Kernel Based Machine Learning Framework for Neural Decoding

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Kernel Based Machine Learning Framework for Neural Decoding
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Li, Lin
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Electrical and Computer Engineering
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Principe, Jose C
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Harris, John G
Shea, John M
Banerjee, Arunava

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linlicnel
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Abstract:
Brain machine interfaces (BMI) have attracted intensive attention as a promising technology to aid disabled humans. However, the neural system is highly distributed, dynamic and complex system containing millions of neurons that are functionally interconnected. How to best interface the neural system with human engineeried technology  is a critical and challenging problem. These issues motivate our research in neural decoding that is a significant step to realize useful BMI. In this dissertation, we aim to design a kernel-based machine learning framework to address a set of challenges in characterizing neural activity, decoding the information of sensory or behavioral states and controlling neural spatiotemporal patterns. Our contributions can be summarized as follows. First, we propose a nonlinear adaptive spike train decoder based on the kernel least mean square (KLMS) algorithm applied directly on the space of spike trains. Instead of using a binned representation of spike trains, we transform the vector of spike times into a function in reproducing kernel Hilbert space (RKHS), where the inner product of two sets of spike times is defined by the Schoenberg kernel, which encapsulates the statistical description of the point process that generates the spike trains, and bypasses the curse of dimensionality-resolution of the other spike representations. The simulation results indicate that our decoder has advantages in both computation time and accuracy, when the application requires fine time resolution. Secondly, the  precise control of the firing pattern in a population of neurons via applied electrical stimulation is a challenge due to the  sparseness of spiking responses and neural system plasticity. In this work, we propose a multiple-input-multiple-output (MIMO) adaptive inverse control scheme that operates on spike trains in a RKHS. The control scheme uses an inverse controller to approximate the neural circuit's inverse. The proposed control system takes advantage of the precise timing of the neural events  using the Schoenberg kernel based decoding methodology we proposed before. During operation, the adaptation of the controller minimizes a difference defined in the spike train RKHS between the system output and the target response and keeps the inverse controller close to the inverse of the current neural circuit, which enables adapting to neural perturbations. The results on a realistic synthetic neural circuit show that the inverse controller based on the Schoenberg kernel can successfully drive the elicited responses close to the original target responses even when significant perturbations occur. Thirdly, the spike train variability causes fluctuations in the neural decoder. Local field potentials (LFPs) are an alternate manifestations of neural activity with a more common continuous amplitude representation and longer spatiotemporal scales that can be recorded simultaneously from the same electrode array and contain complementary information about stimuli or behavior.  We propose a tensor product kernel based decoder for multi-scale neural activity, which allows modeling the sample from different sources individually and mapping them onto the same RKHS defined by the tensor product of the individual kernels for each source. A single linear model is adapted as done before  to identify the nonlinear mapping from the multiscale neural responses to the stimuli. It enables us decoding of more complete and accurate information from heterogeneous multiscale neural activity with only with an implicit assumption of independence on their relationship.  The decoding results in the rat sensory stimulation experiment show that the decoder outperforms the decoders with either single-type neural activities. In addition the multiscale decoding methodology is also used in the adaptive inverse control mode. Due to the accuracy and robustness of the decoder,  the control diagram with open-loop mode is applied to control the spatiotemporal pattern of neural response in the rat somatosensory cortex with micro-stimulation  in order to emulate the tactile sensation and obtained promising results. Finally, we quantify and comparatively validate the temporal functional connectivity between neurons by measuring the statistical dependence between their firing patterns. Temporal functional connectivity provides a quantifiable representation of the transient joint information of multi-channel neural activity, which is important to completely characterize the neural state but is normally overlooked in the temporal decoding due to its complexity. The functional connectivity pattern is represented by a graph/matrix, which is again not a conventional input for machine learning algorithms. Therefore, we propose two approaches to decode the stimulation information from the neural assembly pattern. One is to use graph theory to extract topology feature vector as the model input, which makes conventional machine learning approach applicable but dismisses the information of structural details. Therefore, we also proposed a matrix kernel that is able to map the connectivity matrix into RKHS and enable kernel based machine learning approaches directly to operate on the connectivity matrix, which bypasses the information reduction induced by the feature extraction.
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In the series University of Florida Digital Collections.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Principe, Jose C.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30
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by Lin Li.

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KERNELBASEDMACHINELEARNINGFRAMEWORKFORNEURALDECODINGByLINLIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012LinLi 2

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Tomybelovedparentsandfriends 3

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ACKNOWLEDGMENTS IwouldliketosincerelythankmyPh.D.advisorProf.Principeforhisinvaluableguidance,understanding,patience,andmostimportantly,hiscontinualfaithandcondenceinmeduringmyPh.D.studiesattheUniversityofFlorida.Ifeelextremelyfortunatetohavehimasmyadvisor,whoisalwayswillingtohelpmebothacademicallyandpersonally.IalsoowemywholeheartedgratitudetoProf.JustinSanchezandProf.JosephFrancisforenrichingmyknowledgeonneuroscienceandsupportingmyresearch.IwouldliketothankmyPh.D.committeemembers,Prof.Shea,Prof.Harris,andProf.Banerjee,fortheirconstructivesuggestionsandvaluablecommentsonmyPh.D.researchanddissertation.IalsothankourDARPAteam:AustinBrockmeier,JihyeBae,JohnChoi,MatthewEmigh,andKanLi.Ithasbeenawonderfulcollaborationexperience.IhavebeenfortunatetohavemanyfriendsinCNEL,withwhomIsharedmanygreatmemoryduringmyfouryearPh.D.journey.IspeciallythankSonglinZhao,JihyeBae,AustinBrockmeier,LuisSanchezGiraldo,StefanCraciun,ILParkMemming,SohanSeth,AlexanderAlvarado,Dr.BadongChen,PingpingZhu,ErionHasanbelliu,BilalFadlallah,MiguelTeixeira,RakeshChalasani,KittipatKampa,RoshaPokharel,GoktugCinar,EvanKriminger,MatthewEmigh,KanLi,JongminLee,InJunPark,GavinPhilips,andGabrielNallathambi,formanyvaluablediscussions(andespeciallytheEnglishclasses!)andtheirconsistentsupportandwonderfulfriendship.Lastbutnotleast,Iamindebtedtomyparentsfortheirsupportandloveineverydayofmylife. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 13 1.1DecodingfromSpikeTrainsandControl ................... 14 1.2DecodingfromMulti-ScaleNeuralActivity .................. 16 1.3DecodingfromTemporalFunctionalConnectivityPattern ......... 17 2AKERNEL-BASEDFRAMEWORKFORSPIKETRAINDECODINGANDCONTROL ...................................... 19 2.1BackgroundandMotivation .......................... 19 2.2NeuralDecoding ................................ 22 2.2.1NeuralDecoderwithDiscretizedSpikeTrainRepresentation .... 24 2.2.1.1Generalized-linear-model-baseddecoder ......... 24 2.2.1.2Spikernel-baseddecoder .................. 25 2.2.2KernelbasedDecoderwithSpikeTimingRepresentation ..... 26 2.2.2.1Kerneldesignforspiketrains ................ 27 2.2.2.2Kernelbasedadaptiveregressor .............. 30 2.2.2.3Multiple-input-multiple-output(MIMO)decodingmodel .. 34 2.2.2.4Comparisonoverkernels .................. 35 2.2.3SyntheticExperiment .......................... 36 2.2.3.1Neuralcircuit ......................... 36 2.2.3.2Comparisonoverdecoders ................. 41 2.2.4AnimalExperiment ........................... 42 2.2.4.1Ratdata ........................... 42 2.2.4.2Resultsoftactilestimulation ................ 44 2.2.4.3Resultsofmicro-stimulationstimulation .......... 44 2.3AdaptiveInverseControlofNeuralSpatiotemporalSpikePatterns .... 46 2.3.1Filtered-LMSalgorithm ........................ 46 2.3.2ControldesigninRKHS ........................ 48 2.3.3Syntheticexperiment .......................... 50 2.3.3.1Controllingneuralringpatternwithoutperturbations .. 50 2.3.3.2Controllingneuralringpatternwithperturbations .... 51 2.4Discussion ................................... 55 5

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3NEURALDECODINGFROMMULTI-SCALENEURALACTIVITY ........ 58 3.1TensorProductKernelforMulti-ScaleNeuralActivity ............ 60 3.2SensoryStimulationExperiment ....................... 63 3.2.1ExperimentMotivationandFramework ................ 63 3.2.2TimeScaleEstimation ......................... 65 3.2.3DecodingResults ............................ 66 3.2.3.1Resultsoftactilestimulation ................ 66 3.2.3.2Resultsofmicro-stimulation ................. 68 3.3OpenLoopAdaptiveInverseControl ..................... 69 4FUNCTIONALCONNECTIVITYANALYSISANDMODELING .......... 74 4.1FunctionalConnectivityMeasures ...................... 77 4.1.1StatisticalMeasures .......................... 78 4.1.2PhaseSynchronizationMeasures ................... 83 4.1.3PerformanceComparisonoverMeasures .............. 84 4.1.3.1Simulateddata ........................ 84 4.1.3.2Criterionforcomparingdifferentmeasures ........ 85 4.1.3.3Simulationresults ...................... 87 4.2TemporalFunctionalConnectivityAnalysisandInteractiveVisualization 89 4.2.1CorticalNeuralData .......................... 90 4.2.2KinematicStateAnalysis ........................ 91 4.2.2.1State-relatedassembly ................... 92 4.2.2.2Evolutionofconnectionstrength .............. 94 4.2.2.3Activationdegreeofstate-relatedassemblies ....... 97 4.2.2.4Dynamicsoflocalfunctionalconnectivity ......... 100 4.3NeuralModelingviaFunctionalConnectivityTemporalPattern ...... 103 4.3.1MeasuresofNeuralNetworkTopology ................ 103 4.3.2KernelforFunctionalConnectivityMatrix ............... 105 4.3.3Experiment ............................... 106 4.4Discussion ................................... 109 5CONCLUSIONS ................................... 114 5.1BMIApplication ................................. 114 5.2MethodologyApplication ............................ 117 REFERENCES ....................................... 119 BIOGRAPHICALSKETCH ................................ 128 6

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LISTOFTABLES Table page 2-1Comparisonamongneurondecoders.NiandMirepresentthespikenumberandthewindowlength,respectively. ........................ 42 3-1Comparisonamongneuraldecoders. ....................... 67 4-1Statisticalpowersofthemethodswithrespecttodifferentaveragesynapticweightw,whenthewindowsizeis10samples. .................. 89 4-2Classicationcomparisonoverdifferentinputfeatures. .............. 109 7

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LISTOFFIGURES Figure page 2-1RelationbetweentheoriginalspaceofspiketrainsandtheRKHSdenedbythestrictlypositivedenedkernel. ......................... 29 2-2Syntheticstimulation ................................. 35 2-3Comparisonamongthekernelsdenedinthespaceofspiketrains:CI,NCI,andSchoenbergkernel ............................... 37 2-4Decoderperformancesurfacewithrespecttothekernelfreeparameters .... 38 2-5Abiologicalplausiblesyntheticneuralcircuit. ................... 39 2-6Thestimulationcongurationandthestatisticsoftheneuronresponse. .... 40 2-7Thepatternofthe3Dstimulationdenedbytheelectricaleld(theupperplot)andtheresultingneuralresponse(thebottomplot). ............... 40 2-8Micro-stimulationpatterns .............................. 43 2-9Dataoftheratsensorystimulationexperiment .................. 44 2-10ResultsoftheSchoenbergkernelbaseddecoderoftherststimulationtrial. 45 2-11Resultsofreconstructingmicro-stimulation. .................... 46 2-12AFiltered-LMSadaptiveinversecontroldiagram. ................ 47 2-13AnadaptiveinversecontroldiagraminRKHSofspiketrain. ........... 49 2-14Performanceofstimulationoptimizationusingadaptiveinversecontrol. .... 52 2-15Comparisonofspike-triggeredaverages(STA)ofthethreedimensionallteredelectriceldbetweentheoriginalneuralcircuitandtheperturbedneuralcircuit. ............................................. 53 2-16Theperformanceevolutionduringadaptationoftheinversecontrollertofollowtheperturbedneuroncircuit. ............................ 54 2-17Performanceofstimulationoptimizationafteraperturbationusingadaptiveinversecontrol. .................................... 57 3-1Neuralelementsintactilestimulationexperiments ................ 64 3-2AutocorrelationofLPFsandspiketrains ...................... 66 3-3ResultsoftheLFP&spikedecoderoftherststimulationtrial. .......... 67 8

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3-4Learningcurves(NMSEofthetestset)oftheLFP&spikedecoderoftherststimulationtrial. .................................... 68 3-5Comparisonamongspikedecoder,LFPdecoderandspikeandLFPdecoder. 69 3-6Resultsofreconstructingmicro-stimulation. .................... 70 3-7Qualitativecomparisonofexcerptsbetweentargetandtrainedsystemoutput. 72 3-8Correlationsbetweentargetspiketrainsandneuralsystemoutputofeachchannelforeachtactilestimulationlocation(6digits:d1d2d3d4p3andp1). 73 4-1Thestimulatednetworkof10neurons. ....................... 85 4-2TherasterplotsofneuronAandBunderDISandCONstate.Undereachstate,neuronsAandBreatareasonableringraterange(1Hzto10Hz). 86 4-3Smallsamplesizeperformancecomparisonamongfourdependencemeasures:MSC,CC,MI,andPhS,fordetectingfunctionalconnectivitybetweenspikingneuronAandB. ................................... 88 4-4Exampleofreachingmovementtrajectory.Trajectoryissegmentedintorest(R),rest-to-food(Mv1),food-to-mouth(Mv2),andmouth-to-rest(Mv3)states. 92 4-5104104matricesplotthreefunctionalconnectivitygraphsassessedbyKS-testresults. ........................................ 94 4-6Time-varyingconnectionstrengthofneuron93intheMv3assembly. ..... 96 4-7Averageactivationdegreeofstate-relatedassemblies. .............. 98 4-8ThetemporalevolutionoffunctionalactivationofthreeassembliescorrespondingtoMv1,Mv2,andMv3states. ............................ 100 4-9Localfunctionalconnectivitygraphsin4corticalareas:PP-contra,M1-contra,PMD-contra,andM1/PMD-ipsifor5movementsub-states. ........... 102 4-10Functionalconnectivitymatricesforeachstimulationpattern .......... 108 4-11PairwiseFrobeniusdistanceoffunctionalconnectivitypatterns ......... 108 4-12Classicationresultsoffunctionalconnectivitymatrixkernel ........... 110 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyKERNELBASEDMACHINELEARNINGFRAMEWORKFORNEURALDECODINGByLinLiDecember2012Chair:JoseC.PrincipeMajor:ElectricalandComputerEngineering Brainmachineinterfaces(BMI)haveattractedintensiveattentionasapromisingtechnologytoaiddisabledhumans.However,theneuralsystemishighlydistributed,dynamicandcomplexsystemcontainingmillionsofneuronsthatarefunctionallyinterconnected.Howtobestinterfacetheneuralsystemwithhumanengineeriedtechnologyisacriticalandchallengingproblem.TheseissuesmotivateourresearchinneuraldecodingthatisasignicantsteptorealizeusefulBMI.Inthisdissertation,weaimtodesignakernel-basedmachinelearningframeworktoaddressasetofchallengesincharacterizingneuralactivity,decodingtheinformationofsensoryorbehavioralstatesandcontrollingneuralspatiotemporalpatterns. Ourcontributionscanbesummarizedasfollows.First,weproposeanonlinearadaptivespiketraindecoderbasedonthekernelleastmeansquare(KLMS)algorithmapplieddirectlyonthespaceofspiketrains.Insteadofusingabinnedrepresentationofspiketrains,wetransformthevectorofspiketimesintoafunctioninreproducingkernelHilbertspace(RKHS),wheretheinnerproductoftwosetsofspiketimesisdenedbytheSchoenbergkernel,whichencapsulatesthestatisticaldescriptionofthepointprocessthatgeneratesthespiketrains,andbypassesthecurseofdimensionality-resolutionoftheotherspikerepresentations.Thesimulationresultsindicatethatourdecoderhasadvantagesinbothcomputationtimeandaccuracy,whentheapplicationrequiresnetimeresolution. 10

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Secondly,theprecisecontroloftheringpatterninapopulationofneuronsviaappliedelectricalstimulationisachallengeduetothesparsenessofspikingresponsesandneuralsystemplasticity.Inthiswork,weproposeamultiple-input-multiple-output(MIMO)adaptiveinversecontrolschemethatoperatesonspiketrainsinaRKHS.Thecontrolschemeusesaninversecontrollertoapproximatetheneuralcircuit'sinverse.TheproposedcontrolsystemtakesadvantageoftheprecisetimingoftheneuraleventsusingtheSchoenbergkernelbaseddecodingmethodologyweproposedbefore.Duringoperation,theadaptationofthecontrollerminimizesadifferencedenedinthespiketrainRKHSbetweenthesystemoutputandthetargetresponseandkeepstheinversecontrollerclosetotheinverseofthecurrentneuralcircuit,whichenablesadaptingtoneuralperturbations.TheresultsonarealisticsyntheticneuralcircuitshowthattheinversecontrollerbasedontheSchoenbergkernelcansuccessfullydrivetheelicitedresponsesclosetotheoriginaltargetresponsesevenwhensignicantperturbationsoccur. Thirdly,thespiketrainvariabilitycausesuctuationsintheneuraldecoder.Localeldpotentials(LFPs)areanalternatemanifestationsofneuralactivitywithamorecommoncontinuousamplituderepresentationandlongerspatiotemporalscalesthatcanberecordedsimultaneouslyfromthesameelectrodearrayandcontaincomplementaryinformationaboutstimuliorbehavior.Weproposeatensorproductkernelbaseddecoderformultiscaleneuralactivity,whichallowsmodelingthesamplefromdifferentsourcesindividuallyandmappingthemontothesameRKHSdenedbythetensorproductoftheindividualkernelsforeachsource.Asinglelinearmodelisadaptedasdonebeforetoidentifythenonlinearmappingfromthemultiscaleneuralresponsestothestimuli.Itenablesusdecodingofmorecompleteandaccurateinformationfromheterogeneousmultiscaleneuralactivitywithonlywithanimplicitassumptionofindependenceontheirrelationship.Thedecodingresultsintheratsensorystimulationexperimentshowthatthedecoderoutperformsthedecoderswitheithersingle-type 11

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neuralactivities.Inadditionthemultiscaledecodingmethodologyisalsousedintheadaptiveinversecontrolmode.Duetotheaccuracyandrobustnessofthedecoder,thecontroldiagramwithopen-loopmodeisappliedtocontrolthespatiotemporalpatternofneuralresponseintheratsomatosensorycortexwithmicro-stimulationinordertoemulatethetactilesensationandobtainedpromisingresults. Finally,wequantifyandcomparativelyvalidatethetemporalfunctionalconnectivitybetweenneuronsbymeasuringthestatisticaldependencebetweentheirringpatterns.Temporalfunctionalconnectivityprovidesaquantiablerepresentationofthetransientjointinformationofmulti-channelneuralactivity,whichisimportanttocompletelycharacterizetheneuralstatebutisnormallyoverlookedinthetemporaldecodingduetoitscomplexity.Thefunctionalconnectivitypatternisrepresentedbyagraph/matrix,whichisagainnotaconventionalinputformachinelearningalgorithms.Therefore,weproposetwoapproachestodecodethestimulationinformationfromtheneuralassemblypattern.Oneistousegraphtheorytoextracttopologyfeaturevectorasthemodelinput,whichmakesconventionalmachinelearningapproachapplicablebutdismissestheinformationofstructuraldetails.Therefore,wealsoproposedamatrixkernelthatisabletomaptheconnectivitymatrixintoRKHSandenablekernelbasedmachinelearningapproachesdirectlytooperateontheconnectivitymatrix,whichbypassestheinformationreductioninducedbythefeatureextraction. 12

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CHAPTER1INTRODUCTION Brainmachineinterfaces(BMI)providenewmeanstocommunicatewiththebrainbydirectlyaccessing,interpretingandevencontrollingneuralstates.Theyhaveattractedhugeattentionasapromisingtechnologytoaidthedisabled(i.e.spinalcordinjury,movementdisability,stroke,hearingloss,andblindness).Quantifyingtheinformationcontainedinneuralactivity,modelingtheneuralsystemanddecodingtheintentionofmovementorstimulationarethefundamentalstepstodesignneuralprostheticsandbrainmachineinterfaces. However,theneuralsystemishighlydistributed,dynamicandcomplex,creatingchallengesforthemodelingtask.First,neuronscommunicatethroughelectricalpulses,calledspikes.Thus,informationisrepresentednotintheusualsignalamplitudebutinspiketimings.Thesequenceofspikesorspiketrainrepresentstheactivityofanindividualneuron,forwhichtheconventionalsignalprocessingapproachbasedonsignalamplitudevariationisnotwellsuited.Inaddition,besidestheactivityofindividualneurons(spiketrains),thedevelopmentofrecordingtechnologyallowsustoaccessthebrainactivityalsofromotherfunctionallevelincludinglocaleldpotential(LFP),electrocorticogram(ECG),andelectroencephalogram(EEG),whichcontainscomplementaryinformationofbrainintentionalstates.Thoughitiscleartherearerelationshipsamongthosebrainactivityfactors,itisunknownhowthesefactorscanbeexploitedtobetterdecodeneural-response-stimulusmappings.Thechallengeishowtohandletheheterogenoussignalswithmultiplespatiotemporalscalesforneuraldecoding.Moreover,theneuralmechanismsthatselectandcoordinatethedistributedcorticalactivitytoproducemovementshavebeenthefocusofsystemneuroscience[ 61 ].Thedevelopmentofnewtoolstoassessthisactivationiscriticalbecauseneuralassembliesprovideaconceptualframeworkfortheintegrationofdistributedneuralringpattern.However,duetothespatiallyspecicandtransient 13

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natureoftheinterneuronalcommunication,quantifyingtheassemblyanddecodinginformationfromitsfunctionalconnectivitypatternisadifculttask.Inourwork,weaddressallthechallengingissuesinspecicBMItasksanddevelopnewkernel-basedmachinelearningapproachestoovercomethem,whicharefurtherdiscussedinthefollowingsections. 1.1DecodingfromSpikeTrainsandControl Intracranialstimulationhasthepossibilitytodeliverinformationdirectlytothebrain,andthecapacityofthisinformationchannelisdirectlylinkedtotheabilitytodriveneuralresponsestomatchmeaningfulspikingresponsescreatednaturallysensory.Controllingneuralresponseviastimulationhasraisedtheprospectofmimickingtheresponsestonaturalsensorystimuli,therebycreatingasensoryprothesis.Thisallowsprovidingmorenaturalisticsensoryfeedbackaspartofabrain-machineinterfaceandthusimprovingperformanceovertheuseofarbitraryspatiotemporalelectricalstimulation[ 63 ].Usingelectrodearraysforelectricalmicrostimulationandothersforrecordingallowssimultaneousstimulationandrecordingfrompopulationsofhundredsofneurons.Thetemporalprecisionofthestimulationandrecordingisinthemillisecondrange,whereasthespatialprecisionisonlylimitedbythemicroelectrodearrays.Thisenablestheprecisecontrolofneuralactivityattheneuronandpopulationlevels. Fromacontroltheoryperspective,theneuralcircuitistreatedastheplant,wheretheappliedmicrostimulationisthecontrolsignalandtheplantoutputistheelicitedspiketrains.Thereareseveraluniquechallengesthathavetobeconsideredbeforeweaddressthisproblem.First,thetransferfunctionbetweenthemicrostimulationandthetargetneuralresponseisunknown;thus,thetraditionalapproachofaplantwithknowndynamicscannotbedirectlyappliedonaneuralsystem.Inaddition,theperturbationsinducedbylong-termlearningandstimulationwillcausechangestotheneuralsystemtransferfunction;thisrequiresthecontrollerparameterstoautomaticallytrackneuralplasticity.Allthesefactorssuggestthatadaptivecontrolisanappropriateapproachto 14

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addressthisproblem.However,thecontrolledsystemoutputisaspiketrainsthatisnotconventionalsmoothamplitudesignal,whichisnotcompatiblewiththeexistingcontrolframework,unlessweuseratecodingthatdestroysthetimeresolution. Inonlineadaptivecontrolbasedonsignalprocessingmethodology,thecontrollermodelperformslikeaneuraldecoder:giventheobservedspiketrainsasinputs,thecontrollerisusedtoestimatethestimulationsuchthattheplantoutputapproximatesthetargetspiketrains.However,alltheexcitingdecodingtechniqueshavebeenappliedtodiscretizedrepresentationsofspiketrains,whichhaslimitedapplicabilityforsystemsrequiringanetimeresolution.Forexample,inthecaseofasomatosensoryprosthesis,thetimeresolutionofthemicro-stimulationusedtocreatetactilesensationissmall(around1ms),whichrequiresasmallbinsize.Withsuchasmalldiscretizationstep,theinputspacebecomessparseandthedimensionalityoftheinputspacefrommulti-electrodearraysalsobecomesaproblem(curseofdimensionality).Instead,thesetofspikeoccurrencesprovidesamoreeffectiveandaccuratedescriptionofspiketrains,butthespaceofspiketrainsisnotaconventionalL2functionalspaceenjoyedbytypicaldiscretetimecontinuousamplitudesignals.Algorithmsthatrelyonrealorcomplexvaluescannotdirectlyoperateonthesetofspiketimes.Indeed,thespaceofspiketimesdoesnotpossessanalgebraicstructure,i.e.,operationssuchasadditionandmultiplicationarenotdenedforspiketrains. Inordertoeffectivelyapplymachinelearningalgorithmsforstimulationdecoding,basedonthemathematicaltheoryofRKHSandfunctionalrepresentationofspiketrains,aSchoenbergkernelbetweentwospiketrainsenableskernel-basedregressionalgorithmstobedirectlyappliedinthespaceoftheneuraleventtimings[ 44 67 68 ].Thisdecreasesthecomputationtimeincontrastwiththekernelsbasedontheconventionalraterepresentationandavoidssparsenessandhigh-dimensionalityofthebinnedspiketrains.InChapter 2 ,weproposeaMIMOadaptiveinversecontrolschemewithSchoenbergkernel-basedregressorbuiltinthespiketrainRKHStocontrolthe 15

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neuralactivity.Theadaptationofthecontrollermodelalsoallowsthecontrolsystemtotracktheplasticityoftheunderlyingneuralsystemorganization.Thecontrollerbasedonthenonlinearkernel-basedneuralsystemidenticationmodelhasalinearstructureinRKHS,whichhasasingleglobaloptimalsolution. 1.2DecodingfromMulti-ScaleNeuralActivity Theproblemofextractinginformationfromspiketrain,localeldpotential(LFP),electrocorticogram(ECG),andelectroencephalogram(EEG)isrootedintheirunknownunderlyingrelation,multiplespatialandtemporalscaleandheterogenoussignalformat.Inourwork,wemainlyaddressthemodelingtaskofspiketrainsandLFPsasanexampleofmultiscaleneuralmodeling,sinceitisabletocoverallthechallengesmentionedabove.AnotherreasonofselectingspiketrainandLFPisthattheycanberecordedfromthesamemicro-electrodearrays,whichguaranteesthetimealignmentandthussimplifythedatapreprocessingprocedure. Accordingtotheliterature,spiketrainsandlocaleldpotentials(LFPs)encodecomplementaryinformationofthestimuliorbehaviors[ 8 37 95 ].Inmostrecordings,spiketrainsareobtainedbyahigh-passlterwiththecutofffrequencyabout300)]TJ /F1 11.955 Tf 9.3 0 Td[(500Hz,whileLFPsareobtainedusingalow-passlterwiththecutofffrequencyabout300Hz[ 77 ].Aspiketrainrepresentsthesingle-unitneuralactivitywithanetemporalresolution.However,itsstochasticpropertiesinduceaconsiderabletrial-to-trialvariability,especiallywhenthestimulationamplitudeissmall.Incontrast,LFPsreectthesumofalllocalcurrentsnearthesurfaceoftheelectrode,whichlimitsspecicitybutprovidesrobustnessforcharacterizingthemodulationinducedbystimuli,evenatlowamplitudes.Therefore,anappropriatecombinationofLFPsandspiketrainsindecodingmodelsmakeitpossibletomoreaccuratelydecodethestimuliorbehaviorsfromtheneuralresponse.Forexample,thedecodercancoordinateLFPsorspikepatternstotagparticularlysalienteventsorextractdifferentstimulationfeaturescharacterizedbydifferenttypesignals.However,thedifferentsignalpropertiesbetweenLFPsand 16

