Interactions between Intrinsic and StimulusEvoked ...
Transcript of Interactions between Intrinsic and StimulusEvoked ...
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InteractionsbetweenIntrinsicandStimulusEvoked
ActivityinRecurrentNeuralNetworks
L.F.AbbottandKanakaRajan
DepartmentofNeuroscienceDepartmentofPhysiologyandCellularBiophysics
ColumbiaUniversityCollegeofPhysiciansandSurgeonsNewYork,NY10032‐2695USA
and
HaimSompolinsky
RacahInstituteofPhysicsInterdisciplinaryCenterforNeuralComputation
HebrewUniversityJerusalem,Israel
Introduction
Trial‐to‐trialvariabilityisanessentialfeatureofneuralresponses,butitssourceisasubjectofactivedebate.Responsevariability(MastandVictor,1991;Arielietal.,1995&1996;Andersonetal.,2000&2001;Kenetetal.,2003;Petersenetal.,2003a&b;Fiser,ChiuandWeliky,2004;MacLeanetal.,2005;Yusteetal.,2005;Vincentetal.,2007)isoftentreatedasrandomnoise,generatedeitherbyotherbrainareas,orbystochasticprocesseswithinthecircuitrybeingstudied.Wecallsuchsourcesofvariability“external”tostresstheindependenceofthisformofnoisefromactivitydrivenbythestimulus.Variabilitycanalsobegeneratedinternallybythesamenetworkdynamicsthatgeneratesresponsestoastimulus.Howcanwedistinguishbetweenexternalandinternalsourcesofresponsevariability?Hereweshowthatinternalsourcesofvariabilityinteractnonlinearlywithstimulus‐inducedactivity,andthisinteractionyieldsasuppressionofnoiseintheevokedstate.Thisprovidesatheoreticalbasisandpotentialmechanismfortheexperimentalobservationthat,inmanybrainareas,stimulicausesignificantsuppressionofneuronalvariability(WernerandMountcastle,1963;Fortier,SmithandKalaska,1993;Andersonetal.,2000;FriedrichandLaurent,2004;Churchlandetal.,2006;Finn,PriebeandFerster,2007;Mitchell,SundbergandReynolds,2007;Churchlandetal.,2009).Thecombinedtheoreticalandexperimentalresultssuggestthatinternallygeneratedactivityisasignificantcontributortoresponsevariabilityinneuralcircuits.
Weareinterestedinuncoveringtherelationshipbetweenintrinsicandstimulus‐evokedactivityinmodelnetworksandstudyingtheselectivityofthesenetworkstofeaturesofthestimulidrivingthem.Therelationshipbetweenintrinsicand
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extrinsicallyevokedactivityhasbeenstudiedexperimentallybycomparingactivitypatternsacrosscorticalmaps(Arielietal.,1995&1996).Wedeveloptechniquesforperformingsuchcomparisonsincaseswherethereisnoapparentsensorymap.Inadditiontorevealinghowthetemporalandspatialstructureofspontaneousactivityaffectsevokedresponses,thesemethodscanbeusedtoinferinputselectivity.Historically,selectivitywasfirstmeasuredbystudyingstimulus‐drivenresponses(HubelandWiesel,1962),andonlylaterweresimilarselectivitypatternsobservedinspontaneousactivityacrossthecorticalsurface(Arielietal.,1995&1996).Wearguethatitispossibletoworkinthereverseorder.Havinglittleinitialknowledgeofsensorymapsinournetworks,weshowhowtheirspontaneousactivitycaninformusabouttheselectivityofevokedresponsestoinputfeatures.Throughoutthisstudy,werestrictourselvestoquantitiesthatcanbemeasuredexperimentally,suchasresponsecorrelations,soouranalysismethodscanbeappliedequallytotheoreticalmodelsandexperimentaldata.
Webeginbydescribingthenetworkmodelandillustratingthetypesofactivityitproduces,usingcomputersimulations.Inparticular,weillustrateanddiscussatransitionbetweentwotypesofresponses;oneinwhichintrinsicandstimulus‐evokedactivitycoexist,andtheotherinwhichintrinsicactivityiscompletelysuppressed.Next,weexplorehowthespatialpatternsofspontaneousandevokedresponsesarerelated.Byspatialpattern,wemeanthewaythatactivityisdistributedacrossthedifferentneuronsofthenetwork.Spontaneousactivityisausefulindicatorofrecurrenteffects,becauseitiscompletelydeterminedbynetworkfeedback.Therefore,westudytheimpactofnetworkconnectivityonthespatialpatternofinput‐drivenresponsesbycomparingthespatialstructureofevokedandspontaneousactivity.Finally,weshowhowthestimulusselectivityofthenetworkcanbeinferredfromananalysisofitsspontaneousactivity.
