CellularMolecular Dynamics and Effective Topology Underlying Synchronization in Networks of

Cellular/Molecular

DynamicsandEffectiveTopologyUnderlyingSynchronizationinNetworksofCorticalNeurons

DannyEytanandShimonMarom

DepartmentofPhysiologyandBiophysics,FacultyofMedicine,Technion–IsraelInstituteofTechnology,Haifa32000,Israel

Cognitiveprocessesdependonsynchronizationandpropagationofelectricalactivitywithinandbetweenneuronalassemblies.Invivomeasurementsshowthatthesizeofindividualassembliesdependsontheirfunctionandvariesconsiderably,butthetimescaleofassemblyactivationisintherangeof0.1–0.2sandisprimarilyindependentofassemblysize.Hereweuseaninvitroexperimentalmodelofcorticalassembliestocharacterizetheprocessunderlyingthetimescaleofsynchronization,itsrelationshiptotheeffectivetopologyofconnectivitywithinanassembly,anditsimpactonpropagationofactivitywithinandbetweenassemblies.Weshowthatthebasicmodeofassemblyactivation,“networkspike,”isathreshold-governed,synchronizedpopulationeventof0.1–0.2sdurationandfollowsthelogisticsofneuronalrecruitmentinaneffectivelyscale-freeconnectednetwork.Accordingly,thesequenceofneuronalactivationwithinanetworkspikeisnonrandomandhierarchical;asmallsubsetofneuronsisconsistentlyrecruitedtensofmillisecondsbeforeothers.Theorypredictsthatscale-freetopologyallowsforsynchronizationtimethatdoesnotincreasemarkedlywithnetworksize;ourexperi-mentswithnetworksofdifferentdensitiessupportthisprediction.Theactivityofearly-to-fireneuronsreliablyforecastsanupcomingnetworkspikeandprovidesmeansforexpeditedpropagationbetweenassemblies.Wedemonstratethiscapacitybyobservingthedynamicsoftwoartificiallycoupledassembliesinvitro,usingneuronalactivityofoneasatriggerforelectricalstimulationoftheother.Keywords:neuron;network;synchronization;scale-free;multielectrodearray;cortex

Introduction

Behaviors,fromsimpletomostcomplex,involveorchestratedactivationofneuralcellassemblies.FunctionallydefinedbyHebb,neuronalassemblyisagroupofcellsthatsharesimilarstaticanddynamicresponsepropertieswhenactivatedthroughspecificreceptors,constituting“...thesimplestinstanceofarepresentativeprocess(imageoridea)”(Hebb,1949).Inaseriesofclassicalelectrophysiologicalstudies(forreview,seeMount-castle,1998),aswellasinlaterexperimentsinwhichlarge-scaleimagingtechnologieswereapplied(Slovinetal.,2002;Ohkietal.,2005),theabstractnotionofcellassemblywasmappedtoactualneuralentities.Thesestudiesshowthat,dependingonthenatureandcomplexityofthestimulus,thenumbersofneuronsconsti-tutinganassemblyrangefromhundredstomanyhundredsofthousands(Roland,2002;Derdikmanetal.,2003).Neuronscon-stitutinganassemblysynchronizeduringpresentationofamatchingstimulus,aswellasduring“ongoingactivity,”intheabsenceofstimulation(Kenetetal.,2003).Regardlessofthestimulusmodality,stimuluscomplexity,orcorticalareain-volved,thecharacteristictimescaleofassemblyactivationisintheorderof0.1–0.2s.Thistimescaleemergeswhetherassemblyac-ReceivedJan.22,2006;revisedJuly12,2006;acceptedJuly12,2006.

ThisworkwaspartiallysupportedbygrantsfromtheIsraelScienceFoundation,theNationalInstituteforPsy-chobiologyinIsrael,andtheMinervaFoundation.WethankEllaandVladimirLyakhovfortechnicalassistanceandDr.NoamZivandAmirMinerbiforpreparationofphotoimages.

CorrespondenceshouldbeaddressedtoDr.ShimonMarom,DepartmentofPhysiologyandBiophysics,FacultyofMedicine,Technion–IsraelInstituteofTechnology,P.O.Box9697,Haifa31096,Israel.E-mailmarom@tx.technion.ac.il.

DOI:10.1523/JNEUROSCI.1627-06.2006

Copyright©2006SocietyforNeuroscience0270-6474/06/268465-12$15.00/0tivationismeasuredinthesensory(Superetal.,2001;Slovinetal.,2002),somatosensory(Derdikmanetal.,2003),ormotor(Riehleetal.,1997)areas.Thesametimescalecharacterizestheactivationof“higher”corticalareasduringcategorizationtasks(Keysersetal.,2001)andduringpureinternalevents(Riehleetal.,1997).Thus,the0.1–0.2stimescaleofassemblyactivationisafundamentalfactorthatmightconstrainthetemporalaspectsofcognition.

Becausethetimescaleofassemblyactivationseemscommontovariouscorticalstructures,itisconceivablethatitsbiophysicalcharacteristicsmaybeobtainedbyexperimentallyanalyzinga“generic”corticalneuronalassembly.Alargesetofexperimentaldataandtheoreticalanalysesindicatesthatanetworkofsparselycoupledcorticalneuronsdevelopinginvitromightbeusefulasanexperimentalmodelforagenericassembly,keepinginmindtheobviousconstraintsonextrapolationsfrominvitrotoinvivoconditions(forreview,seeCorneretal.,2002;MaromandSha-haf,2002).Whendevelopingonarraysofmicroelectrodes(Gross,1979;StengerandMcKenna,1994;Morinetal.,2005)(seeFig.1a),thenetworkscanbesampledsimultaneouslyatmanypointsandinterrogatedbysite-specificstimuliwithvary-ingtemporalandspatialstructures,allowingacontrolledbio-physicalexamination.

Inthepresentstudy,weusenetworksofcorticalneuronsde-velopinginvitroonarraysofmicroelectrodestodescribethebiophysicalprocessunderlyingassemblyactivationintermsofpopulationdynamicsandtopology.Weshowthatassemblyacti-vationisathreshold-governedphenomenon.Wedenotethephe-nomenon“networkspike”(NS).Wepointattheoriginofthe0.1–0.2scharacteristictimescaleandshowthattheunderlying

8466•J.Neurosci.,August16,2006•26(33):8465–8476topologyofeffectiveconnectivityisscalefree.Theimpactofthedynamicsandtopologyatthesingleassemblylevelonthesyn-chronizationandpropagationofactivitywithinandbetweenas-sembliesisdemonstratedexperimentally.

