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AlthoughIstillhavefartogo,Iwouldnotevenbe1/100thmycurrentdistancewithoutmyadviserDr.Principe.Hispassionforknowledgeinspiresmetoconstantlyimproveandlearn.Ihavegrownasastudent,researcher,andpersonbecauseofhim.ThesesmallwordscannotexpressthelargedebtofgratitudeIowehim.Thankyoutomycommitteemembersfortheirguidanceandpatience:Dr.Harris,Dr.Rangarajan,Dr.Sanchez.IespeciallythankDr.Slattonforhistimeunderthecircumstances.Itrulywishyouandyourfamilywell.IamalsogratefulfortheenvironmentDr.PrincipehasfosteredinCNEL.TheCNELstudentspastandpresenthaveprovidedgreatopportunitiesfordiscussions,laughterandgrowth.AlthoughtherearemanystudentsinCNELthathaveimpactedme,JeremyAndersonhasbeentheresinceundergradtakingtheridewithme(upsanddowns).IalsoappreciatethefourmusketeersalongtheBMIridewithme:Dr.AntonioPaiva,Dr.AysegulGunduz,Dr.YiwenWangandDr.JackDiGiovanna.Thankyouallforthehelpfuldiscussionsandcollaborationthroughtheyears.ThankstothenewbatchofCNELstudentsfortheirdiscussionsandlaughter,SohanSeth,AlexSingh,ErionHasanbelliu,LuisGiraldo,MemmingPark.ThankyoutoJulieforkeepingCNELrunningsmoothly.ThankyoualsotoMarcus(evenifheisarepublican).ThankyoutoShannonforyearsofhelpandadvicewiththegraduatedepartment.AspecialthankstoalongtimeenemyGiovannielevencents!Montroneforyearsofconstructivepessimismandencouragingwords.Hehasbeentheresincethebeginningandhopefullytilltheend.TheyearsduringmyPhDwerealsobrightenedbysomespecialladies,Sarah,Melissa,andGrisel.Whetheropeningmyeyestovegetariandishes,tattoosorLasVegasCasinos,Iappreciatethetimewespenttogether.Youhelpedtolightenmystressandexposemetodifferentworlds.Thankyou. 4 PAGE 5 5 PAGE 6 page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1OverviewofBMIs ................................ 15 1.2ExperimentalData ............................... 17 1.2.1MonkeyFoodGraspingTask ..................... 17 1.2.2MonkeyCursorControl ......................... 18 1.2.3RatLeverExperiments ......................... 19 1.3ReviewofModelingParadigmsforBMIs ................... 20 1.4DissertationObjectives ............................ 23 2GENERATIVEMODELS ............................... 26 2.1Motivation .................................... 26 2.2RelatedWork .................................. 28 2.3BackgroundonGraphicalModels ....................... 28 2.3.1HiddenMarkovModels ......................... 30 2.3.2MovingBeyondSimpleHMMs ..................... 32 3BRAINMACHINEINTERFACEMODELING:THEORETICAL .......... 35 3.1Motivation .................................... 35 3.2IndependentlyCoupledHMMs ........................ 37 3.3BoostedMixturesofHMMs .......................... 41 3.3.1Boosting ................................. 41 3.3.2ModelingFramework .......................... 43 3.4LinkedMixturesofHMMs ........................... 46 3.4.1ModelingFramework .......................... 46 3.4.2TrainingwithExpectationMaximization ................ 48 3.4.3UpdatingVariationalParameter .................... 51 3.5DependentlyCoupledHMMs ......................... 52 3.5.1ModelingFramework .......................... 53 3.5.2Training ................................. 55 6 PAGE 7 ............. 57 4.1MonkeyFoodGraspingTask ......................... 59 4.1.1BoostedandLinkedMixtureofHMMs ................ 59 4.1.2DependentlyCoupledHMMs ..................... 64 4.2RatSingleLeverTask ............................. 68 4.3MonkeyCursorControlTask ......................... 72 4.3.1PopulationVectors ........................... 72 4.3.1.1A-prioriclasslabelingbasedonpopulationvectors .... 74 4.3.1.2Simplenaveclassiers ................... 75 4.3.2ResultsfortheCursorControlMonkeyExperiment ......... 76 5GENERATIVECLUSTERING ............................ 83 5.1GenerativeClustering ............................. 83 5.2Simulations ................................... 86 5.2.1SimulatedDataGeneration ...................... 86 5.2.2IndependentNeuralSimulationResults ............... 90 5.2.3DependentNeuralSimulationResults ................ 111 5.3ExperimentalAnimalData ........................... 115 5.3.1RatExperiments ............................ 115 5.3.2MonkeyExperiments .......................... 120 5.3.3Discussion ................................ 129 6CONCLUSIONANDFUTUREWORK ....................... 131 6.1FutureWork ................................... 133 6.1.1TowardsClusteringModelStructures ................. 135 6.1.2PreliminaryResults ........................... 137 6.2Contributions .................................. 143 AWIENERFILTER ................................... 145 BPARTIALDERIVATIVES ............................... 148 CSELFORGANIZINGMAP .............................. 150 REFERENCES ....................................... 153 BIOGRAPHICALSKETCH ................................ 160 7 PAGE 8 Table page 3-1Classicationperformanceofexamplesingle-channelHMMchains ....... 36 4-1Classicationresults(BM-HMMselectedchannels) ................ 59 4-2Classicationresults(LM-HMMselectedchannels) ................ 59 4-3Classicationresults(randomBM-HMMselectedchannels) ........... 62 4-4Classicationresults(randomLM-HMMselectedchannels) ........... 62 4-5CorrelationcoefcientusingDC-HMMon3Dmonkeydata ............ 65 4-6NMSEon3Dmonkeydata ............................. 65 4-7Classicationresults(BM-HMMselectedchannels) ................ 69 4-8Classicationresults(LM-HMMselectedchannels) ................ 69 4-9Classicationresults(randomBM-HMMselectedchannels) ........... 71 4-10Classicationresults(randomLM-HMMselectedchannels) ........... 71 4-11CorrelationcoefcientusingdifferentBMImodels ................. 78 4-12CorrelationcoefcientusingDC-HMMon2Dmonkeydata ............ 80 5-1CorrelationcoefcientusingLM-HMMon2Dmonkeydata ............ 123 5-2CorrelationcoefcientusingDC-HMMon3Dmonkeydata ............ 124 5-3CorrelationcoefcientusingDC-HMMoncursorcontroldata .......... 126 6-1CorrelationcoefcientusingDC-HMMon3Dmonkeydata ............ 139 6-2CorrelationcoefcientusingDC-HMMon2Dmonkeydata ............ 142 8 PAGE 9 Figure page 1-1BMIoverview ..................................... 16 1-2Discritizedexampleofcontinuoustrajectory .................... 19 2-1VariousHMMstructures ............................... 32 3-1Probabilisticratiosof14neurons .......................... 37 3-2Zoomedinversionoftheprobabilisticratios .................... 37 3-3IC-HMMgraphicalmodel .............................. 39 3-4LM-HMMgraphicalmodel .............................. 47 3-5DC-HMMtrellisstructure .............................. 55 4-1Multiplemodelmethodology ............................. 58 4-2Correlationcoefcientsbetweenchannels(monkeymoving) ........... 60 4-3CorrelationCoefcientsbetweenchannels(monkeyatrest) ........... 61 4-4Correlationcoefcientbetweenchannels(randomlyselectedformonkey) ... 62 4-5Monkeyexpertaddingexperiment ......................... 63 4-6Parallelperi-eventhistogramformonkeyneuraldata ............... 64 4-7Supervisedmonkeyfoodgraspingtaskreconstruction(position) ........ 66 4-83Dmonkeyfoodgraspingtruetrajectory ...................... 67 4-9Hiddenstatespacetransitionsbetweenneuralchannels(formoveandrest) .. 67 4-10Couplingcoefcientbetweenneuralchannels(3Dmonkeyexperiment) .... 68 4-11Parallelperi-eventhistogramforratneuraldata .................. 70 4-12Ratexpertaddingexperiment ............................ 71 4-13Neuraltuningdepthoffoursimulatedneurons ................... 73 4-14Histogramof30angularvelocitybins ........................ 74 4-152Dangularvelocities ................................. 75 4-16A.ParalleltuningcurvesB.Winningneuronsforparticularangles ........ 76 4-17Histogramof10angularbins ............................ 77 9 PAGE 10 ....... 77 4-19Truetrajectoryandreconstructedtrajectory(DC-HMM) ............. 81 4-20Hiddenstatetransitionsperclass(cursorcontrolmonkeyexperiment) ..... 81 4-21Couplingcoefcientbetweenneuronsperclass(cursorcontrolmonkeyexperiment) 82 5-1Bipartitegraphofexemplars(x)andmodels .................... 85 5-2Neuraltuningdepthoffoursimulatedneurons ................... 89 5-3LM-HMMclusteriterations(twoclasses,k=2) ................... 90 5-4Tuningpreferencefortwoclasses(initialized) ................... 91 5-5Tunedclassesafterclustering(twoclasses) .................... 92 5-6LM-HMMclusteriterations(fourclasses,k=4) ................... 93 5-7Tuningpreferenceforfourclasses(initialized) ................... 94 5-8Tunedclassesafterclustering(fourclasses) .................... 95 5-9LM-HMMclusteriterations(twoclasses,k=4) ................... 96 5-10Classicationdegradationwithincreasedrandomrings ............. 96 5-11Neuraltuningdepthwithhighrandomringrate ................. 97 5-12Surrogatedatasetdestroyingspatialinformation ................. 98 5-13Tunedpreferenceafterclustering(spatialsurrogate) ............... 98 5-14Surrogatedatasetdestroyingtemporalinformation ................ 99 5-15Tunedpreferenceafterclustering(temporalsurrogate) .............. 99 5-16DC-HMMclusteringresults(class=2,K=2) ..................... 100 5-17DC-HMMclusteringhiddenstatetransitions(class=2,K=2) ........... 101 5-18DC-HMMclusteringcouplingcoefcient(class=2,K=2) ............. 101 5-19DC-HMMclusteringlog-likelihoodreductionduringeachround(class=2,K=2) 102 5-20DC-HMMclusteringsimulatedneurons(class=4,K=4) .............. 103 5-21DC-HMMclusteringhiddenstatespacetransitionsbetweenneurons ...... 104 5-22DC-HMMclusteringcouplingcoefcientbetweenneurons(perClass) ..... 105 5-23SOMclusteringonindependentneuraldata(2Classes) ............. 107 10 PAGE 11 ....... 108 5-25SOMclusteringonindependentneuraldataspatialsurrogate(2classes) .... 108 5-26NeuralselectionbySOMonspatialsurrogatedata(2classes) .......... 109 5-27SOMclusteringonindependentneuraldatatemporalsurrogate(2classes) ... 110 5-28Outputfromfoursimulateddependentneuronswith100noisechannels(Class=2) 111 5-29Neuraltuningfordependentneuronsimulation .................. 112 5-30LM-HMMclusteringsimulateddependentneurons(class=2,K=2) ....... 113 5-31DC-HMMclusteringsimulateddependentneurons(class=2,K=2) ....... 113 5-32SOMclusteringondependentneuraldata(2classes) ............... 114 5-33Ratclusteringexperiment,onelever,twoclasses ................. 116 5-34Ratclusteringexperimentzoomed,onelever,twoclasses ............ 117 5-35Ratclusteringexperiment,twolever,twoclasses ................. 117 5-36Ratclusteringexperiment,twolever,threeclasses ................ 118 5-37Ratclusteringexperiment,twolever,fourclasses ................. 119 5-38LM-HMMclusteriterations(Ivy2Ddataset,k=4) ................. 121 5-39ReconstructionusingunsupervisedLM-HMMclusters(blue)vs.realtrajectory(red) .......................................... 122 5-40DC-HMMclusteringonmonkeyfoodgraspingtask(2classes) .......... 125 5-41CouplingcoefcientfromDC-HMMclusteringonmonkeyfoodgraspingtask(2classes) ....................................... 125 5-42CouplingcoefcientfromDC-HMMclusteringonmonkeycursorcontroltask(4classes) ....................................... 127 5-43Averageringrateperclass(4classes,6neurons) ................. 127 5-44Averagevelocityperclass(4classes,6neurons) .................. 128 6-1Bipartitegraphofexemplars(x)andmodels .................... 134 6-2HiddenstatetransitionsDC-HMM(simulationdata2classes) .......... 138 6-3HiddenstatetransitionsDC-HMM(simulationdata2classes) .......... 139 6-4HistogramofstatemodelsfortheDC-HMM(foodgraspingtask) ........ 140 11 PAGE 12 ............... 141 6-6AlphascomputedperstateperchannelDC-HMM(foodgraspingtask) .... 141 6-7Alphasacrossstate-spaceoftheDC-HMM(cursorcontroltask) ........ 142 A-1Topologyofthelinearlterforthreeoutputvariables ............... 147 C-1Self-Organizing-Maparchitecturewith2Doutput ................. 151 12 PAGE 13 13 PAGE 14 14 PAGE 15 1 2 ].Duringevensimplemovement,thehumancentralnervoussystemmusttranslategeneratedringsofmillionsofneuronswhilealsocommunicatingtotheperipheralnervoussystem[ 3 ].Thiscomplexbiologicalsystemalsocontinuouslymanageschemicalandelectricalinformationfromthedifferentcortices,andpaleocortexstructuretocontrolunconsciousandconsciousactionstakenwiththebody[ 3 ].Thebrainmusthandlealloftheseactionswhilecontinuouslyprocessingvisual,tactileandotherinternalsensoryfeedback[ 3 4 ].Unfortunately,thousandsofpeoplehavesufferedtragicaccidentsordebilitatingdiseasesthathaveeitherpartiallyorfullyremovedtheirabilitytoeffectivelyinteractintheexternalworld[ 5 ].Somedevicesexisttoaidthesetypesofpatients,butoftenlacktherequirementstoliveanormallife.EssentiallytheideabehindmotorBrainMachineInterfaces(BMIs)istobridgethegapbetweenthebrainandtheexternalworldtoprovidethesepatientswitheffectiveworld-interaction. 