TheModel
Neuronsinthemodelweconsideraredescribedbyfiring‐rates,theydonotfireindividualactionpotentials.Suchfiring‐ratenetworksareattractivebecausetheyareeasiertosimulatethanspikingnetworkmodelsandareamenabletomoredetailedmathematicalanalyses.Ingeneral,aslongasthereisnolarge‐scalesynchronizationofactionpotentials,firing‐ratemodelsdescribenetworkactivityadequately(Shriki,HanselandSompolinsky,2003;WongandWang,2006).WeconsideranetworkofNinterconnectedneurons,withneuronicharacterizedbyanactivationvariablexisatisfying
€
τdxidt
= −xi + g Jijrj + Iij=1
N
∑ .
Thetimeconstantτissetto10ms.Forallofthefigures,except5e,N=1000.TherecurrentsynapticweightmatrixJhaselementJijdescribingtheconnectionfrompresynapticneuronjtopostsynapticneuroni.Excitatoryconnectionscorrespond
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topositivematrixelements,inhibitoryconnectionstonegativeelements.Theinputterm,Iiforneuroni,takesvariousformsthatwillbedescribedasweusethem.
Thefiringrateofneuroniisgivenby
€
ri = R0 + φ(x i) with
€
φ(x) = R0 tanh(x /R0)for
€
x ≤ 0 and
€
φ(x) = (Rmax − R0)tanh(x /(Rmax − R0)) for
€
x > 0 .Here,R0isthebackgroundfiringrate(thefiringratewhenx=0),andRmaxisthemaximumfiringrate.Thisfunctionallowsustospecifyindependentlythemaximumfiringrate,Rmax,andthebackgroundrate,R0,andsetthemtoreasonablevalues,whileretainingthegeneralformofthecommonlyusedtanhfunction.ThisfiringratefunctionisplottedinFigure1forR0=0.1Rmax,thevalueweuse.Tofacilitatecomparisonwithexperimentaldatainavarietyofsystems,wereportallresponsesrelativetoRmax.Similarly,wereportallinputcurrentsrelativetothecurrentI1/2requiredtodriveanisolatedneurontohalfofitsmaximalfiringrate(seeFigure1).
Figure1.ThefiringratefunctionusedinthenetworkmodelversustheinputIforR0=0.1RmaxnormalizedbythemaximumfiringrateRmax.TheparameterI1/2isdefinedbythedashedlines.
Althoughaconsiderableamountisknownaboutthestatisticalpropertiesof
recurrentconnectionsincorticalcircuitry(Holmgrenetal.,2003;Songetal.,2005),wedonothaveanythinglikethespecificneuron‐to‐neuronwiringdiagramwewouldneedtobuildatrulyfaithfulmodelofacorticalcolumnorhypercolumn.Instead,weconstructtheconnectionmatrixJofthemodelnetworkonthebasisofastatisticaldescriptionoftheunderlyingcircuitry.WedothisbychoosingelementsofthesynapticweightmatrixindependentlyandrandomlyfromaGaussiandistributionwithzeromeanandvariance1/N.Wecoulddividethenetworkintoseparateexcitatoryandinhibitorysubpopulations,butthisdoesnotqualitativelychangethenetworkpropertiesthatwediscuss(vanVreeswijkandSompolinsky,1996&1998;RajanandAbbott,2006).
Theparametergcontrolsthestrengthofthesynapticconnectionsinthemodel,butbecausethesestrengthsarechosenfromadistributionwithzeromeanandnonzerovariance,gactuallycontrolsthesizeofthestandarddeviationofthesynapticstrengths(seeDiscussion).Withoutanyinput(Ii=0foralli)andforlargenetworks(largeN),twospontaneouspatternsofactivityareseen.Ifg<1,thenetworkisinatrivialstateinwhichxi=0andri=R0forallneurons(alli).Thecaseg>1ismoreinterestinginthatthespontaneousactivityofthenetworkischaotic,meaningthatitisirregular,non‐repeatingandhighlysensitivetoinitialconditions
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(Sompolinsky,Crisanti,andSommers,1988;vanVreeswijkandSompolinsky,1996&1998).Wetypicallyuseavalueofg=1.5,meaningthatournetworksareinthischaoticstatepriortoactivatinganyinputs.