MaterialsandMethods

Culturingthecorticalneuronsonmultielectrodearrays.Corticalneuronswereobtainedfromnewbornratswithin24hafterbirth,followingstan-dardprocedures(ShahafandMarom,2001;MaromandShahaf,2002;Eytanetal.,2003,2004).Theneuronswereplateddirectlyontoasubstrate-integratedmultielectrodearray(MEA).ThecultureswerebathedinMEMsupplementedwithheat-inactivatedhorseserum(5%),glutamine(0.5mM),glucose(20mM),andgentamycin(10␮g/ml)andweremaintainedinanatmosphereof37°C,5%COduringtherecordingphases.2/95%airinatissuecultureincubatoraswellasExperimentswereperformedduringthethirdweekafterplating,thusallowingfunc-tionalandstructuralmaturationoftheneurons.MEAsof60Ti/Au/TiNelectrodes,30␮mindiameter,andspaced200␮mfromeachother(MultiChannelSystems,Reutlingen,Germany)wereused.Theinsula-tionlayer(siliconnitride)waspretreatedwithpoly-D-lysine.Experi-mentslastingϾ3hwereconductedusingaslowperfusionsystemwithperfusionratesofϳ100␮l/h.

Electrophysiologicalrecordings.Acommercial60-channelamplifier(B-MEA-1060;MultiChannelSystems)withfrequencylimitsof1–5000Hzandagainof1024ϫwasused.TheB-MEA-1060wasconnectedtoMCP-Plusvariablegainfilteramplifiers(AlphaOmega,Nazareth,Israel)foradditionalamplification.CurrentstimulationthroughtheMEAwasper-formedusingadedicatedeight-channelstimulusgenerator(MultiChan-nelSystems).Thepairofelectrodesforcurrentpassingwerechosenbasedonpracticalconsiderations;inparticular,weuseelectrodesthatarenotusefulforrecordingsbecauseoftheirnoisycharacter.StimulationparametersaredetailedinthelegendsofFigures6,7,and11.Datawasdigitizedusingtwoparallel5200a/526analog-to-digitalboards(Mi-crostarLaboratories,Bellevue,WA).Eachchannelwassampledatafre-quencyof24,000HzandpreparedforanalysisusingtheAlphaMapin-terface(AlphaOmega).Thresholds(8ϫrootmeansquareunits;typicallyintherangeof10–20␮V)weredefinedseparatelyforeachoftherecordingchannelsbeforethebeginningoftheexperiment.Thedatapresentedinthetextisnotspikesorted.Thebasicobservationofnon-randomrecruitmentiscompletelypreservedalsoafterspikesortingus-ingprincipalcomponentanalysis(supplementalFig.3,availableatwww.jneurosci.orgassupplementalmaterial).Thesupplementalfigurealsoshowsthateachelectrodeinoursetupsensesapproximatelyonetotwoneuronsandrarelythreeneurons.Therobustnessofthemainfind-ingtospike-sortingprocedureandthesmallnumberofneuronssensedbyasingleelectrodeledustoofteninterchangetheterms“electrodeactivity”and“neuronalactivity.”

Detectingnetworkspikes.Networkspikesweredetectedusinganactiv-itythreshold.AtimestampisdefinedwhenactivitycrossesathresholdofNnumberofactionpotentialsrecordedthroughouttheelectrodearraywithinaTmillisecondtimebin.Inmostcases(dataofFigs.1,3a,4b,c,7,9),Nisequaltoone-fourthofthe(active)electrodes,andTϭ3ms.AnactiveelectrodeisdefinedasdemonstratinganaveragefiringrateofϾ0.02sϪ1throughouttherecordingsession.Inseveralcasesinwhichanalysesrequireddeviationfromtheabovedefinition,thresholdparam-etersarespecified.Thedataaroundeachtimestamp(Ϯ300ms)areex-tractedandstored.TemporaloverlapbetweenNSswasnotallowed.Inourdata,obtainedbyMEAwithdimensions1.96mm2,nowave-likedirectionalpropagationisobserved.Thisisconsistentwithdataobtainedbyothersthatusethatpreparation(summarizedbyMaromandShahaf,2002)andissimilartoprestructuredpreparations(slicesandculturedslices)inwhichactivityisnotoftenwavelike(BeggsandPlenz,2004,theirFig.2).

Dynamics

Results

Corticalnetworksinvitroarespontaneouslyactive(Fig.1b).Thebasicfiringrateofindividualneuronsis0.1–5Hz,similarto

EytanandMarom•DynamicsandTopologyofSynchronization

figuresobtainedfromcorticalneuronsinvivo.Thesevaluesforspontaneousfiringratesholdundercontinuousperfusioninvitro(seeMaterialsandMethods);ifperfusionisdiscontinued,thenetworkgraduallyconvergestoasynchronicmodeofaction,withlittle(ifany)activitybetweensynchronizations.Thespon-taneousactivityiscompletelyabolishedifexcitatorysynaptictransmissionisblocked,indicativeofthefactthatspontaneousnetworkactivityisinitiatedbyinteractionsatthepopulationlevel(sporadicsynapticactivity)ratherthandrivenbyself-pacingneurons(datanotshown)(forreview,seeMaromandShahaf,2002).Overthepast20years,physiologistsrepeatedlydemon-stratedthat,likeinvivoneurons(Kenetetal.,2003),corticalnetworksinvitrospontaneouslysynchronizeonceevery1–20s(Fig.1b,c),generatingassemblyactivityevents(Habetsetal.,1987;Ramakersetal.,1990;CornerandRamakers,1991,1992;Muramotoetal.,1993;Maedaetal.,1995,1998;Kamiokaetal.,1996;NakanishiandKukita,1998;Ben-Ari,2001;Corneretal.,2002;MaromandShahaf,2002;BeggsandPlenz,2003,2004;Wagenaaretal.,2005,2006).Synchronizationswithsimilarki-neticsmayalsobeevokedbysite-specificelectricalstimuli(Jimboetal.,1999;Eytanetal.,2003).Thesesynchronousactivities,representedintermsofthenumberofactionpotentialsrecordedinmillisecondtimebins,areherereferredtoasnetworkspikes(Fig.1c,d).Resultsobtainedbysystematicallyvaryingthethresh-oldfornetworkspikedetectionindicatethat,inmostcases,onceanetworkspikestartstoassemble,thetotalityofactiveelectrodesinthepreparationiseventuallyrecruited.Inthisrespect,andbasedonthenatureofresponsestoexternalstimulation(dem-onstratedlaterinFig.6c,inset)(Jimboetal.,1999;Eytanetal.,2003),networkspikeshavetheflavorof“all-or-none”phe-nomena(Crain,1976).Figure2,aandb,demonstratesthispoint,showingthenumberofdifferentelectrodesparticipat-inginnetworkspikes(colorcoded)asafunctionofthethresh-oldforNSdetection(thetwopanelssummarizeresultsfromtwodifferentnetworks).Note,however,thatacloserexami-nationofthesehistograms(Fig.2binparticular)revealstheexistenceof“aborted”networkspikes,mostreadilydetectedusingreducedthresholds.Tofurtherclarifythepicture,ϳ900networkspikesthatservedfortheconstructionofFigure2baresortedandplottedinFigure2c:eachcolumndepictstheidentityoftheelectrodes(y-axis)thatparticipatedinagivennetworkspike(x-axis).Theeventsareorderedsuchthatthegroupofabortednetworkspikesisclearlyseen.Notethattheelectrodesrecruitedbyabortedspikesconstituteasubpopu-lationofthoserecruitedbyfullydevelopedspikes;wewillreturntothispointlater.

