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A Kernel Approach to Learning a Neuron Model from Spike Train Data

Permanent Link: http://ufdc.ufl.edu/UFE0042012/00001

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Title: A Kernel Approach to Learning a Neuron Model from Spike Train Data
Physical Description: 1 online resource (90 p.)
Language: english
Creator: Fisher, Nicholas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: classification, data, kernel, model, neuron, spike, spiking, time, train
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: A spiking neuron is a principal component of the brain and the nervous system. Understanding the characteristics of a single neuron as well as the interactions of a population of neurons is essential to neuroscience. At the micro-level, neurons are very complex devices that control the ow of ions in and out of the membrane. However an abstracted view of the neuron sees it as a mechanism which receives electrical signals from other neurons as input and produces and transmits electrical signals to other neurons. Much work has been done to model the neuron at varying levels of complexity in order to explain its dynamics. Competitions exist which compare the accuracy of submitted neurons models. Here we propose a methodology which learns an equivalent mathematical mapping from input spike trains to the output spike train by only considering the timing of all afferent(incoming) and efferent(outgoing) spikes within a bounded finite past. This is done by instantiating a classification problem which uses kernels to dichotomize input spike trains which cause the neuron to generate a spike from those that do not cause the neuron to generate a spike. The kernel used is one that has been generated from a dictionary of functions which are similar to those used in existing neuron models. By using an intuitive dictionary, we produce a kernel which is tailored to the problem of learning spiking neuron models. By the representer theorem we know that only a finite number of training data points will be needed to produce the classification solution. By considering the number of data points used to produce the solution, we are able to assess the complexity of the modeled neuron. Those neurons which need more spike time inputs to reproduce their behavior could be considered more complex than those which need fewer spike time inputs for the given kernel.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Nicholas Fisher.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Banerjee, Arunava.

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042012:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042012/00001

Material Information

Title: A Kernel Approach to Learning a Neuron Model from Spike Train Data
Physical Description: 1 online resource (90 p.)
Language: english
Creator: Fisher, Nicholas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: classification, data, kernel, model, neuron, spike, spiking, time, train
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A spiking neuron is a principal component of the brain and the nervous system. Understanding the characteristics of a single neuron as well as the interactions of a population of neurons is essential to neuroscience. At the micro-level, neurons are very complex devices that control the ow of ions in and out of the membrane. However an abstracted view of the neuron sees it as a mechanism which receives electrical signals from other neurons as input and produces and transmits electrical signals to other neurons. Much work has been done to model the neuron at varying levels of complexity in order to explain its dynamics. Competitions exist which compare the accuracy of submitted neurons models. Here we propose a methodology which learns an equivalent mathematical mapping from input spike trains to the output spike train by only considering the timing of all afferent(incoming) and efferent(outgoing) spikes within a bounded finite past. This is done by instantiating a classification problem which uses kernels to dichotomize input spike trains which cause the neuron to generate a spike from those that do not cause the neuron to generate a spike. The kernel used is one that has been generated from a dictionary of functions which are similar to those used in existing neuron models. By using an intuitive dictionary, we produce a kernel which is tailored to the problem of learning spiking neuron models. By the representer theorem we know that only a finite number of training data points will be needed to produce the classification solution. By considering the number of data points used to produce the solution, we are able to assess the complexity of the modeled neuron. Those neurons which need more spike time inputs to reproduce their behavior could be considered more complex than those which need fewer spike time inputs for the given kernel.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Nicholas Fisher.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Banerjee, Arunava.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042012:00001


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AKERNELAPPROACHTOLEARNINGANEURONMODELFROMSPIKETRAINDATAByNICHOLASK.FISHERADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010 1

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c2010NicholasK.Fisher 2

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ACKNOWLEDGMENTS Ithankmyparents,grandparentsandbrothersfortheirunconditionalloveandsupportonnumerouslevelsthroughoutthisprocessandmylife.TheyarelargelyresponsibleforthepersonIamtoday,whichisnotnecessarilyareasonforpride.Ithankmywifetobeforherlove,supportandinterestaswellasthenumeroustripstoFloridaforshortvisits.Hernumerouscarepackagesalwaysmademesmile.Sheservedasextramotivationtocompletemystudiesandmoveontothenextchapterofmylife.Ithankmysistersin-lawfortheirsupportandmyniecesandnephews,whomIalsothankfortheirunconditionallovewhichbringsmejoyandinspiration.Ithankmyfriendswhokeptmesocialandsanethroughout.IthankmyprimaryandsecondaryteachersforincitinginmeathirstforknowledgewhenIwasyoung.Ioweaspecialdebtofgratitudetomylabmatesandcolleaguesintheeld.Thiswouldnothavebeenpossiblewithoutnumerousdiscussionsonthemathematicsandtheoryinvolvedintheseissues.InadditiontheirsenseofhumorinunderstandingallofthenuancesinvolvedinthePhDprocesswasinvaluable.Ithankmycommitteemembersfortheirknowledgeandinsightrelatingtomydissertation.FinallyIwouldliketogiveaspecialthankstomycommitteechairandadvisorforsharinghisvastamountofknowledgeonacademicandnon-academictopics.Hisundyingoptimism,althoughhetriestohideit,wasamustinthiseld.Hisabilitytounderstandandconveythemostcomplicatedtopicsisunmatchedandhispatienceandunderstandingoflifeoccurrencesbeyondacademiawasanecessity. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1BACKGROUND ................................... 11 1.1Introduction ................................... 11 1.2TheNeuron ................................... 14 1.3NeuronModels ................................. 16 1.3.1Non-PhenomenologicalModels ..................... 17 1.3.2PhenomenologicalModels ........................ 19 1.3.3GeneralModeloftheNeuron ...................... 21 1.3.4Summary ................................. 22 1.4GeneralClassicationProblem ......................... 22 1.5KernelAnalysisofaSpikingNeuron ..................... 24 2AKERNELBASEDNEURONMODEL ...................... 28 2.1Introduction ................................... 28 2.2Model ...................................... 29 2.3ClassicationFramework ............................ 31 2.4Summary .................................... 33 3RECIPROCALEXPONENTIAL-EXPONENTIALFUNCTIONDICTIONARY 35 3.1Approximationofthemembranepotentialfunction ............. 36 3.1.1DiscreteFormulation .......................... 36 3.1.2Continuousformulation ......................... 37 3.1.2.1Representertheorem ..................... 39 3.1.2.2Dualrepresentation ...................... 40 3.1.3SingleSynapse .............................. 41 3.1.3.1Primalproblem ........................ 41 3.1.3.2Dualproblem ......................... 42 3.1.4MultipleSynapses ............................ 43 3.1.4.1Primalproblem ........................ 43 3.1.4.2Dualproblem ......................... 44 3.1.5Summary ................................. 44 4

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4GAUSSIANFUNCTIONDICTIONARY ...................... 46 4.1ApproximationoftheMembranePotentialFunction ............. 46 4.1.1DiscreteFormulation .......................... 47 4.1.2ContinuousFormulation ........................ 47 4.1.2.1Representertheorem ..................... 49 4.1.2.2Dualrepresentation ...................... 50 4.1.3SingleSynapse .............................. 51 4.1.3.1Primalproblem ........................ 51 4.1.3.2Dualproblem ......................... 52 4.1.4MultipleSynapses ............................ 53 4.1.4.1Primalproblem ........................ 53 4.1.4.2Dualproblem ......................... 54 4.1.5Summary ................................. 55 5RECIPROCALEXPONENTIALFUNCTIONDICTIONARY .......... 56 5.1ApproximationoftheConductanceMembranePotentialFunction ..... 57 5.1.1DiscreteFormulation .......................... 57 5.1.2Continuousformulation ......................... 59 5.1.2.1PrimalProblem ........................ 60 5.1.2.2Representertheorem ..................... 61 5.1.2.3Dualrepresentation ...................... 61 5.1.3SingleSynapse .............................. 62 5.1.3.1Primalproblem ........................ 63 5.1.3.2Dualproblem ......................... 64 5.1.4MultipleSynapses ............................ 64 5.1.4.1Primalproblem ........................ 65 5.1.4.2Dualproblem ......................... 65 5.1.5Summary ................................. 66 6KERNELCOMPARISONANDRESULTS .................... 68 6.1GaussianSummationKernel .......................... 68 6.2ReciprocalExponential-ExponentialKernel ................. 69 6.3ConductanceKernel .............................. 70 6.4KernelComparison ............................... 71 6.4.1SingleAerentSynapse ......................... 71 6.4.2MultipleAerentSynapses ....................... 73 6.4.3MultipleAerentSynapseswithEerentSpikes ........... 74 6.4.4ConductanceBasedSynapse ...................... 75 6.5Summary .................................... 76 7CONCLUSIONSANDFUTUREWORK ...................... 85 REFERENCES ....................................... 88 5

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BIOGRAPHICALSKETCH ................................ 90 6

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LISTOFTABLES Table page 6-1Qualitymetricsforsinglesynapsemodel. ...................... 77 6-2Qualitymetricsformultiplesynapsemodelwithoutspikegeneration. ...... 80 6-3Qualitymetricsforconductancesynapsemodel. ................. 85 7

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LISTOFFIGURES Figure page 1-1Ahighleveldiagramofaneuron. ......................... 17 1-2Hodgkin-Huxleyandleakyintegrateandrecircuitdiagrams. .......... 21 1-3Hodgkin-Huxleymembranepotentialdynamics. .................. 23 1-4ModelingfunctionsfortheSRM0. ......................... 26 1-5Classifyinglinearlyseparabledatawiththeoptimalhyperplane. ........ 28 2-1Diagramofspiketrainsusedtotrainourclassier. ................ 36 3-1FiguresdepictingtheformoftheREEFbasis. .................. 38 5-1ComparisonofstereotypicalandscaledPSPs. ................... 60 5-2ScalingeectonthesecondPSPasafunctionoftheinterspikeinterval. .... 60 6-1ComparisonoftheGRBFtotheGSK,inthetwodimensionalcase. ....... 72 6-2ComparisonoftheREEKtotheGSK,inthetwodimensionalcase. ...... 74 6-3Theconductancekernelfortwodimensionalinterspikeintervals. ........ 75 6-4PSPcomparisonforasinglesynapse. ....................... 76 6-5Histogramcomparisonforasinglesynapse. .................... 78 6-6PSPcomparisonforaneuronwithmultipleaerentspiketrains. ........ 79 6-7Histogramcomparisonforaneuronwithmultipleaerentspiketrains. ..... 81 6-8Histogramcomparisonforaneuronwithaerentandeerentspiketrains. ... 83 6-9PSPandAHPcomparisonforaneuronwithaerentandeerentspiketrains. 83 6-10StereotypicalPSPandscalingcomparisonproducedbytheconductancekernel. 84 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyAKERNELAPPROACHTOLEARNINGANEURONMODELFROMSPIKETRAINDATAByNicholasK.FisherAugust2010Chair:ArunavaBanerjeeMajor:ComputerEngineeringAspikingneuronisaprincipalcomponentofthebrainandthenervoussystem.Understandingthecharacteristicsofasingleneuronaswellastheinteractionsofapopulationofneuronsisessentialtoneuroscience.Atthemicro-level,neuronsareverycomplexdevicesthatcontroltheowofionsinandoutofthemembrane.Howeveranabstractedviewoftheneuronseesitasamechanismwhichreceiveselectricalsignalsfromotherneuronsasinputandproducesandtransmitselectricalsignalstootherneurons.Muchworkhasbeendonetomodeltheneuronatvaryinglevelsofcomplexityinordertoexplainitsdynamics.Competitionsexistwhichcomparetheaccuracyofsubmittedneuronsmodels.Hereweproposeamethodologywhichlearnsanequivalentmathematicalmappingfrominputspiketrainstotheoutputspiketrainbyonlyconsideringthetimingofallaerent(incoming)andeerent(outgoing)spikeswithinaboundednitepast.Thisisdonebyinstantiatingaclassicationproblemwhichuseskernelstodichotomizeinputspiketrainswhichcausetheneurontogenerateaspikefromthosethatdonotcausetheneurontogenerateaspike.Thekernelusedisonethathasbeengeneratedfromadictionaryoffunctionswhicharesimilartothoseusedinexistingneuronmodels.Byusinganintuitivedictionary,weproduceakernelwhichistailoredtotheproblemoflearningspikingneuronmodels. 9

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Bytherepresentertheoremweknowthatonlyanitenumberoftrainingdatapointswillbeneededtoproducetheclassicationsolution.Byconsideringthenumberofdatapointsusedtoproducethesolution,weareabletoassessthecomplexityofthemodeledneuron.Thoseneuronswhichneedmorespiketimeinputstoreproducetheirbehaviorcouldbeconsideredmorecomplexthanthosewhichneedfewerspiketimeinputsforthegivenkernel. 10

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CHAPTER1BACKGROUND 1.1IntroductionUnderstandingthestructureandthedynamicsofthebrainandtherestofthenervoussystemisaninherentaimofneuroscienceresearch.Aspikingneuronisthefundamentalcomponentofthenervoussystem.Itspurposeistorelayinformationthroughoutthenervoussystembytransmittingsmallelectricalpulses,knownasspikesoractionpotentials,throughaninterconnectednetworkofneurons.Inordertounderstandhowthebrainfunctions,wemustrstunderstandhowasingleneuronprocessesinformation.Afundamentalgoalofneuroscienceistoproducequantitativedescriptionsofasingleneuronaswellasdescribeitsinteractionswithotherneurons( Squireetal. 2003 ).Atthemicroscopiclevel,neuronsbehaveinaverycomplexmanner.Thecellmembranescontainsvoltagedependentandiondependentproteinswhichcontroltheowofvariousionsintoandoutofthecell,causingthemembranepotentialoftheneurontoincreaseordecrease.Howeveratahigherlevel,aneuroncanbeseenasamuchsimplerdevicethatreceivesincomingspikesasinput,andgeneratesaspikeasoutputdependentupontheinputspiketrains.Since Lapicque ( 1907 )rsttriedtomodelthedataobtainedfromstimulatingthenervesofafrogusingacapacitorcircuitand Hodgkin&Huxley ( 1952 )modeledthevoltagedependentchannelsintheaxonofasquid,manymodelsofatmultiplelevelsofintricacyhavebeenproposedtoattempttoreplicateneuronaldynamics( Gerstner&Kistler 2002 ; Hodgkin&Huxley 1952 ; Izhikevich 2003 ; Naudetal. 2008 ).Althoughthesemodelsvaryincomplexity,intheirtotalitytheyaremechanismswhichtransforminputspiketrainsintoanoutputspiketrain.ThisviewhasfoundexpressionintheQuantitativeSingle-NeuronModelingcompetition(QSNMC)wheresubmittedmodelscompeteonhowaccuratelytheycanpredictanoutputspiketrainofabiologicalneurongivenaninputcurrent( Gerstner&Naud 2009 ).Sincethevastmajorityofneuronsreceive 11

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inputfromchemicalsynapses( Gerstner&Kistler 2002 ),astricterstipulationwouldbetopredictoutputspikesbasedoninputspiketrainsatthevarioussynapsesoftheneuron.Thereareadvantagestothisvariationoftheproblem:complicatedsubthresholductuationsinthemembranepotentialneednotbemodeled,sincemodelsarenowjudgedstrictlyonthebasisoftheirperformanceatpredictingthetimingofoutputspikes.Thisyieldsmodelsthathavethelibertytofocusonthresholdcrossingsattheexpenseofbeinginaccurateinthesubthresholdregime.Suchmodelsbetterrepresentthefunctionalcomplexityoftheinput/outputtransformationofaneuron.Inaddition,comparisonstotherealneuroncanbeconductedinanon-invasivemanner.Inthisdissertation,weproposeaframeworkthatdescribesaspikingneuronanditscomplexityasaspikegeneratingmechanismbyproducingamathematicalmappingfromtheinputspiketrainstotheoutputspiketrain,whichisequivalenttothebiologicalmapping.Thismappingislearnedbyonlyconsideringthetimingofallaerent(incoming)andeerent(outgoing)spikesoftheneuronoveraboundedpast.Thisisinitiallyposedasaregressionproblem,howeverwequicklyndthattheregressionproblemisilldened.Instead,weformulatetheprobleminaclassicationbasedsupervisedlearningframeworkwherespiketraindataislabeledaccordingtowhethertheneuronisabouttospike,orhasjustspiked.Wedemonstratethatoptimizingthemodeltoproperlyclassifythislabeleddatanaturallyleadstoaquadraticprogrammingproblemwhencombinedwithanappropriaterepresentationofthemodelviaadictionaryoffunctions.Wethenderivetwokernels(asdenedintheeldofmachinelearning)onspiketrainstouseinaclassicationframeworkwhichistailoredtothisproblem.TherstkernelwederivedproducesanSRM0approximationoftheneuron.ItiscreatedfromadictionaryofPSP(post-synapticpotential)andAHP(after-hyperpolarizingpotential)likefunctions.ThesearefunctionsusedbyexistingSRM0models( MacGregor&Lewis 1977 )tomodeltheeectsofaerent(incoming)andeerent(outgoing)spikes.Byusingadictionaryoffunctions 12

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whichisusedinneuronmodeling,weproduceakernelwhichiswellsuitedfortheclassicationproblem.WethendiscussasecondkernelwhichisderivedfromasummationofGaussianfunctions.ThekernelproducedissimilarinnotiontotheGaussianradialbasisfunction(GRBF)kernel,however,thekernelresultsfromasummationoffunctionsratherthanaproductasisthecasefortheGRBF.ThepurposeoftheGaussiankernelistodemonstratetheadvantagesofusingakernelwhichisderivedwiththegivenprobleminmind,ratherthanusingamoregeneralkernel.AnSRM0functiondictionarywaschosenforseveralreasons.First,SRM0hasbeenshowntobefairlyversatileandaccurateatmodelingbiologicalneurons( Jolivetetal. 2004 ).Second,SRM0isarelativelysimpleneuronmodel,andthereforeislikelytodisplaybettergeneralizationonunseeninput.Finally,thedisparitybetweenthelearnedneuronmodelandtheactualneuroncouldshedlightonthevariousoperationalmodesofbiologicalneurons.ItmaybethecasethatthelearnedSRM0neuronaccuratelypredictsthebehavioroftherealneuronamajorityofthetime.However,therecouldbestates,burstingforexample,wherethepredictiondiverges.Insuchacase,theactualneuroncanbeseenasoperatingintwodierentmodes,oneSRM0like,andtheothernot.Multiplemodelscouldthenbelearnedtomodeltheneuroninitsvariousoperationalmodes.Oncetheclassierhasbeentrainedusingthesekernels,wearethenabletoproduceanapproximationtothemembranepotentialfunctionwhichislearnedbytheclassier.Thefunctionproducedisdenedoverspiketimesandproducesavalueindicativeofwhetherthespiketrainissubthresholdorsuprathreshold.Asdiscussedintherepresentertheorem( Kimeldorf&Wahba 1971 ),theclassierisabletondtheoptimalsolutioninthefeaturespace,usingonlyanitesubsetofdatapoints.Thenumberofspiketrainsneededforthisapproximationandtheaccuracyoftheapproximationcouldbeusedtoassessthecomplexityoftheneuron.Moredatapoints,wouldindicateamorecomplexmembranepotentialfunctionaroundthethreshold,andthereforeamorecomplicatedneuronwhenitisseenasaspikegeneratingdevice. 13