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spiketrainsmakeintegrationinthesamemodeldifcult.First,theunderlyingstochasticpropertiesaretotallydifferentbetweenthetworepresentations.Aspiketrainthatisasetofspiketimingscanbeinterpretedasarealizationofapointprocess[ 41 93 ],whileLFPsareacontinuousamplitudeprocess.Moreover,theLFPtimescaleissignicantlylongerthanspiketrains.Inchapter 3 ,weproposeatensor-product-kernel-basedregressortodecodethestimulationinformationfrommulti-scaleneuralactivities,whichallowsmodelingheterogeneoussignalsindividuallyandmergetheirinformationinthefeaturespacedenedbythetensorproductoftheindividualkernelsforeachtypesignal[ 85 ]. 1.3DecodingfromTemporalFunctionalConnectivityPattern InBMIs,thegoalistodecodetheintentionofmovement/stimulation,usingmultielectrodeneuraldatafromcortexsynchronizedwithkinematic/stimulationvariablesmeasurements.Sinceneuronsareselectivelycoordinatedwitheachother,fromastatisticalmodelingperspective,thecompleteinformationtosolvethisproblemresidesinthejointprobabilitydensityfunctionofthemultivariateneuraldataandofthemultivariatekinematicvariables,butthisisneverdoneduetothehighdimensionalityofthejointdistributions.Evenwhenwedevelopmodelsthatestimatetheconditionalprobabilityofthekinematicsgiventheneuraldata,wedonotusethejointinformationexpressedinthemultichannelneuraldataforthesamereason.Instead,wemodeltheconditionaldependenceonlywithrespecttothehistoryofavectorofneuralspiketrains.Sinceeachneuralchannelisnothingbutthemarginaldensityofthejointprobabilityofmultivariateneuraldata,thisprocedureisonlyareasonablesolutionwhentheneuronsspikeindependently.Inordertocharacterizethisjointinformationamongneurons,wequantifythetemporalfunctionalconnectivitypatternamongthatcomprisesofpairwisedependenciesbetweenneuronsinvolvedinagivenbehavioral/stimulationtaskandinvestigateitsassociationwithbehavioral/stimulationstatesintimeinChapter 4 17

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However,thefunctionalconnectivitypatternsarerepresentedbygraphsormatricesinsteadofvectors,whichcarriesdifcultieswhenonewantstousegraphsasaninputfeaturefortheneuraldecoder.Insteadofdirectlyaddressingfunctionalconnectivitygraphsormatrices,mostresearchimplementsgraphtheorytoquantifythetopologypropertyofagraphpatternincludingnodedegree,clusteringcoefcient,betweennesscentrality,etc.[ 22 82 ].Thesetopologymetricsextractfeaturevectorsfromthefunctionalconnectivitygraphs/matrices,whichallowstheimplementationofthetraditionalmodelingapproacheswiththevector-basedinputspace.However,theextractingprocessreducesthedetailsofthegraphstructure.Therefore,inchapter 4 ,wealsodesignakernelforfunctionalconnectivitymatrices,whichallowsustoapplykernelbasedmachinelearningalgorithmsdirectlyonthetemporalfunctionalconnectivitypatternwithlessinformationreduction. 18

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CHAPTER2AKERNEL-BASEDFRAMEWORKFORSPIKETRAINDECODINGANDCONTROL 2.1BackgroundandMotivation Applyinglowpowercurrentwaveformstointracranialmicroelectrodes(referredhereinaselectricalmicrostimulation)1isavaluabletooltolearntheunderlyingstructureofneuralcircuitsandtocontroltheiractivity.Modernelectrophysiologicaltechniquesenabletheimplantationofmultiplemicroelectrodearrays.Usingarraysforelectricalmicrostimulationandothersforrecordingallowssimultaneousstimulationandrecordingfrompopulationsofhundredsofneurons.Thetemporalprecisionofthestimulationandrecordingisinthemillisecondrange,whereasthespatialprecisionisonlylimitedbythemicroelectrodearrays.Thisenablestheprecisecontrolofneuralactivityattheneuronandpopulationlevels.Controllingtheneuralactivityviastimulationhasraisedtheprospectofgeneratingspecicspikepatterns,evenmimickingnaturalneuralactivity,whichiscentralbothforourbasicunderstandingofneuralinformationprocessingandforengineeringneuralprostheticdevicesthatcaninteractwiththebraindirectly[ 30 ]. Fromacontroltheoryperspective,theneuralcircuitistreatedastheplant,wheretheappliedmicrostimulationisthecontrolsignalandtheplantoutputistheelicitedspiketrains.Thereareseveraluniquechallengesthathavetobeconsideredbeforeweaddressthisproblem.First,thetransferfunctionbetweenthemicrostimulationandthetargetneuralresponseisunknown;thus,thetraditionalapproachofaplantwithknowndynamicscannotbedirectlyimplementedonaneuralsystem.Inaddition,theperturbationsinducedbylong-termlearningandstimulationitselfwillcausechangestotheneuralsystemtransferfunction;thisrequiresthecontrollerparameterstoautomaticallytrackneuralplasticity.Theexistingapproachesappliedtoneural 1Althoughotherformsofstimulationmaybeconsideredintheparadigm,weconcentrateontheelectricalmicrostimulation. 19

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systemscanbesummarizedintotwomajorclasses:model-freemethods[ 12 30 47 ]andmodel-basedmethods[ 3 14 57 ].Forexampleglobalsearchoptimization[ 30 ]isamodel-freeapproach,whichusesageneticalgorithm(GA)tosearchthestimulationspaceforthecostfunctionminimum,butthecostfunctionisnotdenedintermsofatargetspikingpattern,andbecausethepotentialspatiotemporalspaceofstimuliisusuallyhuge,theglobalsearchisslow.Anothermodel-freemethodappliedtotheneuralsystemsistheproportionalintegralderivativePIDcontroller[ 47 ],whichisafeedbackcontrolschemethatisabletoreducethesystemdisturbanceusingtheoutputerrorasthefeedbacksignal.SincethereisnoeasywaytoautomaticallytunePIDweights,suchcontrollerscannotsuccessfullycontrolaneuralsystemwithtime-varyingperturbations. Inmodel-basedschemes,thecontrollerperformslikeadecoderandisgenerallybuiltintwoways.Oneistotrainaninversemodelofthetransferfunctionfromthestimulationtoneuralactivity[ 14 ].Thetargetneuralactivitypatternisusedastheinitialcontrollerinput,thentheelicitedneuralresponseisreturnedtothecontrollerinputasthefeedback,whichiscapableofcancelingsystemnoise.Anothermodel-basedstrategyreliesontheBayesian`inversion'oftheencodingmodel,suchasGeneralizedLinearModel(GLM),toestimatetheoptimalstimulationwithrespecttothetargetneuralresponse[ 3 ].However,sincethedesiredstimulationforthetargetringpatterninducedbythenaturalstimulationisunavailablefortraining,thecontrollermayconvergetotheoptimalinverseneuralcircuitmodelwithrespecttothetrainingdatabutnotnecessarilytothetargetringpattern;thisstrategymaycauselimitedaccuracyofthecontrolschemesunlessthestimulationspaceisfullyexploredadifcultprobleminitself.Moreover,formodel-basedneuralcontrol,itisessentialtoadaptthemodeltotrackthesystemvariation,whichhasnotbeenformallyaddressedinanyofthemodelsmentionedabove.Becauseoftheuniquechallenges,adaptivecontrolwithsystemidenticationtechnologyisanappropriateapproachtoaddressthisproblem,wheretheneuralcircuitistreatedasthetime-variantplant.Therearemanyclassic 20

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adaptivecontrolmethods[ 5 ]forasystemwithunknowndynamics,suchasStochasticControl,ModelReferenceAdaptiveControllers,etc.,buthowtoappropriatelytthesecontrolschemeforspiketrainsexploitingthespiketiminginformationhasnotbeenwelldeveloped.However,adaptiveinversecontrol[ 105 ]isainput-output(systemidentication)basedcontrolscheme,andthusallowsustoattackadaptivecontrolproblemusingthemethodologyofadaptivesignalprocessing.Withneuralsignalprocessingmethodologies,theadaptiveinversecontrollercanbedirectlyappliedonspiketrains.Therefore,multiple-inputmultiple-output(MIMO)adaptiveinversecontrolisusedinthiswork.Thebasicideaofadaptiveinversecontrolistolearnaninversemodeloftheplantasthecontroller,suchthatthecascadeofthecontrollerandtheplantwillperformlikeanunitarytransferfunction.Thetargetneuralactivityisusedasthecontroller'scommandinput.Thecontrollerparametersareupdatedtominimizethedissimilaritybetweenthetargetneuralactivityandthecascadeoutputduringthewholecontrolprocess,whichenablesthecontrollertotrackanyvariationintheplanttime-variationandcancelanynoise. Thecontrollermodelperformslikeaneuraldecoder:giventheobservedspiketrainsasinputs,thecontrollerisusedtoestimatethestimulationsuchthattheplantoutputapproximatesthetargetspiketrains.Forthedevelopmentofthedecodingalgorithm,mostthedecodingtechniqueshavebeenappliedtodiscretizedrepresentationsofspiketrainsbecauseofitssimplicityandapplicability,whichcanbesummarizedintotwocategories:regressionscheme[ 11 54 55 89 104 ]andBayesianscheme[ 71 101 ].However,thedecoderbasedonthediscretizedrepresentationofspiketrainshaslimitedapplicabilityforsystemsrequiringanetimeresolution.Forexample,inthecaseofasomatosensoryprosthesis,thetimeresolutionofthemicro-stimulationusedtocreatetactilesensationissmall(around1ms),whichrequiresasmallbinsize.Withsuchasmalldiscretizationsize,theinputspacebecomessparseandthedimensionalityoftheinputspacefrommulti-electrodearraysalsobecomesaproblem 21

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(curseofdimensionality).Instead,thesetofthespiketimeoccurrencesprovidesamoreeffectiveandaccuratedescriptionofspiketrains,butthespaceofspiketrainsisnotaconventionalL2functionalspaceenjoyedbytypicaldiscretetimecontinuousamplitudesignals.Algorithmsthatrelyonrealorcomplexvaluescannotdirectlyoperateonthesetofspiketimes.Indeed,thespaceofspiketimesdoesnotpossessanalgebraicstructure,i.e.,operationssuchasadditionandmultiplicationarenotdenedfortospiketrains.Inordertoeffectivelyapplymachinelearningalgorithmsforstimulationoptimization,itisnecessarytodeneanappropriatefunctionalstructureforthespiketrainspacewhereoptimizationalgorithmscanbedeveloped.Positivedenitefunctionsinthespiketrainspacearedenedin[ 67 ].BasedonthemathematicaltheoryofRKHSandfunctionalrepresentationofspiketrains,aSchoenbergkernelbetweentwospiketrainsenableskernel-basedregressionalgorithmstobedirectlyappliedinthespaceoftheneuraleventtimings[ 44 ].Thisdecreasesthecomputationtimeincontrastwiththekernelsbasedontheconventionalraterepresentationandavoidssparsehigh-dimensionalvectorscorrespondingtobinnedspiketrains. Inthischapter,weproposeaMIMOadaptiveinversecontrolschemewithSchoenbergkernel-basedregressorbuiltinthespiketrainRKHStocontroltheneuralactivity.Thegoalofthisworkistoelicitneuralresponsesthatmimicthoseinducedbythenaturalstimulation.Theadaptationofthecontrollermodelalsoallowsthecontrolsystemtotracktheplasticityoftheunderlyingneuralsystemorganization.Thecontrollerbasedonthenonlinearkernel-basedneuralsystemidenticationmodelhasalinearstructureinRKHS,whichhasasingleglobaloptimalsolution.Arealisticsyntheticneuralcircuitisbuilttotestthecontrolsystem. 2.2NeuralDecoding Inneurophysiologyitiswidelyacceptedthatneuronscommunicatethroughadiscretepulse-likewave,calledanactionpotential.Actionpotentialsrecordthevoltagedifferentialtoadistantgroundpoint,typicallytheskull.Despitetheinevitablenoisein 22

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theserecordings,itiseasilyobservedthatactionpotentialshaveaverystereotypicalshape,withacharacteristicxedamplitudeandwidthassociatedwithagivenneuron.Fromtheneurophysiologicalperspective,theactualshapeandmagnitudeofanactionpotentialisredundantbecauseitcontainsnoinformation,butthemomentitoccursisimportant.Underthisperspective,theactionpotentialsaresimpliedtospiketrain,asequenceofspikesorderedintime,whichneglectthestereotypicalshapeofanactionpotential,butpreservesonlythetimeinstantsatwhichtheyoccur(i.e.,spiketimes)[ 24 ].Aspiketrainsisasequenceoforderedspiketimesi.es=ftn2T:n=1,...,Ng,intheintervalT=[0,T].whichcanbedenedbys(t)=NXm=1(t)]TJ /F4 11.955 Tf 11.96 0 Td[(tm),ftm2T:n=1,...,Ng, (2) wheredenotestheDiracdelta.Sincethespiketrainisnothingbutasetofeventtimes,theinformationiscodedintheeventtimes.However,thereisnoalgebraicstructureinthespaceofspiketrains.Noconventionalmachinelearningalgorithmcanbedirectlyappliedtoit.Binningisapredominantprocessinneuralsignalprocessing,whichtransformthespiketrainintoamplitudesignal,andthenallowstheapplicationofconventionalmachinelearninglearningtoforneuraldecoding.Inthissection,werstreviewtheexistingneuraldecodingapproachesonbinneddata.Thenweproposesanonlinearadaptivedecoderforsomatosensorymicro-stimulationbasedonthekernelleastmeansquare(KLMS)algorithmapplieddirectlyonthespaceofspiketrains.Insteadofusingabinnedrepresentationofspiketrains,wetransformthevectorofspiketimesintoafunctioninreproducingkernelHilbertspace(RKHS),wheretheinnerproductoftwospiketimevectorsisdenedbyintensitydrivenkernel.Thisrepresentationencapsulatesthestatisticaldescriptionofthepointprocessthatgeneratesthespiketrains,andbypassesthecurseofdimensionality-resolutionofthebinnedspikerepresentations. 23

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2.2.1NeuralDecoderwithDiscretizedSpikeTrainRepresentation Binningisapredominantapproachincurrentspiketrainanalysisandprocessingmethods[ 24 ].Statistically,itismotivatedbythecountingprocessrepresentationofapointprocess,whichisobtainedbycountingthenumberofspikesthatappearduringaunitintervalandnormalizedbythedurationoftheinterval.Thebinneddataisdenotedbyr,wherer(n)=1 Zn(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1)s(t)dt (2) Thebinneddatacanbeeasilydeterminedfromasingletrialandtransformthepointprocessesintodiscreteamplitudetimeseries,whichallowstheanalysisusingtheconventionalsignalprocessingmethod. Becauseofitssimplicity,avarietyofmachinelearningtechniqueshavebeenappliedonthediscreterepresentationsofspiketrain(binneddata),whichcanberoughlydividedintotwocategories:regressionmodelsandbayesianmodels.Forthedevelopmentoftheregressionmodel,theinuentialoptimallineardecoderisposedasaversionoftheWiener-Hopflter[ 11 ].Elaborationsonthisideaincludelinear[ 55 89 ]andnonlinearregressionmodels,suchasneuralnetwork[ 104 ],Volterrakernel[ 54 ],andkernelregressiontechniques[ 78 91 ],wherethekernel-basedmodelwithspikernelallowsestimationofthenonlinearfunctionalrelationshipbetweenstimuliandneuralresponseandfurtherimprovethedecodingperformance.Bayesianmodelingisanotherpromisingscheme[ 71 101 ],inwhichthepriordistributionofthesignaltobedecodediscombinedwithanencodingmodeldescribingtheprobabilityoftheobservedspiketrain.Inthefollowingsections,aSpikernel-basedmodelandGLMasrepresentativemodelsoftwodifferentcategorieswillbefurtherintroduced. 2.2.1.1Generalized-linear-model-baseddecoder Thegeneralizedlinearmodel(GLM)[ 71 ]isadecodingalgorithmbasedonBayes'rule,inwhichthepriordistributionofthesignaliscombinedwithaencodingmodel 24

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describingtheprobabilityoftheobservedspiketraingiventheinputsignal.TheresultingBayesestimateisoptimalinprinciple,assumingthatthepriordistributionandencodingmodelarecorrect.Tobespecic,theGLMpredictsthecurrentnumberofspikesusingtherecentspikinghistoryandtheprecedingstimulus.InGLM,aspiketrainasapointprocessisgeneratedbyi(t)=f(kix(t)+Xj,hij(t)]TJ /F4 11.955 Tf 11.96 0 Td[(tja)+bi), (2) wherex(t)isthestimulationinput,i(t)denotestheinstantaneousringrateoftheithneuron,kiistheithneuronlinearreceptiveeld,biisadditiveconstantdeterminingthebaselineringrate,andhijisapost-spikeeffectfromthejthneurontotheithneuron.Ingeneral,theconcavenonlinearfunctionf()istheexponentialfunction,becauseofPoissonassumptionandalsostheapplicationofcomputationefciency.Themodelparameters=fki,hij,bigareoptimizedbymaximizingthelog-likelihoodfunctionlogp(rjx,)=Pi,nri(n)logi(n))]TJ /F6 11.955 Tf 12.06 0 Td[(Pi,ni(n)+const.Toestimatetheoptimalxmaptheposteriorlog-likelihoodismaximizedoverthestimulusgiventhetargetringpattern:logp(xjr,)=logp(rjx,)+logp(x)+const,wherep(x)iscomputedbasedonapre-knowledgeofitsdistribution. 2.2.1.2Spikernel-baseddecoder TheSpikernelproposedin[ 90 ]isapromisingdecoderduetoitscapabilitytomodelthenonlinearfunctionalrelationshipbetweentheneuralresponseandstimulus.Itappliesthekernel-basedmachinelearningframeworkandmapsspikecountsequencesintoanhighdimensionalabstractvectorspaceinwhichwecanperformlinearoptimizationtoobtainanonlinearmappingwithnolocalminima.IthasbeenshownthattheSpikerneloutperformsalltheotherstandardvectorbasedkernels[ 90 ],becauseitisdesignedaccordingtothefollowingimportantpropertiesofspikepatterns. 1. Similarringpatternsmayhavesmalldifferencesinabin-by-bincomparison,whichisduetonotonlytheinherentnoiseoftheneuralsystembutalsoresponses 25

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toexternalfactorsthatarenotrecordedandarenotdirectlyrelatedthetothetaskperformed.Therefore,differentlengthsubsequencesofspikecountsamplearecomparedinthekerneldenitionwhichisweightedaccordingtothesubsequencelength. 2. Twopatternsmaybequitedifferentinasimplebin-wisecomparisonbutiftheyarealignedbysomenon-lineartimedistortionorshifting,thesimilaritybecomesapparent.Therefore,timealignmentschemeisinducedinthedenitionofkernel. Thekernelbetweentwospikecountsequencesisdenedby(sl,sm)=NXn=1pnn(sl,sm), (2) whichisasumofkernelsfordifferentpatternlengthweightedwithpn,whereslandsmarethelthandmthringratevectorobtainedbyslidingwindow.Thefeaturemapofsisdenedbyu(s)=Xi2Ind(si,u)js)]TJ /F5 7.97 Tf 6.59 0 Td[(i1j, (2) whichrepresentsthesimilarityofstotheprototypeu.urepresentsapossiblespiketrainprototypeofxedlengthn
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sizeisrequiredtopreservethenetimeresolution,butsparsifythesignal,increasingtheartifactvariabilityandcausingthemodeldimensionality,whichrequirelargedatasetsforpropergeneralizationinsignalprocessingtobeaproblem. Wenotethatthemostaccurateandefcientdescriptionofaringpatternisjustthesetofspiketimes.However,thereisnoalgebraicstructureinthespaceofspiketrains,whichmeansthatnoconventionalregressionalgorithmcanbedirectlyappliedtospiketrains.Ourideaiswhetherwecandeneapositivedenitekernelthatisanoperatoronthesetofspiketimes.ThenthesetofspiketimescanbeprojectedintoareproducingHilbertspace(RKHS)denedbythiskernel,thenwecanbuildadecoderbyapplyingregressionalgorithmintheRKHS.Inthisway,anykernel-basedregressionmethodcanbedirectlyappliedtospiketrains. 2.2.2.1Kerneldesignforspiketrains ReproducingkernelHilbertspaces(RKHS)arespecialHilbertspacesoffunctionswhereallevaluationsarenite.RKHSarecompletelydenedbyapositivedenitefunctionthatiscalledakernel[ 4 ],whichspeciestheinnerproductmetric.TheadvantageofdeningaRKHSforspiketrainsistoobtainacontinuousfunctionalspace,wheretheconventionallinearsignalprocessingalgorithmscanbeapplied.Theissueishowtodeneakernelthatwillbeabletopreservethestatisticallawthatdenesthepointprocess.Accordingtotheliterature,sincespiketrainsalwayscorrespondtoanobservationormeasurementassociatedwithsomeunderlyingpointprocess,itisappropriatetoconsiderthespiketraintobearealizationofpointprocess,fromwhichonlythenumberofspikesandthetimeinstancestheyoccurarerelevant.Ingeneral,forapointprocesspitheconditionalintensityfunction(tjHit)isusedtocompletelycharacterizeapointprocess,wheret2=[0,T]denotesthetimecoordinateandHitisthehistoryoftheprocessuptot.Arecentareaofresearch[ 66 68 ]istodeneaninjectivemappingfromspiketrainstoRKHSbasedonthekernelbetweentheir 27

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conditionalintensityfunctionsoftwopointprocesses[ 66 ].Thereareafamilyofpointprocesskernelsdenedasfollows: 1. Cross-intensity(CI)kernelisdenedbytheinnerproduct(correlation)oftheconditionalintensityfunctionsCI(pi,pj)=h(tjHit),(tjHjt)iL2()=Z(tjHit)(tjHjt)dt. (2) whichisalinearoperatorinthespacespannedbytheintensityfunctionanditispositivesemi-denite[ 66 ]. 2. Nonlinearcross-intensity(NCI)kernelisdenedbythenonlinearproductoftheconditionalintensityfunctionsNCI(pi,pj)=Zexp()]TJ /F6 11.955 Tf 9.29 0 Td[(((tjHit))]TJ /F3 11.955 Tf 11.96 0 Td[((tjHjt))2 2)dt, (2) whichisanonlinearandpositivedenitekernelofconditionalintensityfunctions.Infactitisthecorrentropybetweenintensityfunctionsandthuspreservesmoreinformationbeyondcorrelation[ 49 ].isafreeparameter,whichcontrolsthemetricspacebetweentheinstantaneousvaluesoftwointensityfunctions.largecreatesametricspacesimilartotheL2norm,whereassmallmakethemetricspaceapproachingtoL0norm. 3. Schoenbergkernelisdenedbyanonlineartransformationofthedifferencesquareoftheconditionalintensityfunctions((tjHit),(tjHjt))=exp()]TJ 10.5 8.09 Td[(k(tjHit))]TJ /F3 11.955 Tf 11.95 0 Td[((tjHjt)k2 2)=exp()]TJ /F11 11.955 Tf 10.5 18.45 Td[(R((tjHit))]TJ /F3 11.955 Tf 11.95 0 Td[((tjHjt))2dt 2), (2) whereisthekernelsize.SchoenbergkernelisaGaussian-likekerneldenedonintensityfunctions,whichisstrictlypositivedenitekernelandsensitivetothenonlinearcouplingoftwointensityfunction[ 66 ]. 28

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Thelasttwokernelsarestrictlypositivedenitekernels,thusthemappingbetweenthespiketrainspaceandtheRKHSisinjective[ 4 ].Noticethatthiskernelisverydifferentfromtheonesusedinkernelbasedmachinelearning,e.g.forthesupportvectormachine.InfacthereawindowedthespiketrainismappedintoafunctionintheRKHS.DifferentspiketrainswillthenbemappedtodifferentlocationsintheRKHS,asshowninFigure 2-1 .TheRKHSofwindowedspiketrainsprovideasimplywayto Figure2-1. RelationbetweentheoriginalspaceofspiketrainsandtheRKHSdenedbythestrictlypositivedenedkernel. estimatetheconditionalintensityfunctionnonparametrically,butinduceafreeparameterwindowlength,whichaffectsthestructureoftheRKHS.Inordertoinvestigatethepropertyofeachkernelandthusselectthebestcandidateforneuralmodeling,weanalyzeandcomparetheirdecodingperformanceinsection 2.2.2.4 Inordertobeuseful,themethodologymustleadtoasimpleestimationofthequantitiesofinterest(e.g.thekernel)fromexperimentaldata.Apracticalchoiceusedinourworkestimatestheconditionalintensityfunctionusingakernelsmoothingapproach[ 66 76 ],whichallowsestimatingtheintensityfunctionfromasinglerealization.Theestimatedintensityfunctionisobtainedbysimplyconvolvings(t)withthesmoothing 29

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kernelg(t),yielding^(t)=MXm=1g(t)]TJ /F4 11.955 Tf 11.95 0 Td[(tm),ftm2T:m=1,...,Mg, (2) wherethesmoothingfunctiong(t)integratesto1.Here^(t)canbeinterpretedasanestimationoftheintensityfunction.Therectangularandexponentialfunction[ 24 66 ]arebothpopularsmoothingkernels,whichguaranteeinjectivemappingsfromthespiketraintotheestimatedintensityfunction.Inordertodecreasethekernelcomputationcomplexity,therectangularfunctiong(t)=1 T(U(t))]TJ /F17 11.955 Tf 12.64 0 Td[(U(t)-278(T))(Ttheinter-spikeinterval)isusedinourwork,whereU(t)isaHeavisidefunction.Thisrectangularfunctionwithlowcomputationcomplexitycompromisesthelocalityofthespiketrains,i.e.themappinggivesmoreemphasisontheearlyspikesthanthelaterspikesintheestimatedintensityfunction.However,ourexperimentsshowthatthiscompromiseonlycausesaminimalimpactonthekernel-basedregressionperformance. Themainadvantageofthisfunctionalrepresentationcomparedtobinningisthattheprecisionintheeventlocationisbetterpreservedandthelimitationsofthesparsenessandhigh-dimensionalityisalsoavoidedsincekernelsmoothingmapsthespiketrainintoacontinuousfunction.TheaddedadvantageisthattheRKHShasalinearstructure,soanylinearsignalprocessingalgorithmcanbeeasilyimplementedwiththesocalledkerneltrickwhichdecreasessignicantlythecomputationcomplexity(innerproductsofinnitelylongvectorscanbecomputedbyanevaluationofthekernel). 2.2.2.2Kernelbasedadaptiveregressor Forneuraldecoding,ourgoalistobuildaregressionmodelinthespiketrainkernelspacetocontinuousstimulationwaveforms.Thegreatappealofkernel-basedltersistheusageofthelinearstructureofRKHStoimplementwell-establishedlinearadaptivealgorithmsandtoobtainanonlinearlterintheinputspacethatleadstouniversalapproximationcapabilitywithouttheproblemoflocalminima.Thereareseveralcandidatekernel-basedregressionmethods[ 20 ],suchassupportvector 30