Spatialsamplingislimitedinourexperimentalsetupto60substrate-embeddedmicroelectrodes;however,averagingen-hancesanalysisofthekineticsunderlyinganetworkspike.Aver-agingthousandsofspontaneousnetworkspikesfrom21experi-mentsrevealsthattheearlyphaseofthespike,inwhichtheactivityoftheassemblyjuststartstoincrease,fitsanexponentialpopulationgrowthmodel(Fig.3a).Therateofrecruitment,␴,definedasthenumberofactiveelectrodesatagivenpointintimeA(t),dividedbythenumberofactiveelectrodesat(tϪ⌬t),is1.045(rangeof1.02–1.07;SDof0.015;⌬tϭ1ms;21networks,5796NSs).Thus,thepoolofactiveelectrodesincreasesbyϳ5%witheachmillisecond.Therateofrecruitmentseemstobeinde-pendentofthenetworkspikesize;aPearson’scorrelationcoeffi-cientofrϭϪ0.14wascalculatedbetween␴andspikesize(rang-ingfrom7to17activeelectrodes/ms;21experiments).Furthermore,Figure3b(top)showsadistributionofnetworkspikesizesforonegivennetwork,obtainedusingadetection

EytanandMarom•DynamicsandTopologyofSynchronizationJ.Neurosci.,August16,2006•26(33):8465–8476•8467

inhibitorysynaptictransmission.Figure4ashowsthat,inthepresenceof5␮Mbicuculline,thenumberofabortednet-workspikesdramaticallydecreases,andtheactivityafterthepeakofanindividualnetworkspikebecomesmorevigorous.InFigure4b,anaveragenetworkspikeinthepresenceofbicuculline(depictedinbrown)issuperposedontopofthecontrolaveragenetworkspike,showingthattheinhibitorysubnetworkispredominantlyaffectingthelatephaseofrelaxationfromanNSandhasacharacteristictimescaleofϳ30ms(Fig.4b,bottomrightinset).In-deed,intheabsenceofaninhibitoryeffect(browntrace),thenetworkspikedoesnotrelaxcompletely;theresidualactivityissustainedforseveralhundredmillisec-onds.Figure4balsoshowsthattheinitialdeclineofactivityfromthepeakofanNStobaselineisdominatedbycellular-levelprocesses:muchoftheforcethatrestoresthelevelofactivityafterthepeakofanNSisinsensitivetobicuculline,suggestiveoftheroleplayedbysynapticdepression,refractoriness,andcellularadaptationinthisphaseoftheNS.Analysisoftheim-pactofbicucullineontherateofrecruit-ment,␴,wasenhancedbycollectingre-cruitmentphasesbeforeandafteradditionof5␮Mbicucullinetothebathmedium.Thisprocedure,repeatedin16differentnetworksandsummarizedinFigure4c,revealsanaccelerationintherateofrecruitmentwhentheinhibitorysubnetworkispharmacologicallyneu-tralized,leadingto␴valuesintherangeof1.2/ms.

Thekineticsofrecoveryfromtheef-fectofrestoringforces(“refractorype-riod”ofanNS)occursonasecondstimescaleandisnotmarkedlyaffectedbyblockadeofinhibitorysynapses.Theserecoverykineticswereestimatedbyob-servingindicatorsfornetworkexcitabil-ityafteranetworkspike.Inparticular,

Figure1.a,Corticalnetworkonsubstrate-embeddedmultielectrodearray.Thedarkcircleisa30-␮m-diameterelectrode.weusedelectrodesthatfiresteadilyataNeuronsaretaggedusinggreenfluorescentprotein.b,Exampleofspontaneousactivitysimultaneouslyrecordedfromeightrelativelyhighfrequencybetweennet-differentchannels.Top,At500s.Bottom,Highertemporalresolutionof30sfromthetoppanel(extractedsectionisdepictedbyworkspikes(Fig.5);immediatelyafteraadarkbar).Aboxmarksasingleeventofsynchronousactivity.c,Topthreetracesshowexamplesofindividualsynchronouseventsnetworkspike,theybecomesilent(Fig.intermsofnumberofspikesrecordedin60electrodes(1mstimebins).Theaverageof273suchevents(NSs)isshown.d,Example5a,b)asaresultoftheeffectsofcellular-ofaverageNSsrecordedover1hfromdifferentnetworks(normalizedamplitudes).andnetwork-levelrestoringforces.

Thus,thekineticsofrecoverytothe“on-thresholdoffouractionpotentialsper10mstimebin;recruit-going”firingrateintheseelectrodes,afteranetworkspike

mentratescalculatedseparatelyfromtherightandleftsidesof(Fig.5b,arrows),representsthekineticsofrecoveryfromthethedistributionareverysimilar(1.05and1.04,respectively).Theeffectsofrestoringforce.Obtainingaveragerecoverytimesinslighttendencyforfasterrecruitmentinlargernetworkspikethepresenceandabsenceofbicuculline(Fig.5c)providesansizes,manifestedbyboththenegativePearson’scoefficientandestimateofrecoverykineticsofintrinsic(cellularadaptation,theexponentialfunctionsofFigure3b(bottom),isaddressedrefractoryperiod,andsynapticdepression)andextrinsic(in-later.hibition)restoringforces.Wefindthat,underbothconditionsWeestimatedtheratesgoverningthekineticsofnetwork(withandwithoutbicuculline),therecoverytimescaleisspikesbycomparingrecordingsmadeinthepresenceandab-withintherangeofseveralseconds(CornerandCrain,1972):

senceofbicuculline,apharmacologicalagentthatblocksfasttherecoverytimescaleinthecontrolexperimentis5s(average

