CommunicatedbyJonathanVictorTheTime-RescalingTheoremandItsApplicationt - PDF document

NOTE CommunicatedbyJonathanVictorTheTime-RescalingTheoremandItsApplicationtoNeuralSpikeTrainDataAnalysisEmeryN.Brownbrown@srlb.mgh.harvard.eduNeuroscienceStatisticsResearchLaboratory,DepartmentofAnesthesiaandCriticalCare,MassachusettsGeneral 326E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.FrankplementaryeyeŽeldofamacquemonkeyandacomparisonoftemporalandspatialsmoothers,inhomogeneousPoisson,inhomogeneousgamma,andinhomogeneousinversegaussianmodelsofrathippocampalplacecellspikingactivity.Tohelpmakethelogicbehindthetime-rescalingthe-oremmoreaccessibletoresearchersinneuroscience,wepresentaproofusingonlyelementaryprobabilitytheoryarguments.Wealsoshowhowthetheoremmaybeusedtosimulateageneralpointprocessmodelofaspiketrain.Ourparadigmmakesitpossibletocompareparametricandhistogram-basedneuralspiketrainmodelsdirectly.Theseresultssug-gestthatthetime-rescalingtheoremcanbeavaluabletoolforneuralspiketraindataanalysis.1Introduction Thedevelopmentofstatisticalmodelsthataccuratelydescribethestochasticstructureofneuralspiketrainsisagrowingareaofquantitativeresearchinneuroscience.EvaluatingmodelgoodnessofŽt—thatis,measuringquan-titativelytheagreementbetweenaproposedmodelandaspiketraindataseries—iscrucialforestablishingthemodel’svaliditypriortousingittomakeinferencesabouttheneuralsystembeingstudied.AssessinggoodnessofŽtforpointneuralspiketrainmodelsisamorechallengingproblemthanformodelsofcontinuous-valuedprocesses.Thisisbecausetypicaldistancediscrepancymeasuresappliedincontinuousdataanalyses,suchastheaver-agesumofsquareddeviationsbetweenrecordeddatavaluesandestimatedvaluesfromthemodel,cannotbedirectlycomputedforpointprocessdata.Goodness-of-Žtassessmentsareevenmorechallengingforhistogram-basedmodels,suchasperistimulustimehistograms(PSTH)andratefunctionsestimatedbyspiketrainsmoothing,becausetheprobabilityassumptionsneededtoevaluatemodelpropertiesareoftenimplicit.Berman(1983)andOgata(1988)developedtransformationsthat,underagivenmodel,convertpointprocesseslikespiketrainsintocontinuousmeasuresinordertoassessmodelgoodnessofŽt.Oneofthetheoreticalresultsusedtoconstructthesetransformationsisthetime-rescalingtheorem.Aformofthetime-rescalingtheoremiswellknowninelementaryprob-abilitytheory.ItstatesthatanyinhomogeneousPoissonprocessmayberescaledortransformedintoahomogeneousPoissonprocesswithaunitrate(Taylor&Karlin,1994).TheinversetransformationisastandardmethodforsimulatinganinhomogeneousPoissonprocessfromaconstantrate(homo-geneous)Poissonprocess.Meyer(1969)andPapangelou(1972)establishedthegeneraltime-rescalingtheorem,whichstatesthatanypointprocesswithanintegrableratefunctionmayberescaledintoaPoissonprocesswithaunitrate.BermanandOgataderivedtheirtransformationsbyapplyingthegeneralformofthetheorem.Whilethemoregeneraltime-rescalingtheo-remiswellknownamongresearchersinpointprocesstheory(BrÂemaud,1981;Jacobsen,1982;Daley&Vere-Jones,1988;Ogata,1988;Karr,1991),the TheTime-RescalingTheorem327theoremislessfamiliartoneuroscienceresearchers.Thetechnicalnatureoftheproof,whichreliesonthemartingalerepresentationofapointprocess,mayhavepreventeditssigniŽcancefrombeingmorebroadlyappreciated.Thetime-rescalingtheoremhasimportanttheoreticalandpracticalim-plicationsforapplicationofpointprocessmodelsinneuralspiketraindataanalysis.Tohelpmakethisresultmoreaccessibletoresearchersinneu-roscience,wepresentaproofthatusesonlyelementaryprobabilitytheoryarguments.Wedescribehowthetheoremmaybeusedtodevelopgoodness-of-Žttestsforbothparametricandhistogram-basedpointprocessmodelsofneuralspiketrains.Weapplythesetestsintwoexamples:acompar-isonofPSTH,inhomogeneousPoisson,andinhomogeneousMarkovin-tervalmodelsofneuralspiketrainsfromthesupplementaryeyeŽeldofamacquemonkeyandacomparisonoftemporalandspatialsmoothers,inhomogeneousPoisson,inhomogeneousgamma,andinhomogeneousin-versegaussianmodelsofrathippocampalplacecellspikingactivity.Wealsodemonstratehowthetime-rescalingtheoremmaybeusedtosimulateageneralpointprocess.2Theory 2.1TheConditionalIntensityFunctionandtheJointProbabilityDen-sityoftheSpikeTrain.DeŽneaninterval(0,T],andlet0u1u2,...,un¡1un·Tbeasetofevent(spike)timesfromapointprocess.Fort2(0,T],letN(t)bethesamplepathoftheassociatedcountingprocess.Thesamplepathisarightcontinuousfunctionthatjumps1attheeventtimesandisconstantotherwise(Snyder&Miller,1991).Inthisway,N(t)countsthenumberandlocationofspikesintheinterval(0,t].Therefore,itcontainsalltheinformationinthesequenceofeventsorspiketimes.Fort2(0,T],wedeŽnetheconditionalorstochasticintensityfunctionasl(t|Ht)DlimDt!0Pr(N(tCDt)¡N(t)D1|Ht) Dt,(2.1)whereHtDf0u1u2,...,uN(t)tgisthehistoryoftheprocessuptotimetanduN(t)isthetimeofthelastspikepriortot.IfthepointprocessisaninhomogeneousPoissonprocess,thenl(t|Ht)Dl(t)issimplythePoissonratefunction.Otherwise,itdependsonthehistoryoftheprocess.Hence,theconditionalintensitygeneralizesthedeŽnitionofthePoissonrate.Itiswellknownthatl(t|Ht)canbedeŽnedintermsoftheevent(spike)timeprobabilitydensity,f(t|Ht),asl(t|Ht)Df(t|Ht) 