Figure2.Firingratesandresponsevariabilitynormalizedbythemaximumfiringrate,Rmax,forastimulusinputsteppingfromzerotoaconstant,non‐zerovalueatt=1000ms.Leftcolumnshowsthefiringrateofatypicalneuroninanetworkwith1000neurons.Rightcolumnshowstheaveragefiringrate(redtraces)andthesquarerootoftheaveragefiring‐ratevariance(blacktraces)acrossthenetworkneurons.a‐b)Anetworkwithchaoticspontaneousactivityreceivingnonoiseinput.Theresponsevariability(b,blacktrace)dropstozerowhenthestimulusinputispresent.c‐d)Anetworkwithoutspontaneousactivitybutreceivingnoiseinput.Theresponsevariability(d,blacktrace)risesslightlywhenthestepinputisturnedon.Thestochasticinputinthisexamplewasindependentwhite‐noisetoeachneuron,low‐passfilteredwitha500mstimeconstant.ResponsestoStepInputTobeginourexaminationoftheeffectsofinputonchaoticspontaneousnetworkactivity,weconsidertheeffectofastepofinput(from0toapositivevalue),applieduniformlytoeveryneuron(Ii=Iforalli).Beforetheinputisturnedon(Figure2a&b,t<1000ms),atypicalneuronofthenetworkshowsthehighlyirregularactivitycharacteristicofthechaoticspontaneousstate.However,whenasufficientlystrongstimulusisapplied,theinternallygeneratedfluctuationsarecompletelysuppressed(Figure2a&b,t>1000ms).Wecontrastthisbehaviortothatofexternalnoise,byturningofftherecurrentdynamicsandgeneratingfluctuationswithexternal
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stochasticinputs(Figure1c&d,t<1000ms).Inthiscase,thereisnoreductionintheamplitudeoftheneuronalfluctuationswhenastimulusisapplied,infactthereisasmallincrease.Notethattheincreaseinthemeanactivityissimilarinbothcases.Theseresultsrevealacriticaldistinctionbetweeninternallyandexternallygeneratedfluctuations–theformercanbesuppressedbyastimulusandthereforedonotnecessarilyinterferewithsensoryprocessing.
Torevealthenatureofinternallygeneratedvariability,wehaveconsideredanidealizedscenarioinFigure2inwhichtherewasnoexternalsourceofnoise.Inreality,weexpectbothexternalandinternalsourcesofnoisetocoexistinlocalcorticalcircuits.Aslongastheinternalnoiseprovidesasubstantialcomponentoftheoverallvariability,ourqualitativeresultsremainvalid.Wecansimulatethissituationbyaddingexternalnoise(asinFigure2c&d)toamodelthatexhibitschaoticspontaneousactivity(asinFigure2a&b).Theresultshowsasharpdropinvarianceatstimulusonset,butwithonlypartial,ratherthancomplete,suppressionofresponsevariability(Figure3).Thisresultisingoodagreementwithexperimentaldata(Churchlandetal.,2009).
Figure3.Firingratesandresponsevariabilitynormalizedbythemaximumfiringrate,Rmax,forthesamenetwork,stimulusandnoiseasinFigure2,butforanetworkwithbothspontaneousactivityandinjectednoise.a)Theresponseofatypicalneuron.b)Theaveragefiringrate(redtrace)increasesandtheresponsevariability(blacktrace)decreaseswhenthestimulusinputispresent,butitdoesnotgotozeroasinFigure2b.
ResponsetoPeriodicInputForFigures2&3,thestimulusconsistedofastepinputappliedidenticallytoallneurons.Toinvestigatetheeffectofmoreinterestingandrealisticstimulionthechaoticactivityofarecurrentnetwork,weconsiderinputswithnon‐homogeneousspatio‐temporalstructure.Specifically,weintroduceinputsthatoscillateinasinusoidalmannerwithamplitudeIandfrequencyfandexaminehowthesuppressionoffluctuationsdependsontheiramplitudeandfrequency.Inmanycases,neuronsinalocalpopulationhavediversestimulusselectivities,soa
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particularstimulusmayinducelittlechangeinthetotalactivityacrossthenetwork.Tomimicthissituation,wegivetheseoscillatinginputsadifferentphaseforeachneuron(intermsofavisualstimulus,thisisequivalenttopresentingastationary,counterphasegratingtoapopulationofsimplecellswithdifferentspatial‐phaseselectivities).Specifically,Ii=Icos(2πft+θi),whereθiischosenrandomlyfromauniformdistributionbetween0and2π.Therandomlyassignedphasesensurethatthespatialpatternofinputinourmodelnetworkisnotcorrelatedwiththepatternofrecurrentconnectivity.
Figure4.Achaoticnetworkof1000neuronsreceivingsinusoidal5Hzinput.a)Firingratesoftypicalnetworkneurons(normalizedbyRmax).b)Thelogarithmofthepowerspectrumoftheactivityacrossthenetwork.i)Withnoinput(I=0),networkactivityischaotic.ii)Inthepresenceofaweakinput(I/I1/2=0.1),anoscillatoryresponseissuperposedonchaoticfluctuations.iii)Forastrongerinput(I/I1/2=0.5),thenetworkresponseisperiodic.