15 PAGE 16 BMIoverview electrocorticographic(EEG)systemisonemethodthatroughlyrecordsfrommultipleneuronsthroughthescalp(non-invasive).WhileElectroencephalographic(Ecog)andMicroelectrodeArrays,althoughinvasive,provideanerresolutionofindividualneurons(orsingleunits).Inturn,thenerresolutionhasallowedforsignicantadvancesintheeldofBMIstotakeplacerecently[ 6 7 ].Consequently,MicroelectrodeArraydataistheonlytypeofdatausedthroughoutthisdissertation.InordertoacquireMicroelectrodeArrayData,multipleelectrodegridarraysarechronicallyimplantedinoneormorecortices[ 8 ]andrecordanalogvoltagesfrommultipleneuronsneareachindividualelectrode.Theneuralsignals(i.e.analogvoltages)thenprogressthroughthreeprocessingsteps(inatypicalBMI).First,theampliedanalogvoltagesrecordedfromoneormoreneuronisdigitallyconvertedandpassedtoaspikedetecting/sortingalgorithm.Aspikedetectingandsortingalgorithmessentiallyidentiesifaparticularneuronexhibitedaring/voltagepatternonacorrespondingelectrode(therebyclassifyingitasaspike).DuringthesecondBMI 16 PAGE 17 8 ].Finally,subsequenttrajectory/levercalculationsaresenttoarobotarmordisplaydevice.Alloftheseprocessingstepsoccurasananimalengagesinabehavioralexperiment(leverpress,foodgrasping,ngertracing,orjoystickcontrol).Thisdissertationfocusesonthesignalprocessingoftheneuralspikes(thesecondprocessingstepdescribedabove).EarlyapproachestomodelingBMIsusedsimplepopulationvectors,Wienerltersandarticialneuralnetworks[ 8 9 ]betweentheneuraldataandarmtrajectorytolearnafunctionalrelationshipduringtraining.Duringtestingonlytheneuraldataisusedtoreconstructthepredictedtrajectorytheanimalwanted.Mostofthesemodelingapproachesusebinnedspikes,commonlyreferredtoasratecoding,tocreatetheoutput.Explicitrelationshipsbetweentheneuronsareusuallyassumedtobeindependentformodelingpurposesandtheencodingmethodologyisoftennotevenconsidered[ 8 ].Pleasesee[ 10 11 ]togainamoredetailedunderstandingofwhatencompassesaBMI. 4 12 ].Eachringcountrepresentsthenumberofneuralringsina100msspanoftime,whichisconsistentwithmethodsusedwithintheneurologicalcommunity[ 6 13 14 ].Thecorrespondingkinematicdata(i.e.handlocation)isdownsampledfromtheoriginal10Hztomatchthe100msneuralspikebins.Thisparticularmonkeydatasetcontains 17 PAGE 18 6 8 13 ].Withthisparticularexperimentaldata,themonkey'sarmismotionlessinspaceforabriefamountoftimewhilereachingforfoodorplacingfoodinitsmouth.Duringthistimeofactiveholdingitisunknownifthebrainisencodinginformationtocontractthemusclesintheholdingposition.Ourworkaswellasotherresearchhasshownthisactiveholdingencodingislikely[ 15 ].Duetothisbelief,theactiveholdingisincludedaspartofthemovementclassfortheclassiersinthisdissertation[ 16 ].AnexampleofthistypeofdataisshowninFigure 1-2 ,alongwiththesuperimposedgray-scalecolorsrepresentingthemonkey'sarmmovementonthethreeCartesiancoordinateaxes'(x,y,andz).Notethatthemovementandrestclassesarelabeledby'hand'fromthe10Hz(100ms)trajectorydata. 4 8 ].Themonkeyusedahand-heldmanipulandum(joystick)tomovethecursor(smallercircle)sothatitintersectsthetarget.Uponintersectingthetargetwiththecursor,themonkeyreceivedajuicereward.Whilethemonkeyperformedthemotortask,thehandpositionandvelocityforeachcoordinatedirection(XandY)wererecordedinrealtimealongwiththecorrespondingneuralactivity.Microwireelectrodearrayschronicallyimplantedinthedorsalpremotorcortex(PMd),supplementarymotorarea(SMA),primarymotorcortex(M1,bothhemispheres)andprimarysomatosensorycortex(S1),theringtimesofupto185cellsweresimultaneouslycollected. 18 PAGE 19 Discritizedexampleofcontinuoustrajectory Forthemonkeyneuraldata,eachringcountagainrepresentsthenumberofneuralringsina100msspanoftime.Thisparticularmonkeydatasetcontains185neuralchannelsrecordedfor43.33minutes.Thesubsequenttimerecordingcorrespondstoadatasetof26000x185timebins. 17 ].Thetaskrequiresarattopressasingleleverforaminimumof0.5stoachieveawaterrewardonceaLEDvisualstimulusisobserved.Essentiallythisgo-no-goexperimentincludesneuraldataalongwithleverpresses.Thisparticulardatasetcontains16neuronsyielding13000x16timebins.Witheachtimebeingthespikesortedcountper100msofaparticularneuron(sixteeninthisdataset). 19 PAGE 20 18 ].Thetaskrequiresarattopressoneoftwoleversforaminimumof0.5stoachieveawaterreward(alsoafteraLEDvisualstimulus).Essentiallythisgo-no-goexperimentalsoprovidesneuraldataalongwithleverpresses.Thisparticulardatasetcontains42neuronsyielding19000x42timebins(100mseach).Forbothratexperiments,theratisfreetomovearoundthecagewhilethereisnomeasurementofthemovement. 6 8 9 ].Essentially,theactionpotentials(calledspikesforshort),collectedwithmicro-electrodearrays,aresortedbyneuronandcountedintimewindows(calledbins)andfedintoalinearmodel(Wienerlter)withapredenedtapdelaylinedependingontheexperimentandexperimenter[ 19 20 ].Duringtraining,thelinearmodelhasaccesstothedesiredkinematicdata,asadesiredresponse,alongwiththeneuraldatarecording.Onceafunctionalmappingislearnedbetweentheneuraldataandkinematictrainingset,fortestingthelinearmodelisonlyprovidedneuraldatainwhichthekinematicdataisreconstructed[ 19 ].TheWienerlterisexploitedintwowaysforthisdissertation.First,itservesabaselinelinearclassier(withthreshold)tocompareresultswiththemodelsdiscussed.Second,Wienerltersareusedtoreconstructthetrajectoriesfromneuraldataswitched 20 PAGE 21 A .Alongwiththesupervisedlinearmodeling,researchershavealsoengagedinnon-linearsupervisedlearningalgorithmsforBMIs[ 8 20 ].Verysimilartotheparadigmofthelinearmodeling,theneuraldataandkinematicdataarefedtoanon-linearmodelthatsubsequentlyndstherelationshipbetweenthedesiredkinematicdataandtheneuraldata.Duringtesting,onlyneuraldataisprovidedtothemodelsinorderproducekinematicreconstructions.Withrespecttostate-spacemodels,Kalmanltershavebeenusedtoreconstructthetrajectoryofabehavingmonkey'shand[ 21 ].Specically,agenerativemodelisusedforencodingthekinematicstateofthehand.Fordecoding,thealgorithmpredictsthestateestimatesofthehandandthenupdatesthisestimatewithnewneuraldatatoproduceaposterioristateestimate.OurgroupatUFandelsewherefoundthatthereconstructionwasslightlysmootherthanwhatinput-outputmodelswereabletoproduce[ 21 ].Unfortunately,thereareproblemswithalloftheaforementionedmodels.First,trainingwithdesireddataisproblematicsinceparalyzedpatientsarenotableprovidekinematicdata.Second,thesemodelsdonotcaptureneuronsthatreinfrequentlyduringmovement.Theseneuronsareknowntoexistinthebrainandcanprovideusefulinformation.Buttheinformationislostwithfeed-forwardlterssinceaneuronthatresverylittlewillreceivelessweighting.Third,therearemillionsofotherneuronsnotbeingrecordedormodeled.Incorporatingthismissinginformationintothemodelwouldbebenecial.Lastly,allofthesemodelsmustgeneralizeoverawiderangeofmovements.Normally,generalizationisgoodforamodelofthesametask,butgeneralizationacrosstasksisproblematic.Forexample,adatasetconsistingof3Dfoodgraspingmovementsandcircular2Dmovementswouldrequirethemodeltogeneralizeoverbothkinematicsets(producingpoorerresults). 21 PAGE 22 22 ].Digiovannietal,usedreinforcementlearningtoco-adapttheirmodelandtherat'sbehaviorforgoalorientedtasks[ 23 ].WithrespecttounsupervisedlearningforBMIs,Sietal,usePCAasafeatureextractionpreprocessingforSVMandBayesianclassiers[ 24 ].Theirmaingoalistoclassifyactionswhichmovetherat(onamechanicalcart)towardsarewardlocation.Otherresearchdemonstratedthatneuralstatestructuresexistasanimalsubjectsengageinmovement[ 25 ].Thisresearchalsodemonstratedthatbyndingpartitionsintheneuralinputspacethatcorrespondtospecicmotorstates(handmovingoratrest),trajectoryreconstructionisimprovedwhenthemodelsareconstructedwithhomogeneousdatafromonlyonepartition[ 26 ].Specically,a'switching'generativemodelpartitionstheanimal'sneuralringsandcorrespondingarmmovementsintodifferentmotionprimitives[ 25 ].Thisimprovementisprimarilyduetotheabilityofthegenerativemodelstodistinguishbetweenneuralstates,andtherebyallowingthecontinuouslterstospecializeinaparticularpartoftheinputspaceratherthangeneralizeoverthefullspace.Theworkalsodemonstratedthatmodelingisimprovedwhenspatialdependenciesareexploitedbetweenneuralchannels[ 27 28 ].Othergenerativemodel(evengraphicalmodelslikeHMMs)[ 29 31 ]workonBMIsexploitthehiddenstatevariablestodecodestatestakenortransitionedbythebehavinganimal.Specically,Shenoy(etal)foundthatHMMscanproviderepresentationsofmovementpreparationandexecutioninthehiddenstatesequences[ 31 ].Unfortunately,thisworkalsorequirestheuseofsupervisionoratrainingdatathatmustrstbedividedbyhumanintervention(i.e.auser).Thereforetomovebeyondsuperviseddata,whichisnotacquirablefromparaplegics,aclusteringalgorithmisnecessary. 22 PAGE 23 32 ]andsubsequentlyusedinthecontextofdiscoveringsubfamiliesofproteinsequencesbyKroghetal[ 33 ].OtherclusteringworkfocusedonsingleHMMchainsforunderstandingthetransitionmatrices[ 34 ].Theworkdescribedinthisdissertationmovessignicantlybeyondpriorworkintwoways.First,theclusteringmodelndsunsupervisedhierarchaldependenciesbetweenHMMchains(perneuron)whilealsoclusteringtheneuraldata.Essentiallythealgorithmjointlyrenesthemodelparametersandstructuresastheclusteringiterationsoccur.Thehopeisthattheclusteringmethodologywillserveasafrontendforagoal-orientedBMI(withtheclustersrepresentingaspecicgoal,like'forward')orforaCo-adaptivealgorithmthatneedsreliableclusteringoftheneuralinput.Second,theparadigmischangedtoencompassamultiple-modelapproachtoimproveperformance. 13 35 ].Sincetherearenoknownground-truthsforthesetypesofstructuresordenitivepartitions,secondaryevidencemustbeprovided.Specically,wearguethatthebestwaytoshowthatourmethodologiesarediscoveringthesebenecialstructuresisonhowwellthetrajectorypredictionisimproved.Trajectoryimprovementisaccomplishedbyrstusingourmodelslikea'switch'topartitiontheprimate'sneuralringsandcorrespondingarmmovementsintodifferentprimitives.Similartoadivideandconquerstrategy,byswitchingordelegatingtheseisolatedneural/trajectorydatatodifferentlocallinearmodels,predictionofnalkinematictrajectoriesismarkedly 23 PAGE 24 24 PAGE 25 25 PAGE 26 3 ].Additionally,thepatientsthatareinvolvedinBMIresearchareparalyzed,whichpreventskinematicdatafrombeingrecordedduringthetrainingofamodel.Inordertoprovideparalyzedpatientsadirectabilitytointeractintheexternalworld,theBMIsolutionwillneedtoextractasmuchinformationfromtheneuralinputspace.Unfortunately,intheneuralinputspacetherearemanyproblems.First,onlyafewneuronsarebeingsampledfrommillions.Second,informationaboutthephysicalconnectivityamongthesesampledneuronsisnotavailablewithcurrenttechnology[ 6 8 ].Althoughsomehistologicalstudiescanbedonetostainforthetypeofneuronsacquired,theydonotprovideinformationabouttheirinter-communication(andanimalsareeuthanizedforthesestudies)[ 6 ].Third,thesampleofneuronsacquiredinoneexperimentwillnotbethesameneuronsnorrepresentthesamemotorfunctionsinanotherexperimentwithdifferentpatients[ 3 ].Additionally,someneuronsinthesamplemaynotevencontributetothetaskbeingmodeled.Withtheabsenceofinformation,aprobabilisticapproachisthebestapproachtomodelwhatisobservablefromthebrain.Modelingtheunknownhiddenneuralinformationisaccomplishedwithobservableandhiddenrandomprocessesthatareinteractingwitheachother.Specically,wemaketheassumptionthateachneuron'soutputisanobservablerandomprocessthatisaffectedbyhiddeninformation.Sincetheexperimentdoesnotprovidedetailedbiologicalinformationabouttheinteractionsbetweenthesampledneurons,hiddenvariablesareusedtomodelthesehidden 26 PAGE 27 3 6 ].Wefurtherassumethatthecompositionalrepresentationoftheinteractingprocessesoccursthroughspaceandtime(i.e.betweendifferentneuronsatdifferenttimes).Graphicalmodelsarethebestwaytomodelandobservethisinteractionbetweenvariablesinspaceandtime[ 36 ].Anotherbenetofastate-spacegenerativemodelovertraditionalltersisthatneuronsthatrelessduringcertainmovementscanbemodeledsimplyasanotherstateratherthanalowlterweightvalue.NeuroscientistsoftentreatthemultiplechannelsofneuraldataacquiredfromBMIexperimentsasmultivariateobservationsfromasingleprocess[ 10 ].Thisperspectiverequiresfullycoupledstatisticsacrossallofthechannelsatalltimesirrespectiveofpartialindependenceamongthemultipleprocesses.Ourgroup'sownworkhasshownthatmodelingthisdataasasinglemultivariateprocessisnotthemostappropriate[ 16 ].ThemodelsdescribedwithinthisdissertationalsodifferfromotherworkintheBMIeldsincetheyarenotatraditionalregressionapproachofmappingtheneuraldatadirectlytothepatienthandkinematicswithaconventionallinear/non-linearmodel[ 10 ].Instead,generativemodelsareusedtodividetheinputspaceintoregionsthatrepresentcellassemblieswhicharereferredthroughoutthedissertationasneuralstatestructures.Thesestructureshavebeenlooselytoucheduponinotherwork[ 19 37 ].Thebasicideaisthatkinematicinformationorneuraldataisdecomposedintomotionprimitivessimilartophonemesinspeechprocessing.Inspeechprocessing,graphicalmodels(specicallyHMMs)aretheleadingtechnologybecausetheyareabletocaptureverywellthepiecewisenon-stationarityofspeech[ 32 38 ].Sincespeechproductionisultimatelyamotorfunction,graphicalmodelsarepotentiallyusefulformotorBMIs(alsonon-stationary)[ 32 39 ].Byusingsmallersimplercomponents,morecomplicatedarmkinematicsareconstructedthroughthecombinationofthesesimplestructures.Wehypothesizethattherearedecomposablestructuresintheinputspacethatisanalogoustotheprimitivesinthekinematicspace. 27 PAGE 28 29 ].Later,GatetallusedanHMMtodiscovertheunderlyinghiddenstatesduringago-no-gomonkeyexperiment.TheHMMmodelprovidedinsightintotheunderlyingcorticalnetworkactivityofbehavioralprocesses[ 30 ].Specically,theycouldidentifythebehavioralmodeoftheanimalanddirectlyidentifythecorrespondingcollectivenetworkactivity[ 30 ].Additionally,bysegmentingthedataintodiscretestatesRadonsetaldemonstratedthattheremaybedependencyoftheshort-timecorrelationbetweenneuralcells.InrecentyearstherehasbeenarenewedinterestinusinggenerativemodelsforBMIs.Otherresearchersuseahybridgenerativemodeltodecodetrajectories[ 21 ].Intheirmodel,theyincorporateneuralstatesandhandstatessimilartootherlterworkbutsolelyinaprobabilisticframework.Forthecontinuoushandstatetheyusethemixture-of-trajectoriesmodel.AlthoughnotexplicitlyforBMIs,graphicalmodelshavebeenusedtomodelthebrainthroughbeliefpropagation[ 40 ].InparticularthisworkhasfocusedonthevisualcortexanddetectionofmotionusinganHMM. 