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regression(SVR)[ 92 ],radialbasisfunction(RBF),kernelrecursiveleastsquares(KRLS),kernelleastmeansquare(KLMS)[ 50 ]andetc..Consideringthedynamicsofneuralsystem,KLMSisselectedinourworkduetoitsonlineadaptabilityandlowcomputationcost. ThebasicideaofKLMSistotransformthedatasigivenbysi=ftm)]TJ /F6 11.955 Tf 13.1 0 Td[((i)]TJ /F6 11.955 Tf -418.48 -23.9 Td[(1),tm2[(i)]TJ /F6 11.955 Tf 12.69 0 Td[(1),(i)]TJ /F6 11.955 Tf 12.7 0 Td[(1)+T]:m=1,...,Mg(theithwindowofthespiketimesequenceobtainedbyslidingtheT-lengthwindowwithstep)fromtheinputspacetoahighdimensionalfeaturespaceofvectors(si),wheretheinnerproductscanbecomputedusingapositivedenitekernelfunctionsatisfyingMercer'scondition:(si,sj)=h(si),(sj)i[ 50 ].ForourworktheRKHSofspiketrainsisdenedbytheSchoenbergkernel,unliketheworkinadaptivelteringthatalmostexclusivelyusestheGaussiankernel.Howeverthealgorithmstructureisexactlypreserved,i.e.,thelinearleast-mean-square(LMS)algorithmisdirectlyappliedintheRKHSasaspecialcaseofthegeneralmethodologytoformulatelinearltersandgradientdescentalgorithmsintermsofthekernel(si,sj).Letbethelterweightfunction(whichcanbeconsideredaninnitedimensionalvector)inRKHSandsibetheinputspiketrainand(si)thetransformedinputfunctionintheRKHS,thentheregressoroutputintheinputspaceisy=h,(si)i.Duringonlineadaptationbecomes(n)attimen.FollowingthestochasticgradientLMSupdate,theKLMSinthekernelfeaturespaceusingastochasticinstantaneousestimateofthegradientvector,yields(0)=0(n+1)=(n)+en(sn)(n)=n)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xi=0ei(si) (2) 31

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whereei=di)]TJ /F4 11.955 Tf 12.04 0 Td[(yianddiisthedesiredsignal.Givenandtheinput(sn),theoutputisgivenbyyn=h(n),(sn)i=n)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xi=1eih(si),(sn)i=n)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xi=1ei(si,sn) (2) KLMStopologycanberegardedasagrowingradialbasisfunction(RBF)network,whichallocatesanewunitateachsamplewithsnasthecenterandthescaledpredictionerrorenastheweightcoefcient.Thecoefcientsa(n)=[e1,,en]andthecentersC(n)=fs1,...,sngarestoredinmemoryduringtraining.KLMSonspiketrainscomputesadaptivelynonlinearregressionwithoutconvergingtolocalminima,whichenablesbetteradaptiontoneuralsystemperturbations.TheKLMSalgorithmisintrinsicallyregularizedbythestepsize,thereforethisparametershouldbecarefullydeterminedbecauseitalsoaffectstheconvergencerate[ 50 ]. Themainbottleneckofkerneladaptivelteringisthelinearlygrowinglterstructurewitheachnewsample,whichposesbothcomputationalaswellasmemoryissuesespeciallyforcontinuousadaptationscenarios.Toaddressthisproblem,avarietyoftechniques[ 21 29 48 73 ]havebeenproposedtocurbthegrowthofthenetworks,whereonlynonredundantinputdataareacceptedasnewcenters.Inthiswork,weadoptthequantizationapproach[ 19 ]toconstrainthenetworkgrowth,whichdiscretizestheinputspace(andhencecurbsthenetworksize)byquantizingtheinputvectorintheweightupdateequation.Thealgorithmissummarizedbelow(Algorithm1).Algorithm1showsthepseudocodefortheKLMSwithasimpleonlinevectorquantization(VQ)method,wherethequantizationisperformedbasedonthedistancebetweenthenewinputsnandeachexistingcentersi.Inthiswork,thedistancebetweentwospiketimingsequencesisdenedbyksn)]TJ /F3 11.955 Tf 13.49 0 Td[(sik2.If 32

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thesmallestdistanceissmallerthanapre-speciedquantizationsize",thenewcoefcientenadjuststheweightoftheclosestcenter,otherwiseanewcenterisadded.ThesimpleonlineVQmethodisnotoptimalbutisveryefcient.Sinceinourcasethespiketrainsareoflimitedlength,thealgorithmmustbeappliedseveraltimestothesamedata,toallowconvergence.Therefore,wechoose"=0,whichmergestherepeatedcentersandenjoysthesameperformanceasKLMS. Algorithm1.Quantizedkernelleastmeansquare(QKLMS)algorithm Input:fsn,dng,n=1,2,...,N Initialization:Choosestep-sizeparameter>0,kernelsize>0,andquantizationsize"0andinitializetheweightvector(1):codebook(setofcenters)C(0)=fgandcoefcientvectora(0)=[] Computation: forn=1,2,...,N 1)computetheoutput yn=h(n),(sn)i=Psize(C(n)]TJ /F9 7.97 Tf 6.58 0 Td[(1))j=1aj(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)(sn,Cj(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)) 2)computetheerroren=dn)]TJ /F4 11.955 Tf 11.95 0 Td[(yn 3)computethedistancebetweensnandeachsi2C(n)]TJ /F6 11.955 Tf 12.04 0 Td[(1)d(sn,C(n)]TJ /F6 11.955 Tf 12.04 0 Td[(1))=minjksn)]TJ -458.7 -23.91 Td[(Cj(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1)k 4)Ifd(sn,C(n)]TJ /F6 11.955 Tf 14.13 0 Td[(1))",thenkeepthecodebookunchanged:C(n)=C(n)]TJ /F6 11.955 Tf 14.12 0 Td[(1),andupdatethecoefcientofthecenterclosesttosn:ak(n)=ak(n)]TJ /F6 11.955 Tf 12.92 0 Td[(1)+e(n),wherek=argminjks(n))-222(Cj(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)k 5)Otherwise,storethenewcenter:C(n)=fC(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1),sng,a(n)=[a(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1),en] end 33

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2.2.2.3Multiple-input-multiple-output(MIMO)decodingmodel Therapidadvanceofmicroelectrodearraysandelectrophysiologicalrecordingtechniqueshasenabledthesimultaneousstimulationandrecordingofspatiotemporalactivityofhundredsofneurons.TheMIMOmodelisutilizedtodecodemultiple-channelmicrostimulationwithmultiple-channelneuralactivity.Althoughthetransferfunctionfromthespiketraintothemicrostimulationmaybeacomplexnonlinearrelationship,inRKHSitismodeledasalineardynamiclterY=hW,(S)ithatcorrespondstoauniversalmapperintheinputspace.AssumingthedecoderhasK-channelinputsS(spiketrains)andM-channeloutputsy(stimulation),itiseasytoextendtheprevioussingleinputsingleoutput(SISO)modelintoaMIMOlineardynamiclterrepresentedbyaweightarraydenedby 266666664y1y2...yM377777775=266666664W11W12W1KW21W22W2K...WM1WM1WMK377777775266666664(s1)(s2)...(sK)377777775(2) whereeachWijisaweightvectorintheRKHS(in( 2 ))intheisoptimizedbyapplyingthequantizedKLMSregressiononeachpairofsjandyi.Forillustration,theithoutputofthecontroller^C(z)canbecalculatedby 266666664yi1yi2...yiM377777775=266666664a11a12a1Na12a22a2N......aM1aM2aMN377777775266666664(c11)(c12)(c1K)(c21)(c22)(c2K)......(cN1)(cN2)(cNK)377777775266666664(si1)(si2)...(siK)377777775(2) wherecnkisthekthchannelofthenthcenterandamnisthecoefcientassignedtothenthkernelcenterforthemchanneloftheoutput.ThewaythattheSISOdecoderisextendedtotheMIMOdecodertreatsalltheinputchannelsindependentlywithoutusingthejointinformationamongchannels. 34

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2.2.2.4Comparisonoverkernels Inordertoselectanappropriatekernelforneuraldecoderfromthreecandidates:CIkernel,NCIkernel,andSchoenbergkernel,wecomparetheirdecodingperformanceinsyntheticdataasafunctionofstimulationdurationandsignal-noise-ratio(SNR).Thestimulationdurationsprovideawaytocontrolthetransientstationarityoftheneuralresponse.Morepeakythestimuliare,Moredynamictheneuralresponseare.Thesyntheticdataisgeneratedfromaleakyintegrate-and-re(LIF)neuronstimulatedbythecontinuoussinwavesignaladdedwithwhitenoise.Thestimulationpatternshavefourdifferentdurations(150ms100ms50msand20ms)asshownintheFigure 2-2 ,andtwoSNRlevels(0dBand-2dB)thatistunedwithnoisepower. Figure2-2. Syntheticstimulation Thekernelperformanceof50MonteCarlorunsisevaluatedwithrespectivetothestimulationdurationandSNR,whichisshownintheFigure 2-3 .TheresultsillustratethatthedecreaseofstimulationSNRisinverselyproportionaltotheperformanceforallkernels.SinceCIkernel-baseddecodercanonlycapturethelinearrelationshipbetween 35

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spiketrainandstimulation,CIkernel-baseddecoderperformsworst.Fortheothertwononlinearkernels,SchoenbergkernelhasabetterperformancethanNCIkernelinsmallstimulationduration0dBcase.ThereasonisthatNCIkernelmorereliesonthestationarityassumeofthetemporaldata,sinceNCIkernelissimilartoquantifythecorrentropybetweentwospiketrainsthatisbasedontheaverageovertime.Moreover,incontrastwithNCIkernel,SchoenbergkernelislessinuencedbythedecreaseofSNR,becausetheNCIpaymoreattentiontotheinstantaneousvaluedifferenceoftwointensityfunctionthatiseasilyinuencebythenoise,whereasSchoenbergkernelemphasizestheoveralltemporalpatternofintensityfunction,whichmakeSchoenbergkernelmorereasonableforneuraldecoding.Inordertofurtherinvestigatethepropertyofthekernels,Wedrawtheperformancesurfacewithrespecttothefreeparameters:windowsizeandkernelwidth,asshowninFigure 2-4 .Theperformancesurfaceisestimatedforthesetting(stimulationduration1mswithSNR0dB).Inthiscase,allthreekernelsareabletosuccessfullydecodethestimulationsequence.ForNCIkernel,thekernelwidthisnotdependentonthewindowsize,whichonlyrelatedtotheinstantaneousvalueoftheintensityfunction.Incontrast,theperformanceoftheScheonbergkernelismoreat,whichmeansitlesssensitivetotheselectionofthefreeparameters.ThedecodingaccuracyandrobustnessofthefreeparametersconcludethatSchoenbergkernelisbettercandidateforourneuraldecodingtask.Therefore,itisselectedtobuildthethekernel-basedneuraldecoderinthefollowingwork. 2.2.3SyntheticExperiment 2.2.3.1Neuralcircuit Webuildabiologicalplausibleplantmodelofanensembleofneurons,whichisimplementedintheneuralCircuitSIMulator(CSIM)[ 38 ].Wemodeltheprojectionofatime-varyingthreedimensionalelectriceldattheneuronsfrombipolarandmonopolarstimulationsona28electrodeconguration,asshowninFigure 2-6 .Inthismanner 36

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Figure2-3. Comparisonamongthekernelsdenedinthespaceofspiketrains:CI,NCI,andSchoenbergkernel thecurrenttoeachneuronistheprojectionbetweenthethreedimensionalelectriceldontoaneuron-specicunitvector,representingneuronalelementssuchasaxonswhichcarrythecurrenttoneuralcircuitandarepreferentiallyexcitedbyoneoftheprojectionsoftheelectriceld. Theneuralensemblehasatwolayerstructure,asshowninFigure 2-5 .Theleftlayeristheinputlayer,whereeachinputneuroniscoloredbyitsprojectiondirection.Theensembleofneuronsconsistsof135leakyintegrate-and-reneurons,20%ofwhicharerandomlychosentobeinhibitory(orange).Randomcircuitsareconstructedwithsparse,primarilylocalconnectivity.Parametersofneuronsandsynapsesarebasedon[ 51 ],whicharechosentotdatafrommicrocircuitsinratsomatosensorycortex.Randomcircuitsareconstructedwithsparse,primarilylocalconnectivity,bothtotanatomicaldataandtoavoidchaoticeffects. Thestimulationisgeneratedbylteringtheelectricaleldwithalowpasslterexp()]TJ /F5 7.97 Tf 10.5 4.71 Td[(x ),where=10ms,whichisinthenormaltimeconstantrangeofneuroncellbodyanddendrites[ 39 ].Foreachstimulationconguration,whichincludesadiscreteset 37

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Figure2-4. Decoderperformancesurfacewithrespecttothekernelfreeparameters ofamplitudes,thesingleneuralringrateisestimatedinthe50msintervalrightafterthestimulation.Itsmeanandstandarddeviationover10repeatedtrailsareshowninFigure 2-6 ,wheretherstsubgureshowsthebipolarandmonopolarstimulationsona28electrodecongurationswhichisusedtomodelthe3Delectriceldasshowninthesecondsubgure.Thecontinuousstimulationisgeneratedbypassingthese3Delectricaleldcongurationpatternsthroughalowpasslterexp()]TJ /F5 7.97 Tf 10.51 4.71 Td[(x ),where=10 38

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Figure2-5. Abiologicalplausiblesyntheticneuralcircuit. msandimprintedintothesyntheticneuralmodel(Figure 2-5 ).Foreachstimulationconguration,thesingleneuralringrateisestimatedinthe50msintervalrightafterthestimulation.Itsmeanandstandardderivationover10repeatedtrailsareshowninthethirdandfourthsubgures,respectively. Thecontinuousstimulationsequenceisformedbyuniformlydrawingthestimulationcongurationwithi.i.d.intervalsbetweenstimulations.Thedistributionoftheintervalsis 39

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Figure2-6. Thestimulationcongurationandthestatisticsoftheneuronresponse. atruncatednormaldistributionTN(0.05,0.0025),T2[0,0.1](s).AnexampleofthestimulationandthecorrespondingneuralresponseareshowninFigure 2-7 Figure2-7. Thepatternofthe3Dstimulationdenedbytheelectricaleld(theupperplot)andtheresultingneuralresponse(thebottomplot). 40

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2.2.3.2Comparisonoverdecoders Thedecoderinputsareneuraleventtimingvectors,whichareobtainedbyslidinga200mswindowevery2ms(windowsize:[-100ms100ms]).Eachwindowincludestheprevious-100msspiketraintohelpcanceltheambiguityinducedbythelastinginuenceofpreviousstimulusonthecurrentneuralringpattern.Thecorrespondingoutputistheestimatedamplitudeofthestimulationatthetimet. AdecodingmodelbasedontheSchoenbergkernelinducedRKHSisupdatedviaKLMS.Forcomparison,wealsoapplytheothertwoneuraldecodersbasedonthediscretizationrepresentationofspiketrain:GLM[ 71 ]2andspikernel[ 91 ]3(KLMSisalsousedtoupdatethemodelcoefcients).Timediscretizationis2ms,whichisdeterminedbytherequiredtimeresolutionofthemicro-stimulation.Themodelsaredeterminedbythebestresultsafterscanningtheparameters,(fortheSchoenbergKLMSthekernelwidth2=0.3andthelearningrate=0.001arechosen).Thenormalizedmeansquareerror(NMSE)betweenthereconstructedstimulation(y)andthedesiredstimulation(d)isutilizedasanaccuracycriteriadenedbyNMSE=1 NP(d(n))]TJ /F4 11.955 Tf 11.96 0 Td[(y(n))2 var(d(n)) (2) NMSEsareobtainedbyperformingtenfoldcross-validationforeachdecoder,whichareproducedbyrandomlysplittingthedatainto10groups,400samplespergroup.Weusenineoutofthetenfoldsofthedatatotraindecodersandcomputeanindependenttesterrorontheremainingfold.TheresultsareshowninTable1.Schoenberg-kernel-baseddecoderout-performsthespikernel-baseddecoder 2p(x)iscomputedbasedonaGaussiandistributionN(,2)(and2areestimatedfromthedata).3Freeparametersandare0.7and0.99,whicharedeterminedbythebestresultsafterscanningtheparameters. 41

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Table2-1. Comparisonamongneurondecoders.NiandMirepresentthespikenumberandthewindowlength,respectively. XXXXXXXXXXXXPropertyMethod Schoenbergkernel Spikernel GLM NMSE(mean/STD) 0.23/0.15 0.29/0.12 0.75/0.23 Kernelcomputationtime O(NiNj) O(MiMj) X Kernelfreeparameter 1 3 X andtheGLM-baseddecoder.AlthoughtheperformanceofspikernelisclosetothatofSchoenbergkernel,itscomputationtimeO(MiMj)ismuchgreaterthanthatofSchoenbergkernelO(NiNj),whereMi=100andNi4withtheaverageringrateof20Hz. 2.2.4AnimalExperiment 2.2.4.1Ratdata AnimalprocedureswereapprovedbySUNYDownstateMedicalCenterIACUCandconformedtoNationalInstitutesofHealthguidelines.AsinglefemaleLong-Evansrat(Hilltop,Scottsdale,PA)wasimplantedwithtwomicroarrays(16contactseachina28gridNeuronexus).TheelectrodescoveredsomatosensoryareasofthecortexS1,andtheVPLnucleusofthethalamus[ 32 ].Neuralrecordingsweremadeusingamultichannelacquisitionsystem(TuckerDavis).Theratwasplacedintoasmallchamberwithameshoorwhichwassuspendedaboveatable.Theapparatushelpedkeepitcalmandstationaryeventhoughtheyremainedawake.Spikeandeldpotentialdatawaspreamplied1000x(ltercutoffsat0.7and8.8kHz)anddigitizedat25kHz.LFPswerefurtherlteredfrom1to300Hzusinga3rdorderButterworthlter.Spikesortingwasachievedusingk-meansclusteringoftherst2principalcomponentsofthedetectedwaveforms.TheexperimentinvolvesdeliveringmicrostimulationtoVPLandtactilestimulationtorat'sngersinseparatesections. Microstimulationwasadministeredonadjacentpairs(dipoles)ofthethalamicarray.Thestimulationwaveformsweresinglesymmetricbiphasicrectangularcurrentpulses;Eachrectangularpulsewas200slongandhadanamplitudefromtheset10A;20A; 42

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and30A.Inter-stimulusintervalswereexponentiallydistributedwithmeanintervalof100ms.Stimulusisolationusedacustombuiltswitchingheadstage.Thebipolarmicro-stimulationbeingimprintedinThalamus,whichcontains24patterns:8differentlocations,3differentamplitudelevelsforeachlocation,and125eventsforeachpattern,asshowninFigure 2-8 Figure2-8. Micro-stimulationpatterns Theexperimentalprocedurealsoinvolveddelivering30-40short100mstactiletouchestotherat'sngers(repeatedfordigitpads1-4)usingahand-heldprobe.Theratremainedstillfortherecordingdurations,andtrialswerecancelediftheratchangedorientation.Theappliedforcewasmeasuredusingaleverattachedtotheprobethatpressedagainstathin-lmresistiveforcesensor(TrossenRobotics)whentheprobetipcontactedtherat'sbody.Theresistivechangeswereconvertedtovoltageusingabridgecircuitandwerelteredanddigitizedinthesamewayasdescribedabove.Thedigitizedwaveformswerethende-meanedandlteredat1to60Hzusinga3rdorderButterworthlter.Therstderivativeofthissignalisusedasthedesiredstimulationsignal,whichareshowninFigure 2-9 43

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Figure2-9. Dataoftheratsensorystimulationexperiment 2.2.4.2Resultsoftactilestimulation Wenowpresenttheresultsofdecodingthetactilestimuluswaveformandmicro-stimulationusingKLMSoperatingontheSchoenbergkernel.Timediscretizationis5ms.Thelearningrateandkernelwidtharedeterminedbythebestresultsoftestdataafterscanningtheparameters.Thenormalizedmeansquareerror(NMSE)betweentheestimatedstimulation(y)andthedesiredstimulation(d)isutilizedasanaccuracycriteria. NMSEsoftactileareobtainedacross8trialsdatasets.Foreachtrial,weuse20sdatatotrainthedecodersandcomputeanindependenttesterrorontheremaining2.5sdata.Themeanandstandardderivationofnormalizedmeansquareerroris0.63=0.11(mean/standardderivation).Inordertoillustratethedetailsofthedecodingperformance,thetestresultsofthersttrialareillustratedinFigure 3-3 .Thedecoderoutputisabletocapturemainstructureofthedesiredtactilestimulation.Itisobservedthattheoutputofthespikedecodeructuatesenormouslyandmissessomepulses,i.e.,around0.65s,becauseofthesparsityandvariabilityofthespiketrain. 2.2.4.3Resultsofmicro-stimulationstimulation Wealsoimplementthedecodertoreconstructthemicro-stimulation.First,wemapthe8differentstimulationlocationsto8channels.Wedismisstheshapeofeachstimulus,sincethetimescalethestimuluswidthistwoshort(200s).Thedesiredstimulationpatternofeachchannelisrepresentedbyasparsetimeseriesofthe 44

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Figure2-10. ResultsoftheSchoenbergkernelbaseddecoderoftherststimulationtrial. stimulationamplitudewithsamplingrate200Hz.WeimplementShoenbergkernelwithKLMSforMIMOsystemonthemicrostimulationdata.NMSEsareobtainedwithtensubsequencedecodingresults,whereweuse120sdatatotrainthedecodersandcomputeanindependenttesterrorontheremaining20sdata.ThespiketraindecodersuccessfullycapturesthestimulationpatterninChannel123456and8.Thestimulationisnotwelldiscriminatedfromdifferentoutputchannels,whichcausesthefalse-stimulipresentinspikedecoderoutput. 45

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ADesiredstimulationpattern BSpikedecoderoutput Figure2-11. Resultsofreconstructingmicro-stimulation. 2.3AdaptiveInverseControlofNeuralSpatiotemporalSpikePatterns IntheMIMOcontrolsystemscheme,theplantrepresentstheneuralcircuitdrivenbythestimulationcontrolsignalandproducesspikeactivityoutputs.Thebasicideaofadaptiveinversecontrolistooptimizethecontrollertobetheinverseoftheplant 2-12 .Theadaptiveregressionalgorithmattemptstomakethecascadeofthecontrollerandtheplantbehavelikeaunittransferfunction(i.e.aperfectwirewithsomedelay).Therefore,thetargetspiketrainisusedasthecommandinputoftheinversecontrollertogeneratethecontrolsignaloutput,whichisabletodrivetheplantsuchthattheoutputspiketrainfollowsthetargetringpattern. 2.3.1Filtered-LMSalgorithm Theltered-LMSadaptiveinversecontroldiagram[ 105 ]showninFigure 2-12 representstheltered-approachtond^C(z).IftheidealinversecontrollerC(z)istheactualinversecontroller,themeansquareoftheoverallsystemerrorkwouldbeminimized.Theobjectiveistomake^C(z)ascloseaspossibletotheidealC(z).The 46

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differencebetweentheoutputsofC(z)and^C(z),bothdrivenbythecommandinput,isthereforeanerrorsignal0.Sincethetargetstimulationisunknown,insteadof0,alterederror,obtainedbylteringtheoverallsystemerrorkthroughtheinverseplantmodel^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z),isusedforadaptationinplaceof0. Figure2-12. AFiltered-LMSadaptiveinversecontroldiagram. Duringtheadaptionoftheinversecontroller^C(z),becauseoftheneuralsystemresponsetime,amodelingdelayisadvantageous,whichisdeterminedbytheslidingwindowlengththatisusedtoobtaintheinversecontrollerinput.Thereisnoperformancepenaltyfromthedelayaslongastheinputto^C(z)undergoesthesamedelay.Theparametersoftheinversemodel^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)areinitiallymodeledofineandupdatedduringthewholesystemoperation,whichallows^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)toincrementallyidentifytheinversesystemandthusmakeapproach0.Moreover,theadaptationenables^P)]TJ /F9 7.97 Tf 6.58 0 Td[(1(z)totrackneuralplasticity.Thusminimizingtheltererrorobtainedfrom^P)]TJ /F9 7.97 Tf 6.58 0 Td[(1(z)makesthecontrollertofollowthesystemvariation. If^P)]TJ /F9 7.97 Tf 6.58 0 Td[(1(z)istheperfectinverseplantandtheplantisfreeofdisturbance,thenthelterederrorisexactlyequivalentto0.Inspiteoftheselimitations,thelterederrorcanstillperformwellfor^C(z)adaptation,becauserst,theplantdisturbancesarenormally 47

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uncorrelatedwiththeplantinputandthecommandinputandsecond,errorsin^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)maynotbecriticaltothecorrectconvergenceof^C(z)[ 105 ]. 2.3.2ControldesigninRKHS Inthiscontrolscheme,boththecommandinputthetargetspiketrainsandthecontrolledneuralsystemoutputarespiketrains.ThespiketrainkernelsmapthespiketrainsintotheRKHS;thisavoidsthesparsenessanddimensionalityproblemofusingbinneddata.Inaddition,intheRKHS,bothofthemodels(theplantinverse,^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)andthecontroller^C(z))arelinearsothereisnodangerofconvergingtolocalminima.Inourwork,KLMSisusedtomodelthecontroller,whichkeepstheadaptationcomputationcostatO(n),wherenrepresentsthesamplesize.Akernel-basedadaptiveinversecontrolhasbeenproposedin[ 102 ],whichalsoallowsalinearcontrolinfeaturespace.However,thesupportvectorregressionwasusedtomodelthecontroller,whichmakesthecomputationcostbecomelarge(O(n2)). Specically,thecontroller^C(z)andtheplantinverse^P)]TJ /F9 7.97 Tf 6.58 0 Td[(1(z)areseparatelymodeledwiththeSchoenbergkernel,andthemodelcoefcientsareupdatedwithKLMS.ThisstructureisshowninFigure 2-13 .ThemodelcoefcientsW^CandW^P)]TJ /F16 5.978 Tf 5.76 0 Td[(1representtheweightmatrixof^C(z)and^P)]TJ /F9 7.97 Tf 6.58 0 Td[(1(z)obtainedbyKLMS,respectively.x,yandzarethewindowedtargetspiketrain(commandinput)ofthecontroller,theestimatedstimulation,andthewindowedplantoutputspiketrain,respectively.xisdelayedtargetsignal,whichisalignedwiththeplantoutputz.()representsthemappingfunctionfrominputspacetoRKHS. Forthecontroller^C(z),theltererrorisusedasasubstituteof0toestimatekernelcentercoefcients.Thelterederrorisobtainedbylteringtheoverallsystemerrorkthroughaninverseplantmodel^P)]TJ /F9 7.97 Tf 6.58 0 Td[(1(z).Fortheneuralsystem,theoverallsystemerrorkshouldbethedifferencebetweentwospiketrains.Fortunately,intheRKHSforithchannel(xi))]TJ /F3 11.955 Tf 12.26 0 Td[((zi)iswelldened.Therefore,theoverallsystemerrorforchannelicanbedenedbyk(i)=(xi))]TJ /F3 11.955 Tf 12.09 0 Td[((zi),whichmeantheadaptationofthe 48