Intheabsenceofastimulusinput,thefiringratesofindividualneuronsfluctuateirregularly,asseeninFigure2(Figure4a‐i),andthepowerspectrumacrossnetworkneuronsiscontinuousanddecaysexponentiallyasafunctionoffrequency(Figure4b‐i),acharacteristicfeaturesofthechaoticstateofthisnetwork(Sompolinsky,Crisanti,andSommers,1988).Whenthenetworkisdrivenbyaweakoscillatoryinput,thesingle‐neuronresponseisasuperpositionofaperiodicpatterninducedbytheinputandachaoticbackground(Figure4a‐ii).Thepowerspectrumshowsacontinuouscomponentduetotheresidualchaosandpeaksatthefrequencyoftheinputanditsharmonics,reflectingtheperiodicbutnon‐sinusoidalcomponentoftheresponse(Figure4b‐ii).Foraninputwithalargeramplitude,thefiringratesofnetworkneuronsareperiodic(Figure4a‐iii),andthepowerspectrumshowsonlypeaksattheinputfrequencyanditsharmonics,withnocontinuous
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spectrum(Figure4b‐iii).ThisindicatesacompletesuppressionoftheinternallygeneratedfluctuationsasinFigure2a&b.
Wehaveusedamean‐fieldapproachsimilartothatdevelopedbySompolinsky,Crisanti,andSommers(1988)toanalyzepropertiesofthetransitionbetweenchaoticaperiodicresponsestoaperiodicstimulus(Rajan,AbbottandSompolinsky,2009).Thisextendspreviousworkontheeffectofinputonchaoticnetworkactivity(Molgedey,SchuchhardtandSchuster,1992;BertchingerandNatschläger,2004)tocontinuoustimemodelsandperiodicinputs.Wefindthatthereisacriticalinputintensity(acriticalvalueofI)thatdependsonfandg,belowwhichnetworkactivityischaoticthoughdrivenbytheinput(asinFigures4a‐ii&4b‐ii)andabovewhichitisperiodic(asinFigures4a‐iii&4b‐iii).Asurprisingfeatureofthiscriticalamplitudeisthatitisanon‐monotonicfunctionofthefrequencyfoftheinput.Asaresult,thereisa“best”frequencyatwhichitiseasiesttoentrainthenetworkandsuppresschaos.Fortheparametersweuse,the“best”frequencyisaround5Hz,afrequencyweremanysensorysystemstendtooperate,andtherearesomeinitialexperimentalindicationsthatthisisindeedtheoptimalfrequencyforsuppressingbackgroundactivitybyvisualstimulation(WhiteandFiser,2008).Itisinterestingthatapreferredinputfrequencyforentrainmentariseseventhoughthepowerspectrumofthespontaneousactivitydoesnotshowanyresonantfeatures(Figure4b‐i).
PrincipalComponentAnalysisofSpontaneousandEvokedActivityTheresultsoftheprevioustwosectionsrevealedaregimeinwhichaninputgeneratesanon‐chaoticnetworkresponse,eventhoughthenetworkischaoticintheabsenceofinput.Althoughthechaoticintrinsicactivityhasbeencompletelysuppressedinthisnetworkstate,itsimprintcanstillbedetectedinthespatialpatternofthenon‐chaoticactivity.
ThenetworkstateatanyinstantcanbedescribedbyapointinanN‐dimensionalspacewithcoordinatesequaltothefiringratesoftheNneurons.Overtime,activitytraversesatrajectoryinthisN‐dimensionalspace.Principalcomponentanalysiscanbeusedtodelineatethesubspaceinwhichthistrajectorypredominantlylies.Theanalysisisdonebydiagonalizingtheequal‐timecross‐correlationmatrixofnetworkfiringrates,<ri(t)rj(t)>,wheretheanglebracketsdenoteanaverageovertimeTheeigenvaluesofthismatrixexpressedasafractionoftheirsum(denotedby
€
˜ λ a ),indicatethedistributionofvariancesacrossdifferentorthogonaldirectionsintheactivitytrajectory.Inthespontaneousstate,thereareanumberofsignificantcontributorstothetotalvariance,asindicatedinFigure5a.Forthisvalueofg,theleading10%ofthecomponentsaccountfor90%ofthetotalvariance.Thevarianceassociatedwithhighercomponentsfallsoffexponentially.Itisinterestingtonotethattheprojectionsofthenetworkactivityontotheprincipalcomponentdirectionsfluctuatemorerapidlyforhighercomponents(Figure5c),revealingtheinteractionbetweenthespatialandtemporalstructureofthechaoticfluctuations.
