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2006 - Rabinovich et al. - Dynamical principles in neuroscience - Review of Modern Physics

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REVIEWSOFMODERNPHYSICS,VOLUME78,OCTOBER–DECEMBER2006

Dynamicalprinciplesinneuroscience

MikhailI.Rabinovich*

InstituteforNonlinearScience,UniversityofCalifornia,SanDiego,9500GilmanDrive0402,LaJolla,California92093-0402,USA

PabloVarona

GNB,DepartamentodeIngenieríaInformática,UniversidadAutónomadeMadrid,28049Madrid,SpainandInstituteforNonlinearScience,UniversityofCalifornia,SanDiego,9500GilmanDrive0402,LaJolla,California92093-0402,USA

AllenI.Selverston

InstituteforNonlinearScience,UniversityofCalifornia,SanDiego,9500GilmanDrive0402,LaJolla,California92093-0402,USA

HenryD.I.Abarbanel

DepartmentofPhysicsandMarinePhysicalLaboratory(ScrippsInstitutionofOceanography)andInstituteforNonlinearScience,UniversityofCalifornia,SanDiego,9500GilmanDrive0402,LaJolla,California92093-0402,USA

͑Published14November2006͒

Dynamicalmodelingofneuralsystemsandbrainfunctionshasahistoryofsuccessoverthelasthalfcentury.Thisincludes,forexample,theexplanationandpredictionofsomefeaturesofneuralrhythmicbehaviors.Manyinterestingdynamicalmodelsoflearningandmemorybasedonphysiologicalexperimentshavebeensuggestedoverthelasttwodecades.Dynamicalmodelsevenofconsciousnessnowexist.Usuallythesemodelsandresultsarebasedontraditionalapproachesandparadigmsofnonlineardynamicsincludingdynamicalchaos.Neuralsystemsare,however,anunusualsubjectfornonlineardynamicsforseveralreasons:͑i͒Eventhesimplestneuralnetwork,withonlyafewneuronsandsynapticconnections,hasanenormousnumberofvariablesandcontrolparameters.Thesemakeneuralsystemsadaptiveandﬂexible,andarecriticaltotheirbiologicalfunction.͑ii͒Incontrasttotraditionalphysicalsystemsdescribedbywell-knownbasicprinciples,ﬁrstprinciplesgoverningthedynamicsofneuralsystemsareunknown.͑iii͒Manydifferentneuralsystemsexhibitsimilardynamicsdespitehavingdifferentarchitecturesanddifferentlevelsofcomplexity.͑iv͒Thenetworkarchitectureandconnectionstrengthsareusuallynotknownindetailandthereforethedynamicalanalysismust,insomesense,beprobabilistic.͑v͒Sincenervoussystemsareabletoorganizebehaviorbasedonsensoryinputs,thedynamicalmodelingofthesesystemshastoexplainthetransformationoftemporalinformationintocombinatorialorcombinatorial-temporalcodes,andviceversa,formemoryandrecognition.Inthisreviewtheseproblemsarediscussedinthecontextofaddressingthestimulatingquestions:Whatcanneurosciencelearnfromnonlineardynamics,andwhatcannonlineardynamicslearnfromneuroscience?DOI:10.1103/RevModPhys.78.1213

PACSnumber͑s͒:87.19.La,05.45.Ϫa,84.35.ϩi,87.18.Sn

CONTENTS

I.WhatarethePrinciples?A.Introduction

B.Classicalnonlineardynamicsapproachforneural

systems

C.Newparadigmsforcontradictoryissues

II.DynamicalFeaturesofMicrocircuits:Adaptabilityand

Robustness

A.Dynamicalpropertiesofindividualneuronsand

synapses

1.Neuronmodels

2.Neuronadaptabilityandmultistability

12181218121912181215121712141214

3.Synapticplasticity

4.Examplesofthecooperativedynamicsof

individualneuronsandsynapses

B.RobustnessandadaptabilityinsmallmicrocircuitsC.IntercircuitcoordinationD.ChaosandadaptabilityIII.InformationalNeurodynamics

A.Timeandneuralcodes

1.Temporalcodes

2.Spatiotemporalcodes3.Coexistenceofcodes

4.Temporal-to-temporalinformation

transformation:Workingmemory

B.Informationproductionandchaos

1.Stimulus-dependentmotordynamics2.Chaosandinformationtransmission

C.Synapticdynamicsandinformationprocessing

122212231224122812291231123112311232123312341237123712391240

*Electronicaddress:mrabinovich@ucsd.edu

0034-6861/2006/78͑4͒/1213͑53͒

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D.Bindingandsynchronization1242IV.TransientDynamics:Generation

andProcessingofSequences1244A.Whysequences?

1244B.Spatiallyorderednetworks

12441.Stimulus-dependentmodes12442.Localizedsynﬁrewaves

1247C.Winnerlesscompetitionprinciple

12481.Stimulus-dependentcompetition12482.Self-organizedWLCnetworks12493.Stableheteroclinicsequence12504.Relationtoexperiments1251D.Sequencelearning

1252E.Sequencesincomplexsystemswithrandom

connections

12F.Coordinationofsequentialactivity1256V.Conclusion1258Acknowledgments1259Glossary1259References

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“Williteverhappenthatmathematicianswillknowenoughaboutthephysiologyofthebrain,andneuro-physiologistsenoughofmathematicaldiscovery,forefﬁ-cientcooperationtobepossible?”

—JacquesHadamard

I.WHATARETHEPRINCIPLES?A.Introduction

Buildingdynamicalmodelstostudytheneuralbasisofbehaviorhasalongtradition͑Ashby,1960;Block,1962;Rosenblatt,1962;Freeman,1972,2000͒.Theun-derlyingideagoverningneuralcontrolofbehavioristhethree-stepstructureofnervoussystemsthathaveevolvedoverbillionsofyears,whichcanbestatedinitssimplestformasfollows:Specializedneuronstransformenvironmentalstimuliintoaneuralcode.Thisencodedinformationtravelsalongspeciﬁcpathwaystothebrainorcentralnervoussystemcomposedofbillionsofnervecells,whereitiscombinedwithotherinformation.Adecisiontoactontheincominginformationthenre-quiresthegenerationofadifferentmotorinstructionsettoproducetheproperlytimedmuscleactivitywerecog-nizeasbehavior.Successinthesestepsistheessenceofsurvival.

Giventhepresentstateofknowledgeaboutthebrain,itisimpossibletoapplyarigorousmathematicalanalysistoitsfunctionssuchasonecanapplytootherphysicalsystemslikeelectroniccircuits,forexample.Wecan,however,constructmathematicalmodelsofthephenom-enainwhichweareinterested,takingaccountofwhatisknownaboutthenervoussystemandusingthisinforma-tiontoinformandconstrainthemodel.Currentknowl-edgeallowsustomakemanyassumptionsandputthemintoamathematicalform.Alargepartofthisreviewwilldiscussnonlineardynamicalmodelingasaparticu-larlyappropriateandusefulmathematicalframeworkthatcanbeappliedtotheseassumptionsinorderto

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FIG.1.͑Coloronline͒Illustrationofthefunctionalpartsandelectricalpropertiesofneurons.͑a͒Theneuronreceivesinputsthroughsynapsesonitsdendritictree.Theseinputsmayormaynotleadtothegenerationofaspikeatthespikegenera-tionzoneofthecellbodythattravelsdowntheaxonandtrig-gerschemicaltransmitterreleaseinthesynapsesoftheaxonaltree.Ifthereisaspike,itleadstotransmitterreleaseandactivatesthesynapsesofapostsynapticneuronandtheprocessisrepeated.͑b͒Simpliﬁedelectricalcircuitforamembranepatchofaneuron.Thenonlinearionicconductancesarevolt-agedependentandcorrespondtodifferentionchannels.Thistypeofelectricalcircuitcanbeusedtomodelisopotentialsingleneurons.Detailedmodelsthatdescribethemorphologyofthecellsuseseveralisopotentialcompartmentsimple-mentedbythesecircuitscoupledbyalongitudinalresistance;thesearecalledcompartmentalmodels.͑c͒Atypicalspikeeventisoftheorderof100mVinamplitudeand1–2msinduration,andisfollowedbyalongerafter-hyperpolarizationperiodduringwhichtheneuronislesslikelytogeneratean-otherspike;thisiscalledarefractoryperiod.

simulatethefunctioningofthedifferentcomponentsofthenervoussystem,tocomparesimulationswithexperi-mentalresults,andtoshowhowtheycanbeusedforpredictivepurposes.

Generallytherearetwomainmodelingapproachestakeninneuroscience:bottom-upandtop-downmodels.•Bottom-updynamicalmodelsstartfromadescrip-tionofindividualneuronsandtheirsynapticconnec-tions,thatis,fromacknowledgedfactsaboutthede-tailsresultingfromexperimentaldatathatareessentiallyreductionistic͑Fig.1͒.Usingtheseana-tomicalandphysiologicaldata,theparticularpatternofconnectivityinacircuitisreconstructed,takingintoaccountthestrengthandpolarity͑excitatoryorinhibitory͒ofthesynapticaction.Usingthewiringdiagramthusobtainedalongwiththedynamicalfea-turesoftheneuronsandsynapses,bottom-upmodelshavebeenabletopredictfunctionalpropertiesof

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neuralcircuitsandtheirroleinanimalbehavior.•Top-downdynamicalmodelsstartwiththeanalysisofthoseaspectsofananimal’sbehaviorthatarero-bust,reproducible,andimportantforsurvival.Thetop-downapproachisamorespeculativebig-pictureviewthathashistoricallyledtodifferentlevelsofanalysisinbrainresearch.Whilethishierarchicaldi-visionhasputthedifferentlevelsonanequalfoot-ing,theuncertaintyimplicitinthetop-downap-proachshouldnotbeminimized.Theﬁrststepinbuildingsuchlarge-scalemodelsistodeterminethetypeofstimulithatelicitspeciﬁcbehaviors;thisknowledgeisthenusedtoconstructhypothesesaboutthedynamicalprinciplesthatmightberespon-siblefortheirorganization.Themodelshouldpre-dicthowthebehaviorevolveswithachangingenvi-ronmentrepresentedbychangingstimuli.Itispossibletobuildasufﬁcientlyrealisticneuralcir-cuitmodelthatexpressesdynamicalprinciplesevenwithoutknowledgeofthedetailsoftheneuroanatomyandneurophysiologyofthecorrespondingneuralsys-tem.Thesuccessofsuchmodelsdependsontheuniver-salityoftheunderlyingdynamicalprinciples.Fortu-nately,thereisasurprisinglylargeamountofsimilarityinthebasicdynamicalmechanismsusedbyneuralsys-tems,fromsensorytocentralandmotorprocessing.Neuralsystemsutilizephenomenasuchassynchroni-zation,competition,intermittency,andresonanceinquitenontraditionalwayswithregardtoclassicalnonlin-eardynamicstheory.Onereasonisthatthenonlineardynamicsofneuralmodulesormicrocircuitsisusuallynotautonomous.Thesecircuitsarecontinuouslyorspo-radicallyforcedbydifferentkindsofsignals,suchassen-soryinputsfromthechangingenvironmentorsignalsfromotherpartsofthebrain.Thismeansthatwhenwedealwithneuralsystemswehavetoconsiderstimulus-dependentsynchronization,stimulus-dependentcompe-tition,etc.Thisisadeparturefromtheconsiderationsofclassicalnonlineardynamics.Anotherveryimportantfeatureofneuronaldynamicsisthecoordinationofneu-ralactivitieswithverydifferenttimescales,forexample,thetarhythms͑4–8Hz͒andgammarhythms͑40–80Hz͒inthebrain.

Oneofourgoalsinthisreviewistounderstandwhyneuralsystemsareveryspeciﬁcfromthenonlineardy-namicspointofviewandtodiscusstheimportanceofsuchspeciﬁcitiesforthefunctionalityofneuralcircuits.Wewilltalkabouttherelationshipbetweenneuro-scienceandnonlineardynamicsusingspeciﬁcsubjectsasexamples.Wedonotintendtoreviewherethemethodsorthenonlineardynamicaltoolsthatareimportantfortheanalysisofneuralsystemsastheyhavebeendis-cussedextensivelyinmanyreviewsandbooks͑e.g.,GuckenheimerandHolmes,1986;Crawford,1991;Abarbaneletal.,1993;Ott,1993;KaplanandGlass,1995;Abarbanel,1997;Kuznetsov,1998;Arnoldetal.,1999;Strogatz,2001;Izhikevich,2006͒.

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B.Classicalnonlineardynamicsapproachforneuralsystems

Letussayafewwordsabouttheroleofclassicaldy-namicaltheory.Itmightseematﬁrstsightthattheap-parentlyinﬁnitediversityofneuralactivitymakesitsdy-namicaldescriptionahopeless,evenmeaningless,task.However,hereonecanexploittheknowledgeaccumu-latedinclassicaldynamicaltheory,inparticular,theideasputforthbyAndronovin1931concerningthestructuralstabilityofdynamicalmodelsandtheinvesti-gationoftheirbifurcations͑Andronov,1933;AndronovandPontryagin,1937;Andronovetal.,1949͒.Theessen-tialpointsoftheseideascanbetracedbacktoPoincaré͑Poincaré,12;Goroff,1992͒.InhisbookLaValeurdelaScience,Poincaré͑1905͒wrotethat“themainthingforustodowiththeequationsofmathematicalphysicsistoinvestigatewhatmayandshouldbechangedinthem.”Andronov’sremarkableapproachtowardunder-standingdynamicalsystemscontainedthreekeypoints:•Onlymodelsexhibitingactivitythatdoesnotvarywithsmallchangesofparameterscanberegardedasreallysuitabletodescribeexperiments.Hereferredtothemasmodelsordynamicalsystemsthatarestructurallystable.•Toobtaininsightintothedynamicsofasystemitisnecessarytocharacterizeallitsprincipaltypesofbe-haviorunderallpossibleinitialconditions.ThisledtoAndronov’sfondnessforthemethodsofphase-space͑state-space͒analysis.•Consideringthebehaviorofthesystemasawholeallowsonetointroducetheconceptoftopologicalequivalenceofdynamicalsystemsandrequiresanunderstandingoflocalandglobalchangesofthedy-namics,forexample,bifurcations,ascontrolparam-etersarevaried.Conservingthetopologyofaphaseportraitforady-namicalsystemcorrespondstoastablemotionofthesystemwithsmallvariationofthegoverningparameters.Partitioningparameterspaceforthedynamicalsystemintoregionswithdifferentphase-spacebehavior,i.e.,ﬁndingthebifurcationboundaries,thenfurnishesacom-pletepictureofthepotentialbehaviorsofadynamicalmodel.Isitpossibletoapplysuchabeautifulapproachtobiologicalneuralnetworkanalysis?Theanswerisyes,atleastforsmall,autonomousneuralsystems.However,eveninthesesimplecaseswefacesomeimportantre-strictions.

Neuraldynamicsisstronglydissipative.Energyde-rivedfrombiochemicalsourcesisusedtodriveneuralactivitywithsubstantialenergylossinaction-potentialgenerationandpropagation.Nearlyalltrajectoriesinthephasespaceofadissipativesystemareattractedbysometrajectoriesorsetsoftrajectoriescalledattractors.Thesecanbeﬁxedpoints͑correspondingtosteady-stateactivity͒,limitcycles͑periodicactivity͒,orstrangeat-tractors͑chaoticdynamics͒.Thebehaviorofdynamicalsystemswithattractorsisusuallystructurallystable.Strictlyspeakingastrangeattractorisitselfstructurally

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FIG.2.Sixexamplesoflimitcyclebifurcationsobservedinlivingandmodelneuralsystems͓seeChay͑1985͒;Canavieretal.͑1990͒;Guckenheimeretal.͑1993͒;Huertaetal.͑1997͒;Cre-vierandMeister͑1998͒;Maedaetal.͑1998͒;CoombesandOs-baldestin͑2000͒;Feudeletal.͑2000͒;GavrilovandShilnikov͑2000͒;MaedaandMakino͑2000͒;Mandelblatetal.͑2001͒;Bondarenkoetal.͑2003͒;Guetal.͑2003͒;ShilnikovandCym-balyuk͑2005͒;Soto-Trevinoetal.͑2005͔͒.

unstable,butitsexistenceinthesystemstatespaceisastructurallystablephenomenon.ThisisaveryimportantpointfortheimplementationofAndronov’sideas.

Thestudyofbifurcationsinneuralmodelsandininvitroexperimentsisakeystoneforunderstandingthedynamicaloriginofmanysingle-neuronandcircuitphe-nomenainvolvedinneuralinformationprocessingandtheorganizationofbehavior.Figure2illustratessometypicallocalbifurcations͓theirsupportconsistsofanequilibriumpointoraperiodictrajectory—seethede-taileddeﬁnitionbyArnoldetal.͑1999͔͒andsomeglobalbifurcations͑theirsupportcontainsaninﬁnitesetofor-bits͒ofperiodicregimesobservedinneuralsystems.Manyofthesebifurcationsareobservedbothinexperi-mentsandinmodels,inparticularintheconductance-basedHodgkin-Huxley–typeequations͑HodgkinandHuxley,1952͒,consideredthetraditionalframeworkformodelingneurons,andintheanalysisofnetworkstabil-ityandplasticity.

Themoststrikingresultsinneurosciencebasedonclassicaldynamicalsystemtheoryhavecomefrombottom-upmodels.Theseresultsincludethedescription

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ofthediversityofdynamicsinsingleneuronsandsynapses͑Koch,1999;Vogelsetal.,2005͒,thespatiotem-poralcooperativedynamicsofsmallgroupsofneuronswithdifferenttypesofconnections͑Selverstonetal.,2000;Selverston,2005͒,andtheprinciplesofsynchroni-zationinnetworkswithdynamicalsynapses͑LoebelandTsodyks,2002;Elhilalietal.,2004;Persietal.,2004͒.Sometop-downmodelsalsohaveattemptedaclassi-calnonlineardynamicsapproach.Manyofthesemodelsarerelatedtotheunderstandinganddescriptionofcog-nitivefunctions.Nearlyhalfacenturyago,Ashbyhy-pothesizedthatcognitioncouldbemodeledasady-namicalprocess͑Ashby,1960͒.Neuroscientistshavespentconsiderableeffortimplementingthedynamicalapproachinapracticalway.Themostwidelystudiedexamplesofcognitive-typedynamicalmodelsaremulti-attractornetworks:modelsofassociativememorythatarebasedontheconceptofanenergyfunctionorLyapunovfunctionforadynamicalsystemwithmanyattractors͑Hopﬁeld,1982͓͒seealsoCohenandGross-berg͑1983͒;Waughetal.͑1990͒;Dobolietal.͑2000͔͒.Thedynamicalprocessinsuchnetworksisoftencalled

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“computationwithattractors.”Theideaistodesigndur-ingthelearningstage,inamemorynetworkphasespace,asetofattractors,eachofwhichcorrespondstoaspeciﬁcoutput.Neuralcomputationwithattractorsin-volvesthetransformationofagiveninputstimulus,whichdeﬁnesaninitialstateinsidethebasinofattrac-tionofoneattractor,leadingtoaﬁxeddesiredoutput.Theideathatcomputationorinformationprocessinginneuralsystemsisadynamicalprocessisbroadlyacceptedtoday.Manydynamicalmodelsofbothbottom-upandtop-downtypethataddresstheencodinganddecodingofneuralinformationastheinput-dependentdynamicsofanonautonomousnetworkhavebeenpublishedinthelastfewyears.However,therearestillhugegapsinourknowledgeoftheactualbiologicalprocessesunderlyinglearningandmemory,makingac-curatemodelingofthesemechanismsadistantgoal.ForreviewsseeArbibetal.͑1997͒andWilson͑1999͒.

Classicalnonlineardynamicshasprovidedsomebasisfortheanalysisofneuralensemblesevenwithlargenumbersofneuronsinnetworksorganizedaslayersofnearlyidenticalneurons.Oneoftheelementsofthisformulationisthediscoveryofstablelow-dimensionalmanifoldsinaveryhigh-dimensionalphasespace.Thesemanifoldsaremathematicalimagesofcooperativemodesofactivity,forexample,propagatingwavesinnonequilibriummedia͑Rinzeletal.,1998͒.Modelsofthissortarealsointerestingfortheanalysisofspiralwavesincorticalactivityasexperimentallyobservedinvivoandinvitro͑Huangetal.,2004͒.Manyinterestingquestionshavebeenapproachedbyusingthephasepor-traitandbifurcationanalysisofmodelsandbyconsider-ingattractorsandotherasymptoticsolutions.Neverthe-less,newdirectionsmayberequiredtoaddresstheimportantcomplexityofnervoussystemfunctions.

C.Newparadigmsforcontradictoryissues

Thehumanbraincontainsapproximately1011neuronsandatypicalneuronconnectswithϷ104otherneurons.Neuronsshowawidediversityintermsoftheirmor-phologyandphysiology͑seeFig.3͒.Awidevarietyofintracellularandnetworkmechanismsinﬂuencetheac-tivityoflivingneuralcircuits.Ifwetakeintoaccountthatevenasingleneuronoftenbehaveschaotically,wemightarguethatsuchacomplexsystemmostlikelybe-havesasifitwereaturbulenthydrodynamicﬂow.How-ever,thisisnotwhatisobserved.Braindynamicsaremoreorlessregularandstabledespitethepresenceofintrinsicandexternalnoise.Whatprinciplesdoesnatureusetoorganizesuchbehavior,andwhatmathematicalapproachescanbeutilizedfortheirdescription?Thesearetheverydifﬁcultquestionsweneedtoaddress.

Severalimportantfeaturesdifferentiatethenervoussystemfromtraditionaldynamicalsystems:

•Thearchitectureofthesystem,theindividualneuralunits,thedetailsofthedynamicsofspeciﬁcneurons,aswellastheconnectionsamongneuronsarenot

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FIG.3.Examplesof͑a͒theanatomicaldiversityofneurons,and͑b͒thesingle-neuronmembranevoltageactivityassoci-atedwiththem.͑1͒Lobsterpyloricneuron;͑2͒neuroninratmidbrain;͑3͒catthalamocorticalrelayneuron;͑4͒guineapiginferiorolivaryneuron;͑5͒aplysiaR15neuron;͑6͒cattha-lamicreticularneuron;͑7͒sepiagiantaxon;͑8͒ratthalamicreticularneuron;͑9͒mouseneocorticalpyramidalneuron;͑10͒ratpituitarygonadotropin-releasingcell.Inmanycases,thebehaviordependsonthelevelofcurrentinjectedintothecellasshownin͑b͒.ModiﬁedfromWangandRinzel,1995.

usuallyknownindetail,sowecandescribethemonlyinaprobabilisticmanner.

•Despitethefactthatmanyunitswithinacomplexneuralsystemworkinparallel,manyofthemhavedifferenttimescalesandreactdifferentlytothesamenonstationaryeventsfromoutside.However,forthewholesystem,timeisuniﬁedandcoherent.Thismeansthattheneuralsystemisorganizedhierarchi-cally,notonlyinspace͑architecture͒butalsointime:eachbehavioraleventistheinitialconditionforthenextwindowoftime.Themostinterestingphenom-enonforaneuralsystemisthepresencenotofat-

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tractordynamicsbutofnonstationarybehavior.At-tractordynamicsassumeslong-timeevolutionfrominitialconditions;wemustconsidertransientre-sponsesinstead.

•Thestructureofneuralcircuitsis—inprinciple—geneticallydetermined;however,itisneverthelessnotﬁxedandcanchangewithexperience͑learning͒andthroughneuromodulation.

Wecouldexpandthislist,butthefactsmentionedal-readymakethepointthatthenervoussystemisaveryspecialﬁeldfortheapplicationofclassicalnonlineardy-namics,anditisclearnowwhyneurodynamicsneedsnewapproachesandafreshview.

Weusethefollowingargumentstosupportanopti-misticviewaboutﬁndingdynamicalprinciplesinneuro-science:

•Complexneuralsystemsaretheresultofevolution,andthustheircomplexityisnotarbitrarybutfollowssomeuniversalrules.Onesuchruleisthattheorga-nizationofthecentralnervoussystem͑CNS͒ishier-archicalandbasedonneuralmodules.

•Itisimportanttonotethatmanymodulesareorga-nizedinaverysimilarwayacrossdifferentspecies.Suchunitscanbesmall,likecentralpatterngenera-tors͑CPGs͒,ormuchmorecomplex,likesensorysystems.Inparticular,thestructureofoneoftheold-estsensorysystems,theolfactorysystem,ismoreorlessthesameininvertebratesandvertebratesandcanbedescribedbysimilardynamicalmodels.•Thepossibilityofconsideringthenervoussystemasanensembleofinterconnectedunitsisaresultofthehighlevelofautonomyofitssubsystems.Thelevelofautonomydependsonthedegreeofself-regulation.Self-regulationofneuralunitsoneachlevelofthenervoussystem,includingindividualneu-rons,isakeyprincipledetermininghierarchicalneu-ralnetworkdynamics.•Thefollowingconjectureseemsreasonable:Eachspeciﬁcdynamicalbehaviorofthenetwork͑e.g.,travelingwaves͒iscontrolledbyonlyafewofthemanyparametersofasystem͑likeneuromodulators,forexample͒,andtheserelevantparametersinﬂu-encethespeciﬁccellornetworkdynamicsindependently—atleastinaﬁrstapproximation.Thisideacanbeusefulforthemathematicalanalysisofnetworkdynamicsandcanhelptobuildanapproxi-matebifurcationtheory.Thegoalofthistheoryistopredictthetransformationofspeciﬁcdynamicsbasedonbifurcationanalysisinalow-dimensionalcontrolsubspaceofparameters.•Fortheunderstandingofthemainprinciplesofneu-rodynamics,phenomenologicaltop-downmodelsareveryusefulbecauseevendifferentneuralsystemswithdifferentarchitecturesanddifferentlevelsofcomplexitydemonstratesimilardynamicsiftheyex-ecutesimilarfunctions.

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Inthemainpartofthisreviewwediscusstwocriticalfunctionalpropertiesofneuralsystemsthatatﬁrstglanceappearincompatible:robustnessandsensitivity.Findingsolutionstosuchapparentcontradictionswillhelpusformulatesomegeneraldynamicalprinciplesofbiologicalneuralnetworkorganization.Wenotetwoex-amples.

Manyneuralsystems,especiallysensorysystems,mustberobustagainstnoiseandatthesametimemustbeverysensitivetoincominginputs.Anewparadigmthatcandealwiththeexistenceofthisfundamentalcontra-diction͑neuralRabinovichisthenetworketwinnerlessal.competition͑WLC͒principlewith,2001nonsymmetric͒.Accordingtoinhibitorythisprinciple,connec-ationsisabletoexhibitstructurallystabledynamicsifthestimulusisﬁxed,andqualitativelychangeitsdynamicsifthestimulusischanged.Thisabilityisbasedondifferentfeaturesofthesignalandthenoise,andthedifferentwaystheyinﬂuencethedynamicsofthesystem.

Anotherexampleistheremarkablereproducibilityoftransientbehavior.Becausetransientbehavior,incon-trasttothelong-termstablestationaryactivityofattrac-tors,dependsoninitialconditions,itisdifﬁculttoimag-inehowsuchbehaviorcanbereproduciblefromexperimenttoexperiment.Thesolutiontothisparadoxisrelatedtothespecialroleofglobalandlocalinhibi-tion,whichsetsuptheinitialconditions.

Thelogicofthisreviewisrelatedtothespeciﬁcityofneuralsystemsfromthedynamicalpointofview.InSec.IIweconsiderthepossibledynamicaloriginofrobust-nessandsensitivityinneuralmicrocircuits.Thedynam-icsofinformationprocessinginneuralsystemsisconsid-eredinSec.III.InSec.IV,togetherwithotherdynamicalconcepts,wefocusonanewparadigmofneu-rodynamics:thewinnerlesscompetitionprincipleinthecontextofsequencegeneration,sensorycoding,andlearning.

II.DYNAMICALFEATURESOFMICROCIRCUITS:ADAPTABILITYANDROBUSTNESS

A.Dynamicalpropertiesofindividualneuronsandsynapses1.Neuronmodels

Neuronsreceivepatternedsynapticinputandcom-puteandcommunicatebytransformingthesesynapticinputpatternsintoanoutputsequenceofspikes.Whyspikes?Asspikewaveformsaresimilar,informationen-codedinspiketrainsmainlyreliesontheinterspikein-tervals.Relyingontimingratherthanonthedetailsofaction-potentialwaveformsincreasesthereliabilityandreproducibilityininterneuralcommunication.Disper-sionandattenuationintransmissionofneuralsignalsfromoneneurontootherschangesthewaveformoftheactionpotentialsbutpreservestheirtiminginformation,againallowingforreliabilitywhendependingoninter-spikeintervals.

Thenatureofspiketraingenerationandtransforma-tiondependscruciallyonthepropertiesofmanyvoltage-gatedionicchannelsinneuroncellmembranes.

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Thecellbody͑orsoma͒oftheneurongivesrisetotwokindsofprocesses:shortdendritesandoneormorelong,tubularaxons.Dendritesbranchoutliketreesandreceiveincomingsignalsfromotherneurons.Insomecasesthesynapticinputsitesareondendriticspines,thousandsofwhichcancoverthedendriticarbor.Theoutputprocess,theaxon,transmitsthesignalsgeneratedbytheneurontootherneuronsinthenetworkortoaneffectororgan.Thespikesarerapid,transient,all-or-none͑binary͒impulses,withadurationofabout1ms͑izedseeFig.region1͒.Inatmostthecases,origintheyoftheareaxoninitiatedandatpropagateaspecial-alongtheaxonwithoutdistortion.Nearitsend,thetu-bularaxondividesintobranchesthatconnecttootherneuronsthroughsynapses.