8468•J.Neurosci.,August16,2006•26(33):8465–8476EytanandMarom•DynamicsandTopologyofSynchronization

of96recoveryevents).Inthreebicucul-lineexperiments,theobtainedrecoverytimescalesare2,2.6,and8.6s(1069,828,and726recoveryevents,respectively).Theabovekineticanalysesprovidemuchofwhatisrequiredtoheuristicallymodelthenetworkspikephenomenonintermsofstandardpopulationdynam-ics.Westartbypointingoutthatthe0.1–0.2stimescaleofassemblyactivationre-emergeswhentherecruitmentrate,␴ϭ1.04msϪ1,isembeddedintoanormal-izedlogisticgrowthequation,dA/dtϭ(␴Ϫ1)A(1ϪA),thatdescribesthefractionofactiveelectrodes(A)asafunctionoftime.Theterm(1ϪA)con-strainsthemodeltoafinitepopulationsize.Thelogisticequationiswidelyusedinpopulationmodelsinwhichtherateofgrowthislimitedbythecarryingca-pacity(i.e.,maximumsize)ofthepopu-lation;thus,itcanonlyaccountfortherisingphaseofthenetworkspike.Toac-countfortheentireshapeofanetworkspike,includingitsrelaxation,theim-pactofrestoringforcesatboththecellu-Figure2.a,b,Color-codedrepresentationsofthenumberofdifferentelectrodesparticipatinginanetworkspike

(x-axis)asafunctionofthresholdfordetectionofnetworkspikes(y-axis).Coloredscalebar(ontheright)depictsnumber

larlevel(e.g.,adaptation,refractoriness,ofoccurrences.Thetwopanelsshowdistributionsobtainedovera2hperiodintwodifferentnetworksthatfaithfullyandsynapticdepression)andthenet-representthebehavioroftheentiresetofϳ40networksservedforthisstudy.Thresholdisexpressedintermsoftheworklevel(activationofinhibitorysub-numberofelectrodesthatarerequiredtobeactive(withina3mstimebin)fortime-stampinganNS.Timebinwidthwasnetwork)needbeconsidered.Tothatalsosubjectedtosystematicvariations(from3to30ms)withnoqualitativeeffect(datanotshown).c,Approximately900end,wedescribethepopulationgrowthNSsobtainedover1hofrecordingfromthenetworkshowninb,usingaverylowthreshold(4actionpotentialsin10msbyamodifiedversionofthelogistictimebin).TheidentityoftheelectrodesparticipatingineachNSisshown(black).AlthoughinthemajorityoftheNSsallequationdA/dtϭ(s(A,t)␴Ϫ1)i(A,t)Aactiveelectrodesappear,thereexistsasubpopulationofabortedNSsinwhichonlyasubsetoftheactiveelectrodes(1ϪA).s(A,t)andi(A,t)arekineticparticipate.NotethattheNSsarenotdepictedaccordingtothechronologyoftheirappearance;rather,forpurposesofvariablesrangingfrom0to1,embody-clarity,theyaresortedusingahierarchicalclusteringalgorithm.ingcellularandnetwork-levelrestoringforces,respectively.Theassumedkinetic

Effectivetopology

schemesfors(A,t)andi(A,t)aresimilartotheformusedby

Measurementsinvitro(Ikegayaetal.,2004;vanPeltetal.,

HodgkinandHuxley(1952)intheirdescriptionofaneuronal

2004)aswellasinvivo(Abeles,1991;Tsodyksetal.,1999;

actionpotential;thatis,(1Ϫi)Niand(1Ϫs)Ns,with

Buzsakietal.,2004)indicatethatthesequenceofneuronal

transitionratesbeingexponentialfunctionsofA(t).Theun-activationwithinasynchronizationeventisnonrandomand

derlyingassumptionisthattherecruitmentofinhibitionand

stronglyconstrainedbythepatternofpopulationactivity.Fig-theprocessesofcellularandsynapticadaptationarenonlin-ure7showsthat,indeed,therecruitmentofneuronalactivity

earlydependentonactivity.ToavoiddiscontinuitiesinthewithintheNSishierarchicallystructured;thereexistsastablemodel,weusedexponentialterms;attemptstophysicallyin-subsetofprivilegedneuronsthatreliablyincreasetheirfiring

terprettheseexponentialtermswillbepresumptuous,atratestensofmillisecondsbeforethepeakofthenetworkspike.present.Figure6summarizesourkineticanalysesintheformThetwoplotsofFigure7asummarizedatacollectedover1hofanumericsolution.InFigure6a,thetimeconstantsusedforeach(averagesofϾ200NSsineach).Notethatthepatternofsimulatingi(A)ands(A)areshown.TherateequationsforiactivityobtainedbyaveragingNSsduringthefirsthourofandsweredevisedtoapproximatelymatchthetimescalesofrecordingisverysimilartothepatternobtainedoverthesev-activationandrecoveryoftherestoringforcesattheextremesenthhour;suchstabilityisinagreementwithdatapublishedofAϭ1andAϭ0,basedondatashowninFigures3–5.FigurebyvanPeltetal.(2004).Particularlyrelevantforthesubject6,bandc,showsspontaneousandevokednetworkspikes,matterofthepresentstudyisthefactthatthesameprivilegedrespectively,generatedbythemodelwithGaussiannoiseneuronsappearinbothrecordingsessions.AsshowninFigureadded.TheinsettoFigure6cshowsdatafromanexperiment7b,theprivilegedneuronswithinagivenassemblypredictaninwhichanetworkwasexcitedbyapplyingshortbiphasicupcomingnetworkspikeregardlessofitsactivationsource:currentpulsesbetweenapairofelectrodes.Takenasawhole,whetherthenetworkspikestartsspontaneously(leftpanel)orthemodifiedcontinuousversionofthelogisticequationfaith-isevokedexternallybyappliedelectricalstimuliatdifferent

fullyreconstructskeyfeaturesoftheexperimentalrecords,sites(rightpanel),thesameprivilegedneuronsarethefirstyieldinganetworkspikewitha0.1–0.2scharacteristictime-onestobeactivated.Furthermore,assuggestedbytheresults

scale,whichisfairlyinsensitivetoinhibitioninitspresentofFigure8,thesameprivilegedneuronsarethefirstonestobe

activewhetherthesynchronyevolvestoafullydevelopednet-form.

EytanandMarom•DynamicsandTopologyofSynchronizationFigure3.a,Inset,Grandaveragenetworkspike,calculatedbyaveraging21averageNSssimilartothoseshowninFigure1d(totalof5796NSs).Markedinbrownistheinitialsegment,enlarged(browndots)inthemainfigure.Anexponentialgrowthequation,A(t)ϭaϩbϫe(␴Ϫ1)t,wasfittedtothisinitialsegment;theresultingfunction(aϭ0.05,bϭ0.01,␴ϭ1.045)isdepictedbyacontinuousblackline.b,Top,DistributionofNSpeakactivityforonenetworkthatdemonstratedarelativelybroadspectrumofNSamplitudes.AverylowthresholdforNSdetectionwasusedhere(4actionpotentialsin10msbin).Bottom,Earlyrecruitmentphasesfor207low-amplitudeNSs(Յ6actionpotentials/ms;depictedinblack)and208high-amplitudeNSs(Ն10actionpotentials/ms;depictedingray),fittedwithexponentialgrowthequations,yielding␴valuesof1.04and1.05,respectively.

workspikeoranabortedone.Theactivityofprivilegedneu-ronsreliablypredictsanupcomingnetworkspikeasearlyas100msbeforethepeakofthespike.Figure9ademonstratesthispointbyshowingthecumulativenumberofspikes

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emittedbysixneuronsinagivennetwork,someofwhichareprivileged.