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Figure5:Principalcomponentanalysisofthechaoticspontaneousstateandnon‐chaoticdrivenstate.a)Percentvarianceaccountedforbydifferentprincipalcomponentsforchaoticspontaneousactivity.b)Sameasa,butfornon‐chaoticdrivenactivity.c)Projectionsofthechaoticspontaneousactivityontoprincipalcomponentvectors1,10and50(indecreasingorderofvariance).d)Projectionsofperiodicdrivenactivityontoprincipalcomponents1,3,and5.Projectionsontocomponents2,4,and6aresimilarexceptforbeingphaseshiftedbyπ/2.e)Theeffectivedimension,Neff,ofthetrajectoryofchaoticspontaneousactivity(definedinthetext)asafunctionofgfornetworkswith1000(solidcircles)or2000(opencircles)neurons.Parameters:g=1.5fora‐d,andf=5andI/I1/2=0.7forbandd.
Thenon‐chaoticdrivenstateisapproximatelytwodimensional(Figure5b),with
thetwodimensionsdescribingacircularoscillatoryorbit.Projectionsofthisorbitcorrespondtotheoscillationsπ/2apartinphase.Theresidualvarianceinthehigherdimensionsreflectshigherharmonicsarisingfromnetworknonlinearity,asillustratedbytheprojectionsinFigure5d.
Toquantifythedimensionofthesubspacecontainingthechaotictrajectoryinmoredetail,weintroducethequantity
€
Neff = ˜ λ a2
a=1
N
∑
−1
.
Thisprovidesameasureoftheeffectivenumberofprincipalcomponentsdescribingatrajectory.Forexample,ifnprincipalcomponentssharethetotalvarianceequally,andtheremainingNnprincipalcomponentshavezerovariance,Neff=n.Forthechaoticspontaneousstateinthenetworkswestudy,Neffincreaseswithg(Figure5e),duetothehigheramplitudeandfrequencycontentofthechaoticactivityforlargeg.NotethatNeffscalesapproximatelywithN,whichmeansthatlargenetworkshaveproportionallyhigher‐dimensionalchaoticactivity(comparethetwotracesinFigures5e).Thefactthatthenumberofactivatedmodesisonly2%ofthesystemdimensionality,evenforgashighas2.5,isanothermanifestationofthedeterministicnatureofthefluctuations.Forcomparison,wecalculatedNeffforasimilarnetworkdrivenbyexternalwhitenoise,withgsetbelowthechaotictransitionatg=1.Inthiscase,Neffonlyassumesuchlowvalueswhengiswithinafewpercentofthecriticalvalue1.TheresultsinFigure5illustrateanotherfeature
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ofthesuppressionofspontaneousactivitybyinput,whichisthatthePCAdimensionNeffisreduceddramaticallybythepresenceoftheinput.
NetworkEffectsontheSpatialPatternofEvokedActivityInthenon‐chaoticregime,thetemporalstructureofnetworkresponsesislargelydeterminedbytheinput;theybothoscillateatthesamefrequency,althoughthenetworkactivityincludesharmonicsnotpresentintheinput.Theinputdoesnot,however,exertnearlyasstrongcontrolonthespatialstructureofthenetworkresponse.Thephasesofthefiring‐rateoscillationsofnetworkneuronsareonlypartiallycorrelatedwiththephasesoftheinputsthatdrivethem,andtheyarestronglyinfluencedbytherecurrentfeedback.
Wehaveseenthattheorbitdescribingtheactivityinthenon‐chaoticdrivenstateconsistsprimarilyofacircleinatwo‐dimensionalsubspaceofthefullN‐dimensionsdescribingneuronalactivities.Wenowaskhowthiscirclealignsrelativetosubspacesdefinedbydifferentnumbersofprincipalcomponentsthatcharacterizethespontaneousactivity.Thisrelationshipisdifficulttovisualizebecauseboththechaoticsubspaceandthefullspaceofnetworkactivitiesarehighdimensional.Toovercomethisdifficulty,wemakeuseofthenotionof“principalangles”betweensubspaces(IpsenandMeyer,1995).