Whenthespikeemittedbyapresynapticneuronreachestheterminalofitsaxon,ittriggerstheemissionofchemicaltransmittersinthesynapticcleft͑thesmallgap,oforderafewtensofnanometers,separatingthetwoneuronsatasynapse͒.Thesetransmittersbindtoreceptorsinthepostsynapticneuron,causingadepolar-izationorhyperpolarizationinitsmembrane,excitingorinhibitingthepostsynapticneuron,respectively.Thesechangesinthepolarizationofthemembranerelativetotheextracellularspacespreadpassivelyfromthesyn-apsesonthedendritesacrossthecellbody.Theireffectsareintegrated,and,whenthereisalargeenoughdepo-larization,anewactionpotentialisgenerated͑Kandeletal.,2000͒.OthertypesofsynapsescalledgapjunctionsfunctionasOhmicelectricalconnectionsbetweenthemembranesoftwocells.Aspikeistypicallyfollowedbyabriefrefractoryperiod,duringwhichnofurtherspikescanbeﬁredbythesameneuron.

Neuronsarequitecomplexbiophysicalandbiochemi-calentities.Inordertounderstandthedynamicsofneu-ronsandneuralnetworks,phenomenologicalmodelshavetobedeveloped.TheHodgkin-Huxleymodelisforemostamongsuchphenomenologicaldescriptionsofneuralactivity.Thereareseveralclassesofneuralmod-elspossessingvariousdegreesofsophistication.Wesum-marizetheneuralmodelsmostoftenconsideredinbio-logicalnetworkdevelopmentinTableI.Foramoredetaileddescriptionofthesemodelssee,forexample,Koch͑2004͑1999͒,GerstnerandKistler͑2002͒,andDetailed͒.

Izhikevichconductance-basedneuronmodelstakeintoaccount͑typesKoch,of1994ionicvoltage-dependent͒.Thecurrentsneuralﬂowingmembraneacrossthemembranesodium,maypotassium,containandseveralcal-ciumchannels.Thedynamicsofthesechannelscanalsodependontheconcentrationofspeciﬁcions.Inaddi-tion,thereisaleakagecurrentofchlorideions.Theﬂowofthesecurrentsresultsinchangesinthevoltageacrossthemembrane.Theprobabilitythatatypeofionicchan-nelisopendependsnonlinearlyonthemembranevolt-ageandthecurrentstateofthechannel.Thesedepen-denciesresultinasetofseveralcouplednonlineardifferentialequationsdescribingtheelectricalactivityofthecell.Theintrinsicmembraneconductancescanen-ableneuronstogeneratedifferentspikepatterns,in-Rev.Mod.Phys.,Vol.78,No.4,October–December2006

cludinghigh-frequencyburstsofdifferentdurationswhicharecommonlyobservedinavarietyofmotorneu-ralcircuitsandbrainregions͓seeFig.3͑b2͔͒.Thebio-physicalmechanismsofspikegenerationenableindi-vidualneuronstoencodedifferentstimulusfeaturesintodistinctspikepatterns.Spikes,andburstsofspikesofdifferentdurations,codefordifferentstimulusfeatures,whichcanbequantiﬁedwithoutaprioriassumptionsaboutthosefeatures͑KepecsandLisman,2003͒.

Howdetaileddoesthedescriptionofneuronsorsyn-apseshavetobetomakeamodelofneuraldynamicsbiologicallyrealisticwhilestillremainingcomputation-allytractable?Itisreasonabletoseparateneuronmod-elsintotwoclassesdependingonthegeneralgoalofthemodeling.Ifwewishtounderstand,forexample,howtheratioofinhibitorytoexcitatorysynapsesinaneuralensemblewithrandomconnectionsinﬂuencestheactiv-ityofthewholenetwork,itisreasonabletouseasimplemodelthatkeepsonlythemainfeaturesofneuronbe-havior.Theexistenceofaspikethresholdandthein-creaseoftheoutputspikeratewithanincreaseintheinputmaybesufﬁcient.Ontheotherhand,ifourgoalistoexplaintheﬂexibilityandadaptabilityofasmallnet-worklikeaCPGtoachangingenvironment,thedetailsoftheionicchanneldynamicscanbeofcriticalimpor-tance͑Prinzetal.,2004b͒.Inmanycasesneuralmodelsbuiltonsimpliﬁedparadigmsleadtomoredetailedconductance-basedmodelsbasedonthesamedynamicalprinciplesbutimplementedwithmorebiophysicallyre-alisticmechanisms.Agoodindicationthatthelevelofthedescriptionwaschosenwiselycomesifthemodelcanreproducewiththesameparametersthemainbifur-cationsobservedintheexperiments.

2.Neuronadaptabilityandmultistability

Multistabilityinadynamicalsystemmeansthecoex-istenceofmultipleattractorsseparatedinphasespaceatthesamevalueofthesystem’sparameters.Insuchasystemqualitativechangesindynamicscanresultfromchangesintheinitialconditions.Awell-studiedcaseisthebistabilityassociatedwithasubcriticalAndronov-Hopfbifurcation͑Kuznetsov,1998͒.Multistablemodesofoscillationcanariseindelayed-feedbacksystemswhenthedelayislargerthantheresponsetimeofthesystem.Inneuralsystemsmultistabilitycouldbeamechanismformemorystorageandtemporalpatternrecognitioninbothartiﬁcial͑SompolinskyandKanter,1986͒andliving͑Canavieretal.,1993͒neuralcircuits.Inabiologicalnervoussystemrecurrentloopsinvolvingtwoormoreneuronsarefoundquiteoftenandarepar-ticularlyprevalentincorticalregionsimportantformemory͑TraubandMiles,1991͒.Multistabilityemergeseasilyintheseloops.Forexample,theconditionsunderwhichtime-delayedrecurrentloopsofspikingneuronsexhibitmultistabilitywerederivedbyFossetal.͑1996͒.Thestudyusedbothasimpleintegrate-and-ﬁreneuronandaHodgkin-Huxley͑HH͒neuronwhoserecurrentinputsaredelayedversionsoftheiroutputspiketrains.Theauthorsshowedthattwokindsofmultistabilitywith

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TABLEI.Summaryofmanyfrequentlyusedneuronalmodels.

ModelIntegrate-and-ﬁreneurons

dv͑t͒dt

Example

Variablesv͑t͒istheneuron

membranepotential;␪isthethresholdforspikegeneration.Iextisanexternalstimuluscurrent;Isynisthesumofthesynapticcurrents;and␶1and␶2aretimeconstantscharacterizingthesyn-apticcurrents.ai͑t͒Ͼ0isthespikingrateoftheithneuronorcluster;␳ijistheconnectionmatrix;andF,G,Qare

polynomialfunctions.

Remarks

Aspikeoccurswhentheneuronreachesthethreshold␪inv͑t͒afterwhichthecellisresettotherestingstate.

ReferencesLapicque,1907

Ά−

v͑t͒

+Iext+Isyn͑t͒,0Ͻv͑t͒Ͻ␪␶v͑t−0͒=␪spike͒

v͑t+0͒=0,

spikes

Isyn͑t͒=g

͚f͑t−t

and

f͑t͒=A͓exp͑−t/␶1͒−exp͑−t/␶2͔͒

Ratemodels

˙i͑t͒=Fi͑ai͑t͓͒͒Gi„ai͑t͒…a

−͚j␳ijQj„aj͑t͒…͔

Thisisageneral-izationoftheLotka-Volterramodel͓seeEq.͑9͔͒.FukaiandTanaka,1997;Lotka,1925;Volterra,1931

McCullochandPitts

xi͑n+1͒=⌰͚͑jgijxj͑n͒−␪͒

1,xϾ0

⌰͑x͒=

0,xഛ0

ͭ␪istheﬁring

threshold;xj͑n͒aresynapticinputsatthediscrete“time”n;xi͑n+1͒istheoutput.Inputsandoutputsarebinary͑oneorzero͒;thesynapticconnectionsgijare1,−1,or0.

v͑t͒isthemembranepotential,m͑t͒,andh͑t͒,andn͑t͒

representempiricalvariablesdescribingtheactivationandinactivationoftheionicconductances;Iisanexternalcurrent.Thesteady-statevaluesofthe

conductancevariablesmϱ,hϱ,nϱhaveanonlinearvoltagedependence,typicallythroughsigmoidalorexponentialfunctions.x͑t͒isthemembranepotential,andy͑t͒describesthedynamicsoffastcurrents;Iisan

externalcurrent.Theparametervaluesa,b,andcareconstantschosentoallowspiking.

Theﬁrst

computationalmodelforanartiﬁcialneuron;itisalsoknownasalinearthresholddevicemodel.Thismodelneglectstherelativetimingofneuralspikes.

TheseODEsrepresentpointneurons.Thereisalargelistofmodelsderivedfromthisone,andithasbecometheprincipaltoolincomputational

neuroscience.Otherioniccurrentscanbeaddedtothe

right-handsideofthevoltageequationtobetterreproducethedynamicsand

bifurcationsobservedintheexperiments.Areducedmodeldescribingoscillatoryspikingneural

dynamicsincludingbistability.

McCullochandPitts,1943

˙͑t͒=g͓v−v͑t͔͒Hodgkin-HuxleyCvLL

+gNam͑t͒3h͑t͓͒vNa−v͑t͔͒+gKn͑t͒4͑vK͒−v͑t͒+I,mϱ„v͑t͒…−m͑t͒m˙͑t͒=

␶m„v͑t͒…h„v͑t͒…−h͑t͒˙t͒=ϱh͑

␶h„v͑t͒…n„v͑t͒…−n͑t͒˙t͒=ϱn͑

␶n„v͑t͒…

HodgkinandHuxley,1952

FitzHugh-Nagumo

˙␮x−cx3−y+I,x˙=x+by−ay

FitzHugh,

1961;

Nagumoetal.,1962

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TABLEI.͑Continued.͒

ModelWilson-Cowan

Example

Variables͕E͑x,t͒,I͑x,t͖͒arethenumberdensityofactiveexcitatoryandinhibitoryneuronsatlocationxofthecontinuousneuralmedia.„wee͑x͒,wie͑x͒,wei͑x͒,wii͑x͒…are

connectivitydistribu-tionsamongthepopu-lationsofcells.͕Le,Li͖arenonlinearre-sponsesreﬂectingdif-ferentpopulationsofthresholds.Theoper-ator󰀁isaconvolu-tioninvolvingthecon-nectivitydistributions.v͑t͒isthemembranepotential;n͑t͒

describestherecoveryactivityofacalciumcurrent;Iisanexternalcurrent.

Remarks

Theﬁrst“mean-ﬁeld”model.Itisan

attempttodescribeaclusterofneurons,toavoidtheinherentnoisydynamical

behaviorofindividualneurons;byaveragingtoadistributionnoiseisreduced.

ReferencesWilsonandCowan,1973

␮ץE͑x,t͒ץtץI͑x,t͒ץt

=−E͑x,t͒+͓1−rE͑x,t͔͒ϫLe͓E͑x,t͒󰀁wee͑x͒−I͑x,t͒󰀁wei͑x͒+Ie͑x,t͔͒

␮=−I͑x,t͒+͓1−rI͑x,t͔͒ϫLi͓E͑x,t͒󰀁wie͑x͒−I͑x,t͒󰀁wii͑x͒+Ii͑x,t͔͒

Morris-Lecar

˙t͒=g͓v−v͑t͔͒+n͑t͒gv͑LLn

ϫ͓vn−v͑t͔͒

˙͑t͒…͓vm−v͑t͔͒+I,+gmmϱ„v

˙t͒=␭„v͑t͒…͓n„v͑t͒…−n͑t͔͒n͑ϱ

1v−vm

mϱ͑v͒=1+tanh0

2vm1v−vn

nϱ͑v͒=1+tanh0

2vn

v−vn

␭͑v͒=␾ncosh0

2vn

͑͑͒͒SimpliﬁedmodelthatreducesthenumberofdynamicalvariablesoftheHHmodel.Itdisplaysaction

potentialgenerationwhenchangingIleadstoasaddle-nodebifurcationtoalimitcycle.

MorrisandLecar,1981

˙t͒=y͑t͒+ax͑t͒2−bx͑t͒3−z͑t͒+IHindmarsh-Rosex͑

˙t͒=C−xx͑t͒2−y͑t͒y͑

˙t͒=rˆs͓x͑t͒−x͔−z͑t͒‰z͑

x͑t͒isthemembrane

potential;y͑t͒describesfast

currents;z͑t͒describesslowcurrents;andIisanexternalcurrent.

Simpliﬁedmodelthatusesapolynomialapproximationtotheright-handsideofaHodgkin-Huxleymodel.Thismodelfailstodescribethehyperpolarized

periodsafterspikingofbiologicalneurons.Firstintroducedforchemicaloscillators;goodfordescribingstronglydissipativeoscillatingsystemsinwhichtheneuronsareintrinsicperiodicoscillators.

Oneofaclassof

simplephenomenologi-calmodelsforspiking,burstingneurons.Thiskindofmodelcanbecomputationallyveryfast,buthaslittlebio-physicalfoundation.

HindmarshandRose,1984

Phaseoscillatormodels

d␪i͑t͒

=␻+dt

Hij͑␪i͑t͒−␪j͑t͒͒

␪͑t͒isthephaseoftheithneuronwithapproximately

periodicbehavior;andHijistheconnectivityfunctiondetermininghowneuroniandjinteract.

xtrepresentsthe

spikingactivityandytrepresentsaslowvariable.Adiscretetimemap.

Cohenetal.,1982;

ErmentroutandKopell,1984;

Kuramoto,1984

Mapmodels

xt+1͑i͒=

␣+yt͑i͒

1+xt͑i͒2⑀+xt͑j͒Nj

Cazellesetal.,2001;Rulkov,2002

yt+1͑i͒=yt͑i͒−␴xt͑i͒−␤Rev.Mod.Phys.,Vol.78,No.4,October–December2006

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respecttoinitialspikingfunctionsexist,dependingonwhethertheneuronisexcitableorrepetitivelyﬁringintheabsenceoffeedback.

FollowingHebb’s͑1949͒ideasmoststudiesofthemechanismsunderlyinglearningandmemoryfocusonchangingsynapticefﬁcacy.Learningisassociatedwithchangingconnectivityinanetwork.However,thenet-workdynamicsalsodependsoncomplexinteractionsamongintrinsicmembraneproperties,synapticstrengths,andmembrane-voltagetimevariation.Fur-thermore,neuronalactivityitselfmodiﬁesnotonlysyn-apticefﬁcacybutalsotheintrinsicmembranepropertiesofneurons.PapersbyMarderetal.͑1996͒andTurri-gianoetal.͑1996͒presentexamplesshowingthatbistableneuronscanprovideshort-termmemorymechanismsthatrelysolelyonintrinsicneuronalprop-erties.Whilenotreplacingsynapticplasticityasapow-erfullearningmechanism,theseexamplessuggestthatmemoryinnetworkscouldresultfromanongoinginter-playbetweenchangesinsynapticefﬁcacyandintrinsicneuronproperties.

Tounderstandthebiologicalbasisforsuchcomputa-tionalpropertieswemustexamineboththedynamicsoftheioniccurrentsandthegeometryofneuronalmor-phology.

3.Synapticplasticity

Synapsesaswellasneuronsaredynamicalnonlineardevices.AlthoughsynapsesthroughouttheCNSsharemanyfeatures,theyalsohavedistinctproperties.Theyoperatewiththefollowingsequencesofevents:Aspikeisinitiatedintheaxonnearthecellbody,itpropagatesdowntheaxon,andarrivesatthepresynapticterminal,wherevoltage-gatedcalciumchannelsadmitcalcium,whichtriggersvesiclefusionandneurotransmitterre-lease.Thereleasedneurotransmitterthenbindstore-ceptorsonthepostsynapticneuronandchangestheirconductance͑Nichollsetal.,1992;Kandeletal.,2000͒.Thisseriesofeventsisregulatedinmanyways,makingsynapsesadaptiveandplastic.

Inparticular,thestrengthofsynapticconductivitychangesinrealtimedependingontheiractivity,asKatzobservedmanyyearsago͑FattandKatz,1952;Katz,1969͒.Adescriptionofsuchplasticitywasmadein1949byHebb͑1949͒.Heproposedthat“WhenanaxonofcellAisnearenoughtoexciteacellBandrepeatedlyorpersistentlytakespartinﬁringit,somegrowthprocessormetabolicchangetakesplaceinoneorbothcellssuchthatA’sefﬁciency,asoneofthecellsﬁringB,isin-creased.”Thisneurophysiologicalpostulatehassincebecomeacentralconceptinneurosciencethroughase-riesofclassicexperimentsdemonstratingHebbian-likesynapticplasticity.Theseexperimentsshowthattheef-ﬁcacyofsynaptictransmissioninthenervoussystemisactivitydependentandcontinuouslymodiﬁed.Examplesofsuchmodiﬁcationarelong-termpotentiationandde-pression͑LTPandLTD͒,whichinvolveincreasedorde-creasedconductivity,respectively,ofsynapticconnec-tionsbetweentwoneurons,leadingtoincreasedor

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

decreasedactivityovertime.Long-termpotentiationanddepressionarepresumedtoproducelearningbydif-ferentiallyfacilitatingtheassociationbetweenstimulusandresponse.TheroleofLTPandLTD,ifany,inpro-ducingmorecomplexbehaviorsislesscloselytiedtospeciﬁcstimuliandmoreindicativeofcognition,andisnotwellunderstood.

Long-termpotentiationwasﬁrstreportedinthehip-pocampalformation͑BlissandLomo,1973͒.ChangesinducedbyLTPcanlastformanydays.Long-termpo-tentiationhaslongbeenregarded,alongwithitscoun-terpartLTD,asapotentialmechanismforshort-term-memoryformationandlearning.Infact,thehypothesisiswidelyacceptedinlearningandmemoryresearchthatactivity-dependentsynapticplasticityisinducedatap-propriatesynapsesduringmemoryformationandisbothnecessaryandsufﬁcientfortheinformationstorageunderlyingthetypeofmemorymediatedbythebrainareainwhichthatplasticityisobserved͓seeforareviewMartinetal.͑2000͔͒.HebbdidnotanticipateLTDin1949,butalongwithLTPitisthoughttoplayacriticalrolein“rewiring”biologicalnetworks.

ThenotionofacoincidencerequirementforHebbianplasticityhasbeensupportedbyclassicstudiesofLTPandLTDusingpresynapticstimulationcoupledwithprolongedpostsynapticdepolarization͓see,forexample,MalenkaandNicoll͑1999͔͒.However,coincidencetherewaslooselydeﬁnedwithatemporalresolutionofhun-dredsofmillisecondstotensofseconds,muchlargerthanthetimescaleoftypicalneuronalactivitycharac-terizedbyspikesthatlastforacoupleofmilliseconds.Inanaturalsetting,presynapticandpostsynapticneuronsﬁrespikesastheirfunctionaloutputs.Howpreciselymustsuchspikingactivitiescoincideinordertoinducesynapticmodiﬁcations?Experimentsaddressingthiscriticalissueledtothediscoveryofspike-timing-dependentsynapticplasticity͑STDP͒.Spikesinitiateasequenceofcomplexbiochemicalprocessesinthepostsynapticneuronduringtheshorttimewindowfol-lowingsynapticactivation.Identifyingdetailedmolecu-larprocessesunderlyingLTPandLTDremainsacom-plexandchallengingproblem.Thereisgoodevidencethatitconsistsofacompetitionbetweenprocessesre-moving͑LTD͒andprocessesplacing͑LTP͒phosphategroupsfromonpostsynapticreceptors,orincreasing͑LTP͒ordecreasing͑LTD͒thenumberofsuchreceptorsinadendriticspine.ItisalsowidelyacceptedthatN-methyl-D-aspartate͑NMDA͒receptorsarecrucialforthedevelopmentofLTPorLTDandthatitiscalciuminﬂuxontothepostsynapticcellthatiscriticalforbothLTPandLTD.

Experimentsonsynapticmodiﬁcationsofexcitatorysynapsesbetweenhippocampalglutamatergicneuronsinculture͑BiandPoo,1998,2001͒͑seeFig.4͒indicatethatifapresynapticspikearrivesattimetpreandapostsyn-apticspikeisobservedorinducedattpost,thenwhen␶=tpost−tpreispositivetheincrementalpercentagein-creaseinsynapticstrengthbehavesas

Rabinovichetal.:Dynamicalprinciplesinneuroscience

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FIG.4.Spike-timing-dependentsynapticplasticityobservedinhippocampalneurons.Eachdatapointrepresentstherelativechangeintheamplitudeofevokedpostsynapticcurrentafterrepetitiveapplicationofpresynapticandpostsynapticspikingpairs͑1Hzfor60s͒withﬁxedspiketiming⌬t,whichisde-ﬁnedasthetimeintervalbetweenpostsynapticandpresynap-ticspikingwithineachpair.Long-termpotentiation͑LTP͒anddepression͑LTD͒windowsareeachﬁttedwithanexponentialfunction.ModiﬁedfromBi,2002.

⌬g

ϷaPe−␤g

P␶,͑1͒

with␤creasePinϷsynaptic1/16.8ms.strengthWhenbehaves␶Ͻ0,theas

percentagede-⌬g

Ϸ−aDe␤D␶g

,͑2͒

with␤DϷ1/33.7ms.aPandaDareconstants.ThisisillustratedinFig.4.

ManybiochemicalfactorscontributedifferentlytoLTPandLTDindifferentsynapses.Herewediscussaphenomenologicaldynamicalmodelofsynapticplastic-ity͑Abarbaneletal.,2002͒whichisveryusefulformod-elingneuralplasticity;itspredictionsagreewithseveralexperimentalresults.Themodelintroducestwodynami-calvariablesP͑t͒andD͑t͒thatdonothaveadirectre-lationshipwiththeconcentrationofanybiochemicalcomponents.Nonlinearcompetitionbetweenthesevari-ablesimitatestheknowncompetitioninthepostsynapticcell.Thesevariablessatisfythefollowingsimpleﬁrst-orderkineticequations:

dP͑t͒

=f„Vpre͑t͒…͓1−P͑t͔͒−␤PP͑t͒,dD͑t͒

=g„Vpost͑t͒…͓1−D͑t͔͒−␤DD͑t͒,͑3͒

wherethefunctionsf͑V͒andg͑V͒aretypicallogisticorsigmoidalfunctionswhichrisefromzerototheorderofunitywhentheirargumentexceedssomethreshold.Thesedrivingorinputfunctionsareasimpliﬁcationof

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

thedetailedwayinwhicheachdynamicalprocessisforced.TheP͑t͒processisassociatedwithaparticulartimeconstant1/␤PwhiletheD͑t͒processisassociatedwithadifferenttimeconstant1/␤D.Experimentsshowthat␤P󰀅␤D,andthisistheprimaryembodimentofthetwodifferenttimescalesseeninmanyobservations.Thetwotimeconstantsareacoarse-grainedrepresentationofthediffusionandleakageprocesseswhichdampenandterminateactivities.PresynapticvoltageactivityservestoreleaseneurotransmittersintheusualmannerandthisinturninducesthepostsynapticactionofP͑t͒,whichhasatimecoursedeterminedbythetimeconstant␤−1P.Similarly,thepostsynapticvoltage,constantortimevarying,canbeassociatedwiththeinductionoftheD͑t͒process.

P͑t͒andD͑t͒competetoproduceachangeinsynapticstrength⌬g͑t͒as

d⌬g͑t͒

=␥͓P͑t͒D␩͑t͒−D͑t͒P␩͑t͔͒,͑4͒

where␩Ͼ1and␥Ͼ0.ThisdynamicalmodelreproducessomeofthekeySTDPexperimentalresultslike,forex-ample,thoseshowninFig.4.Italsoaccountsforthecasewherethepostsynapticcellisdepolarizedwhileapresynapticspiketrainispresentedtoit.

4.Examplesofthecooperativedynamicsofindividualneuronsandsynapses

Toillustratethedynamicalsigniﬁcanceofplasticsyn-apsesweconsiderthesynchronizationoftwoneurons:alivingneuronandanelectronicmodelneuroncoupledthroughaSTDPorinverseSTDPelectronicsynapse.Usinghybridcircuitsofmodelelectronicneuronsandbiologicalneuronsisapowerfulmethodforanalyzingneuraldynamics͑Pintoetal.,2000;Szücsetal.,2000;LeMassonetal.,2002;Prinzetal.,2004a͒.Therepresen-tationofsynapticinputtoacellusingacomputertocalculatetheresponseofthesynapsetospeciﬁedpresynaptic͑beenRobinsonshownandinputinmodelingKawai,goesunder1993and;theSharpnameinexperimentset“dynamical.,1993͑Nowotny,͒.clamp”IthasZhigulin,etal.,2003;Zhigulinetal.,2003͒thatcouplingthroughplasticelectronicsynapsesleadstoneuralsyn-chronizationor,morecorrectly,entrainmentthatismorerapid,moreﬂexible,andmuchmorerobustagainstnoisethansynchronizationmediatedbyconnectionsofconstantstrength.Intheseexperimentstheneuralcir-cuitconsistsofaspeciﬁedpresynapticsignal,asimulatedsynapse͑viathedynamicclamp͒,andapostsynapticbio-logicalneuronfromtheAplysiaabdominalganglion.Thepresynapticneuronisaspikegeneratorproducingspikesofpredeterminedformatpredeterminedtimes.Thesynapseanditsplasticityaresimulatedbydynamicclampsoftware͑Nowotny,2003͒.Ineachupdatecycleofϳgenerator100␮sthevoltagepresynapticisupdated,voltagethesynapticisacquired,strengththespikeisde-terminedaccordingtothelearningrule,andtheresult-ingsynapticcurrentiscalculatedandinjectedintothelivingneuronthroughacurrentinjectionelectrode.As

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onepresentsthepresynapticsignalmanytimes,thesyn-apticconductancechangesfromoneﬁxedvaluetoan-otherdependingonthepropertiesofthepresynapticsignal.

Thecalculatedsynapticcurrentisafunctionofthepresynapticandpostsynapticpotentialsofthespikegen-eratorVtively.Itpreis͑calculatedt͒andtheaccordingbiologicaltoneuronthefollowingVpost͑t͒,respec-model.ThesynapticcurrentdependslinearlyonthedifferencebetweenthepostsynapticpotentialVpostanditsreversalpotentialVrev,onanactivationvariableS͑t͒,andonitsmaximalconductanceg͑t͒:

Isyn͑t͒=g͑t͒S͑t͓͒Vpost͑t͒−Vrev͔.

͑5͒

TheactivationvariableS͑t͒isanonlinearfunctionofthepresynapticmembranepotentialVpercentageofneurotransmitterdockedpreandrepresentstheonthepostsyn-apticcellrelativetothemaximumthatcandock.Ithastwotimescales:adockingtimeandanundockingtime.Wetakeittosatisfythedynamicalequation

dS͑t͒Sϱ„Vpre͑t͒…−S͑t͒dt=␶S.͑6͒

syn͓S1−ϱ„V1͑t͒…͔

Sϱ͑V͒isasigmoidfunctionwhichwetaketobe

SVϱ͑V͒=

ͭtanh͓͑V−Vth͒/Vslope͔forVϾth0

otherwise.

͑7͒

Thetimescaleis␶syn͑S1−1͒forneurotransmitterdock-ingand␶synS1forundocking.ForAMPAexcitatoryre-ceptors,thedockingtimeisabout0.5ms,andtheun-dockingtimeisabout1.5ms.Themaximalconductanceg͑t͒isdeterminedbythelearningrulediscussedbelow.Intheexperiments,thesynapticcurrentisupdatedatϳ10kHz.

Todeterminethemaximalsynapticconductanceg͑t͒ofthesimulatedSTDPsynapse,anadditiveSTDPlearn-ingrulewasused.Thisisaccurateifthetimebetweenpresentedspikepairsislongcomparedtothetimebe-tweenspikesinthepair.Toavoidrunawaybehavior,theadditiverulewasappliedtoanintermediategthenﬁlteredthroughasigmoidfunction.Inparticu-rawthatwaslar,thechange⌬grawinsynapticstrengthisgivenby

A⌬t−␶0+

␶e

−͑⌬t−␶0͒/␶+for⌬tϾ␶0⌬g+

raw͑⌬t͒=

Ά⌬t−␶͑8͒

A0−␶e

͑⌬t−␶0͒/␶−for⌬tϽ␶0,

−

where⌬t=tpost−tpreisthedifferencebetweenpostsynap-ticandpresynapticspiketimes.Theparameters␶␶+and−determinethewidthsofthelearningwindowsforpo-tentiationanddepression,respectively,andtheampli-tudesA+andA−determinethemagnitudeofsynapticchangeperspikepair.Theshift␶0reﬂectstheﬁnitetimeofinformationtransportthroughthesynapse.

AsonecanseeinFig.5,thepostsynapticneuronquicklysynchronizestothepresynapticspikegeneratorwhichpresentsspikeswithaninterspikeinterval͑ISI͒of255ms͑toppanel͒.Thesynapticstrengthcontinuously

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FIG.5.Exampleofasynchronizationexperiment.Top:The

interspikeintervals͑ISIs͒ofthepostsynapticbiologicalneu-ron.Bottom:Thesynapticstrengthg͑t͒.PresynapticspikeswithISIof255mswerepresentedtoapostsynapticneuronwithperiodicoscillationsatanISIof330ms.Beforecouplingwiththepresynapticspikegenerator,thebiologicalneuronspikestonicallyatitsintrinsicISIof330ms.Couplingwasswitchedonwithg͑t=0͒=15nSattime6100s.Asonecanseethepostsynapticneuronquicklysynchronizestothepresynap-ticspikegenerator͑toppanel,dashedline͒.Thesynapticstrengthcontinuouslyadaptstothestateofthepostsynapticneuron,effectivelycounteractingadaptationandothermodu-lationsofthesystem.Thisleadstoaverypreciseandrobustsynchronizationatanonzerophaselag.TheprecisionofthesynchronizationmanifestsitselfinsmallﬂuctuationsofthepostsynapticISIsinthesynchronizedstate.Robustnessandphaselagcannotbeseendirectly.ModiﬁedfromNowotny,Zhigulin,etal.,2003.

adaptstothestateofthepostsynapticneuron,effec-tivelycounteractingadaptationandothermodulationsofthesystem͑bottompanel͒.Thisleadstoaverypre-ciseandrobustsynchronizationatanonzerophaselag.TheprecisionofthesynchronizationmanifestsitselfinsmallﬂuctuationsofthepostsynapticISIsinthesyn-chronizedstate.RobustnessandphaselagcannotbeseendirectlyinFig.5.Spike-timing-dependentplasticityisamechanismthatenablessynchronizationofneuronswithsigniﬁcantlydifferentintrinsicfrequenciesasonecanseeinFig.6.Thesigniﬁcantincreaseintheregimeofsynchronizationassociatedwithsynapticplasticityisawelcome,perhapssurprising,resultandaddressestheissueraisedaboveaboutrobustnessofsynchronizationinneuralcircuits.