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Figure2-13. AnadaptiveinversecontroldiagraminRKHSofspiketrain. controllerparametersseekstominimizethedistancebetweenthetargetspiketrainandtheplantoutputintheRKHSoftheplantinverse^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z).Inthisway,sincetheinversemodel^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)hasalinearstructureinRKHS,thelterederrorforstimulationchannelj21,...,Mis(j)=W^p)]TJ /F16 5.978 Tf 5.76 0 Td[(1j1W^p)]TJ /F16 5.978 Tf 5.76 0 Td[(1j2W^p)]TJ /F16 5.978 Tf 5.75 0 Td[(1jK(x)(x)(xK)0)]TJ /F11 11.955 Tf 11.95 16.85 Td[(W^p)]TJ /F16 5.978 Tf 5.76 0 Td[(1j1W^p)]TJ /F16 5.978 Tf 5.76 0 Td[(1j2W^p)]TJ /F16 5.978 Tf 5.75 0 Td[(1jK(z1)(z2)(zK)0 (2) Weassume^C(z)hasK-channelinputsx(spiketrains)andM-channeloutputsy(stimulation).KLMSwithquantizationisusedtomodel^C(z)withNinputsamples.Sinceamongdifferenttrials,thetargetspiketrainisrepeated,whichmeanstherepeatedsampleswillbemergedtothesamekernelcenterbythequantizationandthusthenetworksizeoftheinversecontrollerisxed(Ncenters),onlythecoefcientmatrixaisupdatedwiththelterederrorduringthewholecontroloperation.Theithoutputof^C(z)canbecalculatedby 49

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266666664yi1yi2...yiM377777775=266666664a11a12a1Na12a22a2N......aM1aM2aMN377777775266666664(c^c11)(c^c12)(c^c1K)(c^c21)(c^c22)(c^c2K)......(c^cN1)(c^cN2)(c^cNK)377777775266666664(xi1)(xi2)...(xiK)377777775(2) Wherec^cnkisthekthchannelofthenthcenterandamnisthecoefcientassignedtothenthkernelcenterforthemchanneloftheoutput. 2.3.3Syntheticexperiment 2.3.3.1Controllingneuralringpatternwithoutperturbations Weconductanonlinecontrolofthesyntheticneuralcircuitwiththeadaptiveinversecontrolframework,wheretheinversecontroller^C(z)andtheinversemodel^P)]TJ /F9 7.97 Tf 6.58 0 Td[(1(z)aremodeledbySchoenberg-kernel-baseddecoders.Theinversemodel^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)isinitiallytrainedofinewith6softrainingdata:the135-channelwindowedspiketimingsequencesastheinputandthecorrespondingstimulationasthedesiredsignal.Afterinitialization,anovel2sspiketrains(135channels)areusedasthetargetringpatternandrepeatedlyconcatenatedastheinputoftheinversecontroller.Theresultingstimulationisfeedtotheneuralsystemplant.Theparametersoftheinversecontroller^C(z)areadjustedsample-by-samplewiththelearningrate=0.001byminimizingthecurrentlterederror(i),whichisobtainedbypassingtheoutputoftheplant(z(i))andtargetspiketrainswithdelay=100ms(x(i))throughtheinversemodel^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z).Tomaintainthesystemstability,weupdatetheinversemodel^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)every2swiththeplantoutputyrecordedinthelast2sasinputandthecorrespondingplantstimulationasthedesiresignal(learningrate=0.001).Weimplementthesameadaptationprocedureinthefollowingpartsofthischapter.Figure 2-14 illustratesthecomparisonbetweentheneuralresponsedrivenbythecontrolsignalandthetargetspiketrains. Themaingoalofcontrolistominimizethedissimilaritybetweenthesystemoutputandthetargetringpattern.Therefore,thecontrolperformanceisevaluatedusingthe 50

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similaritymeasureproposedin[ 65 88 ]betweentwospiketrains.Thetwospiketrains1(t)ands2(t)arerstconvolvedwiththesmoothingkernelh(t)=exp()]TJ /F4 11.955 Tf 9.3 0 Td[(t=)U(t),yieldingthetimeseriesof^s1(t)and^s2(t),respectively,wherethetimeconstant=100msthatoverlaps2spikesonaverage(meanringrateofallneurons:20Hz).Thenthepairwisecorrelationcoefcientbetween^s1(t)and^s2(t)iscomputed:Similarity=RT^s1(t)^s2(t)dt q RT^2s1(t)dtq RT^2s2(t)dt (2) whereSimilarity2[0,1].Ifandonlyifs1(t)ands2(t)areidentical,Similarity=1.Iftwospiketrainsarebothempty,thesimilarityisdenedby0. Thesimilaritybetweenthesystemoutputandtargetspiketrainsovereachchannelsiscalculatedtoevaluatethecontrollerperformancetodrivetheneuralcircuittoproducethetargetspatiotemporalringpattern,asshowninFigure 2-14B .Thesimilarityperchannelissortedbyneurons'ringrate.Theboxplotofthesimilarityoverthechannelsareestimated,whichdemonstratesthatthesimilarityof50%neuronsareover0.83.Becauseofthesparse,primarilylocalconnectivityandtheexistinginhibitorysynapticconnection,neuronswithlowaverageringrateareunabletobemodulatedbythestimulation,whichexplainsthelowsimilarityofneuronswithlowringrates. Inthiscase,sincethetargetneuralpatternisgeneratedfromthecurrentneuralcircuit,theplantmodelisabletoreproducethetargetspatiotemporalringpatternwell,whenthecontrollermodelstheinverseoftheneuralcircuitwithhighaccuracy,sothisisabestcasescenario.Inanycase,ourresultsshowthatundertheseconditions,thiscontrolschemeisabletondthehiddendesiredmicrostimulationandforcetheoutputneuronstoreinasimilarpattern. 2.3.3.2Controllingneuralringpatternwithperturbations Consideringneuronalplasticityinducedbyinjury,environmentalchanges,ormicro-stimulation,itisnecessarytotesttheabilityoftheinversecontrollertotrack 51

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AComparisonbetweentheneuralsystemoutputandthetargetspatiotemporalringpattern. BSinglechannelsimilaritybetweentheneuralsystemoutputandthetargetspatiotemporalringpattern.Thesimilarityboxplotisforthepopulationofneurons. Figure2-14. Performanceofstimulationoptimizationusingadaptiveinversecontrol. 52

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underlyingchangesofthefunctionalrelationshipbetweenstimulationandneuralresponses.Theperturbationinducedbyrandomlyresettingthesynapticconnectivityandthelocalizationoftheinhibitoryneuronsinneuralcircuitisusedtotesttheadaptingcapabilityofourcontrolsystem.Inordertoquantifythedifferencebetweentheoriginalneuralcircuitandtheperturbedone,thespike-triggeredaverage(STA)ofeachneuronforthreedimensionstimulationisestimatedtodemonstratethedifferenceoftheneuralresponsepropertyinFigure 2-15 ,whereeachsubgureshowsthe100msSTAsforonedimensionstimulation,whereSTAsof135neuronsaredemonstratedina15(x-axis)9(y-axis)array. Figure2-15. Comparisonofspike-triggeredaverages(STA)ofthethreedimensionallteredelectriceldbetweentheoriginalneuralcircuitandtheperturbedneuralcircuit. Weusetheinversecontroller^C(z)andtheinversemodel^P)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)thatconvergedtotheinverseoftheoriginalneuralcircuitwiththepreviousdataastheinitialcondition. 53

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Thesameadaptationprocessisimplementedonthecontrolsystemtocontroltheperturbedneuralcircuit.Theneuralperturbationadaptingprocedureisillustratedbythesample-by-sampleevolutionofthesimilaritybetweentheneuralsystemoutputandthetargetspiketrainsinFigure 2-16A ,whichisestimatedbyslidinga2swindowby2msstepsandsortedbytheaverageringrateofeachchannel.Wealsodrawthesimilarityboxplotforthepopulationofneuronsevery2sasshowninFigure 2-16B Variationsintheunderlyingneuralorganizationinitiallycausedissimilaritybetweenthesystemoutputandtargetspiketrains(leftoftheFigure 2-16A .Theadaptationallowstheinversecontrollertoconvergetotheinverseofthecurrentneuralcircuitandthusprogressivelyincreasesthesimilaritybetweenthesystemoutputandtarget. APerformanceevolutionduringadaptation. BBoxplotofsimilarityoverchannelsduringadaptation. Figure2-16. Theperformanceevolutionduringadaptationoftheinversecontrollertofollowtheperturbedneuroncircuit. 54

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Thecontrolleradaptationconvergesafter9trialswiththefollowingstoppingcriterion:thedifferenceofthemeansimilarityofallneuronsbetweentwoadjacentadaptationsislessthan0.015.Atthispoint,thesimilarityboxplotinFigure 2-17B showsthat50%ofneuronshaveasimilarityofatleast0.64.ThesimilarityperneuronsortedbytheirringrateisshowninFigure 2-17B ,whichrevealsthatthecontrolschemeiscapableoftrackingneuralperturbationsbyadaptingthecontrollertotheinverseoftheperturbedneuralcircuit.InFigure 2-17B ,theperturbationofneuroncircuithappensat0s.Figure 2-16A showsthesinglechannelsimilaritybetweentheneuronsystemoutputandthetargetspatiotemporalringpattern.Figure 2-16B showsthesimilarityboxplotoverallchannels,whichisestimatedevery2s.Therstboxplotisestimatedimmediatelybeforetheperturbation.TherstboxplotinFigure 2-16B isequaltotheoneinFigure 2-14B (beforeperturbation)andthelastboxplotinFigure 2-16B isequaltotheoneinFigure 2-17B (afterconvergence). However,theresultsoftheperturbedneuralcircuitbecomepoorerthantheoneswiththeoriginalneuralcircuit.Thisisexpected,becausethetargetringpatternisnotgeneratedbytheperturbedneuralcircuit.Therefore,thereisnoguaranteethattheperturbedneuralcircuitiscapableofproducingthetargetneuralpattern,i.e.theremaynotexistastimulationthatcandrivetheperturbedneuralcircuittoreproducethetargetringpattern.Inanycase,thisisactuallyamorerealisticsituation,whenweusetheelicitedneuralresponsetomimictheringpatterninducedbythenaturalstimulation,sincethemicro-stimulationmaychangetheoriginalunderlyingneuralcircuitstructurethatgeneratesthetargetringpatternwithnaturalstimulation.Nevertheless,theinversecontrollerwillalwaysconvergetominimumofthecost,becausethewholecontrolschemehasalinearstructureinRKHSanditistrainedwithaconvexcostfunction. 2.4Discussion Thesparsenessofneuralsignalsandtheneuralsystemplasticityinducedbyhabituationtostimulation,injury,environmentchanges,etc.presentachallengewhen 55

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attemptingtopreciselycontroltheneuralresponsetomimicthetargetspatiotemporalringpatterninducedbynaturalstimulation.ThischapterproposesaneuraldecodingmethodologywithhightimeresolutionbasedonaSchoenbergkernelregressordeneddirectlyinthespaceoftheneuraleventtimingsinsteadofthemoreconventionalintensityraterepresentation.Thismethodologybypassesthesparsenessofconventionalintensityraterepresentation,efcientlydecreasesthecomputationtimebyusingonlytheprecisetimingoftheneuraleventsinsteadofhighmodelorders,andtheresultsonthesyntheticdatashowthatthisdecodingmethodologyoutperformsthemethodsbasedonthediscretizedrepresentationofspiketrain:SpikernelandGLM.Withthisnoveldecodingmethodology,aMIMOadaptiveinversecontrolschemeisbuiltonspiketrainRKHStocontrolneuralresponsetomatchthetargetspatiotemporalringpattern.TheexperimentalresultsarepresentedinasyntheticneuralsystembuiltfromLIFneuronsbuttheexcitationisdiverseandbroadtomimicmorerealisticconditions.Theyshowthatthecontrolleradaptationallowsthecontrolsystemtoprovidethedesiredringpatternswithhighaccuracy.Moreover,thecontrollerisalsoabletoadapttoperturbationoftheunderlyingneuralsystemorganization,whichisanessentialqualityforthecontrolsystem,otherwisethecontrolsystemwillfailtotrackthetargetsignalwhenperturbationshappen.Inthismorerealisticcase,thedesiredstimulationpatternmaynotexisttoreproducethetargetsignalbutthelinearstructureofthecontrolleranditsconvexcostfunctionguaranteethatthiscontrolschemeisabletondtheglobaloptimalspatiotemporalpatternofstimulationthatcanminimizethedissimilaritybetweenthesystemoutputandthetargetsignal.ThesepreliminaryresultsshowthatthisRKHSbasedmethodologyforsomatosensotystimulationmaybeapplicableinrealstimulations.Ourmajorconcernistherequirementtohavesufcientneuralspikingtodrivetheadaptationandtheinuenceofthespiketrainvariabilityandthiswillbeaddressedinrealneuralsystem. 56

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AComparisonbetweentheneuralsystemoutputandthetargetspatiotemporalringpatternaftertheinversecontrollerconvergedtotheinverseoftheperturbedneuroncircuit. BSinglechannelsimilaritybetweentheneuralsystemoutputandthetargetspatiotemporalringpatternaftertheinversecontrollerconvergedtotheinverseoftheperturbedneuroncircuit.Thesimilarityboxplotisforthepopulationofneurons. Figure2-17. Performanceofstimulationoptimizationafteraperturbationusingadaptiveinversecontrol. 57

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CHAPTER3NEURALDECODINGFROMMULTI-SCALENEURALACTIVITY Theproblemofextractinginformationfromspiketrainsandlocaleldpotentials(LFPs),electrocorticogram(ECG),andelectroencephalogram(EEG)isrootedintheirunknownunderlyingrelation,multiplespatialandtemporalscaleandheterogenoussignalformat.Inourwork,wemainlyaddressthemodelingtaskofspiketrainsandLFPsasexampleofmultiscaleneuralmodeling,sinceitisabletocovermostchallengesinmultiscaleneuraldecoding.AnotherreasonofselectingspiketrainsandLFPsisthattheyarerecordedformthesamemicro-electrodearrays,whichguaranteesthetimealignmentandthussimplifythedatapreprocessingprocedure. SpiketrainsandLFPsencodecomplementaryinformationofthestimuliorbehaviors[ 8 37 ].Inmostrecordings,spiketrainsareobtainedbyusingahigh-passlterwiththecutofffrequencyabout300)]TJ /F1 11.955 Tf 9.29 0 Td[(500Hz,whileLFPsareobtainedbyusingalow-passlterwiththecutofffrequencyabout300Hz[ 77 ].Spiketrainrepresentthesingle-unitneuralactivitywithanetemporalresolution.However,itsstochasticpropertiesinduceconsiderablevariability,especiallywhenthestimulationamplitudeissmall,thatis,samestimulicannotelicitthesameringpatterninrepeatedtrails.Theexperimenterstypicallyaddressthisproblemthroughmultipletrialrepetitionsandaveragingtheneuralresponseinordertodimmishthevariability.However,inBMIdecodingthekinematic/stimulationpatternhastobeinstantaneousfromsingletrial.Therefore,trailaveragingisimpossibleinthiscontext. Incontrast,LFPsreecttheaveragesynapticinputtoaregionneartheelectrode[ 16 ],whichlimitsspecicationbutprovidesrobustnessforcharacterizingthemodulationinducedbystimuli,evenatlowamplitudes.Furthermore,itisknownthatLFPsprovidetheinformationofpopulationneuralactivity,whichcannotbecompletedmeasuredonlywithspiketrainsfromalimitedsubsetofthetotalneuronsinaregion.Moreover, 58

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thepopulationactivitystatehasbeenshowntobeassociatedwithuctuationsinspiketrains[ 16 ]. Therefore,anappropriatecombinationofLFPsandspiketrainsindecodingmodelsmakeitpossibletomoreaccuratelydecodethestimuliorbehaviorsfromtheneuralresponse.Forexample,thedecodercancoordinateLFPorspikepatternstotagparticularlysalienteventsorextractdifferentstimulusfeaturescharacterizedbydifferenttypesignals. However,differentsignalpropertiesandformatsbetweenLFPsandspiketrainsmakemerginginthesamemodeldifcult.First,thesignalformatsaretotallydifferentbetweenthetworepresentations.Aspiketrainthatisasetoforderedspiketimings,whichcanbeinterpretedasarealizationofapointprocess[ 41 93 ],whileLFPisacontinuousamplitudeprocess.Moreover,thetimescaleofLFPsissignicantlongerthanspiketrains.Therefore,onlysmallnumberofworkswithsimplyassumptionoftherelationshipbetweentwosignalshavebeendoneformodelingwithbothspiketrainsandLFPsasinputfeatures[ 42 ].However,thefullrelationshipbetweenLFPsandspiketrainsisstillasubjectofsomecontroversy[ 9 46 106 ].ThemodelassumptionsabouttherelationshipbetweenspiketrainandLFPlimitstheitsability. Inthischapter,weproposeatensor-product-kernel-basedregressortodecodethestimulusinformationfrombothspiketrainandLFPasaexampleofmultiscaleneuraldecoding.Thetensorproductkernelallowsmodelingdifferenttypesofsignalsindividuallyandmergetheirinformationinthefeaturespacedenedbythetensorproductoftheindividualkernelsforeachtypesignal[ 87 ]withnoassumptionbetweentwodatasource. Spiketrainscharacterizestimulusinformationwithanetimeresolutionbutalargevariability.WemodelsspiketrainsbyusingtheSchoenbergkerneldenedinthespiketimingspaceintroductioninChapter 2 ,whichdecreasesthecomputationtimeincontrastwiththekernelsbasedontheconventionalraterepresentationandavoids 59

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sparsehigh-dimensionalvectorscorrespondingtobinnedspiketrains.Incontrast,LFPs,ascontinuousprocesses,aremodeledbytheSchoenbergkerneldenedinthecontinuousspaceonthetimestructure(nonlinearcorrelation),whichisabletoenhancetherobustnessofstimulusestimation.Thetensorproductofthetwokernelsmapdataintoajointkernelfeaturespace,wherethekernelleastmeansquare(KLMS)algorithmestimatesthenonlinearlymappingfromtheneuralresponsetothestimuliasnecessaryinsomatosensorystimulationstudies. Inaddition,controllingneuralactivitywithadaptiveinversecontrolisalsoextendtothemultiscaleneuralactivityusingthetensorproductkernelbaseddecodingmethodology.Multiscaleneuralactivityprovidemorecompleteinformationoftheoptimalstimulationpatternwithamorerobustway,whichwillalsoenhancetherobustnessandaccuracyofthecontrolperformance. 3.1TensorProductKernelforMulti-ScaleNeuralActivity WeutilizedistinctivekernelstomapthespiketrainsiandLFPxiintofeaturefunctions'(si)and (xi)intwoRKHSsdenedbyx(xi,xj)ands(si,sj),respectively.InordertomergetheinformationfromLFPsandspiketrains,thetensorproductkernelssandxincorporatetwoindividualRHKSsssandxxintoajointkerneldenedfeaturespace(ss)(xx).Thetensorproductkernelisdenedby(xi,si,xj,sj)=(sx)(xi,si,xj,sj)=x(xi,xj)s(si,sj) (3) Thisconstructionhasseveraladvantagesthatarelistedhere 1. ifbothsandxarepositivedenitekernels,thentheirtensorproduct=sxisalsoapositivedenitekernel[ 87 ]. 2. Sincesubsequentprocessingandestimationareconductedinthekernelspace,therearenoconstrainsontheformatoftheinputsignalsandtheircorrespondingkernelfunctions. 60

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3. Thekernelimposesnoassumptionoftherelationshipbetweentwoinputs. Thoseadvantagessolvetheissuethattwosignalhavedifferentsignalformats,timescales,andspatialscales.ThekernelsforspiketrainsandLFPscanbeselectedindividuallybasedontheirownproperties.Forspiketrainsthatcanbeinterpretedasobservationsofapointprocess,theSchoenbergkerneldenedinthespiketimingspaceisapplied,whichweinvestigatedinChapter 2 .LFPsareacontinuousamplitudestochasticprocess.Therefore,thestandardvector-basedkernelsareapplicableforLFP,whichwillbediscussedinthefollowing. 1. Kernelforspiketrains ThissectionutilizestheSchoenbergkernelonspiketrainsintroductioninChapter 2 ,whichinterpretsaspiketrainasarealizationofanunderlyingpointprocessanddenesaninjectivemappingbasedonastrictlypositivedenitekernelbetweentheconditionalintensityfunctionsoftwopointprocessesdenedby[ 66 ].s(si,sj)=exp()]TJ /F11 11.955 Tf 10.5 19.44 Td[(RTs(^si(t))]TJ /F6 11.955 Tf 12.23 2.66 Td[(^sj(t))2dt 2s), (3) whereisthekernelsize.Theestimatedintensityfunctionisobtainedbysimplyconvolvings(t)withthesmoothingkernelg(t),yielding^s(t)=MXm=1g(t)]TJ /F4 11.955 Tf 11.95 0 Td[(tm),ftm2Ts:m=1,...,Mg, (3) Inordertodecreasethekernelcomputationcomplexity,therectangularfunctiong(t)=1 Ts(U(t))]TJ /F17 11.955 Tf 11.95 0 Td[(U(t)]TJ /F4 11.955 Tf 11.95 0 Td[(Ts))(Tstheinterspikeinterval)isusedhere,whereU(t)isaHeavisidefunction. IndividualLFPorspiketrainchannelsarearelativelypoordecodersofstimulusorbehavioronasingle-trialbasis.Therefore,wedenedthekernelformulti-channel 61

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spiketrainsasfollows[ 85 ]s(si(t),sj(t))=NXn=1s(sni(t),snj(t)) (3) WhereNisthenumberofchannelsofspiketrains. 2. KernelforLFP Incontrastwithspiketrains,thefeatureselectivityoftheLFPisbroader[ 46 ].Intimedomain,thefeatureofspiketraincanbeobtainedbyslidingthewindow,whichdescribethetemporalshapeinformationofLFP.Infrequencydomain,spectralpowerandphaseindifferentfrequencybandarealsoknownastheinformativefeaturefordecoding.Inourwork,weonlyselectthestandardfeatureofLFPintimedomain,asanexample,sinceourworkismorefocusedonprovidingthemachinelearningframeworkformultiscaleneuraldecodinginsteadofthefeatureselection.ThefeatureselectionofLFPhasalsoattractedmassiveattentionintheBMIarea,butitisnotthemainpointofourwork. Inthetimedomain,wecansimplytreatLFPsasatimeseries,andapplytheSchoenbergkerneltocontinuoustimesignals.TheSchoenbergkerneldenedincontinuousspacetmapsthecorrelationtimestructureoftheLFPx(t)intoafunctioninRKHS,x(xi(t),xj(t))=exp()]TJ 10.49 8.09 Td[(kxi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(xj(t)k2 2x)=exp()]TJ /F11 11.955 Tf 10.49 19.44 Td[(RTx(xi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(xj(t))2dt 2x)=exp()]TJ /F11 11.955 Tf 10.49 19.44 Td[(RTxxi(t)xi(t)+xj(t)xj(t))]TJ /F6 11.955 Tf 11.96 0 Td[(2xi(t)xj(t)dt 2x) (3) whereTx=[0Tx]. Similarlytospiketrains,thekernelformulti-channelLFPsisdenedbyx(xi(t),xj(t))=NXn=1x(xni(t),xnj(t)) (3) 62

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WhereNisthenumberofchannelsofLFPs.ThetimescaleoftheanalysisforLFPsandspikesisveryimportantandneedstobedenedbythecharacteristicofeachsignalaswillbeexplainedbelow. 3.2SensoryStimulationExperiment 3.2.1ExperimentMotivationandFramework MostBrainMachineInterface(BMI)systemsthatperformscomplexlimbmovementsrelyonvisualfeedbackalonetoprovidesubjectsthepositionofthedeviceintheexternalworld[ 17 18 26 89 97 100 ].However,somatosensoryfeedbackremainsunderdevelopedinBMI,whichisimportantformotorandsensoryintegrationduringmovementexecution,suchasproprioceptiveandtactilefeedbackaboutlimbstateduringinteractionwithexternalobjects[ 40 58 ].Anumberofearlyexperimentshaveshownthatspatiotemporallypatternedmicro-stimulationdeliveredtosomatosensorycortexcanbeusedtoguidethedirectionofreachingmovements[ 31 62 64 ].InordertoeffectivelyapplythearticialsensoryfeedbackinBMI,itisessentialtondouthowtousethemicro-stimulationtoreplicatethespatiotemporalpatterninsomatosensorycortexthatarenecessarytoemulatesensorystimulation.SomatosensoryinformationoriginatesintheperipheryandascendsthroughVPLthalamusonitswaytothesomatosensorycortices.SincethisinformationistransmittedvianeuronresponsesinVPL,weexpectthatasuitablyneelectrodearraycouldbeusedtoselectivelystimulatealocalgroupofVPLneuronsandtransmitsimilarinformationintosomatosensorycortex.Electrophysiologicalexperiments[ 32 ]suggestthattherostralportionoftheratVPLnucleuscarriesalargeamountofproprioceptiveinformation.Caudaltothisregionisazonewherethecutaneousreceptiveeldsarefocalwithanetopographicmapoftheforeandhindlimbs.SincethebodymapintheVPLisarrangedwithsuchremarkablesomatotopicprecisionthatneuronsresponsibleforrelayingsensoryinformationuptotheS1cortexcanbeeasilylocatedandstimulatedwithimplanted 63

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electrodes.Thus,suitablyprecisespatiotemporalmicrostimuluspatternoftheselocalareasareabletoreproduceS1corticalresponsesinducedbythenaturalstimulus. Figure3-1. Neuralelementsintactilestimulationexperiments Figure 3-1 depictstheneuralelementsthatwillbeinvolvedinourexperiments.Totheleftisourrat'shandwithsomerepresentativecutaneousreceptiveeldsshownaswellaspositionswhereourtactorwilltouchthehand.Whenthetactortouchesaparticularreceptiveeldlocationonthehand,VPLthalamuswillreceivethisinformationandsenditontotheS1cortex.Inordertoemulatenaturaltouchwithmicrocirculation,weaddressinthissectionthefollowingcomputationalproblem:howtogenerateoptimalspatiotemporalmicrostimulationpatternsinventralposterolateralnucleus(VPL)ofthalamussothattheelicitedspiketrainsfromsomatosensoryregions(S1)isableconveythenaturalsensationtotheanimal.WehypothesizethatiftheringpatternsinS1producedbytactile(tactor)andthearticial(electrical)stimulationaresimilartheanimalshouldbehaveinthesameway. 64

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Inourwork,weusekernel-basedadaptiveinversecontroldiagramtosolvethisproblem,whichisproposeinChapter 2 .Beforeimplementingthewholediagram,thedecodingmethodshouldbetestedrst,inordertoguaranteetheitisabletoreconstructbothtactilestimulationandmicro-stimulationwithareasonablegoodaccuracy.IntheChapter2,wetestthespikedecoderonbothtactiledatasetandmicro-stimulationdataset.Becauseofthevariabilityofspiketrains,uctuationindecoderoutputoccur.Sinceourspiketrainprocessingextractedonlypartialinformation,thestimulationoutputchannelcannotbediscriminatedwell.Therefore,we'dliketondoutwhetherthemulti-scaleneuraldecoderisabletoenhancethedecodingperformance.Inthefollowingpartwersttestthemulti-scaledecoderinbothtactileandmicro-stimulationdatasets,whichwementionedinChapter 2 3.2.2TimeScaleEstimation Thetensorproductkernel( 3 )allowsmodelingthesamplefromdifferentsourcesindividually.Therefore,thetimescalesofLFPsandspiketrainscanbespeciedbasedontheirownproperties.Inordertondreasonabletimescales,weestimatetheautocorrelationcoefcientsofLFPsandspiketrains,whichindicatetheresponsedurationinducedbythestimulation.Forthispurpose,spiketrainsarebinnedwithbinsize1ms.Thelocaleldpotentialarealsoresampledwithsamplingrate1000Hz.Theautocorrelationcoefcientsofeachsignalaverageoverchannelsarecalculatedby ^h=PTt=h+1(yt)]TJ /F6 11.955 Tf 12.24 0 Td[(y)(yt)]TJ /F5 7.97 Tf 6.59 0 Td[(h)]TJ /F6 11.955 Tf 12.25 0 Td[(y) PTt=1(yt)]TJ /F6 11.955 Tf 12.24 0 Td[(y)2.(3) Its90%condenceboundsofthehypothesisthattheautocorrelationcoefcientiseffectivelyzeroareapproximatelyestimatedby2SE,where SE=vuut (1+2h)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xi=12i)=N.(3) TheaveragecondenceboundsforLFPandspiketrainsare[)]TJ /F6 11.955 Tf 9.29 0 Td[(0.0320.032]and[)]TJ /F6 11.955 Tf 9.3 0 Td[(0.0310.031],respectively.TheautocorrelationcoefcientsofLFPfallintothe 65