Thefirstprincipalangleistheanglebetweentwounitvectors(calledprincipalvectors),oneineachsubspace,thathavethemaximumoverlap(dotproduct).Higherprincipalanglesaredefinedrecursivelyastheanglesbetweenpairsofunitvectorswiththehighestoverlapthatareorthogonaltothepreviouslydefinedprincipalvectors.Specifically,fortwosubspacesofdimensiond1andd2definedbytheorthogonalunitvectorsV1a,fora=1,2,...,d1andV2b,forb=1,2,...,d2,thecosinesoftheprincipalanglesareequaltothesingularvaluesofthed1byd2matrixformedfromallthepossibledotproductsofthesetwovectors.Theresultingprincipalanglesvarybetween0andπ/2withzeroanglesappearingwhenpartsofthetwosubspacesoverlapandπ/2correspondingtodirectionsinwhichthetwosubspacesarecompletelynon‐overlapping.Theanglebetweentwosubspacesisthelargestoftheirprincipalangles.ThisdefinitionisillustratedinFigure6awhereweshowtheirregulartrajectoryofthechaoticspontaneousactivity,describedbyitstwoleadingprincipalcomponents(blackcurveinFigure6a).Thecircularorbitoftheperiodicactivity(redcurveinFigure6a)hasbeenrotatedbythesmallerofitstwoprincipalangles.Theanglebetweenthesetwosubspaces(theangledepictedinFigure6a)isthentheremaininganglethroughwhichtheperiodicorbitwouldhavetoberotatedtobringitintoalignmentwiththehorizontalplanecontainingthetwo‐dimensionalprojectionofthechaotictrajectory.
Figure6ashowstheanglebetweenthesubspacesdefinedbythefirsttwoprincipalcomponentsoftheorbitofperiodicdrivenactivityandthefirsttwoprincipalcomponentsofthechaoticspontaneousactivity.Wenowextendthisideatoacomparisonofthetwo‐dimensionalsubspaceoftheperiodicorbitandsubspacesdefinedbythefirstmprincipalcomponentsofthechaoticspontaneousactivity.ThisallowsustoseehowtheorbitliesinthefullN‐dimensionalspaceof
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neuronalactivitiesrelativetothetrajectoryofthechaoticspontaneousactivity.Theresults(Figure6b,reddots)showthatthisangleisclosetoπ/2forsmallm,equivalenttotheanglebetweentworandomlychosensubspaces.However,thevaluedropsquicklyforsubspacesdefinedbyprogressivelymoreoftheleadingprincipalcomponentsofthechaoticactivity.Ultimately,thisangleapproacheszerowhenallNofthechaoticprincipalcomponentvectorsareconsidered,asitmust,becausethesespantheentirespaceofnetworkactivities.
Figure6:Spatialpatternofnetworkresponses.a)Definitionoftheanglebetweenthesubspacedefinedbythefirsttwocomponentsofthechaoticactivity(blackcurve)andatwo‐dimensionaldescriptionoftheperiodicorbit(redcurve).b)Relationshipbetweentheorientationofperiodicandchaotictrajectories.Anglesbetweenthesubspacedefinedbythetwoprincipalcomponentsofthenon‐chaoticdrivenstateandsubspacesformedbyprincipalcomponents1throughmofthechaoticspontaneousactivity,wheremappearsonthehorizontalaxis(reddots).Blackdotsshowtheanalogousanglesbutwiththetwo‐dimensionalsubspacedefinedbyrandominputphasesreplacingthesubspaceofthenon‐chaoticdrivenactivity.c)Effectofinputfrequencyontheorientationoftheperiodicorbit.Theangle(verticalaxis)betweenthesubspacesdefinedbythetwoleadingprincipalcomponentsofnon‐chaoticdrivenactivityatdifferentfrequencies(horizontalaxis)andthesetwovectorsfora5Hzinputfrequency.d)Networkselectivitytodifferentspatialpatternsofinput.Signal(dashedcurvesandopencircles)andnoise(solidcurvesandfilledcircles)amplitudesinresponsetoinputsalignedtotheleadingprincipalcomponentsofthespontaneousactivityofthenetwork.Theinsetshowsalargerrangeonacoarserscale.Parameters:I/I1/2=0.7andf=5Hzforb,I/I1/2=1.0forc,andI/I1/2=0.2andf=2Hzford.
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Intheperiodicstate,thetemporalphasesofthedifferentneuronsdeterminetheorientationoftheorbitinthespaceofneuronalactivities.Therapidlyfallinganglebetweenthisorbitandthesubspacesdefinedbyspatialpatternsdominatingthechaoticstate(Figure6b,reddots)indicatesthatthesephasesarestronglyinfluencedbytherecurrentconnectivitythatinturndeterminesthespatialpatternofthespontaneousactivity.Asanindicationofthemagnitudeofthiseffect,wenotethattheanglesbetweentherandomphasesinusoidaltrajectoryoftheinputtothenetworkandthesamechaoticsubspacesaremuchlargerthanthoseassociatedwiththeperiodicnetworkactivity(Figure6b,blackdots).