28 PAGE 29 41 ].Withthistypeofrepresentation,edgesofthegraphrepresentdirectdependenciesbetweenvariables.Asstatedearlier,theabsenceofanedgeallowstheassumptionofconditionalindependencebetweenvariables.Ultimately,theseconditionalindependenciesallowamorecomplicatedmultivariatedistributiontobedecomposed(orfactorized)intosimpleandtractabledistributions[ 41 ]. 29 PAGE 30 32 ].ConsiderthegraphicalmodelinFigure 2-1 E,whichrepresentsahiddenMarkovmodel(HMM).Thistypeofstructuredecomposesthejointdistribution.ForasequenceoflengthT,wesimplyunrollthemodelforTtimesteps.TheMarkovpropertystatesthatthefutureisindependentofthepastgiventhepresent.ThisMarkovchainisparameterizedwiththetriplet,=fA;B;g,whereAistheprobabilisticNXNstatetransitionmatrix,BistheLXNoutputprobabilitymatrix(withLdiscreteoutputsymbols),andistheN-lengthinitialstateprobabilitydistributionvector[ 32 42 ].Iftheseparametersaretreatedasrandomvariables(asintheBayesianapproach),parameterestimationbecomesequivalenttoinference.Iftheparametersaretreatedasunknownquantities,parameterestimationrequiresaseparatelearningprocedure.InordertomaximizetheprobabilityoftheobservationsequenceO,themodelparameters(A;B;)mustbeestimated.Maximizingtheprobabilityisadifculttask;rst,thereisnoknownwaytoanalyticallysolvefortheparametersthatwillmaximizetheprobabilityoftheobservationsequence[ 32 ].Second,evenwithaniteamountofobservationsequencesitisunlikelytondtheglobaloptimumfortheparameters[ 32 ].Inordertocircumventthisissue,theBaum-Welchmethodcaniterativelychoose=fA;B;gthatwilllocallymaximizeP(Oj))[ 42 ]. 30 PAGE 31 Similarly,theelementsfortheoutputprobabilitymatrixB, andnallythevector, where, (2) Pleasenote,isthebackwardvariable,whichissimilartotheforwardvariableexceptthatnowthevaluesarepropagatedbackfromtheendoftheobservationsequence,ratherthanforwardfromthebeginningofO[ 42 ].Specicallythequantityisrecursivelycalculatedbysetting Thewellknownbackwardprocedureissimilar 31 PAGE 32 VariousHMMstructures thiscomputestheprobabilityoftheendingpartialsequenceot+1;:::oTgiventhestartatstatejattimet.Recursively,t(j)isdenedas 2-1 )thatarerelatedtondingdependencieswithinthehiddenstatespace.Thestandardfully-coupledHMMs(Figure 2-1 A)generallyrefertoagroupofHMMmodelsinwhich 32 PAGE 33 43 ].Parameterlearningisverydifcultwiththistypeofstructure.Someresearchershavecreatedvariationsofthefully-coupledHMMsinordertodecreasethemodelsizeandeasethecomplexityofinference.CoupledHMMs[ 44 ]proposedbyMatthewBrandmodelsthejointconditionaldependencyastheproductofallmarginalconditionalprobabilities,i.e. Thissimplicationreducedthetransitionprobabilityparameterspace.Ithasbeenusedforrecognizingcomplexhumanactions/behaviors[ 44 45 ].ButBranddidnotgiveanyassumptionorconditionunderwhichthisequationcanhold[ 43 ].Additionally,therearenoparameterstodirectlycapturetheinteractionbetweenchannels.AnothervariationofthefullycoupledHMMsapproximatestheEMalgorithmwithparticleltering.Thisparticularmodelwasusedtomodelfreewaytrafc[ 46 ].Figure 2-1 BisaspeciccoupledHMMscalledanevent-coupledHMMs[ 47 ].Thismodelisusedforaclassoflooselycoupledtimeserieswhereonlytheonsetofeventsiscoupledintime.ThefactorialHMM[ 48 ](Figure 2-1 C)representstheoppositeoftheCHMMbyusingmultiplehiddenstatechainstorepresentasinglechainofobservables.Thismodelisnotexploitingtheinteractionordependencybetweenmultiplemodels.InFigure 2-1 D,theIO-HMMisusedformodelingtheinput-outputsequencepair.AlthoughtheIO-HMMissimilartothecoupledHMMtheinputusedinIO-HMMandhiddenstatefromprevious 33 PAGE 34 49 ]isonlyforoneHMMmodelandagainnotforgeneralmultiplecoupledHMMs[ 43 ].Chapter3willdiscussanalternativeandmorereasonableformulationoftheCHMMwhichalsoreducestheparameterspacebutincludesparametersthatcapturethecouplingstrengthbetweenHMMs. 34 PAGE 35 16 ]usingtheLinde-Buzo-Gray(LBG)VQalgorithm[ 50 ].ThenasingleHMMwastrainedwiththesesymbolscorrespondingtoneuraldataofaparticularclass(movementvs.restforthemonkeyfoodgraspingtask).Unfortunately,thisswitchingclassieronlymaximallyachieved87%classicationaccuracy[ 16 ].Although,theseresultslentsupporttousingHMMsonBMIdatasincetrajectoryreconstructionwasimprovedoverasinglelinearmodel,theresultswereonlyfairandmotivatedfurtherinvestigation.Intryingtoimproveuponthisperformance,therelevancyofparticularneuronstoarespectivetask(i.e.movementorrest)wasexploredinordertocongregatedifferentneuronsintocorrespondingsubsets.Toquantifythedifferentiation,weexaminedhowwellanindividualneuronclassiesmovementvs.restwhentrainedandtestedonanindividualHMMchain.Sinceeachneuralchannelisbinnedintoadiscretenumberofspikesper100ms,theneuraldatawasdirectlyusedasinput.DuringtheevaluationoftheseparticularHMMchains,theconditionalprobabilitiesforrestormovementarerespectivelycomputedasP(O(i)j(i)r)andP(O(i)j(i)m)forthei-thneuralchannel,whereO(i)istherespectiveobservationsequenceofbinnedringcountsand(i)m;(i)rrepresentthegivenHMMchainparametersfortheclassofmovementandrest,respectively.Togiveaqualitativeunderstandingoftheseweakclassiers,Figure 3-1 presentstheprobabilisticratiosfrom14single-channelHMMchains(shownbetweenthetopandbottommovementsegmentations)thatproducedthebestclassicationsindividually.Specically,thegureillustratesthesimpleratio 35 PAGE 36 3-1 quantitativelyshowsthattheindividualneuralchannelsroughlyclassifythetwoclassesofdatabetterthanchance.Essentiallyifthelikelihoodratiowaslargerthanonethesamplewaslabeledmovementwhilelessthanonebecamealabelforrest.Themodelwastrainedonsetof8000samplesandclassicationwascomputedonaseparatetestsetof3000samples.Overall,Figure 3-1 illustratesthatthesingle-channelHMMsroughlyclassifymovementandrestsegmentsfromtheneuraldata(betterthanrandom).Specically,thegureshowsmorewhitebandsP(Ojr)>>P(Ojm)duringrestsegmentsanddarkerbandsP(Ojm)>>P(Ojr)duringmovementsegments.Figure 3-2 isazoomed-inpictureofFigure 3-1 .Thisgurerevealsthatsomeoftheneuralchannelclassierstunetocertainpartsofthetrajectorylikerest-food,food-mouth,andmouth-rest.Thesesub-segmentationslendfurthersupporttoourhypothesisthatprimitiveswithinthedataexist.Havingobservedtheabilityofsomeneuronstoroughlyclassifythetwoclasseswhileotherneuronswerepoor,theissuenowbecomeshowtocombinetheinformationproperlywhilemaintainingcomputationalsimplicityandtheremovalofVQ.Thenext Table3-1. Classicationperformanceofexamplesingle-channelHMMchains Rest Moving 83.4% 75.0% 62 80.0% 75.3% 8 72.0% 64.7% 29 63.9% 82.0% 72 62.6% 82.6% 36 PAGE 37 Probabilisticratiosof14neurons Figure3-2. Zoomedinversionoftheprobabilisticratios sectionsdetailthestructureandtrainingofdifferentmodelsthatmergetheclassicationperformanceoftheseindividualclassierstoproduceanoverallclassicationdecision. StructureandtrainingfortheICHMM.AsdiscussedinChapter2,theCHMMmodelsmultiplechannelsofdatawithouttheuseofmultivariatepdf'sontheoutputvariables[ 44 ].Unfortunately,thecomplexity(O(TN2D)orO(T(DN)2))andnumberofparametersnecessaryfortheCHMM(and 37 PAGE 38 3 ]duringthecontrolofmovement.Specically,duringmovement,differentmusclesmayactivateforsynchronizeddirectionsandvelocities,yet,arecontrolledbyindependentneuralassembliesorclustersinthemotorcortex[ 3 14 51 ].Conversely,withintheneuralclustersthemselves,temporaldependencies(andco-activations)havebeenshowntoexist[ 3 ].Therefore,thisclassierassumesthatenoughneuronsaresampledfromdifferentneuralclusterstoavoidoverlapordependencies.Thisassumptionisfurtherjustiedbylookingatthecorrelationcoefcients(CC)betweenalltheneuralchannelsinourdataset.ThebestCC's(0.59,0.44,0.42,0.36)occurredbetweenonlyfouroutofthe10,700neuralpairswhiletherestoftheneuralpairswereamagnitudesmaller(abs(CC)<0.01).Additionally,despitetheseweakunderlyingdependencies,thereisalonghistoryofmakingsuchindependenceassumptionsinordertocreatemodelsthataretractableorcomputationallyefcient.TheFactorialHiddenMarkovModelisoneexampleamongstmany[ 44 48 ].AlthoughanindependenceassumptionismadebetweentheneuronstosimplifytheIC-HMM,othermodelswillexploittheseweakdependenciesaswellasmorecomplicated(spatial-temporal)dependenciesinlatersections.Bymakinganindependenceassumptionbetweenneurons,eachneuralchannelHMMistreatedindependently.Thereforethejointprobability 38 PAGE 39 IC-HMMgraphicalmodel becomestheproductofthemarginals oftheobservationsequences(eachlengthT)foreachdthHMMchain.Sincethemarginalprobabilitiesareindependentlycoupled,yettrytomodelmultiplehiddenprocesses,thisnaiveclassier(Figure 3-3 )isnamedtheIndependentlyCoupledHiddenMarkovModel(ICHMM)inordertokeepthenomenclaturesimple.ByusinganICHMMinsteadofaCHMM,theoverallcomplexityreducesfrom(O(TN2D)orO(TD2N2))toO(DTN2)giventhateachHMMchainhasacomplexityofO(TN2).SinceasingleHMMchainistrainedwitharespectiveneuralchannel,thenumberofparametersisgreatlyreduced,consequentlyreducingtherequirementsintheamountoftrainingdata.Specically,theindividualHMMchainsintheICHMMcontainaround70parametersforatrainingsetof10,000samplesasopposedtoalmost18,000parametersnecessaryforacomparableCHMM(duetothedependentstates). 39 PAGE 40 and(d)rand(d)mdenoteHMMchainsthatrepresentthetwostatesofthemonkey'sarm(movingvs.rest).AlloftheHMMchainsarepreviouslytrainedwiththeirrespectiveneuralchannelsusingtheBaum-Welchalgorithm[ 32 42 ]describedinChapter2.Basedonempiricaltesting,threehiddenstatesandanobservationsequencelengthTof10arechosenforthemodel.Withthedatasetdescribed,anobservationlengthof10correspondstoasecondofdata(giventhe100msbins).2.Normally,themonkey'sarmisdecidedtobeatrestif,P(OTjr)>P(OTjm)andismovingif,P(OTjm)>P(OTjr),butinordertocombinethepredictivepowersofalltheneuralchannels,Equation( 3 )isusedtoproducethedecisionboundary ormoreaptly, wherel(O)isthelikelihoodratio,abasicquantityinhypothesistesting[ 38 40 ].Essentially,ratiosgreaterthanthethresholdareclassiedasmovementandthoselessthanastherestclass.Theuseofthresholdsforthelikelihoodratiohasbeenusedinneuralscienceandotherareasofresearch[ 38 40 ].Often,itismorecommontousethelog-likelihoodratioinsteadofthelikelihoodratioforthedecisionrulesothatarelativescalingbetweentheratioscanbefound(aswellassuppressinganyirrationalratios)[ 38 ]: 40 PAGE 41 Byapplyingthelogtotheproductofthelikelihoodratios,essentiallythesumoftheloglikelihoodratiosisfoundtoseeifitislargerorsmallerthanathreshold(log).Insimpleterms,thisdecisionruleposesthequestionofhowlargeistheprobabilityforoneclasscomparedtotheotherandisitoccurringoveramajorityofthesingleclassiers.Notethatbyvaryingthethresholdlog(),classicationperformanceistunedtotanyparticularrequirementsforincreasingtheimportanceofoneclassoveranother.Forthisexperimentequalimportanceisassumedfortheclasses(nobiasforoneoranother).Moreover,optimizationoftheclassierisnownolongerafunctionoftheindividualHMMevaluationprobabilities,butratherafunctionofoverallclassicationperformance.Thenextsectionoutlinesmethodsthatmoveawayfromthenaiveassumptionofindependencebetweentheneuralchannels.Specically,implicitrelationshipsbetweenneuronswillrstbeexploredandthenexplicitrelationships. 3.3.1BoostingBoostingisatechniquethatcreatesdifferenttrainingdistributionsfromaninitialinputdistributionsothatasetofweakclassiersisgenerated[ 52 53 ].Thegeneratedclassiersthenformanensemblevoteforthecurrentdataexample.Thesehierarchicalcombinationsofclassiersarecapableofachievinglowererrorratesthantheindividualbaseclassiers[ 52 53 ].ThereforeboostingwillbeusedtomovebeyondtheIC-HMMandexploitthecomplimentaryinformationprovidedbytheindependentHMMchains.Adaboostisthemostwidelyusedalgorithmtoevolvefromboostingmethods[ 54 ].Thisalgorithmsequentiallygeneratesweakclassiersbasedonweightedtrainingexamples.Essentially,theinitialdistributionoftrainingexamplesisre-sampledeachround(basedonthedistributionoftheweightsWi)inordertotrainthenextclassierup 41 PAGE 42 54 ].Thetrainingexamplesthatfailtobeclassiedonaparticularroundreceiveanincreasedweightingsothatthesubsequentclassiersaremorelikelytobetrainedonthesehardexamples.WiththeinitialvaluesoftheweightsbeingWi=1 52 ].ForfurtherdetailsonAdaboostorboostingandtherelationshiptothemarginorsupportvectormachines,see[ 54 ].WithrespecttoimprovingIC-HMM,Adaboostoffersapromisingwaytoimplicitlynddependenciesamongthechannelsthroughtheprocessofboostingclassiersduringtraining.Inthenextsection,Adaboostisapplieddifferentlytothemultidimensionalneuraldata. 