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condenceintervalafter20ms,whiletheautocorrelationcoefcientsofspiketrainsdieoutafter9ms,asshowninFigure 3-2 .Therefore,thedecoderinputsbuiltfromspiketrainandLFPareobtainedbyslidingthewindowwithwindowsizeTs=9msforspiketrainandTx=20msforLFP. Figure3-2. AutocorrelationofLPFsandspiketrains 3.2.3DecodingResults Wenowpresenttheresultsofdecodingthetactilestimuluswaveformandmicro-stimulationusingKLMSoperatingonthetensorproductkernel.Forcomparison,wealsoapplythekernel-baseddecoderononlyonetypeofsignalstondoutwhatistheperformanceenhancementgainedbyusingmulti-scalesignals.Timediscretizationis5ms.Thelearningratesforeachdecoderaredeterminedbythebestresultsoftestdataafterscanningtheparameters.ThekernelsizesandxaredeterminedbytheaveragedistanceinRKHSofeachpairoftrainingsamples.Thenormalizedmeansquareerror(NMSE)betweentheestimatedstimulation(y)andthedesiredstimulation(d)isutilizedasanaccuracycriteria. 3.2.3.1Resultsoftactilestimulation NMSEsoftactilestimulationareobtainedacross8trialsdatasets.Foreachtrial,weuse20sdatatotrainthedecodersandcomputeanindependenttesterroronthe 66

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Table3-1. Comparisonamongneuraldecoders. XXXXXXXXXXXXPropertyinput LFP&spike LFP spike NMSE(mean/STD) 0.48/0.05 0.55/0.03 0.63/0.11 remaining2.5sdata.TheresultsareshowninTable1,wherewecanobservethattheLFP&spikedecoderout-performsboththeLFPdecoderandthespikedecoder. Figure3-3. ResultsoftheLFP&spikedecoderoftherststimulationtrial. Inordertoillustratethedetailsofthedecodingperformance,thetestresultsofthersttrialareillustratedinFigure 3-3 .Itisobservedthattheoutputofthespikedecodeructuatesalotandmissessomepulses,i.e.,around0.65s,becauseofthesparsityandvariabilityofthespiketrain.Incontrast,theoutputestimatedbyLFPissmooth,becausetheLFPreectsthesumofalllocalcurrentsonthesurfaceoftheelectrodes,whichcausestherobustnessbutisaimitationtothespecicityofthetechnique.TheLFP&spikedecoderpreformsbetterthantheLFPdecoderbygainingtheprecisepulsetiminginformationfromspiketrains.ThelearningcurvesarealsoestimatedbycalculatingthetestingNMSEafterthemodelparameterupdatedaftereachnewsample 67

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entered.Threelearningcurvesofspikedecoder,LFPdecoderandLFP&spikedecoderareshowninFigure 3-4 ,respectively. Figure3-4. Learningcurves(NMSEofthetestset)oftheLFP&spikedecoderoftherststimulationtrial. 3.2.3.2Resultsofmicro-stimulation Wealsoimplementthedecodertoreconstructthemicro-stimulationpattern.First,wemapthe8differentstimulationlocationsto8channels.Wedismisstheshapeofeachstimulus,sincethetimescalethestimuluswidthisonly200s.Thedesiredstimulationpatternofeachchannelisrepresentedbyasparsetimeseriesofthestimulationamplitudewithsamplingrateof200Hz.WeimplementtensorproductkernelwithKLMSforMIMOsystemthatisproposedinChapter 2 onthemicrostimulationdata.NMSEsareobtainedwithtensubsequencedecodingresults.Weuse120sdatatotrainthedecodersandcomputeanindependenttesterrorontheremaining20sdata.TheSpike&LFPdecoderout-performsboththeLFPdecoderandthespikedecoder.ThecomparisonofresultsareshowninFigure 3-5 ,whichindicatesthatforSpike&LFPdecoderisabletoobtainthebestperformanceofallthestimulationchannels.Especiallyforchannel2,4,6,and7,Spike&LFPdecoderperformssignicantlybetterthanboththeLFPdecoderandthespikedecoder.AqualitativecomparisonofanexcerptisshowninFigure 3-6 ,whichexplainssomedetailsaboutthedecoderperformance.First,onlychannel,4,5,6,7,and8stimulationscannotbedecodedfromLFPsonly 68

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modelsatall,sincethenetimeinformationisaverageoutinLFPs.Forspiketrainsonlymodels,thestimulationisnotwelldiscriminatedfromdifferentoutputchannels,whichcausethefalse-stimulipresentinspikedecoderoutput.However,thecombinationofspiketrainsandLFPsmodelsprovidesmorecompletestimulationinformation,whichcontributestoabetterdiscriminationofstimulationpatterncrosschannelsandallowsthemodeltocapturetheprecisestimulationtiming. Figure3-5. Comparisonamongspikedecoder,LFPdecoderandspikeandLFPdecoder. 3.3OpenLoopAdaptiveInverseControl Inordertoemulatenaturaltouchwithmicro-stimulation,weusedmicro-stimulationimprintedinVPLtocontroltheneuralactivityinS1.However,thevariabilityofneuralsignalsandtheneuralsystemplasticityinducedbyinjury,environmentchanges,etcpresentachallengeofpreciselycontrollingtheneuralresponsetomimicthetargetspatiotemporalringpatterninducedbythenaturalstimulation.InChapter 2 ,weillustratetheabilityofadaptiveinversecontroldiagramtotracktheneuralplasticityandthepromisingresultsofmulti-scaleneuraldecoderindicatesthatthedecoderisrobustforinterfacingwiththerealanimalbrain.Consideringtheseuniquechallenges, 69

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ADesiredstimulationpattern BSpikedecoderoutput CLFPdecoderoutput DSpikeandLFPdecoderoutput Figure3-6. Resultsofreconstructingmicro-stimulation. thekernelbasedadaptiveinversecontroldiagramandtensorproductkernelbasedmulti-scaleneuraldecodingmethodologyiscombinedtoaddressthisproblem. 70

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Inourcurrentstep,consideringtheexperimentalsetting,onlyopenloopadaptiveinversecontrolisimplementedintheratexperiment.First,theinversecontrollerC(z)ispre-trainedwith300sdata.Ineachtrial,60sdatarecordedduringthetactilestimulationisusedasthetargetpattern.Giventhetargetneuralresponseastheinput,thetrainedcontrolledgenerateasequenceofmultiplechannelstimulation.Sincethegeneratedstimulationsequenceissparsebutcontinuoussignal,itdoesnotmeettherestrictionsofbipolarstimulationthatcanbeusedasmicro-stimulation.Therefore,somepost-processingofstimulationisimplementedbeforeinterfacingwiththemicrostimulatorasfollows: 1. Theminimalintervalbetweentwostimuli10msissuggestedbytheexperimentalsetting.Themeanshiftalgorithmisusedtolocatethelocalmaximaofasubsequenceofstimulation(10ms)foreachsinglechannel.Thetimelocationandcorrespondingmaximumamplitudeisusedtosetthetimeandamplitudeofstimuli. 2. Atonetimepoint,onlyonepulseacrosschannelscanbestimulated.Therefore,ateachtimepoint,onlythemaximumvaluecrosschannelsisselectforstimulation.Thevalueatotherchannelsaresettozero. 3. Theconstrainofthemaximum/minimumstimulationamplitudeis[8A30A],whichhasbeensuggestedastheeffectiveandsafeamplituderangeinpreviousexperiments. Toassesstheperformanceoftheopen-loopcontrol,thegeneratedmulti-channelmicrostimulationsequenceisappliedimmediatelyfollowingcomputation.Themicrostimulationresponsesetisrecordedandthencomparedwiththeactualnaturalresponseset.AgraphicalcomparisonofanexcerptisshowninFigure 3-7 .Itisobservedthatthemulti-unitneuralresponseinthetargetsetandtheonegeneratebythemicro-stimulationhavethesimilarspatiotemporalringpatterns.Duetotheshortresponsetimeofthemicro-stimulation,asequenceofmicro-stimuliisusedtoemulate 71

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longtermburstingofspiketraininthetargetringpattern,whosesensoryinuenceisunclearatthismoment. Figure3-7. Qualitativecomparisonofexcerptsbetweentargetandtrainedsystemoutput. Forquantitativeanalysis,thecorrelationcoefcientforeachchanneliscalculatedasshowninFigure 3-8 .Tosimplifytheestimationofcorrelationofspiketrain,webin 72

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thedatawithbinsize10ms.Itisobservedthatthemaximumcorrelationcoefcientliesaround0lagformostchannels.Boththemodelinaccuracyandthejittersinneuralsignalcausetheblurovertime. Figure3-8. Correlationsbetweentargetspiketrainsandneuralsystemoutputofeachchannelforeachtactilestimulationlocation(6digits:d1d2d3d4p3andp1). Thepromisingresultsofopenloopcontrolshowtheeffectivenessofcontrollermodeledbythemulti-scaleneuraldecodingmethodology.Infuturework,thewholecontroldiagramwillbeimplementedtosolvethisemulationofnaturaltouch.Weexpectthattheonlinefeedbackisabletoenhancethecontrolperformance. 73

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CHAPTER4FUNCTIONALCONNECTIVITYANALYSISANDMODELING Neuronsselectivelycoordinatewithotherneuronsdistributedintheneighboringcorticalarea,whichcontributestothemovement[ 61 ].However,itisverychallengingtoquantifythenatureofthislocalfunctionalcoordination.Accordingtotheliterature,neuralassembliesprovideaconceptualframeworkfortheintegrationofdistributedandindividualneuralrings,whicharedenedasdistributedlocalnetworksofneuronstransientlylinkedbyactivedynamiccouplings[ 56 ].Thegeneralhypothesisisthattheemergenceofspecicneuronalassembliesunderlycognitiveactions[ 61 99 ].Therearesomedifcultissuestocharacterizesuchneuralassembliesinducedbythespatiallyspecicandtransientnatureoftheinterneuroncommunication. Inourwork,wefocusondecodingtheintentionofmovementorthesensorystimulation,usingmultielectrodeneuraldatafromcorticalcortexsynchronizedwithkinematic/stimulationvariablesmeasurements.Fromastatisticalmodelingperspective,thecompleteinformationtosolvethisproblemresidesinthejointprobabilitydensityfunctionofthemultivariateneuraldataandofthemultivariatekinematicvariables,butthisisneverdoneduetothehighdimensionofthejointdistributions.Evenwhenwedevelopmodelsthatestimatetheconditionalprobabilityofthekinematics/stimulationgiventheneuraldata,wedonotusethejointinformationexpressedinthemultichannelneuraldataforthesamereason.Instead,wemodeltheconditionaldependenceonlywithrespecttothehistoryofavectorofneuralspiketrains.Sinceeachneuralchannelisnothingbutthemarginaldensityofthejointprobabilityofmultivariateneuraldata,thisprocedureisonlyareasonablesolutionwhentheneuronsspikeindependently.Inourwork,weaimtoquantifyandanalyzepairwisedependenciesbetweencorticalneuronsinvolvedinagivenbehavioraltaskusingtheinformationfromneuraldataconditionedonthekinematic/stimulaitoninformation.Pairwisedependencedoesnotfullyquantifythejointdistribution,buttheinformationcontainedinthecombinedpairsbetterdescribes 74

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thedynamicsinanassembly.Inthisworkthedependenciesthatoccurtransientlyintimebetweenpairsofneuronswillbenamedinteractions,andtheirspatiotemporalpatternsareinterpretedassignaturesoffunctionalconnectivity.Ourhypothesisisthatgivenakinematicstateorstimulationpatterndenedbytheexperiment,specicneuralassembliesaretransientlyactivated,whichcanbequantiedbyasignicantenhancementofthedependenciesbetweenpairsofneurons,i.e.theirfunctionalconnectivity. Thefunctionalconnectivityiscommonlyquantiedbycorrelation,coherence,andsynchrony[ 1 13 35 36 72 ].Correlationandcoherencecapturelinearrelationsintimeandfrequencydomainrespectively,while(statistical)synchronymeasuresthesimultaneousringofneuronsinexcessofcoincidentalrings.However,ingeneral,theinteractionbetweenneuronscanbehighlynon-linearandnotexclusivelyintheformofsynchrony[ 27 ].Thegeneralformofinteractioncanbequantiedbystatisticalmeans,namelybydependencemeasures,whichestimatearbitraryrelationshipsbetweentworandomvariables.Forexamplessee[ 10 45 75 ]. Neuralassemblieshaveatransient,dynamicexistencethatspansthetimescaleofbehavior.Theactivationofaneuralassemblyislongenoughforneuralactivitytopropagatethroughtheassembly.Apropagationthatnecessarilyinvolvescyclesofreciprocalspikeexchangeswithtransmissiondelayslaststensofmilliseconds.Inordertoquantifyandstudythisinteractionduringmovements,weneedtofocusonthetemporaldynamicsofneuralnetworksinthehundredstothousandsofmillisecondrange.Duetothesparseringofcorticalneurons,inthischapterwerstconcentrateoncomparingstatisticalmethodsthatsupportsmallsampleestimatorsofdependence.Fourmethods,meansquarecontingence(MSC),mutualinformation(MI),phasesynchronization(PhS),andcrosscorrelation(CC)weretestedonsyntheticspiketrainsgeneratedbyanetworkofleakyintegrate-and-reneurons[ 38 ].Weselecttwobaselinemethodswithdifferentproperties:CCmeasuresthelinearrelationbetweentwoneuron 75

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activitiesintime,whilePhSmeasuresthepairwisesynchronyofneurons.Incontrast,MIandMSCquantifynon-lineardependenceoftwonominalvariablesbasedontwomajorindependencetests:Chi-squaretestandG-testofindependence.MIandMSCcalculatetheKullback-Leiblerdivergenceand2divergenceofthejointprobabilitydistributionwithrespecttotheproductofthetwomarginalprobabilitydistributions,respectively.AlthoughbothMIandMSCarewidelyusedinappliedstatisticsmeasurements,theircapabilityinestimatingfunctionalconnectivitywithsmallsamplesizehasnotbeenfullyinvestigated.See[ 2 ]fortheuseofMIinneuralassemblies.Weusestatisticalpoweranddispersioncriteriatoevaluatetheperformanceofthesefourmethodsasafunctionofthesamplesize. Furthermore,inSection 4.2 thedependencemeasureisappliedtomonkeycorticalneuralactivityrecordedduringafoodreachingtaskwiththreewelldenedkinematicstates.Theestimatedfunctionalconnectivitydescribestheassembliesassociatedwiththetimevaryingkinematicstates.Thenodes(neurons)andedges(thefunctionalconnectionsbetweenpairsofneurons)inanassemblygraphrepresentdifferentaspectsofthetimespatialneuronalinteractions.Specically,wevisualizethelocalassemblygraphwithinfourcorticalareasduringreachingtasks.Moreover,weestimatetheactivationdegreeofaspecicassemblybasedonthenumberofactivatededgesinthegraph.Theactivationdegreeofastate-relatedassemblyreachesapeakwhenthecorrespondingkinematicstatebegins,whichalsorevealsthatthenetworkofinteractionsamongtheneuronsisthekeyfactorfortheoperationofaspecicbehavior.Theevolutionofthiscomplexlocalfunctionalconnectivityoverthecorticalspaceandovertimecanbereadilyvisualizedinamovie(http://cnel.u.edu/research/linli.php)orinstaticgraphsaswedemonstrateintheresultssection. Inaddition,theinferenceofthefunctionalconnectivitydynamicswithcognitivebehavior,disease,stimulationhasalsobeenexploredin[ 7 15 22 25 34 82 ],whichallindicatesthetemporalfunctionalconnectivitypatternisagoodwaytodescribethe 76

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spatiotemporalneuralactivitypattern.Afterinvestigatinghowtoquantifythetemporalfunctionalconnectivitypattern,ourideaiswhetherwecanextractmoreinformationofthestimuliorkinematicstatesfromthetemporalconnectivitypatterncomparingwiththetemporalpatternofneuralsignalwithchannelindependenceassumption.However,thefunctionalconnectivitypatternsarerepresentedbygraphsormatricesinsteadofvectors,whichmakeitchallengingtouseasinputfeaturesforneuralsystemmodeling.Insteadofdirectlyaddressingthefunctionalconnectivitygraphormatrix,mostresearchimplementsgraphtheorytoquantifythetopologyofgraphpropertiesincludingnodedegree,clusteringcoefcient,betweennesscentrality,andetc.[ 22 82 ].Thesetopologymetricsarenotcompletebutabstractfeaturevectorfromthefunctionalconnectivitygraph/matrix,whichallowstheimplementationofthetraditionalmodelingwiththevector-basedinputspace[ 22 ].However,theabstracttopologiesonlycharacterizetheimportanceofaneuroninanetworkcommutationfromonecertainperspective,butreducesthedetailsofthegraphstructure,forexamplehowtheneuronsarefunctionaldistributedinthenetworkorwhatistheirdirectneighbors.Therefore,inSection 4.3 wenotonlyusethetopologyvectorastheinputvector,butalsodesignakernelforthefunctionalconnectivitymatrix,whichallowustoimplementkernelbasedmachinelearningalgorithmdirectlyonthetemporalfunctionalconnectivitypatternfordecodingwithnofurtherpre-featureextraction.ThedecodingperformanceusinggraphtopologyorfunctionalconnectivitymatrixwithmatrixkernelarecomparedinSection 4.3 4.1FunctionalConnectivityMeasures Theexistenceofmassiveanatomicalconnectionsamongcorticalneuronsdoesnotmakethemfunctionallyconnectedallthetime.Evenintheabsenceofneuralcodeknowledge,exchangeofinformationacrosstheneuropilcanbespottedandquantiedexternallybytheco-occurrenceintimeofactionpotentialsproducedbyneurons.Inthissection,wequantifyfunctionalconnectivitybythestatisticaldependencebetweenneuralringrates.Firstwebrieyreviewdifferentneuralfunctionalconnectivitymeasures. 77

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Considertwodiscreteringratetimeseriesxnandyn,n=1,...,N,recordedsimultaneouslyandproducedbytwoneurons.Eachdependencemeasurehasparticularcharacteristicsandwedividethemintotwogroups:statistical-basedandphase-based. 4.1.1StatisticalMeasures Secondordermeasures Themostwidelyusedtechniquetomeasurethesimilaritybetweentwotimeseriesofringratexandyisthecross-correlation,denedinthetimedomainasafunctionofthetimelag=)]TJ /F6 11.955 Tf 9.3 0 Td[((N)]TJ /F6 11.955 Tf 11.95 0 Td[(1),...,0,...,N)]TJ /F6 11.955 Tf 11.96 0 Td[(1: CYX()8><>:=1 N)]TJ /F13 7.97 Tf 6.58 0 Td[(PN)]TJ /F13 7.97 Tf 6.59 0 Td[(n=1(xn+)]TJ ET q .359 w 270.66 -248.38 m 275.32 -248.38 l S Q BT /F5 7.97 Tf 270.66 -253.39 Td[(x x)(yn)]TJ ET q .359 w 301.86 -248.58 m 306.67 -248.58 l S Q BT /F5 7.97 Tf 301.86 -253.6 Td[(y y)0=CYX()]TJ /F3 11.955 Tf 9.3 0 Td[()0 (4) Thecrosscorrelationisobtainedbynormalizingthecrosscovarianceandthusrangesfromminusone(anti-phasecorrelation)toone(in-phasecorrelation),whilevaluesclosetozeroareattainedforuncorrelatedtimeseries.IntimeseriesanalysisusingalineargenerativemodelwithGaussianinputs,thecrosscorrelationcoefcientquantiestotallythedependencebetweenthetimeseries. Asameasureoffunctionalconnectionbetweentwoneurons,thecrosscorrelationcoefcientiscalculatedbetweentworingratetimeserieswithacertainbinsize.Inthisstudyweselectedthebinsizeat100ms,notbecauseitisthebestchoicebutbecauseitisareasonablecompromisefortheapplication.FrommultiscalestudiesonspiketrainsforBMIs[ 83 ]andalsofromtheanalysisoffunctionconnectivityinneuralensembles[ 28 ]itisknownthatthebestapproachtofunctionalconnectivityistousemultipletimescalestoanalyzespiketrains.However,thismethodologyisratherexpensivecomputationallyanditisincompatiblewithonlinestudiesastheonesweenvisageinthiswork.Therefore,wehavetoselectasinglebinsizethatisagoodcompromisebetweenspecicityofneuralringandsufcientsimilaritywith 78

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behavior.From[ 83 ]100msisthetimescalethatbridgesthegapbetweenreasonablecorrelationswiththemovementsandareasonablespecicityregardingneuralrings.Moreover,sincetheaverageringrateofcorticalneuronsisaround10Hz,with100msbinsize,theaveragespikenumberperbinis1,whichyieldsasuitablesparsenessforestimatingdependency.Becausetheneuraldataweusediscollectedfromneighboringcorticalregionsandwequantifyringrateswithbinsizeof100ms,wecanexpectthatthedependenceinducedbythefunctionalconnectionbetweenapairofneuronsusuallyhappenswithinthe100mslag.Therefore,onlythecorrelationwithlagzeroiscalculated. Thecorrelationoftheringpatterncanalsobedirectlycalculatedinthespaceofspiketrainwithnobinningprocess.ThequanticationofcorrelationdenedinthespaceofspiketraincallSchreibergetal.SimilarityMeasure,whichisalsobasedonthefunctionalrepresentationofaspiketrain^(t),whichisdenedbySs=R^(t)^0(t)dt q R^(t)^0(t)dtq R^(t)^0(t)dt (4) wherethesmoothinglterh(t)mayforexampleexponentialorGaussian.Consideringthecomputationefciency,exponentialsmoothfunctionisusedinthiswork.ThisSimilaritymeasureprovidesanestimationofthereectivecorrelationcoefcientbetweentwopointprocesses.ThevalueSSisscaledinto[01].ifandonlyiftwospiketrainss(t)ands0(t)areidentical,thenSs=1.Throughthecorrelationbypassthelimitsinducedbythebinningprocess,italsointroducethefreeparameterstothecalculation,whichneedtobespecied.Therefore,thissimilaritymeasureissuggestedtobeusedinthecontextrequestingnetimeresolution. However,crosscorrelationonlyindicatesthestrengthofalineardependencebetweentwovariables,butitsvaluecannotcompletelycharacterizetheirrelationshipthatmayappearinhigherorderstatisticalmoments(skewness,kurtosis,etc.). Dependencemeasures 79

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Incontrasttocross-correlation,bothMIandMSCquantifythestatisticaldependenciesbetweentworandomvariablesusingthefullstatisticalinformationintheirjointspace,withnoassumptionabouttheformoftheirrespectivedensitiesandimplicitlytheirgeneratingprocesses,whichcanbelinearornonlinear.Itisthereforemoreappropriatewhenthedatagenerationmechanismisnotfullyunderstood,butitismuchmoredifculttoestimateinpractice. MeansquarecontingencyMeanSquareContingency(MSC)isameasureofdependencebasedonanindependencetest,anditwasrstdenedbyPearsonfortwodiscreterandomvariables[ 69 ].MSCcalculatesthenormalized2divergencebetweenthejointprobabilitydistributionandtheproductofthetwomarginalprobabilitydistributionsoftworing-ratetimeseries[ 45 ]. Foratestofindependence,anobservationconsistsofapairofvaluesandthenullhypothesisisthattheoccurrencesofthesevaluesarestatisticallyindependent.Eachobservedprobabilityisallocatedtoonecellofatwo-dimensionalarray(knownascontingencytable)accordingtoitsvalues.Iftherearerrowsandccolumnsinthetable,thetheoreticalprobabilityforacellgiventheindependentassumptionisEi,j=Pi+P+j,Pi+=Pck=1Pi,kandP+j=Prk=1Pk,jarethemarginalprobability,andPi,jistheobserved(joint)probabilityofthecellinrowiandcolumnjincontingencytable. TheMSCvalue,alsocalledthePearsoncontingencycoefcient,denotedbyP,isgivenby P=2 1+21=2, (4) 80

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where2=rXi=1cXj=1(Pi,j)]TJ /F4 11.955 Tf 11.96 0 Td[(Ei,j)2 Ei,j=rXi=1cXj=1P2i,j Pi+P+j)]TJ /F6 11.955 Tf 11.95 0 Td[(1 (4) Notethatthisstatisticbasicallyscalesthe2statistictoavaluebetween0(independence)and1(maximumdependence).Ithasthedesirablepropertyofscaleinvariance.Thatis,thevalueofPearson'scontingencycoefcientdoesnotchangeaslongastheprobabilitiesremainconstant.Unfortunately,theestimationisbiasedwhenthesamplesizeissmall.William'scorrectionisusedtodecreasetheinaccuracy[ 94 ]andthebiascorrectedexpressionP1is P1=2 q+21=2, (4) whereq=1+Pri=11 Pi+Pcj=11 P+j 6N(r)]TJ /F6 11.955 Tf 11.96 0 Td[(1)(c)]TJ /F6 11.955 Tf 11.96 0 Td[(1), (4) whereNisthetotalsamplesize. MutualinformationIncontrastwithMSC,MIquantiestheKL-divergencebetweenthejointprobabilitydistributionandtheproductofthetwomarginalprobabilitydistributions,whichisusedtodeterminethesynapticconnectionstructureofneuronalnetworks[ 107 ].Accordingtothecontingencytable,theShannonentropyoftherow(X)andcolumn(Y)vectoraredenedrespectivelyas ^H(X)=)]TJ /F5 7.97 Tf 18 14.95 Td[(rXi=1Pi+lnPi+,^H(Y)=)]TJ /F5 7.97 Tf 17.61 14.95 Td[(cXi=1P+jlnP+j, (4) 81

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wherePi+andP+jarethemarginalprobabilityandPijistheobserved(joint)probabilityofthecellinrowiandcolumnjincontingencytable.TheestimationofH(row)andH(column)isheavilybiasednegativelywhenthecontingencytablecontainszeroentries(P+j=0orPi+=0)althoughthetrueprobabilityisnon-zero,whichislikelytooccurwithasmallsamplesizecomparedtothenumberofentriesofthetable.TherstorderbiascorrectedentropyH1[ 81 ],isthendenedas: H1(X)=^H(X)+bX)]TJ /F6 11.955 Tf 11.95 0 Td[(1 2N, (4)H1(Y)=^H(Y)+bY)]TJ /F6 11.955 Tf 11.95 0 Td[(1 2N, (4) wherebXandbYarethenumberofstatesforwhichPi+6=0andP+j6=0,respectively.ThejointentropybetweenXandYisdenedas H(XY)=)]TJ /F11 11.955 Tf 11.29 11.36 Td[(Xi,jPi,jlnPi,j, (4) ThemutualinformationbetweenXandYisdenedas ^MI(X,Y)=^H(X)+^H(Y))]TJ /F6 11.955 Tf 13.52 2.66 Td[(^H(X,Y) (4) anditsbiascorrectedexpressionMI1is MI1(X,Y)=^MI(X,Y)+bX+bY)]TJ /F4 11.955 Tf 11.96 0 Td[(bXY)]TJ /F6 11.955 Tf 11.95 0 Td[(1 2N, (4) wherebXYarethenumberofcellsincontingencytablegreaterthanzero.Inthefollowing,themutualinformationisbiascorrectedandalsonormalizedbydividingitbyln(N). 82