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Wedelveintothedetailsandtheaccuracyofthisframeworkandthediscussedkernelsinthefollowingchapters,howeverrstwegiveanoverviewofthebackgroundneededtoproducethismodel.Westartbydiscussingtheintricaciesofthebiologicalspikingneuronandthendiscussexistingtypesofneuronmodelswithspecicexamples.Finallywegiveageneraloverviewoftheclassicationformulationusedtodichotomizethespiketrains. 1.2TheNeuronAneuroncanbebrokendownintothreemainsections,thedendrites,thesomaandtheaxonasshowninFigure 1-1 .Thedendritesreceiveaerentspikesfromotherneuronsandtransferthesespikestothesoma.Theseincomingspikeseitherincreaseordecreasethemembranepotentialoftheneuron.Thesomageneratesanoutputspikewhenthemembranepotentialexceedstheneuron'sthreshold.Thisoutputspikeisthentransmittedtootherinterconnectedneuronsviatheaxon.Thesiteatwhichtheaxonofagivenneuronconnectstoadendrite(orsoma)ofanotherneuronisknownasthesynapse.Thisisthecommunicationlinkbetweenneurons.Theneuronwhichemitsthesignalisknownasthepresynapticneuron,whiletheneuronthatreceivesthesignalisknownasthepostsynapticneuron.Onetypeofsynapseisanelectricalsynapse,alsoknownasagapjunction( Hormuzdi,Filippov,Mitropoulou,Monyer,&Bruzzone 2004 ).Anelectricalsynapsecontainsproteinswhichmakeadirectelectricalconnectionbetweenthepresynapticandthepostsynapticneuron( Gerstner&Kistler 2002 ).Studiesshowthatelectricalsynapsesplayaroleinsynchronizingthesignalssentbyaneuron( Hormuzdietal. 2004 ).Themostcommontypeofsynapseinthehumanbrainisachemicalsynapse( Gerstner&Kistler 2002 ).Achemicalsynapseusesneurotransmitterstorelaytheeectsoftheactionpotentialfromthepresynapticneurontothepostsynapticneuron.Duetothevastabundanceofneuronswithchemicalsynapses,werestrictourmodeltothoseneuronswhichhaveachemicalsynapse. 14

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Neuronscontainaplethoraofvarioustypesofionsandmolecules.Manyofthesemoleculescarrysomecharge,themajorityofwhicharenegative.Thesenegativeionsrepeleachotherandgatheraroundthemembranewall.Thisbuildupofnegativechargealongthewalloftheneuronattractspositiveionstotheexteriorsideofthecellmembrane.Theionscannoteasilypenetratethecellmembranecausingthecellmembranetoactasacapacitor( Dayanetal. 2001 ).Thisvoltagedierenceacrossthecellmembraneisknownasthemembranepotential.Thecellmembranealsocontainsmanyionconductingchannelswhichallowtheionstoowinandoutoftheneuron.Certainchannelsonlyallowionsofacertaintypetopassthrough.Theabilityofachanneltoallowneuronstopasscanalsovarydependingonthemembranepotential,theinternalconcentrationofcertainions,orthepresenceofcertainneurotransmittersoutsidethecell.Inadditiontothesechannels,someofwhichalwaysremainopenandsomeofwhicharedependentupontheconditionofthecell,thecellmembranealsocontainsionpumpswhichattempttomaintaintheconcentrationofcertainionsinsideandoutsidethecell.Conventionallytheexteriorofthecellisdenedas0mV.Thisgivesanegativemembranepotential,usuallyaround-65mV,whentheneuronisatrestandinitsequilibriumstate.Thismembranepotentialcanbealteredbytheopeningandclosingofionchannelsduetoanelectricalstimulusorneurotransmittersreceivedfromanotherneuron.Whenapresynapticneuronemitsaspike,thepostsynapticneuronreceivesneurotransmitterswhichwillopencertainionchannelsaroundthesynapse.Thiswillcauseachangeinthemembranepotentialofthethepostsynapticneuron.Thischangeisknownasapostsynapticpotential(PSP).IfthePSPincreasesthemembranepotential,itissaidtobeexcitatory;ifitdecreasesthemembranepotential,itisinhibitory.AnexcitatoryPSPiscausedbyaninitialinuxofcations(usuallyNa+).Thischangeinthemembranepotentialandtheintracellularconcentrationcausesotherchannelstoopenwhichcausesaneuxofothercations(predominantlyK+).IftheeuxofK+ionsoverpowerstheinuxofNa+ions,thenaslightincreaseinthemembranepotential 15

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isseenbeforetheneuronreturnstoitsequilibriumstate.IfinsteadtheinuxofNa+ionsoverpowerstheeuxofK+,thenthemembranepotentialwillcontinuetoincreaseuntiltheneuronreachesitsthreshold.Atthispoint,theneighboringNa+channelsareopenedcausingaowofionsalongthecell.Thisisknownasanactionpotential,oraspike,andisthemeansofneuronalcommunication.Theactionpotentialwillreleaseneurotransmittersattheendoftheneuron'saxonwhichcausesaPSPinotherneurons.Aftertheactionpotentialoccurs,theNa+channelsareclosedowhichallowsaneuxofpotassiumionsfromthecellmembrane.Thiseuxofpotassiumions,knownastheafter-hyperpolarizingpotential(AHP),decreasesthemembranepotential,toapointwheretheneuronisunabletosendanotherspikeforacertainperiodoftime,knownastherefractoryperiod.Duringtherefractoryperiod,theionpumpsinthecellmembranebringtheneuronbacktoitsequilibriumstateandtherestingpotential. 1.3NeuronModelsAsmentionedpreviouslynumerousmodelshavebeenproposedinaneorttorepresentandexplainthespikingneuron( Gerstner&Kistler 2002 ; Hodgkin&Huxley 1952 ; Izhikevich 2003 ; Naudetal. 2008 ).Allofthesemodelsareimplementedatvaryinglevelsofcomplexity.Everyneuronmodelcanbecategorizedaseitheranon-phenomeno-logicalneuronmodeloraphenomenologicalneuronmodel.Non-phenomenologicalneuronmodelsareverydetailed,biophysicallyaccuratemodels.Theymodeltheinteractionsoftheneuronthatareseenatthemicroscopiclevel;theowofionsinandoutofthecellmembraneviathenumerousionchannels,someofwhicharevoltagedependent.Thesemodelsareusedonlyforasingleneuronduetotheircomplexity.Althoughtheycouldbeextendedtomodelinganetworkofneurons,themodel'scomplexityisprohibitiveofthisduetotheamountoftimeneededtomodelasingleneuron.Phenomenologicalmodelsconsideranabstractedviewoftheneuronandmodelthemembranepotentialasafunctionofsomeinput,beitspikesorastimulatingcurrent.TheymanuallycomparethemembranepotentialtothethresholdandthenproduceAHPeectsusing 16

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aresetvalueorafunctionifthemembranepotentialcrossesthethreshold.Althoughphenomenologicalmodelscouldbeusedforasingleneuron,theyaremoreoftenusedtomodelpopulationsofneurons.Duetotheseabstractions,phenomenologicalmodelsarecomputationallytractableforalargenumberofneurons.However,theyarelessaccuratethannon-phenomenologicalmodelswhicharenotcomputationallytractableforlargescalemodels,butbehavemorelikethebiologicalneuron.Herewediscusssomeoftheexistingmodelsineachcategoryaswellasageneralmodeloftheneuroninwhichthemembranepotentialfortheneuronisdependentupontheinputandoutputspikesoccurringwithinsomenitepast.Thisideaofonlyconsideringaboundedpastisthebasisforcertainphenomenologicalmodels,suchasSRM0. 1.3.1Non-PhenomenologicalModelsNon-phenomenologicalmodelsaremoredetailedmodelsofaneuronwhichmodelthedynamicsofneuronsusingtheowofionsthroughthechannelsofthecellmembraneasseenatthemicroscopiclevel.Thesearemorebiologicallycorrectthanphenomenologicalmodelshowevertheamountofdetailrequiredforsuchmodelsrestrictsthemtomodelingsingleneuronsduetocomputationalconstraints.Themostwellknownnon-phenomenologicalmodelisarguablytheHodgkin-Huxleyneuronmodel( Hodgkin&Huxley 1952 ).TheHodgkin-Huxleymodelisaconductancebasedmodelwhichisthesimplestkindofbiophysicallyrealisticmodels.Basedoasetofexperimentsperformedonthegiantaxonofasquid,theydeneavoltagedependentsodiumandpotassiumcurrent,representativeofsodiumandpotassiumchannelsrespectively.Inaddition,theydenealeakcurrentwhichaccountsfortheremainingionchannels.TheirmodelcanbesummarizedbytheelectricalcircuitshowninFigure 1-2 A.HereCisthemembranecapacitance,KandNaarerespectivelythetimevaryingpotassiumandsodiumconductance,representativeofvoltagedependentpotassiumandsodiumchannels.Ristheleakconductancerepresentativeoftheremainingionchannels 17

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andIistheinputstimuluscurrentwhichwillbedistributedamongthecapacitorandthe3resistorsdependingupontheamountofresistanceacrosseachchannel.UsingtheHodgkinHuxleymodel,wecanseehowthesomaofaneuronreactstoinputstimuli.InFigure 1-3 weseehowthemembranepotentialisaectedwhenstimulatingaHodgkinHuxleyneuronwithaninputcurrent.Figures 1-3 A, 1-3 B,and 1-3 C,showsubthresholdstimulation.Eachplotshowstheresultsofstimulatingtheneuronwitha10Acurrentfor0.5ms.Figure 1-3 Dshowstheresultofstimulatingthecellwitha12.92Acurrentfor0.5ms.InFigure 1-3 A,weseetheeectofasmallstimulusonthemembranepotentialofaneuron.Thisincreaseinthemembranepotentialiscausedbyaninuxofsodiumions.Afterthestimulustheneuronthenreenterstheequilibriumstatebyexpungingthesesodiumions.Thisplotalsoshowshowthemembranepotentialdropsbelowtherestingpotentialforaslightamountoftimebeforereturningtotherestingpotential.Notethatalthoughthestimuluscurrentwasappliedfor0.5ms,itseectsonthemembranepotentiallastedmuchlonger.Figure 1-3 Bshowssimilarbehaviorwithtwoseparatestimulispaced40msapart.Weseeeachstimulushasthesameindividualeectonthemembranepotential.InFigure 1-3 Cweseetheeectsoftwodierentstimuliwhichareplaced3msapart.Inthisgureyoucanseethatinsubthresholdcases,theeectsofindividualPSPsonthemembranepotentialappeartobeadditive.FinallyinFigure 1-3 D,weseewhatoccurswhentheneuronreachesthethreshold.Firstweseeanextremejumpinthemembranepotentialfollowedbyasharpdecrease,whichistheAHP.Themembranepotentialdropspasttherestingpotentialuntilthevoltagegatedchannelsreturntonormalcausingthemembranepotentialtoreturntotherestingpotentialwhentheneuronisinitsequilibriumstate.TheHodgkinHuxleymodelisatypeofsinglecompartmentmodel.Thesearemodelsthatdescribethemembranepotentialofaneuronusingasingleportionoftheneuron.Othersinglecompartmentnon-phenomenologicalmodelsexistaswell.TheConnor-Stevensmodel( Connor&Stevens 1971 )andtheMorris-Lecarmodel( Morris& 18

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Lecar 1981 )areotherconductance-basedneuronmodels.ThesemodelstendtosimulatespecicionchannelsintheneuroninthesamemannerastheHodgkin-Huxleymodel.Themaindierenceisinthespecicchannelsmodeled.Onethingthatsinglecompartmentmodelsdonotaccountforarespatialvariationsinthemembranepotential.Themembranepotentialcanvarythroughouttheneuron.Multi-compartmentmodelsaccountforthisbyhavingmultiplesinglecompartmentconductancebasedcircuits,liketheHodgkinHuxleycircuit,connectedinseries.Thissuccessionofcompartmentsisabletomodeltheowoftheactionpotentialthroughouttheaxon. 1.3.2PhenomenologicalModelsPhenomenologicalmodelstendtomodelasinglecompartmentastheHodgkinHuxleymodeldoes.However,ratherthanmodelingindividualvoltagegatedchannels,phenomenologicalmodelsgenerallysimulatethecumulativeeectofallchannelsonthemembranepotential.TheymodeltheeectsofinputcurrentsandinputspikesonthemembranepotentialmodelaswellastheeectsoftheAHPwhentheneuronspikes.Duetotheabstractionsinthemodeling,phenomenologicalmodelsaremuchsimplerandquicker.Inadditiontobeingcomputationallytractable,phenomenologicalmodelsarealsoeasiertoanalyze.Theyaremoreoftenthannotusedtoexplainthedynamicsofaninterconnectedpopulationofneuronsratherthanasingleneuron.Oneofthemorewidelyusedphenomenologicalmodelsistheleakyintegrateandre(LIF)modelorsomevariationofit.Itrepresentsallchannelsinthecellmembranebyusingasingleresistor.TheleakyintegrateandreneuronmodelisrepresentedusingacapacitorinparallelwitharesistorasshowninFigure 1-2 B.FromthiscircuitadierentialequationtocalculatethemembranepotentialcanbederivedasshowninEquation 1{1 .Inthisequationuisthemembranepotentialandmisthetimeconstantfortheneuronbeingmodeled.Thisequationwillnotcreateanactionpotential.Tosimulateanactionpotential,theLIFmodelmanuallycomparesthecurrent 19

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membranepotential,u(t),tothethreshold.Whenthemembranepotentialexceedsthethreshold,theLIFmodelsetsthemembranepotentialtosomepredenedconstantwhichislessthanthethresholdoftheneuron.AlthoughhedidnotformalizetheLIFmodel,Lapicquecouldbecreditedwithinspiringitduetohisattemptsintryingtouseasimplecapacitorcircuittomodelthedataseenbystimulatingthenervesofafrog( Lapicque 1907 ). mdu dt=)]TJ /F5 11.955 Tf 9.3 0 Td[(u(t)+RI(t)(1{1)Anotherphenomenologicalneuronmodelwasproducedby Izhikevich ( 2003 ).InthismodelIzhikevichusesbifurcationtechniquestoreducetheHodgkin-HuxleymodeltothesystemofordinarydierentialequationsshowninEquation 1{2 and 1{3 .Intheseequationsuandvaredimensionlessvariables;a,b,canddaredimensionlessparameters.Equation 1{4 showshowtheafterspikeresettingisimplemented,inamannersimilartoLIF. dvj dt=0:04v2j+5vj+140)]TJ /F5 11.955 Tf 11.95 0 Td[(uj+I(1{2) duj dt=a(bvj)]TJ /F5 11.955 Tf 11.96 0 Td[(uj)(1{3) ifvj30mV,then8>>>><>>>>:vj cuj uj+dvi vi+gij,8i(1{4)Thespikeresponsemodelzero(SRM0)isaphenomenologicalmodelrstformalizedbyGerstner( Gerstner&Kistler 2002 )andcalculatesthemembranepotentialusingasummationoffunctionswhicharedependentonthetimethathaselapsedsincetheinputandoutputspikeswerereceivedandsent.Forsimplicityinthisexplanation,weassumethereisnoinputcurrent.UnderthisassumptionthemembranepotentialofneuroniatagiventimetcanbecalculatedusingEquation 1{5 .Here,tfkisthetimeatwhichthefthspikewasgeneratedbyneuronk.representstheeectsofeerentspikeswhileijdemonstratestheeectofanaerentspikefromneuronj.ThesecondterminEquation 20

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1{5 showsthecumulativePSPeectfromallspikessentbyallpresynapticneurons.Thismodelassumesthatthemembranepotentialfunctionisadditivelyseparable( Stanisz 1969 )meaningthecontributionofeachspiketothetotalmembranepotentialisindependentoftheotherspikesinthespiketrain.InFigures 1-4 Aand 1-4 B,weshowtheformsofandrespectively.istheeectanincomingspikehasonthemembranepotentialandthereforemodelsthePSPcausedbythatspike.ontheotherhandmodelstheAHPwhichistheeectofaneerentspikeonthemembranepotential. ui(t)=Xf(t)]TJ /F5 11.955 Tf 11.95 0 Td[(tfi)+Xj3j6=iXfij(t)]TJ /F5 11.955 Tf 11.95 0 Td[(tfj)+urest:(1{5) 1.3.3GeneralModeloftheNeuronBanerjeehasshown,thatifoneassumesaneurontobeaniteprecisiondevicewithfadingmemoryandarefractoryperiod,thenthemembranepotentialoftheneuron,P,canbemodeledasafunctionofthetimingofthegivenneuron'saerentandeerentspikeswhichhaveoccurredwithinaboundedpast( Banerjee 2001 ).Spikesthathaveagedpastthisbound,denotedby,areconsideredtohaveanegligibleeectonthepresentvalueofP.ModelsoftheneuroninwhichPSPs/AHPsdecayexponentiallyfasttotherestinglevel,suchasthestandardintegrate-and-re( Abbott&vanVreeswijk 1993 ; Stein 1967 )andspike-responsemodels( Jolivetetal. 2000 ),belongtothiscategoryifonealsoassumesthatthethresholdisnoisy,irregardlessofhowsmallthatnoisemightbe.Wedenotethearrivaltimesofspikesatsynapsejusingthevectortj=htj1;tj2:::tjNji.t0representstheoutputspiketrainoftheneuronandvectorst1:::tmrepresentspiketrainsontheinputsynapses.tjirepresentsthetimethathaselapsedsincethatspikewasgeneratedorreceivedbytheneuron.Spikesareonlyconsiderediftheyoccurredwithintime.WecanthenformalizethemembranepotentialfunctionP:RN!R,whereN=Pmi=0Ni.P(t0;:::;tm)isdenedoverthespaceofallspiketrainsandproducesthemembranepotentialoftheneuron.TheneurongeneratesaspikewhenP(t0;:::;tm)=anddP dt0,whereisthethresholdoftheneuron.Fornotationalsimplicity,wedene 21

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thespikewindow,s2RN,whichrepresentsthetimingofallaerentandeerentspikeswithinthewindowoflength.sisthevectorofvectors,s=ht0;:::;tmi.TheneuronwillgenerateaspikewhenP(s)=anddP dt0. 1.3.4SummaryOneofthemaingoalsofneuroscienceistoproduceaccurate,quantitativedescrip-tionsofasingleneuron( Jolivetetal. 2008a ; Squireetal. 2003 ).Manymodelsexistforsimulatingthedynamicsofaspikingneuron.Withthechoiceofamodeltherewillbeatradeobetweenaccuracyandspeed.Non-phenomenologicalmodelsaredetailedandbiologicallyaccuratehowevertheytendtobecomputationallyandanalyticallyintractableduetotheircomplexity.Phenomenologicalmodelsontheotherhandarelesscomplexbutalsolessaccurate.Themodelproposedhereisonethatonlyconsidersthetimingofspikesthathavebeengeneratedorreceivedwithinanitepast.Thisenablesourmodeltoignorecomplicatedsubthresholductuationsinthemembranepotentialthatneednotbemodeled.Inaddition,comparisonstothemodeledneuroncanbeconductedinanon-invasivemanner.Themodelisbasedinasupervisedlearningclassicationframeworkandsowerstreviewthegeneralclassicationproblembeforegoingfurther. 1.4GeneralClassicationProblemInthegeneralclassicationproblem,wearegivenasetofdatainsomehighdimensionalspace(usuallyRN).Eachdatapointiscategorizedasbeingamemberofoneoftwoclasses.Thegoalistondsomefunction,inthiscaseahyperplane,denedoverRNwhichcanbeusedtoseparatethedatapointswhicharemembersoftherstclassfromthosedatapointsthataremembersofthesecondclass.ThedatasetisdenedasinEquation 1{6 ,wherexiisanNdimensionalvectorwitharespectiveclassicationofciwhichiseither-1or1.Theclassierndsthehyperplane,wwhichseparatesthepointswithaclassicationof-1fromthosepointswithaclassicationof1. D=(xi;ci)jxi2RN;ci2f)]TJ /F1 11.955 Tf 26.56 0 Td[(1;1gi=1:::l(1{6) 22