42 PAGE 43 55 ].TherstmajordeparturefromAdaboost,comesfromhowtheensembleisgenerated.Insteadofformingoneexpertatatime,theCindependentHMMchainsaretrainedinparallelusingtheBaum-Welchformulation(C=thenumberofneuralchannels)[ 42 ].Thisdividesthejointlikelihoodintomarginalssothatindependentprocessesareworkinginsimplersubspacesoftheinput[ 25 ].Thenacompetitivephaseisinitiated.Specically,arankingisperformedwiththeexpertsandawinnerischosenbasedontheclassicationperformanceforthecurrentdistributionofinputexamples.Thewinnerthatminimizestheerrorwithrespecttothedistributionofsamplesischosen.TheminimalerroriscalculatedusingaEuclideandistance(alsoadeparturefromAdaboost)inordertoavoidbiasingclassassignments(sinceclassesmaynothaveequalpriors)[ 25 ].Next,theremainingexpertsaretrainedwithintheirrespectivesubspacebutrelativetotheerrorsofthepreviouswinner.Finally,theWiareusedtoselectthenextdistributionofexamplesfortheremainingexperts.SimilartoAdaboost,theremainingexpertsaretrainedonthehardexamplesfromdifferentsubspaces.Inturn,ahierarchicalstructureisformedasthewinningexpertsaffectthetrainingonthe 43 PAGE 44 56 ].WithBME,improvedperformanceisgainedthroughtheuseofacondencemeasurefortheindividualexperts[ 56 ].Althoughmanydifferentcondencemeasuresexist,themajorityuseascalarfunctionoftheexpert'soutputwhichisthenusedasastaticgatingfunctionormixturecoefcient[ 55 56 ].ThealgorithmusesasimplemeasureforeachexpertbasedontheL2-Normoftheclasserrors(insteadoftheoneoutlinedinequation 3 ) whereerrManderrRaretherespectiveerrorsofthetwoclassesinourproblem,MoveandRest(whichcouldgeneralizetomoreclasses).ThevariablecissubstitutedforthenormalAdaboostformulationofc(Equation 3 )toupdatetheWi'sinEquation 3 .Sincethereisaconditionplacedduringtheboostingphasetodiscardexpertswithlessthan50%classication,negativealphaswillnotoccur[ 54 ].Noticethatastheerrorsbetweenthetwoclassesaresmaller,theweightsfortheexpertsbecomelarger.TheproposedAdaboosttrainingalgorithmispresentedbelow.Given:(x1;y1);:::;(xn;yn)wherexi2X;yi2Y=f1;+1gInitializeWi=1 44 PAGE 45 63 ].Additionally,thetemporalMarkoviandynamicscoupledwiththehierarchicalstructureandmixturemodelingcanbethoughtofasasimpleapproximationtotreestructuredHMMs[ 41 ].Otherworkhasfocusedonsolvingthisproblemofboostingmultipleparallelclassiers[ 64 ].Otherauthorshaveproposedboostingsolutionsthatreducethedimensionalityoftheinputdata[ 64 69 70 ].Fromtheirperspective,themultidimensionalinputsaretreatedassimplefeaturesofasingle 45 PAGE 46 64 ].Ourmodelisdifferentsincetheinputspaceistreatedasmultiplerandomprocessesthatareinteractingwitheachotherinsomeunknownway.Bydecomposingtheinputspaceintomultiplerandomprocesses,thelocalcontributionsoftheindividualprocessesareexploitedinacompetitivefashionratherthanusingtheglobaleffectofasingleprocess.ThistypeofalgorithmalsohascomponentssimilartotheMixtureofExperts(MOE)algorithm.SinceasingleHMMchainistrainedonasingleneuralchannel,thenumberofparametersisverysmallandcansupporttheamountoftrainingdata.AsdiscussedwiththeIC-HMM,theindividualHMMchainsintheBM-HMMcontainaround70parameters.Inthenextsection,thecomplexityoftheBM-HMMisslightlyincreasedinordertoexplicitlymodeldependenciesbetweentheneurons. 3.4.1ModelingFrameworkTomovebeyondtheIC-HMMandimplicitdependenciesintheBM-HMM,anotherlayerofhiddenorlatentvariablesisestablishedtolinkandexpressthespatialdependenciesbetweenthelowerlevelHMMstructures(Figure 3-4 ),thuscreatingacliquetreestructureT(sincetherearecycles),wherethehierarchicallinksexistbetweenneuralchannels.Theloglikelihoodofthedynamicneuralringsfromalloftheneuronsforthisstructure(Figure 3-4 )is 46 PAGE 47 LM-HMMgraphicalmodel WherethedependencybetweenthetreecliquesarerepresentedbyahiddenvariableMinthesecondlayer, 3-4 ,showshowthelowerobservablevariablesOiareconditionallyindependentfromthesecondlayerhiddenvariableMi,aswellasthe 47 PAGE 48 3 isinterpretableasamixturevariable(whenexcludingthehierarchicallinks).TheLM-HMMimplementsamiddlegroundbetweenmakinganindependenceassumptionandafulldependenceassumption.Sincealayerofhiddenvariablesisadded,thecomputationalcostincreases.Thenextsectionwilldetailanapproximationthateasescomputationalcostsbutstillmaintainstherichnessofmodelingtheinterrelationshipsbetweenneurons. 41 79 ].Forthismodel,ameaneldapproximationisusedtoallowinteractionsassociatedwithtractablesubstructurestobetakenintoaccount[ 79 ].Thebasicideaistoassociatewiththeintractabledistributionasimplieddistributionthatretainscertaintermsoftheoriginaldistributionwhileneglectingothers,replacingthemwithparametersuithatoftenreferencedasvariationalparameters.Graphically,themethodcanbeviewedasdeletingedgesfromtheoriginalgraphuntilaforestoftractablestructuresisobtained.Edgesthatremaininthesimpliedgraphcorrespondtotermsthatareretainedintheoriginaldistributionandedgesthataredeletedcorrespondtovariationparameters[ 79 81 ].ApproximationsareusedtondtheexpectationofEquation 3 .Inparticular,P(SitjSit1;Mi;i))isrstapproximatedbytreatingMiasindependentfromSmakingconditionalprobabilityequaltothefamiliarP(SitjSit1;i)).Twoimportantfeatures 48 PAGE 49 55 ].Becausethelower-levelHMMshavebeendecoupled,theBaum-WelchformulationcannowbeusedtocomputesomeofthecalculationsintheE-step,leavingestimationofthevariationalparameterforlater.Asaresult,theforwardpassiscalculatedasEstep: Thisquantityiscalculatedrecursivelybysetting: Thewellknownbackwardprocedureissimilar thiscomputestheprobabilityoftheendingpartialsequenceot+1;:::oTgiventhestartatstatejattimet.Recursively,wedenej(t)as Additionally,theajkandbj(ot)matricesarethetransitionandemissionmatricesdenedforthemodelwhichareupdatedintheM-step.ContinuingintheE-steptheposteriorsarerearrangedintermsoftheforwardandbackwardvariables.Let 49 PAGE 50 andnowwiththeconditionalindependenciestheposteriorisdenedintermsof'sand's Wealsodene Whenexpanded TheM-stepdepartsfromtheBaum-Welchformulationandintroducesthevariationalparameter[ 79 ].Specically,theM-stepinvolvestheupdateoftheparametersj,ajk,bL(wewillsaveuiforlater)Mstep: Therearetwoissueslefttosolve.First,howcanthevariationalparameterbeestimatedandmaximizedgiventhedependencies.Second,ifexperimentallyitisnotknownwhichneuronsareaffectingotherneurons(ifatall),howcanthedependenciesbetweenneuronsbedenedinthemodel. 50 PAGE 51 34 80 81 ] GiventhesetofsequencesandcurrentestimatesoftheparameterstheE-stepconsistsofcomputingtheconditionalexpectationofhiddenvariableM TheproblemwiththisconditionalexpectationisthedependencyonMi1.SinceMi1isindependentfromOiandiwedecomposethisinto Therstterm,awell-knownexpectationforMixtureofExperts,iscalculatedbyusingBayesruleandtheprioriprobabilitythatM=1 82 ].Weapproximatetheintegrationwith 51 PAGE 52 TocomputetheM-step BorrowingfromthecompetitivenatureoftheBM-HMM,winnersarechosenbasedonthesamecriterionofminimizingtheEuclideandistancefortheclassesfortheLM-HMM. 44 ])willbeused,whichwillbereferredtoasDependentlyCoupledHiddenMarkovModel(DC-HMM)inordertomaintainnomenclature.Withthisformulationthejointconditionalprobabilityformulationismodeledasalinearcombinationofmarginalconditionalprobabilitieswiththeweightsrepresentedbycouplingcoefcients.AlthoughsomeDC-HMMformulationshaveshowntobecomputationallyexpensiveandnearlyintractable[ 44 46 ]thisnewformulationalleviatessomeofthoseobstacleswithasimplisticapproximation.AlthoughcomputationalcomplexityisincreasedbeyondtheLM-HMMs,abenecialinsightisgainedsincetheunderlyingstructurewithintheneuraldataisrevealedthroughthemodel'sparameters.Consequentlytheremaybeanopportunitytoexploittheunderlyingstructureformodelingpurposesorgainanunderstandingintotheneurophysiologicinteractions. 52 PAGE 53 43 ]formulation.Fromthere,themodelingframeworkisdetailedaswellastheforwardprocedure.ThisforwardprocedureisimportantforthediscussiononclusteringanalysisandNeurophysiologicunderstandinginChapter6.Finally,thelearningalgorithmforthisDC-HMMformulationispresentedwithadiscussiononthecouplingcoefcientsandhowtheycharacterizethecouplingbetweenchannels. where_c;cisthecouplingweightfrommodel_ctomodelc,i.e.howmuchS(_c)t1affectsthedistributionofS(c)t.WhichiscontrolledbyP(S(c)tjS(_c)t1).Essentially,thejointdependencyismodeledasalinearcombinationofallmarginaldependencies[ 43 ].ThisformulationreducesthenumberofparameterscomparedtothestandardDC-HMM[ 43 ].OfcoursethismodelstillcontainsmoreparametersthanmultiplestandardHMMsandcomputationallymoreexpensivethantheLM-HMMsbutbycomputingthecouplingrelationshipsbetweenneuralchannelstheremaybebenecialinsightintothemicrostructureintheneuraldata.TheadditionalparametersareincurredwiththeC2transitionprobabilitymatricescomparedtoonlyCinCstandardHMMs.Thereisalsoanadditionalcouplingmatrix.Sincetheoutputsymbolsnecessaryfortheneuraldataismuchlargerthanthenumberofhiddenstates,theincreaseinthenumberoftransitionmatricesdoesnotincreasethemodelcomplexitydramatically.Ascomparedtothestandardformulationofthefully-coupledHMMs,thisformulationiseasiertoimplement.TheDC-HMMmodelischaracterizedbythequadruplet=(;A;B;),whereistheinteractionparametersbetweenchannels(asshowninFigure 3-5 ).Assumethere 53 PAGE 54 1. priorprobability=((c)j);1cC;1jN(c) 2. transitionprobabilityA=(a(_c;c)ih);1_c;cC;1iN(_c);1jN(c) 3. observationprobabilityB=(b(c)j(k));1cC;1jN(c);1kM 4. couplingcoefcient=(_c;c);1_c;cC Theforward-backwardcomputationisillustratedinFigure 3-5 .AteachtimeslicethereareNC(ifassumeN(c)=N)'s,whichisanexponentialnumberwithrespecttoC.ThecomputationcomplexitywouldbeTNCtherebymakingitimpracticaltocomputetheforward-backwardvariablesforanyrealisticnumberofchannels.ToreducethecomputationalcomplexityamodiedforwardvariableiscalculatedforeachHMMmodelseparately.ThemodiedforwardvariablereducescomplexitytoO(TCN2)andiscalculatedinductivelyasa)Initialization: 54 PAGE 55 DC-HMMtrellisstructure b)Induction: c)Termination: ExperimentalresultsshowthatcalculatedP(Oj)isclosetothetruevalueofP(Oj)[ 43 ]. 55 PAGE 56 L=P+Xi;_c;c(_c;c)i(NXj=1a(_c;c)ij1) wherethe(_c;c)iaretheundeterminedLagrangemultipliers.Pislocallymaximizedwhen WhilethelikelihoodfunctionP(Oj)ismorecomplicatedinthisDC-HMMformulationthaninstandardHMMcase,Pisstillahomogeneouspolynomialwithrespecttoeachtypeofparameters(namely,,A,Band)andeachtypeoftheparametersisstillsubjecttostochasticconstraints[ 43 ].Overall,there-estimationformulaisguaranteedtoconvergetoalocalmaxima[ 43 66 ].TheonlydifferencefromthestandardHMMcaseisincalculatingtherstderivativeofthelikelihoodfunction.WithstandardHMMs,thesederivativesreducetoaforminwhichonlytheforwardandbackwardvariablesareneeded[ 43 66 ].Intheaboveformulation,thederivativesarecalculatedusingbackpropagationthroughtime.Fortunately,computationalcomplexityisnotsignicantlyincreasedsincesimilarforwardproceduresareapplicabletothecalculationofthederivatives.ThedetailedcomputationoftheserstderivativesispresentedinAppendixB. 56 PAGE 57 4-1 ):rsttheinputdatais 57 PAGE 58 Figure4-1. Multiplemodelmethodology 58 PAGE 59 Classicationresults(BM-HMMselectedchannels) #channels %correct IC-HMM 104 92.4% IC-HMM 9 87.1% 9 92.0% 104 88.3% Linear 9 86.9% Table4-2. Classicationresults(LM-HMMselectedchannels) #channels %correct IC-HMM 104 92.4% IC-HMM 9 89.5% 9 92.1% 104 88.3% Linear 9 87.8% 4.1.1BoostedandLinkedMixtureofHMMsTheBM-HMMsandLM-HMMarebothinitializedthreehiddenstatesandanobservationsequencelengthT=10,whichcorrespondstoonesecondofdata(giventhe100msbins).Thesechoicesarebasedonpreviouseffortstooptimizeperformance[ 16 ].Forthelinearclassier,aWienerlterwitha10-tapdelay(thatcorrespondstoonesecondofdata)isusedfollowedbyathresholdforclassication.TentapsareusedtobeconsistentwiththeHMMwork.Forthemonkeyfoodgraspingtask,8000samplesareusedtotrainthemodelswhile2000samplesareusedasacrossvalidationsettoselectthechannels.Aseparatetestsetof5000samplesisusedfortheclassicationresultsabove.Theseparametersandthresholds(forthelinearmodel)werechosenempiricallyfrommultipleMonte-Carloruns,basedonpreviouswork.Inearlierwork,Leave-K-Outmethodologieswereemployedtoensurethattheresultsonthesedatasetsaregeneral[ 25 ].Toprovideafaircomparisonbetweenthemethods,thesameneuralchannelsthatwerechosenbytheBM-HMMarealsousedwiththelinearmodelandtheIC-HMM.Similarly,fortheLM-HMM,adifferentsetofneurons(althoughsomeoverlaptheBM-HMM)wereselected.Theselectionofneuronsisbasedonthealgorithmwhilethenumberofneuralchannelsisimposedinordertohaveequalnumberofchannelsforcomparison. 59 PAGE 60 4-1 and 4-2 ,showsacomparisoninresultsbetweentheBM-HMMandLM-HMMversusthefullIC-HMMandthelinearclassiercreated.TheBM-HMMandLM-HMMperformonparwiththeIC-HMM,butwiththeaddedbenetofdimensionalityreduction.Essentially,thesameperformanceisobtainedwithafractionofthenumberofneuralchannels.ThisdimensionalityreductionisveryimportantwhenconsideringthehundredstothousandsofchannelsthatwillbeacquiredinfutureBMIexperiments[ 10 ].Inordertounderstandhowthetwomethodsdifferinexploitingthechannels,Figures 4-2 and 4-3 showthecorrelationcoefcientsbetweenthesubsetofchannelschosenbytheBM-HMMandLM-HMM.Interestingly,moreofthechannelsintheLM-HMMhavealargepositiveandlargenegativecorrelation(withrespecttothemoveclass).Incontrast,theBM-HMMhassomepositivecorrelationamongstitssubset,butfewchannelsexhibitnegativecorrelation. Figure4-2. Correlationcoefcientsbetweenchannels(monkeymoving) Tables 4-3 and 4-4 showtheeffectofrandomlyselectingtheexperts.NoticehowtheresultsinthesetablesarepoorincomparisontoTables 4-1 and 4-2 .Inparticular,Tables 4-1 and 4-2 showasignicantdecreaseinperformancewhenmodelingthemonkeydata.Interestingly,mconvergestosimilarvalues(likeanaveraging)forbothmodels.Thisaveragingeffectcouldbeduetothedependencyofrandomlyselectedneurons,(i.e.moreindependent).Sincemanyofthechannelshavelowringrates 60 PAGE 61 CorrelationCoefcientsbetweenchannels(monkeyatrest) ormaynotevencorrespondtotheparticularmotions,someoftheseunimportantneuronsmayalsoreduceperformance.TheaveragingeffectisfurtherconrmedbytheresultsfromtheIC-HMMsincesimilarperformanceisachievedwithrespecttobothmodelsinTables 4-3 and 4-4 .SincetheIC-HMMisanaveragingmodelusedforindependentneurons,similarperformanceontherandomsubsetswouldindicatethatdependenciesarenotbeingexploitedbythemodels.Figure 4-4 showsthecorrelationcoefcientbetweentherandomlyselectedchannels.Sincethecolorsarealllightgreen,thecorrelationbetweenthechannelsislowtozero,furtheralludingtoindependencebetweentheseparticularchannels.TheresultsfurthersuggeststhattheLM-HMMisbetteratexploitingstrongercorrelatedchannels(i.e.dependent)thantheBM-HMM,andmuchstrongerthanrandomlyselectingchannels(whichmayexhibitmoreindependencebetweenchannels).Additionally,Figure 4-5 demonstratesanexpertaddingexperimentinwhichthebestrankedexpertsareaddedonebyonetotheensemblevote.AstheBM-HMMchainsareaddedinFigure 4-5 ,aninterestingresultemerges,theerrorratequicklydecreasesbelowtheIC-HMMerrorwhenappliedtothemonkeyneuraldata.TheLM-HMMshowsafasterdropinerror,butdoesnotachievethesameresultastheBM-HMM.Overall, 61 PAGE 62 