TemporalFrequencyModulationofSpatialPatternsAlthoughrecurrentfeedbackinthenetworkplaysanimportantroleinthespatialstructureofdrivennetworkresponses,thespatialpatternoftheactivityisnotfixedbutinsteadisshapedbyacomplexinteractionbetweenthedrivinginputandtheintrinsicnetworkdynamics.Itisthereforesensitivetoboththeamplitudeandthefrequencyofthisdrive.Toseethis,weexaminehowtheorientationoftheapproximatelytwo‐dimensionalperiodicorbitofdrivennetworkactivityinthenon‐chaoticregimedependsoninputfrequency.Weusethetechniqueofprincipalanglesdescribedintheprevioussection,toexaminehowtheorientationoftheoscillatoryorbitchangeswhentheinputfrequencyisvaried.Forcomparisonpurposes,wechoosethedominanttwo‐dimensionalsubspaceofthenetworkoscillatoryresponsestoadrivinginputat5Hzasareference.Wethencalculatetheprincipalanglesbetweenthissubspaceandthecorrespondingsubspacesevokedbyinputswithdifferentfrequencies.TheresultshowninFigure6cindicatesthattheorientationoftheorbitforthesedrivenstatesrotatesastheinputfrequencychanges.
ThefrequencydependenceoftheorientationoftheevokedresponseislikelyrelatedtotheeffectseeninFigure6cinwhichhigherfrequencyactivityisprojectedontohigherprincipalcomponentsofthespontaneousactivity.Thiscausestheorbitofdrivenactivitytorotateinthedirectionofhigher‐orderprincipalcomponentsofthespontaneousactivityasthestimulusfrequencyincreases.Inaddition,thelargerthestimulusamplitude,theclosertheresponsephasesoftheneuronswillbetotherandomphasesoftheirexternalinputs(resultsnotshown).
NetworkSelectivityWehaveshownthattheresponseofanetworktorandom‐phaseinputisstronglyaffectedbythespatialstructureofspontaneousactivity(Figure6b).Wenowaskifthespatialpatternsthatdominatethespontaneousactivityinanetworkcorrespondtothespatialinputpatternstowhichthenetworkrespondsmostvigorously.Ratherthanusingrandom‐phaseinputs,wenowalignedtheinputstoournetworkalongthedirectionsdefinedbydifferentprincipalcomponentsofitsspontaneousactivity.Specifically,theinputtoneuroniissettoIViacos(2πft),whereIistheamplitudefactorandViaistheithcomponentofprincipalcomponentvectoraofthe
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spontaneousactivity.Theindexaisorderedsothata=1correspondstotheprincipalcomponentwithlargestvarianceanda=Ntheleast.Toanalyzetheresultsofusingthisinput,wedividetheresponseintoasignalcomponentcorrespondingtothetrial‐averagedresponse,andanoisecomponentconsistingofthefluctuationsaroundthisaverageresponse.Wecalltheamplitudeofthesignalcomponentoftheresponsethe“signalamplitude”andthestandarddeviationofthefluctuationsthe“noiseamplitude”.
AsseeninFigure6dtheamplitudeofthesignalcomponentoftheresponsedecreasesslowlyasafunctionofwhichprincipalcomponentisusedtodefinetheinput.Amoredramaticeffectisseenonthenoisecomponentoftheresponse.FortheinputamplitudeusedinFigure6d,inputsalignedtothefirst5principalcomponentsofthespontaneousactivitycompletelysuppressthechaoticnoise,resultinginperiodicdrivenactivity.Forhigher‐orderprincipalcomponents,thenetworkactivityischaotic.Thus,the“noise”showsmoresensitivitytothespatialstructureoftheinputthanthesignal.
DiscussionOurresultssuggestthatexperimentsthatstudythestimulus‐dependenceofthetypicallyignorednoisecomponentofresponsesshouldbeinterestingandcouldprovideinsightintothenatureandoriginofactivityfluctuations.Responsevariabilityandongoingactivityissometimesmodeledasarisingfromastochasticprocessexternaltothenetworkgeneratingtheresponses.Thisstochasticnoiseisthenaddedlinearlytothesignaltocreatethetotalneuronalactivityintheevokedstate.Ourresultsindicatethatrecurrentdynamicsofthecorticalcircuitislikelytocontributesignificantlytotheemergenceofirregularneuronalactivity,andthattheinteractionbetweensuchdeterministic“noise”andexternaldriveishighlynonlinear.Inourwork(Rajan,AbbottandSompolinsky,2009),wehaveshownthatthestimuluscausesastrongsuppressionofactivityfluctuationsandfurthermorethatthenonlinearinteractionbetweentherelativelyslowchaoticfluctuationsandthestimulusresultsinanon‐monotonicfrequencydependenceofthenoisesuppression.