1¡RtuN(t)f(u|Ht)du,(2.2)fort�uN(t)(Daley&Vere-Jones,1988;Barbieri,Quirk,Frank,Wilson,&Brown,2001).Wemaygaininsightintoequation2.2andthemeaningofthe 328E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.FrankconditionalintensityfunctionbycomputingexplicitlytheprobabilitythatgivenHt,aspikeukoccursin[t,tCDt)wherekDN(t)C1.Todothis,wenotethattheeventsfN(tCDt)¡N(t)D1|Htgandfuk2[t,tCDt)|Htgareequivalentandthattherefore,Pr(N(tCDt)¡N(t)D1|Ht)DPr(uk2[t,tCDt)|uk�t,Ht)DPr(uk2[t,tCDt)|Ht) Pr(uk�t|Ht)DRtCDttf(u|Ht)du 1¡RtuN(t)f(u|Ht)du¼f(t|Ht)Dt 1¡RtuN(t)f(u|Ht)duDl(t|Ht)Dt.(2.3)DividingbyDtandtakingthelimitgiveslimDt!0Pr(uk2[t,tCDt)|Ht) DtDf(t|Ht) 1¡RtuN(t)f(u|Ht)duDl(t|Ht),(2.4)whichisequation2.1.Therefore,l(t|Ht)Dtistheprobabilityofaspikein[t,tCDt)whenthereishistorydependenceinthespiketrain.Insurvivalanalysis,theconditionalintensityistermedthehazardfunctionbecauseinthiscase,l(t|Ht)Dtmeasurestheprobabilityofafailureordeathin[t,tCDt)giventhattheprocesshassurviveduptotimet(Kalbeisch&Prentice,1980).Becausewewouldliketoapplythetime-rescalingtheoremtospiketraindataseries,werequirethejointprobabilitydensityofexactlyneventtimesin(0,T].Thisjointprobabilitydensityis(Daley&Vere-Jones,1988;Barbieri,Quirk,etal.,2001)f(u1,u2,...,un\N(T)Dn)Df(u1,u2,...,un\unC1�T)Df(u1,u2,...,un\N(un)Dn)Pr(unC1�T|u1,u2,...,un)DnYkD1l(uk|Huk)exp»¡Zukuk¡1l(u|Hu)du¼¢exp(¡ZTunl(u|Hu)du),(2.5) TheTime-RescalingTheorem329wheref(u1,u2,...,un\N(un)Dn)DnYkD1l(uk|Huk)exp»¡Zukuk¡1l(u|Hu)du¼(2.6)Pr(unC1�T|u1,u2,...,un)Dexp(¡ZTunl(u|Hu)du),(2.7)andu0D0.Equation2.6isthejointprobabilitydensityofexactlyneventsin(0,un],whereasequation2.7istheprobabilitythatthenC1steventoccursafterT.Theconditionalintensityfunctionprovidesasuccinctwaytorepresentthejointprobabilitydensityofthespiketimes.Wecannowstateandprovethetime-rescalingtheorem.Time-RescalingTheorem.Let0u1u2,...,unTbearealizationfromapointprocesswithaconditionalintensityfunctionl(t|Ht)satisfying0l(t|Ht)forallt2(0,T].DeŽnethetransformationL(uk)DZuk0l(u|Hu)du,(2.8)forkD1,...,n,andassumeL(t)1withprobabilityoneforallt2(0,T].ThentheL(uk)’sareaPoissonprocesswithunitrate.Proof.LettkDL(uk)¡L(uk¡1)forkD1,...,nandsettTDRTunl(u|Hu)du.Toestablishtheresult,itsufŽcestoshowthatthetksareindependentandidenticallydistributedexponentialrandomvariableswithmeanone.Becausethetktransformationisone-to-oneandtnC1&#x 00;tTifandonlyifunC1&#x 00;T,thejointprobabilitydensityofthetk’sisf(t1,t2,...,tn\tnC1&#x 00;tT)Df(t1,...,tn)Pr(tnC1&#x 00;tT|t1,...,tn).(2.9)Weevaluateeachofthetwotermsontherightsideofequation2.9.Thefollowingtwoeventsareequivalent:ftnC1&#x 00;tT|t1,...,tngDfunC1&#x 00;T|u1,u2,...,ung.(2.10)HencePr(tnC1&#x 00;tT|t1,t2,...,tn)DPr(unC1&#x 00;T|u1,u2,...,un)Dexp(¡ZTunl(u|Hun)du)Dexpf¡tTg,(2.11) 330E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.FrankwherethelastequalityfollowsfromthedeŽnitionoftT.Bythemultivariatechange-of-variableformula(Port,1994),f(t1,t2,...,tn)D|J|f(u1,u2,...,un\N(un)Dn),(2.12)whereJistheJacobianofthetransformationbetweenuj,jD1,...,nandtk,kD1,...,n.Becausetkisafunctionofu1,...,uk,Jisalowertriangularmatrix,anditsdeterminantistheproductofitsdiagonalelementsdeŽnedas|J|D|QnkD1Jkk|.Byassumption0l(t|Ht)andbyequation2.8andthedeŽnitionoftk,themappingofuintotisone-to-one.Therefore,bytheinversedifferentiationtheorem(Protter&Morrey,1991),thediagonalelementsofJareJkkD@uk @tkDl(uk|Huk)¡1.(2.13)Substituting|J|andequation2.6intoequation2.12yieldsf(t1,t2,...,tn)DnYkD1l(uk|Huk)¡1nYkD1l(uk|Huk)¢exp»¡Zukuk¡1l(u|Hu)du¼DnYkD1expf¡[L(uk)¡L(uk¡1)]gDnYkD1expf¡tkg.(2.14)Substitutingequations2.11and2.14into2.9yieldsf(t1,t2,...,tn\tnC1�tT)Df(t1,...,tn)Pr(tnC1�tT|t1,...,tn)D³nYkD1expf¡tkg´expf¡tTg,(2.15)whichestablishestheresult.Thetime-rescalingtheoremgeneratesahistory-dependentrescalingofthetimeaxisthatconvertsapointprocessintoaPoissonprocesswithaunitrate.2.2AssessingModelGoodnessofFit.Wemayusethetime-rescalingtheoremtoconstructgoodness-of-Žttestsforaspikedatamodel.Oncea TheTime-RescalingTheorem331modelhasbeenŽttoaspiketraindataseries,wecancomputefromitsestimatedconditionalintensitytherescaledtimestkDL(uk)¡L(uk¡1).