B.Robustnessandadaptabilityinsmallmicrocircuits

Thepreciserelationshipbetweenthedynamicsofin-dividualneuronsandthemammalianbrainasawholeremainsextremelycomplexandobscure.Animportantreasonforthisisalackofknowledgeonthedetailedcell-to-cellconnectivitypatternsaswellasalackofknowledgeonthepropertiesoftheindividualcells.Al-thoughlarge-scalemodelingofthissituationisat-temptedfrequently,parameterssuchasthenumberandkindofsynapticconnectionscanonlybeestimated.By

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FIG.6.͑Coloronline͒Thepresynapticsignalgeneratorpre-sentsaperiodicspiketrainwithISIofT1toapostsynapticneuronwithISIofT02,beforecoupling.Whenneuronsare0

coupled,T2→T2.Weplottheratiooftheseperiodsaftercou-plingasafunctionoftheratiobeforecoupling͑a͒,forasyn-apsewithconstantgand͑b͒forasynapticconnectiong͑t͒followingtheruleinthetext.Theenlargeddomainofone-to-onesynchronizationinthelattercaseisquiteclearand,asshownbythechangeintheerrorbarsizes,thesynchronizationismuchbetter.Thisresultpersistswhennoiseisaddedtothepresynapticsignalandtothesynapticaction͑notshown͒.ModiﬁedfromNowotny,Zhigulin,etal.,2003.

usingthelesscomplexmicrocircuits͑MCs͒ofinverte-brates,amoredetailedunderstandingofneuralcircuitdynamicsispossible.

CentralpatterngeneratorsaresmallMCsthatcanproducestereotypedcyclicoutputswithoutrhythmicsensoryorcentralinput͑MarderandCalabrese,1996;

Steinetal.,1997͒.ThusCPGsareoscillators,andtheimageoftheiractivityinthecorrespondingsystemstatespaceisalimitcyclewhenoscillationsareperiodicandastrangeattractorinmorecomplexcases.Centralpatterngeneratorsunderlietheproductionofmostmotorcom-mandsformusclesthatexecuterhythmicanimalactivitysuchaslocomotion,breathing,heartbeat,etc.TheCPGoutputisaspatiotemporalpatternwithspeciﬁcphaselagsbetweenthetemporalsequencescorrespondingtothedifferentmotorunits͑seebelow͒.

ThenetworkarchitectureandthemainfeaturesofCPGneuronsandsynapsesareknownmuchbetterthananyotherbraincircuits.Examplesoftypicalinverte-brateCPGnetworksareshowninFig.7.CommontomanyCPGcircuitsareelectricalandinhibitoryconnec-tionsandthespiking-burstingactivityoftheirneurons.Thecharacteristicsofthespatiotemporalpatternsgener-atedbytheCPG,suchasburstfrequency,phase,length,etc.,aredeterminedbytheintrinsicpropertiesofeachindividualneuron,thepropertiesofthesynapses,andthearchitectureofthecircuit.

ThemotorpatternsproducedbyCPGsfallintotwocategories:thosethatoperatecontinuouslysuchasres-piration͑Ramirezetal.,2004͒orheartbeat͑Cymbalyuketal.,2002͒,andthosethatareproducedintermittentlysuchaslocomotion͑Getting,19͒orchewing͑Selver-ston,2005͒.AlthoughCPGsautonomouslyestablishcor-rectrhythmicﬁringpatterns,theyareunderconstantsupervisionbydescendingﬁbersfromhighercentersandbylocalreﬂexpathways.Theseinputsallowtheanimaltoconstantlyadaptitsbehaviortotheimmediateenvi-ronment,whichsuggeststhatthereisconsiderableﬂex-ibilityinthedynamicsofmotorsystems.Inadditionthereisnowaconsiderablebodyofinformationshowingthatanatomicallydeﬁnedsmallneuralcircuitscanbereconﬁguredinamoregeneralwaybyneuromodulatorysubstancesintheblood,orreleasedsynapticallysothattheyarefunctionallyalteredtoproducedifferentstablespatiotemporalpatterns,whichmustalsobeﬂexibleinresponsetosensoryinputsonacycle-by-cyclebasis;see

FIG.7.ExamplesofinvertebrateCPGmicro-circuitsfromarthropod,mollusk,andannelidpreparations.Allproducerhythmicspa-tiotemporalmotorpatternswhenactivatedbynonpatternedinput.Theblackdotsrepresentchemicalinhibitorysynapses.Resistorsrepre-sentelectricalconnections.Trianglesarechemicalexcitatorysynapses,anddiodesarerectifyingsynapses͑electricalsynapsesinwhichthecurrentﬂowsonlyinonedirection͒.Individualneuronsareidentiﬁablefromonepreparationtoanother.

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SimmersandMoulins͑1988͒,forexample.

CentralpatterngeneratorshavesimilaritywithneuralMCsinthebrain͑Silberbergetal.,2005;SolisandPer-kel,2005;Yusteetal.,2005͒andareoftenstudiedasmodelsofneuralnetworkfunction.Inparticular,thereareimportantsimilaritiesbetweenvertebratespinalcordCPGsandneocorticalmicrocircuitswhichhavebeenemphasizedbyYusteetal.͑2005͒:͑i͒CPGinterac-tions,whicharefundamentallyinhibitory,dynamicallyregulatetheoscillations.Furthermore,subthreshold-activatedvoltage-dependentcellularconductancesthatpromotebistabilityandoscillationsalsopromotesyn-chronizationwithspeciﬁcphaselags.Thesamecellularpropertiesarealsopresentinneocorticalneurons,andunderlietheobservedoscillatorysynchronizationinthecortex.͑ii͒NeuronsinspinalcordCPGsshowbistablemembranedynamics,whicharecommonlyreferredtoasplateaupotentials.Acorrelateofbistablemembranebe-havior,inthiscasetermed“up”and“down”states,hasalsobeendescribedinthestriatumandneocortexbothinvivoandinvitro͑Sanchez-VivesandMcCormick,2000;Cossartetal.,2003͒.Itisstillunclearwhetherthisbistabilityarisesfromintrinsicorcircuitmechanismsoracombinationofthetwo͑Egorovetal.,2002;Shuetal.,2003͒.͑iii͒BothCPGsandcorticalmicrocircuitsdemon-strateattractordynamicsandtransientdynamics͓see,forexample,Abelesetal.͑1993͒;Ikegayaetal.͑2004͔͒.͑iv͒ModulationsbysensoryinputsandneuromodulatorsarealsoacommoncharacteristicthatissharedbetweenCPGsandcorticalcircuits.ExamplesinCPGsincludethemodulationofoscillatoryfrequency,oftemporalco-ordinationamongdifferentpopulationsofneurons,oftheamplitudeofnetworkactivity,andofthegatingofCPGinputandoutput͑Grillner,2003͒.͑v͒Switchingbe-tweendifferentstatesofCPGoperation͑forexample,switchingcoordinatedmotorpatternsfordifferentmodesoflocomotion͒isundersensoryafferentandneu-rochemicalmodulatorycontrol.ThismakesCPGsmul-tifunctionalanddynamicallyplastic.Switchingbetweencorticalactivitystatesisalsoundermodulatorycontrol,asshown,forexample,bytheroleoftheneurotransmit-terdopamineinworkingmemoryinmonkeys͑Goldman-Rakic,1995͒.Thusmodulationreconﬁguresmicrocircuitdynamicsandtransformsactivitystatestomodifybehavior.

TheCPGconceptwasbuiltaroundtheideathatbe-haviorallyrelevantspatiotemporalcyclicpatternsaregeneratedbygroupsofnervecellswithouttheneedforrhythmicinputsfromhighercentersorfeedbackfromstructuresthataremoving.Ifactivated,isolatedinverte-bratepreparationscangeneratesuchrhythmsformanyhoursandasaresulthavebeenextremelyimportantintryingtounderstandhowsimultaneouscooperativein-teractionsbetweenmanycellularandsynapticparam-eterscanproducerobustandstablespatiotemporalpat-terns͓seeFig.8͑d͔͒.Anexampleofathree-neuronCPGphaseportraitisshowninFigs.8͑a͒–8͑c͒.TheeffectofahyperpolarizingcurrentleadstochangesinthepatternasreﬂectedbythephaseportraitinFigs.8͑b͒and8͑c͒.

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

FIG.8.͑Color͒PhaseportraitoftypicalCPGoutput.ThedatawererecordedinthepyloricCPGofthelobsterstomatogastricganglion.Eachaxisrepresentstheﬁringrateofoneofthreepyloricneurons:LP,VD,andPD͑seeFig.7͒.͑a͒Theorbitoftheoscillatingpyloricsystemisshowninblueandtheaverageorbitisshowninred;͑b͒thesamebutwithahyperpolarizingdccurrentinjectedintothePD;͑c͒thedifferencebetweentheaveragedorbits;͑d͒timeseriesofthemembranepotentialsofthethreeneurons.FigureprovidedbyT.Nowotny,R.Levi,andA.Szücs.

Neuraloscillationsariseeitherthroughinteractionsamongneurons͑network-basedmechanism͒orthroughinteractionsamongcurrentsinindividualneurons͑pace-makermechanism͒.SomeCPGsusebothmechanisms.Inthesimplestcase,oneormoreneuronswithintrinsicburstingactivityactsasthepacemakerfortheentireCPGcircuit.Theintrinsiccurrentsmaybeconstitutivelyactiveortheymayrequireactivationbyneuromodula-tors,so-calledconditionalbursters.Synapticconnectionsacttodeterminethepatternbyexcitingorinhibitingotherneuronsattheappropriatetime.Suchnetworksareextremelyrobustandhavegenerallybeenthoughttobepresentinsystemsinwhichtherhythmicactivityisactiveallormostofthetime.Inthesecondcase,itisthesynapticinteractionsbetweennonburstyneuronsthatgeneratetherhythmicactivityandmanyschemesforthetypesofconnectionsnecessarytodothishavebeenpro-posed.Usuallyreciprocalinhibitionservesasthebasisforgeneratingburstsinantagonisticneuronsandtherearemanyexamplesofcellsinpattern-generatingmicro-circuitsconnectedinthisway͑seeFig.7͒.Circuitsofthistypeareusuallyfoundforbehaviorsthatareintermit-tentinnatureandrequireagreaterdegreeofﬂexibilitythanthosebasedonpacemakercells.

Physiologistsknowthatreciprocalinhibitoryconnec-tionsbetweenoscillatoryneuronscanproduce,asare-sultofthecompetition,sequentialactivityofneuronsandrhythmicspatiotemporalpatterns͑Szekely,1965;StentandFriesen,1977;RubinandTerman,2004͒.How-ever,evenforarathersimpleMC,consistingofjustthreeneurons,thereisnoquantitativedescription.Iftheconnectionsaresymmetric,theMCcanreachanattrac-

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tor.Itisreasonabletohypothesizethatasymmetricin-hibitoryconnectionsarenecessarytopreservetheorderofpatternswithmorethantwophasespercycle.Thecontradiction,notedearlier,betweenrobustnessandﬂexibilitycanthenberesolvedbecauseexternalsignalscanmodifytheeffectivetopologyofconnectionssoonecanhavefunctionallydifferentnetworksfordifferentstimuli.

TheoreticalanalysisandcomputerexperimentswithMCsbasedonthewinnerlesscompetitionprinciple͑dis-cussedinSec.IV.C͒showthatsufﬁcientconditionsforthegenerationofsequentialactivitydoexistandtherangeofallowednonsymmetricinhibitoryconnectionsisquitewide͑Rabinovichetal.,2001;Varona,Rabinovich,etal.,2002;Afraimovich,Rabinovich,etal.,2004͒WeillustratethisusingaLotka-Volterraratedescriptionofneuronactivity:

daN

i͑t͒

dt=ai͑t͒ͩ1−͚␳ij͑Si͒aj͑t͒+Si,...,N,

j=1

ͪi=1,͑9͒

wheretherateofeachNneuronisai͑t͒,theconnectionmatrixis␳ij,andthestimuliSiareconstantshere.Thismodelcanbejustiﬁedasarateneuralmodelasfollows͑FukaiandTanaka,1997͒.Theﬁringrateai͑t͒andmem-branepotentialvby

i͑t͒oftheithneuroncanbedescribedai͑t͒=G͑vi−␪͒,͑10͒dvi͑t͒

=−␭vi͑t͒+Ii͑t͒,͑11͒

whereG͑vi−␪͒isagainfunction,␪and␭areconstants,andtheinputcurrentIi͑t͒toneuroniisgeneratedbytheratesaj͑t͒oftheotherneurons:

Ii͑t͒=Si−͚␳ijaj͑t͒.

͑12͒

jHereSiistheinputand␳ijisthestrengthoftheinhibi-torysynapsefromneuronjtoneuroni.WesupposethatG͑x͒isasigmoidalfunction:

G͑x͒=G0/͓1+exp͑−␤x͔͒.

͑13͒

Letusthenmaketwoassumptions:͑i͒theﬁringrateisalwaysmuchsmallerthanitsmaximumvalueGstronglydissipative͑thisisreasonable0;and͑ii͒thesystemisbe-causeweareconsideringinhibitorynetworks͒.Basedonthese͑͑10͒assumptions,aftercombiningandrescalingEqs.can9͒with–͑13͒an,weadditionalobtainthepositiveLotka-Volterraratedescription͑1997bereplacedbyaconstantterm͓seeonFukaitherightandsideTanakathatThe͒fortestsdetailsofwhether͔.

WLCisoperatinginareducedpyloricCPGcircuitareshowninFig.9.ThisstudyusedestimatesofthesynapticstrengthsshowninFig.9͑a͒.Someofthekeyquestionsherearethese.͑i͒Whatistheminimalstrengthfortheinhibitorysynapsefromthepy-Rev.Mod.Phys.,Vol.78,No.4,October–December2006

FIG.9.͑Coloronline͒CompetitionwithoutwinnerinamodelofthepyloricCPG.͑a͒Schematicdiagramofthethree-neuronnetworkusedforratemodeling.Blackdotsrepresentchemicalinhibitorysynapseswithstrengthsgiveninnanoseconds͑XϾ160͒.͑b͒Phaseportraitofthemodel:Thelimitcyclecorre-spondingtotherhythmicactivityisinthe2Dsimplex͑ZeemanandZeeman,2002͒.͑c͒Robustnessinthepresenceofnoise:Noiseintroducedintothemodelshowsnoeffectontheorderofactivationforeachtypeofneuron.FigureprovidedbyR.Huerta.

loricdilator͑PD͒neuronorABgrouptotheVDneu-ronsuchthatWLCexists?͑ii͒DoestheconnectivityobtainedfromthecompetitionwithoutwinnerconditionproducetheorderofactivationobservedinthepyloricCPG?͑iii͒Isthisdynamicsrobustagainstnoise,inthesensethatstrongperturbationsofthesystemdonotal-terthesequence?Ifthestrengthsof␳ijaretakenas

␳ij=΂11.250

0.875

11.25,X/800.6251

΃theWLCformulasimplythatthesufﬁcientconditionsforareliableandrobustcyclicsequencearesatisﬁedifXϾ160.TheactivationsequenceoftheratemodelwithnoiseshowninFig.9͑c͒issimilartothatobservedex-perimentallyinthepyloricCPG.WhenadditiveGauss-iannoiseisintroducedintotherateequations,theacti-vationorderofneuronsisnotbroken,buttheperiodofthelimitcycledependsonthelevelofperturbation.ThereforethecycliccompetitivesequenceisrobustandcanberelatedtothesynapticconnectivityseeninrealMCs.IfindividualneuronsinaMCarenotoscillating,onecanconsidersmallsubgroupsofneuronsthatmayformoscillatoryunitsandapplytheWLCprincipletotheseunits.

Animportantquestionaboutmodelingtherhythmicactivityofsmallinhibitorycircuitsishowthespeciﬁcdynamicsofindividualneuronsinﬂuencesthenetworkrhythmgeneration.Figure10representsthethree-dimensional͑3D͒projectionofthemany-dimensionalphaseportraitofacircuitwiththesamearchitectureas

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FIG.10.͑Coloronline͒Three-dimensionalprojectionofthemany-dimensionalphaseportraitofacircuitwiththesamearchitectureastheoneshowninFig.9,usingHodgkin-Huxleyspiking-burstingneuronmodels.

showninFig.9͑a͒butusingHodgkin-Huxleyspiking-burstingneuronmodels.TheswitchingdynamicsseenintheratemodelisshowninFig.9͑c͒,andthiscircuitisrobustwhennoiseisaddedtoit.

Pairsofneuronscaninteractviainhibitory,excitatory,orelectrical͑gapjunction͒synapsestoproducebasicformsofneuralactivitywhichcanserveasthefounda-tionforMCdynamics.Perhapsthemostcommon͑andwell-studied͒CPGinteractionconsistsofreciprocalin-hibition,anarrangementthatgeneratesarhythmicburstingpatterninwhichneuronsﬁreapproximatelyoutofphasewitheachother͑WangandRinzel,1995͒.Thisiscalledahalf-centeroscillator.Itoccurswhenthereissomeformofexcitationtothetwoneuronssufﬁcienttocausetheirﬁringandsomeformofdecaymechanismtoslowhighﬁringfrequencies.Thedynamicalrangeoftheburstingactivityvarieswiththesynapsestrengthandinsomeinstancescanactuallyproducein-phasebursting.Usuallyreciprocalexcitatoryconnections͑unstableiftoolarge͒orreciprocalexcitatory-inhibitoryconnectionsareabletoreducetheintrinsicirregularityofneurons͑Varona,Torres,Abarbanel,etal.,2001͒.

Modelingstudieswithelectricallycoupledneuronshavealsoproducednonintuitiveresults͑Abarbaneletal.,1996͒.Whileelectricalcouplingisgenerallythoughttoprovidesynchronybetweenneurons,undercertainconditionsthetwoneuronscanburstoutofphasewitheachother͑ShermanandRinzel,1992;Elsonetal.,1998,2002͒;seeFig.11andalsoChowandKopell͑2000͒andLewisandRinzel͑2003͒.Aninterestingmodelingstudyofthreeneurons͑Soto-Trevinoetal.,2001͒withsyn-apsesthatareactivitydependentfoundthatthesynapticstrengthsself-adjustedindifferentcombinationstopro-ducethesamethree-phaserhythm.Therearemanyex-amplesofvertebrateMCsinwhichacollectionofneu-ronscanbeconceptuallyisolatedtoperformaparticularfunctionortorepresentthecanonicalormodularcircuitforaparticularbrainregion͓seeShepherd͑1998͔͒.

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FIG.11.Artiﬁcialelectricalcouplingbetweentwolivingcha-oticPDcellsofthestomatogastricganglionofacrustaceancandemonstratebothsynchronousandasynchronousregimesofactivity.InthiscasetheartiﬁcialelectricalsynapsewasbuiltontopoftheexistingnaturalcouplingbetweentwoPDcells.Thisshowsdifferentsynchronizationlevels͑a͒–͑d͒asafunctionoftheartiﬁcialcouplinggaandadccurrentIinjectedinoneofthecells.͑a͒Withtheirnaturalcouplingga=0thetwocellsaresynchronizedanddisplayirregularspiking-burstingactivity.͑b͒Withanartiﬁcialelectricalcouplingthatchangesthesignofthecurrentga=−200nS,andthuscompensatesthenaturalcoupling,thetwoneuronsbehaveindependently.͑c͒Increasingthenegativeconductanceleadstoaregularizedantiphasespik-ingactivity͑bymimickingmutualinhibitorysynapses͒.͑d͒Withnoartiﬁcialcouplingbutaddingadccurrenttwoneuronsaresynchronized,displayingtonicspikingactivity.ModiﬁedfromElsonetal.,1988.

C.Intercircuitcoordination

ItisoftenthecasethatmoreorlessindependentMCsmustsynchronizeinordertoperformsomecoordinatedfunction.Thereisagrowingliteraturesuggestingthatlargegroupsofneuronsinthebrainsynchronizeoscilla-toryactivityinordertoachievecoherence.Thismaybeamechanismforbindingdisparateaspectsofcognitivefunctionintoawhole͑Singer,2001͒,aswewilldiscussinSec.III.D.However,itismorepersuasivetoexamineintercircuitcoordinationinmotorcircuitswherethephasesofdifferentsegmentsorlimbsactuallycontrolmovements.Forexample,thepyloricandgastriccircuitscanbecoordinatedinthecrustaceanstomatogastricsys-tembyahigher-levelmodulatoryneuronthatchannelsthefasterpyloricrhythmtoakeycellinthegastricmillrhythm͑Fig.12͑͒Bartos.IncrabandstomatogastricNushbaum,1997MCs,;Bartosthegastricetal.,1999mill͒cyclehasaperiodofapproximately10swhilethepy-loricperiodisapproximately1s.Whenanidentiﬁedmodulatoryprojectionneuron͑MCN1͓͒Fig.12͑a͔͒isac-tivated,thegastricmillpatternislargelycontrolledby

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FIG.12.͑a͒SchematiccircuitdiagramunderlyingMCN1acti-vationofthegastricmillrhythmofacrustacean.Thecircuitrepresentstwophasesoftherhythm,retraction͑left͒andpro-traction͑right͒.Lighterlinesrepresentinactiveconnections.LG,Int1,andDGaremembersofthegastricCPGandABandPDaremembersofthepyloricCPG.ArrowsrepresentfunctionaltransmissionpathwaysfromtheMCN1neuron.Barsareexcitatoryanddotsareinhibitory.͑b͒Thegastricmillcycleperiod;thetimingofeachcycleisafunctionofthepy-loricrhythmfrequency.Withthepyloricrhythmturnedoff,thegastricrhythmcyclesslowly͑LG͒.ReplacingtheABinhibitionofInt1withcurrentintoLGusingadynamicclampreducesthegastricmillcycleperiod.ModiﬁedfromBarotsetal.,1999.

interactionsbetweenMCN1andgastricneuronsLGandInt1͑Bartosetal.,1999͒.WhenInt1isstimulated,theABtoLGsynapse͓seeFig.12͑b͔͒playsamajorroleindeterminingthegastriccycleperiodandcoordinationbetweenthetworhythms.Thetworhythmsbecomeco-ordinatedbecauseLGburstonsetoccurswithaconstantlatencyaftertheonsetofthetriggeringpyloricinput.Theseresultssuggestthatintercircuitsynapsescanen-ableanoscillatorycircuittocontrolthespeedofasloweroscillatorycircuitaswellasprovideamechanismforintercircuitcoordination͑Bartosetal.,1999͒.

AnothertypeofintercircuitcouplingoccursamongsegmentalCPGs.Inthecrayﬁsh,abdominalappendages

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

calledswimmeretsbeatinametachronalrhythmfromposteriortoanteriorwithafrequency-independentphaselagofabout90°.Likemostrhythmsofthiskind,thephaselagmustremainconstantoverdifferentfre-quencies.IntheoreticalandexperimentalstudiesbyJonesetal.͑2003͒,itwasshownthatsuchphasecon-stancycouldbeachievedbyascendinganddescendingexcitatoryandinhibitorysynapses,iftherightconnec-tionsweremade.ItappearsrealistictolookatrhythmicMCsasrecurrentnetworkswithmanyintrinsicfeedbackconnectionssothattheinformationonacompletespa-tiotemporalpatterniscontainedinthelong-termactiv-ityofjustafewneuronsinthecircuit.ThenumberofintercircuitconnectionsnecessaryforcoordinationoftherhythmsisthereforemuchsmallerthanthetotalnumberofneuronsintheMC.

Toinvestigatecoordinatingtwoelementsofapopula-tionofneurons,onemayinvestigatehowvariouscou-plings,implementedinadynamicalclamp,mightoper-ateinthecooperativebehavioroftwopyloricCPGs.Thisisahybridandsimpliﬁedmodelofthemorecom-plexinterplaybetweenbrainareaswhosecoordinatedactivitymightbeusedtoachievevariousfunctions.Wenowdescribesuchasetofexperiments.

ArtiﬁciallyconnectingneuronsfromthepyloricCPGoftwodifferentanimalsusingadynamicclampcouldleadtodifferentkindsofcoordinationdependingonwhichneuronsareconnectedandwhatkindofsynapsesareused͑Szücsetal.,2004͒.Connectingthepacemakergroupwithelectricalsynapsescouldachievesame-phasesynchrony;connectingthemwithinhibitorysynapsesprovidedmuchbettercoordinationbutoutofphase.Thetwopyloriccircuits͑Fig.13͒arerepresentativeofcircuitsdrivenbycoupledpacemakerneuronsthatcom-municatewitheachotherviabothgradedandconven-tionalchemicalinteractions.ButwhiletheunitCPGpatternisformedinthisway,coordinatingﬁbersmustusespike-mediatedpostsynapticpotentialsonly.Itthereforebecomesimportanttoknowwhereinthecir-cuittoinputtheseconnectionsinordertoachievemaxi-mumeffectivenessintermsofcoordinatingtheentirecircuitandensuringphaseconstancyatdifferentfre-quencies.SimplycouplingthePDstogetherelectricallyisratherineffectivealthoughthebursts͑notspikes͒dosynchronizecompletelyevenathighcouplingstrengths.ThefactthatthetwoPDsareusuallyrunningatslightlydifferentfrequenciesleadstoboutsofchaosinthetwoneurons,i.e.,areductioninregularity.Moreeffectivesynchronizationoccurswhenthepacemakergroupsarelinkedtogetherwithmoderatelystrongreciprocalin-hibitorysynapsesintheclassichalf-centerconﬁguration.BurstsintwoCPGsareofcourse180°outofphase,butthefrequenciesarevirtuallyidentical.Thebestin-phasesynchronizationisobtainedwhentheLPsarecoupledtothecontralateralPDswithinhibitorysynapses͑Fig.13͒.

D.Chaosandadaptability

Overthepastdecadestherehavebeenmanyreportsoftheobservationofchaosintheanalysisofvarious

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FIG.13.CouplingoftwobiologicalpyloricCPGsPyl1andPyl2bymeansofdynamicclampartiﬁcialinhibitorysynapses.ThedynamicclampisindicatedbyDCL.ReciprocalinhibitorycouplingbetweenthepacemakergroupsABandPDleadstoantiphasesynchronizationwhilenonreciprocalcouplingfromtheLPsproducesin-phasesynchronization.FigureprovidedbyA.Szücs.

timecoursesofdatafromavarietyofneuralsystemsrangingfromthesimpletothecomplex͑Glass,1995;KornandFaure,2003͒.Perhapstheoutstandingfeatureoftheseobservationsisnotthepresenceofchaosbuttheappearanceoflow-dimensionaldynamicalsystemsastheoriginofspectrallybroadband,nonperiodicsig-nalsobservedinmanyinstances͑RabinovichandAbar-banel,1998͒.Allchaoticoscillationsoccurinaboundedstate-spaceregionofthesystem.Thisstatespaceiscap-turedbythemultivariatetimecourseofthevectorofdynamicaldegreesoffreedomassociatedwithneuralspikegeneration.Thesedegreesoffreedomarecom-prisedofthemembranevoltageandthecharacteristicsofthevariousioncurrentsinthecell.Usingnonlineardynamicaltoolsonecanreconstructamathematicallyfaithfulproxystatespacefortheneuronbyusingthemembranevoltageanditstime-delayedvaluesascoor-dinatesforthestatespace͑seeFig.14͒.

Chaosseemstobealmostunavoidableinnaturalsys-temscomprisedofnumeroussimpleorslightlycomplexsubsystems.Aslongastherearethreeormoredimen-sions,chaoticmotionsaregenericinthebroadmath-ematicalsense.Soneuronsaredealtachaotichandbynatureandmayhavelittlechoicebuttoworkwithit.Acceptingthatchaosismoreorlesstheonlychoice,wecanaskwhatbeneﬁtsaccruetotherobustnessandadaptabilityofneuralactivity.

Chaositselfmaynotbeessentialforlivingsystems.However,themultitudeofregularregimesofoperationthatcanbeaccomplishedindynamicalsystemscom-posedofelementswhichthemselvescanbechaoticgivesrisetoabasicprinciplethatnaturemayusefortheor-Rev.Mod.Phys.,Vol.78,No.4,October–December2006

FIG.14.Chaoticspiking-burstingactivityofisolatedCPGneutrons.Toppanel:ChaoticmembranepotentialtimeseriesofasynapticallyisolatedLPneuronfromthepyloricCPG.Bottompanel:State-spaceattractorreconstructedfromthevoltagemeasurementsoftheLPneuronshowninthetoppanelusingdelayedcoordinates͓x͑t͒,y͑t͒,z͑t͔͒=͓V͑t͒,V͑t−T͒,V͑t−2T͔͒.ThisattractorischaracterizedbytwopositiveLyapunovexponents.ModiﬁedfromRabinovichandAbar-banel,1998.

ganizationofneuralassemblies.Inotherwords,chaosisnotresponsiblefortheworkofvariousneuralstruc-tures,butratherforthefactthatthosestructuresfunc-tionattheedgeofinstability,andoftenbeyondit.Byrecognizingchaoticmotionsinasystemstatespaceasunstable,butbounded,thisgeometricnotiongivescre-dencetotheotherwiseunappealingideaofsysteminsta-bility.Theinstabilityinherentinchaoticmotions,ormorepreciselyinnonlineardynamicsofsystemswithchaos,facilitatestheextraordinaryabilityofneuralsys-temstoadapt,maketransitionsfromonepatternofbe-haviortoanotherwhentheenvironmentisaltered,andconsequentlycreatearichvarietyofpatterns.Thuschaosgivesameanstoexploretheopportunitiesavail-abletothesystemwhentheenvironmentchanges,andactsasaprecursortoadaptive,reliable,androbustbe-haviorforlivingsystems.

