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Bayesian nonparametric analysis of neuronal intensity rates

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We propose a flexible hierarchical Bayesian nonparametric modeling approach to compare the spiking patterns of neurons recorded under multiple experimental conditions. In particular, we showcase the application of our statistical methodology using
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   JournalofNeuroscienceMethods 203 (2012) 241–253 ContentslistsavailableatSciVerseScienceDirect  Journal   of    Neuroscience   Methods  journal   homepage:www.elsevier.com/locate/jneumeth Bayesian   nonparametric   analysis   of    neuronal   intensity   rates Athanasios   Kottas a , Sam   Behseta b , ∗ , David   E.   Moorman c , Valerie   Poynor a ,   Carl   R.Olson d a DepartmentofAppliedMathematicsandStatistics,Universityof    California,SantaCruz,   CA   95064,USA b DepartmentofMathematics,CaliforniaStateUniversityFullerton,Fullerton,CA   92834,USA c DepartmentofNeurosciences,MedicalUniversityof    SouthCarolina,Charleston,SC29425,USA d CenterfortheNeuralBasisofCognition,CarnegieMellonUniversity,Pittsburgh,PA15213,USA a   r   t   i   c   le   i   nf   o  Articlehistory: Received13December2010Receivedinrevisedform5August2011Accepted20September2011 Keywords: BayesiannonparametricsDependentDirichletprocesspriorDirichletprocessmixturemodelsMultipleexperimentalconditionsPSTHSensorymotorneuronsSupplementaryeyefield a   b   s   t   ra   ct Weproposea   flexible   hierarchicalBayesian   nonparametric   modeling   approachtocompare   the   spikingpatternsof    neurons   recorded   under   multiple   experimental   conditions.In   particular,weshowcase   theapplication   of    ourstatistical   methodology   usingneuronsrecorded   fromthe   supplementary   eyefieldregion   of    the   brains   of    twomacaque   monkeys   trainedto   make   delayedeyemovements   to   three   differenttypes   of    targets.   Theproposed   Bayesianmethodology   canbe   usedto   perform   either   a   global   analysis,allowing   for   the   construction   of    posterior   comparative   intervals   overthe   entireexperimental   time   win-dow,   or   apointwise   analysis   for   comparing   the   spiking   patternslocally,   ina   predetermined   portion   of the   experimentaltime   window.   Bydeveloping   ournonparametric   Bayesianmodel   we   areabletoana-lyzeneuronaldatafrom   three   or   more   conditionswhile   avoiding   the   computationalexpenses   typicallyassociatedwith   more   traditional   analysis   of    physiological   data. © 2011 Elsevier B.V. All rights reserved. 1.Introduction Thefundamentalstatisticalquestionin   thiswork,andin   awiderangeofsimilarstudiesinvolvingdistinctexperimentalconditions,revolvesaroundthecalibrationofthesimilaritiesof    multiplefiringpatternsalongwiththeidentificationof    sharpdifferencesbetweenthem.Thisideais   motivatedfromtheobservationthatthepresenceortheabsenceof    firingactivityis   consideredasthemainmarkerof thedegreeof    involvementof    theneuronin   thestudiedbehavior.Ourinterestisin   developingamethodwherebytheoverallpatternofactivityacrossmultipleconditionscanbedescribedwhichwillallowustobetterdefineexactlywhatroletheseneuronsplayinvariablesensorimotorcontexts.Consequently,itwouldbeusefultodeviseasuitablestatisticalmethodologyto   addresssuchcompara-tiveinquiries,mainlyontwofronts:first,a   