Recallthatnoneoftheneuronsinthenetworkfiresunlessitisdrivenbyotherneurons.Thismeansthatthefiringrateofanindividualneuroningeneral,andduringtherecruitmentphaseinparticular,reflectsitssensitivitytotheactivityofotherneuronsinthenetwork.Notethatthissensitivityismediatedbyacombinationoffactors,including,forinstance,thenumberofsynapticinputsthattheneuronreceives,thedistributionofsynapticweightsinitsdendritictree,restingpotential,firingthreshold,dendriticconductances,anddy-namicsofcellularadaptation.Thusweusethetermeffectiveconnectivity(ratherthansimply“connectivity”)todesignatethesensitivityofaneurontonetworkactivityasreflectedinitsfiringrate.Histogramsoffiringratesobtainedfrom1200ac-tiveelectrodes(20networks)throughout600msthatsur-roundanNS,aswellaswithinthetimewindowofϪ100toϪcircles,75msrespectively).beforeitspeakBothareshowndistributionsinFigureare9b(graybroad,andandblackthelatterisfittedbyapowerlawoverϳ2ordersoffiringratemagnitudes(coveringthephysiologicalrangeoffiringrates).Thepower-lawdistributionshowninFigure9doesnotartifi-ciallyresultfromthevarianceinaveragefiringratesofthenetworksfromwhichneuronalactivitieswerepooled.Toes-tablishthispoint,welookedatfiringratedistributionsob-tainedfromtwosubgroupsofnetworks.Inonesubgroup(sixnetworks),theaverageelectrodefiringratewas9.9–13.7Hzduringa600mstimewindowsurroundingthenetworkspike.Intheothersubgroup(sevennetworks),theaveragefiringratewaslower(3.4–5.2Hz).One-wayANOVAsuggeststhatthenetworks,withineachofthetwosubgroups,donotsignifi-cantlydifferinthefiringratedistributions(all60electrodesfromeachnetworkweretakenintoaccount,includingthosethatdidnotshowanyactivity;Fratioϭ1.25,pϭ0.28andFratioϭ1.01,pϭ0.41,forthehigh-andlow-firing-ratenet-works,respectively).SupplementalFigure1(availableatwww.jneurosci.orgassupplementalmaterial)showsthatthefiringratedistributionsofelectrodesfrombothlow-andhigh-firing-ratesubgroupsarebroadanddescribedbyapower-lawfunction(dϭϪ1.85,R2R2ϭ0.87anddϭϪ2.01,(Fig.ϭ90.91,b,inset)respectively).indicatesthatAnalysisthecoefficientoffiringrateofvariancefluctuationsforstronglyactiveelectrodesisreducedcomparedwithactiveelectrodesthatfirelessduringtherecruitmentphase.Assum-ingthatthedynamicsofindividualsynapsesaswellasthemembranepropertiesofneuronsthatfiremorearenotdiffer-entfromthosethatfireless,theinsetofFigure9bimpliesthatneuronsthatfiremoreintegrateovermoresynapticinputcomparedwithneuronsthatfireless.

Broadlydistributedconnectivityingeneral,andpower-lawdistributedconnectivityinparticular,areusuallyassociatedwithrandomgraphsinwhichtheaveragetopologicaldistancebetweennodesincreasesveryslowlywiththenumberofnodes,despitealargelocalinterconnectedness(Strogatz,2001).Inaccordance,undercertainconditionsthatseemap-plicabletoneuronalnetworks(Nishikawaetal.,2003;Motteretal.,2005),comparedwithErdo¨s-Re´nyitypeofconnectivity,dynamicsystemscoupledinthiswaydisplayenhancedsignalpropagationspeedthatincreasesveryslowly(Strogatz,2001;Newman,2003)orevendecreases(Barthelemyetal.,2004)asthenetworksizegetsbigger.Totestthisprediction,wepre-parednetworksattwodifferentneuronaldensities(0.1ϫand2ϫthestandarddensity).Thedifferencebetweenthedensities

8470•J.Neurosci.,August16,2006•26(33):8465–8476Figure4.a,Networkspikesrecordedoveraperiodof2hintheabsence(left)andpresence(right)of5␮Mbicuculline.TheseNSswereobtainedusingathresholdofthreeactionpotentialswithina10mstimebin.EachblackdotmarksanactionpotentialdetectedinanyoftheelectrodesduringtheϮ250mssurroundingtheNSthreshold.NSsareorderedusingacluster-ingalgorithmtoenhancevisualizationofbicucullineeffects:areducednumberofabortedNSsandamorevigorousactivitywithineachNS.b,AnaverageNSobtainedinthepresenceof5␮Mbicuculline(brown).AverageNSincontrolsolutionisshowninblack.Toprightinset,Fittedfastdecliningphaseinthepresenceofbicuculline,representingthetimescaleofcellular-level(in-trinsic)restoringforces.Bottomrightinset,FittedslowdecliningphaseofthecontrolNS,rep-resentingthetimescaleofrestoringforceactingthroughtheinhibitorysubnetwork.Notethetimescaleseparationbetweenthetwotypesofrestoringforcesinvolved(6.5msfortheeffectofcellular-levelforces;33.7msfortheeffectoftheinhibitorysubnetwork).c,Averagedphasesofrecruitmentsextractedfrom16experimentsbefore(black)andafter(brown)theadditionof5␮Mbicucullinetothebathingsolution.Eachrecordingepisodelasted1–2h.Brokenlinesdepict1SD(one-sidedforclarity).Thicklinesdepictsegmentsforwhichasingle-exponentialrecruit-mentfunctioncouldbereliablyfittedforbothconditions;␴valuesobtainedwere1.06and1.22forcontrolandbicucullineconditions,respectively.