Correlationcoefcientbetweenchannels(randomlyselectedformonkey) theboostedmixturesandlinkedmixturesareexploitingmoreusefulandcomplimentaryinformationforthenalensemblethanthesimpleIC-HMM.Figure 4-6 presentstheperi-eventtimehistogramforalltheneuralchannelsinparallel.Thesehistogramspresenttheringrateofthedifferentneuronsaveragedacrossasingleevent(i.ethestartofamovementtrial)[ 71 ].Forthesegures,theringrateisnormalizedsothatthelightcolorsillustratedecreasedringratesandthedarkercolorsindicateincreasedringrates.Overlaidandstretchedoneachimage Table4-3. Classicationresults(randomBM-HMMselectedchannels) #channels %correct IC-HMM 104 92.4% IC-HMM 9 69.1% 9 69.4% 104 88.3% Linear 9 68.1% Table4-4. Classicationresults(randomLM-HMMselectedchannels) #channels %correct IC-HMM 104 92.4% IC-HMM 9 63.5% 9 65.4% 104 88.3% Linear 9 63.2% 62 PAGE 63 Monkeyexpertaddingexperiment istheaverageofthemovementtrials.Interestingly,somechannelshavenopatternassociatedwiththemwhereasothershaveaconsistentpattern.AlsonoticehowtheBM-HMMselectsneuronsthatdisplaybothanincreaseinringactivityaswellasneuronsthatdecreasetheirringduringtheonsetofmovement.TheLM-HMMhasalsobeenobservedtouseneuronsthatreducetheirringrateduringmovement.Overall,linearornon-linearltersmaynotbeabletotakeadvantageofthesetypesofneuronswhereasthesegraphicalmodelsincorporatetheinformationassimplydifferentstates.Overalltheresultsdemonstratethreeinterestingpoints.First,nineBM-HMMandLM-HMMchainsoutperformthelinearclassierthatusesthefullinputspaceonthemonkeydata.Second,thesubsetofexpertsthatarechosenbytheBM-HMMperformwellonthelinearmodel.ThisresultisexpectedsincetheBM-HMMchainsselectneuralchannelswithimportantcomplimentaryinformation.Third,whencomparingtheBM-HMMandLM-HMMtothelinearclassierandtheIC-HMMusingthesamesubsetofneuralchannels,theresultsshowthatthehierarchicaltrainingoftheBM-HMM 63 PAGE 64 Parallelperi-eventhistogramformonkeyneuraldata andLM-HMMprovidesasignicantincreaseinperformance.Thisincreaseisrelatedtothedependenciesthatarebeingexploitedduringeachroundoftraining,wheretheothermodelssimplytrytouniformlycombinealltheneuralinformationintoasinglehypothesis.Tables 4-3 and 4-4 supportthishypothesissincetheLM-HMMandBM-HMMdefaulttoanindependentapproximation.Finally,otherBMIresearchersapplysensitivityanalysistounderstandtheimportanceofaneuralchannelrespectivetothekinematicsperformedbythesubject.IncontrasttheBM-HMMandLM-HMMchannelselectionistryingtoimproveclassicationresultsbyexploitingdependenciesbetweenchannelsaswellaskinematics.Interestingly,someofthechannelsselectedinbothdatasetsdooverlapsomeofthesameneuronsselectedduringsensitivityanalysis[ 20 67 ]. 64 PAGE 65 4-5 showsthecorrelationcoefcientresultsafterusingtheDC-HMMtopartitiontheinputspacefortrajectoryreconstructionbyrespectiveWienerlters.ThetableshowsthattheDC-HMMwith6neuronsdoesnotperformaswellastheLM-HMMbutisfarbetterthanusingrandomlabelsorasingleWienerlter.Theresultsinthetablealsoshowthataddingmoreneuronsimprovesthecorrelationcoefcient(butnotbeyondtheLM-HMM). Table4-5. CorrelationcoefcientusingDC-HMMon3Dmonkeydata CC .81:18 .82:15 .84:13 .76:19 .75:20 .77:17 NMSEon3Dmonkeydata NMSE .26 DC-HMM(29Neurons) .23 LM-HMM .22 SingleWiener .36 Figure 4-7 showstheDC-HMMreconstructionofthekinematicoutofthefoodgraspingtaskwhereasFigure 4-8 showsthetruetrajectory.Qualitativelythetrajectoriesaresimilar,conrmingthecorrelationcoefcientsshownabove.Additionally,thenormalizedmeansquarederror(NMSE)wascomputedforthedifferentmodelsshowing 65 PAGE 66 Figure4-7. Supervisedmonkeyfoodgraspingtaskreconstruction(position) Figure 4-9 showstheViterbihiddenstatepathforthedifferentneuronsacrossthreehiddenstates.Essentially,thethreedimensions(X,Y,andZ)ofthemonkey'sarmisoverlaidoverthestatepaths.OntheYaxisthereare24states(6neuronswithfourstates)ateachtimebin.Thedarklinesindicatethelargest(equation 3 )ateachtimebin.TheleftgureistheViterbipathfortheDC-HMMtrainedonmovementdatawhiletherightgureisforthedatawhenthemonkeyisatrest.Theguresdepictrepeatingpatternsthatcorrespondtothekinematics.Alsonoticethatthestatesfromdifferentneuronsaredominatingacrossthedatasetforeachoftheclasses.Althoughnotpertinenttotheresultsinthischapter,theseViterbipathsinspirefurtherexplorationinChapter6. 66 PAGE 67 3Dmonkeyfoodgraspingtruetrajectory Figure4-9. Hiddenstatespacetransitionsbetweenneuralchannels(formoveandrest) 67 PAGE 68 4-10 showsthecouplingcoefcientsforeachclass(move/rest)forthesixneuronsinthesubset.Thecouplingcoefcientisfromthepreviousneuron'sstatetothenext.Brightercolorsindicatealargervalueforthecouplingcoefcient.TheleftgurepresentstheDC-HMMtrainedwithmovementdatawhiletherightisforthemodelofrest.Thediagonalthatformsinthegureisthecouplingcoefcientbetweenthesameneurons(indicatingpreferenceforstayinginthesamestateorneuron).Despitesomeneuronsillustratingweakdependency,thereisstrongerevidenceforindependencebetweenneurons.Thisresultmayhelptoexplainwhyeventhelinearltersareabletoworkwellwiththeseneurons(sinceDC-HMMhaslesstoexploit).AlsonoticeafewneuronsdominatethemodelswhichreinforceswhatisshowninthepreviousFigure 4-9 Figure4-10. Couplingcoefcientbetweenneuralchannels(3Dmonkeyexperiment) 68 PAGE 69 Table4-7. Classicationresults(BM-HMMselectedchannels) #channels %correct IC-HMM 16 62.5% IC-HMM 6 58.3% 6 64.0% 16 61.8% Linear 6 56.9% Table4-8. Classicationresults(LM-HMMselectedchannels) #channels %correct IC-HMM 16 62.5% IC-HMM 6 56.5% 6 62.3% 16 61.8% Linear 6 55.2% Tables 4-7 and 4-8 showsacomparisonoftheclassicationresultsfromtheBM-HMMandLM-HMMversusthefullIC-HMMandasimplelinearclassiercreatedbyaregressionmodelfollowedbyathreshold[ 25 ].Comparingthesetableswith 4-1 and 4-2 ,theclassicationperformanceonmonkeydataisbetterthanontheratdata.ThemoreimportantpointisthatagaintheBM-HMMandLM-HMMperformonparwiththeIC-HMM,butwiththeaddedbenetofdimensionalityreduction.Effectively,thesameperformanceisobtainedwithafractionofthenumberofneuralchannels. 69 PAGE 70 4-11 ,theparallelperi-eventhistogram,showsthatthemodelsalsocaptureratneuronsthatdramaticallyreducetheirringrateduringmovement. Figure4-11. Parallelperi-eventhistogramforratneuraldata Tables 4-9 and 4-10 showtheeffectofrandomlyselectingtheexperts.NoticetheresultsinthesetablesarepoorincomparisontoTables 4-7 and 4-8 .TheperformancereductionislesspronouncedforthedatacollectedfromtheRatthanfromtheMonkeyfoodgraspingtask.Thediscrepancyislikelyduetotheavailablenumberofchannelsthatarerandomlyselected.Sincethemonkeydataiscollectedfromagreaternumberofchannels,thereisahigherlikelihoodofselectinglessimportantneuronsfortraining.Figure 4-12 presentstheresultsfromanexpertaddingexperimentinwhichthebestrankedexpertsareaddedonebyonetotheensemblevote.ThegureshowsthatastheBM-HMMchainsareadded,aninterestingresultemerges,theerrorratequicklydecreasesbelowtheIC-HMMerrorwhenappliedtotheratneuraldata.TheLM-HMMhasanevenfasterdropinerror,butdoesnotachievethesameresultastheBM-HMM.Similartotheresultsforthemonkeyfoodgraspingtask,theboostedmixtures 70 PAGE 71 Classicationresults(randomBM-HMMselectedchannels) #channels %correct IC-HMM 16 62.5% IC-HMM 6 56.5% 6 56.9% 16 61.8% Linear 6 55.1% Table4-10. Classicationresults(randomLM-HMMselectedchannels) #channels %correct IC-HMM 16 62.5% IC-HMM 6 55.4% 6 54.3% 16 61.8% Linear 6 54.8% andlinkedmixturesareexploitingmoreusefulandcomplimentaryinformationforthenalensemblethanthesimpleIC-HMMfortheRatlevertask. Figure4-12. Ratexpertaddingexperiment 71 PAGE 72 4.3.1PopulationVectorsThelasttwosectionsdemonstratedthatpartitioningtheinputisbenecialwithsimplego-no-goexperimentsforbothanimalexperiments(ratandmonkey).Thefocusnowturnstomorecomplicatedmotionprimitivesotherthanmovementandrest.Toaccomplishthisgoal,itisrstimportanttounderstandwhatinformationtheneuronsprovideaboutthedifferentarmkinematics.Specically,motorcortexneuronsareknowntoinitiatemotorcommandsthatthendictatethelimbkinematics.Thereforethissectionwillfocusonmovementdirectionsinceitisthemostnaturalwaytonavigatetoagoal[ 9 ].Thepopulationvectormethod[ 14 ]wasoneoftherstmethodsthattriedtoaddressthecomplicatedrelationshipbetweenmovementdirectionandmotorcorticalactivity.Thismethodlinkshighneuralmodulationwithpreferredmovementdirection.Inordertoimplementthepopulationvector,tuningcurvestatisticsarecomputedforeachoftheneurons.Thesetuningcurvesprovideareferenceofactivityfordifferentneurons.Inturn,theneuralactivityrelatestoakinematicvector,suchashandposition,handvelocity,orhandacceleration,oftenusingadirectionoranglebetween0and360degrees.Adiscretenumberofbinsarechosentocoarselyclassifyallthemovementdirections.Foreachdirection,theaverageneuralringrateisobtainedbyusinganon-overlappingwindowof100ms.Thepreferreddirectioniscomputedusingcircularstatisticsas 4-13 showsanexamplepolarplotoffoursimulatedneuronsandtheaveragetuninginformationwithstandarddeviationacross100MonteCarlotrialsevaluatedfor16minduration.Thecomputedcircularmean,estimatedastheringrateweighteddirection,isshownasasolidredlineonthepolarplot.Thegureclearly 72 PAGE 73 Figure4-13. Neuraltuningdepthoffoursimulatedneurons Afterpreferreddirectionsareacquired,thevectorsofanglesfrompredictingneuronsareaddedtocreatethetrajectoryinthepopulationvectormethod.Althoughtheresultsarefarfromgreat[ 14 ],thistypeofanalysismaylendsomeinsightintoabettersegmentationoftheinputspace(relativetoangulardirections). 73 PAGE 74 Histogramof30angularvelocitybins Sinceangularvaluesexistontherealline,itisnecessarytoquantizeorbintheangles.Figure 4-14 showsthehistogramforthe30differentangularbinsandthenumberofexamplesperbin(i.e.thepolarplotisstretchedhorizontally).Eachbinrepresentsarangeofangularvelocities(12).Thegurealsopresentstwodistinguishablepeaksinthehistogram,thisisduetomonkey'shandmovementmakingpredominantlydiagonalmovements(gure 4-15 ). 4-16 providesacomparisonbetweenthetuningcurvesoftheneuronsdrawninparallel.Essentiallyeachneuron'stuningdepthisdrawnhorizontally(x-axis)ratherthanonapolarplot(angularbinsstartfromleft0toright360).Thedepthsarethenplottedin 74 PAGE 75 2Dangularvelocities parallelalongthey-axis(185neurons).Thedarkerpixelsindicateahigherringrateforaparticularangle.Sinceallthetuningdepthsarenormalized,thegureclearlyshowsthatsomeneuronsaretunedtodifferentanglesrelativetootherneurons.Figure 4-16 Bisaplotofthemaximumdepthataparticularangularbin.Interestingly,Figure 4-16 Bshowsthatasmallsubsetofneuronshaveaveryhighringrate(normalized)duringparticularangles.Interestingly,someneuronsmodulateacrossmultiplebins(i.e.awiderangeofangles). 75 PAGE 76 A.ParalleltuningcurvesB.Winningneuronsforparticularangles theringrateforeachneuron(withGaussianassumption).Departingfromexperimentone,theprobabilityofeachangularbinistreatedasamarginalforeachneuron(ateachtimeinstance).Thelargestjointprobabilityisthenfoundforeachangletobasetheclassicationdecision.Forbothexperiments,classicationperformanceistheonlymetricconsidered.Unfortunatelytheresultsfrombothexperimentsarepoor.Inexperimentone;theangularbinscouldonlybecorrectlyclassied10%ofthetime.Similarly,experimenttwoproducespoorresults.Additionally,classicationdoesnotsignicantlyimprovewhenmorequantizedbinsareused.Figure 4-17 showsthehistogramfor10Bins(36).Figures 4-18 A,and 4-18 Bdemonstratesimilartuningbutatamorequantizedlevel.NoticeinFigure 4-18 Bthatfewerneuronsarecoveringalloftheangles(whichistobeexpectedwithlargerquantizedbins).Overall,theseresultsshowthatsimplemodulationsarenotenoughtoclassifytheneuralinputandthatmorecomplicatedmodelingstructuresarenecessary. 76 PAGE 77 Histogramof10angularbins Figure4-18. A.ParalleltuningcurvesB.WinningneuronsforparticularAngles angularvelocities.Althoughsegmentingtheneuralinputisknowntoimprovetrajectoryreconstructionwithmultiplemodelsonfoodreachingtasks,thismethodologyhasnotbeentestedwithcontinuous2Dtrajectories.Byusingangularvelocitiesforsegmentation,theLM-HMMsandDC-HMMSshouldcapturetheneuronsthataremodulatingforparticularangles.Withthesemodelsisolatedforaparticularangular 77 PAGE 78 Table4-11. CorrelationcoefcientusingdifferentBMImodels CC(X) CC(Y) .67:03 .67:02 .66:02 .68:03 .70:02 .70:03 .71:03 .71:03 .65:03 4-11 showsthecorrelationcoefcientsforthedifferentBMImodelsthathavebeenusedonthisparticularcursorcontroldataset.Sincethefollowingexperimentsusethesamedataset,itisappropriatetocomparethesecorrelationcoefcientstothoseproducedbyourmethods.OfparticularinterestistheNon-LinearMixtureofCompetitiveLinearModels(NMCLM)sincethemethodologyforconstructingthetrajectoryissimilartoabovediscussedmultiple-modelapproach.Specically,theNMCLMdividestheinputspaceandappliesaswitchingmechanismtoselectawinningWienerltertoconstructthetrajectoryinapiecewisefashion[ 19 ]. 78 PAGE 79 4-11 .ThecorrelationcoefcientofthereconstructedtrajectoryprovidesametricandbaselinecomparisonfortheunsupervisedresultsinChapter5.Fortheexperimentsbelow,atrainingsetof5000samplesisusedalongwithatestsetof3000samples.Trainingwiththeclassesisatwo-stepprocess.FirsttheLM-HMMandDC-HMMaretrainedonthesegmenteddataandthentheparametersarefrozen.ThenthelinearmodelsaretrainedwithneuraldatathathasbeenclassiedasaparticularclassbytheLM-HMMorDC-HMM.Theweightsarethenfrozenforthelinearmodels.Duringtesting,theLM-HMMandDC-HMMarecomputedwiththetestsetandswitchestheinputtothecorrespondinglinearlterwhichthencomputestheoutputtrajectory.Asmentioned,theangularbinscanbequantizedintomanyangularbins,throughempiricaltestingfourclassesorangularbinswereselected.Withrespecttotheothermodelsusedonthiscursorcontroltask,Table 4-12 demonstratesthatagaintheDC-HMMisnotquiteasgoodastheLM-HMMbutbotharestillbetterthanthesingleWienerlterandTDNN(whiledramaticallybetterthantheNMCLM).Figure 4-19 showsthereconstruction(red)versusthetruetrajectories(blue)ofthemonkeycursorcontroltask.Overall,thereissomebenetinusingtheDC-HMM.Perhapsthesubduedresultsareduetoanindependentsetofneuronsacquiredinthedataset.Withsuchdata,theDC-HMMwouldnotbeasmuchbenetsinceitcouldnotexploitasmuchinformation. 