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GivenalinearlyseparablesetofdataasshowninFigure 1-5 A,multiplehyperplanescouldbeusedtotrytodichotomizethedataset.HyperplaneH3doesnotgiveasuccessfulclassicationastherearepointsfromeachclassononesideofthedata.HyperplanesH1andH2bothsuccessfullyseparatethegivendataset.HoweverusingH1orH2couldleadtoover-ttingthetrainingdata.InordertogeneralizeourmodelandreducetheriskboundasdiscussedbyVapnik( Vapnik&Kotz 2006 ),wewishtondthehyperplanewhichnotonlysuccessfullyseparatesthedatabutalsomaximizesthemargin(shownbythedashedlines),asisdonebyhyperplaneH0.Figure 1-5 Bshowsanexampleoflinearlyseparabledatawhichisseparatedbythehyperplane,wx+b=0,whichmaximizesthemarginandwillbereferredtoasH.OneachsideofH,wealsoseethemarginhyperplanes,H+andH)]TJ /F1 11.955 Tf 7.08 -4.33 Td[(,representedbywx)]TJ /F5 11.955 Tf 10.99 0 Td[(b=1andwx)]TJ /F5 11.955 Tf 10.08 0 Td[(b=)]TJ /F1 11.955 Tf 9.3 0 Td[(1,respectively.H+liesalongthetrainingpointsfromthepositiveclass(ci=1)whichareclosesttoH.H)]TJ /F1 11.955 Tf 10.99 -4.34 Td[(liesalongthetrainingpointsfromthenegativeclass(ci=)]TJ /F1 11.955 Tf 9.3 0 Td[(1)whichareclosesttoH.TheoptimalseparatinghyperplanewillbethehyperplanethathasthegreatestdistancefrombothH+andH)]TJ /F1 11.955 Tf 7.09 -4.33 Td[(.Inaddition,itmustalsobeenforcedthattherearenotrainingpointsbetweenH+andH)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(.Thisleadstoaconvexoptimizationproblem.GivenascaledversionofthedatashowninEquation 1{6 ,theclassierwouldliketondthehyperplaneHwhichmaximizesthemargin,orthedistancebetweenHandtheclosestdatapointsfromeachclass.FromFigure 1-5 B,wecanseethatafterscalingthedatasothattheclosestpointshaveaunitdistancefromH,thatthedistancefromHtoH+andfromHtoH)]TJ /F1 11.955 Tf 10.98 -4.34 Td[(areboth1 jjwjj.Therefore,thesizeofthemarginis2 jjwjj.Inordertomaximizethesizeofthemargin,wemustminimizejjwjj.However,weadditionallymustadheretotheconstraintthattherearenopointswithinthemarginitself.Therefore,wemustenforcetheconstraintswxi)]TJ /F5 11.955 Tf 12.1 0 Td[(b)]TJ /F1 11.955 Tf 22.27 0 Td[(1foreveryxiinthenegativeclass(ci=)]TJ /F1 11.955 Tf 9.3 0 Td[(1)andwxi)]TJ /F5 11.955 Tf 12.4 0 Td[(b1foreveryxiinthepositiveclass(ci=1).ThisleadsustotheoptimizationproblemshowninEquation 1{7 23

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Minimize1 2jjwjj2withrespecttow;bsubjectto:ci(wxi)]TJ /F5 11.955 Tf 11.96 0 Td[(b)1(1{7)Inthisproblemwecombinetheconstraintsforeachclassintoasingleconstraint,andwechoosetominimize1 2jjwjj2ratherthanjjwjjforsimplicity,butbothwouldyieldthesameanswer.Thisyieldsaquadraticprogrammingoptimizationproblemwhichcanbeeasilysolvedusingquadraticprogrammingalgorithms( VandePanne 1975 ). 1.5KernelAnalysisofaSpikingNeuronHeretoforewehavediscussedtheneedformathematicaldescriptionsofaspikingneuronaswellasmodelsthatarecurrentlyusedforsuchdescriptions.Wehavealsodiscussedthebackgroundneededforourclassicationframeworks.Inthefollowingchapters,wediscussthedetailsofourframeworkwhichproducesanSRM0approximationofaneuronbyonlyconsideringthetimingofallaerentandeerentspikesoftheneuronoveraboundedpast.InChapter 2 wediscussandderivetheclassicationproblemusedforourmodelaftershowingwhyaregressionapproachwouldnotwork.InChapter 3 wediscussandshowthederivationforakernelwhichwastailoredtotheproblemoflearningthemembranepotentialfunctionfromspiketimes.InChapter 4 wediscussandderiveakernelwhichisproducedfromadictionaryofGaussianfunctions.ThiskernelisusedtoshowtheadvantagesofusingakernelwhichisproducedwithaspecicprobleminmindasopposedtousingageneralkernelsuchastheGaussiankernel.InChapter 6 wecompareandcontrastthetwokernelsforneuronmodelsofvaryingcomplexity. 24

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Figure1-1. Ahighleveldiagramofaneuron.Aneuronismadeupofdendrites,whichreceivetheinput,thesomawhichprocessestheinputandproducestheoutputandtheaxonwhichtransmitstheoutput. Figure1-2. Hodgkin-Huxleyandleakyintegrateandrecircuitdiagrams.A)showstheHodgkinHuxleycircuit.Cisthemembranecapacitance,KandNaarerespectivelythetimevaryingpotassiumandsodiumconductance,representativeofvoltagedependentpotassiumandsodiumchannels.RistheleakconductancerepresentativeoftheremainingionchannelsandIistheinputstimuluscurrentwhichwillbedistributedamongthecapacitorandthe3resistorsdependingupontheamountofresistanceacrosseachchannel.B)showstheleakyintegrateandrecircuit.Theleakyintegrateandremodelisaphenomenologicalmodelthatrepresentstheconductancefromallchannelsinthecellmembranewithasingleresistor.ThatcombinedwiththecapacitanceofthecellmembraneyieldsaparallelR-Ccircuit. 25

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Figure1-3. Hodgkin-Huxleymembranepotentialdynamics.EectsonthemembranepotentialwheninjectingaHodgkinHuxleyneuronwithavaryinginputcurrents.A)showstheresultfrominjectingtheneuronwitha10Acurrentfor0.5ms.Weseeasmallincreaseinthemembranepotential,untiltheeectsofthestimuluswearsoandthemembranepotentialreturnstotherestingpotential.B)showstheeectsofinjectingtheneuronwithtwoidenticalstimulispaced40msapart.Weseeeachstimulushasthesameeectonthemembranepotentialatthetimeseachstimuluswasapplied.C)showstheadditiveeectofthePSPwhentheinjectedcurrentsareplacedonly3msapart.FinallyD)showstheeectofinjectingtheneuronwithacurrentwhichisstrongenoughtogenerateanactionpotential.WeseealargeincreaseinthemembranepotentialfollowedbyasharpdecreasewhencomparedtothesubthresholdplotsshowninA),B),andC). Figure1-4. ModelingfunctionsfortheSRM0.ExamplesofwhatandmightlooklikeforSRM0whenmodelingtheeectsofaerentandeerentspikesrespectively.A)showstheformof,whichrepresentstheindividualeectofanincomingspikeandmodelstheformofaPSP.B)shows,whichrepresentstheeectsofanAHPonthemembranepotentialcausedbyaneerentspike. 26

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Figure1-5. Classifyinglinearlyseparabledatawiththeoptimalhyperplane.A)showsmultiplehyperplaneswhichcouldattempttoclassifythegivendataset.HyperplaneH3doesnotgiveasuccessfulclassication.HyperplanesH1andH2bothsuccessfullyseparatethegivendataset,butarenotoptimal.HyperplaneH0notonlysuccessfullyseparatesthedatabutalsomaximizesthemargin(shownbythedashedlines).B)showstheoptimalseparatinghyperplane,H,whichmaximizesmargin.ThemarginisthedistancebetweenHandtheplanecreatedbytheclosestdatapointsfromeachclass,representedbyH+andH)]TJ /F1 11.955 Tf 10.98 -4.34 Td[(. 27

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CHAPTER2AKERNELBASEDNEURONMODEL 2.1IntroductionSingleneuronsarethepredominantcomponentofthenervoussystem.Thecharacteristicsandinteractionsofneuronscomprisealargeareaofneuroscienceresearch.Tofurtherunderstandthedynamicsofanetworkofneurons,manyspikingneuronmodelshavebeenproposed( Gerstner&Kistler 2002 ; Izhikevich 2003 ; Markram 2006 ; Naudetal. 2008 ).Thesespikingneuronmodelscanbeclassiedaseitheradetailedbiophysicalmodelorasimplerphenomenologicalmodel( Jolivetetal. 2008b ).Biophysicallyrealisticmodelssimilarto( Hodgkin&Huxley 1990 )attempttoreplicatethemostdetailedcharacteristicsofaneuron.Howeverthesemodelsarecomputationallyexpensiveanddiculttoanalyze,especiallywhenappliedtolargescalenetworksofneurons.Thephenomenologicalneuronmodelsarelesscomputationallyextensiveandanalyticallytractable,buttheirisdoubtastohowrealisticthemodeliswhencomparedtoanactualneuron( Jolivetetal. 2008a ).Inthisdissertation,weproposeaframeworkthatdescribesaspikingneuronanditscomplexityasaspikegeneratingmechanismbyproducingamathematicalmappingfromtheinputspiketrainstotheoutputspiketrain,whichisequivalenttothebiologicalmapping.Thismappingislearnedbyonlyconsideringthetimingofallaerent(incoming)andeerent(outgoing)spikesoftheneuronoveraboundedpast.Weformulatetheprobleminaclassicationbasedsupervisedlearningframeworkwherespiketraindataislabeledaccordingtowhethertheneuronisabouttospike,orhasjustspiked.Wederivevariouskernelsonspiketrainstouseintheclassicationframework.TherstkernelwederivedproducesanSRM0approximationoftheneuron.ItiscreatedfromadictionaryofPSP(post-synapticpotential)andAHP(after-hyperpolarizingpotential)likefunctions.Wediscussasecondkernel,derivedfromasummationofGaussianfunctions,whosepurposeistodemonstratetheadvantagesofusingakernel 28

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whichisderivedwiththegivenprobleminmind,ratherthanusingamoregeneralkernel.Thelastkerneldiscussedisonewhichtakesscalingintoaccount.Thersttwokernelsmaketheassumptionthatallspikestravelingacrossthesamesynapsewillhavestereotypicaleectsonthemembranepotential.Aswillbediscussed,thisisnotalwaysthecase.Thereforewecreateanextensionoftheclassicationframeworkwhichtakesscalingeectsintoaccount.AnSRM0functiondictionarywaschosenforseveralreasons.First,SRM0hasbeenshowntobefairlyversatileandaccurateatmodelingbiologicalneurons( Jolivetetal. 2004 ).Second,SRM0isarelativelysimpleneuronmodel,andthereforeislikelytodisplaybettergeneralizationonunseeninput.Finally,thedisparitybetweenthelearnedneuronmodelandtheactualneuroncouldshedlightonthevariousoperationalmodesofbiologicalneurons.ItmaybethecasethatthelearnedSRM0neuronaccuratelypredictsthebehavioroftherealneuronamajorityofthetime.However,therecouldbestates,burstingforexample,wherethepredictiondiverges.Insuchacase,theactualneuroncanbeseenasoperatingintwodierentmodes,oneSRM0like,andtheothernot.Multiplemodelscouldthenbelearnedtomodeltheneuroninitsvariousoperationalmodes. 2.2ModelAsdiscussedpreviously, Banerjee ( 2001 )hasshown,thatifaneuronhasanoisythresholdthenthemembranepotentialoftheneuron,P,canbederivedfromabounded-timepasthistoryofallaerentandeerentspikesofthatneuron.Spikesthathaveagedpastthisbound,denotedby,areconsideredtohavenegligibleeectonthepresentvalueofP.Werepresentthetimingofspikeswithinthisboundusingaseriesofvectors,t1;t2;:::;tm.Hereti=)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(ti1;:::;tiNiisavectoroflengthNiwhereeachelement,tijisthetimethathaselapsedsincethejthspikewasgenerated(eerentspikes)orreceived(aerentspikes)bytheneuronbeingmodeled.t0representstheeerentspikeswhiletii6=0representstheaerentspikesfromneuroni.ThisallowsustodeneamembranepotentialfunctionP:RN!R,whereN=Pmi=0Ni.P(t0;t1;:::;tm) 29

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takesaseriesofspiketrainsmadeupofanitenumberofspiketimesasinputandgivesthemembranepotentialfortheneuronatthatpointintime.Nisthetotalnumberofpossiblespikesinthespikewindowboundedby.TheneuronwillthenproduceaspikewhenP(t0;t1;:::;tm)>anddP dt0whereisthethresholdoftheneuron.Wenotethatthesetofvectorsht0;t1;:::;tmirepresentstheentirespiketrainseenbytheneuronanddetermineswhetherornottheneuronwillspike.Therefore,fornotationalsimplicity,wedeneanewvectors2RN,suchthatsisavectorofvectorsands=ht0;t1;:::;tmi. min^P(s)ZRNh^P(s))]TJ /F5 11.955 Tf 11.95 0 Td[(P(s)i2ds(2{1)Theidealmodelofsuchaneuronwouldbetoproduceafunction^P:RN!RoverthesamespacewhichapproximatesthetruemembranepotentialfunctionP.ThisleadstotheregressionproblemshowninEquation 2{1 .Asolutiontothisproblemwouldapproximatethetruemembranepotentialfunctionforthegivenneuronatallpointsintime.Thisincludestimeswhentheneurondoesnotspike.Therearetwoissueswiththissolution.First,themembranepotentialisa\private"quantitywhichwecannotaccess.Themodeledneuronneverconveysitsmembranepotentialtosurroundingneurons,itonlyrevealswhenthemembranepotentialexceedsthethresholdbygeneratingaspike.Weareonlyconcernedwiththeneuronasaspikegeneratingmechanism,sincethatishowitconveysinformation.Forthisreason,weonlyneedtoconsiderspiketrainsthatcausethemodeledneurontospike.Inaddition,asolutiontoEquation 2{1 wouldperturbtheapproximatingsurface,^P,totthemembranepotentialsurfaceatallpoints,includingthosethatdon'tcauseittospike.Thiswouldmostlikelycausetheapproximatingsurfacetoloseaccuracyatthepointsthatmattermost:whenthetruemembranepotentialcrossesthethreshold().Amoreapplicableapproachwouldbetoonlyconsiderthespiketimeswheretheneuronspikes(P(s)=)asshowninEquation 2{2 min^P(s)ZZZP(s)=h^P(s))]TJ /F5 11.955 Tf 11.95 0 Td[(P(s)i2ds(2{2) 30

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min^P(s)ZZZP(s)=h^P(s))]TJ /F1 11.955 Tf 11.95 0 Td[(i2ds(2{3)Atthethreshold,themembranepotentialisalways.Therefore,theregressionprobleminEquation 2{2 wouldleadtotheregressionprobleminEquation 2{3 ,whichisilldened.Theoptimalsolutionwouldbetoset^P(s)=8s.ThereforewemustuseanothertechniquetolearnthehypersurfacecreatedbythepointswhereP(s)=.RatherthanconsideringthespiketrainswhichareonthehypersurfacecreatedbyP,wecreatetwoclassesofspiketrains.Wecreateoneclassofspiketrainswhichyieldmembranepotentialsimmediatelybelowthethreshold,andanotherclassofspiketrainswhichyieldmembranepotentialsimmediatelyabovethethreshold.ByTaylor'stheorem( Rudin 1964 ),weknowthatifthetruemembranepotentialfunction,Piscontinuousandithasaderivativethenthemembranepotentialatagivenspiketrainscanbecalculatedusingarstorderapproximation.Thismeansthatforsomespiketrainssands0,giventhatsands0aresucientlyclose,wecancalculateP(s)usingalinearapproximationfroms0asshowninEquation 2{4 .Thisimpliesthatiftwospiketrainsfromtheaforementionedclassesaresucientlyclosetoeachother,thenthemembranepotentialsproducedbythesespiketrainsarelinearlyseparable. P(s)=P(s0)+(s)]TJ /F4 11.955 Tf 11.95 0 Td[(s0)P0(s0)(2{4) 2.3ClassicationFrameworkFromthisformulation,wecanderiveourclassicationproblem.Inordertolearnanapproximationofaneuroninanon-invasivemanner,wehaveposedasupervisedlearningclassicationproblemwhichlabelsthegivenspiketraindataaccordingtowhethertheneuronisabouttospikeorhasrecentlyspiked.WedenotetheformerS)]TJ /F1 11.955 Tf 10.98 -4.34 Td[(andthelatterS+.Thisproblemisequivalenttoclassifyingsubthresholdspiketrains(^P(s)<)fromsuprathresholdspiketrains(^P(s)),whichleadstotheclassicationproblemshowninEquation 2{5 .Itshouldbenotedthatthemembranepotentialfunction,P,wouldbethe 31

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optimalsolutiontothisproblemsinceP(s)<8s2S)]TJ /F1 11.955 Tf 10.98 -4.34 Td[(andP(s)8s2S+.Minimize^P(s)2 (2{5)s.t.^P(s))]TJ /F1 11.955 Tf 11.96 0 Td[(1s2S+^P(s))]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F1 11.955 Tf 21.92 0 Td[(1s2S)]TJ /F1 11.955 Tf -361.99 -40.8 Td[(TogeneratespiketrainswhichbelongtoS+andS)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,weprovidethespiketrainswhichoccurataxedinnitesimaltimedierentialbeforeandaftertheneurongeneratesaspike,asillustratedinFigure 2-1 .Thespiketrainthatoccurredwhentheneurongeneratedaspikeisshownbythesolidlines.WethenmovethespikewindowintothepasttoproducespiketrainsthatbelongtoS)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(,shownbythecirculararrows.WealsomovethespikewindowintothefuturetoproducespiketrainsthatbelongtoS+,denotedbythetriangulararrows.Noticethatintheoutputspiketrain,t0,thespikewhichiscurrentlygeneratedisnotconsideredinthespiketrainwhichbelongstoS+orS)]TJ /F1 11.955 Tf 7.08 -4.33 Td[(.ThereasonitisnotusedintheS)]TJ /F1 11.955 Tf 10.99 -4.34 Td[(spiketrainisbecauseitsimplyhasnotbeengeneratedatthatpointintime.However,therearetworeasonswhyitisnotconsideredintheS+spiketrain.First,itwouldinduceanAHPeectwhichwouldcausethemembranepotentialtofallbelowthethreshold.Inaddition,ifitwereincluded,thiswouldcausetheclassiertoonlyconsiderwhetherornotthatparticularspikeexistedwhenclassifyingagivenspiketrainasamemberofS+orS)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(.Ifitdidexist,itwouldbelongtoS+,andifitdidnotexistitwouldbelongtoS)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(.Althoughthismethodwouldworkwellforthetrainingdata,itwouldnotgeneralizewellwhenconsideringlivespiketraindata.Producingahypersurfacewhichcandividethesupra-thresholdspiketrainsfromthesub-thresholdspiketrainswithinthespiketimefeaturespace,wouldbeextremelydicult.Asdiscussedabove,ifwecouldmapagivenspiketrainstoitscorrespondingmembranepotentialP(s),thentheclassicationproblemistrivial.Althoughwedonothaveaccesstothemembranepotentialfunction,wecanusemembersfromafunctiondictionarytotryandreproduceanapproximationtothemembranepotentialfunctionP. 32