(2.16)Ifthemodeliscorrect,then,accordingtothetheorem,thetksareindepen-dentexponentialrandomvariableswithmean1.IfwemakethefurthertransformationzkD1¡exp(¡tk),(2.17)thenzksareindependentuniformrandomvariablesontheinterval(0,1).Be-causethetransformationsinequations2.16and2.17arebothone-to-one,anystatisticalassessmentthatmeasuresagreementbetweenthezksandauni-formdistributiondirectlyevaluateshowwelltheoriginalmodelagreeswiththespiketraindata.Herewepresenttwomethods:Kolmogorov-Smirnovtestsandquantile-quantileplots.ToconstructtheKolmogorov-Smirnovtest,weŽrstorderthezksfromsmallesttolargest,denotingtheorderedvaluesasz(k)s.Wethenplottheval-uesofthecumulativedistributionfunctionoftheuniformdensitydeŽnedasbkDk¡1 2 nforkD1,...,nagainstthez(k)s.Ifthemodeliscorrect,thenthepointsshouldlieona45-degreeline(Johnson&Kotz,1970).ConŽdenceboundsforthedegreeofagreementbetweenthemodelsandthedatamaybeconstructedusingthedistributionoftheKolmogorov-Smirnovstatistic.Formoderatetolargesamplesizesthe95%(99%)conŽdenceboundsarewellapproximatedasbk§1.36/n1/2(bk§1.63/n1/2)(Johnson&Kotz,1970).WetermsuchaplotaKolmogorov-Smirnov(KS)plot.Anotherapproachtomeasuringagreementbetweentheuniformprob-abilitydensityandthezksistoconstructaquantile-quantile(Q-Q)plot(Ventura,Carta,Kass,Gettner,&Olson,2001;Barbieri,Quirk,etal.,2001;Hogg&Tanis,2001).Inthisdisplay,weplotthequantilesoftheuniformdistribution,denotedherealsoasthebks,againstthez(k)s.AsinthecaseoftheKSplots,exactagreementoccursbetweenthepointprocessmodelandtheexperimentaldataifthepointslieona45-degreeline.PointwiseconŽ-dencebandscanbeconstructedtomeasurethemagnitudeofthedepartureoftheplotfromthe45-degreelinerelativetochance.Toconstructpoint-wisebands,wenotethatifthetksareindependentexponentialrandomvariableswithmean1andthezksarethusuniformontheinterval(0,1),theneachz(k)hasabetaprobabilitydensitywithparameterskandn¡kC1deŽnedasf(z|k,n¡kC1)Dn! (n¡k)!(k¡1)!zk¡1(1¡z)n¡k,(2.18)for0z1(Johnson&Kotz,1970).Wesetthe95%conŽdenceboundsbyŽndingthe2.5thand97.5thquantilesofthecumulativedistribution 332E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.Frankassociatedwithequation2.18forkD1,...,n.Theseexactquantilesarereadilyavailableinmanystatisticalsoftwarepackages.Formoderatetolargespiketraindataseries,areasonableapproximationtothe95%(99%)conŽdenceboundsisgivenbythegaussianapproximationtothebino-mialprobabilitydistributionasz(k)§1.96[z(k)(1¡z(k))/n]1/2(z(k)§2.575[z(k)(1¡z(k))/n]1/2).Toourknowledge,theselocalconŽdenceboundsfortheQ-Qplotsbasedonthebetadistributionandthegaussianapproxi-mationarenew.Ingeneral,theKSconŽdenceintervalswillbewiderthanthecorre-spondingQ-Qplotintervals.Toseethis,itsufŽcestocomparethewidthsofthetwointervalsusingtheirapproximateformulasforlargen.Fromthegaussianapproximationtothebinomial,themaximumwidthofthe95%conŽdenceintervalfortheQ-Qplotsoccursatthemedian:z(k)D0.50andis2[1.96/(4n)1/2]D1.96n¡1/2.Fornlarge,thewidthofthe95%conŽdenceintervalsfortheKSplotsis2.72n¡1/2atallquantiles.TheKSconŽdenceboundsconsiderthemaximumdiscrepancyfromthe45-degreelinealongallquantiles;the95%bandsshowthediscrepancythatwouldbeexceeded5%ofthetimebychanceiftheplotteddataweretrulyuniformlydistributed.TheQ-QplotconŽdenceboundsconsiderthemaximumdiscrepancyfromthe45-degreelineforeachquantileseparately.Thesepointwise95%con-Ždenceboundsmarktheamountbywhicheachvaluez(k)woulddeviatefromthetruequantile5%ofthetimepurelybychance.TheKSboundsarebroadbecausetheyarebasedonthejointdistributionofallndeviations,andtheyconsiderthedistributionofthelargestofthesedeviations.TheQ-Qplotboundsarenarrowerbecausetheymeasurethedeviationateachquantileseparately.Usedtogether,thetwoplotshelpapproximateupperandlowerlimitsonthediscrepancybetweenaproposedmodelandaspiketraindataseries.3Applications 3.1AnAnalysisofSupplementaryEyeFieldRecordings.FortheŽrstapplicationofthetime-rescalingtheoremtoagoodness-of-Žtanalysis,weanalyzeaspiketrainrecordedfromthesupplementaryeyeŽeld(SEF)ofamacaquemonkey.NeuronsintheSEFplayaroleinoculomotorprocesses(Olson,Gettner,Ventura,Carta,&Kass,2000).Astandardparadigmforstudyingthespikingpropertiesoftheseneuronsisadelayedeyemovementtask.Inthistask,themonkeyŽxates,isshownlocationsofpotentialtargetsites,andisthencuedtothespeciŽctargettowhichitmustsaccade.Next,apreparatorycueisgiven,followedarandomtimelaterbyagosignal.Uponreceivingthegosignal,theanimalmustsaccadetothespeciŽctargetandholdŽxationforadeŽnedamountoftimeinordertoreceiveareward.BeginningfromthepointofthespeciŽctargetcue,neuralactivityisrecordedforaŽxedintervaloftimebeyondthepresentationofthegosignal.Afterabriefrestperiod,thetrialisrepeated.Multipletrialsfromanexperiment