EytanandMarom•DynamicsandTopologyofSynchronization

Figure5.a,b,Firingrateofarelativelyactiveneuron;therateincreasesduringanNS,decreasesdramaticallyaftertheNS,andthengraduallyrecovers(arrowsinb).Thesekineticsareusedforestimationofrecoveryfromanetworkspike.c,Fittedrecoveries,suchasthosedepictedbyarrowsinb,incontrol(black)andbicuculline(brown)solutions.[Notethattheinter-NSintervalsinbicucullinesolutionareshort;pointsatwhichtheaveragefiringrateduringrecoverywas“contaminated”bythepresenceofNSswereomitted(bins7–9,11,14s)].

ofthetwogroupsastheymatured(thirdweekinvitro)didnotremainaslargeaswaswhenthenetworkswereplated,proba-blybecauseofcompensatorymechanismsthatarebeyondthescopeofthepresentstudy.However,therewasstillaverycleardifferencebetweentheneuronaldensitiesofthetwogroups,observedstructurally(Fig.10a)andfunctionally(Fig.10b).Figure10cshowsthatindeedthetimeittakesforthenetwork

EytanandMarom•DynamicsandTopologyofSynchronizationFigure6.NumericalsolutionofthemodifiedlogisticequationdA/dtϭ(s(A,t)␴Ϫ1)i(A,t)A(1ϪA)forspontaneousandevokedNSs.a,Timeconstantsofi(A)ands(A).Exponen-tialrateequationsforiandsweredevisedtoapproximatelymatchthetimescalesofactivationandrecoveryoftherestoringforcesattheextremesofAϭ1andAϭ0,basedondatashown)ϩ0.5)inFigures3–5.Theequationsusedthefollowing(timeinmillisecondunits):(t)ϩ0.5)␤sϭ30eϪ20(A(t;␣sϭ0.00001e10A(t);␤iϭ10eϪ20(A;␣iϭ0.000001e10A(t).0ՅA(t)Յ1;␴wassetto1belowathresholdofA(t)ϭ0.05.Oncethethresholdiscrossed,␴issettoavaluerangingfrom1.04to1.06.Noisewasgeneratedusingnormal(Gaussian)distributionaroundmeanA(t),withSDof0.001and0.0002forbandc,respectively.Forthecaseofevokednetworkspikes(c),afractionofthepopulationisexcited(i.e.,stimulated)fromaroundthresholdandhigherindiscretesteps.Insetincshowsdatafromanexperimentinwhichanetworkwasexcitedbyapplyingshort(0.4ms)biphasiccurrentpulsesbetweenapairofelectrodes[step-pingfrom10␮A(bottomtrace)to80␮A(toptrace)].Averageevokedresponsesto15presen-tationsofstimuliateachamplitudeareshown(eachtraceis300mslong).Missedresponseswerenotincludedexceptforthecaseof10␮Astimulationamplitude.Weobserved15of15missedresponsesfor10␮Astimuli,5of15for20␮A,4of15for30␮A,and2of15for40␮A;beyond40␮Astimulation,nomissedresponseswereobserved.

J.Neurosci.,August16,2006•26(33):8465–8476•8471

Figure7.a,FiringprobabilityofneuronsasafunctionoftimesurroundinganNS.Colorscalerangesfrom0to1spikeper5ms(thefewcasesinwhichϾ1spikeoccurredduringa5msbinarerepresentedas1spike/bin).b,Inagivennetwork,theearly-to-fireneuronsaresimilarforspontaneous(left)andevoked(right;3differentstimulationsources)NSs.ForthecaseofspontaneousNS,arrowspointtotimesforwhichfiringprobabilitiesarepresented.Probabilitiesoffiring0–5and25–75msafterstimulationareshownforthreedifferentstimulationsites(S1,S2,S3).Short(0.4ms)biphasic30␮Acurrentpulsesbetweeneachofthreedifferentpairsofelectrodeswereapplied.Averageresponsesto60presentationsofstimuliforeachstimulationpointareshown.Horizontallinesdepictfourexamplesofearly-to-fireneurons.Poststimulitimehistogramsforthethreestimulationsitesareshownatthebottomright.

tosynchronizedoesnotincreaseasthenumberofactiveelec-trodesparticipatinginthesynchronyincreases;ifanything,thedenserthenetworkis,moreactiveelectrodesparticipateinthesynchronyandsynchronizationbecomesfaster.

8472•J.Neurosci.,August16,2006•26(33):8465–8476Figure8.Anetworkwitharelativelylargenumberofabortednetworkspikesissubjectedtoanalysesaimedatcomparingearly-to-fireneuronsinabortedversusfull-blownNSs.a,All1087NSsdetectedover1hofrecordingareshowninarasterplot,Ϯ250mssurroundingadetectionthresholdoffouractionpotentialin10mstimebin.AblackdotdepictsanactionpotentialrecordedinanyoftheelectrodesduringtheNS.Responseswerereorderedusingaclusteringalgorithmtoenhancevisualseparationbetweenfull-blownandabortednetworkspikes.b,ThedistributionofpeaknumberofactionpotentialsineachNS(1mstimebin;sameprocedureasinFig.3b).Distributionsaroundthresholdfor537NSswithapeakofՅ5actionpotentials(c)and437NSswithapeakofՆ10actionpotentials(d)areshownusingcolorscalerangesfrom0to0.33actionpotentialsper5ms(thecasesinwhichϾ0.33actionpotentialsoccurredduringa5msbinarerepresentedas0.33actionpotentialsperbin).Notethatthesameearly-to-fireneuronsareactiveforbothsubsetsofNSs.ThecorrelationbetweenfiringratesoftheelectrodesofbothsubsetsattimesrangingfromϪ300toϪ50msbeforeNSthresholdis0.98.

Hierarchicalrecruitmentwithinanassemblyprovidesapowerfulmechanismformodulationoftimedelaysbetweencoupledassemblies.Todemonstratethisfeature,weelectricallycoupledapairofassemblies,X3Y,usingastimulusgenerator.Obviously,undersuchartificialconditionsthe“actual”timede-laysarearbitrarilydictatedbysettingthecouplingstimuluspa-rameters.Itwouldbeexpectedthat“ranking”thecapacityofdifferentneuronsfromassemblyXtospeedupthesynchroniza-tionbetweenXandYisindependentoftheartificialstimulationparameters.Figure11,aandb,showsthat,byclampingtheamplitudeandlocusoftheinputstimulustoY,whilechangingtheidentityofthetriggeringneuron(inX),thetimedelaysbetweennetworkspikesinXandYmaybestronglyaffected;differentneurons,byvirtueoftheireffectiveconnectivity,may