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Thechoiceofthedictionaryiscrucial.Bychoosingadictionarywhichistailoredtothemembranepotentialfunction,weincreasethechanceofsuccessfullymodelingthegivenneuron. 2.4SummaryTheframeworkwehavediscussedtriestolearnthetruemembranepotentialfunction,Pforagivenneuron.Sincewearemodelingneuronswhichareconsideredtobeaniteprecisiondevice,theinputtothemembranepotentialfunctionisaspiketrainoversomenitepast.OurmodelismotivatedbyusingaregressionproblemtondthebestapproximationofthetruemembranepotentialfunctionP.Howeverwearenotconcernedwiththeinternaldynamicsoftheneuron.Weonlywanttomodeltheneuronwhenitconveysinformation,i.e.whenitgeneratesaspike.Thereforewerestrictthespiketrainspacetothosespiketrainswhichcausetheneurontoproduceaspike(P(s)=).Wendthattheregressionprobleminthisrestrictedspaceisill-dened.Forthisreason,weconverttheregressionproblemintoaclassicationproblem.Withintheclassicationproblem,wetrytoproducethemembranepotentialapproximationusingadictionaryoffunctionswhicharetailoredtotheproblemofneuronmodeling.TheSRM0modelisanadditivelyseparablemodel( Stanisz 1969 ).Thismeansthemembranepotentialisasumoffunctionsoftheindividualspikesofthespiketrain(P(s)=Pmk=0PNki=1Pik(tki)).Forthisreason,weformulateourprobleminamannerwheretheeectsofsinglespikeswithinthespiketrainareconsideredindividually.Tobegin,wemakeafurtherassumptionthatspikestravelingalongthesamesynapsewillhaveidenticaleectsonthemembranepotential.TheseideasofadditiveseparabilityandstereotypedeectsaretakenintoconsiderationwhenformulatingourclassicationframeworkinChapter 3 andChapter 4 .Therstdictionary,discussedinChapter 3 ,isafamilyofPSPandAHPlikefunctions,similartothoseusedby MacGregor&Lewis ( 1977 )intheirSRMmodel.InChapter 4 ,wediscusstheframeworkwhenusingafamilyofGaussianfunctionswithxedstandarddeviationandvariablemean.Thisisamoregeneraldictionarywhich 33

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doesnotworkaswellasthedictionaryofPSPandAHPlikefunctions.Ourlastkerneladdressestheissueofstereotypedeects.Spikeswhichtravelacrossthesamesynapsedonotnecessarilyhavethesameeectonthemembranepotential.Themagnitudeoftheeectistimingdependent.Toaddressthisissue,weprovideathirdkernelwhichaccountsforPSPsofvaryingmagnitudeforspikestravelingacrossagivensynapse.ThiskernelwillbediscussedinChapter 5 .FinallyacomparisonofthekernelsaswellasresultswillbegiveninChapter 6 Figure2-1. Diagramofspiketrainsusedtotrainourclassier.Thespiketrainthatoccurredwhentheneurongeneratedaspikeisshownbythesolidlines.ThespikewindowismovedintothepasttoproducespiketrainsthatbelongtoS)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,shownbythecirculararrows.ItisthenmovedintothefuturetoproducespiketrainsthatbelongtoS+,denotedbythetriangulararrows. 34

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CHAPTER3RECIPROCALEXPONENTIAL-EXPONENTIALFUNCTIONDICTIONARYTheformulationaswellasthechoiceofthefunctiondictionaryareessentialtothesuccessofourmodel.Ifweweretousealinearclassier,thanourmodelwouldhaveaverysmallchanceofsuccessduetothenonlinearnatureofthemembranepotentialfunction.Thedictionaryweusedhasbeenchosenduetoitsinherentutilityinneuronmodeling,particularlyinmodelingthecontributionsofaerentandeerentspikestothecumulativemembranepotential.WehavealsoformulatedtheclassicationproblemtotakeintoaccounttheadditiveseparabilityoftheneuronaswellasthestereotypedeectsofPSPsandAHPs.Thedictionarychosenisthefamilyofreciprocalexponential-exponentialfunctions(REEF).Thesearefunctionsoftheformf(t)=1 exp()]TJ /F5 11.955 Tf 9.3 0 Td[(=t)exp()]TJ /F5 11.955 Tf 9.3 0 Td[(t=)forparametersand.Thisfunctionfort=1scanbeseeninFigure 3-1 Aasafunctionofand.InFigure 3-1 B,weshowtheresultingREEFsasafunctionoftimewhenholdingconstantandvarying,aswellasthevariousREEFsthatoccurwhenholdingconstantandvarying.Itshouldbepointedoutthatwhen=0thattheresultingfunctionissimilartothatofanAHPeect;when6=0,theresultingfunctionissimilartothatofaPSPeect.Intheadditivelyseparablecaseweattempttoapproximatetheindividualeectofeachspikeonthemembranepotential.Weassumethatthetotalmembranepotentialcanbecalculatedbyasumofindividualeectsofeachspike(P(s)=Pmk=0PNki=1Pik(tki)).Wethentrytolearntheindividualeects,Pik(tki).Ifk=0,thenwearelookingforthecontributionoftheitheerentspiketothemembranepotential.Putanotherway,wearetryingtomodeltheAHPacrossthemodeledneuronsaxon.Ifk6=0,thenwearemodelingthecontributiontothemembranepotentialoftheithaerentspiketravelingacrosssynapsek.ThiswouldbethePSPcausedbyaspiketravelingacrossthekthsynapse.Wemakethefurtherassumptionofstereotypedeectsforspikestraveling 35

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acrossthesamesynapse.Thismeansthatallspikestravelingacrossaparticularsynapsewillhaveidenticaleectsonthemembranepotential.ThisimpliesthatratherthanlearningfunctionsPikforeveryspiketravelingacrosseverysynapse,weonlyneedtolearnfunctionsPkwhichrepresentstheeectofaspiketravelingacrossthekthsynapse. 3.1ApproximationofthemembranepotentialfunctionWewouldliketocombinemembersofthechosendictionarytocreateanapproximationofthemembranepotentialfunction,P,whichwillyieldasolutiontotheclassicationproblemposedinEquation 2{5 .Wewillrstdiscusshowthiscanbeachievedinadiscreteframework,wherewecombineanitenumberofparameterizedfunctionsfromthedictionarytomodelthemembranepotentialfunction.Followingthediscreteformulation,wewilldiscussacontinuousformulation,inwhichwecombineaninniterangeofparameterizedfunctionsinordertoproduceamembranepotentialapproximation.Inthecontextofthecontinuousformulation,wewillshowaspecicinstanceoftherepresentertheorem,whichwasrstshownby Kimeldorf&Wahba ( 1971 ).Therepresentertheoremshowsthattheoptimalsolutiontotheposedclassicationproblemmustlieinthespanofthedatapointswhichwereusedtotraintheclassier.Inthediscreteandcontinuousformulation,wewillassumeasinglespikeforsimplicity.Wewillconcludebyextendingthecontinuousformulationtothecaseofmultiplespikesonasinglesynapse,andthecaseofmultiplespikesonmultiplesynapses. 3.1.1DiscreteFormulationInthediscreteformulation,wewouldliketoapproximatethemembranepotentialfunctionusingalinearcombinationofapredenedsetoffunctionsfromtheREEFdictionary.Inthesinglespikecase,weonlyneedtomodeltheeectofasinglespikeonthemembranepotential.Wedenotethiseectonthemembranepotentialby^PanditisdenedasalinearcombinationofparameterizedREEFsasshowninEquation 3{1 .ft(;)=1 exp()]TJ /F5 11.955 Tf 9.3 0 Td[(=t)exp()]TJ /F5 11.955 Tf 9.3 0 Td[(t=)arethefunctionsdenedbyand.Specicparametersf(1;1);:::;(M;1);(1;2);:::;(M;N)gareusedtocreateafunction^P 36

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thatcanbestreproducetheeectofthespikeonthemembranepotential.Finitelymanyfunctions,ft(i;j),willbescaledappropriatelyinordertoapproximateP. ^P(t)=MXi=1NXj=1i;jft(i;j)(3{1)Theapproximation^Pcanbeimprovedbyincreasingthenumberoffunctionsused.However,asthenumberoffunctionsincreases,sodoesthedimensionalityofourfeaturespace.Increasingthedimensionalityofthefeaturespacecanleadtoover-ttingofthetrainingdata.Toresolvethisproblem,wedeneakernelwhichcanbeusedinacontinuousformulationofthisproblem.Thisallowsallpossiblefunctionstobeconsideredwhenproducingthe^Papproximationwhilekeepingthedimensionalityofthefeaturespaceconstant. 3.1.2ContinuousformulationInthecontinuousformulation,weconsiderL2,theHilbertspaceofsquareLebesgueintegrablefunctionsonthedomainf;g2[0;1)2.Weareconcernedwithndingsomethresholddependentclassicationfunction^P,suchthat^P(t)+1whenthespiket2S+and^P(t))]TJ /F1 11.955 Tf 12.87 0 Td[(1whent2S)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(.ThisfunctionisdenedinEquation 3{2 .Inthisformulation,(;)isdenedtobeamemberofL2.Therefore,ifft(;)2L2,then^P(t)2L2bytheCauchy-Schwartzinequalitysinceh(;);ft(;)ik(;)kkft(;)k<1ifbothk(;)k<1andkft(;)k<18t.Toshowthatft(;)2L2wemustshowhft(;);ft(;)i<1. ^P(t)=h(;);ft(;)i=Z10Z10(;)ft(;)dd(3{2) 37

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Proof:hft(;);ft(;)i<1. hfx(;);fy(;)i=Z10Z101 exp)]TJ /F5 11.955 Tf 10.49 8.08 Td[( xexp)]TJ /F5 11.955 Tf 10.49 8.08 Td[(x 1 exp)]TJ /F5 11.955 Tf 10.5 8.08 Td[( yexp)]TJ /F5 11.955 Tf 10.64 8.08 Td[(y dd (3{3)=Z10"1 2exp)]TJ /F5 11.955 Tf 10.49 8.09 Td[(x )]TJ /F5 11.955 Tf 13.29 8.09 Td[(y )]TJ /F5 11.955 Tf 17.7 8.09 Td[(xy x+yexp)]TJ /F5 11.955 Tf 10.49 8.09 Td[( x)]TJ /F5 11.955 Tf 13.15 8.09 Td[( y1=0!#d (3{4)=Z101 2exp)]TJ /F5 11.955 Tf 10.49 8.09 Td[(x )]TJ /F5 11.955 Tf 13.3 8.09 Td[(y xy x+yd (3{5)=xy x+y1 x+yexp)]TJ /F5 11.955 Tf 10.49 8.09 Td[(x+y 1=0 (3{6)=xy (x+y)2 (3{7)Thereforeweseethathft(;);ft(;)i=1 4<18t2[;]forsome>0whereisthelengthofthenitepastforwhichweconsiderspikes.Bydeningthemembranepotentialfunctioninthismanner,wehaveproducedaproblemwhichyieldsasolutionwhichisdierentthanthesolutiontothediscreteproblem.Thedeltafunctionwhichiscenteredatsomepoint(;)doesnotbelongtoL2.Thisimpliesthattheapproximatingmixingfunction(;)cannotbemadeupofalinearcombinationofthesedeltafunctions,asisthecaseinthediscreteformulation.Inaddition,wearenotworkingwithareproducingkernelHilbertspacesinceweareconsideringL2.However,ourdenitioninEquation 3{2 denesthe\pointevaluation"ofourmembranepotentialfunction.Since^P(t)isdenedusingthestandardinnerproductinL2withrespecttoparticularmembersofL2,wecanreformulatetheclassicationprobleminEquation 2{5 asshowninEquation 3{8 ,whereMisthenumberofdatapoints,andymisthecorrespondingclassicationforspiketimetm(ym=+1iftm2S+andym=)]TJ /F1 11.955 Tf 9.3 0 Td[(1iftm2S)]TJ /F1 11.955 Tf 7.08 -4.33 Td[().Minimizek(;)k2 (3{8)s.t.ym(h(;);ftm(;)i)]TJ /F1 11.955 Tf 19.26 0 Td[()1m=f1:::Mg 38

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WenowcanuseaspecicinstanceoftheRepresentertheoremwhichwasrstprovenby Kimeldorf&Wahba ( 1971 )toshowthattheoptimalsolutiontotheoptimizationprobleminEquation 3{8 canbeexpressedintermsofaweightedsumofkernelevaluations,whenthekernelisevaluatedatanitenumberofdatapointsfromourtrainingset((;)=PMk=1kftk(;)).WecanthensubstitutethisequalitybackintoEquation 3{8 toproduceadualformulationoftheproblem,whichleadstoastandardquadraticprogrammingproblem. 3.1.2.1RepresentertheoremForsome1;2;:::M2R,thesolutiontoEquation 3{8 canbewrittenintheform (;)=MXk=1kftk(;)(3{9) Proof. WeconsiderthesubspaceofL2spannedbyourREEFsevaluatedatthetimesofthegivendatapoints(spanfftk(;):1kMg).Wethenconsidertheprojectionk(;)of(;)onthissubspace.Bynoting(;)=k(;)+?(;)andrewritingEquation 3{8 initsLagrangianform,weareleftwithEquation 3{10 .However,bydenitionof?(;),h?(;);ftk(;)i=0,sothissimpliesthesummationtermofEquation 3{10 toonlydependuponk(;)asshowninEquation 3{11 .min(;)2L2k(;)k2+ (3{10)MXk=1k1)]TJ /F5 11.955 Tf 11.95 0 Td[(yk)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(hk(;);ftk(;)i+h?(;);ftk(;)i)]TJ /F1 11.955 Tf 19.26 0 Td[(min(;)2L2k(;)k2+MXk=1k1)]TJ /F5 11.955 Tf 11.96 0 Td[(yk)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(hk(;);ftk(;)i)]TJ /F1 11.955 Tf 19.26 0 Td[( (3{11)Inaddition,byconsideringtherelationsshowninEquation 3{12 ,wendthatthersttermisminimizedwhen(;)=k(;).Andso,wendthattheoptimalsolutiontoEquation 3{8 willlieintheaforementionedsubspaceandthereforehavetheformofEquation 3{9 k(;)k2=kk(;)k2+k?(;)k2kk(;)k2(3{12) 39

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3.1.2.2DualrepresentationWiththeproofoftherepresentertheorem,wecannowsubstitutetheoptimalsolutionformshowninEquation 3{9 backintotheoriginaloptimizationproblem,showninEquation 3{8 .ThisleadstotheprobleminEquation 3{13 andisequivalenttoEquation 3{14 ,aquadraticprogrammingproblem,whichissolvablegiventhatwehaveaccesstothematrixK,whichwasderivedintheproofinSection 3.1.2 andshowninEquation 3{15 .InordertoshowthattheproblemgiveninEquation 3{14 isconvex,wemustshowthatKispositivedenite.Asdiscussedin( Berlinet&Thomas-Agnan 2004 ),weknowthatsinceKisdenedasaninnerproductinsomeHilbertspace,L2specically,weknowthatKispositivedenitesincePni=1Pnj=1aiajK(ti;tj)=Pni=1Pnj=1haifti(;);ajftj(;)i=kPni=1aifti(;)k22R+MinimizeMXk=1kftk(;)2 (3{13)s.t.ym *MXk=1kftk(;);ftm(;)+)]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgMinimizeMXi=1MXj=1ijK(ti;tj) (3{14)s.t.ym MXk=1kK(tk;tm))]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgK(ti;tj)=hfti(;);ftj(;)i (3{15)=Z10Z10fti(;)ftj(;)dd=titj (ti+tj)2 40

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3.1.3SingleSynapseInthecaseofasinglespike,thekerneldidnotconsidertheeectofmultiplespikesonthemembranepotential.However,sincewearemakinganSRM0approximation,weassumethattheeectsofspikesareadditivelyseparable( Stanisz 1969 )andthateachspike'seectonthemembranepotentialforthegivensynapsewillbeidentical.Withoutthelatterassumption,theproblemproducedisverygeneralandmuchmorediculttosolve.WerstdenethethresholddependentclassicationfunctionforasinglespikeinanidenticalmannertothesinglespikeformulationasshowninEquation 3{2 .Thiswillbethe\stereotyped"eectthataspikearrivingatthissynapsehasonthemembranepotential. 3.1.3.1PrimalproblemWenowconsidertheadditiveeectsofmultiplespikesarrivingatasynapse.Wedenethevectortj=htj1;tj2;:::;tjNjitobethejthdatapoint,whichconsistsofNjspikes,representedbytheirspiketimes.Theprimaloptimizationproblem,denedinEquation 3{16 ,isequivalenttoEquation 3{17 .Byposingtheconstraintsoftheoptimizationprobleminthismanner,weforceourmodeltoassumethattheeectofspikesonthissynapsearestereotyped,ratherthanallowingeachspiketohaveadierenteectonthemembranepotential.Minimizek(;)k2 (3{16)s.t.ym NmXi=1(;);ftmi(;))]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgMinimizek(;)k2 (3{17)s.t.ym *(;);NmXi=1ftmi(;)+)]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgTherepresentertheoremstatesthattheoptimal(;)mustlieinthespanofourbasisevaluatedatthegivendatapoints(spanfPNki=1ftki(;):1kMg).We 41

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omittheformalproof,butwiththeexceptionofthebasisusedtoproduce(;),theargumentsareidenticaltothosemadeinSection 3.1.2.1 .Weseethattheoptimal(;)toEquation 3{17 willbeoftheformofEquation 3{18 (;)=MXj=1jNjXi=1ftji(;)(3{18) 3.1.3.2DualproblemWeonceagainsubstituteEquation 3{18 intotheoptimizationproblemgiveninEquation 3{17 toproducethedualproblemshowninEquation 3{19 .ThisisequivalenttotheprobleminEquation 3{20 ,whichcanbesolvedgiventhekernelinEquation 3{21 .WealsoknowthattheprobleminEquation 3{20 isconvexsinceKispositivedenite.WeknowKispositivedenitebecausethesumofpositivedenitekernelsisalsopositivedenite.MinimizeMXj=1jNjXi=1ftji(;)2 (3{19)s.t.ym0@*MXj=1jNjXi=1ftji(;);NmXk=1ftmk(;)+)]TJ /F1 11.955 Tf 11.96 0 Td[(1A1m=f1:::MgMinimizeMXi=1MXj=1ijK(ti;tj) (3{20)s.t.ym MXk=1kK(tk;tm))]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgK(tp;tq)=*NpXi=1ftpi(;);NqXk=1ftqk(;)+ (3{21)=NpXi=1NqXk=1Dftpi(;);ftqk(;)E=NpXi=1NqXk=1tpitqk (tpi+tqk)2 42

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3.1.4MultipleSynapsesInthemultiplesynapsecase,theprinciplesareidenticaltothesinglesynapsecase,withtheexceptionthatthespikesarrivingatdierentsynapsescouldhavedierenteectsonthemembranepotential,dependingontheweightandthetypeofthesynapticjunction.Therefore,weneedtokeeptheeectsofeachsynapseonthemembranepotentialseparate.Todothis,eachsynapseisassigneditsown(;)function. 3.1.4.1PrimalproblemSinceeachsynapsewillhaveitsown(;)function.ThissimplyaddsanothersummationtermovertheSsynapses.TheprimaloptimizationproblemisdenedinEquation 3{22 whichisequivalenttoEquation 3{23 .Sisthenumberofsynapses,Nm;sisthenumberofspikesonthesthsynapseofthemthdatapoint,andtm;sisthespiketrainthatexistsonthesthsynapseofthemthdatapoint.MinimizeSXs=1ks(;)k2 (3{22)s.t.ym SXs=1Nm;sXi=1Ds(;);ftm;si(;)E)]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgMinimizeSXs=1ks(;)k2 (3{23)s.t.ym SXs=1*s(;);Nm;sXi=1ftm;si(;)+)]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgInasimilarmannertotheprevioustwoformulations,therepresentertheoremstatesthattheoptimals(;)forthesthsynapsesmustlieinthespanofthebasis(spanfPNm;si=1ftm;si(;):1mMg).Weagainomittheformalproof,butwiththeexceptionofthebasisusedtoproduces(;),theargumentsareidenticaltothosemadeinSection 3.1.2.1 .Weseethattheoptimals(;)toEquation 3{23 willbeoftheformofEquation 3{24 s(;)=MXj=1jNj;sXi=1ftj;si(;)(3{24) 43