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4.1.2PhaseSynchronizationMeasures Phasesynchronization(PhS)isanubiquitousphenomenoninmanyphysicaloscillatorysystems,whichalsooccursinneuralstructures.However,themethodologyisbasedondeterministicdynamicalsystemprinciplesthatmaynotapplytoneuralstructuresduetotheirintrinsicstochasticnature.Nevertheless,phasesynchronizationhasbeenwidelyappliedtoneuralsystems[ 70 ].Inthiswork,weapplyPhStothetimeseriesofringrate.Weviewthetimeseriesofringrateasacontinuoussignalx(t)=xn,nt<(n+1),where=100msrepresentsthebinsize.Thislongbinwidthissufcientlylongtosmoothshorttermuctuationsproducedbyneuralstochasticity.Phasesynchronizationindicatesthefollowingphaselockingconditionappliedforanytimet,'(t)=jnx(t))]TJ /F4 11.955 Tf 12.69 0 Td[(my(t)jconstant,wherex(t)andy(t)representthephaseofthesignalrecordedfromtheneuronxandy,respectively[ 70 ].Inthefollowingcomputationswetakethetermswithjmj=jnj=1.Basically,phasesynchronizationanalysisproceedsintotwosteps:(i)estimationoftheinstantaneousphasesand(ii)quanticationofthephaselocking. EstimationofinstantaneousphasesInthisstudy,thephaseisextractedviaHilberttransform,althoughtherearemanyothermethods[ 53 ](forasurveyofmethods,wereferto[ 70 ]).Theanalyticsignal(t)oftheunivariatemeasurementisacomplexfunctionofcontinuoustimex(t)denedas (t)=x(t)+ixh(t)=a(t)ej(t), (4) wherethefunctionxh(t)istheHilberttransformofx(t). xh(t)=1 P.V.Z+1x(t) t)]TJ /F3 11.955 Tf 11.96 0 Td[(d. (4) 83

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P.V.indicatesthattheintegralistakeninthesenseofCauchyprincipalvalue.a(t)and(t)aretheinstantaneousamplitudeandphaseoftheanalyticsignal(t)ofx(t).Theinstantaneousphasex(t)istakenequalto(t),whilethephasey(t)isestimatedfromy(t)followingthesameprocedureforphi(x). Phase-lockingquanticationForneuralsignals,thephaselockingconditioncanbebetterunderstoodinastatisticalsense,byestimatinginthedistributionofcyclicrelativephase'0(t)='(t)mod2[ 79 ].Forindependenttimeseriesx(t)andy(t),thedistributionoftherelativephase'0(t)isuniformwithinagiventimewindow.Thedetectionofphaselockinginvolvesquantifyinghowfarfromuniformistherelativephasedistribution.Severalmeasureshavebeenproposedforthispurpose[ 43 96 ].Theoneweuseis[ 43 ] =k1 NNXj=1ei'0(tj)k, (4) whereNisthenumberofsampleinawindowandisthemeanphasecoherenceoftheangulardistribution[ 59 ].Byconstruction,itisboundedby0(nosynchronization)and1(perfectsynchronization). 4.1.3PerformanceComparisonoverMeasures 4.1.3.1Simulateddata Wetestthecapabilityofthefourmethodsinestimatingfunctionalconnectivityasafunctionofthewindowsizeofobservations.TheexperimentisperformedinaneuralCircuitSIMulator(CSIM)[ 38 ],usinganetworkof10neuronsorganizedasacellassembly.Giventheparametersofthesimulation,thisnumberissufcienttogeneralizetheconclusionsoftheirperformanceforlargernetworksizes[ 80 ].Twoneurons,AandBfromthecellassemblyareselectedformeasurement.NeuronAandneuronBareexcitedbyindependentspiketrains.Thespiketrainsarerecordedintwostates:thecellassemblyiseitherfullyconnected(CONstateasshowninFigure 4-1A )ordisconnected 84

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(directandindirect)betweenneuronAandB(DISstateasshowninFigure 4-1B ).TheinteractionbetweenneuronAandBisestimatedbasedondependencemeasures. ACONstate(thecellassemblyisfullyconnected) BDISstate(thecellassemblyisdis-connected(directandindirect)betweenneuronAandB) Figure4-1. Thestimulatednetworkof10neurons. AllneuronsaremodeledasLeakyIntegrate-and-Fire(LIF)units,sincetheintegrate-and-reneuronmodelisoneofthemostwidelyusedspikingneuronmodelsforanalyzingthebehaviorofneuralsystems.Itprovidespracticalandrelativelyrealisticdescriptionsoftheneuronmembranepotential(spike)intermsofthesynapticinputsandtheinjectedcurrentthatitreceives.Neuronparametersare[ 52 ]:membranetimeconstant30ms,absoluterefractoryperiod3ms(excitatoryneurons),threshold15mV(forarestingmembranepotentialassumedtobe0),resetvoltage14.3mV,constantnonspecicbackgroundcurrent13.5nA,inputresistance1M,andinputnoise9nA.ThepostsynapticcurrentismodeledasanexponentialdecayWexp()]TJ /F4 11.955 Tf 9.3 0 Td[(t=s)withs=3ms.TheaveragesynapticweightWis1.Allneuronsreatareasonableringraterange(1Hzto10Hz)asshowninFigure 4-2 .Foranalysis,neuronalspikeeventsarebinnedinnon-overlappingwindowsof100ms. 4.1.3.2Criterionforcomparingdifferentmeasures Tocomparethedifferentmeasuresoffunctionalconnectivityintermsoftheirtimeresolution,thestatisticalpoweranalysisandthevarianceanalysisareutilized 85

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Figure4-2. TherasterplotsofneuronAandBunderDISandCONstate.Undereachstate,neuronsAandBreatareasonableringraterange(1Hzto10Hz). toevaluatetheircapabilitytodetectconnectivityandtheprecisionoftheconnectionstrengthestimation,respectively. Statisticalpowerestimatestheprobabilityofdetectingafunctionalconnectiongiventhatafunctionalconnectionexistswithaxedtype-Ierror[ 108 ].WeassumeDISstateandCONstatetobethenull-hypothesisandthealternative-hypothesis,respectively.SincethemeanvalueofdependenceinCONstateislargerthanthatinDISstate,weperformaone-tailhypothesistest.ThethresholdisdenedbythedependenceofDISstatewiththesignicanceof0.05.ThestatisticalpoweristheprobabilitythatthedependenceofCONstateislargerthanthethreshold. Wecomparetheperformanceoffourmethodsasafunctionofwindowsizesbycomputingthestatisticalpowers.Withthesamewindowsize,thelargerstatisticalpowerindicatesthebettercapabilityofdetectingfunctionalconnectivity. 86

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Inaddition,wealsoconsiderthedependencevarianceasacriterion.Becausethedependencevalueisusedtodescribetheconnectionstrength,themethodwithasmallervarianceispreferredasamorepreciseconnectivityestimator. 4.1.3.3Simulationresults Moresamplesusuallyenablethemeasureoffunctionalconnectivitywithabetteraccuracy.However,becausethetimescaleofbehaviorissoshort,100-1000ms,astatisticalmethodthatyieldsarobustestimatorfor10samplesisdesired.Tocomparetheperformanceamongdifferentstatisticalestimatorswithrespecttothewindowsize,wevarythelengthofthetimewindowappliedtothesignalfrom1sto10s(non-overlappingwindowswith10to100samplescollectedwithaperiodof100ms). Werunthesimulationin100randomsettingsofthesynapticweightsbetweenpairofneuronsbasedonuniformdistributedrealizationofthetruncatednormaldistributionWN(1,0.1),W2[0,2].Ineachsetting,thesimulationgenerates200subsetsofneuralactivityinDISstateandCONstate,respectively.Thedependenceismeasuredforeachsubsetwithdifferentwindowsizes.Thestatisticalpowerofdependenceiscalculatedforeachrandomsettingwithrespecttothewindowsize.Figure 4-3A showsthatthestatisticalpowerofthecouplingdetectionisamonotonicallyincreasingfunctionofwindowsize.Thephasesynchronizationincreasesthemost,whichsuggeststhatphasesynchronizationisprogressivelymoreefcientwithlongertimewindowsthatcharacterizestableregimes.Incontrast,themeansquarecontingencyandMIhaveamoregradualincreaseindicatingthattheirperformanceistheleastsensitivetothewindowlength.ThestatisticalpowerofMIisthebestoverallwindowsizes,whilethestatisticalpowerforcross-correlationisalwayslowerthanthatofotherdependenceestimators,becausethenonlinearcouplingisnotdetectedwellbycross-correlation. Moreover,aKolmogorov-SmirnovTest(KS-test)isperformedonstatisticalpowersofeachpairofmeasures,whichindicatesthatthestatisticalpowerofMIandMSCissignicantlygreaterthanthatofcross-correlationandphasesynchronizationwhen 87

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AStatisticalpowersofthemethodswithrespecttodifferentesti-mationwindowsizes.Werunthesimulationmodelin100randomsettings.Theerrorbarrepresentsthestandardderivationofthestatisticalpowerover100randomsettings. BThemeanandstandarddeviationofeachmeasurewithrespecttodifferentestimationwindowsizes. Figure4-3. Smallsamplesizeperformancecomparisonamongfourdependencemeasures:MSC,CC,MI,andPhS,fordetectingfunctionalconnectivitybetweenspikingneuronAandB. 88

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Table4-1. Statisticalpowersofthemethodswithrespecttodifferentaveragesynapticweightw,whenthewindowsizeis10samples. MethodWeight 1 1.5 2 2.5 MSC 0.34 0.40 0.64 0.77 CC 0.17 0.17 0.37 0.51 MI 0.44 0.64 0.83 0.84 PhS 0.18 0.38 0.38 0.64 thewindowsizeissmall,e.g.,10and20samples.With10-samplewindowsize,whenwedecreasetheestimationdifcultybyincreasingtheaveragesynapticweightW,thestatisticalpowerofMIandMSCissignicantlygreaterthanthatofcross-correlationandphasesynchronization,asshowninTable1.ThisreectstherobustnessofMIandMSCwithsmallwindowestimation.ThisrobustnesssuggeststhatMIandMSCarebettercandidatestoestimatethedynamicfunctionalconnectivity. Tofurtherinvestigateeachmeasure'scapabilityforsmallwindowestimation,wepresentthevarianceoftheestimatedconnectionbasedonfourmethodsinFigure 4-3B .Thestandarddeviationdecreaseswithincreasingwindowsize,asitcanbeexpected.However,ifoneisinterestedinsmallwindowsizestoquantifythefastdynamicsofneuralassemblies,weshouldcomparethemethodsinthesmallwindowsizecase.ThestandarddeviationofMSC(0.10forCONstateand0.18forDISstate)isthesmallestwhenwindowsizeis10samples,whilethestandardderivationofMI(0.25forCONstateand0.18forDISstate)isrelativelylarger,whichimpliesthatMSCisabletoestimatethestrengthofthefunctionalconnectionmoreprecisely. Insummary,thecomparisonrevealsthatMSCisarelativelyreliableestimatoroffunctionalconnectivitywithshorttimeresolutionandwillbeourchoiceinthiswork. 4.2TemporalFunctionalConnectivityAnalysisandInteractiveVisualization Inthissection,thedependencemeasureisappliedtomonkeycorticalneuralactivityrecordedduringafoodreachingtaskwiththreewelldenedkinematicstates.Thetemporalfunctionalconnectivityischaracterizedbyadependencygraph,where 89

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thenodes(neurons)andedges(thefunctionalconnectionsbetweenpairsofneurons)representdifferentaspectsofthetimespatialneuronalinteractions.Thefunctionconnectivityprovideadescriptionoftheassembliesassociatedwiththetimevaryingkinematicstates. Inthissection,wecharacterizethetemporalfunctionalconnectivitypatternbythetransientdependencegraphamongmultiplechannels,whichisquantifybymeasuringthestatisticaldependence.Theevolutionoftransientdependencegraphprovidesaquantiablerepresentationofthedynamicnatureofneuralprocessingandfurtherinvestigateitsassociationwiththeunderlyingbehavioralstate.ThedynamicdependencegraphisalsointeractivelyvisualizedwithavisualizationJavatoolkit(Flare),whichhelpstoobservethedetailconnectionofthefunctionalconnectivitypattern,itstemporalevolution,anditsassociationwithbehavioralstate. 4.2.1CorticalNeuralData ThedatafortheseexperimentswascollectedinDr.Nicolelis'primatelaboratoryatDukeUniversity(see[ 60 ]fordetails).Forourexperiments,neuraldataarerecordedfromanowlmonkey'scortexwhentheanimalisperformingafoodreachingtask.Multiplemicro-wirearraysareusedtorecordthisdatafrom104neuralcellsinthefollowingfourcorticalareas:posteriorparietalcortex(PP)]TJ /F4 11.955 Tf 11.18 0 Td[(contra),primarymotorcortex(M1)]TJ /F4 11.955 Tf 12.77 0 Td[(contra),dorsalpremotorcortex(PMD)]TJ /F4 11.955 Tf 12.77 0 Td[(contra)andprimarymotor&dorsalpremotor(M1=PMD)]TJ /F4 11.955 Tf 12.7 0 Td[(ipsi).Intandemwiththeneuraldatarecording,the3-Dhandpositionsaredigitizedwhenthemonkeyisperformingthetask. Fordependenceanalysis,neuronalspikeeventsarebinnedinnon-overlappingwindowsof100ms.Thehandpositiondatasetsaredigitallylow-pass-lteredtoavoidaliasinganddownsampledto10Hztomatchthebinningutilizedintheneuraldata.Takingtheprimate'sreactiontimeintoaccount,thespiketrainsweredelayedby0.230swithrespecttothehandposition[ 103 ].Ourparticulardatasetcontains104neuralchannelsrecordedfor38.33min.Thistimerecordingcorrespondstoadatasetof 90

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23000104timebins,forwhichabout250reachesareperformed.Thetimespentinmovementandrestingarearound11minand27min,respectively. 4.2.2KinematicStateAnalysis Ourtestingwithrealdatawillelucidatetheadequacyofthedependencemeasurestoquantifyandanalyzeneuraldatagiventheknowledgeofthekinematicvariables.Conditioninginthekinematicvariablesisjustawaytosegmentthedataforourstatisticaltestsanddoesnotinvalidatetheassumptionthatonlyneuraldatainteractionsarebeingutilizedintheanalysis.Iftheresultswiththissegmentationshowstatisticaldifferentvaluesforthedifferentphasesofthemovements,thenwehavedemonstratedthatthedependencemeasuresmaybeusefulformotorcontrolandotherbehaviorexperiments,evenwhennomeasuredexternalvariablesareavailable.Theonlyextradecisiontobemadebytheexperimenterinthismoreabstractframeworkistheselectionofthebaseline,thestartoftheanalysisandthesegmentlength.Bybruteforceonecanalwaysuseasmallwindowandrepeattheanalysisineachandthenperformclusteringonthevaluesobtained.Forkinematicstateanalysis,ourhypothesisisthatspecicneuralassembliesareactivatedgivenakinematicstateandthegoaloftheanalysisistocharacterizeassembliesbyasetofneuronswhosefunctionalconnectivityisenhancedsignicantlytodeneastate-relatedneuralassembly.Specically,thefunctionalconnectivityanalysisofstate-relatedassemblyisbasedonahypothesistest.Givenakinematicstate,thepairwiseneuronconnectionscomposingthestate-relatedassemblyaresignicantlyhigherthanthoseinthereststate,whichisagainstthenullhypothesisthattheconnectionsbetweenapairofneuronssharethesamedistributioninthereststateandthegivenkinematicstate.Moreover,agraphbasedonthosepairwisefunctionalconnectionsarebuilttodescribeastate-relatedassembly.Inthegraph,thenodesandtheedgesrepresenttheindexofneuronsandthefunctionalconnectivitybetweenapairofneurons. 91

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4.2.2.1State-relatedassembly Fortheexperiments,theneuraldataaresegmentedintothereststateandthreemovementstates:rest-to-food(Mv1),food-to-mouth(Mv2),mouth-to-rest(Mv3)accordingtotheprevioussegmentationofthe3-Dhandtrajectory[ 84 ],asshowninFigure 4-4 .TheMSC-basedmeasureisappliedtoestimatethefunctionalconnectivityofpairwiseneuronsineachsegment.Sincetheaveragedurationofeachmovementstateisonly1s,theestimationwindowsizeisselectedas1s(10samples),andwefurtherassumethattheinteractionwithinthestate-relatedassemblyduringeachstateremainsstationary.MSCbetweenpairwiseneuronsisestimatedfor50trialsbelongingtoeachmovementstateand100trialsduringreststate,whichissufcienttodeterminethestate-relatedassembly. Figure4-4. Exampleofreachingmovementtrajectory.Trajectoryissegmentedintorest(R),rest-to-food(Mv1),food-to-mouth(Mv2),andmouth-to-rest(Mv3)states. Theestimateddependenceforeachtrialistreatedasanobservationoftheconditionalprobabilitydistributiongivenakinematicstate.WeperformaKolmogorov-SmirnovTest(KS-test),whichgivesanindicationoftheseparabilityoftwomeasurementsets,tocheckwhetherthepairwiseneurondependenceinamovementstateandthe 92

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reststatearesignicantlydifferent.Thenullhypothesisisthattheconnectionbetweenapairofneuronssharethesamedistributioninthereststateandthegivenmovementstate,whichistranslatedfortheKStestonnonsignicantdifferenceofmeasuredfunctionalconnectionbetweenthesetwostates.Therefore,arejectionofnullhypothesisatthe0.0001signicancelevelindicatesthattheconnectionbetweenapairofneuronsissignicantlydifferentduringthegivenstate.Thesmallsignicancelevel0.0001isdeterminedbyBonferroni'scorrectionformultiplecomparisonswherewedivideafamilywiseerrorrateof0.05bythenumberofcomparisons.Givenamovementstate,allactivatedconnections(edges)andtheconnectedneurons(nodes)comprisetheassemblygraph.Thedegreeofaneuronisdenedasthenumberofedgesincomingtotheneuron.Threeconditionalassemblygraphsgiventhreemovementstates(rest-to-food(Mv1),food-to-mouth(Mv2),andmouth-to-rest(Mv3)states)areplottedbasedonKS-testresultsinFigure 4-5 .Foreachgraph,theblackdotsdenotetheactivatedfunctionalconnectionsinthismovementstatewithrespecttothereststate.Thereare104neuronsintheneuraldataset.Eachpoint(x,y)inthematrixrepresentsthefunctionalconnectionbetweenneuronxandneurony.Thebackgroundcolorofeachgraphisusedtoindicatethecorticalarea.Thebottomplotsineachsub-gureshowtheneurondegree(thenumberofedgesincomingtotheneuron)intheassemblygraphgivenamovementstate. Foreachgure,theblackdotsdenotetheactivatedfunctionalconnections(edges)atthemovementstate,comparedtothereststate.Infact,inanyofthemovementstatesthedependenceisalwayshigherthaninthereststate(Figuresnotshown),whichisconsistentwiththehypothesisthatreciprocalinteractionsamongneuronscontributetotheemergenceofthebehavior[ 99 ].Whatismoreinterestingisthattheringrateofneuronsduringrestingandmovementisnotsignicantlydifferent,asshowninFigure 4-4 93

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ARest-to-food(Mv1)state BFood-to-mouth(Mv2)state CMouth-to-rest(Mv3)state Figure4-5. 104104matricesplotthreefunctionalconnectivitygraphsassessedbyKS-testresults. Thevariabilityamongthesethreegraphsdemonstratesthespecicityofthestate-relatedassemblies.Inotherwords,thestate-relatedassembliesprovideasubsetofthecorticalneuronsthatareespeciallyimportanttoemergenceandimplementationofaspecicbehavior.Forexample,intheinitiationoftheMv2movementthereisfunctionalconnectivityamongposteriorparietalneurons.Thisconnectivityspreadstomultiplecortices(PMD-contraandM1-contra)duringthelargeexcursionreachbetweenthefoodandmouth.Tofurtherinvestigatethecorrelationbetweenthefunctionalconnectivityinstate-relatedassembliesandbehavioralstates,westudytheevolutionoftheconnectionstrengthandtheactivationdegreeofthestate-relatedassembliesinthecortex. 4.2.2.2Evolutionofconnectionstrength Theevolutionoffunctional-connectivitystrengthisestimatedbyslidinga1sor3swindowby0.1sstepsoverthedata.Theresultsrevealthatthestrengthofensembleconnectionsisenhancedatthemovementstates.Toillustratethisconnectivityevolution,theneuronsconnectedtoneuron93(seeFigure 4-5C )areselectedfromtheassemblygraphoftheMv3assembly,whichisoneofthemostconnectedneuron.Thetime-varyingstrengthsoftheseconnectionsaredepictedinFigure 4-6A and 4-6B 94

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Ineachcolumnsub-gure,therstpanelshowsthe3-Dhandpositiontimeseries,followedbytheconnectionstrengthestimatedbyMSC,MI,PhS,andCCwitha1or3smovingwindows(slidingby0.1ssteps).Thecolorrepresentstheconnectionstrengthofneuron93toitsfunctionallyconnectedneurons.ThebottomplotinFigure 4-6A showstheringratesofneurons93anditsfunctionalconnectedneurons. 95

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AWindowsize:1s BWindowsize:3s Figure4-6. Time-varyingconnectionstrengthofneuron93intheMv3assembly. Figure 4-6A demonstratesthedynamicsoftheconnectivityassociatedwiththeMv3kinematicstatewitha1smovingwindow.Thedelay,theduration,andthepeaksoftheconnection'sstrengthvaryamongdifferentneuronpairs.However,theconnectivitypatternrepeatsateveryreachingtask,withthevarianceinducedbyslightmovement 96

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differencesacrosstrials.Theseresultssupportthehypothesisthattheconnectionstrengthinstate-relatedassemblybasedonKS-testismodulatedbythekinematicstate. InFigure 4-6A ,itisworthnotingthatMIandMSCareabletodetecttheassociationbetweenthefunctionalconnectivityandthekinematicstatewithanetimeresolution.Incontrast,cross-correlationandphasesynchronizationfailtodetectsuchassociationwiththe1sestimationwindow.Whenthewindowincreasesto3s(Figure 4-6B ),theconnectionstrengthbasedonphasesynchronizationandcrosscorrelationalsoshowstheassociationwiththetransitionfromresttomovementstates,butwithrelativelylowerlevelandworsetimeresolution.ThecomparisonfurtherdemonstratestherobustnessofMSCandMIindetectingthefunctionalconnectivitywithshortestimationwindow. 4.2.2.3Activationdegreeofstate-relatedassemblies Sofarwehavefocusedonindividualneurons,nowwemovefocustoneuralassemblies.Itisnon-trivialtoextendthepairwisedependenciestoameasurepertainingtothetotalassemblyactivity.Hence,weareinterestedinhowtoquantifythedegreeofactivationofanassemblybycombiningtheestimatedpairwisefunctionalconnectionsreportedabove,whichprovidesawaytoquantifythegraphisomorphismbetweentemporalfunctionalconnectivityandstate-relatedassembly.Tocomparethevalidityofthequalications,weinvestigatethecorrelationbetweentheactivationdegreeofstate-relatedassembliesandthespecickinematicstates. Specically,thethresholdforfunctionalconnectionbetweeneachpairofneuronsisestimatedfromthereststatesetofpairwisedependencewithap-valueof0.05.Ifthedependenceofapairofneuronsisoverthethreshold,thecorrespondingedge(linkingthenodepair)andnodesintheassemblygraphareactivated.Therearetwopossiblewaystodenethedegreeoftheassemblyactivation:1)theprobabilityoftheactivatededge(edgemethod)denedas: 97

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pedge=nedge=Nedge (4) 2)theprobabilityoftheactivatednodesintheassemblygraph(nodemethod)denedas: pnode=nnode=Nnode (4) wherenedgeandnnoderepresentthenumberoftheactivatededgesandnodesandNedgeandNnodearethetotalnumberoftheedgesandnodesintheassemblygraph.Theedge-basedactivationdegreemeasurestherelativestrengthofthevalueoffunctionalconnections(interactions)inthegivenassembly.Thenode-basedactivationdegreemeasurestherelativenumberofneuronsinthegivenassemblywithenhancedfunctionalconnection. AAverageactivationdegreeoftheassemblybasedontheactivatededge BAverageactivationdegreeoftheassemblybasedontheactivatednode Figure4-7. Averageactivationdegreeofstate-relatedassemblies. Forthesethreestate-relatedassemblies,theiraverageactivationdegreeisestimatedoveralltrailsduringRest-to-food,food-to-mouth,mouth-to-rest,andrestkinematicstates,respectively,asshowninFigure 4-7 .Figure 4-7A showsthe 98

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edge-basedactivationdegreeresults,wheregivenastate-relatedassembly,itsactivationdegreealwayshasthehighestvalueduringitscorrespondingkinematicstate.Fromanotheraspect,duringeachmovementstate,itscorrespondingassemblyalwayshasthehigheractivationdegreethantheothers.Moreover,theactivationdegreesofallthreeassembliesaredecreasedinthereststate.Incontrast,therearenosuchresultsobtainedbythenode-basedactivationdegree,asillustratedinFigure 4-7B Inaddition,thetemporalevolutionoffunctionalactivationestimatedwiththeKS-testofthreeassembliescorrespondingtoMv1,Mv2,andMv3usingtheedgeandnodemethodsareshowninFigure 4-8 ,wheretheupperpanelshowsthereachingmovementtrajectoryof3Dhandposition.Thesecondpanelistheactivationdegreeofthreeneuralassembliesbasedontheactivatededgemethod,estimatedbyamovingwindow(sliding1swindowby0.1ssteps).They-axisisthethreeassemblyindexcorrespondingtherest-to-food,food-to-mouthandmouth-to-reststate.Thethirdpaneldemonstratestheactivationdegreebasedontheactivatednodemethod.Theassemblyactivationbasedontheedgemethoddetectsconsistentlytherepeatedfood-reachingtasks.Moreover,thehighestactivationoftheMv1assemblyisonlypresentinthebeginningoftheMv1state.TheMv2assemblyactivationpeaksduringMv2andMv3stateoccurs.Theseresultsshowthattheactivationoftheassembliesisabletoreachthepeakrepeatedlywhenthespecicmovementstatearrives.Theseresultsalsosuggestthatthespecicassemblyexplainstheemergenceandoperationofaspecicbehavior.Incontrast,theassemblyactivationaccessedbythenodeprobabilityislesseffectivetodetectthespecickinematicstatebecausethesamesetofneuronsisinvolvedindifferentmovements,inspiteoftheincreasedactivationduringthemovementstates.However,theorganizationoftheinteractionnetworkappearsmoreconsistentgivenaspecickinematicstate,andthisisthereasonwhyedgesarebettersuitedfordiscriminationoffunctionalconnectivity.Thiscomparisonrevealsthattheoperationofaspecicbehaviorismorecloselyassociatedwiththenetworkofinteractionsamongtheneurons. 99