globalanalysisovertheentireexperimentaltimewindow,enablingneurophysiologiststodecidewhethertheneuronshouldbe   consideredforfurtherstudy;second,apointwiseanalysis,topinpointdifferentialpatternsatspecifictimepointsin   theexperimentaltimeinterval.Themethodproposedinthisworkis   wellsuitedto   addressthesescientificgoals. ∗ Correspondingauthor.Tel.:+16572788560;fax:+16572781392. E-mailaddresses: thanos@ams.ucsc.edu(A.Kottas),sbehseta@fullerton.edu(S. Behseta),moorman@musc.edu(D.E.Moorman),vpoynor@ams.ucsc.edu(V.Poynor), colson@cnbc.cmu.edu(C.R.Olson). Theneedfora   comparativestudyofspikingpatternsinmultipleconditionsmay   bejustifiedbyinvestigatingtheneuronalactivitiespresentedin   Fig.1,   wherea   peri-stimulustime   histogram(PSTH)fora   singleneuronrecordedfromanawake,behavingmonkeyisplottedunderthreeexperimentalconditionsdescribedbelow.A4000ms   windowis   considered.Thetimeisalignedona   sac-cadiceyemovement.Condition1(“Space”conditiononthetoppanel)reflectsatrimodalfiringpattern:apeakin   firingactivityin   about1000mspriortothesaccade,followedbya   significantlyless-pronouncedpeakatthesaccadetime,yieldingtoyetanotherstrongpeakatabout1500ms   afterthesaccadetime.Theresponsein   condition2(“Dot”conditiononthemiddlepanel)is   inherentlydifferent:aseriesofcomparablyweakerburstsof    activity,morelikea   randomnoise,nonethelesswitha   seeminglynoticeabledeclinein   firingactivityatthesaccadetime.Finally,in   condition3(“Ring”conditiononthelowerpanel)a   multimodalpatternissuggestedwiththemostnoticeablepeakatthesaccadetime.Wenowcanrestatethemainobjectivesofthisworkinthecontextof    thedatapresentedin   thisfigure:1–Arethedifferencesandsimilaritiesinthespikingpatternsof    thethreeconditionsinFig.1statisti-callysignificantorshouldtheybeinterpretedasperturbationsdueto   chanceandhencebeignored?2–Whenstudiedin   predeter-minedslicesoftheentiretimesegment,couldsuchdifferencesshedlightonthesensorimotorproperties?Theseare   amongtheques-tionsthatmotivatethestatisticalstrategiesweadoptinordertocomparethefiringpatternsofeachneuronin   multipleconditions.Here,weaddresstheabovetwoquestionsbydevelopingaBayesian 0165-0270/$–seefrontmatter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jneumeth.2011.09.017  242  A.Kottasetal./JournalofNeuroscienceMethods  203 (2012) 241–253 -2000-10000   1000   20000204060 sp259a.1c1, Ring Condition -2000-10000   1000   20000204060 sp259a.1c1, Dot Condition -2000-10000   1000   20000204060 sp259a.1c1, Space Condition     F    i   r    i   n   g     R   a   t   e    (   s   p    /   s    )  Time (ms) Fig.1. Neuronsp259a.1.PSTHis   shownforthreeexperimentalconditions(condi-tions   1,2,and3).A4000ms   windowis   considered.Thetimeisalignedonthespatialcue   onset.Conditions1(Space,toppanel),2   (Dot,middlepanel),and3(Ring,lowerpanel)demonstratedifferentresponses. nonparametricmodelthat   allowsusto   compareneuronalintensityratesunderanumberofdistinctexperimentalconditionswithinacoherentprobabilisticframeworkforinference.Theoutlineofthepaperisasfollows.Section2developsthe methodologywithtechnicaldetailsonimplementationincludedinanAppendixA.InSection3,weprovidemoredetailsfortheexper- imentusedtoillustratetheproposedmethodology,andin   Section4,wepresenttheresultsfromtheanalysisof    thecorrespondingdata.Finally,Section5concludeswithanoverviewanddiscussion. 2.Models InSection2.1,   wediscussthestochasticmodelunderlyingourapproachandprovidea   briefreviewoftheclassofmodelsfromthefieldofBayesiannonparametricsthatprovidesthefoundationfortheproposedmethodology.Section2.2developsthemodel-ing   approachforcomparisonof    neuronalspikingpatternsundermultipleexperimentalconditions.  