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3.1.4.2DualproblemWesubstituteEquation 3{24 intoEquation 3{23 toproducethedualproblemshowninEquation 3{25 .ThisisequivalenttotheproblemseeninEquation 3{26 wherewehaveintroducedanewvectorwi,whichrepresentstheorderedcollectionofspiketrainsoneachsynapsefortheithdatapoint(wi=hti;1;ti;2:::ti;si).ThiscanbesolvedgivenaccesstothekerneldenedinEquation 3{27 .WealsoknowthattheprobleminEquation 3{26 isconvexsinceKispositivedeniteforthesamereasonsdiscussedinthesinglesynapsecase.MinimizeSXs=1MXj=1jNj;sXi=1ftj;si(;)2 (3{25)s.t.ym0@SXs=1*MXj=1jNj;sXi=1ftj;si(;);Nm;sXk=1ftm;sk(;)+)]TJ /F1 11.955 Tf 11.95 0 Td[(1A1m=f1:::MgMinimizeMXi=1MXj=1ijK(wi;wj) (3{26)s.t.ym MXk=1kK(wk;wm))]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgK(wp;wq)=SXs=1*Np;sXi=1ftp;si(;);Nq;sXk=1ftq;sk(;)+ (3{27)=SXs=1Np;sXi=1Nq;sXk=1tp;sitq;sk (tp;si+tq;sk)2 (3{28) 3.1.5SummaryWiththeabovekernels,whichwedenotethereciprocalexponential-exponentialkernels(REEK),weareabletoformulatequadraticprogrammingproblemswhichcaneasilybesolvedwithSVMLight( Joachims 1999 ).Thechoiceofthefunctiondictionaryusedtoderivethekerneliscriticaltothesuccessofthistechnique.Here,weuseaREEFdictionarywhichisadditiveinnatureandismadeupofPSPandAHPlikefunctions. 44

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AdictionarywhichissimilartotheformsofPSPandAHPfunctionswillperformbetterthanamoregeneralclassoffunctions.InthenextchapterweshowanidenticalformulationoftheproblemwhenusingaGaussiandictionary.WewillthenshowinChapter 6 theadvantagesofusingadictionarywhichistailoredtotheproblemathand. Figure3-1. FiguresdepictingtheformoftheREEFdictionary.A)showstheREEFasafunctionofandfort=1s.B)showstheresultingREEFswhensetting=0:5anddierentvaluesasafunctionoftime.B)alsoshowsthevariousREEFsthatoccurwhen=20forvaryingvaluesasafunctionoftime. 45

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CHAPTER4GAUSSIANFUNCTIONDICTIONARYToreiteratewhatwehavesaidbefore,theclassicationformulationandthefunctiondictionaryusedtoderivethekernelarecriticalindeterminingthesuccessofourmodel.TheREEFdictionarywaschosenspecicallywiththeproblemoflearningthecontributionsofaspiketothemembranepotentialinmind.Theclassicationproblemwasalsoformulatedtoaccountfortheadditiveseparabilityassumptionofthemembranepotentialaswellasthestereotypedeectsofspikestravelingacrossasynapse.Todemonstratetheimportanceofthedictionarychoice,wenowderiveakernelfromadictionaryofGaussianfunctionswithxedstandarddeviation.ThesearenormalizedGaussianfunctionsoftheformf(t)=1 p 2exp)]TJ /F3 7.97 Tf 6.58 0 Td[((t)]TJ /F6 7.97 Tf 6.59 0 Td[()2 22whichareparameterizedbymeanandstandarddeviation.Weformulatetheclassicationprobleminthesamemanner,theonlydierenceisthedictionaryfromwhichthekernelisderived.AcomparisonofthesekernelswillbegiveninChapter 6 4.1ApproximationoftheMembranePotentialFunctionInamannersimilartotheREEFformulation,wewouldliketocombinemembersofthechosendictionarytocreateanapproximationofthemembranepotentialfunction,P,whichwillyieldasolutiontotheclassicationproblemposedinEquation 2{5 .Again,wewillrstdiscusshowthiscanbeachievedinadiscreteframeworkandthendiscussasimilarcontinuousformulation.Usingtherepresentertheorem,weshowthattheoptimalsolutiontotheposedclassicationproblemmustlieinthespanofthedatapointswhichwereusedtotraintheclassierandweproduceanequivalentdualformulationoftheproblem.Inthediscreteandcontinuousformulation,wewillassumeasinglespikeforsimplicity.Weconcludebyextendingthecontinuousformulationtothecaseofmultiplespikesonasinglesynapse,andthecaseofmultiplespikesonmultiplesynapses. 46

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4.1.1DiscreteFormulationInthediscreteformulation,wewouldliketoapproximatethemembranepotentialfunctionusingalinearcombinationofapredenedsetoffunctionsfromaGaussiandictionary.Inthesinglespikecase,weonlyneedtomodeltheeectofasinglespikeonthemembranepotential.Wedenotethiseectonthemembranepotentialby^PanditisdenedasalinearcombinationofparameterizedGaussianfunctionsasshowninEquation 4{1 .ft(;)=1 p 2exp)]TJ /F3 7.97 Tf 6.59 0 Td[((t)]TJ /F6 7.97 Tf 6.59 0 Td[()2 22arethefunctionsdenedbyand.Specicparametersf(1;1);:::;(M;1);(1;2);:::;(M;N)gareusedtocreateafunction^Pthatcanbestreproducetheeectofthespikeonthemembranepotential.Finitelymanyfunctions,ft(i;j),willbescaledappropriatelyinordertoapproximateP. ^P(t)=MXi=1NXj=1i;jft(i;j)(4{1)Asisthecasewithanydictionary,theapproximation,^P,canbeimprovedbyincreasingthenumberoffunctionsused.However,thiswillalsoincreasethedimensionalityofourfeaturespacewhichcanleadtoover-ttingofthetrainingdata.Instead,wedeneakernelwhichcanbeusedinacontinuousformulationofthisproblem.Thisallowsallpossiblefunctionstobeconsideredwhenproducingthe^Papproximationwhilekeepingthedimensionalityofthefeaturespaceconstant. 4.1.2ContinuousFormulationInthecontinuousformulation,weconsiderL2,theHilbertspaceofsquareintegrablefunctionsonthedomain2(;1).willbeaparameterandtreatedasaconstant.Weareconcernedwithndingsomethresholddependentclassicationfunction^P,suchthat^P(t)+1whenthespiket2S+and^P(t))]TJ /F1 11.955 Tf 12.87 0 Td[(1whent2S)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(.ThisfunctionisdenedinEquation 4{2 .Inthisformulation,()isdenedtobeamemberofL2.Therefore,ifft(;)2L2,then^P(t)2L2bytheCauchy-Schwartzinequalitysinceh();ft(;)ik()kkft(;)k<1ifbothk()k<1andkft(;)k<18t.To 47

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showthatft(;)2L2wemustshowhft(;);ft(;)i<1. ^P(t)=h();ft(;)i=Z1()ft(;)d(4{2) Proof:hft(;);ft(;)i<1. Intheproof,wewillusethepropertyshowninEquation 4{3 ,whichcanbefoundin( Jerey&Dai 2008 ),inordertointegrateEquation 4{6 intoaclosedformsolution.Fromthatpointweusealgebraicmanipulationtoproducethenalsolution. Z1exp)]TJ /F1 11.955 Tf 9.3 0 Td[((ax2+bx+c)dx=r aexpb2)]TJ /F1 11.955 Tf 11.96 0 Td[(4ac 4a(4{3)hfx(;);fy(;)i=Z11 p 2exp)]TJ /F1 11.955 Tf 9.3 0 Td[((x)]TJ /F5 11.955 Tf 11.95 0 Td[()2 221 p 2exp)]TJ /F1 11.955 Tf 9.3 0 Td[((y)]TJ /F5 11.955 Tf 11.95 0 Td[()2 22d (4{4)=1 22Z1exp)]TJ /F5 11.955 Tf 9.3 0 Td[(x2+2x)]TJ /F5 11.955 Tf 11.96 0 Td[(2)]TJ /F5 11.955 Tf 11.95 0 Td[(y2+2y)]TJ /F5 11.955 Tf 11.96 0 Td[(2 22d (4{5)=1 22Z1exp)]TJ /F7 11.955 Tf 11.29 16.86 Td[(2 222)]TJ /F1 11.955 Tf 13.15 8.09 Td[((2x+2y) 22+(x2+y2) 22d (4{6)=1 22r 22 2exp0B@()]TJ /F3 7.97 Tf 6.59 0 Td[(2x)]TJ /F3 7.97 Tf 6.59 0 Td[(2y) 222)]TJ /F1 11.955 Tf 11.95 0 Td[(42 22(x2+y2) 22 42 221CA (4{7)=p 22exp ()]TJ /F1 11.955 Tf 9.29 0 Td[(2x)]TJ /F1 11.955 Tf 11.95 0 Td[(2y) 42)]TJ /F1 11.955 Tf 13.15 8.08 Td[((x2+y2) 22! (4{8)=1 2p exp )]TJ /F7 11.955 Tf 11.29 16.85 Td[(x)]TJ /F5 11.955 Tf 11.96 0 Td[(y 22! (4{9)Thereforeweseethathft(;);ft(;)i=1 2p <18t2[;]forsome>0whereisthelengthofthenitepastforwhichweconsiderspikes.Bydeningthemembranepotentialfunctioninthismanner,wehaveproducedaproblemwhichyieldsasolutionwhichisdierentthanthesolutiontothediscreteproblem.Thedeltafunctionwhichiscenteredatsomepoint(;)doesnotbelongtoL2.Thisimpliesthattheapproximatingmixingfunction()cannotbemadeupofalinearcombinationofthesedeltafunctions,asisthecaseinthediscreteformulation.Inaddition,wearenotworkingwithareproducingkernelHilbertspacesinceweare 48

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consideringL2.However,ourdenitioninEquation 4{2 denesthe\pointevaluation"ofourmembranepotentialfunction.Since^P(t)isdenedusingthestandardinnerproductinL2withrespecttoparticularmembersofL2,wecanreformulatetheclassicationprobleminEquation 2{5 asshowninEquation 4{10 ,whereMisthenumberofdatapoints,m=1:::M,andymisthecorrespondingclassicationforspiketimetm(ym=+1iftm2S+andym=)]TJ /F1 11.955 Tf 9.3 0 Td[(1iftm2S)]TJ /F1 11.955 Tf 7.09 -4.34 Td[().Minimizek()k2 (4{10)s.t.ym(h();ftm(;)i)]TJ /F1 11.955 Tf 19.26 0 Td[()1m=f1:::MgWenowcanuseaspecicinstanceoftheRepresentertheoremwhichwasrstprovedby Kimeldorf&Wahba ( 1971 )toshowthattheoptimalsolutiontotheoptimizationprobleminEquation 4{10 canbeexpressedintermsofaweightedsumofkernelevaluations,whenthekernelisevaluatedatanitenumberofdatapointsfromourtrainingset(()=PMk=1kftk(;)).WecanthensubstitutethisequalitybackintoEquation 4{10 toproduceadualformulationoftheproblem,whichleadstoastandardquadraticprogrammingproblem. 4.1.2.1RepresentertheoremForsome1;2;:::M2R,thesolutiontoEquation 4{10 canbewrittenintheform ()=MXk=1kftk(;)(4{11) Proof. ThisproofisalmostidenticaltotheoneshowninChapter 3 ,butwereiterateithereforcompleteness.WeconsiderthesubspaceofL2spannedbyourGaussianfunctionsevaluatedatthetimesofthegivendatapoints(spanfftk(;):1kMg).Wethenconsidertheprojectionk()of()onthissubspace.Bynoting()=k()+?()andrewritingEquation 4{10 initsLagrangianform,weareleftwithEquation 4{12 .However,bydenitionof?(),h?();ftk(;)i=0,sothissimpliesthesummation 49

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termofEquation 4{12 toonlydependuponk()asshowninEquation 4{13 .min()2L2k()k2+MXk=1k1)]TJ /F5 11.955 Tf 11.95 0 Td[(yk)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(hk();ftk(;)i+h?();ftk(;)i)]TJ /F1 11.955 Tf 19.27 0 Td[( (4{12)min()2L2k()k2+MXk=1k1)]TJ /F5 11.955 Tf 11.95 0 Td[(yk)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(hk();ftk(;)i)]TJ /F1 11.955 Tf 19.26 0 Td[( (4{13)Inaddition,byconsideringtherelationsshowninEquation 4{14 ,wendthatthersttermisminimizedwhen()=k().Andso,wendthattheoptimalsolutiontoEquation 4{10 willlieintheaforementionedsubspaceandthereforehavetheformofEquation 4{11 k()k2=kk()k2+k?()k2kk()k2(4{14) 4.1.2.2DualrepresentationWiththeproofoftherepresentertheorem,wecannowsubstitutetheoptimalsolutionformshowninEquation 4{11 backintotheoriginaloptimizationproblem,showninEquation 4{10 .ThisleadstotheprobleminEquation 4{15 andisequivalenttoEquation 4{16 ,whichleadstoaquadraticprogrammingproblem,whichissolvablegiventhatwehaveaccesstothematrixK,whichwasderivedintheproofinSection 4.1.2 andshowninEquation 4{17 .AsdiscussedinChapter 3 weknowthatKispositivedenitesinceitisdenedasaninnerproductinL2.ThisimpliesthattheproblemgiveninEquation 4{16 isconvex.MinimizeMXk=1kftk(;)2 (4{15)s.t.ym *MXk=1kftk(;);ftm(;)+)]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::Mg 50

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MinimizeMXi=1MXj=1ijK(ti;tj) (4{16)s.t.ym MXk=1kK(tk;tm))]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgK(ti;tj)=hfti(;);ftj(;)i (4{17)=Z1fti(;)ftj(;)d=1 2p exp )]TJ /F7 11.955 Tf 11.29 16.86 Td[(ti)]TJ /F5 11.955 Tf 11.95 0 Td[(tj 22! 4.1.3SingleSynapseInthecaseofasinglespike,thekerneldidnotconsidertheeectofmultiplespikesonthemembranepotential.However,sincewearemakinganSRM0approximation,weassumethattheeectsofspikesareadditivelyseparable( Stanisz 1969 )andthateachspike'seectonthemembranepotentialforthegivensynapsewillbeidentical.Withoutthelatterassumption,theproblemproducedisverygeneralandmuchmorediculttosolve.WerstdenethethresholddependentclassicationfunctionforasinglespikeinanidenticalmannertothesinglespikeformulationasshowninEquation 4{2 .Thiswillbethe\stereotyped"eectthataspikearrivingatthissynapsehasonthemembranepotential. 4.1.3.1PrimalproblemWenowconsidertheadditiveeectsofmultiplespikesarrivingatasynapse.Wedenethevectortj=htj1;tj2;:::;tjNjitobethejthdatapoint,whichconsistsofNjspikes,representedbytheirspiketimes.Theprimaloptimizationproblem,denedinEquation 4{18 ,isequivalenttoEquation 4{19 .Byposingtheconstraintsoftheoptimizationprobleminthismanner,weforceourmodeltoassumethattheeectofspikesonthissynapsearestereotyped,ratherthanallowingeachspiketohaveadierenteectonthe 51

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membranepotential.Minimizek()k2 (4{18)s.t.ym NmXi=1();ftmi(;))]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgMinimizek()k2 (4{19)s.t.ym *();NmXi=1ftmi(;)+)]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgTherepresentertheoremstatesthattheoptimal()mustlieinthespanofourbasis(spanfPNki=1ftki(;):1kMg).Weomittheformalproof,butwiththeexceptionofthebasisusedtoproduce(),theargumentsareidenticaltothosemadeinSection 4.1.2.1 .Weseethattheoptimal()toEquation 4{19 willbeoftheformofEquation 4{20 ()=MXj=1jNjXi=1ftji(;)(4{20) 4.1.3.2DualproblemWeonceagainsubstituteEquation 4{20 intoEquation 4{19 toproducethedualproblemshowninEquation 4{21 .WerewritethisintheformshowninEquation 4{22 ,whichcanbesolvedgiventhekernelinEquation 4{23 .Inaddition,Equation 4{22 isknowntobeconvextothepositivedenitenessofK.MinimizeMXj=1jNjXi=1ftji(;)2 (4{21)s.t.ym0@*MXj=1jNjXi=1ftji(;);NmXk=1ftmk(;)+)]TJ /F1 11.955 Tf 11.95 0 Td[(1A1m=f1:::Mg 52

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MinimizeMXi=1MXj=1ijK(ti;tj) (4{22)s.t.ym MXk=1kK(tk;tm))]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgK(tp;tq)=*NpXi=1ftpi(;);NqXk=1ftqk(;)+ (4{23)=NpXi=1NqXk=1Dftpi(;);ftqk(;)E (4{24)=NpXi=1NqXk=11 2p exp )]TJ /F7 11.955 Tf 11.29 16.86 Td[(tpi)]TJ /F5 11.955 Tf 11.95 0 Td[(tqk 22! 4.1.4MultipleSynapsesInthemultiplesynapsecase,theprinciplesareidenticaltothesinglesynapsecase,withtheexceptionthatthespikesarrivingatdierentsynapsescouldhavedierenteectsonthemembranepotential,dependingontheweightandthetypeofthesynapticjunction.Therefore,weneedtokeeptheeectsofeachsynapseonthemembranepotentialseparate.Todothis,eachsynapseisassigneditsown()function. 4.1.4.1PrimalproblemSinceeachsynapsewillhaveitsown()function,thissimplyaddsanothersummationtermovertheSsynapses.TheprimaloptimizationproblemisdenedinEquation 4{25 whichisequivalenttoEquation 4{26 .Sisthenumberofsynapses,Nm;sisthenumberofspikesonthesthsynapseofthemthdatapoint,andtm;sisthespiketrainthatexistsonthesthsynapseofthemthdatapoint.MinimizeSXs=1ks()k2 (4{25)s.t.ym SXs=1Nm;sXi=1Ds();ftm;si(;)E)]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::Mg 53

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MinimizeSXs=1ks()k2 (4{26)s.t.ym SXs=1*s();Nm;sXi=1ftm;si(;)+)]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgInasimilarmannertotheprevioustwoformulations,therepresentertheoremstatesthattheoptimals()forthesthsynapsesmustlieinthespanofthebasis(spanfPNm;si=1ftm;si(;):1mMg).Weagainomittheformalproof,butwiththeexceptionofthebasisusedtoproduces(),theargumentsareidenticaltothosemadeinSection 4.1.2.1 .Weseethattheoptimals()toEquation 4{26 willbeoftheformofEquation 4{27 s()=MXj=1jNj;sXi=1ftj;si(;)(4{27) 4.1.4.2DualproblemWesubstituteEquation 4{27 intoEquation 4{26 toproducethedualproblemshowninEquation 4{28 .ThisisequivalenttotheproblemseeninEquation 4{29 wherewehaveintroducedanewvectorwi,whichrepresentstheorderedcollectionofspiketrainsoneachsynapsefortheithdatapoint(wi=hti;1;ti;2:::ti;si).ThiscanbesolvedwiththekerneldenedinEquation 4{30 .AswiththepreviousformulationthisoneisalsoconvexsinceKispositivedenite.MinimizeSXs=1MXj=1jNj;sXi=1ftj;si(;)2 (4{28)s.t.ym0@SXs=1*MXj=1jNj;sXi=1ftj;si(;);Nm;sXk=1ftm;sk(;)+)]TJ /F1 11.955 Tf 11.96 0 Td[(1A1m=f1:::MgMinimizeMXi=1MXj=1ijK(wi;wj) (4{29)s.t.ym MXk=1kK(wk;wm))]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::Mg 54