EytanandMarom•DynamicsandTopologyofSynchronization

Figure9.a,Cumulativenumberofspikesemittedbysixneuronsinanetwork,asafunctionoftimerelativetotheoccurrenceofanNS(fordefinitionofNStimestamp,seeMaterialsandMethods).Inset,DistributionofintervalsbetweenspontaneouslyoccurringNSs(491synchro-nouseventsrecordedfromthisnetwork).b,Firingratedistribution100–75msbeforetheNS(black)andthroughouttheNS(gray;1200neuronsfrom20networks;9400NSs).Thedistribu-tionisfittedbyapower-lawfunctionwithascalingpowerdϷϪ2.Inset,Atotalof387differentneurons(12differentnetworks)thatwereactiveasearlyasϪ100toϪ50msbeforeanNSthresholdcrossingpointwereselected.Fortheseneurons,thecoefficientofvariance(CV)ofinstantaneousfrequency(manifestedbyinterspikeinterval)during50msafterthresholdcrossingwascalculatedandplottedasafunctionoftheirfiringrateϮ300mssurroundingtheNSdetectionthreshold.Notethatthecoefficientofvarianceisnegativelycorrelatedtotheneuronalfiringrate(rϭϪ0.22;pϽ0.0001).

consistentlyyielddifferenttimedelays.WhenassemblyY“reads”theactivityofassemblyXthroughpoorlyconnectedneurons,thetimedelaybetweenthenetworkspikesinXandYislarge.However,whenhighlyconnectedneuronsareread,thenetwork

EytanandMarom•DynamicsandTopologyofSynchronizationFigure10.Effectofneuronaldensitiesonsynchronizationtime.a,Photographsofexemplarmaturenetworks(3rdweekinvitro)platedat0.1ϫ(left)and2ϫ(right)thestandarddensity.Exemplarsoftheresultingnetworkspikeamplitudesareshowninbandnormalizedintheinsetinc,demonstratingthatthetimeittakesforthenetworktosynchronizedoesnotincreaseasthenumberofneuronsparticipatinginthesynchronyincreases.Toallowanalysesunderconditionsoflowdensity,a10mstimebinforthresholddetectionwasusedforboth(highandlow)densities.c,Asummaryof13differentnetworks,showingthatthetime-to-peakoftheaveragenetworkspikedecreasesasthenetworkbecomesmoredense.Time-to-peakwascalculatedfrom10%activitytopeakactivity.

spikeinYcanappearsimultaneouslywiththatofXorevenprecedeit.

Discussion

Usingasubstrate-integratedmultielectrodearray,wemea-suredandcharacterizedthekineticsofneuronalassemblyac-tivationinvitro.Expressedintermsofpopulationfiringrate,assemblyactivationisanall-or-none-likethreshold-governedphenomenonthatisveryreminiscentofanactionpotentialin

J.Neurosci.,August16,2006•26(33):8465–8476•8473

Figure11.a,NSsofartificiallycoupledassemblies.ActivityofdifferentneuronsinassemblyX(brown)isusedtotriggerthedeliveryofafixedcurrentamplitudestimulus(0.4ms,biphasic50␮A)toYassembly(black).DifferenttracesrepresenttheimpactofusingdifferenttriggeringneuronsfromX.b,Summaryofresultsfromfourartificiallycoupledpairsofnetworks(4differ-entsymbols).TimedelaysbetweenaverageNSsinXandYareshownasafunctionofthetriggeringneuroneffectiveconnectivity.Asanindexfortheeffectiveconnectivityinthisfigure,weusedtheaveragenumberofactionpotentialsemittedbyaneuronoverthetimeperiodfromϪ300toϪ15msrelative(i.e.,before)toanNStimestamp.NotethatvaluesϽ1indicatethatatriggeringneurondoesnotemitactionpotentialsbeforeeachnetworkspikeduringthedesignatedperiod,inwhichcase,itisitsparticipationintheNSatlaterstagesthatcausesassemblyYtobeignited.

asingleneuron,hencethetermnetworkspike.Therecruit-mentofneuronsduringtheearlyphasesofanetworkspikefollowsasingle-rateexponentialprocessthatamountstoacharacteristictimescaleof0.1–0.2sforachievingsynchroni-zation.Themeasuredrecruitmentrate(␴)isclosetounity.BeggsandPlenz(2003,2004)usedarelatedmeasuretode-scribethepropagationoflocalfieldpotentialsincorticalslices.Basingtheirelegantanalysesonequationsthatgovern

8474•J.Neurosci.,August16,2006•26(33):8465–8476avalanches,theyalsoreportabranchingparameterclosetounity,avaluethatoptimizesinformationtransmissioninfeedforwardnetworksbutpreventsrunawaynetworkexcita-tion.Althoughthefitofasingle-exponentialrecruitmenttotheaveragedatadoesseemtight(Fig.3),smallvariationsaroundthevalueof␴(SDof0.015)arenotnegligible,asonemightbetemptedtoassume.Incorporating␴valuesrangingfrom1.025to1.055(Ϯ1SDaround␴ϭ1.04msϪ1)intoalogisticgrowthequationyieldsarangeofϳ80toϳ180msinthetimetohalf-maximumactivation,respectively.Indeed,examinationofthetracesshowninFigure1dsuggestthatsucharangeisrealistic.

Theall-or-nonenatureofnetworkspikesreportedhereseemsincongruentwithBeggsandPlenz(2003)power-lawsizedistributionofavalanches(i.e.,burstsofactivities).More-over,analysisofourdatausingtheprocedureofBeggsandPlenzresultsinbimodaldistributionsforbothavalanchesize(numberofparticipatingelectrodes)andavalancheduration(supplementalFig.2,availableatwww.jneurosci.orgassup-plementalmaterial).Ourbimodaldistributionsareverysim-ilartothebimodalavalanchesizedistributionsseeninthepresenceofpicrotoxin,afastinhibitionblocker(Plenz,2005).WebelievethatthedifferencebetweenthetwoobservationsisattributabletothefactthatBeggsandPlenzwereusingaprep-arationcutoutfromaprestructurednetworkwithconservedlayerorganization;we,conversely,useanexvivospontane-ouslydevelopingpreparation.Indeed,arecentabstractofPlenz’sgroup(Stewartetal.,2005)indicatesthattheava-lanchestheyobserveoriginateinsuperficiallayersandarealsoconfinedtotheselayersmostofthetime.

Therelativeimpactofcellular-andnetwork-levelrestoringforceswasestimatedusingpharmacologicalmanipulations.Al-thoughlaterphasesofthenetworkspikeareaffectedtosomeextentbyactivationoftheinhibitorysubnetwork,themajorre-storingforcethatoperatesthroughoutanetworkspikearisesfromcellularprocesses[mostprobablysynapticdepression(Eytanetal.,2003)].Alogisticequationwithkineticvariablesthatrepresentthevariousrestoringforcesaccountsforthe0.1–0.2stime-amplitudetrajectoryofanetworkspikeandisofferedasaframeworkformathematicalmodelingofassemblysynchro-nizationevents.