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Figure4-8. ThetemporalevolutionoffunctionalactivationofthreeassembliescorrespondingtoMv1,Mv2,andMv3states. 4.2.2.4Dynamicsoflocalfunctionalconnectivity Thedifferentorganizationofthestate-relatedassemblies(Figure 4-5 )motivatesthequestionaboutthevaryingroleofcorticalregionsinvoluntarymovement.Tofurtherinvestigatetherelationshipbetweencorticalneuralassembliesandbehavior,theFlaretoolkit[ 98 ]isappliedtovisualizethelocalassemblygraphsoffourcorticalareas(PP-contra,M1-contra,PMD-contra,M1/PMD-ipsi). Inordertoshowtheevolutionofthegraphsintime,wefurtherseparatethedatainto5sub-states,whichareMv1,Mv2,Mv3andtwomiddlestates(theonebetweenMv1andMv2andtheonebetweenMv2andMv3),asshowninFigure 4-9 .Foreachsub-gure,thevefunctionalconnectivitygraphscorrespondtothe5movementsub-states.Theneuronsarepacedalongacircleandalinkindicatesthattwoneuronsareconnected.Thestate-relatedassemblygraphsarebasedontheKS-testresultsbetweeneachsub-stateandthereststate,similarlytoSection 4.2.2.1 .IntheinteractivevisualizationofpairwiseneuraldependencegraphcreatedbyFlare,theneuronsareplacedalongacircle.Alinkindicatesthattwoneuronsareconnected. 100

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AsshowninFigure 4-9 ,thenetworkinthePP-contrabecomeshighlyinterconnectedbeforeandatthebeginningofthereachingtask.ItsupportstheroleofPP-contrainproducingplannedmovementandthelinkagebetweeninternaltoexternalcoordinatesystems. Inourresults,thelocalfunctionalconnectivitywithinM1-contraisactivatedbythemovement.Figure 4-9C showsthatwhenthevelocityapproacheszeroinsub-state3,littleinteractionappearswithinthisregion(Figure 4-9 ).Moreover,forthemovementswithdifferentjointanglesinsub-state1andsub-state5,mostconnectionsinassemblynetworkarechangedinM1-contra,whichfurthersupportsthehypothesisthatthespecicassemblycontributestotheemergenceofthemovementwithspecickinematicparameters. Inaddition,considerablefunctionalconnectionsarecreatedduringthemouthtoreststateinM1/PMD-ipsi,butnotduringotherkinematicstates,whichisconsistentwiththeresultsofthehandtrajectoryreconstructionin[ 84 ].TheyshowthatM1/PMD-ipsiaccuratelycapturesthemouth/restregions,butmissesthebeginningofmovement.Althoughthemainorganizationalprincipleofprimatemotorsystemsiscorticalcontrolofcontralaterallimbmovement,motorareasalsoappeartoplayaroleinipsilaterallimbmovements.OurresultsrevealthattheinteractioninM1ismodulatedwithfunctionalconnectivitybytheipsilaterallimbmovements,whichsupporttheargumentthatthemotorareasarealsoabletocorrelateipsilaterallimbkinematicswithhighprecision[ 33 ]. 101

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A5sub-stateofthereachingtasktrajectory. BLocalfunctionalconnectivitygraphsinPP-contragiventhe5movementsub-states CLocalfunctionalconnectivitygraphsinM1-contragiventhe5movementsub-states DLocalfunctionalconnectivitygraphsinPMD-contragiventhe5movementsub-states ELocalfunctionalconnectivitygraphsinM1/PMD-ipsigiventhe5movementsub-states Figure4-9. Localfunctionalconnectivitygraphsin4corticalareas:PP-contra,M1-contra,PMD-contra,andM1/PMD-ipsifor5movementsub-states. 102

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4.3NeuralModelingviaFunctionalConnectivityTemporalPattern Intheprevioussections,weinvestigatetheestimationandvisualizationofthetemporalfunctionalconnectivity,whichillustratesitsunderlyingassociationwithbehavioralstates.Theinferenceofthefunctionalconnectivitydynamicswithcognitivebehavior,disease,stimulationhasalsobeenexploredin[ 7 15 22 25 34 82 ],whichindicatesthatthetemporalfunctionalconnectivitypatternisagoodwaytodescribethespatiotemporalneuralactivitypattern.However,thisfunctionalconnectivitypatternisrepresentedbyagraphinsteadofavector,whichmakeitchallengingtouseitasinputfeatureinneuralsystemmodeling.Insteadofdirectlyaddressingfunctionalconnectivitygraphesormatrices,mostresearchesimplementgraphtheorytoquantifythetopologypropertyofgraphpatternincludingnodedegree,clusteringcoefcient,betweennesscentrality,andetc.[ 22 82 ].Thesetopologymetricsabstractfeaturevectorsfromthefunctionalconnectivitygraphs/matrices,whichallowstheimplementationofthetraditionalmodelingwiththevector-basedinputspace.However,theabstractprocesscanonlycapturehowimportantaneuronplaysaroleintheunderlyingfunctionalcommunicationsfromcertainperspective,butreducestheglobalgraphstructure.Therefore,inourworkwedesignakernelforthefunctionalconnectivitymatrix,whichallowustoimplementkernelbasedmachinelearningalgorithmdirectlyonthetemporalfunctionalconnectivitypattern.Inthissection,werstreviewthestatistictopologyattributemeasurethatarewidelyusedtoinvestigatefunctionalconnectivitypattern.Thenweproposeakerneldesignedforfunctionalconnectivitymatrices.Inaddition,weinvestigateandcomparethemodelingperformanceusinggraphtopologiesorfunctionalconnectivitymatriceswithmatrixkernel. 4.3.1MeasuresofNeuralNetworkTopology Graphtopologycanbequantitativelydescriedbyawidevarietyofmeasures,whichhasbeenimplementedtocharacterizethefunctionalconnectivitypatterninanabstractwayandthenusedasanfeaturevectorinmodeling. 103

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1. NodedegreeThedegreeofanodeisthenumberofconnectionsthatlinkittotherestofthenetwork,whichisthemostfundamentalnetworkmeasure. 2. ClusteringcoefcientsTheweightedclusteringcoefcient[ 6 ]iscalculatedindividuallyforeachnode,whichmeasurethedegreetowhichnodesinnetworktendtoclustertogether.Ifthenearestneighborsofanodearealsodirectlyconnectedtoeachother,thentheyconsistacluster.Theclusteringcoefcientquantiesthenumberofconnectionsthatexistbetweenthenearestneighborsofanodeasproportionofmaximumnumberofpossibleconnections,whichdenedbyCn=2en kn(kn)]TJ /F6 11.955 Tf 11.95 0 Td[(1), (4) whereknisthenumberorneighborsofthenoden,enisthenumberofconnectedpairsbetweenallneighborsofthenoden,andkn(kn)]TJ /F6 11.955 Tf 12.15 0 Td[(1)isthemaximumnumberofpossibleconnectionamongallneighbors. 3. BetweennesscentralityThecentralityofanodemeasurehowmanyoftheshortestpathsbetweenallothernodepairsinthenetworkpassthroughit.Anodewithhighcentralityindicatesthatitiscrucialtoefcientcommunication.ThebetweennesscentralityiscalculatedBn=Xi6=j6=nnij(n) nij, (4) wherenijisthenumberofshortestpathsbetweeniandj,andnij(n)isthenumberofshortestpathbetweeniandjthattraversen. AllthesegraphtopologymeasuresareabletoabstractavectorfromtheNNfunctionalconnectivitygraph,whereNthenumberofthenodes.ThefeaturevectorsizeisN.Thenweareabletousethemasinputfeatureinneuralmodelbyapplyingtraditionalvector-basedmachinelearningalgorithm.However,itisnotestablishedwhethertheseattributescansufcientlycharacterizedthefunctionalconnectivitypattern. 104

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4.3.2KernelforFunctionalConnectivityMatrix Inordertoeffectivelyusetemporalfunctionalconnectivitypatternastheinputfeatureforneuralsystemmodeling,itisimportanttoselectasuitablerepresentation.Inourcase,theglobalconnectionpattern,theconnectionweightofeachpairofneurons,andalsotheneuron'slabelsinthisnetworkareconsideredastheeffectiveinformation.Therefore,werepresentthetemporalpatternasmatrixdenedby s=0BBBB@s11s1n.........sn1snn1CCCCA,fornneurons wheresijisthefunctionalconnectionweightbetweenneuroniandneuronj.Thefunctionalconnectivitymatrixissymmetricwithrespecttothemaindiagonal. Conceptually,thiskerneloffunctionalconnectivitymatrixoperatesinthesamewayasthekernelsoperatingondatasamplesinmachinelearning[ 86 ]andinformationtheoreticlearning[ 74 ].TotakeadvantageoftheframeworkforstatisticalsignalprocessingprovidedbyRKHStheory,isrequiredtobeasymmetricpositivedenitefunction.BytheMoore-Aronszajntheorem[ 4 ],thisensuresthatanRKHSHmustexistforwhichisareproducingkernel.Sinceweonlycareabouttheinterdependencepattern,thekernelisfurtherrequiredtobeinvariantwiththearticialorderoftheneurons,thatis,exchangingrowandcorrespondingcolumnshouldnotimpactthekernelvalue. ByanalogytohowtheGaussiankernelisobtainedfromtheEuclideannorm,wecandeneasimilarkernelforfunctionalconnectivitymatrixas (s,s0)=exp()]TJ 10.5 8.09 Td[(ks)]TJ /F8 11.955 Tf 11.95 0 Td[(s0k2 2)=exp()]TJ 10.5 8.09 Td[(kDk2 2) (4) 105

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whereisthenonlinearweightingkernelsizeparameterandD=s)]TJ /F8 11.955 Tf 12.5 0 Td[(s0.ThematrixnormusedinthisworkisFrobeniusnormdenedbykDkF=p trace(D,D)=vuut nXi=1nXj=1jdijj2, (4) Weinvestigatepropertiesofthekernelthatwedesignedforfunctionalconnectivitymatrix,whichisessentialtoindicatethatthiskerneliswelldenedandabletoinduceanRKHSwithnecessarymathematicalstructureforcomputation. Property1. Thekerneloffunctionalconnectivitymatrixisasymmetricpositivedenitekernelofmatrix. Proof:Thefunction'ssymmetryisinheriteddirectlyfrompropertyofthenorm.Inaddition,thekernelwiththematrixnormdenition(Equation 4 )canberewrittenas(s,s0)=nYinYjexp()]TJ 10.49 9.1 Td[(ksij)]TJ /F4 11.955 Tf 11.95 0 Td[(s0ijk 2), (4) whichistheproductofGaussiankernel.SinceGaussiankernelispositivedenitekernel,theproductofgaussiankernelisalsopositivedenite.ThepositivedenitenessofthekernelindicatesthatthekernelinduceanRKHS. Property2. Thekerneloffunctionalconnectivitymatrixisinvariantwiththerow/columnpermutation. Proof:Thekernelisonlysensitivetothedifferencebetweentwomatrices.Consequently,wecanwrite(s,s0)=(s)]TJ /F8 11.955 Tf 9.88 0 Td[(s0)=(D)=(Pni=1Pnj=1jdijj2),whichisaentrywiseoperator.Therefore,itisunrelatedtotherow/columnorder. 4.3.3Experiment Inordertoinvestigatethekernelcapabilitytoidentifythemappingfromthefunctionalconnectivitytoitsunderlyingstimulationoritsreasoningbehavioralstate,temporalpatternoffunctionalconnectivityisusedastheinputfeaturetoclassifythe 106

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multiplemicro-stimulationpatterns.Hereweusetwocategoryrepresentationsoffunctionalconnectivitypattern,thatistheconnectivitymatrixandthetopologyvectorincluding:nodedegree,clusteringcoefcientandbetweennesscentrality.Forthetopologyvector,standardvector-basedGaussiankernelisused.Thematrixkernelproposedintheprevioussectionisappliedtotheconnectivitymatrix.Inthisway,alltheinputfeaturesaremappedintotheRKHSs,respectively.ThestandardsupportvectormachineclassicationalgorithmisusedtoidentifythemapfromasampleinRKHStotheircorrespondingstimulationclass. ThedatasetisthesamedataweusedinChapter 3 ,whichrecordedfromsomatosensorycortexduringthebipolarmicro-stimulationbeingimprintedinThalamus,whichcontains24patterns:8differentlocations,3differentamplitudelevelsforeachlocation,and125eventsforeachpattern,asshowninFigure 2-8 .Thespikedataarecollectedfrom50Channels.Theneuralresponseisselectedrightaftereachstimuluswiththe20mswindowlength. Consideringtheshorttimescaleofneuronresponse,Schreibergetal.similaritymeasuredenedinEquation 4 isusedtoestimatethefunctionalconnectionweightbetweeneachpairofneurons,whichconstituteafunctionalconnectivitymatrix.InSchreibergetal.similaritymeasureforspiketrains,thefunctionalrepresentationofaspiketrainisdenedby^=NXn=1exp()]TJ /F4 11.955 Tf 10.49 8.09 Td[(t)]TJ /F4 11.955 Tf 11.96 0 Td[(tn )U(t)]TJ /F4 11.955 Tf 11.95 0 Td[(tn), (4) Inthisfunctionalrepresentation,isafreeparameter,whichisimportantfortherepresentationaccuracy.Therefore,weoptimizebyminimizingE(^)]TJ /F3 11.955 Tf 12.41 0 Td[()2=E(^2+2)]TJ /F6 11.955 Tf 12.35 0 Td[(2^),whichcanbeequalizedtomaximizeE(^2)]TJ /F6 11.955 Tf 12.34 0 Td[(2^),where(t)=E(s(t))=1=NPNn=1sn(t).Inthisdataset,weestimate=0.01,whichisaverageoverallthechannels.TheaveragefunctionalconnectivitymatrixforeachstimulationpatternisshowninFigure 4-10 ,wherethedifferenceamongdifferentstimulationpatternsis 107

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observable.Inordertoquantifythisdifference,weestimatedtheFrobeniusdistanceofeachpairoffunctionalconnectivitymatrices(3000graphs)denedinEquation 4 ,whichisshowninFigure 4-11 Figure4-10. Functionalconnectivitymatricesforeachstimulationpattern Figure4-11. PairwiseFrobeniusdistanceoffunctionalconnectivitypatterns 108

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Table4-2. Classicationcomparisonoverdifferentinputfeatures. Matrix GraphTopology Temporal kernel Clusteringcoefcient Nodedegree Betweennesscentrality data Testerror 0.16 0.19 0.34 0.51 0.23 InFigure 4-11 ,thex-axisandy-axisweresortedbytheindexofstimulationpattern.Itwasobservedthatthedistancevalueissmallwithin125125squarelyingonthediagonal,whichindicatesthatthedistancebetweenthesampleswithinthesameclassisrelativelysmallerthanthecross-classdistanceinRKHS.Forclassicationtask,weperformaten-foldcross-validationtest,where2700samplesareusedasthetrainingdataandtherest300samplesareusedasthetestdata.ThetestingresultsofdifferentinputfeaturesareshowninTable 4-2 ,whichillustratesthatthematrixkernelbasedmethodhasthebestclassicationperformance.Comparisonamongdifferentgraphtopologyvectorsshowsthatclusteringcoefcientshavethebestclassicationaccuracy.Theclusteringcoefcientsprovideadescriptionofthelocalityoffunctionalconnectivitypattern,whichisimportantinneuralcoordinatemechanism[ 61 ].However,theclusteringcoefcientvectorlosestheinformationofwhichassemblyacertainneuronbelongsto,whichchangescrossdifferentstimuliandisinformativetothestimulationpattern. WefurtherinvestigatethetestresultsofthematrixkernelasshowninFigure 4-12 ,wherethemostclassicationmistakeshappensamongthestimulationpatternsthatsharedthesamestimulationlocationbuthavedifferentstimulationamplitude.Moreover,thesamplescorrespondingtothestimulationwithsmallamplitudeareeasiertobemisclassied,sincethevariabilityofspiketrainsislargeespeciallyforsmallstimulithatcorrespondtosmallsignalnoiseratio. 4.4Discussion Inthischapter,weutilizedependencemeasurestoquantifyandstudythedynamicsoffunctionalconnectivityinmotor/sensorycortexexclusivelyfromtheneuralactivity. 109

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Figure4-12. Classicationresultsoffunctionalconnectivitymatrixkernel Becausetheestimationwindowislimitedbythetimescaleofkinematic/stimulationstates,therobustnessofthefunctional-connectivityestimationwithasmallwindowsizeiskeytodetectthedynamicnatureofneuralactivity.ThecomparisonamongfourdependencemeasuresdemonstratesthatMSCandMIarethemostreliableestimatorsforonesecondwindowswhichcontain10samplesofbinnedneuralactivity(100ms).ThisbinsizehasbeenpreferredformotorBMIs,butfuturestudiesshouldbeconductedtoverifythatitisappropriateforfunctionalstudiesastheonepresentedinthisworkandifnot,determineitsoptimalvalue.Inaddition,bothMSCandMIaredependencemeasuresthatcanbegeneralizedtostudymultichanneldata,sothisextensionwillbeinvestigatedinourfuturework. Inthisstudy,wetestthisalgorithmwithmicroelectrodearraydataforafoodreachingtask.Threestate-relatedassembliesareassessedcorrespondingtotherest-to-food,food-to-mouth,andmouth-to-restkinematicstate,respectively.Inspiteofthecoarsesamplingofneuralstructuresandtherelativelycoarsetimeresolution 110

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ofthemethod,whichimposedthegroupingofallactivatedneuronsinasingleneuralassembly,thegraphsofconditionalfunctionconnectivityshowdistinctassemblies. Ourapproachdetectsacloseassociationbetweentheincreaseoffunctionalconnectionstrengthinstate-relatedassemblyandthetransitionfromresttomovement.Moreover,theactivationdegreeofthestate-relatedassemblyrepeatedlyreachesthepeakwhenthecorrespondingmovementstateoccurs.Infuturework,theactivationdegreecanbefurtherinvestigatedasaquantiablewaytoestimatethespecicmovementstatewithfunctionalconnectivity.Theseresultsalsorevealthatthenetworkofinteractionsamongneuronsseemsthekeyfactorfortheoperationofaspecicbehavior.Althoughintheseexperimentsweconditionedonthekinematicvariables,themethodextractstheinformationsolelyfromneuraldata,transcendsmotortasksandcanbeappliedtoanybrainarea.Inthismoregeneralscenario,theexperimenterwillhavetoselectabaselinestate,astartingpointandadatalengthforstatisticalrobustness(1seccurrently),andapplythedependencemeasuresforeachwindowofneuraldata.Clusteringofthedependencymeasures,orbuildingthegraphstondconsistencybetweendependenciesacrosssequentialsegmentscanbeusedtoabstractdifferentinteractionsthatarethesignaturefordifferentkinematicstates.Therelationshiptokinematicsismerelyusedtosupportourconclusions. ThecircletreegraphsofthelocalfunctionalconnectivitywithineachcorticalareaPP-contra,M1-contra,PMD-contra,andM1/PMD-ipsihelpunderstandthetimevaryingspatialassembliesthatareinthecortex.Moreover,thedynamicsofthelocalassemblygraphsforeachcorticalareacorroboratetheregionalcontributiontothemovementimplementation.SincePP-contralocalizesthebodyinspaceandplansthemovement,neuronsarehighlyinteractiveatthebeginningofthereachingtask.TheinteractioninM1-contraisonlyexpressedwhenthemovementvelocityislarge.Theseresultsalsoshowthatthefunctionalconnectivityisassociatedwithipsilaterallimbmovement. 111

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Althoughthiswasunexpected,wehaveshownbydifferentanalysis[ 84 ]thatthisonlyoccursduringthelatterportionofthetrajectory. Insummary,themeansquarecontingencyseemstobeavaluabletechniquetoinquireaboutfunctionalactivationofneuralassembliesduringbehaviorbecauseofitsstatisticalpoweratsmallsamplesizeanditseasycomputation.Onebigchallengeishowtoshortentheobservationwindow(nowatleast1s)toincreasethetemporalresolutionneededtostudybehavior.Theotherbigchallengeishowtofullyutilizetheinformationcontainedinthepairwiseactivations.Ourapproachwastoapplyastatisticalsignicantthresholdtosimplytheconnectivitymatrixandprovidearstlevelanalysisoffunctionalconnectivity.However,thereispotentiallymuchmoreinformationintheconnectionmatrixtobequantiedbyforinstanceperformingspectralclusteringonthismatrixtondandtrackclustersoffunctionalactivationovertime.Thiswillbepursuedinfurtherstudies. Thedependenceanalysisofneuronalfunctionalconnectivitydynamicscanprovideanunsupervisedestimationofthehiddenneuronalstatesandtheirtransitionsrelatedtothekinematicstates.Inaddition,thestate-relatedassemblyalsorevealstheinternalfunctionalstructurepresentinthemultichannelofneuralsignals,whichallowsamoreefcientutilizationoftheneuraldatainBMIapplications,sinceneuralmodelingusingmultichanneldatanormallyrequireanindependenceassumptionamongstneuralsignals. However,functionalconnectivitypatterncanonlybecharacterizedmatricesorgraphes,whichinducesthechallengewhenwedecodethekinematic/stimulationinformationfromit,thatis,noconventionalvector-basedmachinelearningcannotbedirectlyappliedonit.Thegraphtheoryprovideawaytoextractfeaturevectorfromagraph/matrixwithlosingsomedistributionandglobalstructureoftheconnectivitypattern.Thematrixkernelproposedinthisworkenabledecodingkinematic/stimulationstatesfromthefunctionalconnectivitypatternwithnofeatureextraction.Itbypass 112

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thereduceoftheinformationinducedbyfeatureextraction,whichcontributestotheenhancementofthedecodingaccuracyinthestimulationpatternclassicationtest.Inourclassicationcost,bothmatrixkernel-basedapproachortopology-basedapproachisabletoobtainbetterclassicationperformancethanthetemporalpatternbasedapproach,whichsuggestedthatsomeimportantkinesmetic/stimulationstateiscodedinthedistributedfunctionalnetworkstructures. 113

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CHAPTER5CONCLUSIONS Brainisacomplexsystem,whichcontainsalargenumberoffunctionalinteractingneurons.Theneuronscommunicatethroughspiketrains.Theyselectandcoordinatetheirdistributedactivity,whichisassociatedwithmovements,stimuli,andsomeothercognitiveactions[ 61 ].Inthiswork,ourgoalistodecodethebehavior/stimulationinformationfromtheneuralactivity.Consideringthecomplexityofthesystem,insteadofmodelingeachcomponentintheneuralsystemandcombiningthemtobuildawholemodel,wetreattheneuralsystemasablackboxwithobservableinputs(neuralactivity)andoutputs(correspondingbehavioral/stimulationstate).Amachinelearningframeworkisdesignedtoidentifythefunctionalmappingfromtheneuralactivityandbehaviorstates.Ascanbeexpected,itischallengingtoextractinformationfromtheactivityinahighlydistributed,dynamicandcomplexsystemusingtraditionalmodelingapproaches. 5.1BMIApplication FromBMIapplicationperspective,ourworkproposesthekernel-basedmachinelearningframeworkto,whichprovidesabetterwaytodecodeinformationfromthetemporalandspatialpatternofneuralactivity.Thekernel-basedframeworkovercomeschallengesinducedbythespecialsignalformatandthusallowedustomodelandcontrolaneuralsystem.Becausetheneuronscommunicatewithspiketrains,theinformationisnotcodedintheamplitudebutinthetimingofspikes.Therefore,noconventionalmachinelearningapproachcanbedirectlyappliedtoit.Theexistingapproachistotransformspiketrainstoamplitudedatabycountingthenumberofspikesinashortbinandusethisnumbertorepresentthevalueatthistime.Aftertransformingaspiketraintoadiscretizedvector,anumberofexistingsignalprocessingtechnologieshavebeenextendedandthenappliedonthediscretizedringdata.However,theoutstandingdrawbacksarewhenanetimeresolution(lessthan10ms)isrequired.Thenetimeresolutionrequiresasmallbinsize,whichcausesthesparsity 114

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andhigh-dimensionalityofthedecoderinputandthenworsenthemodelingdifcultyandthecomputationcost.Inourwork,weinvestigatekernelsdesignedonthespaceofspiketrains,whichareabletoprojectionthespiketrainintoaRKHS.Althoughthereisnoalgebraicstructureintheoriginalspiketrainspace,theregressionmethodappliedinRKHScanbeimplementedbymanykernel-basedregressor.Therefore,wetakeadvantageofthekerneldesignedforspiketrainandkernel-basedregressor,andproposeadecoderthatcandirectlyworkonthespiketimes,whichbypasseslimitationsofthediscretizedrepresentationbasedapproaches.TheexperimentindicatesthatourapproachoutperformsGLMandthespikernelwithrespecttobothdecodingperformanceandcomputationtimewhenthetimeresolutionisrequiredtobesmall. However,intheratsensorystimulationexperiment,itisobservedthattheringpatternofneuronshasmarkedvariability,thatis,forthesamestimuli,theneuralresponseisnotrepeated,whichcausestheuctuationsinthedecoderoutput.Inaddition,afunctionalunitofthebraincontainsthousandsofneurons.Onlytheactivityofasmallsubsetneuronscanberecordedintheexperiment,whichfailstocoverthecompleteinformationthatgivesrisetotheintentionalstates.Thisisillustratedbylackofabilitytodiscriminatedifferentstimulationchannelwhenwedecodethemulti-channelmicro-stimulationpattern.Incontrastwithspiketrains,LFPsasanalternativeneuralsignalwithlargerspatialandtimescalearerecodedsimultaneouslywithspiketrainfromthesameelectrodearray,whichcontainscomplementaryinformationoftheintentionalstateofneuralsystem.Therefore,itisworthtocombinetheminthesamemodeltoenhancetherobustnessandaccuracyofthedecoder.Consideringthechallengesposedbythedifferentsignalformat,spatiotemporalscale,andunknownrelationship,tensorproductkernelbaseddecoderisproposedforthemulti-scaleneuraldecoding.Theresultsofreconstructingthetactile/microstimulationshowthatthisdecoderisabletoeffectivelyextractthecomplementaryinformationfrommulti-scaleneuralresponses. 115

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Theultimategoalofourworkistoemulatenaturetouchwithmicrostimulation.Withthisnoveldecodingmethodology,aMIMOadaptiveinversecontrolschemeisbuiltinthespiketrainRKHStocontrolneuralresponsestomatchthetargetspatiotemporalringpattern.TheexperimentalresultsarepresentedinasyntheticneuralsystembuiltfromLIFneuronsbuttheexcitationisdiverseandbroadtomimicmorerealisticconditions.Theyshowthatthecontrolleradaptationallowsthecontrolsystemtoprovidethedesiredringpatternswithhighaccuracy.Moreover,thecontrollerisalsoabletoadapttoperturbationoftheunderlyingneuralsystemorganization,whichisanessentialqualityforthecontrolsystem,otherwisethecontrolsystemwillfailtotrackthetargetsignalwhenperturbationshappen.Inthismorerealisticcase,thedesiredstimulationpatternmaynotexisttoreproducethetargetsignalbutthelinearstructureofthecontrollerinRKHSanditsconvexcostfunctionguaranteethatthiscontrolschemeisabletondtheglobaloptimalspatiotemporalpatternofstimulationthatcanminimizethedissimilaritybetweenthesystemoutputandthetargetsignal.Thesepreliminaryresultsshowthatthiskernel-basedcontroldiagrammaybeapplicableinrealstimulationcontrolwiththemulti-scaleneuraldecodingmethodology.Inthecurrentmode,onlyopenloopcontrolisappliedontheratbecauseoftheanimalexperimentsettingup.Itisobservedthatthecontrolledsystemoutputhavethesimilarspatiotemporalpatternwiththetargetoneinducedbythetactilestimulation.Closeloopcontrolwillbeimplementedinfuture. Inaddition,mostexistingdecodingframeworkarebasedtemporalpatternwithoutconsideringthejointinformationcrosschannels,becauseofitscomplexity.Analternativewaytocharacterizethejointinformationistoestimatefunctionalconnectivitypattern,wherethedependencebetweeneachfairofneuronsisquantied.Inourwork,weestimatedtemporalfunctionalconnectivitypattern.Theassociationbetweenthetemporalfunctionalconnectivitypatternandtheanimalintentionalstatesisobserved,whichisnotdirectlyvisiblebetweenthetemporalpatternofneuralactivityandintentionalstates.Theassociationisalsoestablishedinthepastusingasupervised 116