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K(wp;wq)=SXs=1*Np;sXi=1ftp;si(;);Nq;sXk=1ftq;sk(;)+ (4{30)=SXs=1Np;sXi=1Nq;sXk=11 2p exp )]TJ /F7 11.955 Tf 11.29 16.86 Td[(tp;si)]TJ /F5 11.955 Tf 11.96 0 Td[(tq;sk 22! 4.1.5SummaryWiththeabovekernels,whichwedenotetheGaussianSummationKernel(GSK),weareabletoformulatequadraticprogrammingproblemswhichcaneasilybesolvedwithSVMLight( Joachims 1999 ).Thechoiceofthedictionaryusedtoderivethekerneliscriticaltothesuccessofthistechnique.Here,weuseaGaussiandictionarywhichisadditiveinnature.ThekernelsdenedinthissectionaresimilartotheGaussianradialbasisfunction(GRBF)kernel.However,thekernelsderivedtakethesummationofGaussianswithxedstandarddeviationratherthantakingtheirproduct.HencethekernelsderivedherearemoresuitedfortheproblemoflearningthemembranepotentialofaneuronthantheGRBFbutlesssuitedthantheREEK.WeprovideamorethoroughcomparisonofthekernelsinChapter 6 ,butrstwediscussaclassicationformulationandkernelwhichwerederivedfornonstereotypicalPSPs. 55

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CHAPTER5RECIPROCALEXPONENTIALFUNCTIONDICTIONARYUptothispointwehavemadetheassumptionofstereotypedPSPs.Thatis,weassumethatallspikestravelingacrossaparticularsynapsewillhavethesameeectonthemembranepotential,butthisisnotalwaysthecase.Inanactualneuron,neurotransmittersandvoltagechangescausegatedionchannelstoopen.Thesegatedchannelseventuallyclosewhentheeectsoftheneurotransmitterorvoltagechangeareexhausted.Iftwospikesarereceivedconsecutivelythesecondspike'seectonopeningthechannelswillnotbeaslargeasthatoftherstsincethechannelsarealreadypartiallyopen.ThisisdemonstratedinFigure 5-1 whichshowsthemembranepotentialforaneuronafterreceivingtwoconsecutivespikeinputswithvaryinginterspikeintervals(thedierenceintimebetweenthetwospikes).Themodelusedwasdesignedby Destexheetal. ( 1994 )totakesynapticconductancesintoaccount.InFigure 5-1 A,weseethatwhenthespikeshavearelativelylargeinterspikeinterval,thattheindividualPSPshavethesamemagnitudewhichisthatofastereotypicalPSP.However,wecanseeinFigure 5-1 B,thatwhentheinterspikeintervalissmall,themagnitudeofthesecondspikediminisheswhencomparedtothestereotypicalPSP.Intheconductanceformulationweassumetheexistenceofsomestereotypicaleectonthemembranepotential.Wethencantreattheeectsofconsecutivespikesassomescaledversionofthestereotypicaleect.IfweconsiderthescalingeectonthesecondPSPasafunctionoftheinterspikeintervalfortheaforementionedmodel( Destexheetal. 1994 ),weseecurvessimilartothoseshowninFigure 5-2 .Wenoticethatwhentheinterspikeintervalissmall,thevaluebywhichthestereotypicalPSPisscaledisalsosmall.However,whentheinterspikeintervalislarge,thesecondPSPisscaledbyavalueof1andthereforeequivalenttothestereotypicalPSP.Theexactmagnitudeandrisetimeofthecurvesaredependentonthemodelparameters;theonesprovidedwerefoundempiricallyfromthemodel. 56

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Thesecurvesserveasthemotivationforourdictionarychoiceinproducingakernel.WeassumetheexistenceofsomestereotypicalPSPandthencalculatetheeectofagivenspikebyconsideringittobesomescaledversionofthestereotypicalPSPwheretheamountofthescaleisdependentontheinterspikeinterval.WeagainusetheREEFdictionarytomodelthestereotypicalPSP,howevertomodelscalingeectseeninFigure 5-2 ,weusereciprocalexponentialfunctionsoftheformexp()]TJ /F5 11.955 Tf 9.3 0 Td[(=t),whereistheparameterandtistheinterspikeinterval. 5.1ApproximationoftheConductanceMembranePotentialFunctionAswiththepreviousderivations,wewouldliketocombinemembersofthechosendictionarytocreateanapproximationofthemembranepotentialfunction,P,whichwillyieldasolutiontotheclassicationproblemposedinEquation 2{5 .However,aswillbecomeclear,wedividethemembranepotentialapproximationintotwoparts,theapproximationofthestereotypicalPSPandtheapproximationofthescalingfunction.Wewillrstdiscusshowthiscanbeachievedinadiscreteframework;wecombineanitenumberfunctionstomodelthestereotypicalPSPasafunctionofasinglespike.Wealsocombinemembersofaseparatedictionarytomodelthescalingeectasafunctionoftheinterspikeinterval.Followingthediscreteformulation,wewilldiscussacontinuousformulation,inwhichwecombineaninniterangeofparameterizedfunctionsinordertoproduceamembranepotentialapproximation.Weagainwillusetherepresentertheoremtoproduceadualformulationintheformofaquadraticprogrammingproblem.Wethenconcludebyextendingthecontinuousformulationtothecaseofmultiplespikesonasinglesynapse,andthecaseofmultiplespikesonmultiplesynapses. 5.1.1DiscreteFormulationInthediscreteformulation,wewouldliketoapproximatethemembranepotentialfunctionusingalinearcombinationofapredenedsetoffunctionsfromtheREEFdictionary.Inderivingtheformulationfortheconductanceframework,wemustaddanadditionalpieceofinformationtoourspiketraindata,theinterspikeinterval.Therefore, 57

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weconsider,thespiketimetaswellasitscorrespondinginterspikeinterval,t.Althoughitisnotintuitivethatasinglespikeshouldhaveaninterspikeinterval,fornowwecanthinkoftheinterspikeintervalbeinganextremelylargevaluesothattherewouldbenoscalingeecttothestereotypicalPSP.Thereasonforincludingtwillbecomeapparentwhenwegeneralizetomultiplespikes.WerstdenethestereotypicalPSPwiththeREEFdictionaryasshowninEquation 5{1 .ThisisidenticaltothemethodusedforthesinglespikedenitioninSection 3.1.1 .Inasimilarmanner,wedenotethescalingeectonthestereotypicalPSPby^SanditisdenedasalinearcombinationofparameterizedreciprocalexponentialfunctionsasshowninEquation 5{2 .~ft(i)=exp()]TJ /F5 11.955 Tf 9.29 0 Td[(=t)arethefunctionsdenedby.Specicparametersf1;2;:::;~Mgareusedtocreateafunction^SthatcanbestreproducethescalingeectonthestereotypicalPSP.Finitelymanyfunctions,~ft(i),willbescaledappropriatelyinordertoapproximatethetruescalingfunction,S. ^P(t)=MXi=1NXj=1i;jft(i;j)(5{1) ^S(t)=~MXi=1~i~ft(i)(5{2)Withthesedenitionswecannowdenethefunction^Q,whichisthescaledeectofaspikeonthemembranepotentialasshowninEquation 5{3 ^Q(t;t)=^P(t)^S(t)= MXi=1NXj=1i;jft(i;j)!0@~MXk=1~k~ft(k)1A(5{3)Theapproximation^Qcanbeimprovedbyincreasingthenumberoffunctionsusedfromeitherdictionary.However,asthenumberoffunctionsincreases,sodoesthedimensionalityofourfeaturespace;thisleadstoover-ttingofthetrainingdata.Toresolvethisproblem,weproduceakernelusingacontinuousformulationofthisproblem. 58

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Thisallowsallpossiblefunctionstobeconsideredwhenproducing^Qwhilekeepingthedimensionalityofthefeaturespaceconstant. 5.1.2ContinuousformulationInthecontinuousformulationfortheconductancekernel,wewillremaininL2.Wewillconsiderthedomainf;;g2[0;1)3.Weareconcernedwithndingsomethresholddependentclassicationfunction^Q,suchthat^Q(t;t)+1whenthedatapoint(t;t)2S+and^Q(t;t))]TJ /F1 11.955 Tf 12.94 0 Td[(1when(t;t)2S)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(.ThisfunctionisdenedinEquation 5{4 .Inthisformulation,both(;)and~()aredenedtobeamemberofL2.Wehavealreadyshownthat^P(t)=h(;);ft(;)iisalsoamemberofL2inSection 3.1.2 .If~ft()2L2,then^S(t)2L2bytheCauchy-Schwartzinequalitysinceh~();~ft()ik~()kk~ft()k<1ifbothk~()k<1andk~ft()k<18t.If^S(t),iswelldenedand^P(t)iswelldenedthenweknow^Q(t;t)=^S(t)^P(t)isalsowelldened.Toshowthat~ft()2L2wemustshowh~ft();~ft()i<1.^Q(t;t)=^P(t)^S(t) (5{4)=h(;);ft(;)ih~();~ft()i=Z10Z10(;)ft(;)ddZ10~()~ft()d Proof:h~ft();~ft()i<1. h~fx();~fy()i=Z10exp()]TJ /F5 11.955 Tf 9.3 0 Td[(=x)exp()]TJ /F5 11.955 Tf 9.3 0 Td[(=y)d (5{5)=Z10exp)]TJ /F5 11.955 Tf 9.3 0 Td[(x+y xyd (5{6)=)]TJ /F1 11.955 Tf 16.04 8.08 Td[(xy x+yexp)]TJ /F5 11.955 Tf 9.3 0 Td[(x+y xy1=0 (5{7)=xy x+y (5{8)Thereforeweseethath~ft();~ft()i=t 2<18t2[;]forsome>0,whereisthelengthofthenitepastforwhichweconsiderspikes. 59

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5.1.2.1PrimalProblemSince^Q(t;t)isdenedusingstandardinnerproductsinL2withrespecttoparticularmembersofL2,wecanreformulatetheclassicationprobleminEquation 2{5 asshowninEquation 5{9 .Inthisformulationweassumethatwehavealreadyproduced^P(t),theapproximationtothestereotypicalPSP,P(t).ThiscanbeeasilydonebyusingtheclassicationframeworkdiscussedinChapter 3 andtrainingwithspiketraindatathathaslargeenoughinterspikeintervalssuchthatnoscalingeectsexist.Wethenusetheresulting^Pinthisformulation.Misthenumberofdatapoints,andymisthecorrespondingclassicationfordatapoint(tm;tm)(ym=+1if(tm;t)2S+andym=)]TJ /F1 11.955 Tf 9.3 0 Td[(1if(tm;t)2S)]TJ /F1 11.955 Tf 7.09 -4.34 Td[().Minimizek~()k2 (5{9)s.t.ym^P(tm)h~();~ftm()i)]TJ /F1 11.955 Tf 19.26 0 Td[(1m=f1:::MgMinimizek~()k2 (5{10)s.t.ymh~();^P(tm)~ftm()i)]TJ /F1 11.955 Tf 19.26 0 Td[(1m=f1:::MgSince^Pisnotdependenton,wecanbringitintotheinnerproducttogetEquation 5{10 .FromEquation 5{10 wewillagainuseaspecicinstanceoftheRepresentertheorem( Kimeldorf&Wahba 1971 )toshowthattheoptimalsolutiontotheoptimizationprobleminEquation 5{10 canbeexpressedintermsofaweightedsumofdictionaryfunctionevaluations,whenthekernelisevaluatedatanitenumberofdatapointsfromourtrainingset~()=PMk=1k^P(tk)~ftk().WecanthensubstitutethisequalitybackintoEquation 5{10 toproduceadualformulationoftheproblem,whichleadstoastandardquadraticprogrammingproblem. 60

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5.1.2.2RepresentertheoremForsome1;2;:::M2R,thesolutiontoEquation 5{10 canbewrittenintheform ~()=MXk=1k^P(tk)~ftk()(5{11) Proof. WeconsiderthesubspaceofL2spannedbyourreciprocalexponentialfunctionsevaluatedatthetimesofthegivendatapoints(spanf^P(tk)~ftk():1kMg).Wethenconsidertheprojection~k()of~()onthissubspace.Bynoting~()=~k()+~?()andrewritingEquation 5{10 initsLagrangianform,weareleftwithEquation 5{12 .However,bydenitionof~?(),h~?();^P(tk)~ftk()i=0,sothissimpliesthesummationtermofEquation 5{12 toonlydependupon~k()asshowninEquation 5{13 .min~()2L2k~()k2+ (5{12)MXk=1kh1)]TJ /F5 11.955 Tf 11.96 0 Td[(ykD~k();^P(tk)~ftk()E+D~?();^P(tk)~ftk()E)]TJ /F1 11.955 Tf 11.96 0 Td[(imin~()2L2k~()k2+MXk=1kh1)]TJ /F5 11.955 Tf 11.95 0 Td[(ykD~k();^P(tk)~ftk()E)]TJ /F1 11.955 Tf 11.95 0 Td[(i (5{13)Inaddition,byconsideringtherelationsshowninEquation 5{14 ,wendthatthersttermisminimizedwhen~()=~k().Andso,wendthattheoptimalsolutiontoEquation 5{10 willlieintheaforementionedsubspaceandthereforehavetheformofEquation 5{11 k~()k2=k~k()k2+k~?()k2k~k()k2(5{14) 5.1.2.3DualrepresentationWiththeproofoftherepresentertheorem,wecannowsubstitutetheoptimalsolutionformshowninEquation 5{11 backintotheoriginaloptimizationproblem,showninEquation 5{10 .ThisleadstotheprobleminEquation 5{15 .Itisequivalentto 61

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Equation 5{16 ,whichleadstoaquadraticprogrammingproblem,whichissolvablegiventhatwehaveaccesstothematrixK,whichwasderivedintheproofinSection 5.1.2 andshowninEquation 5{17 .AsdiscussedinChapter 3 andChapter 4 ,weknowthatKispositivedenitesinceitisdenedasaninnerproductinL2andthereforetheproblemgiveninEquation 5{16 isconvex.MinimizeMXk=1k^P(tk)~ftk()2 (5{15)s.t.ym *MXk=1k^P(tk)~ftk();^P(tm)~ftm()+)]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgMinimizeMXi=1MXj=1ijK((ti;ti);(tj;tj)) (5{16)s.t.ym MXk=1kK((tk;tk);(tm;tm)))]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgK((ti;ti);(tj;tj))=h^P(ti)~fti();^P(tj)~ftj()i (5{17)=^P(ti)^P(tj)Z10Z10~fti()~ftj()d=^P(ti)^P(tj)titj ti+tj 5.1.3SingleSynapseWecannowextendtheaboveframeworktoasinglesynapse.Weagainmaketheadditiveseparabilityassumption,andthatthereexistssomestereotypedPSPforasinglespike.WeaimtolearnthetotaleectagivenspikehasonthemembranepotentialbylearningtheamountthestereotypedPSPshouldbescaledduetothetimethathaselapsedsincethelastspikewasreceived.Withmultiplespikes,wenowhaveanintuitivenotionofwhattshouldbe,thedierenceintimingbetweenthecurrentspikeandthelastonereceived. 62

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5.1.3.1PrimalproblemWedenethevector~tj=Dtj1;tj1;tj2;tj2;:::tjNjtjNj;Etobethejthdatapoint,whichconsistsofNjspikes,representedbytheirspiketimes,tjiandtherecorrespondinginterspikeintervaltji.Wenotethattheinterspikeintervalcanbeeasilycalculatedfromthespiketimes)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(tji=tji+1)]TJ /F5 11.955 Tf 11.95 0 Td[(tji,weexplicitlyaddthemforsimplicityinthederivation.WearegivenMdatapoints~t1:::~tMwithrespectiveclassicationsy1:::yM.Belowwedividethevector~tjintoitsspiketimeandinterspikeintervalcomponentsdenotedbythevectorstj=Dtj1;tj2;:::tjNjEandtj=Dtj1;tj2;:::tjNjE.Wecanthenproducetheprimaloptimizationproblem,denedinEquation 5{18 ,whichisequivalenttoEquation 5{19 .Byposingtheconstraintsoftheoptimizationprobleminthismanner,weforceourmodeltoassumethattheeectofspikesonthissynapsearesomescaledversionofastereotypedPSP.Minimizek~()k2 (5{18)s.t.ym NmXi=1^P(tmi)D~();~ftmi()E)]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgMinimizek~()k2 (5{19)s.t.ym *~();NmXi=1^P(tmi)~ftmi()+)]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgTherepresentertheoremstatesthattheoptimal~()mustlieinthespanofourbasisPNmi=1^P(tmi)~ftmi().Weomittheformalproof,butwiththeexceptionofthebasisusedtoproduce~(),theargumentsareidenticaltothosemadeinSection 5.1.2.2 .Weseethattheoptimal~()toEquation 5{19 willbeoftheformofEquation 5{20 ~()=MXj=1jNjXi=1^P(tji)~ftji()(5{20) 63

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5.1.3.2DualproblemWeonceagainsubstituteEquation 5{20 toproducethedualproblemshowninEquation 5{21 .WerewritethisintheformshowninEquation 5{22 ,whichcanbesolvedgiventhekernelinEquation 5{23 .Thisproducesaconvexoptimizationproblemforreasonsdiscussedpreviously.MinimizeMXj=1jNjXi=1^P(tji)~ftji()2 (5{21)s.t.ym0@*MXj=1jNjXi=1^P(tji)~ftji();NmXk=1^P(tmk)~ftmk()+)]TJ /F1 11.955 Tf 11.96 0 Td[(1A1m=f1:::MgMinimizeMXi=1MXj=1ijK)]TJ /F4 11.955 Tf 4.74 -7.4 Td[(~tk;~tm (5{22)s.t.ym MXk=1kK)]TJ /F4 11.955 Tf 4.73 -7.4 Td[(~tk;~tm)]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgK)]TJ /F4 11.955 Tf 4.73 -7.41 Td[(~tp;~tq=*NpXi=1^P(tpi)~ftpi();NqXk=1^P(tqk)~ftqk()+ (5{23)=NpXi=1NqXk=1^P(tpi)^P(tqk)D~ftpi();~ftqk()E (5{24)=NpXi=1NqXk=1^P(tpi)^P(tqk)tpitqk tpi+tqk 5.1.4MultipleSynapsesInthemultiplesynapsecase,theprinciplesareidenticaltothesinglesynapsecase,withtheexceptionthatthespikesarrivingatdierentsynapsescouldhaveadierentstereotypicalPSPandadierentscalingeect,dependingontheweightandthetypeofthesynapticjunction.Therefore,weneedtokeeptheeectsofeachsynapseonthemembranepotentialseparate.Todothis,eachsynapseisassigneditsown~()functionaswellasitsown^Pfunction. 64

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5.1.4.1PrimalproblemSinceeachsynapsewillhaveitsown~()and^Pfunction,thissimplyaddsanothersummationtermovertheSsynapses.TheprimaloptimizationproblemisdenedinEquation 5{25 whichisequivalenttoEquation 5{26 .Sisthenumberofsynapses,Nm;sisthenumberofspikesonthesthsynapseofthemthdatapoint,andand~tm;sisthevectorofspiketimesandinterspikeintervalsrepresentingthespiketrainthatexistsonthesthsynapseofthemthdatapoint.MinimizeSXs=1k~s()k2 (5{25)s.t.ym SXs=1Nm;sXi=1D~s();^Ps(tm;si)~ftm;si()E)]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgMinimizeSXs=1k~s()k2 (5{26)s.t.ym SXs=1*~s();Nm;sXi=1^Ps(tm;si)~ftm;si()+)]TJ /F1 11.955 Tf 11.95 0 Td[(!1m=f1:::MgInasimilarmannertotheprevioustwoformulations,therepresentertheoremstatesthattheoptimal~s()forthesthsynapsesmustlieinthespanofthebasis(spanfPNm;si=1^Ps(tm;si)~ftm;si():1mMg).Weagainomittheformalproof,butwiththeexceptionofthebasisusedtoproduce~s(),theargumentsareidenticaltothosemadeinSection 5.1.2.2 .Weseethattheoptimal~s()toEquation 5{26 willbeoftheformofEquation 5{27 ~s()=MXj=1jNj;sXi=1^Ps(tj;si)~ftj;si()(5{27) 5.1.4.2DualproblemWesubstituteEquation 5{27 intoEquation 5{26 toproducethedualproblemshowninEquation 5{28 .ThisisequivalenttotheproblemseeninEquation 5{29 wherewehaveintroducedanewvector~wi,whichrepresentstheorderedcollectionofspiketrainson 65