TheTime-RescalingTheorem333suchasthisarejointlyanalyzedusingaPSTHtoestimateŽringrateforaŽniteintervalfollowingaŽxedinitiationpoint.Thatis,thetrialsaretimealignedwithrespecttoaŽxedinitialpoint,suchasthetargetcue.ThedataacrosstrialsarebinnedintimeintervalsofaŽxedlength,andtherateineachbinisestimatedastheaveragenumberofspikesintheŽxedtimeinterval.KassandVentura(2001)recentlypresentedinhomogeneousMarkovin-terval(IMI)modelsasanalternativetothePSTHforanalyzingmultiple-trialneuralspiketraindata.ThesemodelsuseaMarkovrepresentationfortheconditionalintensityfunction.OneformoftheIMIconditionalintensityfunctiontheyconsideredisl(t|Ht)Dl(t|uN(t),h)Dl1(t|h)l2(t¡uN(t)|h),(3.1)whereuN(t)isthetimeofthelastspikepriortot,l1(t|h)modulatesŽr-ingasafunctionoftheexperimentalclocktime,l2(t¡uN(t)|h)repre-sentsMarkovdependenceinthespiketrain,andhisavectorofmodelparameterstobeestimated.KassandVenturamodeledlogl1(t|h)andlogl2(t¡uN(t)|h)asseparatepiecewisecubicsplinesintheirrespectiveargumentstandt¡uN(t).Thecubicpieceswerejoinedatknotssothattheresultingfunctionsweretwicecontinuouslydifferentiable.Thenumberandpositionsoftheknotswerechoseninapreliminarydataanalysis.Inthespecialcasel(t|uN(t),h)Dl1(t|h),theconditionalintensityfunctioninequation3.1correspondstoaninhomogeneousPoisson(IP)modelbecausethisassumesnotemporaldependenceamongthespiketimes.KassandVenturausedtheirmodelstoanalyzeSEFdatathatconsistedof486spikesrecordedduringthe400msecfollowingthetargetcuesignalin15trialsofadelayedeyemovementtask(neuronPK166afromOlsonetal.,2000,usingthepatterncondition).TheIPandIMImodelswereŽtbymaximumlikelihood,andstatisticalsigniŽcancetestsonthesplineco-efŽcientswereusedtocomparegoodnessofŽt.Includedintheanalysisweresplinemodelswithhigher-orderdependenceamongthespiketimesthantheŽrst-orderMarkovdependenceinequation3.1.TheyfoundthattheIMImodelgaveastatisticallysigniŽcantimprovementintheŽtrelativetotheIPandthataddinghigher-orderdependencetothemodelgavenofurtherimprovements.TheŽtoftheIMImodelwasnotimprovedbyin-cludingtermstomodelbetween-trialdifferencesinspikerate.TheauthorsconcludedthattherewasstrongŽrst-orderMarkovdependenceintheŽr-ingpatternofthisSEFneuron.KassandVenturadidnotprovideanoverallassessmentofmodelgoodnessofŽtorevaluatehowmuchtheIMIandtheIPmodelsimprovedoverthehistogram-basedratemodelestimatedbythePSTH.UsingtheKSandQ-Qplotsderivedfromthetime-rescalingtheorem,itispossibletocomparedirectlytheŽtsoftheIMI,IP,andPSTHmodelsandtodeterminewhichgivesthemostaccuratedescriptionoftheSEFspiketrainstructure.TheequationsfortheIPratefunction,lIP(t|hIP)Dl1(t|hIP), 334E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.FrankandfortheIMIconditionalintensityfunction,lIMI(t|uN(t),hIMI)Dl1(t|hIMI)l2(t¡uN(t),hIMI),aregivenintheappendix,alongwithadis-cussionofthemaximumlikelihoodprocedureusedtoestimatetheco-efŽcientsofthesplinebasiselements.Theestimatedconditionalinten-sityfunctionsfortheIPandIMImodelsare,respectively,lIP(t|,OhIP)andlIMI(t|uN(t),OhIMI),whereOhisthemaximumlikelihoodestimateofthespeciŽcsplinecoefŽcients.ForthePSTHmodel,theconditionalintensityestimateisthePSTHcomputedbyaveragingthenumberofspikesineachof4010msecbins(the400msecfollowingthetargetcuesignal)acrossthe15trials.ThePSTHistheŽtofanotherinhomogeneousPoissonmodelbecauseitassumesnodependenceamongthespiketimes.TheresultsoftheIMI,IP,andPSTHmodelŽtscomparedbyKSandQ-QplotsareshowninFigure1.FortheIPmodel,thereislackofŽtatlowerquantiles(below0.25)becauseinthatrange,itsKSplotliesjustout-sidethe95%conŽdencebounds(seeFigure1A).Fromquantile0.25andbeyond,theIPmodeliswithinthe95%conŽdencebounds,althoughbe- TheTime-RescalingTheorem335yondthequantile0.75,itslightlyunderpredictsthetrueprobabilitymodelofthedata.TheKSplotofthePSTHmodelissimilartothatoftheIPexceptthatitliesentirelyoutsidethe95%conŽdencebandsbelowquan-tile0.50.Beyondthisquantile,itiswithinthe95%conŽdencebounds.TheKSplotofthePSTHunderpredictstheprobabilitymodelofthedatatoagreaterdegreeintheupperrangethantheIPmodel.TheIMImodeliscompletelywithinthe95%conŽdenceboundsandliesalmostexactlyonthe45-degreelineofcompleteagreementbetweenthemodelandthedata.TheQ-Qplotanalyses(seeFigure1B)agreewithKSplotanalyseswithafewexceptions.TheQ-QplotanalysesshowthatthelackofŽtoftheIPandPSTHmodelsisgreateratthelowerquantiles(0–0.50)thansug-gestedbytheKSplots.TheQ-QplotsfortheIPandPSTHmodelsalsoshowthatthedeviationsofthesetwomodelsnearquantiles0.80to0.90arestatisticallysigniŽcant.Withtheexceptionofasmalldeviationbelowquantile0.10,theQ-QplotoftheIMIliesalmostexactlyonthe45-degreeline. Figure1:Facingpage.(A)Kolmogorov-Smirnov(K-S)plotsoftheinhomoge-neousMarkovinterval(IMI)model,inhomogeneousPoisson(IP),andperstim-ulustimehistogram(PSTH)modelŽtstotheSEFspiketraindata.Thesolid45-degreelinerepresentsexactagreementbetweenthemodelandthedata.Thedashed45-degreelinesarethe95%conŽdenceboundsforexactagree-mentbetweenthemodelandexperimentaldatabasedonthedistributionoftheKolmogorov-Smirnovstatistic.The95%conŽdenceboundsarebk§1.36n¡1 2,wherebkD(k¡1 2)/nforkD1,...,nandnisthetotalnumberofspikes.TheIMImodel(thick,solidline)iscompletelywithinthe95%conŽdenceboundsandliesalmostexactlyonthe45-degreeline.TheIPmodel(thin,solidline)haslackofŽtatlowerquantiles(0.25).Fromthequantile0.25andbe-yond,theIPmodeliswithinthe95%conŽdencebounds.TheKSplotofthePSTHmodel(dottedline)issimilartothatoftheIPmodelexceptthatitliesoutsidethe95%conŽdencebandsbelowquantile0.50.Beyondthisquantile,itiswithinthe95%conŽdencebounds,yetitunderpredictstheprobabilitymodelofthedatatoagreaterdegreeinthisrangethantheIPmodeldoes.TheIMImodelagreesmorecloselywiththespiketraindatathaneithertheIPorthePSTHmodels.