79 PAGE 80 4-20 showstheViterbistatetransitionsforthedifferentneuronschosenfortheDC-HMM.AlongtheY-axisarethedifferentneuralstates(24total:fourstatesforeachsixneurons).Eachsubplotrepresentoneofthefourdifferentclasses.Noticehowdifferentneuronsarepreferredoverotherswithrespecttoeachclass.Someneuronspredominatelytransitiontothemselvesratherthanotherneurons.Thesepredominanttransitionswerealsoobservedinthelastsectionwiththefoodgraspingtask.Unfortunately,discerningarepeatingpatternisdifcultwiththisdatasetsincethecursortaskisnotrepetitivelikethefoodgraspingtask.Figure 4-21 furtherconrmsthatmostoftheneuronsareonlycoupledtothemselvesmoresothantheotherneurons.Thegureshowsthefourdifferentclassesandthecouplingcoefcientsbetweenneurons. Table4-12. CorrelationcoefcientusingDC-HMMon2Dmonkeydata CC(X) CC(Y) .74:08 .79:04 .66:02 .65:03 .67:03 .67:02 PAGE 81 Truetrajectoryandreconstructedtrajectory(DC-HMM) Figure4-20. Hiddenstatetransitionsperclass(cursorcontrolmonkeyexperiment) 81 PAGE 82 Couplingcoefcientbetweenneuronsperclass(cursorcontrolmonkeyexperiment) 82 PAGE 83 13 35 ]andthattheswitchingbehaviorisexploitableforBMIs.Unfortunately,topartitiontheneuralinputspace,a-prioriclasslabelsareneededforseparation.However,underrealconditionswithparaplegics,therearenokinematiccluestoseparatetheneuralinputintoclasslabels(orclusters).Currently,mostofthebehavinganimalsengagedinBMIexperimentsarenotparalyzed,allowingthekinematicinformationtobeusedfortrainingthemodels.ThisAchillesheelplaguesmostBMIalgorithmssincetheyrequirekinematictrainingdatatondamappingtotheneuraldata.Sincekinematiccluesarenotavailablefromparaplegics,neuraldatamustbeexclusivelyusedtondaseparation.Findingneuralassembliesorstructuresmayofferasolution.Thehypothesisarguedthroughoutthisdissertationisthattherearemultipleneuralstructurescorrespondingtomotionprimitives.Initialsupervisedresultssupportthishypothesis[ 16 ].Thereforethegoalistondamodelthatcanlearnthesetemporal-spatialstructuresorclustersandsegmenttheneuraldatawithoutkinematiccluesorfeatures(i.e.unsupervised).Inthischapter,theLM-HMMandDC-HMMmodelsarecombinedwithaclusteringmethodologyinordertoclusterneuraldata.Thesemodelsarechosenfortheirabilitytooperatesolelyintheinputspaceandtheirabilitytocharacterizethetemporalspatialspaceatareducedcomputationalcost.Themethodologydescribedinthenextsectionwillexplainhowthemodelslearnthe 83 PAGE 84 74 ].Thebipartitegraphview(Figure 5-1 )assumesasetofNdataobjectsD(e.g.,exemplars,representedbyS1;S2;::;SN,andKprobabilisticgenerativemodels(e.g.,LM-HMMsorDC-HMMs),1;2;:::;K,eachcorrespondingtoaclusterofexemplars(i.ewindowsofdata)[ 75 ].Thebipartitegraphisformedbyconnectionsbetweenthedataandmodelspaces.Themodelspaceusuallycontainsmembersfromaspecicfamilyofprobabilisticmodels.Amodelycanbeviewedasthegeneralized'centroid'ofclustery,thoughittypicallyprovidesamuchricherdescriptionoftheclusterthanacentroidinthedataspace.AconnectionbetweenanobjectSandamodelyindicatesthattheobjectSisbeingassociatedwithclustery,withtheconnectionweight(closeness)betweenthemgivenbythelog-likelihoodlogp(Sjy).Astraightforwarddesignofamodel-basedclusteringalgorithmistoiterativelyretrainmodelsandre-partitiondataobjects.Essentially,clusteringisachievedbyapplyingtheEMalgorithmtoiterativelycomputethe(hidden)clusteridentitiesofdataexemplarsintheE-stepandestimatethemodelparametersintheM-step.Althoughthemodelparametersstartoutaspoorestimates,eventuallytheparametersconvergetotheirtruevaluesastheiterationsprogress.Thelog-likelihoodsareanaturalwaytoprovidedistancesbetweenmodelsasopposedtoclusteringintheparameterspace(whichisunknown).Basically,duringeachround,eachtrainingexemplarisre-labeled 84 PAGE 85 Bipartitegraphofexemplars(x)andmodels bythewinningmodelwiththenaloutcomeretainingasetoflabelsthatrelatetoaparticularclusterorneuralstatestructureforwhichspatialdependencieshavealsobeenlearned.Thedependencystructureislearnedduringtheinner-loopoftheLM-HMMorDC-HMMtraining.Settingtheparametersisdauntingsincetheexperimentermustchoosethenumberofstates,thelengthoftheexemplar(windowsize)andthedistancemetric(e.g.log-likelihood).Toalleviatesomeofthesemodelinitializationproblems,previousparametersettingsfoundduringearlyworkareusedfortheseexperiments[ 16 ].Specically,ana-prioriassumptionismadethattheneuralchannelsareofthesamewindowsizeandsamenumberofhiddenstates.Theclusteringframeworkisoutlinedbelow:LetdatasetDconsistofNsequencesforJneuralchannels,D=S11;:::;SJN,whereSjn=(Oj1;:::OjT)isasequencesofobservableslengthTand=(1;:::K)a 85 PAGE 86 1.RandomlyassignKlabels(withK PAGE 87 76 77 ].Twosimulatedindependentneuraldatasetsaregeneratedinthefollowingexperiments.Onedatasetcontainsfourneuronstunedtotwoclasses(Figure 5-2 )andaseconddatasetcontainseightneuronstunedtofourclasses(onepairofneuronsperclass).Tocreatethesedatasetsrst,velocity,timeseriesisgeneratedwith100Hzsamplingfrequencyand16minduration(1000000samplestotally).Specically,asimple2.5kHzcosineandsinefunctionisusedtoemulatethekinematics(X-YVelocities)forthesimulationexperiments.Thentheentirevelocitytimeseries(forbothdatasets)ispassedthrougha(LNP)modelwiththeassumednonlineartuningfunctioninEquation 5 87 PAGE 88 79 ].Afterallthesamplesaregeneratedforeachrespectiveclass,theexemplarsfromeachclassarearticiallyplacedinanalternatingpattern.Inthefollowingdependentneuralsimulations,fourneuronswerecreatedwitheachclassproducing5000samples.Thedatasetsalsocontain100channelsoffakeneuronsinordertoassesstherobustnessofthemodels(asexplaininpriorsections). 88 PAGE 89 Neuraltuningdepthoffoursimulatedneurons 89 PAGE 90 ClusteringwiththeLM-HMM.Figure 5-3 demonstratestheclusteringresultsusingtheLM-HMMonthetwo-classsimulateddatasetconsistingofindependentneurons.Forthisparticularexperiment,themodelparametersareset(duringtraining)fortwoclasses(k=2)whichisequaltothetruenumberofclassesinthesimulationdata.Additionally,theclasslabelsalternatesincetheyrepresentthealternatingkinematics(shownatthebottomofgure).Asseenfromthegure,themodelisabletocorrectlyclusterthedatainarelativelysmallnumberofiterations(threetofour).Fortherstiteration,eachexemplarinthefulldatasetisrandomlyassignedtooneoftheclusters(indicatedbygreenandbluecolors).Fortheremainingiterations,apatternstartstoemergethatlookssimilartothealternatingkinematics.Althoughthekinematics(cosineandsinewave)areshownbelowtheclasslabels,theclusteringresultswereacquiredsolelyfromtheinputspace. Figure5-3. LM-HMMclusteriterations(twoclasses,k=2) 90 PAGE 91 5-4 showstheclasstuningpreferencewhenthemodelisinitializedwithrandomdata.Theterm'tuningpreference'referstotheangularpreferenceoftheparticularclass(orcluster)label.Thequantityiscalculatedthesamewayasinneuraltuning,exceptthatthedataiscollectedfromthesamplesthathavebeenlabeledbyaparticularclass(i.ecircularstatisticsarecalculatedonthekinematicsfromclass3ratherthananeuron).Thisgureclearlyshowsthatbeforemodelingtherandomclasslabelinghasnotintroducedpreferencesforanyparticularangle. Figure5-4. Tuningpreferencefortwoclasses(initialized) Figure 5-5 showstheangularpreferenceoftheclassesafterclustering.Overlaidinblueistheoriginalangulartuningsofsomeoftheneurons.Clearlythemodelisabletosuccessfullyndtheseparationinneuralrings,sinceeachclassisrepresentedbyadifferentangulartuning(similartorespectiveneurons).Althoughangularvelocityshowsitselftobeausefulkinematicfeaturetoseparatetheclusters,themodelsarenotsolelyrestrictedtothistypeoffeature.Onlyinthisexperimentdidthemostprevalentfeatureappeartobevelocity.Inlaterexperimentswithsimulatedandrealneuraldata,thiskinematicfeaturewillnotbethemostobvious.ForthesimulationinFigure 5-6 ,theLM-HMMisusedtoclusterafour-classsimulateddataset(k=4).Again,thecorrectnumberofclustersk=4issetduringtrainingtomatchthetruenumberofclassesintheinputdata(anotheroscillating 91 PAGE 92 Tunedclassesafterclustering(twoclasses) pattern).Thegureshowsafewissueswithshrinkageandexpansionwithrespecttotheclasslabels.Inotherwordssomeoftheclasslabelsareextendedorshortoftheactualsize.Theseeffectsareduetothetemporal-spatialdata(notstaticclassication)andthemodel'spropensitytostayinaparticularstate.OverallthenalresultdemonstratesthattheclusteringmodelwiththeLM-HMMisabletodiscovertheunderlyingclusterspresentinthesimulatedindependentneuraldata.Figure 5-7 showstheclusteringontheinitialrandomlabelswhileFigure 5-8 showsthetunedpreferenceofthefourclassesafterclustering.Remarkably,themodelisabletodeterminetheseparationfromthefourclassesusingtheneuralinputonly.Next,theclusteringmodelistestedforrobustnesswhenthenumberofclustersisunknownorincreasednoiseisaddedtotheneuraldata.Aswithallclusteringalgorithms,choosingthecorrectnumberofunderlyingclustersisdifcult.ChoosingthenumberofclustersforBMIdataisevenmoredifcultsincetherearenoknownorestablishedgroundtruths(withrespecttomotionprimitives).Figure 5-9 ,illustrateswhentheclusteringmodelisinitializedwithfourclasses(k=4)despitethesimulationonlycontainingtwounderlyingclasses(orclusters)fortheinputspace.Again,theresultsaregeneratedwithinarelativelysmallnumberofiterations.Noticefromthegurethattheextratwoclasslabelsareabsorbedintothetwoclasses 92 PAGE 93 LM-HMMclusteriterations(fourclasses,k=4) showninthepreviousFigure 5-3 (alsoshownbelowthefourclasslabels).Interestingly,arepeatedpatternofconsistentswitchingoccurswiththeclasslabels(asindicatedbythepatternofcolorblocks).Specically,Figure 5-9 showsthatclass1precedesclass3andclass2,whencombined,theycorrespondtoclass1inFigure 5-3 ,whileclass4ingure 5-9 correspondstoclass2inFigure 5-3 .Remarkably,theneuraldatafromsuchasimplesimulationiscomplicatedyettheclusteringmethodndstheconsistentpatternofswitching(perhapsindicatingthatthesimpleclassesarefurtherdivisible).Tofurthertestrobustness,randomspikesareaddedtotheunbinnedspikedtrainsoftheearliertunedneurons(ofFigure 5-2 ).Specically,uniformlyrandomspikesaregeneratedwithaprobabilityofspikingevery1ms.Figure 5-10 showstheclassicationperformanceastheprobabilityofringisincreasedfroma1%chanceofspikingto16%chanceofspikingin1ms.Interestingly,performancedoesnotdecreasesignicantly.Therobustnessisduetothetunedneuronsstillmaintainingtheirunderlyingtemporalstructure.Figure 5-11 showsthetuningpolarplotsofthefouroriginalneuronswiththe 93 PAGE 94 Tuningpreferenceforfourclasses(initialized) addedrandomspikes.Althoughthisgureshowsthattuningbroadensacrossmanyangularbins,therandomspikesdonothaveatemporalstructure.Thereforetheydonotdisplacethetemporalstructureofthetunedneuronssignicantly(asindicatedbyonlyasmallchangeinperformance).Pleasenotethatincreasingtheprobabilityofrandomspikesto16%every1msputsthespikingbeyondtherealisticringrateofrealneurons.Asexplained,thedifferentarticialneuronsaremodulatedsothattheirtuningdepthmonotonicallyincreased(i.e.setfrom1to4).TheLM-HMMclusteringsuccessfullyselectstheneuronsinthecorrectorder(respectivetotuningdepth)fromthe100randomneuralchannels.Theresultisthesamewhenthetunedneuronsarecorruptedwithrandomspikenoise. 94 PAGE 95 Tunedclassesafterclustering(fourclasses) Giventhattheclusteringmodelstillachievesgoodperformanceduringnoise,itisimportantensurethatthesimulationisnottoosimplistic.Therefore,twosurrogatedatasetsaregeneratedfromthesimulationdata.Therstsurrogateisgeneratedbyrandomizingthespatialrelationshipsbetweentheneurons.Specically,ateachtimebinthebincountsforeachneuronarerandomlyswitchwithanotherchannel.ThisprocessisrepeatedthroughthelengthofthedatasetwherethespatialrelationshipofbinNisdifferentthanbinN1.Figure 5-12 showstheclusteringresultsonsuchasurrogatedataset.Theclusteringmodelcorrectlyfailstoclusterthedatasincethespatialinformationisruined. 95 PAGE 96 LM-HMMclusteriterations(twoclasses,k=4) Figure5-10. Classicationdegradationwithincreasedrandomrings 96 PAGE 97 Neuraltuningdepthwithhighrandomringrate Figure 5-13 showsthetunedpreferenceoftheclustersafterclustering.Thisgurecorrectlyshowsthatthereisnotunedpreferencesincethemodelfailed(asishoped).Thesecondsurrogateisgeneratedbyrandomizingthetemporalrelationshipsbetweentheneurons.Specically,ateachtimebinthebincountsforalloftheneuronsarerandomlyswitchedwithanotherbinintime(keepingthesamechannel).Thisprocessisrepeatedthroughthelengthofthedatasetwherethetemporalrelationshipisdestroyedbutthespatialrelationshipiskeptintact.Figure 5-14 showstheclusteringresultsonsuchasurrogatedataset.Itcorrectlyfailstoclusterthedatasincethespatialinformationthroughtimeisruined.Figure 5-15 showsthetunedpreferenceofthe 97 PAGE 98 Surrogatedatasetdestroyingspatialinformation Figure5-13. Tunedpreferenceafterclustering(spatialsurrogate) clustersafterclustering.Thisgurecorrectlyshowsthatthereisnotunedpreferencesincethemodelfailed(asishoped). 