suchthat Theprecedingsectionexamined‘ideal’dynamicalphe- Kn=1τ(cid:1)Hn+1−Hn(cid:2) (5) nomena produced by computer models and defined by known mathematical equations. These models generate whereτisthetimeunit. pure low dimensional chaos which is, however, only If the system is periodic, K equals zero whereas it n foundrarelyinthenaturalworld.Furthermorewhendeal- increaseswithoutinterruption,oritincreasestoaconstant ing with natural phenomena, a ‘reverse’ approach is value, depending whether the system is stochastic or required: the dynamics need to be determined starting chaotic(seefigure12C). from a sequence of measurements and, whenever pos- 2.5.3. Dimension sible,thetypeofappropriateequationshavetobeidenti- fied to model the system. Most often this proves to be a Thetwoabovedescribedinvariantsfocusonthedynam- verydifficulttask. ics(evolutionintime)oftrajectoriesinthephasespace.In The nonlinear methods described above are generally contrast,dimensionemphasizesthegeometricfeaturesof oflimitedhelpwhendealingwithexperimentaltimeseries attractors. due to the lack of stationarity of the recorded signals, Since Descartes, the dimension of a space has been meaningthatalltheparametersofasystem,particularlyof thoughtofintermsofthenumberofcoordinatesneededto abiologicalone,rarelyremainwithaconstantmeanand locate a point in that space. Describing the location of a varianceduringthemeasurementperiod.Thiscreatesan point on a line requires one number, on a plane two inherent conflict between this non stationarity and the numbers,andinourfamiliar3-dimensionalsurroundings need for prolonged and stable periods of observation for itrequiresthreenumbers.Amodernperspectivegeneral- reachingreliableandunambiguousconclusions. izes the idea of dimension in terms of scaling laws. For A second problem is that in contrast to computer out- example,theamountofspaceenclosedbyacirclegiven by the familiar formula πr2. Its numerical value depends puts,puredeterminismandlowdimensionalchaos(which can be modeled with a small number of variables, ontheunitswithwhichrismeasured.Acircleofradius1 menclosesareaπwhenmeasuredinmetres,104πwhen i.e.<4–5) are unlikely in real world systems. Natural measuredincentimetres,andarea1012πwhenmeasured systems interact with their surroundings so that there is inmicrons.Intheexpressionsπr2,thedimensioncanbe generally a mixture of fluctuations (or noise): those produced by the environment, those by the systems them- readoffastheexponentonrorastheslopeofalog–log selvesandthosebytherecordingtechniques.Thusspecial plotoftheareaversusthelengthscale. andsophisticatedproceduresareneededtodistinguish,if Definingdimensioninsuchmannerprovidesawayto possible, between nonlinear deterministic or linear sto- specifyanewclassofgeometricalobjectcalledfractal,the chastic(orGaussian)behaviour[31](seealsosection4). dimensionofwhichisnoninteger. Despite initial expectations, most statistical measures Asimpleself-similarshapeistheCantorset(figure8C). maynotbeadequateforsignalprocessinginthecontextof It is constructed in successive steps, starting with a line nonlineardynamics.Forexamplebroadbandpowerspec- segmentoflength1.Thislengthcanbecoveredbya‘box’ tra with superimposed peaks have often been associated ofsidee.Forthefirststageofconstruction,onedeletesthe withchaoticdynamics.Thisconclusionisoftenpremature middle third of that segment. This leaves two segments, because similar power spectra can also be produced by eachoflength1/3.Forthesecondstage,themiddlethird noisysignals[8,20]. of each of those segments is deleted, resulting in four Nevertheless, given that the dynamical properties of a segments, each of length 1/9. Increasing the depth of systemaredefinedinphasespaces,itisalsohelpfulwhen recursion, and for the Mth step, one removes the middle analysingexperimentaldatatostartinvestigationsbycon- third of each of the remaining segments to produce 2M structing a phase description of the phenomenon under segments, each of length (1/3)M. Continuing as M→∞, study. thesizeeofeachenclosingbox→0. Onecanthencalculatethebox-countingdimensionof 3.1. Reconstructionofthephaseplaneandembedding thissetkeepinginmindthatasM→∞,thereremainsonly Sinceatimeseriesconsistsofrepeatedmeasurementsof a series of points. Then if N(e) is the number of boxes of asinglevariable,theproblemistoestablishmultidimen- lengthewhichcoversentirelytheset(ortheattractor),the sional phase spaces without knowing in advance the fractaldimensionthatcannotbeinteger,becomes: (cid:2) (cid:3) numberofdegreesoffreedomthatneedtoberepresented, 1 i.e. the number of variables of the system. This difficulty log N(cid:1)e(cid:2) can be bypassed because even for a phenomenon that D= lim (6) comprisesseveraldimensions,thetimeseriesinvolvinga e→0 log(cid:1)e(cid:2) singlevariablecanbesufficienttodetermineitsfulldynam- Strangeattractorsarefractalobjectsandtheirgeometryis ics[14,32].Theprocedureusedinpracticediffersaccord- invariantagainstchangesinscale,orsize.Theyarecopies ingtowhetheroneisdealingwithacontinuousorwitha ofthemselves[30,26]. discretetimeseries. 782