(B)Quantile-quantile(Q-Q)plotsoftheIMI(thick,solidline),IP(thin,solidline),andPSTH(dottedline)models.Thedashedlinesarethelocal95%conŽdenceboundsfortheindividualquantilescomputedfromthebetaprobabilitydensitydeŽnedinequation2.18.Thesolid45-degreelinerepresentsexactagreementbetweenthemodelandthedata.TheQ-QplotssuggeststhatthelackofŽtoftheIPandPSTHmodelsisgreateratthelowerquantiles(0–0.50)thansuggestedbytheKSplots.TheQ-QplotsfortheIPandPSTHmodelsalsoshowthatthedeviationsofthesetwomodelsnearquantiles0.80to0.90arestatisticallysigniŽcant.Withtheexceptionofasmalldevia-tionbelowquantile0.10,theQ-QplotoftheIMIliesalmostexactlyonthe45-degreeline. 336E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.FrankWeconcludethattheIMImodelgivesthebestdescriptionoftheseSEFspiketrains.InagreementwiththereportofKassandVentura(2001),thisanalysissupportsaŽrst-orderMarkovdependenceamongthespiketimesandnotaPoissonstructure,aswouldbesuggestedbyeithertheIPorthePSTHmodels.ThisanalysisextendstheŽndingsofKassandVenturabyshowingthatofthethreemodels,thePSTHgivesthepoorestdescriptionoftheSEFspiketrain.TheIPmodelgivesabetterŽttotheSEFdatathanthePSTHmodelbecausethemaximumlikelihoodanalysisoftheparametricIPmodelismorestatisticallyefŽcientthanthehistogram(methodofmo-ments)estimateobtainedfromthePSTH(Casella&Berger,1990).Thatis,theIPmodelŽtbymaximumlikelihoodusesallthedatatoestimatetheconditionalintensityfunctionatalltimepoints,whereasthePSTHanal-ysisusesonlyspikesinaspeciŽedtimebintoestimatetheŽringrateinthatbin.TheadditionalimprovementoftheIMImodelovertheIPisduetothefactthattheformerrepresentstemporaldependenceinthespiketrain.3.2AnAnalysisofHippocampalPlaceCellRecordings.Asasecondexampleofusingthetime-rescalingtheoremtodevelopgoodness-of-Žttests,weanalyzethespikingactivityofapyramidalcellintheCA1regionoftherathippocampusrecordedfromananimalrunningbackandforthonalineartrack.Hippocampalpyramidalneuronshaveplace-speciŽcŽring(O’Keefe&Dostrovsky,1971);agivenneuronŽresonlywhentheanimalisinacertainsubregionoftheenvironmenttermedtheneuron’splaceŽeld.Onalineartrack,theseŽeldsapproximatelyresembleone-dimensionalgaussiansurfaces.Theneuron’sspikingactivitycorrelatesmostcloselywiththean-imal’spositiononthetrack(Wilson&McNaughton,1993).Thedataseriesweanalyzeconsistsof691spikesfromaplacecellintheCA1regionofthehippocampusrecordedfromaratrunningbackandforthfor20minutesona300cmU-shapedtrack.Thetrackwaslinearizedforthepurposesofthisanalysis(Frank,Brown,&Wilson,2000).Therearetwoapproachestoestimatingtheplace-speciŽcŽringmapsofahippocampalneuron.OneapproachistousemaximumlikelihoodtoŽtaspeciŽcparametricmodelofthespiketimestotheplacecelldataasinBrown,Frank,Tang,Quirk,andWilson(1998)andBarbieri,Quirk,etal.(2001).Ifx(t)istheanimal’spositionattimet,wedeŽnethespatialfunctionfortheone-dimensionalplaceŽeldmodelasthegaussiansurfaces(t)Dexp»a¡b(x(t)¡m)2 2¼,(3.2)wheremisthecenterofplaceŽeld,bisascalefactor,andexpfagisthemaximumheightoftheplaceŽeldatitscenter.Werepresentthespiketimeprobabilitydensityoftheneuronaseitheraninhomogeneousgamma(IG) TheTime-RescalingTheorem337model,deŽnedasf(uk|uk¡1,h)Dys(uk) C(y)µZukuk¡1ys(u)du¶y¡1exp»¡Zukuk¡1ys(u)du¼,(3.3)orasaninhomogeneousinversegaussian(IIG)model,deŽnedasf(uk|uk¡1,h)Ds(uk) µ2phRukuk¡1s(u)dui3¶1 2exp8�&#x 00;:¡1 2±Rukuk¡1s(u)du¡y²2 y2Rukuk¡1s(u)du9�=�;,(3.4)wherey�0isalocationparameterforbothmodelsandhD(m,a,b,y)isthesetofmodelparameterstobeestimatedfromthespiketrain.IfwesetyD1inequation3.3,weobtaintheIPmodelasaspecialcaseoftheIGmodel.Theparametersforallthreemodels—theIP,IG,andtheIIG—canbeestimatedfromthespiketraindatabymaximumlikelihood(Barbieri,Quirk,etal.,2001).Themodelsinequations3.3and3.4areMarkovsothatthecurrentvalueofeitherthespiketimeprobabilitydensityortheconditionalintensity(rate)functiondependsononlythetimeofthepreviousspike.Becauseofequation2.2,specifyingthespiketimeprobabilitydensityisequivalenttospecifyingtheconditionalintensityfunction.IfweletOhdenotethemaximumlikelihoodestimateofh,thenthemaximumlikelihoodestimateoftheconditionalintensityfunctionforeachmodelcanbecomputedfromequation2.2asl(t|Ht,Oh)Df(t|uN(t),Oh) 1¡RtuN(t)f(u|uN(t),Oh)du.(3.5)fort�uN(t).Theestimatedconditionalintensityfromeachmodelmaybeusedinthetime-rescalingtheoremtoassessmodelgoodnessofŽtasdescribedinSection2.1.Thesecondapproachistocomputeahistogram-basedestimateoftheconditionalintensityfunctionbyusingeitherspatialsmoothing(Muller&Kubie,1987;Franketal.,2000)ortemporalsmoothing(Wood,Dudchenko,&Eichenbaum,1999)ofthespiketrain.Tocomputethespatialsmooth-ingestimateoftheconditionalintensityfunction,wefollowedFranketal.