Throughoutevolutionneuralsystemshavedevelopeddifferentmethodsofself-controlorself-organization.Ontheonehand,suchmethodspreservealladvantagesofthecomplexbehaviorofindividualneurons,suchasallowingregulationofthetimeperiodoftransitionsbe-tweenoperatingregimes,aswellasregulationoftheoperationfrequencyinanygivenregime.Theyalsopre-servethepossibilityofarichvarietyofperiodicandnonperiodicregimesofbehavior;seeFig.11andElsonetal.͑1988͒andVarona,Torres,Huerta,etal.͑2001͒.Ontheotherhand,thesecontrolororganizationaltech-niquesprovidetheneededpredictabilityofbehavioralpatternsinneuralassemblies.

Organizingchaoticneuronsthroughappropriatewir-ingassociatedwithelectrical,inhibitory,andexcitatoryconnectionsappearstoallowforessentiallyregularop-erationofsuchanassembly͑Huertaetal.,2001͒.Asanexamplewementionthedynamicsofanartiﬁcialmicro-

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FIG.15.AverageburstingperiodofthemodelheartbeatCPGactivityasafunctionoftheinhibitorycoupling⑀.ModiﬁedfromMalkovetal.,1996.

circuitthatmimicstheleechheartbeatCPG͑Calabreseetal.,1995͒.ThisCPGmodelconsistsofsixchaoticneu-ronsimplementedwithHindmarsh-Roseequationsre-ciprocallycoupledtotheirneighborsthroughinhibitorysynapses.Themodelingshowedthatinspiteofchaoticoscillationsofindividualneuronsthecooperativedy-namicsisregularand,mostimportantly,theperiodofburstingofthecooperativedynamicssensitivelyde-pendsonthevaluesoftheinhibitoryconnections͑Malkovetal.,1996͒͑seeFig.15͒.Thisexampleshowsthehighlevelofadaptabilityofanetworkconsistingofchaoticelements.

Chaoticsignalshavemanyofthetraditionalcharac-teristicsattributedtonoise.Inthepresentcontextwerecognizethatbothchaosandnoiseareabletoorganizetheirregularbehaviorofindividualneuronsorneuralassemblies,buttheprincipaldifferenceisthatdynamicalchaosisacontrollableirregularity,possessingstructureinstatespace,whilenoiseisanuncontrollableactionofdynamicalsystems.Thisdistinctionisextremelyimpor-tantforinformationprocessingasdiscussedbelow͑seeSec.III.B.2anditsﬁnalremarks͒.

Thereareseveralpossiblefunctionsfornoise͑Lind-neretal.,2004͒,evenseenashigh-dimensionalessen-tiallyunpredictablechaoticmotion,inneuralnetworkstudies.Inhigh-dimensionalsystemscomposedhereofmanycouplednonlinearoscillators,theremaybesmallbasinsofattractionwhere,inprinciple,thesystemcouldbecometrapped.Noisewillblurthebasinboundariesandremovethepossibilitythatthemainattractorscouldaccidentallybemissedandhighlyfunctionalsynchro-nizedstateslosttoneuronalactivity.Somenoisemaypersistinthedynamicsofneuronstosmoothouttheactionsofthechaoticdynamicsactiveincreatingrobust,adaptablenetworks.

Chaosshouldnotbemistakenfornoise,astheformerhasphase-spacestructurewhichcanbeutilizedforsyn-chronization,transmissionofinformation,andregular-izationofthenetworkforperformanceofcriticalfunc-tions.Inthenextsectionwediscusstheroleofchaosininformationprocessingandinformationcreation.

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III.INFORMATIONALNEURODYNAMICS

Theﬂowofinformationinthebraingoesfromsen-sorysystems,whereitiscapturedandencoded,tocen-tralnervoussystems,whereitisfurtherprocessedtogenerateresponsesignals.Inthecentralnervoussystemcommandsignalsaregeneratedandtransportedtothemusclestoproducemotorbehavior.Atallthesestageslearningandmemoryprocessesthatneedspeciﬁcrepre-sentationstakeplace.Thusitisnotsurprisingthatthenervoussystemhastousedifferentcodingstrategiesatdifferentlevelsofthetransport,storage,anduseofin-formation.Differenttransformationsofcodeshavebeenproposedfortheanalysisofspikingactivityinthebrain.Thedetailsdependontheparticularsystemunderstudybutsomegeneralizationispossibleintheframeworkofanalyzingthespatial,temporal,andspatiotemporalcodes.Therearemanyunknownfactorsrelatedtothecooperationbetweenthesedifferentformsofinforma-tioncoding.Somekeyquestionsareasfollows:͑i͒Howcanneuralsignalsbetransformedfromonecodingspacetoanotherwithoutlossofinformation?͑ii͒Whatdy-namicalmechanismsareresponsibleforstoringtimeinmemory?͑iii͒Canneuralsystemsgeneratenewinforma-tionbasedontheirsensoryinputs?Inthissection,wediscusssomeimportantexperimentalresultsandnewparadigmsthatcanhelptoaddressthesequestions.

A.Timeandneuralcodes

Informationfromsensorysystemsarrivesatsensoryneuronsasanalogchangesinlightintensityortempera-ture,orchemicalconcentrationofanodorant,orskinpressure,etc.Theseanalogdataarerepresentedinin-ternalneuralcircuitdynamicsandcomputationsbyaction-potentialsequencespassedfromsensoryreceiv-erstohigher-orderbrainprocesses.Neuralcodesguar-anteetheefﬁciency,reliability,androbustnessofthere-quiredneuralcomputations͑Machens,Gollisch,etal.,2005͒.

1.Temporalcodes

Twoofthecentralquestionsinunderstandingthedy-namicsofinformationprocessinginthenervoussystemarehowinformationisencodedandhowthecodingspacedependsontime-dependentstimuli.

Acodeinthebiophysicalcontextofthenervoussys-temisaspeciﬁcrepresentationoftheinformationoper-atedonorcreatedbyneurons.Acoderequiresaunitofinformation.However,thisisalreadyacontroversialis-suesince,aswehavepreviouslydiscussed,informationisconveyedthroughchemicalandelectricalsynapses,neuromodulators,hormones,etc.,whichmakesitdifﬁ-culttopointoutasingleuniversalunitofinformation.Aclassicalassumptionatthecellularlevel,validformanyneuralsystems,isthataspikeisanall-or-nothingeventandthusagoodcandidateforaunitofinformation,atleastinacomputationalsense.Thisisnottheonlysim-

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FIG.16.͑Coloronline͒Twopossiblecodesfortheactivityofasingleneuron.Inaratecode,differentinputs͑A–D͒aretrans-formedintodifferentoutputspikingrates.Inatimingcode,differentinputsaretransformedintodifferentspikingse-quenceswithprecisetiming.

pliﬁcationneededtoanalyzeneuralcodesforaﬁrstap-proach.Acodingschemeneedstodetermineacodingspaceandtakeintoaccounttime.

Acommonhypothesisistoconsiderauniversaltimeforallneuralelements.Althoughthisistheapproachwediscusshere,weremindthereaderthatthisisalsoanarguableissue,sinceneuronscansensetimeinmanydifferentways:bytheirintrinsicactivity͑subcellulardy-namics͒orbyexternalinput͑synapticandnetworkdy-namics͒.Internalandexternal͑network͒clocksarenotnecessarilysynchronizedandcanhavedifferentdegreesofprecision,timescales,andabsolutereferences.Somedynamicalmechanismscancontributetomakeneuraltimeuniﬁedandcoherent.

Ontheonehand,whenweconsiderjustasingleneu-ron,aspikeastheunitofinformation,andauniversaltime,wecantalkabouttwodifferenttypesofencoding:thefrequencyofﬁringcanencodeinformationaboutthestimulusinaratecode;ontheotherhand,theexacttemporaloccurrenceofspikescanencodethestimulusanditsresponseinaprecisetimingcode.Thetwocod-ingalternativesareschematicallyrepresentedinFig.16.Inthiscontext,aprecisetimingortemporalcodeisacodeinwhichrelativespiketimings͑ratherthanspikecounts͒areessentialforinformationprocessing.Severalexperimentalrecordingshaveshownthepresenceofboth͑typesofsingle-cellcodinginthe1991Adrian;McClurkinandZotterman,etal.,19911926;Softky,;Barlow,nervous19951972;Shadlen;Abeles,systemandNewsome,1998͒.Inparticular,ﬁnetemporalprecisionandreliabilityofspikedynamicsarereportedinmanycelltypes͑SegundoandPerkel,1969;MainenandSejnowski,1995;deCharmsandMerzenich,1996;deRytervanStevenincketal.,1997;Segundoetal.,1998;Mehtaetal.,2002;ReinagelandReid,2002͒.Singleneu-ronscandisplaythesetwocodesindifferentsituations.

2.Spatiotemporalcodes

Apopulationofcoupledneuronscanhaveacodingschemedifferentfromthesumoftheindividualcodingmechanisms.Interactionsamongneuronsthroughtheirsynapticconnections,i.e.,theircooperativedynamics,al-lowformorecomplexcodingparadigms.Thereismuchexperimentalevidencewhichshowstheexistenceofso-calledpopulationcodesthatcollectivelyexpressacom-plexstimulusbetterthantheindividualneurons͓see,

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e.g.,Georgopoulusetal.͑1986͒;WilsonandMcNaugh-ton͑1993͒;Fitzpatricketal.͑1997͒;Pougetetal.͑2000͔͒.Theefﬁcacyofpopulationcodinghasbeenassessedmainlyusingmeasuresofmutualinformationinmodel-ingefforts͑SeungandSompolinsky,1993;Panzerietal.,1999;Sompolinskyetal.,2001͒.

Twoelementscanbeusedtobuildpopulationcodes:neuronalidentity͑i.e.,neuronalspace͒andthetimeoc-currenceofneuralevents͑i.e.,thespikes͒.Accordingly,informationaboutthephysicalworldcanbeencodedintemporalorspatial͑combinatorial͒codes,orcombina-tionsofthesetwo:spiketimecanrepresentphysicaltime͑apuretemporalcode͒,spiketimecanrepresentphysicalspace,neuronalspacecanrepresentphysicaltime͑apurespatialcode͒,andneuronalspacecanrep-resentphysicalspace͑Nádasdy,2000͒.Whenwecon-siderapopulationofneurons,informationcodescanbespatial,temporal,orspatiotemporal.

Populationcodingcanalsobecharacterizedasinde-pendentorcorrelated͑deCharmsandChristopher,1998͒.Inanindependentcode,eachneuronrepresentsaseparatesignal:allinformationthatisobtainablefromasingleneuroncanbeobtainedfromthatneuronalone,withoutreferencetotheactivitiesofotherneurons.Foracorrelatedorcoordinatedcodingmessagesarecarriedatleastinpartbytherelativetimingofthesignalsfromapopulationofneurons.

Thepresenceofnetworkcoding,i.e.,aspatiotemporaldynamicalrepresentationofincomingmessages,hasbeenconﬁrmedinseveralexperiments.Asanexample,wediscussherethespatiotemporalrepresentationofepisodicexperiencesinthehippocampus͑Linetal.,2005͒.Individualhippocampalneuronsrespondtoawidevarietyofexternalstimuli͑WilsonandMcNaugh-ton,1994;Dragoietal.,2003͒.Theresponsevariabilityatthelevelofindividualneuronsposesanobstacletotheunderstandingofhowthebrainachievesitsrobustreal-timeneuralcodingofthestimulus͑Lestienne,2001͒.Reliableencodingofsensoryorothernetworkinputsbyspatiotemporalpatternsresultingfromthedy-namicalinteractionofmanyneuronsundertheactionofthestimuluscansolvethisproblem͑HamiltonandKauer,1985;Laurent,1996;Vaadiaetal.,1999͒.

Linetal.͑2005͒showedthatmnemonicshort-timeepisodes͑aformofone-triallearning͒cantriggerﬁringchangesinasetofCA1hippocampalneuronswithspe-ciﬁcspatiotemporalrelationships.Toﬁndsuchrepresen-tationsinthecentralnervoussystemofananimalisanextremelydifﬁcultexperimentalandcomputationalproblem.Becausetheindividualneuronsthatpartici-pateintherepresentationofaspeciﬁcstimulusandformatemporalneuralclusterindifferenttrialscanbediffer-ent,itisnecessarytomeasuresimultaneouslytheactiv-ityofalargenumberofneurons.Inaddition,becauseofthevariabilityintheindividualneuronresponses,thespatiotemporalpatternsofdifferenttrialsmayalsolookdifferent.Thus,toshowthefunctionalimportanceofthespatiotemporalrepresentationofthestimulus,thereaderhastousesophisticatedmethodsofdataanalysis.Linetal.͑2005͒developeda96-channelarraytorecord

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FIG.17.͑Coloronline͒Tempo-raldynamicsofindividualCA1neuronsofthehippocampusinresponseto“startling”events.Spikerasterplots͓͑a͒–͑d͒up-per,sevenrepetitionseach͔andcorrespondingperieventhisto-gram͓͑a͒–͑d͒lower,binwidth500ms͔forunitsexhibitingthefourmajortypesofﬁringchangesobserved:͑a͒transientincrease,͑b͒prolongedin-crease,͑c͒transientdecrease,͑d͒andprolongeddecrease.FromLinetal.,2005.

simultaneouslytheactivitypatternsofasmanyas260individualneuronsinthemousehippocampusduringvariousstartlingepisodes͑airblow,elevatordrop,andearthquakeshake͒.Theyusedmultiple-discriminantanalysis͑Dudaetal.,2001͒andshowedthat,eventhoughindividualneuronsexpressdifferenttemporalpatternsindifferenttrials͑seeFig.17͒,itispossibletoidentifyfunctionalencodingunitsintheCA1neuronassembly͑seeFig.18͒.

Therepresentationofnonstationarysensoryinforma-tion,say,avisualstimulus,canusethetransformationofatemporaltoaspatialcode.Therecognitionofaspe-ciﬁcneuralfeaturecanbeimplementedthroughthetransformationofaspatialcodeintoatemporalonethroughcoincidencedetectionofspikes.Aspatialrep-resentationcanbetransformedintoaspatiotemporalonetoprovidethesystemwithhighercapacityandro-bustnessandsensitivityatthesametime.Finally,aspa-tiotemporalcodecanbetransformedintoaspatialcodeinprocessesrelatedtolearningandmemory.Thesepos-sibilitiesaresummarizedinFig.19.

Morphologicalconstraintsofneuralconnectionsinsomecasesimposeaparticularspatialortemporalcode.Forexample,projectionneuronstransferinformationbetweenareasofthebrainalongparallelpathwaysbypreservingtheinputtopographyasneuronalspeciﬁcityattheoutput.Inmanycasestheinputtopographyistransformedtoadifferenttopographythatispreserved;forexample,theretinotopicmapoftheprimaryvisualareasandsomatotopicmapsofthesomatosensoryandmotorareas.Othertransformationsdonotpreserveto-pology.Theseincludetransformationsinplacecellsinthehippocampus,andthetonotopicrepresentationintheauditorycortex.Thereisahighdegreeofconver-genceanddivergenceofprojectionsinsomeofthesetransformationsthatcanbeacomputationallyoptimaldesign͑Garcia-SanchezandHuerta,2003͒.Inmostofthesetransformations,thetemporaldimensionofthestimulusisencodedbyspiketimingorbytheonsetofﬁring-ratetransients.

Anexampleoftransformingaspatiotemporalcodetoapurespatialcodewasfoundintheolfactorysystemof

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locusts,andhasbeenmodeledbyNowotny,Rabinovich,etal.͑2003͒andNowotnyetal.͑2005͒.Figure20givesagraphicalexplanationoftheconnectionsinvolved.Thecomplexspatiotemporalcodeofsequencesoftransientlysynchronizedgroupsofprojectionneuronsintheanten-nallobe͑Laurentetal.,2001͒isslicedintotemporalsnapshotsofactivitybyfeedforwardinhibitionandcoin-cidencedetectioninthenextprocessinglayer,themush-roombody͑Perez-Oriveetal.,2002͒.Thissnapshotcodeispresumablyintegratedovertimeinthenextstagesofthemushroomlobes,completingthetransformationofthespatiotemporalcodeintheantennallobetoapurelyspatialcode.Itwasshowninsimulationsthatthetem-poralinformationonthesequenceofactivityinthean-tennallobethatcouldbelostindownstreamtemporalintegrationcanberestoredthroughslowlateralexcita-tioninthemushroombody͑Nowotny,Rabinovich,etal.,2003͒.Thishasbeenreportedexperimentally͑LeitchandLaurent,1996͒.Withthisextrafeaturethetransfor-mationfromaspatiotemporalcodetoapurespatialcodebecomesfreeofinformationloss.

3.Coexistenceofcodes

Differentstagesofneuralinformationprocessingaredifﬁculttostudyinisolation.Inmanycasesitishardtodistinguishbetweenwhatisanencodingofaninputandwhatisastaticordynamic,perhapsnonlinear,responsetothatinput.Thisisacrucialobservationthatisoftenmissed.Encodinganddecodingmayormaynotbepartofadynamicalprocess.However,thecreationofinfor-mation͑discussedinthenextsection͒andthetransfor-mationofspatialcodestotemporalorspatiotemporalcodesarealwaysdynamicalprocesses.

Another,butlessfrequentlyaddressed,issueaboutcodingisthepresenceofmultipleencodingsinsingle-cellsignals͑Latorreetal.,2006͒.Thismayoccursincemultifunctionalnetworksmayneedmultiplecoexistingcodes.TheneuralsignaturesininterspikeintervalsofCPGneuronsprovideanexample͑Szücsetal.,2003͒.Individualﬁngerprintscharacteristicoftheactivityofeachneuroncoexistwiththeencodingofinformationin

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FIG.18.͑Coloronline͒Classiﬁcation,visualization,anddy-namicaldecodingofCA1ensemblerepresentationsofstartleepisodesbymultiple-discriminantanalysis͑MDA͒methods.͑a͒Firingpatternsduringrest,airblow,drop,andshakeep-ochsareshownafterbeingprojectedtoathree-dimensionalspaceobtainedbyusingMDAformouseA;MDA1–MDA3denotethediscriminantaxes.Bothtraining͑darksymbols͒andtestdataareshown.Aftertheidentiﬁcationofstartletypes,asubsequentMDAisfurtherusedtoresolvecontexts͑fullvsemptysymbols͒inwhichthestartleoccurredforair-blowcon-text͑b͒andforelevatordrop͑c͒.͑d͒Dynamicalmonitoringofensembleactivityandthespontaneousreactivationofstartlerepresentations.Three-dimensionalsubspacetrajectoriesofthepopulationactivityinthetwominutesafteranair-blowstartleinmouseAareshown.Theinitialresponsetoanairblow͑blackline͒isfollowedbytwolargespontaneousexcur-sions͑blue/darkandred/lightlines͒,characterizedbycoplanar,geometricallysimilarlower-amplitudetrajectories͑directional-ityindicatedbyarrows͒.͑e͒Thesametrajectoriesasin͑a͒fromadifferent3Dangle.͑f͒Thetiming͑t1=31.6sandt2=.8s͒ofthetworeactivations͑markedinblue/darkandred/light,respectively͒aftertheactualstartle͑inblack͒͑t=0s͒.Theverticalaxisindicatestheair-blowclassiﬁcationprobabil-ity.FromLinetal.,2005.

thefrequencyandphaseofthespiking-burstingrhythms.Thisisanexamplethatshowsthatcodescanbenonexclusive.Inburstingactivity,codingcanexistinslowwaves,butalso,andsimultaneously,inthespikingactivity.

Inthebrain,speciﬁcneuralpopulationsoftensendmessagesthroughprojectionstoseveralinformation“users.”Itisdifﬁculttoimaginethatallofthemdecodetheincomingsignalsinthesameway.Inneurosciencetherelationshipbetweentheencoderanddecoderisnotaone-to-onemapbutcanbemanysimultaneousmapsfromthesenderstodifferentreceivers,basedondiffer-entdynamics.ThisdepartsfromShannon’sclassical

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formulationofinformationtheory͑Fano,1961;Gallager,1968͒.Forexample,cochlearafferentsinbirdsbifurcatetotwodifferentareasofthebrainwithdifferentdecod-ingproperties.Oneareaextractsinformationaboutrelativetimingfromaspiketrain,whereastheotherex-tractstheaverageﬁringrate͑Konishi,1990͒.

4.Temporal-to-temporalinformationtransformation:Workingmemory

Thereisanotherimportantcodetransformationofin-teresthere:thetransformationofaﬁniteamountoftemporalinformationtoaslowtemporalcodelastingforseconds,minutes,orhours.Weareabletorememberaphonenumberfromsomeonewhojustcalledus.Persis-tentdynamicsisoneofthemechanismsforthisphenom-enon,whichisusuallynamedshort-termmemory͑STM͒orworkingmemory;itisabasicfunctionofthebrain.Workingmemory,incontrasttolong-termmemorywhichmostlikelyrequiresmolecular͑membrane͒orstructural͑connection͒changesinneuralcircuits,isadynamicalprocess.Thedynamicaloriginsofworkingmemorycanvary.

OneplausibleideaisthatSTMsaretheresultofac-tivereverberationininterconnectedneuralclustersthatﬁrepersistently.Sinceitsconceptualization͑deNó,1938;Hebb,1949͒,reverberatingactivityinmicrocircuitshasbeenexploredinmanymodelingpapers͑Grossberg,1973;AmitandBrunel,1997a;Durstewitzetal.,2000;Seungetal.,2000;Wang,2001͒.Experimentswithcul-turedneuronalnetworksshowthatreverberatoryactiv-itycanbeevokedincircuitsthathavenopreexistinganatomicalspecialization͑LauandBi,2005͒.Therever-berationisprimarilydrivenbyrecurrentsynapticexcita-tionratherthancomplexindividualneurondynamicssuchasbistability.Thecircuitrynecessaryforreverber-atingactivitycanbearesultofnetworkself-organization.Persistentreverberatoryactivitycanexisteveninthesimplestcircuit,i.e.,anexcitatoryneuronwithinhibitoryself-feedback͑Connors,2002;Egorovetal.,2002͒.Inthiscase,reverberationdependsonasyn-chronoustransmitterreleaseandintracellularcalciumstoresasshowninFig.21.

Natureseemstousedifferentdynamicalmechanismsforpersistentmicrocircuitactivity:cooperationofmanyinterconnectedneurons,persistentdynamicsofindi-vidualneurons,orboth.Thesemechanismseachhavedistinctadvantages.Forexample,networkmechanismscanbeturnedonandoffquickly͑McCormicketal.,2003͓͒seealsoBrunelandWang͑2001͔͒.Mostdynami-calmodelswithpersistentactivityarerelatedtotheanalysisofmicrocircuitswithlocalfeedbackexcitationbetweenprincipalneuronscontrolledbydisynapticfeedbackinhibition.Suchbasiccircuitsspontaneouslygeneratetwodifferentmodes:relativequiescenceandpersistentactivity.Thetriggeringbetweenmodesiscon-trolledbyincomingsignals.ThereviewbyBrunel͑2003͒considersseveralbasicmodelsofpersistentdynamics,includingbistablenetworkswithexcitationonlyandmultistablemodelsforworkingmemoryofadiscreteset

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FIG.19.͑Coloronline͒Sum-maryofpossiblescenariosforthetransformationofcodes,theirfunctionalimplications,andthedynamicalmechanisminvolved.

ofpictureswithstructuredexcitationandglobalinhibi-tion.

Workingmemoryisusedfortaskssuchasplanning,organizing,rehearsing,andmovementpreparation.Ex-perimentswithfunctionalmagneticresonanceimagingrevealsomeaspectsofthedynamicsofworkingmemory͓see,forexample,Diwadkaretal.͑2000͒andNystrometal.͑2000͔͒.Itisimportanttonotethatworkingmemoryhasalimitedcapacityofaroundfourtosevenitems͑Cowan,2001;VogelandMichizawa,2004͒.Anessentialfeatureattributedtoworkingmemoryisthelabileandtransientnatureofitsrepresentations.Becausesuchrepresentationsinvolvemanycoupledneuronsfromcor-ticalareas͑CurtsandD’Esposito,2003͒,itisnaturaltomodelworkingmemoryasthespatiotemporaldynamicsoflargeneuralnetworks.

Apopularideaistomodelworkingmemorywithat-tractors.Representationofitemsinworkingmemorybyattractorsmayguaranteeitsrobustness.Althoughro-bustnessisanimportantrequisiteforaworking-memorysystem,itstransientpropertiesarealsoimportant.Con-sideraforagingtaskinwhichananimalusesvisualinputtocatchprey͑NakaharaandDoya,1998͒.Itishelpfultostorethelocationofthepreyintheanimal’sworkingmemoryifthepreygoesbehindabushandthesensorycuebecomestemporarilyunavailable.However,thememoryshouldnotberetainedforeverbecausethepreymayhaveactuallygoneawayormayhavebeeneatenbyanotheranimal.Furthermore,ifmorepreyappearsneartheanimal,theanimalshouldquicklyloadthelocationofthenewpreyintoitsworkingmemorywithoutbeingdisturbedbytheoldmemory.

Thisexampleillustratesthattherearemorerequire-mentsforaworking-memorysystemthansolelyrobust

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maintenance.First,theactivityshouldbemaintainedbutnotfortoolong.Second,theactivityshouldberesetquicklywhenthereisanovelsensorycuethatneedstobestored.Inotherwords,theneuraldynamicsinvolvedinworkingmemoryforgoal-directedbehaviorsshouldhavethepropertiesoflong-termmaintenanceandquickswitching.Acorrespondingmodelbasedon“near-saddle-node”bifurcationdynamicshasbeensuggestedbyNakaharaandDoya͑1998͒.Theauthorshaveana-lyzedthedynamicsofanetworkofmodelneuralunitsthataredescribedbythefollowingmap͑seeFig.22͒:

yi͑tn+1͒=Fayi͑tn͒+b+͚␳ijyi͑tn͒+␥iIi͑tn͒,

j󰀅i

ͩͪ͑14͒

whereyi͑tn͒istheﬁringrateoftheithunitattimetn,F͑z͒=1/͓1+exp͑−z͔͒isasigmoidfunction,aistheself-connectionweight,␳ijarethelateralconnectionweights,Ii͑t͒areexternalinputs,bisthebias,and␥iarecon-stantsusedtoscaletheinputsIi͑t͒.Asthebiasbisin-creased,thenumberofﬁxedpointschangessequentiallyfromonetotwo,three,two,andthenbacktoone.Asaddle-nodebifurcationoccurswhenthestabletransi-tioncurvey͑tn+1͒=F͑z͒istangenttotheﬁxedpointy͑tn+1͒=y͑tn͒͑seeFig.22͒.Justnearthesaddle-nodebi-furcationthesystemshowspersistentactivity.Thismeansthatitspendsalongtimeinthenarrowchannelbetweenthebisectrixandthesigmoidactivationcurveandthengoestotheﬁxedpointquickly.Suchdynamicalbehaviorremindsoneofthewell-knownintermittencyphenomenoninphysics͑LandauandLifshitz,1987͒.Be-causetheeffectofthesumofthelateralandexternalinputsinEq.͑14͒isequivalenttoachangeinthebias,themechanismmaysatisfytherequirementsofthedy-

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FIG.20.͑Color͒Illustrationofthetransformationoftemporalintospatialinformation.Ifacoincidencedetectionoccurs,thelocalexcitatoryconnectionsactivatetheneighborsoftheac-tiveneuron͑yellowneurons͒.CoincidencedetectionofinputisnowmoreprobableintheseactivatedneighborhoodsthaninotherKenyoncells͑KCs͒.Whichoftheneighborsmightﬁreaspike,however,dependsontheactivityoftheprojectionneu-rons͑PNs͒inthenextcycle.ItmightbeadifferentneuronforactivegroupBofPNs͑upperbranch͒thanforactivegroupC͑lowerbranch͒.InthiswaylocalsequencesofactiveKCsform.ThesedependontheidentityofactivePNs͑coincidencede-tection͒aswellasonthetemporalorderoftheiractivity͑ac-tivatedneighborhoods͒.ModiﬁedfromNowotny,Rabinovich,etal.,2003.

namicsofworkingmemoryforgoal-directedbehavior:long-termmaintenanceandquickswitching.

Anotherreasonablemodelforworkingmemorycon-sistsofcompetitivenetworkswithstimulus-dependentinhibitoryconnections͓asinEq.͑9͔͒.Oneoftheadvan-tagesofsuchamodelistheabilitytohavebothworkingmemoryandstimulusdiscrimination.Thisideawaspro-posedbyMachens,Romo,etal.͑2005͒inrelationtothefrontal-lobeneuralarchitecture.Thenetworkﬁrstper-ceivesthestimulus,thenholdsitintheworkingmemory,andﬁnallymakesadecisionbycomparingthatstimuluswithanotherone.Themodelintegratesbothworkingmemoryanddecisionmakingsincethenumberofstableﬁxedpointsandthesizeofthebasinsofattractorsarecontrolledbytheconnectionmatrix␳ij͑S͒whichde-pendsonthestimuliS.Theworking-memoryphasecor-respondstothebifurcationboundary,i.e.,␳model,thisij=␳ji=␳thestatespaceofthedynamicalphaseii.Inisrepresentedbyastablemanifoldcalleda“continuous

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FIG.21.͑Coloronline͒Reverberationcanbethedynamicaloriginforworkingmemoryinminimalcircuits.͑a͒Mostneu-ronsrespondtoexcitatorystimuli͓upwardstepsinthelinebelow͑c͔͒byspikingonlyaslongaseachstimuluslasts.͑b͒Veryrareneuronsarebistable:briefexcitationleadstopersis-tentspiking,alwaysatthesamerate;briefinhibition͓down-wardstepsinthelinebelow͑c͔͒canturnitoff.͑c͒Multistableneuronspersistentlyincreaseordecreasetheirspikingacrossarangeofratesinresponsetorepeatedbriefstimuli.͑d͒Inthereverberatorynetworkmodelofshort-termmemorydiscussedinthetext,anexcitatorystimulus͑leftarrow͒leadstorecur-siveactivityininterconnectedneurons.Inhibitorystimuli͑barontheright͒canhalttheactivity.͑e͒Egorovetal.͑2002͒sug-gestthatgradedpersistentactivityinsingleneurons͓asin͑c͔͒mightbetriggeredbyapulseofinternalCa2+ionsthatenterthroughvoltage-gatedchannels;Ca2+thenactivatescalcium-dependentnonspeciﬁccation͑CAN͒channels,throughwhichaninwardcurrent͑largelycomprisingNa+ions͒enters,persis-tentlyexcitingtheneuron.Thepositivefeedbackloop͑brokenarrows͒mayincludetheactivityofmanyionicchannels.Modi-ﬁedfromConnors,2002.

attractor.”Thisisanattractorthatconsistsofcontinuoussetsofﬁxedpoints͓seeAmari͑1977͒andSeung͑1998͔͒.Thusthestimuluscreatesaspeciﬁcﬁxedpointand,atthenextstage,theworkingmemory͑acontinuousat-tractor͒maintainsit.Duringthecomparisonanddeci-sionphase,thesecondstimulusismappedontothesamestatespaceasanotherattractor.Thecriterionofthede-cisionmakerisreﬂectedinthepositionsofthesepara-tricesthatseparatethebasinsofattractionofdifferent

FIG.22.Temporalresponsesofself-recurrentunits:Near-saddle-nodebifurcationwitha=11.11,b=−7.9͑centerpanels͒.Increasedbias,b=−3.0͑leftpanels͒.Decreasedbiasb=−9.0͑rightpanels͒.ModiﬁedfromNakaharaandDoya,1998.