2.1.Motivationandbackground 2.1.1.Poissonprocessmodelingforneuronalfiringintensities Stochasticmodelingandstatisticalestimationtechniquesfortheanalysisofdatafromsingle-recordingneurophysiologicalexperimentshavereceivedconsiderableattentionin   theneuro-science,aswellasthestatisticsliterature(see,e.g.,Brillinger,1992;Venturaetal.,   2002;Kassetal.,2005).Predominantly,thefocusof    statisticalmodelingapproachesis   onthetemporalevolutionoftheneuronalfiringactivity.Historically,thestochasticmodel-ing   of    spiketrainsmay   betracedbacktotheoriginalformsof    theso-calledIntegrateandFire(IF)models(GersteinandMandelbrot,1964;Stein,1965).Inthesestochasticmodels,theoutputis   takenasaone-dimensionalvoltagewhiletheinputsconsistof    currentandmembraneconductance.Theerroris   typicallycapturedbya   Brow-nianmotion,representingthestochasticfeatureof    themodel.AsnotedinPaninskietal.(2010),   fromthestatisticalpointof    view,theIFmodelmay   alsobestudiedviahiddenMarkov(orstatespace)modelsin   whichtheunobserved(hidden)voltageis   mod-eledthroughaMarkovianprocessevaluatedattheobservedspikingtimes(Brownetal.,1998;Volgesteinetal.,2009).Analternativeapproach,leadingeventuallytothemethodologyproposedinthispaper,is   to   viewthespiketrainasarealiza-tionof    apointprocess,a   randomsequenceof    timesassociatedwithspikeoccurrences,andsubsequentlymodelthespikecountswitha   time-varyingintensityfunctionformulatedthrougha   Non-HomogeneousPoissonProcess(NHPP)asdescribedbelow.Reviewsof    theanalysisof    neuronaldatausingpointprocesses,fromeitherasingleneuronorfrommultipleneurons,canbefoundin,e.g.,Brillinger(1992),Brownetal.   (2004),   andKassetal.(2005).Let N  ( t  a ,t  b )  denotethenumberof    spikeoccurrencesin   thetimeinterval( t  a ,   t  b ).   Bydefinition,a   NHPPpointprocessmodeliscon-structedovertwo   conditions:(a)Foranyinterval( t  a ,   t  b ), N  ( t  a ,t  b )  followsa   Poissondistribu-tionwithmean   t  b t  a  ( u ) du .Here,  ( · )is   theNHPPintensityfunction,a   non-negativeandlocallyintegrablefunction(i.e.,   D  ( u ) du < ∞ foranyboundedsubset D of    thepositiverealline).(b)Foranynon-overlappingintervals,( t  a , t  b ),and( t  c  , t  d ),theran-domvariables N  ( t  a ,t  b ) ,and N  ( t  c  ,t  d )  areindependent.Subsequently,ifa   genericsetof    n spiketimes, { s 1 , ... ,   s n } ,observedintimewindow(  A , B ),is   assumedtoarisefromaNHPP,thecorrespondinglikelihoodfortheintensityfunctionisgivenby e −   B A  ( u ) du  ni = 1  ( s i ).Statistically,theproblemof    modelingthespiketrainswillthenrevolvearoundestimatingtheNHPPintensityfunction  ( · )fromwhichinferenceonseveralfeaturesoftheneuronalspikingpatterncanbeobtained.Inthispaper,weadopta   Bayesiannonparamet-ricpointof    viewforsuchanestimationproblemdevelopingapracticallyimportantmethodologicalextensionof    theapproachproposedin   KottasandBehseta(2010).   Thisapproachandtherel-evantclassofnonparametricBayesianpriormodelsarereviewedin   thefollowingsection.  2.1.2.BackgroundonBayesiannonparametricmixturemodels AnonparametricmodelingapproachfortheNHPPintensitytreatstheentirefunction  ( · )astheunknownparameter,whichundertheBayesianparadigm,necessitatesplacinga   prioroveraspaceof    functions(i.e.,overaninfinitedimensionalparameter).ThefieldofBayesiannonparametricsdealswiththeproblemof 
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