(2000)anddividedthe300cmtrackinto4.2cmbins,countedthenumberofspikesperbin,anddividedthecountbytheamountoftimetheanimalspendsinthebin.WesmooththebinnedŽringratewithasix-pointgaus-sianwindowwithastandarddeviationofonebintoreducetheeffectof 338E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.Frankrunningvelocity.Thespatialconditionalintensityestimateisthesmoothedspatialratefunction.Tocomputethetemporalratefunction,wefollowedWoodetal.(1999)anddividedtheexperimentintotimebinsof200msecandcomputedtherateasthenumberofspikesper200msec.Thesetwosmoothingproceduresproducehistogram-basedestimatesofl(t).Botharehistogram-basedestimatesofPoissonratefunctionsbe-causeneithertheestimatedspatialnorthetemporalratefunctionsmakeanyhistory-dependenceassumptionaboutthespiketrain.Asintheanaly-sisoftheSEFspiketrains,weagainusetheKSandQ-QplotstocomparedirectlygoodnessofŽtoftheŽvespiketrainmodelsforthehippocampalplacecells.TheIP,IG,andIIGmodelswereŽttothespiketraindatabymax-imumlikelihood.Thespatialandtemporalratemodelswerecomputedasdescribed.TheKSandQ-Qplotgoodness-of-ŽtcomparisonsareinFigure2.TheIGmodeloverestimatesatlowerquantiles,underestimatesatinter-mediatequantiles,andoverestimatesattheupperquantiles(seeFigure2A). TheTime-RescalingTheorem339TheIPmodelunderestimatesthelowerandintermediatequantilesandoverestimatestheupperquantiles.TheKSplotofthespatialratemodelissimilartothatoftheIPmodelyetclosertothe45-degreeline.Thetemporalratemodeloverestimatesthequantilesofthetrueprobabilitymodelofthedata.ThisanalysissuggeststhattheIG,IP,andspatialratemodelsaremostlikelyoversmoothingthisspiketrain,whereasthetemporalratemodelun-dersmoothsit.OftheŽvemodels,theonethatisclosesttothe45-degreelineandliesalmostentirelywithintheconŽdenceboundsistheIIG.Thismodeldisagreesonlywiththeexperimentaldatanearquantile0.80.BecauseallofthemodelswiththeexceptionoftheIIGhaveappreciablelackofŽtintermsoftheKSplots,theŽndingsintheQ-Qplotanalysesarealmostidentical(seeFigure2B).AsintheKSplot,theQ-QplotfortheIIGmodelisclosetothe45-degreelineandwithinthe95%conŽdenceboundswiththeexceptionofquantilesnear0.80.TheseanalysessuggestthatIIGratemodelgivesthebestagreementwiththespikingactivityofthispyramidalneuron.ThespatialandtemporalratefunctionmodelsandtheIPmodelhavethegreatestlackofŽt.3.3SimulatingaGeneralPointProcessbyTimeRescaling.Asasec-ondapplicationofthetime-rescalingtheorem,wedescribehowthethe-oremmaybeusedtosimulateageneralpointprocess.WestatedintheIntroductionthatthetime-rescalingtheoremprovidesastandardapproachforsimulatinganinhomogeneousPoissonprocessfromasimplePoisson Figure2:Facingpage.(A)KSplotsoftheparametricandhistogram-basedmodelŽtstothehippocampalplacecellspiketrainactivity.TheparametricmodelsaretheinhomogeneousPoisson(IP)(dottedline),theinhomogeneousgamma(IG)(thin,solidline),andtheinhomogeneousinversegaussian(IIG)(thick,solidline).Thehistogram-basedmodelsarethespatialratemodel(dashedline)basedon4.2cmspatialbinwidthandthetemporalratemodel(dashedanddottedline)baseda200msectimebinwidth.The45-degreesolidlinerepresentsexactagree-mentbetweenthemodelandtheexperimentaldata,andthe45-degreethin,solidlinesarethe95%conŽdenceboundsbasedontheKSstatistic,asinFigure1.TheIIGmodelKSplotliesalmostentirelywithintheconŽdencebounds,whereasalltheothermodelsshowsigniŽcantlackofŽt.ThissuggeststhattheIIGmodelgivesthebestdescriptionofthisplacecelldataseries,andthehistogram-basedmodelsagreeleastwiththedata.(B)Q-QplotsoftheIP(dottedline),IG(thin,solidline),andIIG(thick,solidline)spatial(dashedline)temporal(dottedanddashedline)models.The95%localconŽdencebounds(thindashedlines)arecomputedasdescribedinFigure1.TheŽndingsintheQ-QplotanalysesarealmostidenticaltothoseintheKSplots.TheQ-QplotfortheIIGmodelisclosetothe45-degreelineandwithinthe95%conŽdencebounds,withtheexceptionofquantilesnear0.80.ThisanalysisalsosuggeststhatIIGratemodelgivesthebestagreementwiththespikingactivityofthispyramidalneuron.ThespatialandtemporalratefunctionmodelsandtheIPmodelhavethegreatestlackofŽt. 340E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.Frankprocess.Thegeneralformofthetime-rescalingtheoremsuggeststhatanypointprocesswithanintegrableconditionalintensityfunctionmaybesim-ulatedfromaPoissonprocesswithunitratebyrescalingtimewithrespecttotheconditionalintensity(rate)function.Givenaninterval(0,T],thesim-ulationalgorithmproceedsasfollows:1.Setu0D0;SetkD1.2.Drawtkanexponentialrandomvariablewithmean1.3.FindukasthesolutiontotkDRukuk¡1l(u|u0,u1,...,uk¡1)du.4.Ifuk�T,thenstop.5.kDkC16.Goto2.Byusingequation2.3,adiscreteversionofthealgorithmcanbeconstructedasfollows.ChooseJlarge,anddividetheinterval(0,T]intoJbinseachofwidthDDT/J.ForkD1,...,JdrawaBernoullirandomvariableu¤kwithprobabilityl(kD|u¤1,...,u¤k¡1)D,andassignaspiketobinkifu¤kD1,andnospikeifu¤kD0.Whileinmanyinstancestherewillbefaster,morecomputationallyefŽ-cientalgorithmsforsimulatingapointprocess,suchasmodel-basedmeth-odsforspeciŽcrenewalprocesses(Ripley,1987)andthinningalgorithms(Lewis&Shedler,1978;Ogata,1981;Ross,1993),thealgorithmaboveissim-pletoimplementgivenaspeciŽcationoftheconditionalintensityfunction.Foranexampleofwherethisalgorithmiscrucialforpointprocesssimu-lation,weconsidertheIGmodelinequation3.2.ItsconditionalintensityfunctionisinŽniteimmediatelyfollowingaspikeify1.Ifinaddition,yistimevarying(yDy(t)1forallt),thenneitherthinningnorstandardalgorithmsformakingdrawsfromagammaprobabilitydistributionmaybeusedtosimulatedatafromthismodel.Thethinningalgorithmfailsbe-causetheconditionalintensityfunctionisnotbounded,andthestandardalgorithmsforsimulatingagammamodelcannotbeappliedbecauseyistimevarying.Inthiscase,thetime-rescalingsimulationalgorithmmaybeappliedaslongastheconditionalintensityfunctionremainsintegrableasyvariestemporally.4Discussion Measuringhowwellapointprocessmodeldescribesaneuralspiketraindataseriesisimperativepriortousingthemodelformakinginferences.Thetime-rescalingtheoremstatesthatanypointprocesswithanintegrablecon-ditionalintensityfunctionmaybetransformedintoaPoissonprocesswithaunitrate.Berman(1983)andOgata(1988)showedthatthistheoremmaybeusedtodevelopgoodness-of-Žttestsforpointprocessmodelsofseismo-logicdata.Goodness-of-Žtmethodsforneuralspiketrainmodelsbasedon TheTime-RescalingTheorem341time-rescalingtransformationsbutnotthetime-rescalingtheoremhavealsobeenreported(Reich,Victor,&Knight,1998;Barbieri,Frank,Quirk,Wilson,&Brown,2001).Here,wehavedescribedhowthetime-rescalingtheoremmaybeusedtodevelopgoodness-of-Žttestsforpointprocessmodelsofneuralspiketrains.Toillustrateourapproach,weanalyzedtwotypesofcommonlyrecordedspiketraindata.TheSEFdataareasetofmultipleshort(400msec)seriesofspiketimes,eachmeasuredunderidenticalexperimentalconditions.ThesedataaretypicallyanalyzedbyaPSTH.Thehippocampusdataarealong(20minutes)seriesofspiketimerecordingsthataretypicallyanalyzedwitheitherspatialortemporalhistogrammodels.ToeachtypeofdataweŽtbothparametricandhistogram-basedmodels.Histogram-basedmodelsarepopularneuraldataanalysistoolsbecauseoftheeasewithwhichtheycanbecomputedandinterpreted.TheseapparentadvantagesdonotoverridetheneedtoevaluatethegoodnessofŽtofthesemodels.Wepreviouslyusedthetime-rescalingtheoremtoassessgoodnessofŽtforparametricspiketrainmodels(Olsonetal.,2000;Barbieri,Quirk,etal.,2001;Venturaetal.,2001).Ourmainresultinthisarticleisthatthetime-rescalingtheoremcanbeusedtoevaluategoodnessofŽtofparametricandhistogram-basedmodelsandtocomparedirectlytheaccuracyofmodelsfromthetwoclasses.Werecommendthatbeforemakinganinferencebasedoneithertypeofmodel,agoodness-of-Žtanalysisshouldbeperformedtoestablishhowwellthemodeldescribesthespiketraindata.Ifthemodelanddataagreeclosely,thentheinferenceismorecrediblethanwhenthereissigniŽcantlackofŽt.TheKSandQ-QplotsprovideassessmentsofoverallgoodnessofŽt.ForthemodelsŽtbymaximumlikelihood,theseassessmentscanbeappliedalongwithmethodsthatmeasurethemarginalvalueandmarginalcostsofusingmorecomplexmodels,suchasAkaikie’sinformationcriterion(AIC)andtheBayesianinformationcriterion(BIC),inordertogainamorecompleteevaluationofmodelagreementwithexperimentaldata(Barbieri,Quirk,etal.,2001).WeassessedgoodnessofŽtbyusingthetime-rescalingtheoremtocon-structKSandQ-Qplotshaving,respectively,liberalandconservativecon-Ždencebounds.Together,thetwosetsofconŽdenceboundshelpcharacter-izetherangeofagreementbetweenthemodelandthedata.Forexample,amodelwhoseKSplotliesconsistentlyoutsidethe95%KSconŽdencebounds(theIPmodelforthehippocampaldata)agreespoorlywiththedata.Ontheotherhand,amodelthatiswithinallthe95%conŽdenceboundsoftheQ-Qplots(theIMImodelfortheSEFdata)agreescloselywiththedata.AmodelsuchastheIPmodelfortheSEFdata,thatis,withinnearlyalltheKSbounds,maylieoutsidetheQ-Qplotintervals.Inthiscase,ifthelackofŽtwithrespecttotheQ-Qplotintervalsissystematic(i.e.,isoverasetofcontiguousquantiles),thissuggeststhatthemodeldoesnotŽtthedatawell.Asasecondapplicationofthetime-rescalingtheorem,wepresentedanalgorithmforsimulatingspiketrainsfromapointprocessgivenitscondi- 342E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.Franktionalintensity(rate)function.Thisalgorithmgeneralizesthewell-knowntechniqueofsimulatinganinhomogeneousPoissonprocessbyrescalingaPoissonprocesswithaconstantrate.Finally,tomakethereasoningbehindthetime-rescalingtheoremmoreaccessibletotheneuroscienceresearchers,weproveditsgeneralformusingelementaryprobabilityarguments.Whilethiselementaryproofismostcer-tainlyapparenttoexpertsinprobabilityandpointprocesstheory(D.Brill-inger,personalcommunication;Guttorp,1995)itsdetails,toourknowledge,havenotbeenpreviouslypresented.Theoriginalproofsofthistheoremusemeasuretheoryandarebasedonthemartingalerepresentationofpointpro-cesses(Meyer,1969;Papangelou,1972;BrÂemaud,1981;Jacobsen,1982).Theconditionalintensityfunction(seeequation2.1)isdeŽnedintermsofthemartingalerepresentation.Ourproofuseselementaryargumentsbecauseitisbasedonthefactthatthejointprobabilitydensityofasetofpointprocessobservations(spiketrain)hasacanonicalrepresentationintermsofthecon-ditionalintensityfunction.Whenthejointprobabilitydensityisrepresentedinthisway,theJacobianinthechangeofvariablesbetweentheoriginalspiketimesandtherescaledinterspikeintervalssimpliŽestoaproductofthere-ciprocalsoftheconditionalintensityfunctionsevaluatedatthespiketimes.TheproofalsohighlightsthesigniŽcanceoftheconditionalintensityfunctioninspiketrainmodeling;itsspeciŽcationcompletelydeŽnesthestochasticstructureofthepointprocess.Thisisbecauseinasmalltimein-terval,theproductoftheconditionalintensityfunctionandthetimeintervaldeŽnestheprobabilityofaspikeinthatintervalgiventhehistoryofthespiketrainuptothattime(seeequation2.3).Whenthereisnohistorydependence,theconditionalintensityfunctionissimplythePoissonratefunction.AnimportantconsequenceofthissimpliŽcationisthatunlesshistorydepen-denceisspeciŽcallyincluded,thenhistogram-basedmodels,suchasthePSTH,andthespatialandtemporalsmoothersareimplicitPoissonmodels.InboththeSEFandhippocampusexamples,thehistogram-basedmodelsgavepoorŽtstothespiketrain.ThesepoorŽtsarosebecausethesemod-elsusedfewdatapointstoestimatemanyparametersandbecausetheydonotmodelhistorydependenceinthespiketrain.Ourparametricmod-elsusedfewerparametersandrepresentedtemporaldependenceexplicitlyasMarkov.Pointprocessmodelswithhigher-ordertemporaldependencehavebeenstudiedbyOgata(1981,1988),Brillinger(1988),andKassandVentura(2001)andwillbeconsideredfurtherinourfuturework.Paramet-ricconditionalintensityfunctionsmaybeestimatedfromneuralspiketraindatainanyexperimentswherethereareenoughdatatoestimatereliablyahistogram-basedmodel.ThisisbecauseifthereareenoughdatatoestimatemanyparametersusinganinefŽcientprocedure(histogram/methodofmo-ments),thenthereshouldbeenoughdatatoestimateasmallernumberofparametersusinganefŽcientone(maximumlikelihood).UsingtheK-SandQ-Qplotsderivedfromthetime-rescalingtheorem,itispossibletodeviseasystematicapproachtotheuseofhistogram-based TheTime-RescalingTheorem343models.Thatis,itispossibletodeterminewhenahistogram-basedmodelaccuratelydescribesaspiketrainandwhenadifferentmodelclass,temporaldependence,ortheeffectsofcovariates(e.g.,thethetarhythmandtheratrunningvelocityinthecaseoftheplacecells)shouldbeconsideredandthedegreeofimprovementthealternativemodelsprovide.Insummary,wehaveillustratedhowthetime-rescalingtheoremmaybeusedtocomparedirectlygoodnessofŽtofparametricandhistogram-basedpointprocessmodelsofneuralspikingactivityandtosimulatespiketraindata.Theseresultssuggestthatthetime-rescalingtheoremcanbeavaluabletoolforneuralspiketraindataanalysis.Appendix A.1MaximumLikelihoodEstimationoftheIMIModel.ToŽttheIMImodel,wechooseJlarge,anddividetheinterval(0,T]intoJbinsofwidthDDT/J.WechooseJsothatthereisatmostonespikeinanybin.Inthisway,weconvertthespiketimes0u1u2,...,un¡1un·Tintoabinarysequenceu¤j,whereu¤jD1ifthereisaspikeinbinjand0otherwiseforjD1,...,J.BydeŽnitionoftheconditionalintensityfunctionfortheIMImodelinequation3.1,itfollowsthateachu¤jisaBernoullirandomvariablewiththeprobabilityofaspikeatjDdeŽnedasl1(jD|h)l2(jD¡u¤N(jD)|h)D.WenotethatthisdiscretizationisidenticaltotheoneusedtoconstructthediscretizedversionofthesimulationalgorithminSection3.3.Thelogprobabilityofaspikeisthuslog(l1(jD|h))Clog(l2(jD¡u¤N(jD)|h)D),(A.1)andthecubicsplinemodelsforlog(l1(jD|h))andlog(l2(jD¡u¤N(jD)|h))are,respectively,log(l1(jD|h))D3XlD1hl(jD¡j1)lCCh4(jD¡j2)3CCh5(jD¡j3)3C(A.2)log(l2(jD¡u¤N(jD)|h))D3XlD1hlC5(jD¡u¤N(jD)¡c1)lCCh9(jD¡u¤N(jD)¡c2)3C,(A.3)wherehD(h1,...,h9)andtheknotsaredeŽnedasjlDlT/4forlD1,2,3andclistheobserved100l/3quantileoftheinterspikeintervaldistributionofthetrialspiketrainsforlD1,2.WehaveforSection3.1thathIPD(h1,...,h5) 344E.N.Brown,R.Barbieri,V.Ventura,R.E.Kass,andL.M.FrankandhIMID(h1,...,h9)arethecoefŽcientsofthesplinebasiselementsfortheIPandIMImodels,respectively.TheparametersareestimatedbymaximumlikelihoodwithDD1msecusingthegamfunctioninS-PLUS(MathSoft,Seattle)asdescribedinChap-ter11ofVenablesandRipley(1999).Theinputstogamaretheu¤js,thetimeargumentsjDandjD¡u¤N(jD),thesymbolicspeciŽcationofthesplinemod-els,andtheknots.Furtherdiscussiononspline-basedregressionmethodsforanalyzingneuraldatamaybefoundinOlsonetal.(2000)andVenturaetal.(2001).Acknowledgments WearegratefultoCarlOlsonforallowingustousethesupplementaryeyeŽelddataandMatthewWilsonforallowingustousethehippocampalplacecelldata.WethankDavidBrillingerandVictorSoloforhelpfuldiscussionsandthetwoanonymousrefereeswhosesuggestionshelpedimprovetheexposition.ThisresearchwassupportedinpartbyNIHgrantsCA54652,MH59733,andMH61637andNSFgrantsDMS9803433andIBN0081548.References 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