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FIG.23.HallucinationsgeneratedbyLSDareanexampleofadynamicalrepresentationoftheinternalactivityofthevisualcortexwithoutaninputstimulus.Figureshowsexamplesof͑a͒funneland͑b͒spiralhallucinations.ModiﬁedfromBressloff,etal.2001.

stimuli,i.e.,ﬁxedpoints͑seeanalternativeapproachinRabinovichandHuerta,2006͒.

Wethinkthattheintersectionofthemechanismsre-sponsibleforpersistentactivityofsingleneuronswiththeactivityofanetworkwithlocalornonlocalrecur-renceprovidesrobustnessagainstnoiseandperturba-tions,andatthesametimemakesworkingmemorymoreﬂexible.

B.Informationproductionandchaos

Informationprocessinginthenervoussysteminvolvesmorethantheencoding,transduction,andtransforma-tionofincominginformationtogenerateacorrespond-ingresponse.Inmanycases,neuralinformationiscre-atedbythejointactionofthestimulusandtheindividualneuronandnetworkdynamics.Acreativeac-tivitylikeimprovisationonthepianoorwritinganewpoemresultsinpartfromtheproductionofnewinfor-mation.Thisinformationisgeneratedbyneuralcircuitsinthebrainanddoesnotdirectlydependontheenvi-ronment.

Time-dependentvisualhallucinationsareoneex-ampleofinformationproducedbyneuralsystems,inthiscasethevisualcortex,themselves.Suchhallucina-tionsconsistinseeingsomethingthatisnotinthevisualﬁeld.Thereareinterestingmodels,beginningfromthepioneeringpaperofErmentroutandCowan͑1979͒,thatexplainhowtheintrinsiccircuitryofthebrain’svisualcortexcangeneratethepatternsofactivitythatunderliehallucinations.Thesehallucinationpatternsusuallytaketheformofcheckerboards,honeycombs,tunnels,spi-rals,andcobwebs͑seetwoexamplesinFig.23͒.Becausethevisualcortexisanexcitablemediumitispossibletousespatiotemporalamplitudeequationstodescribethedynamicsofthesepatterns͑seethenextsection͒.Thesemodelsarebasedonadvancesinbrainanatomyandphysiologythathaverevealedstrongshort-rangecon-nectionsandweakerlong-rangeconnectionsbetweenneuronsinthevisualcortex.Hallucinationpatternscanbequasistatic,periodicallyrepeatable,orchaoticallyre-peatableasinlow-dimensionalconvectiveturbulence;seeforareviewRabinovichetal.͑2000͒.Unpredictabil-ityofthespeciﬁcpatterninthehallucinationsequences

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FIG.24.͑Coloronline͒Dualsensorynetworkdynamics.Toppanels:Schematicrepresentationofthedualroleofasinglestatocyst,thegravitysensoryorganofthemolluskClione.Dur-ingnormalswimming,astonelikestructure,thestatolith,hitsthemechanoreceptorneuronsthatreacttothisexcitation.InClione’shuntingbehavior,thestatocystreceptorsreceiveaddi-tionalexcitationfromthecerebralhuntingneuron͑H͒whichgeneratesawinnerlesscompetitionamongthem.Bottompan-els:Chaoticsequentialswitchingdisplayedbytheactivityofthestatocystduringhuntingmodeinamodelofasix-receptornetwork.Thispaneldisplaysthetimeintervalsinwhicheachneuronisactive͑aiϾ0.03͒.Eachneuronisrepresentedbyadifferentcolor.Thedottedrectanglesindicatetheactivation-sequencelocksamongunitsthatareactiveatagiventimeintervalwithineachnetworkfortimewindowsinwhichallsixneuronsareactive.

͑principlemovie͒meanscanbethecharacterizedgenerationofbyinformationthevaluethatoftheinKolmogorov-Sinaientropy͑Scott,2004͒.

Thecreationorproductionofnewinformationisathemethathasbeenneglectedintheoreticalneuro-science,butitisaprovocativeandchallengingpointthatwediscussinthissection.Asmentionedbefore,infor-mationproductionorcreationmustbeadynamicalpro-cess.Belowwediscussanexamplethatemphasizestheabilityofneuralsystemstoproduceinformation-richoutputfrominformation-poorinput.

1.Stimulus-dependentmotordynamics

Asimplenetworkwithwhichwecandiscussthecre-ationofnewinformationisthegravity-sensingneuralnetworkofthemarinemolluskClionelimacina.Clioneisablindplanktonicanimal,negativelybuoyant,thathastomaintaincontinuousmotoractivityinordertokeepitspreferredhead-uporientation.ItsmotoractivityiscontrolledbywingCPGsandtailmotorneuronsthatuse͑modelPanchinsignalswithetfromsynaptical.,its1995gravity-sensinginhibition͒.Asix-receptororgans,thestatocystshasbeenbuiltneuraltonetworkdescribeasinglestatocyst͑Varona,Rabinovich,etal.,2002͒͑seeFig.24͒.Thisisasmallsphereinwhichastatolith,astonelikestructure,movesaccordingtothegravitational

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FIG.25.͑Coloronline͒Clioneswimmingtrajectoriesindiffer-entsituations.͑a͒Three-dimensionaltrajectoryofroutineswimming.Hereandinthefollowingﬁgures,differentcolors͑graytones͒areusedtoemphasizethethree-dimensionalper-ceptionofthetrajectoriesandchangeaccordingtothexaxis.Theindicatedtimetisthedurationofthetrajectory.͑b͒Swim-mingtrajectoryofClionewiththestatocystssurgicallyre-moved.͑c͒Trajectoryofswimmingduringhuntingbehaviorevokedbythecontactwiththeprey.͑d͒Trajectoryofswim-mingafterimmersionofClioneinasolutionthatpharmaco-logicallyevokeshunting.ModiﬁedfromLevietal.,2004.

ﬁeld.Thestatolithexcitestheneuroreceptorsbypress-ingdownonthem.Whenexcited,thereceptorssendsignalstotheneuralsystemsresponsibleforwingbeat-ingandtailorientation.

Thestatocystshaveadualrole͑Levietal.,2004,2005͒.Duringnormalswimmingonlyneuronsthatareexcitedbythestatolithareactive,andthisleadstoawinner-take-alldynamicalmodeasaresultofinhibitoryconnectionsinthenetwork.͑Winner-take-alldynamicsisessentiallythesameastheattractor-basedcomputa-tionalideasdiscussedearlier.͒However,whenClioneissearchingforitsfood,acerebralhuntingneuronexciteseachneuronofthestatocyst͑seeFig.24͒.Thistriggersacompetitionbetweenallstatocystneuronswhosesignalsparticipateinthegenerationofacomplexmotionthattheanimalusestoscantheimmediatespaceuntilitﬁndsitsprey͑Levietal.,2004,2005͒͑seeFig.25͒.Thefollow-ingLotka-Volterra-typedynamicscanbeusedtode-scribetheactivityͩofthisnetwork:

daN

i͑t͒

dt=ai͑t͒␴͑H,S͒−͚␳ijaj͑t͒+Hi͑t͒ͪ+Si͑t͒,j=1

͑15͒

whereai͑t͒ജ0representstheinstantaneousspikingrateofthestatocystneurons,Hstimulusfromthecerebralihunting͑t͒representsinterneurontheexcitatorytoneu-Rev.Mod.Phys.,Vol.78,No.4,October–December2006

FIG.26.͑Coloronline͒Irregularswitchinginanetworkofsixstatocystreceptors.Tracesrepresenttheinstantaneousspikingrateofeachneuronai͓neurons1,2,3areshownin͑a͒,neurons4,5,6in͑b͔͒.Notethatafteraneuronissilentforawhile,itsactivityreappearswiththesamesequencerelativetotheoth-ers͑seearrows,andFig.24͒.͑c͒Aprojectionofthephaseportraitofthestrangeattractorin3Dspace;seemodel͑15͒.

roni,Si͑t͒representstheactionofthestatolithonthereceptorthatitispressing,and␳connectionmatrix.Whenijisthenonsymmetricstatocystthereisnostimulusfromthehuntingneuron,Hi=0,orthestatolith,Si=0,then␴͑H,S͒=−1andallneuronsaresilent.WhenthehuntingneuronisactiveHi󰀅0󰀅and/or0,␴͑Hthe,S͒=statolith+1.

ispressingoneofthereceptors,SiDuringhuntingHthehuntingneuroni󰀅0,andweassumethattheactionofoverridestheeffectofthesta-tolithandthusSdisplayiϷ0.Asaresultofthecompetition,thereceptorsahighlyirregular,infactchaotic,switchingactivity.Thephase-spaceimageofthechaoticdynamicsofthestatocystmodelinthisbehavioralmodeisastrangeattractor͓theheteroclinicloopsinthephasespaceofEq.͑15͒becomeunstable;seeSec.IV.C͔.Forsixreceptorswehaveshown͑Varona,Rabinovich,etal.,2002͒thattheobserveddynamicalchaosischaracterizedbytwopositiveLyapunovexponents.

ThebottompanelinFig.24isanillustrationofthenonsteadyswitchingactivityofthereceptors.Aninter-estingphenomenoncanbeseeninthisﬁgureandisalsopointedoutinFig.26.Althoughthetimingofeachac-tivityisirregular,thesequenceofswitchingamongthestatocystreceptorsisthesameforthoseneuronsthatareactiveatagiventimewindow.DottedrectanglesinFig.24pointoutthisfact.Theactivation-sequencelock

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amongthestatocystreceptorneuronsemergesinspiteofthehighlyirregulartimingoftheswitchingdynamicsand͑VenailleisafeatureInthisetthatcanbeusedformotorcoordinationexampleal.,2005the͒.

winnerlesscompetitionistrig-geredbyaconstantexcitationtoallstatocystreceptors͓Hi=ci;seedetailsbyVarona,Rabinovich,etal.͑2002͔͒.Thusthestimulushaslowinformationcontent.None-theless,thenetworkofstatocystreceptorscanusethisactivitytogenerateaninformation-richsignalwithposi-tiveKolmogorov-Sinaientropy.Thisentropyisequaltothevalueofthenewinformationencodedinthedy-namicalmotion.Thestatocystsensorynetworkisthusmultifunctionalandcangenerateacomplexspatiotem-poralpatternusefulformotorcoordinationevenwhenitsdynamicsarenotevokedbygravity,asduringhunt-ing.

2.Chaosandinformationtransmission

Toillustratetheroleofchaosininformationtransmis-sion,weuseasanexampletheinferiorolive͑IO͒,whichisaninputsystemtothecerebellum.NeuronsoftheIOmaychaoticallyrecodethehigh-frequencyinformationcarried͑byitsinputsintochaotic,low-rateaSchweighofersystemthatcontrolsetal.,2004and͒.coordinatesTheIOhasbeendifferentproposedoutputrhythmsasthroughtheintrinsicoscillatorypropertiesofitsneuronsand͑alsoLlinásthebeenandnatureimplicatedWelsh,of1993theirinmotor;deelectricalZeeuwlearningetinterconnections͑al.Ito,,19981982͒͒.andIthasincomparingtasksofintendedandachievedmovementsasageneratoroferrorsignals͑Oscarsson,1980͒.

ExperimentalrecordingsshowthatIOcellsareelec-tricallycoupledanddisplaysubthresholdoscillationsandspikingactivity.Subthresholdoscillationshavearel-evantroleforinformationprocessinginthecontextofasystemwithextensiveelectricalcoupling.Insuchsys-temsthespikingactivitycanbepropagatedthroughthenetwork,and,inaddition,smalldifferencesinhyperpo-larizedmembranepotentialspropagateamongneigh-boringcells.

AmodelingstudysuggeststhatelectricalcouplinginIOneuronsmayinducechaos,whichwouldallowinformation-rich,butlow-ﬁring-rate,errorsignalstoreachindividualPurkinjecellsinthecerebellarcortex.Thiswouldprovidethecerebellarcortexwithessentialinformationforefﬁcientlearningwithoutdisturbingon-goingmotorcontrol.Thechaoticﬁringleadstothegen-erationofIOspikeswithdifferenttiming.BecausetheIOhasalowﬁringrate,anaccurateerrorsignalwillbeavailableforindividualPurkinjecellsonlyafterre-peatedtrials.ElectricalcouplingcanprovidethesourceofdisorderthatinducesachaoticresonanceintheIOnetwork͑Schweighoferetal.,2004͒.Thisresonanceleadstoanincreaseininformationtransmissionbydis-tributingthehigh-frequencycomponentsoftheerrorin-putsoverthesporadic,irregular,andnon-phase-lockedspikes.

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TheIOsingle-neuronmodelconsistsoftwocompart-mentsthatincludealow-thresholdcalciumcurrent͑ICal͒,ananomalousinwardrectiﬁercurrent͑Ih͒,aHodgkin-Huxley-typesodiumcurrent͑INa͒,andade-layedrectiﬁerpotassiumcurrent͑Icompartment͑seeTableI͒.ThedendriticKd͒incompartmentthesomaticcontainsacalcium-activatedpotassiumcurrent͑IKCa͒andahigh-thresholdcalciumcurrent͑IpartmentalsoreceiveselectricalconnectionsCah͒.Thiscom-fromneighboringneurons.Fastionicchannelsarelocatedinthesoma,andslowchannelsarelocatedinthedendriticcompartment.Someofthechannelconductancesde-pendonthecalciumconcentration.Theequationsforeachcompartmentofasingleneuroncanbesumma-rizedas

CdV͑t͒

=−͑Iion+Il+Iinj+Icomp͒,͑16͒

whereCMisthemembranecapacitance,Ilisaleakcur-rent,Iinjistheinjectedstimuluscurrent,Icompconnectsthecompartments,andIionisthesumofthecurrentsaboveforeachcompartment.Inaddition,thedendriticcompartmenthastheelectricalcouplingcurrentI͑t͔͒,wheretheindexirunsovertheneigh-ec=gc͚i͓V͑t͒−Viborsofeachneuron,andgcistheelectricalcouplingconductance.

EachIOneuronisrepresentedbyasystemofordi-narydifferentialequations͑ODEs͒,andthenetworkisasetofthesesystemscoupledthroughtheelectricalcou-plingcurrentsI3,andec.Thenetworksexaminedconsistedof2ϫ2,3ϫ9ϫ3neurons,wherecellsareconnectedtotheirtwo,three,orfourneighborsdependingontheirpositionsinthegrid.

Thisisacomplexnetwork,evenwhenitisonly2ϫ2,andonemustselectanimportantfeatureofthedynam-icstocharacterizeitsbehavior.ThelargestLyapunovexponentofthenetworkisagoodchoiceasitisinde-pendentofinitialconditionsandtellsusaboutinforma-tionﬂowinthenetwork.Figure27displaysthelargestLyapunovexponentforeachnetworkasafunctionoftheelectriccouplingconductancegc.WealsoseeinFig.27thatthegcproducingthelargestLyapunovexponentyieldsthelargestinformationtransferthroughthenet-work,evaluatedastheaveragemutualinformationperspike.

InamoregeneralframeworkthantheIO,itisre-markablethatthechaoticactivityofindividualneuronsunexpectedlyunderlieshigherﬂexibilityand,atthesametime,greateraccuracyandprecisionintheirneuraldynamics.Theoriginofthisphenomenonisthepoten-tialabilityofcoupledneuronswithchaoticbehaviortosynchronizetheiractivitiesandgeneraterhythmswhoseperioddependsonthestrengthofthecouplingorothernetworkparameters͓forareviewseeRabinovichandAbarbanel͑1998͒andAihara͑2002͔͒.Networkswithmanychaoticneuronscangenerateinterestingtransientdynamics,i.e.,chaoticitinerancy͑CI͒͑Tsuda,1991;Rowe,2002͒.Chaoticitinerancyresultsfromweakinsta-bilitiesintheattractors,i.e.,attractorsetsinwhose

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FIG.27.͑Coloronline͒Chaoticdynamicsincreasesinforma-tiontransmissioninIOmodels.Toppanel:LargestLyapunovexponentasafunctionoftheelectricalcouplingstrengthgcfordifferentIOnetworksofnonidenticalcells.Bottompanel:Net-workaveragemutualinformationperspikeasafunctionofgc.ModiﬁedfromSchweighoferetal.,2004.

neighborhoodtherearetrajectoriesthatdonotgototheattractors͑Milnor-typeattractors͒.AdevelopedCImo-tionneedsbothmanyneuronsandaveryhighlevelofinterconnections.Thisisincontrasttothetraditionalconceptofcomputationwithattractors͑Hopﬁeld,1982͒.Chaoticitinerancyyieldscomputationswithtransienttrajectories;inparticular,therecanbemotionalongseparatricesasinwinnerlesscompetitiondynamics͑Sec.IV.C͒.AlthoughCIisaninterestingphenomenon,ap-plyingittoexplainandpredicttheactivityofsensorysystems͑Kay,2003͒,andtoanynonautonomousneuralcircuitdynamics,posesaquestionthathasnotbeenan-sweredyet:HowcanCIbereproducibleandrobustagainstnoiseandatthesametimesensitivetoastimu-lus?

Toconcludethissectionitisnecessarytoemphasizethattheanswertothequestionofthefunctionalroleofchaosinrealneuralsystemsisstillunclear.Inspiteoftheattractivenessofsuchideasas͑i͒chaosmakesneuralcircuitsmoreﬂexibleandadaptive,͑ii͒chaoticdynamicscreateinformationandcanhelptostoreit͑seeabove͒,and͑iii͒thenonlineardynamicalanalysesofphysiologi-caldata͑e.g.,electroencephalogramtimeseries͒canbeimportantforthepredictionorcontrolofpathologicalneuralstates,itisextremelydifﬁculttoconﬁrmtheseideasdirectlyininvivooreveninvitroexperiments.Inparticular,therearethreeobstaclesthatcanfundamen-Rev.Mod.Phys.,Vol.78,No.4,October–December2006

tallyhinderthepowerofdataanalyses:͑i͒ﬁnitestatisti-calﬂuctuations,͑ii͒externalnoise,and͑iii͒nonstation-arityoftheneuralcircuitactivity͓see,forexample,Laietal.͑2003͔͒.

C.Synapticdynamicsandinformationprocessing

Synaptictransmissioninmanynetworksofthener-voussystemisdynamical,meaningthatthemagnitudeofpostsynapticresponsesdependsonthehistoryofpresynapticactivity͑ThompsonandDeuchars,1994;Fuhrmannetal.,2002͒.Thisphenomenonisindepen-dentof͑orinadditionto͒theplasticitymechanismsofthesynapses͑discussedinSec.II.A.3͒.Theroleofsyn-apsesisoftenconsideredtobethesimplenotiﬁcationtothepostsynapticneuronofpresynapticcellactivity.However,electrophysiologicalrecordingsshowthatsyn-aptictransmissioncanimplyactivity-dependentchangesinresponsetopresynapticspiketrains.Themagnitudeofpostsynapticpotentialscanchangerapidlyfromonespiketoanother,dependingontheparticulartemporaldistributionofthepresynapticsignals.Thuseachsinglepostsynapticresponsecanencodeinformationaboutthetemporalpropertiesofthepresynapticsignals.

Themagnitudeofthepostsynapticresponseisdeter-minedbytheinterspikeintervalsofthepresynapticac-tivityandbytheprobabilisticnatureofneurotransmitterrelease.Indepressingsynapsesashortintervalbetweenpresynapticspikesisfollowedbysmallpostsynapticre-sponses,whilelongpresynapticinterspikeintervalsarefollowedbyalargepostsynapticresponse.Facilitatingsynapsestendtogenerateresponsesthatgrowwithsuc-cessivepresynapticspikes.Inthiscontext,severaltheo-reticaleffortshavetriedtoexplorethecapacityofsingleresponsesofdynamicalsynapsestoencodetemporalin-formationaboutthetimingofpresynapticevents.

TheoreticalmodelsfordynamicalsynapsesarebasedonthetimevariationofthefractionofneurotransmitterreleasedfromthepresynapticterminalR͑t͒,0ഛR͑t͒ഛ1.Whenapresynapticspikeoccursattimetsp,thefractionUofavailableneurotransmittersandtherecov-erytimeconstant␶recdeterminetherateofreturnofresourcesR͑t͒totheavailablepresynapticpool.Inadepressingsynapse,Uand␶recareconstant.Asimplemodeldescribesthefractionofsynapticresourcesavail-ablefortransmissionas͑Fuhrmannetal.,2002͒

dR͑t͒1−R͑t͒

dt=

␶−UR͑t͒␦͑t−tsp͒,͑17͒

rec

andtheamplitudeofthepostsynapticresponseattimetspisproportionaltoR͑tsp͒.

Forafacilitatingsynapse,UbecomesafunctionoftimeU͑t͒increasingateachpresynapticspikeandde-cayingtothebaselinelevelwhenthereisnopresynapticactivity:

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FIG.28.Dynamicalsynapsesimplythatsynaptictransmissiondependsonpreviouspresynapticactivity.Thisshowstheaver-agepostsynapticactivitygeneratedinresponsetoapresynap-ticspiketrain͑bottomtrace͒inapyramidalneuron͑toptrace͒andinamodelofadepressingsynapse͑middletrace͒.Postsyn-apticpotentialinthemodeliscomputedusingapassivemem-branemechanism␶m͑dV/dt͒=−V+RiIsyn͑t͒,whereRiisthein-putresistance.ModiﬁedfromTsodyksandMarkram,1997.

dU͑t͒U͑t͒

dt=−

␶+U1͓1−U͑t͔͒␦͑t−tsp͒,͑18͒

facil

whereU1isaconstantdeterminingthestepincreaseinU͑t͒and␶facilistherelaxationtimeconstantofthefa-cilitation.

Otherapproachestomodelingdynamicalsynapsesin-cludeprobabilisticmodelstoaccountforﬂuctuationsinpresynapticreleaseofneurotransmitters.AtasynapticconnectionwithNreleasesiteswecanassumethatateachsitetherecanbe,atmost,onevesicleavailableforrelease,andthatthereleaseateachsiteisanindepen-dentevent.Whenapresynapticspikeisproducedattimetsp,eachsitecontainingavesiclewillreleaseitwiththesameprobabilityU͑t͒.Onceareleaseoccurs,thesitecanbereﬁlledduringatimeintervaldtwithprobabilitydt/␶rec.TheprobabilisticreleaseandrecoverycanbedescribedbytheprobabilityPv͑t͒foravesicletobeavailableforreleaseatanytimet:

dPv͑t͒1−Pv͑t͒

dt=

␶−U͑t͒Pv͑t͒␦͑t−tsp͒.͑19͒

rec

Figure28showshowthisformulationpermitsanaccu-ratedescriptionofadepressingsynapseinresponsetoaspeciﬁedpresynapticspiketrain.

Thetransmissionofsensoryinformationfromtheen-vironmenttodecisioncentersthroughneuralcommuni-cationchannelsrequiresahighdegreeofreliabilityand

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sensitivityfromnetworksofheterogeneous,inaccurate,andsometimesunreliablecomponents.Thepropertiesofthechannelitself,assumingthesensorisaccurate,mustbericherthanconventionalchannelsstudiedinen-gineeringapplications.Thosechannelsarepassiveand,whenofhighquality,canrelayinputsaccuratelytoareceiver.Neuralcommunicationchannelsarecomposedofdynamicallyactiveelementscapableofcomplexau-tonomousoscillations.Individually,chaoticneuronscancreateinformationinawaysimilartothestudyofnon-linearsystemswithunstabletrajectories:Twostatesofthesystem,indistinguishablebecauseonlyﬁnite-resolutionobservationscanoccur,maythroughtheac-tionoftheinstabilitiesofthenonlineardynamicsﬁndthemselvesinthefuturewidelyseparatedinstatespace,andthusdistinguishable.Informationaboutdifferentstatesthatwasunavailableatonetimemaybecomeavailableatalatertime.

Biologicalneuralcommunicationpathwaysareabletorecoverinformationfromahiddencodingspaceandtotransferinformationfromonetimescaletoanotherbe-causeoftheintrinsicnonlineardynamicsofsynapses.Asanexample,wediscussaverysimpleneuralinformationchannelcomposedofsensoryinputintheformofaspiketrainthatarrivesatamodelneuronandthenmovesthrougharealisticdynamicalsynapsetoasecondneuronwheretheinformationintheinitialsensorysig-nalisread͑Eguiaetal.,2000͒.Themodelneuronsarefour-dimensionalgeneralizationsoftheHindmarsh-Roseneuron,andamodelofchemicalsynapsederivedfromﬁrst-orderkineticsisused.Thefour-dimensionalmodelneuronhasarichvarietyofdynamicalbehaviors,includingperiodicbursting,chaoticbursting,continuousspiking,andmultistability.Formanyoftheseregimes,theparametersofthechemicalsynapsecanbetunedsothattheinformationaboutthestimulus,whichisun-readabletotheﬁrstneuroninthepath,canberecov-eredbythedynamicalactivityofthesynapse,andthesecondneuroncanreadit͑seeFig.29͒.

Thequantitativedescriptionofthisunexpectedphe-nomenonwasdonebycalculatingtheaveragemutualinformationI͑S,N1͒betweenthestimulusSandthere-sponseoftheﬁrstneuronN1,andI͑S,N2͒betweenthestimulusandtheresponseofthesecondneuronN2.TheresultintheexampleshowninFig.29isI͑S,NϾandI͑Sneurons,N2͒1͒.Thisactingresultasindicatesinputandhowoutputnonlinearsystemssynapsesalongacommunicationchannelcanrecoverinformationappar-entlyhiddeninearliersynapticconnectionsinthepath-way.Herethemeasureofinformationtransmissionusedistheaveragemutualinformationbetweenelements,andbecausethechannelisactiveandnonlinear,theav-eragemutualinformationbetweenthesensorysourceandtheﬁnalneuronmaybegreaterthantheaveragemutualinformationfoundinanintermediateneuroninthechannel͑butnotgreaterthantheoriginalinforma-tion͒.

Anotherformofsynapticdynamicsinvolvedininfor-mation͑alreadyprocessingdiscussedinandSec.especiallyII.A.3͒.Informationinlearningtransduc-

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FIG.29.Exampleofrecoveryofhiddeninformationinneuralchannels.Apresynapticcellreceivesspeciﬁedinputandcon-nectstoapostsynapticcellthroughadynamicalsynapse.Toppanel:thetimeseriesofsynapticinputtothepresynapticcellJ1͑t͒;middlepanel:themembranepotentialoftheﬁrstburst-ingneuronX1͑t͒;bottompanel:.NotethemembranethatfeaturespotentialoftheofinputthesecondburstingneuronX2͑t͒hiddenintheresponseX1͑t͒arerecoveredintheresponsefollowingadynamicalsynapseX2͑t͒͑notehyperpolarizationregionsforX2͒.ModiﬁedfromEguiaetal.,2000.

tionisinﬂuencedbySTDP͑Chechik,2003;HopﬁeldandBrody,2004͒,whichalsoplaysanimportantroleinbind-ingandsynchronization.

D.Bindingandsynchronization

Wehavediscussedthediversityofneurontypesandthevariabilityofneuralactivity.Neuralprocessingre-quiresthefastinteractionofmanyneuronsindifferentneuralsubsystems.Thereareseveraldynamicalmecha-nismscontributingtothecomplexintegrationofinfor-mationthatneuralsystemsperform.Amongthem,thesynchronizationofneuralactivityistheonethathascapturedthemostattention.Synchronizationofneuralactivityisalsooneoftheproposedsolutionstoawidelydiscussedquestioninneuroscience:thebindingprob-lem,whichwedescribebrieﬂyinthissection.

ThebindingproblemwasoriginallyformulatedasatheoreticalproblembyvonderMalsburgin1981͓seeareviewbyvonderMalsburg͑1999͒,andRoskies͑1999͒;Singer͑1999͔͒.However,examplesofbindinghadal-readybeenproposedbyRosenblatt͑1962͒forthevisualsystem͓forareviewofthebindingprobleminvisionseeSinger͑1999͒,andWolfeandCave͑1999͔͒.Thebindingproblemisformulatedastheneedforacoherentrepre-sentationofanobjectprovidedbytheassociationofall

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itsfeatures͑shape,color,location,speed,etc.͒.Theas-sociationofallfeaturesorbindingallowsauniﬁedper-ceptionoftheobject.Thebindingproblemisageneral-izedtaskofthenervoussystemasitseekstoreconstructanytotalperceptionfromitscomponents.Therearealsocognitivebindingproblemsrelatedtocognitiveidentiﬁ-cationandmemory.Nodoubtthebindingproblem,likemanyotherproblemsinbiology,hasmultiplesolutions.Thesesolutionsaremostlikelyimplementedthroughtheuseofdynamicalmechanismsforthecontrolofneu-ralactivity.