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eachsynapsefortheithdatapoint)]TJ /F4 11.955 Tf 7.07 -9.69 Td[(~wi=~ti;1;~ti;2:::~ti;s.Equation 5{29 canbesolvedgivenaccesstothekerneldenedinEquation 5{30 .Thisisagainaconvexoptimizationproblem.MinimizeSXs=1MXj=1jNj;sXi=1^Ps(tj;si)~ftj;si()2 (5{28)s.t.ym0@SXs=1*MXj=1jNj;sXi=1^Ps(tj;si)~ftj;si();Nm;sXk=1^Ps(tm;sk)~ftm;sk()+)]TJ /F1 11.955 Tf 11.95 0 Td[(1A1m=f1:::MgMinimizeMXi=1MXj=1ijK(~wi;~wj) (5{29)s.t.ym MXk=1kK(~wk;~wm))]TJ /F1 11.955 Tf 11.96 0 Td[(!1m=f1:::MgK(~wp;~wq)=SXs=1*Np;sXi=1^Ps(tp;si)~ftp;si();Nq;sXk=1^Ps(tq;sk)~ftq;sk()+ (5{30)=SXs=1Np;sXi=1Nq;sXk=1^Ps(tp;si)^Ps(tq;sk)tp;sitq;sk tp;si+tq;sk (5{31) 5.1.5SummaryWiththeconductancekernelabove,weareabletoformulatequadraticprogrammingproblemswhichcaneasilybesolvedwithSVMLight( Joachims 1999 ).WehaveproducedawaytoaccountforPSPsofvaryingmagnitudebyassumingtheexistenceofsomestereotypedPSPandlearningthescalingeectonthePSPcausedbyeachspike,whencomparedtothestereotypedPSP.Thisisthelastkernelwhichwehavederived.Wewillnowfurtherdiscussthedierencesbetweenthekernelsaswellasdemonstrateexperimentaldierencesbetweenthekernels. 66

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Figure5-1. ComparisonofstereotypicalandscaledPSPs.Themembranepotentialforaneuronafterreceivingtwoconsecutivespikeinputswithvaryinginterspikeintervals.A)showsPSPsthatareidenticaltoeachotheraswellasthestereotypicalPSPwhentheinterspikeintervalislarge.B)demonstratesthatwhentheinterspikeintervalissmall,themagnitudeofthesecondspikediminisheswhencomparedtothestereotypicalPSP. Figure5-2. ScalingeectonthesecondPSPasafunctionoftheinterspikeinterval.Theexactmagnitudeandrisetimeofthecurvesareparameterdependent,howevertheonesprovidedarefoundempiricallyfromthemodeldevelopedby( Destexheetal. 1994 ). 67

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CHAPTER6KERNELCOMPARISONANDRESULTSTothispoint,wehavediscussedtheclassicationframeworkwehaveusedtolearnthemembranepotentialfunctionofaspikingneuron.Wehavepointedouttheimportanceofthedictionarychoiceandthereforethekernelchoiceinthisframeworkandhaveshownthederivationofkernelswehavecreatedfortheproblemoflearningthemembranepotentialofaspikingneuron.Inthischapterwewillcompareandcontrastthenewkernelsandtheiradvantagesanddisadvantages. 6.1GaussianSummationKernelTheGaussiansummationkernel(GSK)thatwasderivedissimilarinnotiontotheGRBF,withoneexception:theGSKusesasummationofGaussianfunctions,whiletheGRBFismadeupofaproductofGaussianfunctions.Tosimplifythiscomparison,wewillconsideronlythespiketimesonasinglesynapse.Considerspiketimevectorstp=htp1;tp2;:::;tpNpiandtq=htq1;tq2;:::;tqNqiAsshowninChapter 4 ,theGSKwouldevaluatetoEquation 6{1 KGSK(tp;tq)=1 2p NpXi=1NqXk=1exp )]TJ /F7 11.955 Tf 11.29 16.86 Td[(tpi)]TJ /F5 11.955 Tf 11.96 0 Td[(tqk 22!(6{1)TheGRBFisdenedtobeKrbf(tp;tq)=exp()]TJ /F5 11.955 Tf 9.3 0 Td[(jtp)]TJ /F4 11.955 Tf 12.39 0 Td[(tqj2)( Joachims 2002 ).ThereforewhenweexpandthevectorsintheexponentialwendthattheGRBFisaproductofGaussianfunctionsasshowninEquation 6{2 KRBF(tp;tq)=exp )]TJ /F5 11.955 Tf 9.3 0 Td[(NXi=1(tpi)]TJ /F5 11.955 Tf 11.95 0 Td[(tqi)2!=NYi=1exp()]TJ /F5 11.955 Tf 9.3 0 Td[((tpi)]TJ /F5 11.955 Tf 11.96 0 Td[(tqi)2)(6{2)WenotethattheGRBFisnotwelldenedforthisproblemsinceitmakestheassumptionthatallspiketrainswillhavethesamenumberofspikes,N.Asimilarkernelcouldbeproducedwhichaccountsforvariablelengthspiketrains,however,aswewillshowbelow,itstillwouldnotaccountfortheadditivenatureofPSPsandAHPs.InFigure 6-1 weshowtheresultingGRBFandGSKevaluations,whenvaryingtqinthetwodimensionalcase(twospikesoneachsynapse).Wesettp=)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(50;50tobeconstantandvarytqacross 68

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thepossiblespiketimes.Figure 6-1 AshowstheGRBF,whileFigure 6-1 BshowstheGSK.Fromthisimage,youcanseethedierencebetweenthetwokernels.Wenotethatifaspiketrainvariesinonedimension,i.e.onespiketimeisthesameandonespiketimeisdierent,thentheGSKacceptsthesetobesimilarwheretheGRBFdoesnot.ThisallowstheGSKtoaccountfortheadditivenatureofthePSPsandAHPswhichtheGRBFcannot.FromthisdiscussionweareabletoseewhytheGSKismoresuitedforlearningamembranepotentialfunctionthantheGRBF.However,oneissuewhichshouldbeaddressedistheconstantstandarddeviationoftheGSK.IfweweretouseaGaussiandictionary,itwouldbeidealtohaveadictionarywherethestandarddeviationcouldvaryaswell.Forinstance,adictionaryinwhichthestandarddeviationincreasedwithtimewouldbeabettersuitedGaussiandictionary.ThiswouldallowthethinnerGaussianfunctionstocombinetoapproximatetherisingportionofthePSPandthentheGaussianfunctionswithalargerstandarddeviationwouldbebetterforapproximatingthefallingpartofthePSP.Unfortunatelyitisnotpossibletogetsuchasolutioninclosedform.Thereciprocalexponential-exponentialkernel(REEK)providesasuperiorrepresentationincomparisontousingaGaussianwithaconstant(orvarying)standarddeviationbyprovidingtheexactformofanAHPorPSP.WenowcompareandcontrasttheREEKwiththeGSK. 6.2ReciprocalExponential-ExponentialKernelAsshownpreviously,theREEKiswellsuitedforlearningmembranepotentialsbecauseitusesadictionarymadeupofREEFswhichcanapproximatetheformofaPSPoranAHP.InFigure 6-2 ,wecomparetheREEKtotheGSKforatwospikespiketrain.Wesettptobeconstantandvarytqacrossthepossiblespiketimes.Figure 6-2 A,showstheresultsoftheREEKevaluationwhentp=)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(5;20).Figure 6-2 BshowstheresultsoftheGSKevaluationwhentp=)]TJ /F1 11.955 Tf 5.47 -9.69 Td[(10;50).WeseethattheGSKusesthesamemeasureofsimilarityregardlessofthetimingofthespikes.Onlythedistancebetweentwospikesis 69

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considered,ratherthantheirlocation.Thismeansthatwhenconsideringaspikeseectonthemembranepotential,aspikethatoccurred80msinthepastisgiventhesameeectasonethatoccurred10msinthepast.Thisisdisadvantageoussincespikesoccurring80msinthepasthaveasmallercontributiontothemembranepotentialthanthoseoccurring10msinthepast.Thedierenceintheeectonthemembranepotentialbetweenspikesthatoccurredat10msand15msismuchlargerthanthethedierenceintheeectonthemembranepotentialbetweenspikesoccurringat80msand85ms.Forthisreason,spikesoccurringfurtherinthepastshouldnothaveasmuchofaneectonthekernelresult.TheREEKaccommodatesforthisissue.InFigure 6-2 Ayoucanseethatthosespikeswhichoccurmorerecentlyaregivenalargerweightingthanthosewhichoccurfurtherinthepast.Inaddition,therelativeformoftheREEKmatchesthatofaPSP.ThisiswhytheREEKperformsbetterthantheGSKwhenmodelinganeuronusingtheclassicationframework.ItisabletoconstructaccuratereproductionsofthePSPsandAHPsforagivenneuronwithfewersupportvectors. 6.3ConductanceKernelTheconductancekernelisanextensiontoeitherofthetwopreviouskernels.ItassumesthatsomestereotypedPSPhasalreadybeenlearned.ThiscanbedonewitheithertheGSKortheREEKformulation.TheconductancekernelisthenabletousetheformofthestereotypedPSPtolearnanyscalingdierencesbetweenitandtheeectanotherspikehasonthemembranepotential.TheconductancekernelevaluatedfortpandtqinthetwodimensionalcaseareshowninFigure 6-3 .InFigure 6-3 A,wesettp=(1;1)andvarytq.InFigure 6-3 B,wesettp=(10;10).Fromtheseimagesweseethattheconductancekernelproduceslargevalueswhenprovidedlargeinterspikeintervalsandsmallvaluesforsmallinterspikeintervals.ThiscoincidestoadecreaseinthemagnitudeofthestereotypicalPSPasthesizeofinterspikeintervaldecreases.Inaddition,therateofdecreaseissimilartothatmodeledbythedictionaryfunctions. 70

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6.4KernelComparisonAsshownintheabovesections,theREEKismuchbettersuitedtolearnamembranepotentialfunctionthanGSK,whichinturnisbettersuitedtolearnthemembranepotentialfunctionthantheGRBF.Herewediscussresultswhenmodelingvariousadditivelyseparableneuronmodels.Themodelsusedincludeaneuronwithasingleinputsynapse,aneuronwithmultipleinputsynapsesthatdoesnotgeneratespikes,andthenaneuronhavingmultipleinputsynapsesthatcangeneratespikes.WethenconsideraconductancebasedneuroninordertocomparetheconductancekernelandtheREEK.EachmodelistrainedasdiscussedinSection 2.3 .Forallmodels,weusedtheclassicalalphafunctiontomodelthePSP[PSP(t)=Ctexp()]TJ /F5 11.955 Tf 9.3 0 Td[(t=)]andanexponentialfunctiontomodeltheAHP[AHP(t)=Cexp()]TJ /F5 11.955 Tf 9.3 0 Td[(t=)].Thevaluesofthesynapticweightsandthefalltimes()werevariedinsomeofthemodels. 6.4.1SingleAerentSynapseInthecaseofthesinglesynapse,themembranepotentialisdeterminedbyaseriesofspikeswiththesamePSP.InFigure 6-4 weseethetruePSPusedinthemodelcomparedtoapproximatedPSPgeneratedbytheclassierfordierentkernels.Tocalculatetheclassicationmodel'sapproximatedPSPwearticiallysendasinglespikeacrosseachinputsynapseandthenarticiallygenerateaspiketoproducetheAHPapproximation.Byconsideringthedistancetothesinglespikedatapointfromtheclassier'smarginasthespikeages,wecangetascaledandtranslatedversionofthePSPandAHP.Theguresshowtheseapproximationsscaledandtranslatedbytheosetbandthethreshold,,oftheneuron.InFigure 6-4 AweseethetruePSPcomparedtotheapproximationproducedbytheREEKaswellastheapproximationproducedbytheGSKwhenparameterizedbyavalueof=1ms.FromthisgureweseethatboththeREEKandtheGSKwhenparameterizedby=1msproduceanaccurateapproximationofthetruePSP.InFigure 6-4 BweseethetruePSPcomparedtotheapproximationproduced 71

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bytheGSKwhen=5msand=25ms.HereweseethatalargevalueofdoesnotproduceanaccurateapproximationofthePSP.InadditiontoproducingtheapproximatedPSP,wealsodidananalysisofhowaccurateourmodelwasatpredictingwhentheneuronwouldcrossthethreshold.Werstdidasimplecalculationofaccuracycorrectclassications numberofpointsclassied,sensitivitycorrectpositiveclassications numberofpositivepointsandspecicitycorrectnegativeclassications numberofnegativepoints.Wherethepositivepointswerespiketrainsthatwouldcausethemembranepotentialtoexceedthethresholdandnegativepointswerethosespiketrainsthatwouldnotcausethemembranepotential.ThesevaluesareshowninTable 6-1 .Wealsocalculatedahistogramtoseehowfarothepredictedtimeofthresholdcrossingswere.Foranygiventhresholdcrossingintheneuron,wedeterminedhowclose,temporally,ourmodelwastopredictingathresholdcrossingatthistime.Wethencalculatedthefrequencyofeachtimedierence.Figure 6-5 showstwohistogramsdepictingthesecalculationsforeachkernel.Thelargerhistogramcontainspredictionswithtimedierencesvaryingbetween0and70ms,withabinsizeof1ms.Theinlaidhistogramrangesfrom0to10msandhasabinsizeof0.1ms.Bothusealogarithmicscaleonthey-axisduetothelargedierenceinthenumberofcorrectverseincorrectpredictions.Figures 6-5 A, 6-5 Band 6-5 CshowthehistogramswhenusingtheGSKwithparametersof=1ms,=5ms,and=25msrespectively.Figure 6-5 DshowstheresultinghistogramswhenusingtheREEK.Fromthesehistograms,weseethatthevastmajorityofthresholdcrossingswerepredictedcorrectly(withatimedierenceof0ms).Howeverthetimingofallmispredictedthresholdcrossingsfellwithin10msoftheactualtime.Fromtheseresultsitseemsthatbothmodelsdofairlywellwhenpredictingthresholdcrossingsforasingleaerentsynapse.However,ascanbeseeninTable 6-1 ,theREEKrequiresfewersupportvectors.AsshowninSections 4.1.2.1 and 3.1.2.1 ,onlyafractionofthetrainingpointsareusedtoproducetheoptimalclassier.Thesepointsareknown 72

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assupportvectorsandliealongtheedgeofthemargin.Thereforetheyrepresentthedimensionalityoftheclassifyingspace,andsofewersupportvectorsindicateasimplermodel.Asexpected,theREEKturnsouttoproduceverysimilarresultstotheGSK,butinamuchsimplerfashion. 6.4.2MultipleAerentSynapsesIntheneuronmodelwhichhasmultipleaerentsynapses,theformofthePSPcouldvarybetweenthesynapses.Inthisspecicexample,weonlyconsider2aerentsynapses,withdierentsynapticweights.InFigure 6-6 weseethePSPsofthemodelforeachsynapsecomparedtotheapproximatedPSPsgeneratedbytheclassierforthedierentkernels.InFigure 6-6 AweseethePSPforoneofthesynapsescomparedtotheapproximationsproducedbytheREEKandtheGSKwhenparameterizedbyavalueof=1ms.FromthisgureweseethattheREEKproducesanaccurateapproximation,buttheGSKparameterizedby=1doesnot.InFigure 6-6 BweseethetruePSPcomparedtotheapproximationproducedbytheGSKwhen=5msand=25ms.HereweseethatalargevalueofdoesnotproduceanaccurateapproximationofthePSPeither.SimilarcomparisonsaregiveninFigures 6-6 Cand 6-6 Dforthesecondaerentsynapse.Asinthesinglesynapsecase,wealsodidananalysisofhowaccurateourmodelwasatpredictingwhentheneuronwouldcrossthethreshold.Whencalculatingtheaccuracyandmodelcomplexity,wefoundthevaluesshowninTable 6-2 .FromtheseresultswecanseetheREEKisabletoproduceabettermodelthantheGSK.However,itdoesrequiremoresupportvectors.Thisispartiallyaresultofhavingamorecomplicatedneuron,howeveritcouldalsobeexplainedbythelackofaparametertotheREEK.Sincethereisnoparameter,theREEKwillapproximatethemembranepotentialusingPSPlikefunctionswithvaryingriseandfalltimes.TheGSKontheotherhandusesGaussianfunctionswithaxedstandarddeviation,thereforereducingthenumberoffunctionsthatcouldbeconsideredaspartofthedictionary. 73

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Inaddition,wealsocalculatedahistogramtoseehowfarothepredictedtimeofthresholdcrossingswereaswasdoneinthesinglesynapsecase.Figure 6-7 showstwohistogramsdepictingthesecalculationsforeachkernel.Thelargerhistogramcontainspredictionswithtimedierencesvaryingbetween0and70ms,withabinsizeof1ms.Theinlaidhistogramrangesfrom0to10msandhasabinsizeof0.1ms.Bothusealogarithmicscaleonthey-axisduetothelargedierenceinthenumberofcorrectverseincorrectpredictions.TheresultsseeninthehistogramsreectthoseseenintheaccuracyanalysisinTable 6-2 .WeseethattheREEKproducesabettermodelthantheGSK,however,themodeloftheGSK,producesanadequatemodeloftheneuron.Itcorrectlypredicted92%ofthedatapoints,andallthresholdcrossingwerepredictedwithin4msoftheactualthresholdcrossing.ThisseemscounterintuitivewhentheonelooksattherelativelypoorapproximationofthePSPsseeninFigure 6-6 .Thisdierencecouldbeexplainedbythesuppliedtrainingdata.Sincesubthresholdsubtletiesareignoredinthisformulation,partsofthePSPwhichwouldonlycontributetosubthresholdvaluescouldbeanythingaslongasthemembranepotentialstaysbelowthethreshold.However,amorecomprehensivesetoftrainingdatacouldleadtocaseswherethetimingofthecurrentsubthresholdpointsleadtocombinationswherethemembranepotentialdoescrossthethreshold. 6.4.3MultipleAerentSynapseswithEerentSpikesInthisversion,weincreasedthecomplexityoftheneuronbyintroducingAHPeectsaswellasdierenttypes(excitatoryandinhibitory)ofaerentsynapseswithvaryingsynapticweightsandfalltimes.Ithad4excitatoryinputspiketrainsand1inhibitoryinputspiketraintomimictheratioofconnectionsseeninthecortex( Izhikevich 2003 ).Theneuronalsogeneratedanoutputspiketrain.Theweightofeachinputspiketrainwasvariedbyasynapticweightandtheinhibitoryandexcitatoryspiketrainsdieredintheirriseandfalltimes.FortheexcitatoryPSP,C=0:1and=10,wheretisinunitsof 74