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framework(i.e.withtheknowledgeofthedesiredresponse[ 84 ]orusingaHiddenMarkovModeltrainedwithpresegmenteddata[ 23 ]),howeverhereitisdecidedwithastatisticaltestexclusivelyfromneuraldatadependencies.Inaddition,matrixkernelisdesignedforthefunctionalconnectivitymatrix,whichallowstousethetemporalfunctionalconnectivitypatternastheinputfeatureofthedecoder.Theresultsofratexperimentshowthatusingtemporalfunctionalconnectivityisabletoclassifythemicro-stimulationpatternandhavelessmisclassicationerrorthanusingtemporalpatternofthedatawithindependentassumption.However,thereareconsiderablelimitationsofdecodinginformationfromthetemporalfunctionalconnectivitypatterns.Measuringtemporaldependencyrequiresatruckofdata,whichlimitsitstimeresolution.Itmaysuggestthatthefunctionalconnectivityisnotanappropriatefeaturefordecodingtaskthatrequiresshorttimeresolution. 5.2MethodologyApplication Thekernelbasedframeworkwedesignedinthisworkisageneralcomputationalframework,whoseapplicationareaisnotconstrainedtotheneuralengineering.Forexample,theSchoenbergkernelbasedregressorofspiketrainsissuitableforgeneralpointprocessdata.Innanceandeconomicsarea,wecanmodelthearrivalofeventssuchascorporatebankruptcies,mergersandacquisitions,orsecuritytradesaspointprocessdata.Moreover,thecustomerclaimarrivalsforinsurance,thearrivalsordeparturesofcustomersinqueuingproblem,andtheoccurrencesofdiseasesinhealthcareareallpointprocessdata,whichcanbemodeledbytheSchoenbergkernel.Anotherpotentialapplicationareaisspikingneuralnetwork,whichfallsintothethirdgenerationofneuralnetworkmodels.Thespikingneuralnetworktakeadvantageofthepulsecodingandtheprinciplesofreservoircomputing,andthushavebetterabilitytomodelthenonlinearmappingfromthesysteminputtosystemoutputintheory.However,inpracticethereisnosimpleandpracticalwaytotunetheparametersoftheneuralcircuitsbecausetheelementsofneuralcircuitareinteractedwithspike 117

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trainsinsteadofamplitudevectors.Asaresulttherehasbeenfewapplicationsofspikingneuralnetworkstosolvecomputationaltasksthatarecommonlyaddressedbyapplyingconventionalmachinelearningmethodontheratecodedoutputlayerofneuralnetworks.WiththeSchoenbergkernelbasedregressionmethod,thetransformationfromspiketrainstotheratevectorisnotrequiredanymoreintheoutputlayer. Inaddition,kernel-basedadaptiveinversecontroldiagramproposedinthisworkprovideageneralcontrolframeworkthatallowscontrollinganonlineardynamicsystemfullyintheRKHS.TheadvantagesofcontrolinRKHSlieonthefollowingaspects.First,itcanbedirectlyappliedtoanytypeofdataformate,aslongaswecandeneakernelthatcanmapthedatafromtheoriginalinputspacetoRKHS.Evenforthedataspacewithnoalgebraicstructure,thecontrollerparametercanbeadjustedtominimizethedistancebetweentargetsignalandthesystemoutputinRKHS.Inaddition,thecontroldiagramisabletocontroltheoutputofanonlinearsystemtofollowthetargetpatternwithlinearcontrolinRKHS.ItslinearstructureinRKHSguaranteesthatthecontrollerwillnotconvergetolocalminimaduringthecontrolprocess.Inourwork,weshowtwoexamples:spiketrainsandheterogenousmulti-sourcedata. 118

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REFERENCES [1] A.M.Aertsen,G.L.Gerstein,M.K.Habib,andG.Palm,Dynamicsofneuronalringcorrelation:modulationofeffectiveconnectivity,JournalofNeurophysiol-ogy,vol.61,no.5,pp.900,1989. [2] M.Aghagolzadeh,S.Eldawlatly,andK.Oweiss,Synergisticcodingbycorticalneuralensembles,IEEETransactionsonInformationTheory,vol.56,pp.875889,2010. [3] Y.Ahmadian,A.M.Packer,R.Yuste,andL.Paninski,Designingoptimalstimulitocontrolneuronalspiketiming,JournalofNeurophysiology,vol.106,pp.1038,2011. [4] N.Aronszajn,Theoryofreproducingkernels,TransactionsoftheAmericanMathematicalSociety,vol.68,no.3,pp.337,1950. [5] K.J.AstromandB.Wittenmark,AdaptiveControl,2nded.PrenticeHall,Dec.1994. [6] A.Barrat,M.Barthelemy,R.Pastor-Satorras,andA.Vespignani,Thearchitectureofcomplexweightednetworks,PNAS,vol.101,pp.3747,2004. [7] D.S.BassettandE.Bullmore,Small-worldbrainnetworks,Neuroscientist,vol.12,p.512,2006. [8] A.Belitski,S.Panzeri,C.Magri,N.K.Logothetis,andC.Kayser,Sensoryinformationinlocaleldpotentialsandspikesfromvisualandauditorycortices:timescalesandfrequencybands.Journalofcomputationalneuroscience,vol.29,no.3,pp.533,Dec.2010. [9] P.Berens,G.Keliris,A.Ecker,N.Logothetis,andA.Tolias,Featureselectivityofthegamma-bandofthelocaleldpotentialinprimateprimaryvisualcortex,FrontiersinNeuroscience,vol.2,p.199207,2008. [10] P.Berkes,F.Wood,andJ.Pillow,Characterizingneuraldependencieswithcopulamodels,AdvancesinNeuralInformationProcessingSystems,vol.21,pp.129,2009. [11] W.Bialek,F.Rieke,R.deRuytervanSteveninck,andD.Warland,Readinganeuralcode,Science,vol.252,no.5014,pp.1854,1991. [12] A.J.Brockmeier,S.Member,J.S.Choi,M.M.Distasio,andJ.T.Francis,Optimizingmicrostimulationusingareinforcementlearningframework,IEEEEngineeringinMedicineandBiologyMagazine,pp.1069,2011. [13] E.N.Brown,R.E.Kass,andP.P.Mitra,Multipleneuralspiketraindataanalysis:state-ofthe-artandfuturechallenges,natureneuroscience,vol.7,pp.456,2004. 119

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[14] D.Brugger,S.Butovas,M.Bogdan,andC.Schwarz,Real-timeadaptivemicrostimulationincreasesreliabilityofelectricallyevokedcorticalpotentials,BiomedicalEngineering,IEEETransactionson,vol.58,pp.1483,2011. [15] E.BullmoreandO.Sporns,Complexbrainnetworks:graphtheoreticalanalysisofstructuralandfunctionalsystems.Naturereviews.Neuroscience,vol.10,no.3,pp.186,2009. [16] G.BuzsakiandA.Draguhn,Neuronaloscillationsincorticalnetworks,Science,vol.304,no.5679,pp.1926,Jun.2004. [17] J.M.Carmena,M.A.Lebedev,R.E.Crist,J.E.O'Doherty,D.M.Santucci,D.F.Dimitrov,P.G.Patil,C.S.Henriquez,andM.A.L.Nicolelis,Learningtocontrolabrain-machineinterfaceforreachingandgraspingbyprimates,PLoSBiol,vol.1,no.2,pp.193208,2003. [18] J.K.Chapin,K.A.Moxon,R.S.Markowitz,andM.A.Nicolelis,Real-timecontrolofarobotarmusingsimultaneouslyrecordedneuronsinthemotorcortex.Natureneuroscience,vol.2,no.7,pp.664,Jul.1999. [19] B.Chen,S.Zhao,P.Zhu,andJ.C.Principe,Quantizedkernelleastmeansquarealgorithm,TransactionsonNeuralNetworks,vol.23,pp.22,2012. [20] N.CristianiniandJ.Shawe-Taylor,AnIntroductiontoSupportVectorMachinesandOtherKernel-basedLearningMethods.CambridgeUniversityPress,2000. [21] L.CsatoandM.Opper,Sparseonlinegaussianprocesses,NeuralComputation,vol.14,pp.641,2002. [22] I.Daly,S.J.Nasuto,andK.Warwick,Braincomputerinterfacecontrolviafunctionalconnectivitydynamics,PatternRecognition,vol.45,pp.2123,2012. [23] S.Darmanjian,Designandanalysisofgenerativemodelsforbrainmachineinterfaces,Ph.D.dissertation,UniversityofFlorida,2009. [24] P.DayanandL.F.Abbott,TheoreticalNeuroscience:ComputationalandMathe-maticalModelingofNeuralSystems.MITPress,2001. [25] L.Deuker,E.T.Bullmore,M.Smith,S.Christensen,P.J.Nathan,B.Rockstroh,andD.S.Bassett,Reproducibilityofgraphmetricsofhumanbrainfunctionalnetworks,NeuroImage,vol.47,no.4,pp.14601468,2009. [26] J.DiGiovanna,B.Mahmoudi,J.Fortes,J.C.Principe,andJ.C.Sanchez,Co-adaptivebrainmachineinterfaceviareinforcementlearning,IEEETrans-actionsonBiomedicalEngineering(SpecialissueonHybridBionics),vol.56,pp.54,2009. 120

PAGE 121

[27] A.S.Ecker,P.Berens,G.A.Keliris,M.Bethge,N.K.Logothetis,andA.S.Tolias,Decorrelatedneuronalringincorticalmicrocircuits,Science,vol.327,pp.584,January2010. [28] S.Eldawlatly,R.Jin,andK.G.Oweiss,Identifyingfunctionalconnectivityinlarge-scaleneuralensemblerecordings:Amultiscaledataminingapproach,Neuralcomputation,vol.21,pp.450,2009. [29] Y.Engel,S.Mannor,andR.Meir,Thekernelrecursiveleastsquaresalgorithm,IEEETransactionsonSignalProcessing,vol.52,pp.2275,2003. [30] X.Feng,B.Greenwald,H.Rabitz,E.S.Brown,andR.Kosut,Towardclosed-loopoptimizationofdeepbrainstimulationforparkinson'sdisease:conceptsandlessonsfromacomputationalmodel,JournalofNeuralEngineering,vol.4,no.2,p.L14,2007. [31] N.A.Fitzsimmons,W.Drake,T.L.Hanson,M.A.Lebedev,andM.A.L.Nicolelis,Primatereachingcuedbymultichannelspatiotemporalcorticalmicrostimulation,TheJournalofNeuroscience,vol.27,no.21,pp.5593,2007. [32] J.T.Francis,S.Xu,andJ.K.Chapin,Proprioceptiveandcutaneousrepresentationsintheratventralposterolateralthalamus,JournalofNeuro-physiology,vol.99,no.5,pp.2291,2008. [33] K.Ganguly,L.Secundo,G.Ranade,A.Orsborn,E.F.Chang,D.F.Dimitrov,J.D.Wallis,N.M.Barbaro,R.T.Knight,andJ.M.Carmena,Corticalrepresentationofipsilateralarmmovementsinmonkeyandman,TheJournalofNeuroscience,vol.29,pp.12948956,2009. [34] C.E.GinestetandA.Simmons,Statisticalparametricnetworkanalysisoffunctionalconnectivitydynamicsduringaworkingmemorytask,NeuroImage,vol.55,no.2,pp.688704,2011. [35] F.GrammontandA.Riehle,Spikesynchronizationandringrateinapopulationofmotorcorticalneuronsinrelationtomovementdirectionandreactiontime,BiologicalCybernetics,vol.88,pp.360,2003. [36] S.Grun,M.Diesmann,andA.Aertsen,Unitaryeventsinmultiplesingle-neuronspikingactivity:I.detectionandsignicance,NeuralComputation,vol.14,no.1,pp.43,January2002. [37] J.R.Huxter,T.J.Senior,K.Allen,andJ.Csicsvari,Thetaphase-speciccodesfortwo-dimensionalposition,trajectoryandheadinginthehippocampus,NatureNeuroscience,vol.11,pp.587,2008. [38] Csim:aneuralcircuitsimulator.,TheIGILSMGroup,2006. 121

PAGE 122

[39] B.JamesandJ.Ranck,Whichelementsareexcitedinelectricalstimulationofmammaliancentralnervoussystem:Areview,BrainResearch,vol.98,no.3,pp.417440,1975. [40] R.S.JohanssonandG.Westling,Rolesofglabrousskinreceptorsandsensorimotormemoryinautomaticcontrolofprecisiongripwhenliftingrougherormoreslipperyobjects,ExperimentalBrainResearch,vol.56,no.3,pp.550,Oct.1984. [41] R.E.Kass,V.Ventura,andE.N.Brown,Statisticalissuesintheanalysisofneuronaldata,JournalofNeurophysiology,vol.94,no.1,pp.8,2005. [42] R.C.Kelly,M.A.Smith,R.E.Kass,andT.S.Lee,Localeldpotentialsindicatenetworkstateandaccountforneuronalresponsevariability,Journalofcomputa-tionalneuroscience,vol.29,pp.567,2010. [43] J.P.Lachaux,E.Rodriguez,J.Martinerie,andF.J.Varela,Measuringphasesynchronyinbrainsignals,HumanBrainMapping,vol.8,pp.194,1999. [44] L.Li,I.M.Park,S.Seth,J.S.Choi,J.T.Francis,J.C.Sanchez,andJ.C.Principe,Anadaptivedecoderfromspiketrainstomicro-stimulationusingkernelleast-mean-squares(klms),inMachineLearningforSignalProcessing(MLSP),2011IEEEInternationalWorkshopon,Sept.2011,pp.1. [45] L.Li,S.Seth,I.Park,J.C.Sanchez,andJ.Principe,Neuronalfunctionalconnectivitydynamicsincortex:Anmsc-basedanalysis,inIEEEInternationalEMBSConference,2010. [46] J.LiuandW.Newsome,Localeldpotentialincorticalareamt:Stimulustuningandbehavioralcorrelations,JournalofNeuroscience,vol.26,pp.77797790,2006. [47] J.Liu,H.K.Khalil,andK.G.Oweiss,Model-basedanalysisandcontrolofanetworkofbasalgangliaspikingneuronsinthenormalandparkinsonianstates,JournalofNeuralEngineering,vol.8,no.4,p.045002,2011. [48] W.Liu,I.Park,andJ.C.Principe,Aninformationtheoreticapproachofdesigningsparsekerneladaptivelters.IEEETransactionsonNeuralNetworks,pp.1950,2009. [49] W.Liu,P.Pokharel,andJ.Principe,Correntropy:Propertiesandapplicationsinnon-gaussiansignalprocessing,IEEETransactionsonSignalProcessing,vol.55,pp.52865298,2007. [50] W.Liu,J.C.Prncipe,andS.Haykin,Kerneladaptiveltering,S.Haykin,Ed.Johnwileysons,Inc.,2010. 122

PAGE 123

[51] W.Maass,T.Natschlager,andH.Markram,Computationalmodelsforgenericcorticalmicrocircuits,ComputationalNeuroscience:AComprehensiveApproach,2003. [52] W.Maass,T.Natschlager,andH.Markram,Real-timecomputingwithoutstablestates:Anewframeworkforneuralcomputationbasedonperturbations,NeuralComputation,vol.14,pp.2531,2002. [53] S.Mallat,AWaveletTourofSignalProcessing,ThirdEdition:TheSparseWay.AcademicPress,2008. [54] V.Z.Marmarelis,NonlinearDynamicModelingofPhysiologicalSystems.Wiley-IEEEPress,2003. [55] N.Mesgarani,S.V.David,J.B.Fritz,andS.A.Shamma,Inuenceofcontextandbehavioronstimulusreconstructionfromneuralactivityinprimaryauditorycortex.JournalofNeurophysiology,vol.102,no.6,pp.3329,Dec.2009. [56] M.Mesulam,Large-scaleneurocognitivenetworksanddistributedprocessingforattention,language,andmemory,AnnalsofNeurology,vol.28,pp.597,2004. [57] J.Moehlis,E.Shea-Brown,andH.Rabitz,Optimalinputsforphasemodelsofspikingneurons,JournalofComputationalandNonlinearDynamics,vol.1,pp.358,2006. [58] S.Monaco,G.Kroliczak,D.Quinlan,P.Fattori,C.Galletti,M.Goodale,andJ.Culham,Contributionofvisualandproprioceptiveinformationtotheprecisionofreachingmovements,ExperimentalBrainResearch,vol.202,pp.15,2010. [59] F.Mormann,K.Lehnertz,P.David,andC.E.Elger,Meanphasecoherenceasameasureforphasesynchronizationanditsapplicationtotheeegofepilepsypatients,PhysicaD:NonlinearPhenomena,vol.144,no.3-4,pp.358,2000. [60] M.A.L.Nicolelis,Brain-machineinterfacestorestoremotorfunctionandprobeneuralcircuits,NatureReviewsNeuroscience,vol.4,pp.417,2003. [61] M.A.L.NicolelisandM.A.Lebedev,Principlesofneuralensemblephysiologyunderlyingtheoperationofbrainmachineinterfaces,NatureReviewsNeuro-science,vol.10,no.7,pp.530,July2009. [62] J.E.ODoherty,M.A.Lebedev,T.L.Hanson,N.A.Fitzsimmons,andM.A.Nicolelis,Abrain-machineinterfaceinstructedbydirectintracorticalmicrostimulation.Frontiersinintegrativeneuroscience,vol.3,2009. [63] J.E.ODoherty,M.A.Lebedev,P.J.Ifft,K.Z.Zhuang,S.Shokur,H.Bleuler,andM.A.L.Nicolelis,Activetactileexplorationusingabrain-machine-braininterface,Nature,vol.advanceonlinepublication,Oct.2011. 123

PAGE 124

[64] J.E.ODoherty,M.A.Lebedev,P.J.Ifft,K.Z.Zhuang,S.Shokur,H.Bleuler,andM.A.L.Nicolelis,Activetactileexplorationusingabrain-machine-braininterface,Nature,vol.advanceonlinepublication,2011. [65] A.Paiva,I.Park,andJ.Principe,Acomparisonofbinlessspiketrainmeasures,NeuralComputing&Applications,vol.19,pp.405,2010. [66] A.R.C.Paiva,I.Park,andJ.C.Principe,Areproducingkernelhilbertspaceframeworkforspiketrainsignalprocessing,NeuralComputation,vol.21,pp.424,2009. [67] I.M.Park,Capturingspiketrainsimilaritystructure:apointprocessdivergenceapproach,Ph.D.dissertation,UniversityofFlorida,2010. [68] I.M.Park,S.Seth,M.Rao,andJ.C.Principe,Strictlypositivedenitespiketrainkernelsforpointprocessdivergences,NeuralComputation,accepted. [69] K.Pearson,J.Harris,A.Treloar,andM.Wilder,Onthetheoryofcontingency,AmericanStatisticalAssociation,vol.25,no.171,pp.320,Sep1930. [70] E.Pereda,R.Q.Quiroga,andJ.Bhattacharya,Nonlinearmultivariateanalysisofneurophysiologicalsignals,ProgressinNeurobiology,vol.77,pp.1,2005. [71] J.Pillow,Y.Ahmadian,andL.Paninski,Model-baseddecoding,informationestimation,andchange-pointdetectioninmultineuronspiketrains,Statistics,pp.1,2006. [72] G.Pipa,D.W.Wheeler,W.Singer,andD.Nikolic,NeuroXidence:reliableandefcientanalysisofanexcessordeciencyofjoint-spikeevents.Journalofcomputationalneuroscience,vol.25,no.1,pp.64,August2008. [73] J.Platt,Aresource-allocatingnetworkforfunctioninterpolation,NeuralComput.,vol.3,no.2,pp.213,1991. [74] J.Principe,J.W.Fisher,andD.Xu,UnsupervisedAdaptiveFiltering.Wiley,2000,ch.InformationTheoreticLearning,pp.265. [75] R.Q.Quiroga,A.Kraskov,T.Kreuz,andP.Grassberger,Performanceofdifferentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals,PhysicalReviewE,vol.65,no.4,pp.903,Mar2002. [76] H.RamlauHansen,Smoothingcountingprocessintensitiesbymeansofkernelfunctions,TheAnnalsofStatistics,vol.11,no.2,pp.pp.453,1983. [77] M.J.Rasch,A.Gretton,Y.Murayama,W.Maass,andN.K.Logothetis,Inferringspiketrainsfromlocaleldpotentials,journalofneurophysiology,vol.99,pp.1461,2007. 124

PAGE 125

[78] J.Rigosa,D.J.Weber,A.Prochazka,R.B.Stein,andS.Micera,Neuro-fuzzydecodingofsensoryinformationfromensemblesofsimultaneouslyrecordeddorsalrootganglionneuronsforfunctionalelectricalstimulationapplications.JournalofNeuralEngineering,vol.8,no.4,p.046019,2011. [79] M.Rosenblum,A.Pikovsky,J.Kurths,C.Schafer,andP.A.Tass,Phasesyn-chronization:fromtheorytodataanalysis.ElsevierScience,2001,ch.9,pp.279. [80] Y.Roudi,S.Nirenberg,andP.E.Latham,Pairwisemaximumentropymodelsforstudyinglargebiologicalsystems:Whentheycanworkandwhentheycan't,PLoSComputBiol,vol.5,no.5,052009. [81] M.S.Roulston,Estimatingtheerrorsonmeasuredentropyandmutualinformation,PhysicaD:NonlinearPhenomena,vol.125,pp.285,1999. [82] M.Rubinov,S.A.Knock,C.J.Stam,S.Micheloyannis,A.W.Harris,L.M.Williams,andM.Breakspear,Small-worldpropertiesofnonlinearbrainactivityinschizophrenia,HumanBrainMapping,vol.30,p.403416,2009. [83] J.SanchezandJ.C.Principe,BrainMachineInterfaceEngineering.MorganandClaypool,2007. [84] J.C.Sanchez,Fromcorticalneuralspiketrainstobehavior:Modelingandanalysis,Ph.D.dissertation,UniversityofFlorida,2004. [85] B.SchlkopfandA.J.Smola,LearningwithKernels:SupportVectorMachines,Regularization,Optimization,andBeyond,MA,Ed.MITPress,2001. [86] B.Scholkopf,A.Smola,andK.-R.Muller,Nonlinearcomponentanalysisasakerneleigenvalueproblem,NeuralComputation,vol.10,no.5,pp.1299,1998. [87] B.ScholkopfandA.J.Smola,Learningwithkernels:supportvectormachines,regularization,optimization,andbeyond,ser.Adaptivecomputationandmachinelearning.MITPress,2002. [88] S.Schreiber,J.Fellous,D.Whitmer,P.Tiesinga,andT.Sejnowski,Anewcorrelation-basedmeasureofspiketimingreliability,Neurocomputing,vol.52-54,pp.925931,2003. [89] M.D.Serruya,N.G.Hatsopoulos,L.Paninski,M.R.Fellows,andJ.P.Donoghue,Brain-machineinterface:Instantneuralcontrolofamovementsignal,Nature,vol.416,no.6877,pp.141,Mar.2002. [90] L.Shpigelman,Y.Singer,R.Paz,andE.Vaadia,Spikernels:Embeddingspikingneuronsininnerproductspaces,inAdvancesinNeuralInformationProcessingSystems15.MITPress,2003. 125

PAGE 126

[91] ,Spikernels:Predictingarmmovementsbyembeddingpopulationspikeratepatternsininner-productspaces,NeuralComputation,vol.17,no.3,pp.671,March2005. [92] A.J.SmolaandB.Scholkopf,Atutorialonsupportvectorregression,StatisticsandComputing,vol.14,no.3,pp.199,2004. [93] D.L.SnyderandM.I.Miller,RandomPointProcessesinTimeandSpace.Springer-Verlag,1991. [94] R.SokalandF.Rohlf,Biometry:Theprinciplesandpracticeofstatisticsinbiologicalresearch,3rded.NewYork,1995. [95] R.Storchi,A.G.Zippo,G.C.Caramenti,M.Valente,andG.E.M.Biella,Predictingspikeoccurrenceandneuronalresponsivenessfromlfpsinprimarysomatosensorycortex,PLoSONE,vol.7,no.5,p.e35850,52012. [96] P.Tass,M.G.Rosenblum,J.Weule,J.Kurths,A.Pikovsky,J.Volkmann,A.Schnitzler,andH.J.Freund,Detectionofn:mphaselockingfromnoisydata:Applicationtomagnetoencephalography,Phys.Rev.Lett.,vol.81,no.15,pp.3291,Oct1998. [97] D.M.Taylor,S.I.Tillery,andA.B.Schwartz,Directcorticalcontrolof3dneuroprostheticdevices,Science,vol.296,no.5574,pp.1829,Jun.2002. [98] UCBerkeleyVisualizationLab,2009. [99] F.Varela,J.P.Lachaux,E.Rodriguez,andJ.Martinerie,Thebrainweb:phasesynchronizationandlarge-scaleintegration,NatureReviewsNeuroscience,vol.2,pp.229,2001. [100] M.Velliste,S.Perel,M.C.Spalding,A.S.Whitford,andA.B.Schwartz,Corticalcontrolofaprostheticarmforself-feeding.Nature,vol.453,no.7198,pp.1098,2008. [101] F.W.Volterra,R.Kelly,andT.S.Lee,Decodingv1neuronalactivityusingparticlelteringwithvolterrakernels,inAdvancesinNeuralInformationProcessingSystems,2003,pp.15. [102] H.Wang,D.Pi,andY.Sun,Onlinesvmregressionalgorithm-basedadaptiveinversecontrol,Neurocomput.,vol.70,pp.952,2007. [103] Y.Wang,Pointprocessmontecarlolteringforbrainmachineinterface,Ph.D.dissertation,UniversityofFlorida,2008. [104] D.K.Warland,P.Reinagel,M.Meister,D.K,andP.Reinagel,Decodingvisualinformationfromapopulationofretinalganglioncells,J.Neurophysiol,vol.78,pp.2336,1997. 126

PAGE 127

[105] B.WidrowandE.Walach,AdaptiveInverseControl,E.Cliff,Ed.Prentice-Hall,1995. [106] D.Xing,C.Yeh,andR.Shapley,Spatialspreadofthelocaleldpotentialanditslaminarvariationinvisualcortex.JournalofNeuroscience,vol.29,pp.1154011549,2009. [107] S.Yamada,K.Matsumoto,M.Nakashima,andS.Shiono,Informationtheoreticanalysisofactionpotentialtrainsii.analysisofcorrelationamongytneuronstodeduceconnectionstructure,Journalofneurosciencemethods,vol.66,pp.35,1996. [108] J.H.Zar,Biostatisticalanalysis,T.Ryu,Ed.PrenticeHall,1999. 127

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BIOGRAPHICALSKETCH LinLireceivedtheB.E.degreeinelectricalandinformationengineeringfromUniversityofScienceandTechnology,Beijing,China,in2007,theM.E.degreeinelectricalandcomputerengineeringfromUniversityofFlorida,Gainesville,in2010.SheiscurrentlypursuingthePh.D.degreeinelectricalandcomputerengineeringatUniversityofFlorida,Gainesville.ShehasbeenworkingintheComputationalNeuroEnginneringlaboratoryattheUniversityofFloridaunderthesupervisionofDr.J.C.Principesince2009.Herresearchinterestsaremachinelearning,statisticalmodeling,andcomputationalneuroscience. 128