Themostwidelystudiedmechanismproposedtosolvethebindingproblemistemporalsynchrony͑ortemporalcorrelation͒͑SingerandGray,1995͒.IthasbeensuggestedbyvonderMalsburgandSchneider͑binding.1986͒thatHowever,synchronizationthereisstillisthecriticismbasisoffortheperceptualtemporalbindinghypothesis͑GhoseandMaunsell,1999;Riesen-huberandPoggio,1999͒.Obviously,neuraloscillationsandsynchronoussignalsareubiquitousinthebrain,andneuralsystemscanmakeuseofthesephenomenatoencode,learn,andcreateeffectiveoutputs.Thereareseverallinesofexperimentalevidencethatrevealtheuseofsynchronizationandactivitycorrelationforbind-ingtasks.Figure30showsanexampleofhowneuralsynchronizationcorrelateswiththeperceptualsegmen-tationofacomplexvisualpatternintodistinct,spatiallyoverlappingsurfaces͑Castelo-Brancoetal.,2000͒͑seetheﬁgurecaptionfordetails͒.Indeed,modelingstudiesshowthatinvolvingtimeintheseprocessescanleadtothebindingofdifferentfeatures.Theideaistousethecoincidenceofcertaineventsinthedynamicsofdiffer-entneuralunitsforbinding.Usuallysuchdynamicalbindingisrepresentedbysynchronousneuronsorneu-ronsthatareinphasewithanexternalﬁeld.However,dynamicaleventssuchasphaseorfrequencyvariationsusuallyarenotveryreproducibleandrobust.Asdis-cussedinthenextsection,itisreasonabletohypothesizethatbraincircuitsdisplayingsequentialswitchingofneu-ralactivityusethecoincidenceofthisswitchingtoimplementdynamicalbindingofdifferentWLCnet-works.

Anyspatiotemporalcodingneedsthetemporalcoor-dinationofneuralactivityamongdifferentpopulationsofneuronstoprovide͑i͒betterrecognitionofspeciﬁcfeatures,͑ii͒fasterprocessing,͑iii͒higherinformationcapacity,and͑iv͒featurebinding.Neuralsynchroniza-tionhasbeenobservedthroughoutthenervoussystem,particularlyinsensorysystems,forexample,intheolfac-torysystem͑LaurentandDavidowitz,1994͒andthevi-sualsystem͑Grayetal.,19͒.Fromthepointofviewofdynamicalsystemtheory,transientsynchronizationisanidealmechanismforbindingneuronsintoassembliesforseveralreasons:͑i͒thesynchronizedneuronsdonotnec-essarilyhavetobeneighbors;͑ii͒asynchronizationeventdependsonthestateoftheneuronandthestimu-lusandcanbeveryselective,thatis,neuronsfromthesamenetworkcanbetemporalmembersofdifferentcellassembliesatdifferentinstantsoftime;͑iii͒basicbrainrhythmsareabletosynchronizeneuronsresponsiblefor

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FIG.30.Anexampleofbindingshowingdependenceofsyn-chronyontransparencyconditionsandreceptiveﬁeld͑RF͒conﬁgurationinthecatvisualcortex.͑a͒Stimulusconﬁgura-tion.͑b͒Synchronizationbetweenneuronswithnonoverlap-pingRFsandsimilardirectionalpreferencesrecordedfromareasA18andPMLSofthecatvisualcortex.Left,RFconstel-lationandtuningcurves;right,crosscorrelogramsforre-sponsestoanontransparent͑left͒andtransparentplaid͑right͒movinginthecells’preferreddirection.Gratingluminancewasasymmetrictoenhanceperceptualtransparency.Smalldarkcorrelogramsareshiftpredictors.͑c͒Synchronizationbe-tweenneuronswithdifferentdirectionpreferencesrecordedfromA18͑polarandRFplots,left͒.Top,correlogramsofre-sponsesevokedbyanontransparent͑left͒andatransparent͑right͒plaidmovinginadirectionintermediatetothecells’preferences.Bottom,correlogramsofresponsesevokedbyanontransparentplaidwithreversedcontrastconditions͑left͒,andbyasurfacedeﬁnedbycoherentmotionofintersections͑right͒.Scaleonpolarplots:dischargerateinspikespersec-ond.Scaleoncorrelograms:abscissa,shiftintervalinms,binwidth1ms;ordinate,numberofcoincidencespertrial,nor-malized.ModiﬁedfromCastelo-Brancoetal.,2000.

theprocessingofinformationfromdifferentsensoryin-puts;and͑iv͒thesynchronizationispossibleevenbe-tweenneuraloscillatorswithstronglydifferentfrequen-cies͑Rabinovichetal.,2006͒.

Inearlyvisualprocessingneuronsthatencodefea-turesofacomplexvisualperceptareassociatedinfunc-Rev.Mod.Phys.,Vol.78,No.4,October–December2006

tionalassembliesthroughgamma-frequencysynchroni-zation͑Engeletal.,2001͒.Whensensorystimuliareperceptuallyorattentionallyselected,andtherespectiveneuronsareboundtogethertoraisetheirsaliency,thengamma-frequencysynchronizationamongtheseneuronsisalsoenhanced.Gamma-mediatedcouplinganditsmodulationbyattentionarenotlimitedtothevisualsystem:theyarealsofoundintheauditory͑Tiitinenetal.,1993͒andsomatosensorydomains͑DesmedtandTomberg,1994͒.Gammaoscillationsallowvisiomotorbindingbetweenposteriorandcentralbrainregions͑memory.RodriguezAsetaal.means,1999for͒anddynamicallyareinvolvedbindinginshort-termneuronsintoassemblies,gamma-frequencysynchronizationap-pearstobetheprimemechanismforstabilizingcorticalconnectionsamongmembersofaneuralassemblyovertime.Ontheotherhand,neuronscanincreaseorde-creasethestrengthoftheirsynapticconnectionsde-pendingontheprecisecoincidenceoftheiractivation͑videsSTDPthe͒,andrequiredgamma-frequencytemporalprecision.

synchronizationpro-Hatsopoulosetal.͑2003͒andJacksonetal.͑2003͒re-vealedthefunctionalsigniﬁcanceofneuralsynchroniza-tionandcorrelationswithinthemotorsystem.Preemi-nentamongbrainactionsmustbetheaggregationofdisparatespikingpatternstoformspatiallyandtempo-rallycoherentneuralcodesthatthendrivealphamotorneuronsandtheirassociatedmuscles.Essentially,motorbindingseemstodescribeexactlywhatmotorstructuresofthemammalianbraindo:providehigh-levelcoordi-nationofsimpleandcomplexvoluntarymovements.Neuronswithsimilarfunctionaloutputhaveanin-creasedlikelihoodofexhibitingneuralsynchronization.Incontrasttoclassicalsynchronization͑Pikovskyetal.,2001͒,synchronizationintheCNSisalwaystran-sient.Thephase-spaceimageoftransientsynchroniza-tioncanbeasaddlelimitcycleinthevicinityofwhichthesystemspendsﬁnitetime.Alternatively,itcanbealimitcyclewhosebasinofattractiondecreasesintime.Inbothcasesthesystemisabletoleavethesynchroni-zationregionafteraspeciﬁcstageofprocessingiscom-pletedandproceedwiththenexttask.Thisisabroadareawheretheissuesandapproachesarenotsettled,andthusitprovidesanopportunityforinnovativeideastoexplainthephenomenon.

Toconcludethissection,wenotethatthefunctionalroleofsynchronizationintheCNSandtheimportanceofspike-timingcodingingeneralarestillasubjectofdebate.Ontheonehand,itispossibletobuildmodelsthatusedynamicalpatternsofspikesforneuralcompu-tations,e.g.,representation,recognition,anddecisionmaking.Examplesofsuchspike-timing-basedcomputa-tionalmodelshavebeendiscussedbyHopﬁeldandBrody͑HopﬁeldandBrody,2001;BrodyandHopﬁeld,2003͒.Inthisworktheauthorsshowed,inparticular,thatspikesynchronizationacrossmanyneuronscanbeachievedintheabsenceofdirectsynapticinteractionsbetweenneuronsthroughphaselockingtoacommonunderlyingoscillatorypotential͑likegammaoscillation;seeabove͒.Ontheotherhand,therealconnectionsof

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FIG.31.͑Coloronline͒Spontaneousspatiotemporalpatternsobservedintheneocortexinvitroundertheactionofcarba-chol.Imagescomposedofopticalsignalsrecordedbyeightdetectorsarrangedhorizontally.Theopticalsignalfromeachdetectorwasnormalizedtothemaximumonthatdetectordur-ingthatperiodandnormalizedvalueswereassignedcolorsaccordingtoalinearcolorscale͑atthetopright͒.Thetracesaboveimages2and5areopticalsignalsfromtwoopticalde-tectorslabeledwiththiscolorscale.Thexdirectionoftheimagesrepresentstime͑12s͒andtheydirectionofeachimagerepresents2.6mmofspaceincorticaltissue.Notealsothattheﬁrstspikehadahighamplitudebutpropagatedmoreslowlyinthetissue.ModiﬁedfromBaoandWu,2003.

suchtheoreticalmodelswithexperimentsinvivoarenotestablished͑2003͔͒.

͓seealsoFelletal.͑2003͒andO’Reillyetal.IV.TRANSIENTDYNAMICS:GENERATIONANDPROCESSINGOFSEQUENCESA.Whysequences?

Thegenerationandcontrolofsequencesisofcrucialimportanceinmanyaspectsofanimallife.Workingmemory,birdsongs,ﬁndingfoodinalabyrinth,jumpingfromonestonetoanotherontheshore—allthesearetheresultsofsequentialactivitygeneratedbythener-voussystem.Lashleycalledtheproblemofcoordinationofconstituentactionsintoorganizedsequentialspa-tiotemporalpatternstheactionsyntaxproblem͑Lashley,1960͒.Thegenerationofsequencesisalsoimportantforintermediateinformationprocessingaswediscussbe-low.

Thesequencescanbecyclic,likemanybrainrhythmsandspatiotemporalpatternsgeneratedbyCPGs.Theycanalsobeirregular,likeneocorticalthetaoscillations͑͑4–10Hz͒generatedspontaneouslyincorticalﬁniteBaoandintimeWu,2003like͒those͑seeFig.generated31͒.Thebysequencesnetworksaneuralcancircuitbeundertheactionofexternalinputasinsensorysystems.Fromaphysicist’spointofview,anyreproducibleﬁnitesequencethatisfunctionallymeaningfulresultsfromthecooperativetransientdynamicsofthecorresponding

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neuralensembleorindividualneurons.Evenbrainrhythmsdemonstratetransientdynamicsbecausethecircuit’speriodicactivityismodulatedbynonstationarysensoryinputsorsignalsfromtheperiphery.Itisimpor-tanttoemphasizethefundamentalroleofinhibitioninthegenerationandcontrolofsequencesinthenervoussystem.

Inthissectionweconcentrateontheoriginofse-quencegenerationandthemechanismsofreproducibil-ity,sensitivity,andfunctionalreorganizationofMCs.Inthestandardstudyofnonlineardynamicalsystems,at-tentionisfocusedonthelong-timebehaviorofasystem.Thisistypicallynottherelevantquestioninneuro-science.Herewemustaddressthetransientresponsestoastimulusexternaltotheneuralsystemandmustcon-sidertheshort-termbindingofacollectionofresponses,perhapsfromdifferentsensoryinputs,tofacilitateac-tioncommandsdirectedtothemotorsystem.Ifyouat-tempttoswataﬂy,itcannotaskyoutoperformthisactionmanytimessothatitcanaverageoveryourac-tions,allowingittoperformsomestandardoptimalre-sponse.Fewﬂieswantingthisrepetitionwouldsurvive.

B.Spatiallyorderednetworks1.Stimulus-dependentmodes

Manyneuralensemblesareanatomicallyorganizedasslightlyinhomogeneousexcitablemedia.Examplesofsuch͑andLeznikmediaareretina͑Tohyaetal.,2003͒,IOnetworkthalamocorticalandLlinas,2002layers͒,cortex͑Contreras͑Ichinoheetal.et,1996al.,2003͒.All͒,theseareneuronallatticeswithchemicalorelectricalconnectionsoccurringprimarilybetweenneighbors.Therearesomegeneraldynamicalmechanismsofse-quencegenerationinsuchspatiallyorderednetworks.ThesemechanismsareusuallyrelatedtotheexistenceofwavemodessuchasthoseshowninFig.31thataremodulatedbyexternalinputsorstimuli.

Manysigniﬁcantobservationalandmodelingresultsforthissubjectarefoundinthevisualsystem.Visualsystemsareorganizeddifferentlyfordifferentclassesofanimals.Forexample,themammalianvisualcortexhasseveraltopographicallyorganizedrepresentationsofthevisualﬁeldandneuronsatadjacentpointsinthecortexareexcitedbystimulipresentedatadjacentregionsofthevisualﬁeld.Thisindicatesthereisacontinuousmap-pingofthecoordinatesofthevisualﬁeldtothecoordi-natesofthecortex͑vanEssen,1979͒.Incontrasttosuchamappingconnectionsfromthevisualﬁeldtothevisualcortexintheturtle,forexample,aremorecomplex:Alocalspotinthevisualﬁeldactivatesmanyneuronsinthecortexbutinanorderedway.Asaresulttheexcita-tionoftheturtlevisualcortexisdistributedandnotlo-calized,andthissuggeststhetemporaldynamicsofsev-eralinteractingmembranemodes͑seeFig.32͒.Inthemammaliancortexamovingstimulusevokesalocalizedwaveorwavefront,whileintheturtlevisualcortexadifferentiallymovingstimulusmodulatestemporalinter-actionsofthecorticalmodesdifferentlyandisrepre-sentedbydifferentsequentialswitchingsbetweenthem.

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FIG.32.͑Coloronline͒Sequentialchangingofcorticalmodesintheturtlevisualcortex.Comparisonbetweenthespatialor-ganizationofthecorticalactivityintheturtlevisualsystemandthenormalmodesofarectangularmembrane͑drum͒.FromSensemanandRobbins,1999.

Tounderstandthedynamicsofthewavemodes,i.e.,stability,sensitivitytostimuli,dependenceonneuro-modulators,etc.,onehastobuildamodelthatisbasedontheexperimentalinformationaboutthepossibilityofthesemodesmaintainingthetopologicalspacestructureobservedinexperiments.Inmanysimilarsituationsonecanintroducecooperativeorpopulationvariablesthatcanbeinterpretedastheamplitudeofsuchmodesde-pendingontime.Thecorrespondingamplitudeequa-tionsareessentiallythewidelystudiedevolutionequa-tions͑CrossofthedynamicaltheoryofpatternformationForanandanalysisHohenberg,ofthe1993wave;Rabinovichmodedynamicsetal.,2000of͒the.turtlevisualcortexSensemanandRobbins͑1999͒usedtheKarhunen-Loevedecompositionandasnapshotofaspatiotemporalpatternattimet=t0couldberepre-sentedasaweightedsumofbasicmodesMi͑x,y͒withcoordinates͑x,y͒ontheimage:

u͑x,y,t0͒=͚ai͑t0͒Mi͑x,y͒,

͑20͒

iwhereu͑x,y,t͒representsthecooperativedynamicsofthesemodes.Thepresentationofdifferentvisualstimuli,suchasspotsoflightatdifferentpointsinthevisualﬁeld,producedspatiotemporalpatternsrepre-sentedbydifferenttrajectoriesinthephasespaceapossible1͑t͒,a2͑tto͒,...make,an͑ta͒.reductionDuetal.in͑the2005dimensionality͒showedthatofittheiswavemodesbyasecondKarhunen-Loevedecomposi-tion,whichmapsinsometimewindowthetrajectoryin͑Fig.ai͒space33͒.Theintoobservedapointintransientalow-dimensionaldynamicsisspacesimilar͑seetotheexperimentalresultsontherepresentationofdiffer-entodorsintheinvertebrateolfactorysystem͓seeFig.46andGalanetal.͑2004͔͒.Nenadicetal.͑2002͒usedalarge-scalecomputermodelofturtlevisualcortextore-producequalitativelythefeaturesofthecorticalmodedynamicsseenintheseexperiments.

Itisremarkablethatnotonlydospatiotemporalpat-ternsevokedbyadirectstimuluslooklikewavemodes,butevenspontaneousactivityinthesensorycortexiswell͑organizedandverydifferentfromturbulentsumptionArielietaboutal.,1996the͒.stochasticThismeansandthatuncorrelatedthecommonﬂowsponta-as-Rev.Mod.Phys.,Vol.78,No.4,October–December2006

FIG.33.͑Color͒Spacerepresentationofcorticalresponsesintheturtlevisualcortextoleft,center,andrightstimuli.FromDuetal.,2005.

neousactivityofneighboringneuronsinneuralnetworks͓see,forexample,vanVreeswijkandSompo-linsky͑1996͒;AmitandBrunel͑1997b͔͒isnotalwayscorrect.Localﬁeldpotentialsandrecordingsfromsingleneuronsindicatethepresenceofhighlysynchronouson-goingactivitypatternsorwavemodes͑seeFig.34͒.Thespontaneousactivityofasingleneuronconnectedwithothers,inprinciple,canbereconstructedusingtheevokedpatternsofnetworkactivity͑Tsodyksetal.,1999͒.

Therearesomeillustrativemodelsofwavemodesthatwenotehere.In1977Amari͑1977͒foundspatiallylocalizedregionsofhighneuralactivity͑“bumps”͒innetworkmodelsconsistingofasinglelayerofcoupledexcitatory͑andinhibitoryrateneurons.Laingetnection2002͒extendedAmari’sresultstoanonmonotoniccon-al.͑mensions:

showninfunctionFig.35͒͑“Mexicanandaneuralhat”layerwithinoscillatingtwospatialtailsdi-͒ץu͑x,y,t͒

ץt

=−u͑x,y,t͒+

͵͵␻⍀

͑x−q,y−p͒f„u͑q,p,t͒…dqdp,

͑21͒

f͑u͒=2e−␶/͑u−th͒2

⌰͑u−th͒,͑22͒

␻͑x,y͒=e−b

ͱx2+y2͓bsin͑ͱx2+y2͒+cos͑ͱx2+y2͔͒.

͑23͒

Anexampleofatypicallocalizedmodeinsuchneuralmediawithlocalexcitationandlong-rangeinhibitionisrepresentedinFig.36.Differentmodes͑withdifferentnumbersofbumps͒canbeswitchedfromonetoanotherbytransientexternalstimuli.Multipleitemscanbestoredinthismodelbecauseoftheoscillatingtailsofthe

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FIG.34.͑Coloronline͒Relationbetweenthespikingactivityofasingleneuronandthepopulationstateofcorticalnetworks.͑a͒Frombottomtotop:stimulustimecourse;correlationcoefﬁcientoftheinstantaneoussnapshotofpopulationactivitywiththespatialpatternobtainedbyaveragingoverallpatternsobservedatthetimescorrespondingtospikesevokedbytheoptimalorientationofthestimuluscalledtheneuron’spreferredcorticalstate͑PCS͒pattern;observedspiketrainofevokedactivitywiththeoptimalorientationforthatneuron;reconstructedspiketrain.Thesimilaritybetweenthereconstructedandobservedspiketrainsisevident.Also,strongupswingsinthevaluesofcorrelationcoefﬁcientsareevidenteachtimetheneuronemitsburstsofactionpotentials.Everystrongburstisfollowedbyamarkeddownswinginthevaluesofthecorrelationcoefﬁcients.͑b͒Thesameas͑a͒,butforaspontaneousactivityrecordingsessionfromthesameneuron͑eyesclosed͒.͑c͒Theneuron’sPCS,calculatedduringevokedactivityandusedtoobtainboth͑a͒and͑b͒.͑d͒Thecorticalstatecorrespondingtospontaneousactionpotentials.Thetwopatternsarenearlyidentical͑correlationcoefﬁcient0.81͒.͑e͒and͑f͒Anotherexampleofthesimilaritybetweentheneuron’sPCS͑e͒andthecorticalstatecorrespondingtospontaneousactivity͑f͒fromadifferentcatobtainedwiththehigh-resolutionimagingsystem͑correlationcoefﬁcient0.74͒.ModiﬁedfromTsodyksetal.,1999.

effectiveconnectionstrength.Thisistheresultofthecommonactivityoftheexcitatoryandinhibitoryconnec-tionsbetweenneurons.Inhibitionplaysacrucialroleforthestabilityoflocalizedmodes͑Laingetal.,2002͒.

Localizedmodeswithdifferentnumbersofbumpsre-mindoneofcomplexlocalizedpatternsinadissipativenonequilibriummedia͑Rabinovichetal.,2000͒.Based

onthisanalogy,itisreasonabletohypothesizethatdif-ferentmodesmaycoexistinaneurallayerandtheirinteractionandannihilationcanexplainthesequentialeffectivenessofthedifferentevents.Thissuggeststheycouldbeamodelofsequentialworkingmemory͑seebelow͒.

Manyrhythmsofthebraincantaketheformofwaves:spindlewaves͑7–14Hz͒seenattheonsetofsleep͑Kimetal.,1995͒,slowerdeltarhythmsofdeepersleep,thesynchronousdischargeduringanepilepticsei-

FIG.35.Connectionfunction␻͑x,y͒,centeredatthecenterofthedomain.ModiﬁedfromLaingetal.,2002.

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FIG.36.Six-bumpstablesolutionofthemodel͑21͒–͑23͒:b=0.45,␶=0.1,th=1.5.ModiﬁedfromLaingetal.,2002.

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zure͑ConnorsandAmitai,1997͒,wavesofexcitationassociatedwithsensoryprocessing,40-Hzoscillations,andothers.Inthalamocorticalnetworksthesameclus-tersofneuronsareresponsiblefordifferentmodesofrhythmicactivity.Whatisthedynamicaloriginofsuchmultifunctionality?Thereisnouniqueanswertothisquestion,andthereareseveraldifferentmechanismsthatcanberesponsibleforit͑wehavealreadydiscussedthisforsmallinvertebratenetworks;seeSec.II.B͒.Ter-manetal.͑1996͒studiedthetransitionbetweenspin-dlinganddeltasleeprhythms.Theauthorsshowedthatthesetworhythmsmakedifferentusesofthefastinhi-bitionandslowinhibitiongeneratedbythalamicreticu-lariscells.ThesetwotypesofinhibitionaremediatedinthecortexbyGABA͑A͒andGABA͑B͒receptors,re-spectively͑Schutter,2002;Tamsetal.,2003͒.

Thewavemodeequationdiscussedaboveisfamiliartophysicistsandcanbewrittenbothwheninteractionsbetweenneuronpopulationsarehomogeneousandiso-tropic͑Ermentrout,1998͒andwhentheneurallayerispartitionedintodomainsorhypercolumnslikethepri-maryvisualcortex͑V1͒ofcatsandprimates,whichhasacrystallinelike͑Bressloff,structureattheInthenext2002section;BressloffwediscussandCowan,millimeterthepropagation2002length͒.

scaleofpat-ternsofsynchronousactivityalongspatiallyorderedneuralnetworks.

2.Localizedsynﬁrewaves

Auditoryandvisualsensorysystemshaveaveryhightemporalresolution.Forexample,theretinaisabletoresolvesequentialtemporalpatternswithaprecisioninthemillisecondrange.Doesthetransmissionofsensoryinformationfromtheperipherytothecortexmaintainsuchhighresolution?Iftheanswerisyes,whatarethedynamicalmechanismsresponsibleforthis?Theseques-tionsarestillopen.

Thereareseveralneurophysiologicalexperimentsthatshowtheabilityofneuralsystemstotransmittempo-rarilymodulatedresponsesofsensorynetworkswithhighprecisionoverseveralprocessinglevels.Forex-ample,crosscorrelationsbetweensimultaneouslyre-cordedresponsesofretinalcellsrelayneuronswithinthethalamus,andcorticalneuronsshowthattheoscilla-torypatterningisreliablytransmittedtothecortexwitharesolutioninthemillisecondrange͓seeforreviewsSinger͑1999͒andNaseetal.͑2003͔͒.Asimilarphenom-enonwasobservedbyKimpoetal.͑2003͒whoshowedevidenceforthepreservedtimingofspikingactivitythroughmultiplestepsofaneuralcontrolloopinthebirdbrain.Thedynamicaloriginofsuchprecisemessagepropagation,independentoftherateﬂuctuation,isoftenattributedtosynchronizationofthemanyneuronsintheoverallcircuit͑Abeles,1991;Diemannetal.,1999͒.

Wenowdiscussbrieﬂythedynamicsofwavesofsyn-chronousneuralﬁring,i.e.,synﬁrewaves.Onemodelingstudy͑Diesmannetal.,1999͒hasshownthatthestablepropagationoflocalizedsynﬁrewaves,short-lastingsyn-chronousspikingactivity,ispossiblealongasequenceof

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FIG.37.Sequenceofpoolsofexcitatoryneurons,connectedinafeedforwardwaybyso-calleddivergentandconvergentcon-nections.Thenetworkiscalledasynﬁrechainifitsupportsthepropagationofsynchronousspikepatterns.ModiﬁedfromGe-waltigetal.,2001.

layersorpoolsofneuronsinafeedforwardcorticalnet-work͑spikeAbeles,suchtimes1991astheamong͒.TheoneshowninFig.37,asynﬁrechainthedegreepools’oftemporalmembersaccuracydeterminesofwhethersubsequentpoolscanreproduce͑orevenim-prove͒thisaccuracy͓Fig.38͑a͔͒,orwhethersynchronousexcitationdispersesandeventuallydiesoutasinFig.38͑b͒forasmallernumberofspikesinthevolley.Thusinthecontextofsynﬁrenetworkfunctionthequalityoftimingisjudgedonwhethersynchronousspikingissus-tainedorwhetheritdiesout.

Diesmannetal.͑1999͒,CateauandFukai͑2001͒,Kis-tler͑critical2003and͒havedevalueshownZeeuwdeterminedthat͑2002if͒bythe,andNowotnyandHuertathepoolconnectivitysizeismorebetweenthanalayers,thewaveactivityinitiatedattheﬁrstpoolpropa-gatesfromonepooltothenext,formingasynﬁrewave.NowotnyandHuerta͑2003͒havetheoreticallyproventhatnootherstatesexistbeyondsynchronizedorunsyn-chronized͑volleysasshownintheexperimentsbyReyes2003The͒.

synﬁrefeedforwardchain͑Fig.37͒isanoversim-pliﬁedmodelforanalyzingsynﬁrewavesbecauseinre-alityanynetworkwithsynﬁrechainsisembeddedinalargercorticalnetworkthatalsohasinhibitoryneurons

FIG.38.Propagationofﬁringactivityinsynﬁrechains.͑a͒Stableand͑b͒unstablepropagationofsynchronousspikinginamodelofcorticalnetworks.Rasterdisplaysofpropagatingspikevolleyalongfullyconnectedsynﬁrechain.Panelsshowthespikesintensuccessivegroupsof100neuronseach͑syn-apticdelaysarbitrarilysetto5ms͒.Initialspikevolley͑notshown͒wasfullysynchronized,containing͑a͒50or͑b͒48spikes.ModiﬁedfromDiesmannetal.,1999.

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andmanyrecurrentconnections.Thisproblemisdis-cussedindetailbyAvieletal.͑2003͒.

C.Winnerlesscompetitionprinciple1.Stimulus-dependentcompetition

Hereweconsideraparadigmofsequencegenerationthatdoesnotdependonthegeometricalstructureoftheneuralensembleinphysicalspace.Itcan,forexample,beatwo-dimensionallayerwithconnectionsbetweenneighborsorathree-dimensionalnetworkwithsparserandomconnections.Thisparadigmcanbehelpfulfortheexplanationandpredictionofmanydynamicalphe-nomenainneuralnetworkswithexcitatoryandinhibi-torysynapticconnections.Theparadigmiscalledthewinnerlesscompetitionprinciple.WehavetouchedonaspectsofWLCnetworksearlier,andhereweexpandontheirpropertiesandtheirpossibleuseinneuro-science.

“Survivaloftheﬁttest”isaclichéthatisoftenassoci-atedwiththetermcompetition.However,competitionisnotmerelyameansofdeterminingthewinner,asinawinner-take-allnetwork.Itisalsoamultifunctionalin-strumentthatnatureusesatalllevelsoftheneuronalhierarchy.Competitionisalsoamechanismthatmain-tainsthehighestlevelofvariabilityandstabilityofneu-raldynamics,evenifitisatransientbehavior.

OvertwohundredyearsagothemathematiciansBordaanddeCondorcetwereinterestedintheprocessofpluralityelectionsattheFrenchRoyalAcademyofSciences.TheyconsideredvotingdynamicsinacaseofthreecandidatesA,B,andC.IfAbeatsBandBbeatsCinahead-to-headcompetition,wemightreasonablyexpectAtobeatC.Thuspredictingtheresultsoftheelectioniseasy.However,thisisnotalwaysthecase.ItmayhappenthatCbeatsA,resultinginaso-calledCon-dorcettriangle,andthereisnorealwinnerinsuchacompetitiveprocess͑Borda,1781;Saari,1995͒.Thisex-ampleisalsocalleda“votingparadox.”Thedynamicalimage͑ofthisphenomenonisaisseeevenFig.structurally39͒.Insomestablespeciﬁc͑GuckenheimercasesrobusttheheteroclinicheterocliniccycleandHolmes,cycle1988;Krupa,1997;StoneandArmbruster,1999;Ashwinetal.,2003;PostlethwaiteandDawes,2005͒.

Thecompetitionwithoutawinnerisalsoknowninhydrodynamics:BusseandHeikesdiscoveredthatcon-vectiverollpatternsinarotatingplanelayerexhibitse-quentialchangesoftheroll’sdirectionasaresultofthecompetitionbetweenpatternswithdifferentrollorien-tations.Nopatternbecomesawinnerandthesystemexhibitsperiodicorchaoticswitchingdynamics͑Busseand͑Heikes,1980͒.Forreviewseegenetic2000͒.TheRabinovichetal.system,samei.e.,phenomenoninexperimentshasbeenwithadiscoveredsyntheticnet-inaworkofthreetranscriptionalregulators͑ElowitzandLeibler,2000͒.Speciﬁcally,theseauthorsdescribedthreerepressorgenesA,B,andCorganizedinaclosedchainwithunidirectionalinhibitoryconnectionssuchthatA,B,andCbeateachother.Thisnetworkbehaveslikea

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FIG.39.͑Coloronline͒IllustrationofWLCdynamics.Toppanel:PhaseportraitcorrespondingtotheautonomousWLCdynamicsofathree-dimensionalcase.Bottompanel:Projec-tionofanine-dimensionalheteroclinicorbitofthreeinhibitorycoupledFitzHugh-Nagumospikingneuronsinathree-dimensionalspace͑thevariables␰1,␰3,␰3arelinearcombina-tionsoftheactualphasevariablesofthesystem͒.FromRabinovichetal.,2001.

clock:itperiodicallyinducessynthesisofgreenﬂuores-centproteinsasanindicatorofthestateofindividualcellsonatimescaleofhours.