98 PAGE 99 Surrogatedatasetdestroyingtemporalinformation Figure5-15. Tunedpreferenceafterclustering(temporalsurrogate) 99 PAGE 100 5-16 demonstratestheclusteringresultsusingtheDC-HMMonthetwo-classsimulateddataset.Forthisparticularexperimentthenumberofclustersk=2.Thegureshowsthatthemodelisabletocorrectlyclusterthedatainarelativelysmallnumberofiterations(threetofour).Thegurealsoshowsthatontherstiterationthedifferentclasslabelswererandomlyassignedtothetwoclasses(indicatedbygreenandbrowncolors)starttoconvergetoapatternsimilartothekinematicsfromwhichtheinputwasderived(bottomofgurecosineandsinewave).Althoughthekinematicsareshownbelowtheclasslabels,theclusteringresultsareacquiredsolelyfromtheinputspace. Figure5-16. DC-HMMclusteringresults(class=2,K=2) Figure 5-17 illustratesthehiddenstatetransitionsfortheDC-HMMonthesimulateddataset.Interestinglyarepeatingpatternmatchesthecorrespondingkinematics.Additionallythegureshowsthatpairsofneuronsareactivelyinvolvedinthehiddenstatespace.Thispairingisexpectedsincethesimulatedneuronsareinpairsforthe 100 PAGE 101 5-18 whereonlyfourneuronshaveactivecouplings.Furthermore,thecorrectcouplingisobservedbetweenneurons. Figure5-17. DC-HMMclusteringhiddenstatetransitions(class=2,K=2) Figure5-18. DC-HMMclusteringcouplingcoefcient(class=2,K=2) Figure 5-19 presentsthelikelihoodsgeneratedateachiterationoftheclusteringround.Thetwolikelihoodguresareforeachclass.Thisgureclearlyshowsa 101 PAGE 102 Figure5-19. DC-HMMclusteringlog-likelihoodreductionduringeachround(class=2,K=2) ForthesimulationinFigure 5-20 ,thecorrectnumberofclustersk=4matchestheunderlyingnumberofclassesintheinputdata.Theclusteringresultsdemonstratewhenfourclassesareactuallypresentintheneuraldata.Themodelcorrectlymatchestheunderlyingclusterspresentintheneuraldata.Unfortunately,theDC-HMMclusteringproducesafewmoreerrorsthantheLM-HMMclusteringofthesamedata.Figure 5-21 showsthehiddenstatetransitionsfortheDC-HMMonthesimulateddatasetwithfourclasses.Interestingly,repeatingpatternsareobservedthatcorrespondtothekinematicsbutnotasobviousasthetwoclassversion.Additionallyfromthisgure,pairsofneuronsareactivelyinvolvedwithregardtohiddenstatespace.Thispairingisexpectedsincethesimulatedneuronsareinpairsforthetwoclasseswithouttheextranoiseneurons.ThisresultisfurtherconrmedwithFigure 5-22 wherethecorrectcouplingsareobservedbetweenneurons. 102 PAGE 103 DC-HMMclusteringsimulatedneurons(class=4,K=4) 103 PAGE 104 DC-HMMclusteringhiddenstatespacetransitionsbetweenneurons 104 PAGE 105 DC-HMMclusteringcouplingcoefcientbetweenneurons(perClass) 105 PAGE 106 73 ],isatypeofunsupervisedneuralnetwork.Thegoalofthismodelistolearninanunsupervisedwaytherepresentationoftheinputspace.SOMarealsodifferentfromotherneuralnetworkssincetheyuseaneighborhoodfunctiontopreservethetopologicalpropertiesoftheinputspace.FormoredetailsontheSOMpleaseseeAppendixC.Fortheexperimentsdescribedbelow,variousnumbersofprocessingelements(PE's)weretestedempirically.Thebestcompromiseincomputationalcomplexityandperformanceresultedinusing25PE's.Theinitialstepsizeusedfortheorderingphasewas.9whiletheconvergingormappingphasestartedwitha.02stepsize.AlthoughthestaticversionoftheSOMwastested(whichfailedonthistypedata),timeembeddingwasaddedtotheSOM(AppendixC)inordertoprovidethebestcomparison.Figure 5-23 showstheSOMresultsonthetwoclasssimulationwithindependentneurons.Thegureshowsasuccessfulclusteringoftheoscillatingclasses(asdiscussedearlier).Classicationperformancewas96.4%.Overall,thisclusteringmodelisabletoclusterthesimplisticsimulation.InordertotesttherobustnessoftheSOM,noisewasaddedtothesimulation.Figure 5-24 showswhentheprobabilityofrandomlyringisincreasedto16%pertimebin(whichisbeyondrealneuralringrates).Clearlythegureandclassicationresults(92.3%)demonstratethattheSOMissuccessfulandrobustenoughtoclusterthesimulationwithnoisyindependentneurons.Next,spatialandtemporalsurrogatesareusedtofurthertesttherobustnessoftheSOM.Figure 5-25 presentstheclusteringresultswiththespatialsurrogatedata.Theclassicationresultsare85.58%whichiscompletelyincorrectsinceallofthe 106 PAGE 107 5-26 ).Bydefault,thespatialsurrogaterandomlyplacestheringratesofthedifferentneuronswithotherneurons.ThisisinterestingsincetheSOMisfailingandisarticiallycapturingstructurethatdoesnotexistandissimplyselectingtheburstingactivityfromsomeoftheneurons.ThisincorrectresultmayhelptoexplainwhythesomeofthepreviousBMIresultswithnon-linearltersworkedbetterwiththeballisticfoodgraspingtaskssincethosetypesofneuronswouldbemoremodulated(i.e.bursting)atcoincidingpointstothemovement.Incontrast,theneuronsarealwaysringandmodulatingasinthecursorcontroltasktherebydecreasingtheperformanceofthelinear/non-linearmodels(whilethegraphicalmodelsdonotsufferasmuch).Finally,temporalsurrogatedataisclusteredwiththeSOM.InthisinstancetheSOMfails(asexpected)atclusteringthisdataset(gure 5-25 ).Theclusteringresultswere Figure5-23. SOMclusteringonindependentneuraldata(2Classes) 107 PAGE 108 SOMclusteringonindependentneuraldatawithnoise(2Classes) Figure5-25. SOMclusteringonindependentneuraldataspatialsurrogate(2classes) 108 PAGE 109 NeuralselectionbySOMonspatialsurrogatedata(2classes) justaboverandom(53.04%),whichisvalidforadatasetthathastemporalstructuredestroyed. 109 PAGE 110 SOMclusteringonindependentneuraldatatemporalsurrogate(2classes) 110 PAGE 111 5-28 showsthespikeoutput(100msbins)fromthedependentneurons.Inthegure,fourdependentlygeneratedneuronsand100fakeneuronsareshownwithdarkercolorsindicatingahigherringrate.Theguredoesnotprovidediscerniblepatterns,eventhoughanalternatingpatternisunderlyingthedata(similartooriginalindependentneurons).Figure 5-29 providesfurtherevidencethatsimplisticpatternsdonotexistsincetheneuronsarenotspecicallytunedtoanyparticularangle. Figure5-28. Outputfromfoursimulateddependentneuronswith100noisechannels(Class=2) Figure 5-30 showstheclusteringresultsusingtheLM-HMMwiththeclusteringmethodology.Despitenotobservinganyvisualoscillatingpatternintheneuraldata,themodelisabletocorrectlycluster(asseeninthepattern).Thegurealsodemonstratesthatonlyasmallnumberofclusteringiterationsareneededtodiscernthepattern.Withrespecttothe104neuronsthemodelwasabletocorrectlyidentifythefourneuronsthatwerepertinenttotheclusters. 111 PAGE 112 Neuraltuningfordependentneuronsimulation Interestingly,Figure 5-31 showsthattheDC-HMMisalsoabletocorrectlyclustertheneuraldatawithdependencies.ThemostinterestingresultsareshowninFigure 5-32 .Thegureshowstheclusteringresultsfromthetime-embeddedSOM.ClearlytheSOMisnotsuccessfulinclusteringtheoscillatingclasses.Theclassicationresultsareslightlyaboverandom(53.98%).TheSOMisnotcapturingthepatternsinceindividualneuronsarenotburstingormodulatingwithsignicantincreasesinringrate.Althoughastate-basedmodelgeneratedthedata,thissimulationprovidesmoreevidencethatthegraphicalmodelshavetheabilitytocapturethecommunicationbetweenneuronsthroughtime. 112 PAGE 113 LM-HMMclusteringsimulateddependentneurons(class=2,K=2) Figure5-31. DC-HMMclusteringsimulateddependentneurons(class=2,K=2) 113 PAGE 114 SOMclusteringondependentneuraldata(2classes) 114 PAGE 115 5.3.1RatExperimentsFortherstexperiment,5000datapointsofthesingleleverpressexperimentareusedforclustering.Withtheinitialclusteringround,allofthedatapointsarerandomlylabeledforthetwoclasses(leverpressandnon-lever-press).Theseexperimentscallformanyparameterstobeinitialized.Theseinclude1.Observationlength(windowsize)2.Numberofstates3.NumberofclusteringroundsTheobservationlengthwasvariedfrom5to15whichcorrespondsto.5secondsto1.5seconds.Thenumberofhiddenstateswerevariedfrom3to5.Finally,thenumberofroundswerevariedfrom4to10.Afterexhaustingthedifferentcombinations,anobservationlength(timewindow)of10wasselectedalongwith3hiddenstatesand6rounds.Theseparameterswerekeptthesameforallneuralchannels.Figure 5-33 ,illustratesthatthemodelprovidesreasonableclusteringofthetwoclassesinthesingleleverpress.Withrespecttoclassicationperformance,themodelisabletocorrectlyclassifyeachclassaround66%ofthetime.Remarkably,despite66%beingalownumber,theunsupervisedclusteringmodelachievesaclassicationperformancethatrivalsthesupervisedclassicationresults.Ofcoursetocompareclassicationstheclasslabelsmustbeassignedsincetheyareunknownafterclustering.Thelabelsareassignedbasedonkinematicfeatures,likealeverpress,andthedifferentpriors(leverpresseshavefarlesssamplesthannot-leverpresses).Undernormalclustering(withoutclassicationcomparisons),theclasslabelsareunknown.Mostlikelytheexperimentalsetupwillneedtofocusonsimpletasksbywhichthepatientexpressestheirdesiredgoals(movearmleftetc).Thispatient-baseddirectionwouldallowthedifferentclasslabelstobeappropriatelyassigned. 115 PAGE 116 Ratclusteringexperiment,onelever,twoclasses Figure 5-34 presentsaslightlyexpandedpictureofgure 5-33 .Noticehowtheclusteringfailsincertainlocations.Thesefailuresmaybeduetotheactualexperimentsincetheratmovesaroundthecagewithoutbeingrecorded.Inthenextexperiment,10000datapointsofthetwo-leverratexperimentareusedforclustering.Fortheinitialclusteringround,allofthedatapointsarerandomlylabeledforthetwoclasses.Thesameparametersselectedinthepreviousexperimentareusedforthisexperiment.Unlikethepreviousexperiment,thisexperimentincludesthetimelocationforcue-signalsandrewards.Figure 5-35 showswherethecuesignalsandrewardsarelocatedaswellastheleverpresses(redisleft,greenisright).Forexample,onthefthcue-signaltheratwassupposedtopressleftbutinsteadpressedright(asindicatedbycolorsontheplot).Figure 5-35 showsconsistentandrepeatingclusteringpatterns.Unfortunately,theresultsarenotasgoodastheexperimentwithasingleleverpress.Thedifferencemay 116 PAGE 117 Ratclusteringexperimentzoomed,onelever,twoclasses Figure5-35. Ratclusteringexperiment,twolever,twoclasses 117 PAGE 118 Figure5-36. Ratclusteringexperiment,twolever,threeclasses Figure 5-36 showstheresultswhenclusteringthedataforthetwo-leverpressintothreeclasses.Interestinglytherearesomeconsistentresultsbutvisuallytheresultsaredifculttointerpret.Theconsistenciesinvolvethetransitionfromoneclasstoanotherclass(likeredtogreenorgreentoblue).Thesetransitionscoincidewhenakinematiceventisexhibited(i.eleverpressed).Additionally,thesetransitionsaresimilartowhatwasobservedinthesimulations.Figure 5-37 againshowssomeinterestingbehaviorwiththeclusteringresultswhenusingfourclasses.Thereareconsistencieswhenlookingatthetransitionsfromoneclasstoanotherbeforeandaftertheleverpresses.Unfortunately,therearen'teven3or4kinematiceventsthatareapplicableforclassication.Onlyqualitativeinterpretationisappropriateforthisdataset.Thislackofquantiablemetricsiswhythemonkeydataandsimulationsaremorepertinenttotestingthediscussedmethodologies. 118 PAGE 119 Ratclusteringexperiment,twolever,fourclasses Overall,theclusteringresultsfortheratdataaremixed.Infuturework,perhapsisolatingthetimedataaroundtheleverpresseswhentheratismorefocusedonthetaskcouldimprovemodeling.Althoughnotpresentedhere,preliminaryresultsindicatethisimprovementtobethecase. 119 PAGE 120 LM-HMM.Twoimportantquestionsmustbeansweredwiththefollowingexperiments.First,whattypeofclusteringresultsareobtained,i.e.arethererepeatingpatternscorrespondingtothekinematics.Second,howdoesthetrajectoryreconstructionfromtheunsupervisedclusteringcompareagainstthetrajectoryreconstructionfromsupervisedBMIalgorithms.Classicationperformanceisnotconsideredintheseexperimentssincetherearenoknownclassesbywhichtotest.Althoughangularbinsmightservethepurposeforsingleneurons,itisknownthatsomeneuronsdonotsimplymodulateforangularbins.Therefore,correlationcoefcientwillserveasthemetricandallowaconsistentcomparisonbetweentheresultsinthispaperandpreviouswork.MultipleMonteCarlosimulationswerecomputedtoeliminatespuriouseffectsfrominitialrandomconditions(classlabels,parameters,etc).Theparametersthatrequireinitializationinclude:1.Observationlength(windowsize)2.Numberofstates3.Numberofclusteringrounds4.NumberofclassesTodeterminetheparameters,theobservationlength(windowsize)wasvariedfrom5to15timebins(correspondingto.5secondsto1.5seconds).Thenumberofhiddenstateswasvariedfrom3to5.Whilethenumberofclusteringiterationsvariedfrom4to10.Afterexhaustingthedifferentcombinations,theparametersaresetto:anobservationlengthequalto5timebins,3hiddenstatesand6clusteringiterations(sincelessthan5%ofthelabelschanged).Theseparametersarethesameforeachneuralchannel.Themodelwasinitializedwithfourclassesafteranempiricalsearch(usingdifferentparametersets).Sincegroundtruthsareunknown,trajectoryreconstructionserves 120 PAGE 121 5-38 showsthelabelingresultsfromtheLM-HMMclustering.TheY-Axisrepresentsthenumberofiterationsfrominitialrandomlabels(top),tothenalclusteringiteration(bottom).Eachcolorintheclusteringresultscorrespondstoadifferentclass(fourinall).Thekinematics(xandyvelocities)areoverlaidatthebottomoftheimageforthiscursorcontrolexperiment.Figure 5-38 showsrepeatingpatternsforsimilarkinematicproles.Theserepetitiveclasstransitionswerealsoobservedinthesimulateddata.Figure 5-39 showstrajectoryreconstructionmatchesverycloselytotheoriginaltrajectorytherebyindirectlyvalidatingthesegmentationproducedbytheclusteringmethod(qualitatively). Figure5-38. LM-HMMclusteriterations(Ivy2Ddataset,k=4) Foraquantitativeunderstanding,thecorrelationcoefcient(CC)isawaytoshowiftheclusteringresultshavemerit.Interestingly,theCCresultsforthisunsupervisedclusteringareslightlybetterthanthesupervisednon-linearTDNNandlinearWienerlterasshowninTable 5-1 .Asexpectedrandomlabelingoftheclassesproduces 121 PAGE 122 ReconstructionusingunsupervisedLM-HMMclusters(blue)vs.realtrajectory(red) poorresultscomparedtoactualclustering.AdditionallytherandomlabelingresultsaresimilartoothersupervisedBMImodels.Asdiscussedearlier,thesimilarresultsareduetotherandomclustersprovidinggeneralizationofthefullspaceforeachlter(therebybecomingequivalenttoasingleWienerlter).Remarkably,Table 5-1 showsthatthecorrelationcoefcientproducedbytheunsupervisedLM-HMMclusteringisonlyslightlylessthanthecorrelationcoefcientproducedwiththesupervisedversionoftheLM-HMM.ThisresultisunderstandablesincethesupervisedversionoftheLM-HMMconsistentlyisolatestheneuraldatabasedonkinematicfeaturesthatwereimposedbytheuserinlabelingtheclasses. 