Thecontributionofdifferentneuronstothevariousphasesofthenetworkspikeisnotrandom.Thephaseofthenetworkspikewithinwhichaneuronfiresisstronglyconstrained;someneuronsconsistentlyfireattheveryearlyphaseofthenetworkspike,whereasothersstarttofireatlaterphasesofthespike.Nonrandom,sequencedactivationofneuronsthatisstronglyconstrainedbythepatternofpopulationactivitywasdemonstratedinvitro(Ikegayaetal.,2004;vanPeltetal.,2004),aswellasinvivo(Abeles,1991;Tsodyksetal.,1999;Buzsakietal.,2004).GrinsteinandLinsker(2005)addressedtheroleplayedbynetworktopologyindeterminingthenatureofsynchronousneuralactivity.Theyshowthat,incontrasttorandomErdo¨s-Re´nyinetworks,networkshavingapower-lawconnectivitydistributiongeneratelargesynchronousfiringpeaksdominatedbyasmallsubsetofnodes.Hereweusedthefiringratesofindividualneuronsduringtheearlyrecruitmentphaseasindicatorsfortheireffectiveconnectivity.Wejustifyourapproachbyhavingobservedthatneuronsinourprepa-rationfireonlyinresponsetosynapticinput,notspontane-ously;hence,firingthatstartsearliermeansmoresensitivitytonetworkactivity,thatis,moreeffectiveconvergence.FiringthatstartsearlieralsoprobablymeansmoreeffectivedivergentEytanandMarom•DynamicsandTopologyofSynchronization

connectivityinthesensethatthemoreaneuronfires,thehigheritsimpactonthepropagationofactivityandassemblysynchronizability.Weshowthatthedistributionoffiringratesduringtheearlyphasesofthenetworkspikeisbroadanddescribedbyapowerlaw,whichwouldbeconsistentwithanessentiallyscale-freetopologyofconnectivity.

Itistemptingtothinkoftheaboveresultsinthecontextofobservationsinvivo.Thetimetoreachsynchronizationinourinvitropreparation(thatcontainsϳ100,000neurons)is0.1–0.2s.Thisisalsothetimeittakesforasinglecorticalcolumntoreachsynchronization(Derdikmanetal.,2003),aswellasforwholebrainareassuchasprimaryvisualareaV1andsec-ondaryvisualareaV2(Slovinetal.,2002),andinfactfortheentirebrain;indeed,0.1–0.2sisalsotheacceptedfigureforsimplebehavioralreactiontimes.Toalargeextent,itseemsthatthetimetoreachneuralsynchronyissizeinvariant.Inrecentyears,therelationshipsbetweennetworksizeandtimetosynchronizeisintensivelystudiedaspartofthegrowinginterestinthefieldofcomplexrandomgraphs(forreview,seeNewman,2003).Awiderangeofbiologicalandsocialnet-works(graphs)hasverybroadconnectivitydistributions;thesegraphsareoftenreferredtoas“scale-free”networks,designatingthepower-lawdistributionofconnectivitythatischaracteristicofsomeofthem.Insuchcases,theaverageto-pologicaldistancebetweennodesincreasesslowlywiththenetworksize,despitealargelocalinterconnectedness(Stro-gatz,2001).Intheneurobiologicalcontext,scale-freetopol-ogywithinneuronalnetworkswassuggestedtoprovideaneconomicalmeansforglobalsynchronyandoscillationsatmultipletimescales(Buzsakietal.,2004;Spornsetal.,2004;GrinsteinandLinsker,2005).Asexplainedabove,undercer-tainconditionsthatmightbeapplicabletoneuronalnetworks(Nishikawaetal.,2003;Motteretal.,2005),dynamicsystemscoupledinthiswaydisplayenhancedsignalpropagationspeedthatisinvariantorevendecreaseswithnetworksize(Strogatz,2001;Newman,2003;Barthelemyetal.,2004).Wesuggestthinkinginsuchtermsonthemechanismofsizeinvarianceinthetimeittakestosynchronizenetworksinthebrain.Ourcontributiontothislineofthoughtistoshowthatincreasingthenumberofneuronsthatparticipateinanetworkspikecausesadecreaseintimetoreachsynchrony.

Hierarchicalrecruitmentofneuronsduringsynchronyprovidesameansformodulationoftimedelaysbetweense-quentialactivationsofcoupledassemblies.Thereliabilityofforecastinganetworkspiketensofmillisecondsbeforeitspeakbasedontheactivityofearly-to-fireneurons,togetherwiththeslowintegrationtowardanetworkspike,allowsawidetemporalrangeforsuchmodulation.Assemblycoupling,pre-sentedinFigure11,althoughlimitedbytheartificialityofthestimulus,demonstratessuchtemporalmodulation:thetimedelaysbetweentheactivitiesoftwocoupledassembliesmaybedramaticallyreduced.Thisschemeprovidesapossibleexpla-nationfortheenigmaticrapidityofprocessinginpipelinesofneuronalassemblies.Consider,forinstance,thecaseofvisualcategorizationexperiments.Electrophysiologicalandpsycho-physicalanalysesshowthatthevisualsystemrecognizesacat-egoryinlessthan1⁄10ofasecond(Keysersetal.,2001;Van-RullenandKoch,2003).Takingintoaccountthenumberofneuralcellassembliesthroughwhichthesignaltravels,suchrapidprocessingimplies0.01sofactivitywithineachassem-blybeforeitisforwardedtothenextone.Thisnumberisdifficulttoreconcilewiththe10timesslowerprocess(0.1–0.2s)requiredforafull-blownactivationofanindividualassem-

EytanandMarom•DynamicsandTopologyofSynchronizationblyinresponsetoastimulus.Figure11showsthat,inprinci-ple,whenrelyingontheactivityofearly-to-fireneuronsinthehierarchyofrecruitmentwithinanassembly,suchshortre-sponsetimesareaccountedfor(ThorpeandFabre-Thorpe,2001;Thorpe,2002).Figure11alsosuggeststhat,inprinciple,adriven(next-in-line)assemblymaysynchronizesimulta-neouslywith,orevenbefore,itsdriving(forerunner)assem-bly.Thispossibilitysurfacesapotentialmethodologicaldiffi-cultyintheinterpretationsoffunctionalmacroscopicneuraldata(e.g.,electroencephalogram,magnetoencephalography,andfunctionalmagneticresonanceimaging)thatrelyontimedelaysbetweenactivationofassemblies.Forinstance,whentwoassemblies(XandY)seemtobeactivetogether,thepos-sibilityofXdrivingYorviceversawouldberuledout;itgoeswithoutsayingthat,whenmacroscopicactivityfromassemblyXprecedesthatofY,concludingthatthelatterbeingacauseoftheformerwouldseemunreasonable.TheresultsofFigure11implythat,inconstructingschemesofactivationpaths,muchcareshouldbeexercisedwhenrelyingonmacroscopictimedelays.

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