Inneuralsystemssuchclockcompetitivedynamicscanresultfromtheinhibitoryconnectionsamongneu-rons.Forexample,Jefferysetal.͑1996͒showedthathip-pocampalandneocorticalnetworksofmutuallyinhibi-toryinterneuronsgeneratecollective40-Hzrhythms͑amplegammaofoscillationsneuralcompetition͒whenexcitedwithouttonically.awinnerAnotherwasdis-ex-cussedbyErmentrout͑1992͒.Theauthorstudiedthedynamicsofasingleinhibitoryneuronconnectedtoasmallclusteroflooselycoupledexcitatorycellsandob-servedtheemergenceofalimitcyclethroughahetero-cliniccycle.Forautonomousdynamicalsystemscompe-titionwithoutawinnerisawell-knownphenomenon.WeusethetermWLCprincipleforthenonautono-moustransientdynamicsofneuralsystemsreceivingex-ternalstimuliandexhibitingsequentialswitchingamongtemporalwinners.ThemainpointoftheWLCprincipleisthetransformationofincominginputsintospatiotem-poraloutputsbasedontheintrinsicswitchingdynamicsoftheneuronalensemble͑seeFig.40͒.Inthephasespaceofthenetwork,suchswitchingdynamicsarerep-resentedbyaheteroclinicsequencewhosearchitecturedependsonthestimulus.Suchasequenceconsistsof

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FIG.40.Transformationoftheidentityspatialinputintospa-tiotemporaloutputbasedontheintrinsicsequentialdynamicsofaneuralensemblewithWLC.

manysaddleequilibriaorsaddlecyclesandmanyhet-eroclinicorbitsconnectingthem,i.e.,manyseparatrices.Thesequencecanserveasanattractingsetifeverysemistablesethasonlyoneunstabledirection͓seealsoAshwinandTimme͑2005͔͒.

ThekeypointsonwhichWLCnetworksarebasedarethefollowing:͑i͒thestimulus-dependentheteroclinicse-quencecorrespondingtoaspeciﬁcorderofswitchinghasalargebasinofattraction,i.e.,thesequenceisro-bust;and͑ii͒thetopologyoftheheteroclinicsequencesensitivelydependsontheincomingsignals,i.e.,WLCdynamicshaveahighresolution.

Inthismannerstimulus-dependentsequentialswitch-ingofneuronsorgroupsofneurons͑clusters͒isabletoresolvethefundamentalcontradictionbetweensensitiv-ityandrobustnessinsensoryrecognition.Anykindofsequentialactivitycanbeprogrammed,inprinciple,byanetworkwithstimulus-dependentnonsymmetricinhibi-toryconnections.Itcanbethecreationofspatiotempo-ralpatternsofmotoractivity,thetransformationofthespatialinformationintospatiotemporalinformationforsuccessfulrecognition͑seeFig.40͒,andmanyothercomputations.

Thegenerationofsequencesininhibitorynetworkshasalreadybeendiscussedwhenweanalyzedthedy-namicsofCPGs͑seeSec.II.B͒focusingonrhythmicactivity.ThemathematicalimageinphasespaceoftherhythmicsequentialswitchingshowninFigs.8and9isalimitcycleinthevicinityoftheheterocliniccontour͓cf.Fig.39͑a͔͒.

WLCdynamicscanbedescribedintheframeworkofneuralmodelsatdifferentlevels.Thesecouldberatemodels,Hodgkin-Huxley-typemodels,orevensimplemapmodels͑seeTableI͒.Forspikingneuronsorgroupsofsynchronizedspikingneuronsinanetworkwithnon-symmetricallateralinhibitionWLCmayleadtoswitch-ingbetweenactiveandinactivestates.Themathemati-calimageofsuchswitchingactivityisalsoaheteroclinicloop,butinthiscasetheseparatricesdonotconnectsaddleequilibriumpoints͓Fig.39͑a͔͒butsaddlelimitcyclesasshowninFig.39͑b͒.TheWLCdynamicsinamodelnetworkofninespikingneuronswithinhibitoryconnectionsisshowninFig.41.Similarresultsbasedon

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FIG.41.SpatiotemporalpatternsgeneratedbyanetworkofnineFitzHugh-Nagumoneuronswithinhibitoryconnections.Theleftandrightpanelscorrespondtotwodifferentstimuli.FromRabinovichetal.,2001.

a͑2003mapmodelofneuronshavebeenreportedbyCasadoAn͒.

importantadvantageofWLCnetworksisthattheycanproducedifferentspatiotemporalpatternsinresponsetodifferentstimuli,and,remarkably,neuronsspontaneouslyformsynchronizedclustersdespitetheabsenceofexcitatorysynapticconnections.Foradiscus-sionofsynchronizationwithinhibitionseealsovanVreeswijketal.͑1994͒andElsonetal.͑2002͒.

FinallyWLCnetworksalsopossessastrikinglydiffer-entcapacityorabilitytorepresentinadistinguishablemanneranumberofdifferentpatterns.InanattractorcomputationnetworkoftheHopﬁeldvariety,anetworkwithNattractorshasbeenshowntohaveacapacityofapproximatelyN/7.InasimpleWLCnetworkwithNnodes,thiscapacityhasbeenshown͑Rabinovichetal.,2001͒tobeofordere͑N−1͒!,whichisaremarkablegainincapacity.

2.Self-organizedWLCnetworks

Itisgenerallyacceptedthatthereisinsufﬁcientge-neticinformationavailabletoaccountforallthesynap-ticconnectivityinthebrain.Howthencanthefunc-tionalarchitectureofWLCcircuitsbegeneratedintheprocessofdevelopment?

OnepossibleanswerhasbeenfoundbyHuertaandRabinovich.Startingwithamodelcircuitconsistingof100ratemodelneuronsconnectedrandomlywithweakinhibitorysynapses,newsynapticstrengthsarecom-putedfortheconnectionsusingHebblearningrulesinthepresenceofweaknoise.TheneuronratesaaLotka-Volterramodelfamiliarfromourearlieri͑t͒discus-satisfysion.Inthiscasethematrix␳ij͑t͒isadynamicalvariable:

dai͑t͒

dt=ai͑t͒ͩ␴͑S͒−͚␳ij͑t͒aj͑t͒ͪ+␰i͑t͒.͑24͒

␴the͑S͒strengthsisafunctionofthedependentinhibitoryonconnectionsthestimulusdetermined

S,␳ij͑t͒are1250

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FIG.42.͑Color͒Resultofsimulatinganetworkof100neuronssubjecttothelearningruleg͑ai,aj͒=aiaj͓10tanh͑aj−ai͒+1͔.Theactivityofrepresentativeneuronsinthisnetworkisshownindifferentcolors.Thesystemstartsfromrandominitialcon-ditionsfortheconnections.Thenoiselevelis␩=0.01.Forsim-plicity,theswitchingactivityofonlyfourofthe100neuronsisshown.

bysomelearningrules,and␰͗equations

␰͑t−tЈ͒.Thelearningi͑t͒isGaussianisdescribednoisebywithi͑t͒␰j͑tЈ͒͘=␩␦ij␦thed␳ij͑t͒

=␳ij͑t͒g„ai͑t͒,aj͑t͒,S…−͓␳ij͑t͒−␥͔,͑25͒

whereg͑ai,aj,S͒representsthestrengtheningofinterac-tionsfromneuronitoneuronjasafunctionoftheexternalstimulusS.Theparameter␥representsthelowerboundofthecouplingstrengthsamongneurons.Figure42showstheactivityofrepresentativeneuronsinanetworkbuiltwiththismodel.Aftertheself-organizationphase,thisnetworkdisplaysWLCswitch-ingdynamics.

Winnerlesscompetitiondynamicscanalsobethere-sultoflocalself-organizationinnetworksofHHmodelneuronsthatdisplaySTDPwithinhibitorysynapticcon-nectionsasshowninFig.43.Suchmechanismsofself-organization,asshownbyNowotnyandRabinovich,canbeappropriatefornetworksthatgeneratenotonlyrhythmicactivitybutalsotransientheteroclinicse-quences.

3.Stableheteroclinicsequence

Thephase-spaceimageofnonrhythmicWLCdynam-icsisatrajectoryinthevicinityofastableheteroclinicsequence͑SHS͒inthestatespaceofthesystem.Suchasequence͑seeFig.44͒isanopenchainofsaddleﬁxedpointsconnectedbyone-dimensionalseparatriceswhichretainnearbytrajectoriesinitsvicinity.TheﬂexibilityofWLCdynamicsisprovidedbytheirdependenceontheidentityofparticipatingneuralclustersofstimuli.Se-quencegenerationinchainlikeorlayerlikenetworksofneuronsmayresultfromafeedforwardwavelikepropa-gationofspikeslikewavesinsynﬁrechains͑seeabove͒.Incontrast,WLCdynamicsdoesnotneedaspeciﬁcspa-tialorganizationofthenetwork.However,theimageofawaveisausefulone,becauseinthecaseofWLCawaveofneuralactivitypropagatesinstatespacealongtheSHS.Suchawaveisinitiatedbyastimulus.The

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FIG.43.͑Coloronline͒ExampleofWLCdynamicsentrainedinanetworkbyalocallearningrule.Inisolation,thefourHHneuronsinthenetworkarereboundbursters,i.e.,theyﬁreabriefburstofspikesafterbeingstronglyinhibited.Theall-to-allinhibitorysynapsesinthesmallnetworkaregovernedbyaSTDPlearningrulewhichstrengthensthesynapseforpositivetimedelaysbetweenpostsynapticandpresynapticactivityandweakensitotherwise.SuchSTDPofinhibitorysynapseshasbeenobservedintheentorhinalcortexofrats͑Haasetal.,2006͒.͑a͒Beforeentrainmenttheneuronsjustfollowtheinputsignalofperiodiccurrentpulses.͑b͒Theresultingburstsstrengthentheforwardsynapsescorrespondingtotheinputsequencemakingthemeventuallystrongenoughtocausere-boundbursts.͑c͒Afterentrainmentactivatinganyoneoftheneuronsleadstoaninﬁniterepetitionofthetrainedsequencecarriedbythesuccessivereboundburstsoftheneurons.

speedofthesequentialswitchingdependsonthenoiselevel␩.NoisecontrolsthedistancebetweentrajectoriesrealizedbythesystemandtheSHS.FortrajectoriesthatgetclosertotheSHSthetimethatthesystemspendsnearsemistablestates͑saddles͒,i.e.,theintervalbe-tweenswitching,becomeslonger͑seeFig.44͒.

ThemechanismofreproducingtransientsequentialneuralactivityhasbeenanalyzedbyAframovich,Zhigu-lin,etal.͑2004͒͑seeFig.44͒.Itisquitegeneralanddoesnotdependonthedetailsoftheneuronalmodel.Saddlepointsinthephasespaceoftheneuralnetworkcanbereplacedbysaddlelimitcyclesorevenchaoticsetsthatdescribeneuralactivityinmoredetail,asintypicalspik-ingorspiking-burstingmodels.Thisfeatureisimportantforneuralmodelingbecauseitmayhelptobuildabridgebetweentheconceptsofneuralcompetitionandsynchronizationofspikes.

WecanformulatethenecessaryconditionsfortheconnectivityofaWLCnetworkthatmustbesatisﬁedinorderforthenetworktoexhibitreproduciblesequentialdynamicsalongtheheteroclinicchain.Asbefore,webaseourdiscussionontheratemodel

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FIG.44.Astableopenheteroclinicsequenceinaneuralcir-cuitwithWLC.Wsiisastablemanifoldoftheithsaddleﬁxedpoint͑heavydots͒.ThetrajectoriesinthevicinityoftheSHSrepresentsequenceswithdifferenttimings.ThetimeintervalsbetweenswitchesisproportionaltoTϳ͉ln␩͉/␭u,where␭uisapositiveLyapunovexponentthatcharacterizestheone-dimensionalunstableseparatricesofthesaddlepoints͑StoneandHolmes,1990͒.ModiﬁedfromAfraimovich,Zhigulin,etal.,2004.

ai͑t͒dt=ai͑t͒ͩជNl

␴i͑Sl͒−͚␳ij͑Sជl͒aj͑t͒ͪ+␰i͑t͒,͑26͒

where␰i͑t͒isanexternalGaussiannoise.InthismodelisassumedthatthestimulusS

ជit

linﬂuencesthematrix␳andincrements␴ionlyinthesubnetworkNl

.Eachin-crement␴icontrolsthetimeconstantofaninitialexpo-nentialgrowthfromtherestingstateabyAframovich,Zhigulin,etal.͑2004͒toi͑t͒assure=0.Asthatshown

theSHSisinthephasespaceofthesystem͑26͒thefollow-inginequalitiesmustbesatisﬁed:

␴ik−1k−1␴Ͻ␳␴iiik−1ikϽ

k␴+1,͑27͒

␴ik+1␴−1Ͻ␳␴ik+1iik+1ikϽ

␴,͑28͒

␳␴i−␴ik−1

iikϾ␳ik−1ik+

␴.

͑29͒

␴imistheincrementofthemthsaddlewhoseunstablemanifoldisonedimensional;␳ik±1ikisthestrengthoftheinhibitoryconnectionbetweenneighboringsaddlesintheheteroclinicchain.Thecomputermodelingresultofanetworkwithparametersthatsatisfy͑27͒–͑29͒isshowninFig.45.

InthenextsectionwediscusssomeexperimentsthatsupporttheSHSparadigm.

4.Relationtoexperiments

Theolfactorysystemmayserveasoneexampleofaneuralsystemthatgeneratestransient,buttrial-to-trial

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FIG.45.͑Coloronline͒TimeseriesoftheactivityofaWLCnetworkduringtentrials͑only20neuronsareshown͒:simula-tionsofeachtrialwerestartedfromadifferentrandominitialcondition.Inthisploteachneuronisrepresentedbyadiffer-entcoloranditslevelofactivitybythesaturationofthecolor.FromAfraimovich,Zhigulin,etal.,2004.

reproducible,sequencesofneuronalactivitywhichcanbeexplainedwiththeWLCprinciple.Thecomplexin-trinsicdynamicsintheantennallobe͑AL͒ofinsectstransformstaticsensorystimuliintospatiotemporalpat-ternsofneuralactivity͑Laurentetal.,2001͒.SeveralexperimentalresultsaboutthereproducibilityofthetransientspatiotemporalALdynamicshavebeenpub-lished͑Stopferetal.,2003;Galanetal.,2004;MazorandLaurent,2005͒͑seeFig.46͒.InexperimentsdescribedbyGalanetal.͑2004͒beeswerepresentedwithdifferentodors,andneuralactivityintheALwasrecordedusingcalciumimaging.Theauthorsanalyzedthetransienttra-jectoriesintheprojectionneuronactivityspaceandfoundthattrajectoriesrepresentingdifferenttrialsofstimulationwiththesameodorwereverysimilar.Itwasshownthatafteratimeintervalofabout800msdiffer-entodorsarerepresentedinphasespacebydifferentstaticattractors,i.e.,thetransientspatiotemporalpat-ternsconvergetodifferentspatialpatternsofactivity.However,theauthorsemphasizethatduetotherepro-ducibilityofthetransientdynamicssomeodorswererecognizedintheearlytransientstageassoonas300msaftertheonsetoftheodorpresentation.ItishighlylikelythatthetransienttrajectoriesobservedintheseexperimentsrepresentrealizationsofaSHS.

Thegenerationofreproduciblesequencesplaysalsoakeyroleinthehighvocalcenter͑HVC͒ofthesongbirdsystem͑Hahnloseretal.,2002͒.LikeaCPG,thisneuralsystemisabletogeneratesparsespatiotemporalpat-ternswithoutanyrhythmicstimuliinvitro͑SolisandPerkel,2005͒.InitsprojectionstothepremotornucleusRA,HVCinanawakesingingbirdsendssparseburstsofhigh-frequencysignalsonceforeachsyllableofthesong.Theseburstshaveaninterspikeintervalabout2msandlastabout8mswithinasyllabletimescaleof100–200ms.TheburstsareshownforseveralHVC→tainsRAprojectionmanyinhibitoryneuronsininterneuronsFig.47.TheHVC͑Mooneyalsocon-andPrather,2005͒.Theinterneuronsburstdenselythrough-outthevocalizations,incontrasttotheburstingoftheRA-projectingHVCneuronsatsingleprecisetimings.A

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FIG.46.͑Color͒TransientALdynamics.Toppanel:Trajecto-riesoftheantennallobeactivityduringpoststimulusrelaxationinonebee.ModiﬁedfromGalanetal.,2004.Bottompanel:VisualizationoftrajectoriesrepresentingtheresponseofaPNpopulationinalocustALovertime.Time-slicepointswerecalculatedfrom110PNresponsestofourconcentrations͑0.01,0.05,0.1,1͒ofthreeodors,projectedontothreedimensionsusinglocallylinearembedding,analgorithmthatcomputeslow-dimensional,neighborhood-preservingembeddingsofhigh-dimensionalinputs͑RoweisandSaul,2000͒.ModiﬁedfromStopferetal.,2003.

plausiblehypothesisisthatHVC’ssynapticconnectionsarenonsymmetricandWLCcanbeamechanismoftheneuralspatiotemporalpatterngenerationofthesong.Thiswouldprovideabasisforthereproduciblepat-ternedoutputfromtheHVCwhenitreceivesasongcommandstimulus.

D.Sequencelearning

Sequencelearningandmemoryassequencegenera-tionrequiretemporalasymmetryinthesystem.Suchasymmetrycanresultfromspeciﬁcpropertiesofthenet-workconnections,inparticular,asymmetryofthecon-nections,orcanresultfromtemporalasymmetryinthedynamicalfeaturesofindividualneuronsandsynapses,orboth.Thespeciﬁcdynamicalmechanismsofsequencelearningdependonthetimescaleofthesequencethat

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FIG.47.͑Coloronline͒HVCsongbirdpatterns.SpikerasterplotoftenHVC͑RA͒neuronsrecordedinonebirdduringsinging.Eachrowoftickmarksshowsspikesgeneratedduringonerenditionofthesongorcall;roughlytenrenditionsareshownforeachneuron.ModiﬁedfromHahnloseretal.,2002.

thisneuralsystemneedstolearn.Learningoffastse-quences,20–30msandfaster,needsprecisesynchroni-zationofthespikesorphasesofneuralwaves.Onepos-siblemechanismforthiscanbethelearningofsynﬁrewaves.Forslowsequences,likeautonomousrepetitivebehavior,itwouldbepreferabletolearnrelevantbehav-ioraleventsthattypicallyoccuronthetimescaleofhun-dredsofmillisecondsorslowerandtheswitching͑tran-sitions͒betweenthem.NetworkswhosedynamicsarebasedonWLCareabletodosuchajob.Weconsiderhereslowsequencelearningandspatialsequentialmemory͑SSM͒.

TheideaisthatsequentialmemoryisencodedinamultidimensionaldynamicalsystemwithaSHS.Eachofthesaddlepointsrepresentsaneventinasequencetoberemembered.Oncethestateofthesystemapproachesoneﬁxedpointrepresentingacertainevent,itisdrawnalonganunstableseparatrixtowardthenextﬁxedpoint,andthemechanismrepeatsitself.Thenecessaryconnec-tionsareformedinthelearningphasesbydifferentsen-soryinputsoriginatedbysequentialevents.

Seligeretal.͑2003͒havediscussedamodeloftheSSMinthehippocampus.Itiswellacceptedthatthehippo-campusplaysthecentralroleinacquisitionandprocess-inginformationrelatedtorepresentingmotioninphysi-calspace.Themostspectacularmanifestationofthisroleistheexistenceofso-calledplacecellswhichrepeat-edly͑ﬁrewhenananimalisinacertainspatiallocationalsoO’KeefefavorsandanDostrovsky,alternative1971concept͒.Experimentalofspatialresearchmemorybasedonalinkedcollectionofstoredepisodes͑WilsonandMcNaughton,1993͒.Eachepisodecomprisesase-quenceofevents,which,besidesspatiallocations,mayincludeotherfeaturesoftheenvironment͑orientation,odor,sound,etc.͒.Itisplausibletodescribethecorre-spondinglearningwithapopulationmodelthatrepre-sentsneuralactivitybyratecoding.Seligeretal.͑2003͒haveproposedatwo-layerdynamicalmodelofSSMthatcananswerthefollowingkeyquestions:͑i͒Howisacertainevent,e.g.,animageoftheenvironment,re-cordedinthestructureofthesynapticconnectionsbe-

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tweenmultiplesensoryneurons͑SNs͒andasingleprin-cipalneuron͑PN͒duringlearning?͑ii͒WhatkindofcooperativedynamicsforcesindividualPNstoﬁrese-quentially,inawaythatwouldcorrespondtoaspeciﬁcsequenceofsnapshotsoftheenvironment?͑iii͒Howcomplexshouldthisnetworkbeinordertostoreacer-tainnumberofdifferentepisodeswithoutmixingdiffer-enteventsorstoringspuriousepisodes?

Thetwo-layerstructureoftheSSMmodelisreminis-centoftheprojectionnetworkimplementationofthenormalformprojectionalgorithm͑NFPA͒;seeBairdandEeckman͑1993͒.IntheNFPAmodel,thedynamicsofthenetworkiscastintermsofnormalformequationswhicharewrittenforamplitudesofcertainnormalformscorrespondingtodifferentpatternsstoredinthesystem.Thenormalformdynamicscanbechosentofollow͑improved1993͒havecertaincapacityshowndynamicalcanthatrules.BairdandEeckmanbeabuiltHopﬁeld-typeusingthisapproach.networkwithFur-thermore,ithasbeensuggested͑BairdandEeckman,1993͒thatspeciﬁcchoicesofthecouplingmatrixforthenormalformdynamicscanleadtomultistabilityamongmorecomplexattractingsetsthansimpleﬁxedpoints,suchaslimitcyclesorevenchaoticattractors.Forex-ample,quasiperiodicoscillationscanbedescribedbyanormalformthatcorrespondstoamultipleHopfbifur-cation͑GuckenheimerandHolmes,1986͒.Asshownbe-low,amodelofSSMafterlearningiscompletedcanbeviewedasavariantoftheNFPAwithaspeciﬁcchoiceofnormalformdynamicscorrespondingtowinnerlesscompetitionamongdifferentpatterns.

Toillustratetheseideasconsideratwo-levelnetworkofNcanreasonablysSNs͓xi͑t͔͒assumeandNthatpprincipalneurons͓asensoryneuronsdoi͑nott͔͒.haveOnetheirowncomplexdynamicsandareslavedeithertoexternalstimuliinthelearningorstoringregimeortothePNsintheretrievalregime.Inthelearningregime,xi͑t͒isabinaryinputpatternconsistingof0’sand1’s.

Duringtheretrievalphase,xϫNi͑t͒=͚ofconnectionsjN=1pPijaj͑t͒,wherePijis

theNspprojectionmatrixamongSNsandPNs.

ThePNsaredrivenbySNsduringthelearningphase,buttheyalsohavetheirowndynamicscontrolledbyin-hibitoryinterconnections.Whenlearningiscomplete,thedirectdrivingfromSNsisdisconnected.Theequa-tionsforthePNratesai͑t͒read

daNN

i͑t͒

sdt=ai͑t͒−ai͑t͚͒Vijaj͑t͒+␣ai͚PTijxj͑t͒+␰͑t͒,j=1

j=1͑30͒

where␣󰀅0inthelearningphaseand␣=0inthere-trievalphase,andPTijistheprojectionmatrix.Thecou-plingbetweenSNsandPNsisbidirectional.Thelast

termontheright-handsideofEq.͑30͒representssmallpositiveexternalperturbationswhichcaninputsignalsfromotherpartsofthebrainthatcontrollearningandretrievaldynamics.

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Afteracertainpatternispresentedtothemodel,the

sensorystimuliresetthestateofthePNlayeraccording

totheprojectionruleaaccordingtoEq.͑30͒.

i͑t͒=͚jN=1SPTijxj͑t͒,butai͑t͒change

ThedynamicsofSNsandPNsduringthelearningandretrievalphaseshavetwolearningprocesses:͑i͒formingtheprojectionmatrixPofsensoryijwhichisresponsibleforcon-nectingagroupneuronsoftheﬁrstlayercor-respondingtoacertainstoredpatterntoasinglePNwhichrepresentsthispatternatthePNlevel;and͑ii͒learningofthecompetitionmatrixVforthetemporal͑logical͒orderingijwhichisrespon-sibleofthesequentialmemory.

Theslowlearningdynamicsoftheprojectionmatrixiscontrolledbythefollowingequation:

P˙ij=⑀ai͑␤xj−Pij

͒͑31͒

with⑀Ӷ1.WeassumethatinitiallyallP␩ijconnectionsarenearlyidenticalPij=1+ij,where␩ijaresmallran-domperturbations,͚j␩ij=0,͗␩22

thatinitiallythematrixij͘=␩V0Ӷ1.Additionally,weassumefori󰀅j.

ijispurelycompeti-tive:VSupposeii=1andVwewantij=Vto0Ͼ1memorizeacertainpatternAinourprojectionmatrix.WeapplyasetofinputsAspondingtothepatternAoftheSNs.Asbefore,icorre-weassumethatexternalstimulirendertheSNsinoneoftwostates:excited,Ai=1,andquiescent,Ai=0.Theini-tialstateofthePNlayerisfullyexcited:ai͑0͒=͚jPijAj.Accordingtothecompetitivenatureofinteractionsbe-tweenPNsafterashorttransient,onlyoneofthem,theneuronAwhichcorrespondstothemaximumamainsexcitedandtheothersbecomequiescent.i͑Which0͒,re-neuronbecomesresponsibleforthepatternAisactuallyrandom,asitdependsontheinitialprojectionmatrixPij.ItfollowsfromEq.͑31͒thatforsmall⑀synapsesofsuppressedPNsdonotchange,whereassynapsesofthesingleexcitedneuronevolvesuchthatconnectionsbe-tweenexcitedSNsandPNsneuronsamplifytoward␤ϾSNs1,anddecayconnectionstozero.Asbetweenaresult,excitedtheﬁrstPNsinputandpatternquiescentwillberecordedinoneofthematrixPijrows,whileotherrowswillremainalmostunchanged.Nowsupposethatwewanttorecordasecondpatterndifferentfromtheﬁrstone.Wecanrepeattheproceduredescribedabove,namely,applyexternalstimuliassociatedwithpatternBtotheSNs,projectthemtotheinitialstateofthePNlayer,aapticconnectionsi͑0͒=͚jPijBfromj,andletthesystemevolve.Sincesyn-SNssuppressedbytheﬁrstpat-terntoneuronAhavebeeneliminated,anewsetofstimulicorrespondingtopatternBwillexciteneuronAmoreweaklythanmostoftheothers,andcompetitionwillleadtoselectionofonePNBdifferentfromneuronA.InthiswaywecanrecordasmanypatternsastherearePNs.

ThesequentialorderofthepatternsrecordedintheprojectionnetworkisdeterminedbythecompetitionmatrixV󰀅jandVij,Eq.͑30͒.InitiallyitissettoVij=V0Ͼ1foriii=1whichcorrespondstowinner-take-allcom-petition.Thegoalofsequentialspatiallearningisto

Rabinovichetal.:Dynamicalprinciplesinneuroscience

FIG.48.Amplitudesofprincipalneuronsduringthememoryretrievalphaseinatwo-layerdynamicalmodelofsequentialspatialmemory.͑a͒Periodicretrieval,twodifferenttestpat-ternspresented;͑b͒aperiodicretrievalwithmodulatedinhibi-tion͑seetext͒.ModiﬁedfromSeligeretal.,2003.

recordthetransitionofpatternAtopatternBintheformofsuppressingthecompetitionmatrixelementVBA.Wesupposethattheslowdynamicsofthenondi-agonalelementsofthecompetitionmatrixarecon-trolledbythedelay-differentialequation

V˙ij=⑀ai͑t͒aj͑t−␶͒͑V1−Vij

͒,͑32͒

where␶isconstant.Equation͑32͒showsthatonlythematrixelementscorrespondingtoa󰀅0arechangingtowardtheasymptotici͑t͒value󰀅0andVaj͑t−␶͒respondingtothedesiredtransition.Sincemost1Ͻ1cor-ofthetime,exceptforshorttransients,onlyonePNisexcited,onlyoneoftheconnectionsVijischangingatanytime.Asaresult,anarbitrary,nonrepeating,sequenceofpat-ternscanberecorded.

WhenatestpatternTispresentedtothesensorylayer,xi͑0͒=T͑i͒ai͑0͒=͚iPijTTj,andTresemblesoneoftherecordedpatterns,thiswillinitiateaperiodicse-quenceofpatternscorrespondingtothepreviouslyre-cordedsequenceinthenetwork.Figure48showsthebehaviorofprincipalneuronsafterdifferentinitialpat-ternsresemblingdifferentdigitshavebeenpresented.Inbothcases,thesystemquicklysettlesontoacyclicgen-erationofpatternsassociatedwithagiventestpattern.Atanygiventime,exceptforashorttransienttimebe-tweenpatterns,onlyasinglePNison,correspondingtoaparticularpattern.