122 PAGE 123 CorrelationcoefcientusingLM-HMMon2Dmonkeydata CC(X) CC(Y) .77:07 .65:04 .79:04 .66:02 .67:03 .67:02 .68:03 .70:02 .70:03 .71:03 .71:03 .65:03 PAGE 124 5-2 showsthatthecorrelationcoefcientontheunsupervisedDC-HMMclusteringreconstructionisbetterthanusingtheSOM,randomlabelingorasingleWienerlter.Unfortunately,theresultsarenotasgoodasthesupervisedversionoftheDC-HMMorLM-HMM.Supervisedresultsareexpectedtobebettersincethekinematicfeaturescanserveasclassesthatdemarcatepartition-ablepointsintheinputspacethatallowthemodeltospecialize(asopposedtogloballylearningtheclasslabels). Table5-2. CorrelationcoefcientusingDC-HMMon3Dmonkeydata CC .72:19 .70:18 .68:19 .68:20 5-40 demonstratesthatthereisacorrespondingclusteringpatterntothekinematics.Unfortunatelythereareareasoferrororatleastwhatcouldbeperceivedaserrorsincetherearenoknowngroundtruthstotestagainst.Nevertheless,theCCshowstobeslightlydegradedfromasupervisedversion.Thedegradedresultsareattributabletoalackofinformationbetweenchannels.Figure 5-41 showsthereareagainmoreindependentneuronsthandependent.PerhapstheLM-HMMexploitstheindependentneuronsbettersinceitbuildsaconsensusamongthechannelsratherthanmodelinganexplicitjointdistribution. 124 PAGE 125 DC-HMMclusteringonmonkeyfoodgraspingtask(2classes) Figure5-41. CouplingcoefcientfromDC-HMMclusteringonmonkeyfoodgraspingtask(2classes) Forthecursorcontrolmonkeydata,Table 5-3 illustratesthattheDC-HMMproducessimilarCCresultstothatoftheLM-HMM.Again,bothoftheseclusteringresultsarebetterthanthesupervisedversionsofthenon-linearTDNNandsinglelinearWiener 125 PAGE 126 5-42 .ThiswouldhelptoexplainwhytheWienerltersproducebetterresultsonthefoodgraspingtaskovercursorcontrol(i.e.independentneuronsaresimplymodulatingwithtask).Additionally,thesimulationswithdependentneuronsshowedthatthestate-spacemodelwerebetterthantheSOMatclusteringthistypeofdata. Table5-3. CorrelationcoefcientusingDC-HMMoncursorcontroldata CC(X) CC(Y) .71:09 .77:07 .65:04 .66:02 .68:03 .65:03 5-43 showstheeffectofaveragingtheringratesacrossatimewindowequaltotheobservationlength(vetimebins).Eachparticularclassproducesadifferentringpatternacrossthesixselectedchannels.Interesting,someofthechannelsremoreinthebeginningportionofthewindowwhileothersremoretowardstheend.Additionally,Figure 5-44 showsthecorrespondingaveragekinematicsofthefourclasses.Asmentionedearlier,themonkeymakesmostlydiagonalmovementsandthisisclearlyobservedinthisgure(asishoped). 126 PAGE 127 CouplingcoefcientfromDC-HMMclusteringonmonkeycursorcontroltask(4classes) Figure5-43. Averageringrateperclass(4classes,6neurons) 127 PAGE 128 Averagevelocityperclass(4classes,6neurons) 128 PAGE 129 35 ].ThentheclusteringmethodwascomparedtoconventionalBMIsignalprocessingalgorithmsonrealneuraldata.Although,trajectoryreconstructionisusedtoshowthevalidityoftheclusters,themodelcouldbeusedasfrontendforaco-adaptivealgorithmorgoal-orientedtasks(simpleclassicationthatparaplegicscouldselect,i.e.moveforward).Despitetheseencouragingresults,improvementsinperformanceareachievableforthehierarchicalclustering.Forexample,thegenerativemodelsinthehierarchicalclusteringframeworkmaynotbetakingfulladvantageofthedynamicspatialrelationships.AlthoughthehierarchicaltrainingmethodologydoescreatedependenciesbetweentheHMMexperts,perhapstherearebetterwaystoexploitthedependenciesoraggregatethelocalinformation.AsshownwiththecouplingcoefcientsandViterbistatepathsfromtheDC-HMMresults,theremaybeimportantdependenciessincedifferentneuralprocessesareinteractingwithotherneuralprocessesinanasynchronousfashionandthatunderlyingstructurecouldprovideinsightintotheintrinsiccommunicationsoccurringbetweenneurons.Asanalpoint,therewasaninterestingeffectfromtheexperiments(simulatedandrealneuraldata).Lookingcloselyatsomeoftheresults,consistenttransitionsoccurfromdifferentclassestootherclasses.Forexampletheremaybeaconsistenttransitionfromclass1toclass3andclass2toclass1.Investigatingthisphenomenonfurtherwouldbeinteresting.Perhapsthereisaswitchingbehaviorbetweenstationary 129 PAGE 130 130 PAGE 131 131 PAGE 132 35 ].Theresultswerejustiedwiththeimprovementintrajectoryreconstruction.Specically,thecorrelationcoefcientonthetrajectoryreconstructionservedasametrictocompareagainstotherBMImethods.Additionally,simulationswereusedtoshowthemodel'sabilitytoclusterunknowndatawithunderlyingdependencies.Thisisnecessarysincetherearenogroundtruthsinrealneuraldata.OveralltheworkdescribedinthisdissertationdemonstratedimprovedapproachestomodelingBMIs.Thisworkalsoaddressedthemostimportantproblemofpatients 132 PAGE 133 6-1 showstheBipartitegraphinwhichthemodelsarenowsequesteringthedatapoints.Clusteringthemodelstructuresorparametersrequiresastructurelearningalgorithm.Thesetypesofalgorithmsemploysearchingandscoringfunctionsinordertobuildthestructureofthemodel.Sincethenumberofmodelstructuresislarge(exponential),asearchmethodisneededinordertodecidewhichstructures 133 PAGE 134 Bipartitegraphofexemplars(x)andmodels toscore.Evenagraphicalmodelwithasmallnumberofnodescontainstoomanynetworkstoexhaustivelyscore.Agreedysearchcouldbedonebystartingwithaninitialnetwork(with/withoutconnectivity)anditerativelyaddingordeletinganedge,measuringtheaccuracyoftheresultingnetworkateachstage,untilalocalmaximaisfound.Alternatively,amethodsuchassimulatedannealingcouldguidethesearchtotheglobalmaximum.Iteratingthroughallofthestructuresiscomputationallyexpensive.Theproblemisfurthercomplicatedwhenmultiplestructuresmustbedeterminedformultipleclustersorclasses.Unfortunately,whentheclassesareundened(i.e.theexperimentsinthisdissertation),ndingthesestructuresisintractable.Thereforeasimpleiterativemethodmustbeemployedtoapproximatethestructuresforeachrespectiveclass.Ratherthanuseaconventionalsearchmethod,asinglegenerativemodelwillbetrainedoverthefulldataset.Thenastate-pathndingalgorithmsimilartoViterbiwillndthemostlikelypathsforeachdataexemplar.Thesepathsrepresenttheplausibledependencystructurebetweenchannelsthroughtime.Oncethesestructuresare 134 PAGE 135 i.e.theprobabilityofbeinginstateQ(c)attimet,giventheobservationsequenceO,andthemodelinchainc.Equation 6 canbeexpressedsimplyintermsoftheforwardvariables,i.e., since(c)t(i)accountsforthepartialobservationsequenceO(c)1O(c)2:::O(c)tandthecontributionsfromeachofthechannel'spreviousstatesThenormalizationfactorQCc=1(PN(C)j=1(c))T(j)makes(c)t(i)aprobabilitymeasuresothat 135 PAGE 136 Using(c)t(i),themostlikelystateqtattimetcanbesolvedindividually,as Byselectingthemostlikelystateonaparticularneuralchannelforeachtimestep,thesizeofthemodelstructureisgreatlysimplied.EssentiallyachainstructureGkisconstructedthroughtimeandacrossthechannels.SinceeachmodelGkhasthesamenumberofparameters,usingBICtoscorethemodelsisinappropriate.ThereforeadifferentapproachisnecessarytogroupthemodelsGk.Findingadifferentapproachisdifcultsincethereare(CN)Tstructureseveninthesimpliedformulationdiscussedabove.Onecluecomesfromlookingattheempiricalresults.Empiricallytherearefarfewerrealizationsofthestructuresk<(CN)T.Byfurtherlimitingthenumberofobservedmodelstogreaterthantwo,signicantlylessmodelsareempiricallyobserved(mergingthesingleobservationsintoasinglecluster).ThisledtoasimplehistogrammethodtoclusterthemodelsGk.Inordertofurtherreducethenumberofmodels,thetoptwomodelsobservedthemostarechosen(withtherestofthesamplesrelabeledasathirdclass).LetdatasetDconsistofNsequencesforJneuralchannels,D=S11;:::;SJN,whereSjn=(Oj1;:::OjT)isasequencesofobservableslengthT.SpecicallythefulldatasetisusedandallexemplarsStrainasinglemodel.Initially,theparametersarerandomizeduntiltheyconvergetoaninitialguess.Thenthelikelystatepathisfoundforeachtrainingexemplarandahistogramisbuiltfromthefoundstructures.Thealgorithmisasfollows: 136 PAGE 137 Independentneuralsimulationdata.Forthisparticularexperimentthenumberofclustersk=2.Empirically,thenumberofhiddenstateswasvariedfrom2to5withthenalselectionbeing3hiddenstatesforthemodel.Basedonpriorresultsasequencelengthof5ischosenalongwith2to6roundsofclustering.Figure 6-2 demonstratestheclusteringresultsusingtheDC-HMMonthetwo-classsimulateddataset.Thegureclearlyshowsthatthemodelisabletocorrectlyclusterthedata.Theclassicationresult(basedonpriorlabeling)is82%whichiscomparabletothelikelihood-basedclusteringinChapter5.Althoughthekinematicsisshownbelowtheclasslabels,theclusteringresultsareacquiredsolelyfromtheinputspace.Figure 6-3 showstheViterbipathsfortheDC-HMMonthesimulateddataset.Interestingly,therearerepeatingpatternsthatcorrespondtothekinematics.Additionallyfromthisgure,pairsofneuronsareactivelyinvolvedinthestatetransitionsinthehiddenstatespace.Thisisexpectedsincethesimulatedneuronsareinpairsforthetwoclasses. 137 PAGE 138 HiddenstatetransitionsDC-HMM(simulationdata2classes) noknowngroundtruthsfortheinputspace(everythinglikeangularbinsisimposedbytheuser)andunknownkinematicfeaturesmayberepresentedintheneuraldata.Inconsiderationofspuriousresults,multipleMonteCarlosimulationswereexecutedtoconrmempiricalresults.Additionally,therearemanyparameterstobeinitialized.Theseinclude: 1. Observationsequencelength(windowsize) 2. Numberofstates 3. Numberofclusteringrounds 4. NumberofclassesTheobservationlengthwasvariedfrom5to15timebins(100ms),whichcorrespondto.5secondsto1.5seconds.Thenumberofhiddenstateswasvariedfrom3to5.Finally,thenumberofclusteringiterationswasvariedfrom4to10.Afterexhaustingthe 138 PAGE 139 HiddenstatetransitionsDC-HMM(simulationdata2classes) differentcombinationstheparametersweresetasthefollowing:observationlengthof5,3hiddenstatesand6clusteringrounds(sincelessthan5%ofthelabelschanged).Table 6-1 showsthatthecorrelationcoefcientontheunsupervisedDC-HMMmodel-basedclusteringreconstructionisbetterthanusingrandomlabelingorasingleWienerlter.Unfortunately,theresultisonlyasgoodasthelikelihood-basedclustering.Onereasonforthisisthatbothclusteringtechniquesrelyonapproximations.Figure 6-4 showsthehistogramforthestatemodels.Clearlythereareareasonablenumberofmodelsfoundempiricallywithaparticularmodelobservedthemost(correspondingto Table6-1. CorrelationcoefcientusingDC-HMMon3Dmonkeydata CC .70:21 .72:19 .68:19 .68:20 PAGE 140 6-5 showssomeoftheactualViterbipathstakenbythemodel.They-axisrepresentsthedifferentmodelswhilethex-axisisthetimebins(oflengthve).Eachcolorrepresentsaparticularstateobservedbyacorrespondingneuron(similarcolorsrepresentthesameneuroninthesamestate). Figure6-4. HistogramofstatemodelsfortheDC-HMM(foodgraspingtask) Althoughtheunderlyingapproximationsleadthenalresultstobesimilar,Figure 6-6 thecomputed'sshowsthattheunderlyingstructureanddependenciescanbediscerned.Thisinformationmayproveusefulinabiologicalsettingorfuturefront-endanalysisfordifferentalgorithmsandneedsfurtherexploration. 6-2 showsthatthemodel-basedclusteringproducesslightlyworstresultsthanthelikelihood-basedclustering.Bothoftheseclusteringresultsareagainbetterthanthesupervisedversionsofthenon-linearTDNNandsinglelinearWienerlter. 140 PAGE 141 StatemodelsfortheDC-HMM(foodgraspingtask) Figure6-6. AlphascomputedperstateperchannelDC-HMM(foodgraspingtask) 141 PAGE 142 6-7 whichshowsthe'scomputedacrosschannelsandstateswhilethekinematicvariable(velocitydifferencebetweenthetwodimensions)isalsoplottedonthegure.Obviouslytherearedifcultiesincharacterizingtheunderlyingneuralstructurewithrespecttoakinematicstructure.Thisisduetohowcomplicatedarmkinematicsisexecutedwithmillionsofmusclebers. Figure6-7. Alphasacrossstate-spaceoftheDC-HMM(cursorcontroltask) Table6-2. CorrelationcoefcientusingDC-HMMon2Dmonkeydata CC(X) CC(Y) .65:09 .71:09 .77:07 .66:02 .68:03 .65:03 PAGE 143 143 PAGE 144 144 PAGE 145 whereRisthecorrelationmatrixofneuralspikeinputswiththedimensionof(LM)x(LM), Eachsub-blockmatrixrijcanbedecomposedas 145 PAGE 146 59 19 ].Inthiscase,theestimateofcorrelationbetweentwoneurons,rij(mk),canbeobtainedby Thecross-correlationvectorpiccanbedecomposedandestimatedinthesameway.rij()isestimatedusingequation A fromtheneuronalbincountdatawithxi(n)andxj(n)beingthebincountofneuronsiandjrespectively.Fromequation A ,itcanbeseenthatrij(t)isequaltorji(t). 146 PAGE 147 Topologyofthelinearlterforthreeoutputvariables 147 PAGE 148 @w=Xc(P P(c)@P(c) P(c)NXj=1@(c)T(j) Using producestherstorderderivativesof(c)t(j)withrespecttoeachtypeofparameterasfollows 148 PAGE 149 149 PAGE 150 ThelateralinhibitionnetworkisassumedtoproduceaGaussiandistributioncenteredatthewinningPE.ThisallowsthealgorithmtojustndthewinningPEandassumetheotherPEshaveanactivityproportionaltotheGaussianfunctionevaluatedateachPE'sdistancefromthewinner.TheSOMcompetitiverulebecomes wherethefunctionisaneighborhoodfunctioncenteredatthewinningPE.Duringeachiterationtheneighborhoodfunctionandstepsizechange.TheneighborhoodfunctionisaGaussianfortheexperimentsdescribedinthedissertation: withavariancethatdecreaseswitheachiteration.Asrst,thefullmapisalmostcovered,thenateachiterationthevariancereducestoaneighborhoodofzero,nallyallowingonlythewinningPEtobeupdated.Alineardecreaseinneighborhoodradiusisspeciedby 150 PAGE 151 TheupdatesfortheneighborsarereducedexponentiallybythedistancetothewinningPE.Thenetworkmovesfromasoftcompetitiontohardcompetitionastheneighborhoodshrinks.SincethewinningPEanditsneighborsareupdatedateachstep,thewinnerandallofitsneighborsmovetowardthesameposition,althoughtheneighborsmovemoreslowlyastheirdistancefromthewinningPEincreases.Overtime,thisorganizesthePEssothatneighboringPEs(intheSOMoutputspace)sharetherepresentationofthesameareaintheinputspace(areneighborsintheinputspace),regardlessoftheirinitiallocations. 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