Animportantfeatureofthenetworkswestudyisthatthevarianceofthesynapticstrengthsacrossthenetworkcontrolstheemergenceofinterestingcomplexdynamics.Thishasimportantimplicationsforexperimentsbecauseitsuggeststhatthemostinterestingandrelevantmodulatorsofnetworksmaybesubstancesoractivity‐dependentmodulationsthatdonotnecessarilychangepropertiesofsynapsesonaverage,butratherchangesynapticvariance.Synapticvariancecanbechangedeitherbymodifyingtherangeoverwhichsynapticstrengthsvaryacrossapopulationofsynapses,aswehavedonehere,orbymodifyingthereleaseprobabilityandvariabilityofquantalsizeatsinglesynapses.Suchmodulatorsmightbeviewedaslesssignificantbecausetheydonotchangethenetbalancebetweenexcitationandinhibition.However,networkmodelingsuggeststhatsuchmodulationsareofgreatimportanceincontrollingthestateoftheneuronalcircuit.
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Therandomcharacteroftheconnectivityinournetworkprecludesasimpledescriptionofthespatialactivitypatternsintermsoftopographicallyorganizedmaps.Ouranalysisshowsthatevenincorticalareaswheretheunderlyingconnectivitydoesnotexhibitsystematictopography,dissectingthespatialpatternsoffluctuationsinneuronalactivitycanrevealimportantinsightaboutbothintrinsicnetworkdynamicsandstimulusselectivity.Principalcomponentanalysisrevealedthatdespitethefactthatthenetworkconnectivitymatrixisfullrank,theeffectivedimensionalityofthechaoticfluctuationsismuchsmallerthanthenumberofneuronsinthenetwork.Thissuppressionofspatialmodesismuchstrongerthanexpectedfromalinearnetworklow‐passfilteringaspatio‐temporalwhitenoiseinput.Furthermore,asinthetemporaldomain,activespatialpatternsexhibitstrongnonlinearinteractionbetweenexternaldrivinginputsandintrinsicdynamics.Surprisingly,evenwhenthestimulusamplitudeisstrongenoughtofullyentrainthetemporalpatternofnetworkactivity,spatialorganizationoftheactivityisstillstronglyinfluencedbyrecurrentdynamics,asshowninFigures6cand6d.
Wehavepresentedtoolsforanalyzingthespatialstructureofchaoticandnon‐chaoticpopulationresponsesbasedonprincipalcomponentanalysisandanglesbetweentheresultingsubspaces.Principalcomponentanalysishas,beenappliedprofitablytoneuronalrecordings(see,forexample,Broome,JayaramanandLaurent,2006).Theseanalysesoftenplotactivitytrajectoriescorrespondingtodifferentnetworkstatesusingthefixedprincipalcomponentcoordinatesderivedfromcombinedactivitiesunderallconditions.Ouranalysisoffersacomplementaryapproachwherebyprincipalcomponentsarederivedforeachstimulusconditionseparately,andprincipalanglesareusedtorevealnotonlythedifferencebetweentheshapesoftrajectoriescorrespondingtodifferentnetworkstates,butalsothedifferenceintheorientationofthelowdimensionalsubspacesofthesetrajectorieswithinthefullspaceofneuronalactivity.
Manymodelsofselectivityincorticalcircuitsrelyonknowledgeofthespatialorganizationofafferentinputsaswellascorticalconnectivity.However,inmanycorticalareas,suchinformationisnotavailable.Ourresultsshowthatexperimentallyaccessiblespatialpatternsofspontaneousactivity(e.g.fromvoltage‐orcalcium‐sensitiveopticalimagingexperiments)canbeusedtoinferthestimulusselectivityinducedbythenetworkdynamicsandtodesignspatiallyextendedstimulithatevokestrongresponses.Thisisparticularlytruewhenselectivityismeasuredintermsoftheabilityofastimulustoentraintheneuraldynamics,asinFigure6d.Ingeneral,ourresultsindicatethattheanalysisofspontaneousactivitycanprovidevaluableinformationaboutthecomputationalimplicationsofneuronalcircuitry.
AcknowledgmentsResearchofKRandLAsupportedbyNationalScienceFoundationgrantIBN‐0235463andanNIHDirector'sPioneerAward,partoftheNIHRoadmapforMedicalResearch,throughgrantnumber5‐DP1‐OD114‐02.HSispartiallysupportedbygrantsfromtheIsraelScienceFoundationandtheMcDonnell
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Foundation.ThisresearchwasalsosupportedbytheSwartzFoundationthroughtheSwartzCentersatColumbiaandHarvardUniversities.
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