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E.Sequencesincomplexsystemswithrandomconnections

Thelevelofcellularandnetworkcomplexityinthenervoussystemleadsonetoask:Howdoevolutionandgeneticsbuildacomplexbrain?Comparativestudiesoftheneocortexindicatethatearlymammalianneocorti-ceswerecomposedofonlyafewcorticalﬁeldsandinprimatestheneocortexexpandeddramatically;thenum-berofcorticalﬁeldsincreasedandtheconnectivitybe-tweenthembecameverycomplex.Thearchitectureofthemicrocircuitryofthemammalianneocortexremainslargelyunknownintermsofcell-to-cellconnections;however,theconnectionsofgroupsofneuronswithothergroupsarebecomingbetterunderstoodthankstonewanatomicaltechniquesandtheuseofslicetech-niques.Manypartsoftheneocortexdevelopedunderstrictgeneticcontrolasprecisenetworkswithconnec-tionsthatappearsimilarfromanimaltoanimal.Ko-zloskietal.͑2001͒discussedvisualnetworksinthiscon-text.However,thelocalconnectivitycanbeprobabilisticorrandomasaconsequenceofexperience-dependentplasticityandself-organization͑Chklovskiietal.,2004͒.Inparticular,theimagingofindividualpyramidalneu-rons͑drivesMaravallinthetheetmouseformational.,2004barrel͒cortexoveraperiodofweeksandshowedeliminationthatsensoryofsynapsesexperienceandthatthesechangesmightunderlieadaptiveremodelingofneuralcircuits.

Thusthebrainappearsasacompromisebetweenex-istinggeneticconstraintsandtheneedtoadapt,i.e.,net-worksareformedbybothgeneticsandactivity-dependentorself-organizingmechanisms.Thismakesitverydifﬁculttodeterminetheprinciplesofnetworkar-chitectureandtobuildreasonabledynamicalmodelsthatareabletopredictthereactionsofacomplexneuralsystemtochangesintheenvironment;wehavetotakeintoaccountthatevenself-organizednetworksareun-dergeneticcontrolbutinadifferentsense.Forexample,geneticscancontroltheaveragebalancebetweenexci-tatoryandinhibitorysynapticconnections,sparsenessoftheconnections,etc.ThepointofviewthattheinfantcortexisnotacompletelyorganizedmachineisbasedonthesuppositionthatthereisinsufﬁcientstoragecapacityintheDNAtocontroleveryneuronandeverysynapse.This͑Ince,ideawasformulatedﬁrstAsimple1992͒.

byAlanTuringin1948calculationrevealsthatthetotalsizeofthehumangenomecanspecifytheconnectivityofabout105neurons.Thehumanbrainactuallycontainsaround1011neurons.LetussaythatwehaveNneurons.Eachneu-ronrequiresNplog2Nbitstocompletelyspecifyitscon-nections,wherepistheaveragenumberofconnections.ThereforeweneedatleastN2plog2Nbitstospecifytheentireon-offconnectivitymatrixofNneurons.IftheconnectivitydegreepisnotverysparsethenwejustneedN2bits.So,ifwesolvemin͑N2,N2plog109basepairsinthehumangenomeusinga2N͒=3.3ϫconnec-tivitydegreeof1%,weobtainamaximumof105neu-ronsthatcanbecompletelyspeciﬁed.Sincewedonotknowhowmuchofthegenomeisusedforbrainconnec-

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tivity,itisnotpossibletonarrowdowntheestimation.Nevertheless,itdoesnotmakesensetoexpectthewholegenometospecifyallconnectionsinthebrain.Thissimpleestimatemakesclearthatlearningandsyn-apticplasticityhaveaveryimportantroleindetermin-ingtheconnectivityofbraincircuits.

Thedynamicsofcomplexnetworkmodelsaredifﬁculttodissect.Themappingofthecorrespondinglocalandglobalbifurcationsinalow-dimensionalsystemhasbeenextensivelystudied.Toperformsuchanalysisinhigh-dimensionalsystemsisverydemandingifnotimpos-sible.Averagemeasures,suchasmeanﬁringrates,aver-agemembranepotential,correlations,etc.,canhelpustounderstandthedynamicsofthenetworkasafunctionofafewvariables.Oneoftheﬁrstmodelstouseamean-ﬁeldapproachwastheWilson-Cowanmodel͑WilsonandCowan,1973͒.Individualneuronsinthemodelre-sembleintegrate-and-ﬁreneuronswithamembranein-tegrationtime␮andarefractoryperiodr.WilsonandCowan’smainhypothesisisthattheunreliableindi-vidualresponses,whengroupedtogether,canleadtomorereliableoperations.TheWilson-Cowanformalismcanbereducedtothefollowingequations:

␮ץE͑x,t͒

ץt

=−E͑x,t͒+͓1−rE͑x,t͔͒ϫLeͫ͵E͑y,t͒wee͑y,x͒dy

−

͵I͑y,t͒wei͑y,x͒dy+Se͑x,t͒ͬ,

͑33͒

␮ץI͑x,t͒

ץt

=−I͑x,t͒+͓1−rI͑x,t͔͒ϫLiͫ͵E͑y,t͒wie͑y,x͒dy

−

͵I͑y,t͒wii͑y,x͒dy+Si͑x,t͒ͬ,

͑34͒

whereE͑x,t͒andI͑x,t͒aretheproportionsofﬁringneuronsintheexcitatoryandinhibitorypopulation,thecoordinatexisacontinuousvariablethatrepresentsthepositioninthecorticalsurface,wee,wei,wie,andwiiaretheconnectivityweights,andSeandSpopulations,iareexternalin-putstotheexcitatoryandinhibitoryrespec-tively.ThegainfunctionsLexcitatoryeandLibasicallyreﬂecttheexpectedproportionsofandinhibitoryneu-ronsreceivingatleastthresholdexcitationperunitoftime.Onesubtletrickusedinthederivationofthismodelisthatthemembraneintegrationtimeisintro-ducedthroughsynapticconnections.Themodelex-pressedinthisformattemptstoeliminatetheuncer-taintyofsingleneuronsbygroupingthemaccordingtothosewithreliablecommonresponses.Wearestillleftwiththeproblemofwhattoexpectinanetworkofclus-tersconnectedrandomlytoeachother.Herewewilldiscussitinmoredetail.

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Inarandomnetworkofexcitatoryandinhibitoryneu-rons,itisnotuncommontoﬁndoscillatoryactivity͑Jin,2002;HuertaandRabinovich,2004͒.However,itismoreinterestingtostudythetransientbehaviorofneuralre-currentnetworks.Thesearefastbehaviorsandimpor-tantforsensoryprocessingandforthecontrolofmotorcommands.Instudyingthisoneneedstoaddresstwomainissues:͑i͒whetheritispossibletoconsistentlyﬁndnetworkswithrandomconnections,describedbyequa-tionssimilartoEqs.͑33͒and͑34͒,behavingregularly,and͑ii͒whethertransientbehaviorinthesenetworksisreproducible.

HuertaandRabinovich͑2004͒showed,usingtheWilson-Cowan͑controllimitcyclesparameter͒ismoreformalism,spacelikelywheretoperiodicbeinhibitoryfoundsequentialinactivityandregionsexcitatoryofthesynapsesareslightlyoutofbalance.However,reproduc-ibletransientdynamicsismorelikelyfoundinthere-gionofparameterspacefarfrombalancedexcitationandinhibition.Inparticular,theauthorsinvestigatedthemodel

dxNNI

␮i͑t͒

=⌰͚ͩ

EwEE−j=1

ijxj͑t͒

͚wEIijyj͑t͒+SE

−xi͑t͒,j=1

iͪ͑35͒

dyNNi͑t͒

␮dt

=⌰͚ͩ

wIE−j=1

ijxj͑t͒

͚wIIijyj͑t͒+SI

iͪ−yi͑t͒,j=1

͑36͒

wherex͑t͒andyi͑t͒representthefractionsofactiveneu-ronsinclusterioftheexcitatoryandinhibitorypopula-tions,respectively.Thenumbersofexcitatoryandinhibi-toryclustersareNEandNI.ThelabelsEandIareused

todenotequantitiesassociatedwiththeexcitatoryorinhibitorypopulations,respectively.TheexternalinputsSE,Iareinstantaneouskicksappliedtoafractionofthetotalpopulationattimezero.Thegainfunctionis⌰͑z͒=͕tanh͓͑z−b͒/␴͔+1͖/2,withathresholdb=0.1belowtheexcitatoryandinhibitorysynapticstrengthofasingleconnection.Clustersaretakentohaveverysharpthresholdsofexcitabilitybychoosing␴=0.01.Thereisawiderangeofvaluesthatgeneratessimilarresults.ThetimescaleissetasdonebyWilsonandCowan͑1973͒,␮drawn=10ms.fromTheaBernoulliconnectivityprocessmatrices͑HuertawXYij

haveentriesandRabino-vich,2004͒.Themaincontrolparametersinthisproblemaretheprobabilitiesofconnectionsfrompopulationtopopulation.

Nowwecananswerthefollowingquestion:Whatkindofactivitycananetworkwithmanyneuronsandran-domconnectionsproduce?Intuitionsuggeststhattheanswerhastobeacomplexmultidimensionaldynamics.However,thisisnotthecase͑Fig.49͒:mostobservablestimulus-dependentdynamicsaremoresimpleandre-producible;periodic,transient,orchaotic͑alsolowdi-mensional͒.

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FIG.49.͑Coloronline͒Three-dimensionalprojectionsofsimulationsofrandomnetworksof200neurons.Forillustra-tivepurposesweshowthreetypesofdynamicsthatcanbegeneratedbyarandomnetwork:͑top͒chaos,͑middle͒limitcycle͑bothintheareasofparameterspacethatareclosetobalanced͒,and͑bottom͒transientdynamics͑farfrombal-anced͒.

Thisisaveryimportantpointforunderstandingcor-texdynamicsthatinvolvesthecooperativeactivityofmanycomplexnetworks͑unitsormicrocircuits͒.Fromthefunctionalpointofview,thestimulus-dependentdy-namicsofthecortexcanbeconsideredasacoordinatedbehaviorofmanyunitswithlow-dimensionaltransientdynamics.Thisisthebasisofanewapproachtocortexmodelingnamedthe“liquid-statemachine”͑Maassetal.,2002͒.

F.Coordinationofsequentialactivity

Coordinationofdifferentsequentialbehaviorsiscru-ciallyimportantforsurvival.Fromthemodelingpointofviewitisaverycomplexproblem.TheIO͑anetwork

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

FIG.50.͑Color͒Spatiotemporalpatternsofcoordinatedrhythmsinducedbystimuliinamodeloftheinferiorolive.Severalstructureswithdifferentfrequenciescancoexistsimul-taneouslyinacommensuraterepresentationofthespikingfre-quencieswhenseveralstimuliarepresent.Incommensuratestimuliareintroducedintheformofcurrentinjectionsindif-ferentclustersofthenetwork.͑Panelsontherightshowthepositionsoftheinputclusters.͒Thesecurrentinjectionsinducedifferentspikingfrequenciesintheneurons.Colorsinthesepanelsrepresentdifferentcurrentinjections,andthusdifferentspikingfrequenciesintheinputclusters.Toprowshowstheactivityofanetworkwithtwodifferentinputclusters.Bottomrowshowstheactivityofanetworkwith25differentinputclusters.Sequencesdevelopintimefromlefttoright.Regionswiththesamecolorhavesynchronousbehavior.Colorbarmapsthemembranepotential.Redcorrespondstospikingneurons͑−45mVisabovetheﬁringthresholdinthemodel͒.Darkbluemeanshyperpolarizedactivity.Bottompanelshowstheactivityofasingleneuronwithsubthresholdoscillationsandspikingactivity.ModiﬁedfromVarona,Aguirre,etal.,2002.

alreadydiscussedinSec.III.B.2͒hasbeensuggestedasasystemthatcoordinatesmotorvoluntarymovementsin-volvingseveralsimultaneousrhythms͑LlinásandWelsh,1993͒.Hereanexampleofhowsubthresholdoscillationscoordinatedifferentincommensuraterhythmsinacom-mensuratefashionisshown.IntheIO,neuronsareelec-tricallycoupledtotheircloseneighbors.Theiractivityischaracterizedbysubthresholdoscillationsandspikingactivity͑seeFig.50͒.ThecooperativedynamicsoftheIOundertheactionofseveralincommensurateinputshasbeenmodeledbyVarona,Aguirre,etal.͑2002͒.Theresultsoftheselarge-networksimulationsshowthattheelectricalcouplingofIOneuronsproducesquasisyn-chronizedsubthresholdoscillations.Becausespikingac-tivitycanhappenonlyontopoftheseoscillations,in-commensurateinputscanproduceregionswithdifferentcommensuratespikingfrequencies.Severalspikingfre-quenciesareabletocoexistinthesenetworks.Theco-existenceofdifferentrhythmsisrelatedtothedifferentclusterizationofthespatiotemporalpatterns.

Anotherimportantquestionrelatedtocoordinationofseveralsequentialbehaviorsconcernsthedynamicalprinciplesthatcanbeabasisforfastneuronalplanningandreactiontoachangingenvironment.Onemight

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thinkthattheWLCprinciplecanbegeneralizedinor-dertoorganizethesequentialswitchingaccordingto͑i͒thelearnedskilland͑ii͒thedynamicalsensoryinputs.ThecorrespondingmathematicalmodelmightbesimilartoEqs.͑26͒–͑29͒togetherwithalearningrulesimilartoEq.͑25͒.StimuliSlchangesequentiallyandthetimingofeachstep͑thetimethatthesystemspendsmovingfromthevicinityofonesaddletothevicinityofthenextone;seeFig.51͒shouldbecoordinatedwiththetimeofchangeintheenvironment.Inrecurrentnetworks,asaresultoflearning,thestimuluscangosequentiallytothespeciﬁcgoalofanoptimalheteroclinicsequenceamongmanysuchsequencesthatexistinthephasespaceofthemodel.Whatisimportantisthatatthesametime,i.e.,inparallelwiththechoosingoftherestofthemotorplan,thealreadyexistingpartofthemotoractivityplanisexecuted.

Thetwoideasjustdiscussedcanbeappliedtothecerebellarcircuit,whichisanexampleofacomplexre-currentnetwork͑seeFig.52͒.Togiveanimpressionofthecomplexityofthecerebellarcortexwenotethatitisorganizedintothreelayers:themolecularlayer,thePurkinjecelllayer,andthegranulecelllayer.Onlytwosigniﬁcantinputsreachthecerebellarcortex:mossyﬁ-bersandclimbingﬁbers.Mossyﬁbersareinthemajority͑mation4:1͒andofcarrymultipleawealthmodalities.ofsensoryTheyandmakecontextualspecializedinfor-excitatorysynapsesinstructurescalled“glomeruli”withthedendritesofnumerousgranulecells.Granulecellaxonsformparallelﬁbersthatruntransverselyinthemolecularlayer,makingexcitatorysynapseswithPurkinjecells.EachPurkinjecellreceivesϷ150000syn-apses.Thesesynapsesarethoughttobemajorstoragesitesfortheinformationacquiredduringmotorlearning.ThePurkinjecellaxonprovidestheonlyoutputfromthecerebellarcortex.Thisisviathedeepcerebellarnu-clei.EachPurkinjecellreceivesjustoneclimbingﬁberinputfromtheinferiorolive,butthisinputisverypow-erfulbecauseitinvolvesseveralhundredsofsynapticcontacts.Theclimbingﬁberisthoughttohavearoleinteachinginthecerebellum.TheGolgicellisexcitedbymossyﬁbersandgranulecellsandexercisesaninhibi-toryfeedbackcontrolupongranulecellactivity.StellateandbasketcellsareexcitedbyparallelﬁbersinordertoprovidefeedforwardinhibitiontoPurkinjecells.

Thehugenumberofinhibitoryneuronsandthearchi-tectureofthecerebellarnetworks͑deZeeuwetal.,1998͒supportthegeneralizedWLCmechanismforco-ordination.AwidelydiscussedhypothesisisthatthespeciﬁccircuitryoftheIO,cerebellarcortex,anddeepcerebellarnucleicalledtheslowloop͑seeFig.52͒canserveasadynamicalworkingmemoryorasaneuronalclockwithϷ100-mscycletimewhichwouldmakeiteasytoconnectittobehavioraltimescales͑KistleranddeZeeuw,2002;Melamedetal.,2004͒.

Temporalcoordinationand,inparticular,synchroniza-tionofneuralactivityisarobustphenomenon,fre-quentlyobservedacrosspopulationsofneuronswithdi-versemembranepropertiesandintrinsicfrequencies.Inthelightofsuchdiversitythequestionofhowprecise

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

FIG.51.IllustrationofthelearnedsequentialswitchinginarecurrentnetworkwithWLCdynamics:Thinlines,possiblelearnedsequences;thickline,sequentialswitchingchosenon-linebythedynamicalstimulus.

synchronizationcanbeachievedinheterogeneousnet-worksiscritical.Severalmechanismshavebeensug-gestedandmanyofthemrequireanunreasonablyhighdegreeofnetworkhomogeneityorverystrongconnec-tivitytoachievecoherentneuralactivity.Asdiscussedabove͑Sec.II.A.4͒,inanetworkoftwosynapticallycoupledneuronsSTDPatthesynapseleadstothedy-namicalself-adaptationofthesynapticconductancetoavaluethatisoptimalfortheentrainmentofthepostsyn-apticneuron.ItisinterestingtonotethatjustafewSTDPsynapsesareabletomaketheentrainmentofa

FIG.52.Aschematicrepresentationofthemammaliancer-ebellarcircuit.Arrowsindicatethedirectionoftransmissionacrosseachsynapse.Sourcesofmossyﬁbers:Ba,basketcell;BR,brushcell;cf,climbingﬁber;CN,cerebellarnuclei;Go,Golgicell;IO,inferiorolive;mf,mossyﬁber;pf,parallelﬁber;PN,pontinenuclei;sbandsmb,spinyandsmoothbranchesofPcelldendrites,respectively;PC,Purkinjecell;bat,basketcellterminal;pcc,Pcellcollateral;no,nucleo-olivarypathway;nc,collateralofnuclearrelaycell.ModiﬁedfromVoogdandGlickstein,1998.

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heterogeneousnetworkofelectricallycoupledneuronsmoreeffective͑ZhigulinandRabinovich,2004͒.IthasbeenshownthatsuchanetworkoscillateswithamuchhigherdegreeofcoherencethanwhenitissubjecttostimulationthatismediatedbySTDPsynapsesascom-paredwithstimulationthroughstaticsynapses.Theob-servedphenomenondependsonthenumberofstimu-latedneurons,thestrengthofelectricalcoupling,andthedegreeofheterogeneity.Inreality,long-termplastic-itydependsnotonlyonspiketiming͑STDP͒butalsoontheﬁringrateandthecooperativityamongdifferentneuronalinputs͑Sjöströmetal.,2001͒.Thismakesmod-elingself-organizationandlearningmorechallenging.Realbehaviorinnonstationaryorcomplexenviron-ments,asalreadydiscussed,requiresswitchingbetweendifferentsequentialactivities.Janckeetal.͑2000͒haveidentiﬁeddistributedregionsindifferentpartsofthecortexthatareinvolvedintheswitchingamongsequen-tialmovements.Itisimportantfordynamicalmodelingthatthisdifferentialpatternofactivationisnotseenforsimplerepetitivemovements.Thussuchmovementsaretoosimpletoevokeadditionalactivation.Thismeansthatadynamicalmodelthataimstodescribethese-quentialbehavioringeneralhastocorrectlydescribetheswitchingfromalow-dimensionalsubspacetoahigh-dimensionalstatespace,andviceversa.Therearenogeneralmethodsfordescribingmultidimensionaldis-sipativenonlinearsystemswithsuchtransientbutrepro-ducibledynamics.WethinkthattheWLCprinciplemightbetheﬁrststepinthisdirection.

V.CONCLUSION

Physicists,mathematicians,andphysiologistsallagreethatanimportantattributeofanydynamicalmodelofCNSactivityisthatnotonlyshoulditbeabletoﬁttheavailableanatomicalandphysiologicaldata,butitshouldalsobecapableofexplainingfunctionandpre-dictingbehavior.However,thewaysinwhichphysicistsandmathematicians,ononehand,andphysiologists,ontheotherhand,usemodelingarebasedontheirownexperienceandviewsandthusaredifferent.Inthisre-viewwetriedtobringthesedifferentviewpointsclosertogetherand,usingmanyexamplesfromthesensory,motor,andcentralnervoussystems,discussedjustafewprincipleslikereproducibility,adaptability,robustness,andsensitivity.

Letusreturntothequestionsformulatedatthebe-ginningofthereview:

•Whatcannonlineardynamicslearnfromneuro-science?•Whatcanneuroscience

learn

from

nonlinear

dynamics?

Afterreadingthisreview,wehopethereadercanjoinusinintegratingthekeymessagesinourpresentation.Perhapswemayofferourcompactformulation.

Addressingtheﬁrstquestionofwhatnonlineardy-namicscanlearnfromneuroscience:

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

•Themostimportantactivitiesofneuronalsystemsaretransientandcannotbeunderstoodbyanalyzingattractordynamicsalone.Theseneedtobeaug-mentedbyreliabledescriptionsofstimulus-dependenttransientmotionsinstatespaceasthiscomprisestheheartofmostneurobiologicalactivity.Nonetheless,becausethedynamicsofrealisticneu-ronalmodelsarestronglydissipative,theirstimulus-dependenttransientbehaviorisstronglyattractedtosomelow-dimensionalmanifoldsembeddedinthehigh-dimensionalstatespaceoftheneuralnetwork.Itisastrongstimulustononlineardynamicstode-velopatheoryofreasonablylow-dimensionaltran-sientactivityand,inparticular,toconsiderthelocalandglobalbifurcationsofsuchobjectsashomoclinicandheteroclinictrajectories.•Formanydynamicalproblemsofneuroscience,incontrasttotraditionaldynamicalapproaches,theini-tialconditionsdomattercrucially.Persistentneu-ronalactivity͑i.e.,dynamicalmemory͒,stimulus-dependenttransientcompetition,stimulus-dependenttransientsynchronization,andstimulus-dependentsynapticplasticityareallaspectsofthis.Clearly,addressingtheseimportantphenomenawillrequireanexpansioninourapproachestodynamicalsystems.Addressingthesecondquestionofwhatneurosciencecanlearnfromnonlineardynamics:

•Dynamicalmodelsconﬁrmthekeyroleofinhibitioninneuronalsystems.Thefunctionofinhibitionisnotjusttoorganizeabalancewithexcitationinordertostabilizeanetworkbutmuchmore:͑a͒inhibitorynetworkscangeneraterhythms,suchasreproducibleandadaptivemotorrhythmsinCPGs,orgammarhythmsinthebrain;͑b͒theyareresponsibleforthetransformationofanidentitysensorycodetoaspa-tiotemporalcodeimportantforbetterrecognitioninanacousticallyclutteredenvironment;and͑c͒thankstoinhibition,neuralsystemscanbeatthesametimeverysensitivetotheirinputandrobustagainstnoise.•Dynamicalchaosisnotjustafundamentalphenom-enonbutalsoimportantforthesurvivaloflivingor-ganisms.Neuronalsystemsmayusechaosfortheor-ganizationofnontrivialbehaviorsuchastheirregularhunting-swimmingofClioneandfortheor-ganizationofhigherbrainfunctions.

•Theimprovementinyield,stability,andlongevityofmultielectroderecordings,newimagingtechniques,combinedwithnewdataprocessingmethods,haveallowedneurophysiologiststodescribebrainactivi-tiesasthedynamicsofspatiotemporalpatternsinsomevirtualspace.Wethinkthisisabasisforbuild-ingabridgebetweentransientlarge-scalebrainac-tivityandanimalbehavior.

Andﬁnallyaswepursuetheinvestigationofdynami-calprinciplesinneuroscience,wehopethateventuallynottoseethesetwoquestionsapartfromoneanother

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butasanintegratedapproachtodeepandcomplexsci-entiﬁcproblems.

ACKNOWLEDGMENTS

WethankRamonHuertaandThomasNowotnyfortheirhelp,andRafaelLeviandValentinZhigulinforusefulcomments.ThisworkwassupportedbyNSFGrantNo.NSF/EIA-0130708,andGrantNo.PHY0414174;NIHGrantNo.1R01NS50945andGrantNo.NS40110;MECBFI2003-07276,andFundaciónBBVA.

GLOSSARY

antennallobe,theﬁrstsiteofsensoryintegrationfromtheol-factoryreceptorsofinsects.AMPAreceptors

transmembranereceptorfortheneurotransmitterglutamatethatmediatesfastsynaptictransmission.

bumpsspatiallylocalizedregionsofhighneuralactivity.

CA1subsystemofthehippocampuswithaveryactiveroleingen-eralmemory.

carbacholchemicalthatinducesoscilla-tionsininvitropreparations.Clionemarinemolluskwhosenervoussystemisfrequentlyusedinneurophysiologystudies.CNScentralnervoussystem.

CPG

centralpatterngenerator,asmallneuralcircuitthatcanproducestereotypedrhythmicoutputswithoutrhythmicsen-soryorcentralinput.

depolarization

anychangeintheneuronmem-branepotentialthatmakesitmorepositivethanwhenthecellisinitsrestingstate.

dynamicclamp

acomputersetuptoinsertvir-tualconductancesintoaneuralmembranetypicallyusedtoaddsynapticinputtoacellbycalculatingtheresponsecur-renttoaspeciﬁcpresynapticin-put.

GABA

neurotransmitteroftypicallyinhibitorysynapses;theycanbemediatedbyfastGABA͑A͒orslowGABA͑B͒receptors.

heteroclinicloopaclosedchainofheteroclinictrajectories.

heteroclinictrajectory

trajectorythatliessimulta-neouslyonthestablemanifoldofonesaddlepoint͑orlimitcycle͒andtheunstablemani-foldofanothersaddle͑orlimitcycle͒connectingthem.

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

Hodgkin-Huxleyneuronmodel.

HVChighvocalcenterinthebrainof

songbirds.

hyperpolarizationanychangeintheneuronmem-branepotentialthatmakesit

morenegativethanwhenthecellisinitsrestingstate.

interneuronsneuronswhoseaxonsremain

withinaparticularbrainregionascontrastedwithprojectionneurons,whichhaveaxonspro-jectingtootherbrainregions,orwithmotoneurons,whichin-nervatemuscles.

IOinferiorolive,aneuralsystem

thatisaninputtothecerebel-larcortexpresumablyinvolvedinmotorcoordination.

KCsKenyoncells,interneuronsof

themushroombodyofinsects.

Kolmogorov-Sinaiameasureofthedegreeofpre-entropydictabilityoffurtherstatesvis-itedbyachaotictrajectory

startedwithinasmallregioninastatespace.

LPlateralpyloricneuronofthe

crustaceanstomatogastricCPG.

LTDlong-termdepression,activity-dependentdecreaseofsynaptic

efﬁcacytransmission.

LTMlong-termmemory.LTPlong-termpotentiation,

activity-dependentreinforce-mentofsynapticefﬁcacytrans-mission.

Lyapunovexponents␭jtherateofexponentialdiver-gencefromperturbedinitialconditionsinthejthdirectionofthestatespace.Fortrajecto-riesbelongingtoastrangeat-tractorthespectrum␭pendentofinitialconditionsjisinde-andcharacterizesthestablechaoticbehavior.

MCsmicrocircuits;circuitscom-posedofasmallnumberof

neuronsthatperformspeciﬁcoperationaltasks.

mushroombodylobedsubsystemoftheinsect

braininvolvedinclassiﬁcation,learning,andmemoryofodors.

mutualinformationameasureoftheindependence

oftwosignalsXandY,i.e.,theinformationofXthatissharedbyY.Inthediscretecase,ifthejointprobabilitydensityfunc-tionofXandYisp͑x,y͒=P͑X=x,Y=y͒,theprobability

1260

Rabinovichetal.:Dynamicalprinciplesinneuroscience

densityfunctionofXaloneisf͑x͒=P͑X=x͒,andtheprob-abilitydensityfunctionofYaloneisg͑y͒=P͑Y=y͒,thenthemutualinformationofXandYisgivenbyI͑X,Y͒=͚neuromodulators

asubstancex,yp͑x,y͒logother2͓p͑xthan,y͒/f͑ax͒neu-g͑y͔͒.rotransmitter,releasedbyneu-ronsthatcanaffecttheintrinsicandsynapticdynamicsofotherneurons.

neurotransmitterschemicalsthatareusedtorelayatthesynapsesthesignalsbe-tweenneurons.

pacemakerneuronorcircuitthathasen-dogenousrhythmicactivity.PD

pyloricdilatorneuronofthecrustaceanCPG.

phase͑lockingsynchronization͒theshiponsetofacertaincoupledbetweenself-sustainedthephasesrelation-oscilla-oftors.

placecell

atypeofneuronfoundinthehippocampusthatﬁresstronglywhenananimalisinaspeciﬁclocationinanenvironment.plasticitychangesthatoccurintheorga-nizationofsynapticconnec-tionsorintracellulardynamics.PN

projectionorprincipalneurons.Purkinjecellmaincelltypeofthecerebellarcortex.

RApremotornucleusofthesong-birdbrain.

receptor

aproteinonthecellmembranethatbindstoaneurotransmit-ter,neuromodulator,orothersubstance,andinitiatesthecel-lularresponsetotheligand.receptorneuronsensoryneuron.

SHSstableheteroclinicsequence.SNsensoryneuron.

SSMsequentialspatialmemory.

statocyst

balanceorganinsomeinverte-bratesthatconsistsofasphere-likestructurecontainingamin-eralizedmass͑statolith͒andseveralsensoryneuronsalsocalledstatocystreceptors.

STDP

spike-timing-dependentplastic-ity.

STM

short-termmemory.

structuralstability

conditioninwhichsmallchangesintheparametersdonotchangethetopologyofthephaseportraitinthestatespace.

synapse

specializedjunctionthroughwhichneuronssignaltoonean-

Rev.Mod.Phys.,Vol.78,No.4,October–December2006

other.Thereareatleastthreedifferenttypesofsynapses:ex-citatoryandinhibitorychemi-calsynapsesandelectricalsyn-apsesorgapjunctions.

synﬁrechain

propagationofsynchronousspikingactivityinasequenceoflayersofneuronsbelongingtoafeedforwardnetwork.WLC

winnerless-competitionprin-cipleforthenonautonomoustransientdynamicsofneuralsystemsreceivingexternalstimuliandexhibitingsequen-tialswitchingamongtemporal“winners.”

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