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milliseconds.FortheinhibitoryPSP,C=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:39and=5andfortheAHP,C=)]TJ /F1 11.955 Tf 9.3 0 Td[(16:667and=2.Whenlearningthismorecomplexmodel,theREEKwastheonlyonetoconvergeinareasonableamountoftime.Therefore,wedonotprovideresultsfortheGSK.WhenusingtheREEK,wefoundthatthemodelhadanaccuracyof99.474%,asensitivityof95.322%andaspecicityof99.479%.Themodelrequired10,682supportvectors.Fromthenumberofsupportvectors,weseethatthemodelproducedismuchmorecomplexthanthepreviouslydiscussedmodels.Thisisalsoreectedbythedecreasedsensitivityofthemodel.InFigure 6-8 weseetwohistogramsdepictingthetemporalproximityofthepredictedthresholdcrossingsandthetruethresholdcrossings.Inthismodel,sincethereisaneerentsynapse,thisisequivalenttopredictingthetimingofaspike.Thelargerhistogramcontainspredictionswithtimedierencesvaryingbetween0and70ms,withabinsizeof1ms.Theinlaidhistogramrangesfrom0to10msandhasabinsizeof0.1ms.Bothusealogarithmicscaleonthey-axisduetothelargedierenceinthenumberofcorrectverseincorrectpredictions.Fromthehistograms,weseethatthevastmajorityofspikeswerepredictedcorrectly(withatimedierenceof0ms).Howeveroutofthemispredictedspiketimes,thetimingofallspikesfellwithin70msoftheactualspiketime.InFigure 6-9 weseethecomparisonoftheapproximatedPSPsandAHPversusthetruePSPsandAHPfortheREEK.Figure 6-9 AshowstheEPSPsinblackandthetheEPSPapproximationsingreenforthefourexcitatorysynapseswithvaryingsynapticweights.Thelinestylesareindicativeofaparticularsynapse.Figure 6-9 BshowstheIPSPinblackanditsapproximationingreenandFigure 6-9 CshowstheAHP(black)anditsapproximation(green). 6.4.4ConductanceBasedSynapseInadditiontousingneuronmodelswhichassumeallspikesonasynapsehadthesameeect,wealsousedamodelwhichwouldscaletheeectofaspikebasedonthetimingofthepreviouslysentspike.Inordertoeliminatethescalingeectsofspikesthat 75

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occurredbeforethepreviousspike,wemanuallyimplementedthescalingeectsasafunctionoftheinterspikeintervalbetweenthespikeinquestionandthepreviousspikeusingcurvessimilartothoseshowninFigure 5-2 .Figure 6-10 showsthestereotypicalPSPcomparisonaswellasthescalingfunctioncomparisonproducedwiththeconductancekernel.Figure 6-10 AshowsthelearnedPSPincomparisontotheactualPSPwhenusingtheREEKbasedconductancekernel.Inordertoproducethis,wegeneratespiketraindatainwhichtheinterspikeintervalislargeenoughsothatnoscalingeectsareobserved.WethenusethelearnedstereotypicalPSPintheconductancekernelmodeltolearnthescalingfunction.Indoingthese,weconsiderdatathatcontainsinterspikeintervalsofanysize.Figure 6-10 Bshowstheresultingscalingfunctionwhentheclassieristrainedinthismanner.WecomparedtheconductancekerneltotheREEKforaconductanceneuron.Whendoingananalysisoftheaccuracy,sensitivityandspecicity,wendtheresultsshowninTable 6-3 .WenoticethattheconductancekerneldoesmuchbetterthantheREEK.InadditionitusesfewersupportvectorssinceithasaccesstothestereotypicalPSP.ThereisnoneedforittolearntheformofthePSP,justthescalingeects. 6.5SummaryAsshownabove,duetotheGSK'sadditivenature,itprovidesabettermethodtolearnamembranepotentialfunctionthantheGRBF.Also,duetotheREEK'sabilitytotakethetimingofaspikeinadditiontothedistancebetweentwospikes,theREEKisabetterkernelforlearningmembranepotentialfunctionsthantheGSK.ThisisseenintheabovesimulationswheretheREEKconsistentlyproducesequivalentorbetterpredictionresultswithfewersupportvectors.However,theREEKalsohasitshortfallswhenscalingeectsareconsidered.Forthisreason,weproducedtheconductancekernel.InthecomparisonofPSPs/AHPsandthereapproximation,thereaderwillnoticethatsometimesslightdierencesbetweentheapproximationandthetruefunctionsexist.Thisiswhythemodeldidnotclassifyeverydatapointcorrectly.IfthePSPand 76

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AHPapproximationswereexact,thenwewouldhaveseenperfectclassicationresults.However,aswithmostmachinelearningtechniques,wearelimitedbythetrainingdatawhichisgiven.Theclassierproducesasolutionwhichbothperformswellonthegivendataandgeneralizestounseendata.Therefore,thenalsolutionwillbelimitedbythedataprovidedduringthetrainingphase.Inaddition,wewouldalsoliketopointoutthattheAHPwasapproximatedwithaPSPlikefunctioninthemodeloftheneuronwhichimplementedaneerentspiketrainshowninFigure 6-9 C.Whentheclassierproducesasolutionitisonlyabletoutilizethedataprovidedinthetrainingstage.Thereforeifnodataisgivenat0ms,thenitisnotabletoproduceavalidapproximationforthattimeperiod.Therefore,asisthecasethecasehere,itproducesasolutiononlyusingthespiketimesgiven.ThesolutionproducedusingtheformofthePSPmusthavebeenadequatetoapproximatetheAHPforthegivendata. Figure6-1. ComparisonoftheGRBFtotheGSK,inthetwodimensionalcase.A)showstheGRBFevaluationasonespiketrainissetto)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(50;50andtheotherisalteredineitherdimension.B)showsthesame,butfortheGSK.FromthetwowecanseehowtheGSKallowsfortheadditivenatureofPSPsandAHPsbyindicatingsimilarityinspiketrainswhicharedierentbyonlyonespiketime. 77

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Figure6-2. ComparisonoftheREEKtotheGSK,inthetwodimensionalcase.A),showstheresultsoftheREEKevaluationwhentpissetto)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(5;20)andtqvaries.Weseethereektakesthetimingofthespikesintoaccountwhendeterminingthereeectonthemembranepotential.B)showstheresultsoftheGSKevaluationwhentp=(10;50).WhenwecompareB)toFigure 6-1 B,weseethattheGSKusesthesamemeasureofsimilarityregardlessofthetimingofthespikes. Table6-1. Accuracy,sensitivity,specicityandmodelcomplexityvalueswhenmodelinganeuronwithasingleaerentsynapseforvariouskernels. ModelAccuracySensitivitySpecicitySupportVectors GSK=1ms99.99699.88799.997213GSK=5ms99.98999.99399.989503GSK=25ms99.69899.34199.703961REEK99.98999.99399.98929 Table6-2. Accuracy,sensitivity,specicityandmodelcomplexityvalueswhenmodelinganeuronwith2aerentsynapseswithdierentsynapticweights. ModelAccuracySensitivitySpecicitySupportVectors GSK=1ms99.947100.0099.94210,277GSK=5ms99.94999.98699.9457,266GSK=25ms98.32793.74798.7463,845REEK99.94799.87499.9532,426 Table6-3. Accuracy,sensitivity,specicityandmodelcomplexityvalueswhenmodelingaconductancebasedneuronwiththeREEKandconductancekernel. ModelAccuracySensitivitySpecicitySupportVectors ConductanceREEK98.9999.6298.7710REEK87.2477.6090.6948 78

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Figure6-3. Theconductancekernelfortwodimensionalinterspikeintervals.Theconductancekernelevaluatedfortpandtqinthetwodimensionalcaseareshown.InA),wesettp=(1;1)andvarytq.InB),wesettp=(10;10).Fromtheseimagesweseethattheconductancekernelproduceslargevalueswhenprovidedlargeinterspikeintervalsandsmallvaluesforsmallinterspikeintervals. Figure6-4. PSPcomparisonforasinglesynapse.ThetruePSPusedinthemodelcomparedtoapproximatedPSPgeneratedbytheclassierfordierentkernelswhenmodelingasinglesynapse.A)showstruePSPcomparedtotheapproximationproducedbytheREEKaswellastheapproximationproducedbytheGSKwhenparameterizedbyavalueof=1ms.B)showstruePSPcomparedtotheapproximationproducedbytheGSKwhen=5msand=25ms. 79

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Figure6-5. Histogramcomparisonforasinglesynapse.Histogramsdepictingthetemporalproximityofpredictedthresholdcrossingtothetimetheneuron'smembranepotentialactuallycrossedthethresholdwhenusingasingleaerentsynapse.Thelargerhistogramcontainspredictionswithtimedierencesvaryingbetween0and70ms,withabinsizeof1ms.Theinlaidhistogramrangesfrom0to10msandhasabinsizeof0.1ms.Bothusealogarithmicscaleonthey-axisduetothelargedierenceinthenumberofcorrectverseincorrectpredictions.A),B)C)showthehistogramswhenusingtheGSKwithparametersof=1ms,=5ms,and=25msrespectively.D)showstheresultinghistogramswhenusingtheREEK. 80

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Figure6-6. PSPcomparisonforaneuronwithmultipleaerentspiketrains.ComparisonofPSPsandtheirapproximationsforeachsynapseandforeachkernelusedforthe2aerentsynapsemodel.A)showsthePSPforoneofthesynapsescomparedtotheapproximationsproducedbytheREEKandtheGSKwhenparameterizedbyavalueof=1ms.B)givesthetruePSPcomparedtotheapproximationproducedbytheGSKwhen=5msand=25ms.SimilarcomparisonsaregiveninC)andD)forthesecondaerentsynapse. 81

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Figure6-7. Histogramcomparisonforaneuronwithmultipleaerentspiketrains.Histogramsdepictingthetemporalproximityofpredictedthresholdcrossingtothetimetheneuronactuallycrossedthethresholdforthe2aerentsynapsemodel.Thelargerhistogramcontainspredictionswithtimedierencesvaryingbetween0and70ms,withabinsizeof1ms.Theinlaidhistogramrangesfrom0to10msandhasabinsizeof0.1ms.Bothusealogarithmicscaleonthey-axisduetothelargedierenceinthenumberofcorrectverseincorrectpredictions.A),B)C)showthehistogramswhenusingtheGSKwithparametersof=1ms,=5ms,and=25msrespectively.D)showstheresultinghistogramswhenusingtheREEK. 82

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Figure6-8. Histogramcomparisonforaneuronwithaerentandeerentspiketrains.Histogramsdepictingthetemporalproximityofthepredictedspiketimeandthetruespiketimeforspikesgeneratedbythemodeledneuron.Thelargerhistogramcontainspredictionswithtimedierencesvaryingbetween0and70ms,withabinsizeof1ms.Theinlaidhistogramrangesfrom0to10msandhasabinsizeof0.1ms. Figure6-9. PSPandAHPcomparisonforaneuronwithaerentandeerentspiketrains.A)showstheEPSPsinblackandthetheEPSPapproximationsingreenforthefourexcitatorysynapseswithvaryingsynapticweights.B)showstheIPSPinblackandthetheIPSPapproximationingreenfortheinhibitorysynapseandC)showsthetrueAHP(black)anditsapproximation(green).Linestylesareindicativeofcorrespondingsynapses. 83

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Figure6-10. StereotypicalPSPandscalingcomparisonproducedbytheconductancekernel.A)showsthelearnedPSPincomparisontotheactualPSPwhenusingtheREEKbasedconductancekernelondatawithlargeinterspikeintervals.B)showstheresultingscalingfunctionwhentheclassieristrainedusingtheconductancekernelandthelearnedstereotypicalPSP. 84

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CHAPTER7CONCLUSIONSANDFUTUREWORKModelingandquantitativelydescribingaspikingneuronisanessentialeldofresearchinneuroscience.Inthisdissertation,wehaveproposedaframeworkwhichusesaclassiertolearntheresponseofamodeledneurontothetimingsoftheneuron'saerentandeerentspikes.Ashasbeenpreviouslyshown( Banerjee 2001 ),themembranepotentialfunctionforaniteprecisionneuronwithfadingmemoryisdependentuponthetimingoftheaerentandeerentspikeswithinanitepast.Weshowthat,althougharegressionproblemistheintuitivewaytolearnanapproximationtothetruemembranepotentialfunction,theregressionproblemthiscreatesisillposed.Withthisknowledge,weproduceaframeworkwhichusesaclassiertolearnthemembranepotentialfunctionaneuronbasedoitsaerentandeerentspiketimes.Weclassifyspiketrainswhichareabouttoproduceaspike(yieldamembranepotentiallessthanthethreshold)asthenegativeclassandspiketrainswhichhavejustproducedaspike(yieldamembranepotentialgreaterthanthethreshold)asthepositiveclass.Sincewedonothaveaccesstothemembranepotentialfunction,wetrytoapproximateitusingadictionaryoffunctions.WewouldliketondadictionarywhichmostcloselyapproximatesthemembranepotentialfunctionP.Withtwospecicdictionaries,wehavederivedaframeworkwhichapproximatesthemembranepotentialfunctionusingakernelderivedfromeachdictionary.OurformulationthenproducesaquadraticprogrammingproblemusingthederivedkernelwhichwesolveusingSVMLight( Joachims 1999 ).Therstkernelwediscussedwasderivedspecicallyfortheproblemoflearningthemembranepotentialfunction.Itwasformedfromafamilyoffunctionsusedinneuronmodelingby MacGregor&Lewis ( 1977 ).Thisreciprocalexponential-exponentialkernel(REEK)isabletogiveavaliddistancebetweentwospikeswhiletakingintoaccountthetimingofthespikessothatmorerecentspikesareconsideredtohavemoreofaneectonthemembranepotential. 85

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Wealsoderivedanddiscussedamoregeneralkernelforcomparisonpurposes.ItutilizesasummationofGaussianfunctionstoreproducethemembranepotentialfunction.ThiswasamoresuitablekernelthantheGRBFbecauseitsadditivenaturewouldbeabletolearntheadditivelyseparableeectsofspikesonthemembranepotential.Thiskernelwassuccessfulinrecreatingthemembranepotentialfunctionsinbasicmodels,however,itfailedtoconvergetoasolutioninmorecomplexmodels.TheGaussiansummationkernel(GSK)doesnottakeintoaccountthetimingofthespikeswhencomparingtwospiketrains.Itwouldonlyconsiderthedierenceinspiketimes.HoweverduetotheexponentialdecayoftheeectsofPSPsandAHPs,theeectsofmorerecentspikesarelargerthanthosethathaveoccurredfurtherinthepast,whichiswhytheREEKperformsbetter.Finallyweextendedourformulationtoconsiderscalingeects.Thepreviouskernelsmadetheassumptionsthateveryspikeonasynapsewouldhavethesameeect.Thisisnotalwaysthecase.Inanactualneuron,theeectofanaerentspikewillbedepreciatediftheneuronhasrecentlyreceivedanotherspike.Forthisreason,weproducedakernelwhichaccountsforscalingofaspike'seectonthemembranepotentialbyconsideringtheinterspikeintervalbetweenthatspikeandthepreviouslysentspike.ThiskernelusessomestereotypicalPSP,whichcanbelearnedwitheithertheREEKortheGSK.ItthenconsiderstheeectofagivenspiketobesomescaledversionofthatPSP.Nowthatthetheoreticalfoundationhasbeensetandwehaveshownthattheconceptoflearninganapproximationtothemembranepotentialfunctionfromspiketimesispossible,inthefuture,Iwouldlikeinvestigatehowwellthesekernelscandowhenappliedtomorecomplexneuronmodels.Therearemanyphenomenologicalandnon-phenomenologicalmodelswhichdonotspecicallyrespondtospiketimes( Hodgkin&Huxley 1990 ; Izhikevich 2003 ; Naudetal. 2008 ).Itwouldbeinterestingtoseetheaccuracyofthisframeworkinlearningsuchnon-additivelyseparablemodels.Inaddition,I 86

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wouldalsoliketoinvestigatethepossibilityofusingaprobabilisticframeworkratherthanaconvexoptimizationapproachtoseemoreecientsolutionscanbeproduced. 87

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REFERENCES Abbott,L.,&vanVreeswijk,C.(1993).Asynchronousstatesinnetworksofpulse-coupledoscillators.PhysicalReviewE,48(2),1483{1490. Banerjee,A.(2001).Onthephase-spacedynamicsofsystemsofspikingneurons.I:Modelandexperiments.NeuralComputation,13(1),161{193. Berlinet,A.,&Thomas-Agnan,C.(2004).ReproducingkernelHilbertspacesinprobabilityandstatistics.SpringerNetherlands. Connor,J.,&Stevens,C.(1971).Predictionofrepetitiveringbehaviourfromvoltageclampdataonanisolatedneuronesoma.TheJournalofPhysiology,213(1),31{53. Dayan,P.,Abbott,L.,&Abbott,L.(2001).Theoreticalneuroscience:Computationalandmathematicalmodelingofneuralsystems.MITPress. Destexhe,A.,Mainen,Z.,&Sejnowski,T.(1994).Anecientmethodforcomputingsynapticconductancesbasedonakineticmodelofreceptorbinding.NeuralComputa-tion,6(1),14{18. Gerstner,W.,&Kistler,W.(2002).SpikingNeuronModels:AnIntroduction.CambridgeUniversityPressNewYork,NY,USA. Gerstner,W.,&Naud,R.(2009).HowGoodAreNeuronModels?Science,326(5951),379. Hodgkin,A.,&Huxley,A.(1952).ThedualeectofmembranepotentialonsodiumconductanceinthegiantaxonofLoligo.TheJournalofphysiology,116(4),497. Hodgkin,A.,&Huxley,A.(1990).Aquantitativedescriptionofmembranecurrentanditsapplicationtoconductionandexcitationinnerve.BulletinofMathematicalBiology,52(1),25{71. Hormuzdi,S.,Filippov,M.,Mitropoulou,G.,Monyer,H.,&Bruzzone,R.(2004).Electricalsynapses:adynamicsignalingsystemthatshapestheactivityofneuronalnetworks.BBA-Biomembranes,1662(1-2),113{137. Izhikevich,E.(2003).Simplemodelofspikingneurons.IEEETransactionsonNeuralNetworks,14(6),1569{1572. Jerey,A.,&Dai,H.(2008).Handbookofmathematicalformulasandintegrals.AcademicPress. Joachims,T.(1999).Makinglarge-scalesupportvectormachinelearningpractical.InAdvancesinKernelMethods,(p.184).MITPress. Joachims,T.(2002).LearningtoClassifyTextUsingSupportVectorMachines:Methods,TheoryandAlgorithms.Springer,NewYork. 88

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Jolivet,R.,Lewis,T.,&Gerstner,W.(2000).TheSpikeResponseModel:AFrameworktoPredictNeuronalSpikeTrains.time(ms),2050(2100),2150. Jolivet,R.,Lewis,T.,&Gerstner,W.(2004).Generalizedintegrate-and-remodelsofneuronalactivityapproximatespiketrainsofadetailedmodeltoahighdegreeofaccuracy.JournalofNeurophysiology,92(2),959. Jolivet,R.,Roth,A.,Schurmann,F.,Gerstner,W.,&Senn,W.(2008a).Specialissueonquantitativeneuronmodeling.BiologicalCybernetics,99(4),237{239. Jolivet,R.,Schurmann,F.,Berger,T.,Naud,R.,Gerstner,W.,&Roth,A.(2008b).Thequantitativesingle-neuronmodelingcompetition.BiologicalCybernetics,99(4),417{426. Kimeldorf,G.,&Wahba,G.(1971).SomeresultsonTchebycheansplinefunctions*1.JournalofMathematicalAnalysisandApplications,33(1),82{95. Lapicque,L.(1907).Recherchesquantitativessurlexcitationelectriquedesnerfstraiteecommeunepolarisation.J.Physiol.Pathol.Gen,9,620{635. MacGregor,R.,&Lewis,E.(1977).NeuralModeling.PlenumPress,NewYork. Markram,H.(2006).Thebluebrainproject.NatureReviewsNeuroscience,7(2),153{160. Morris,C.,&Lecar,H.(1981).Voltageoscillationsinthebarnaclegiantmuscleber.BiophysicalJournal,35(1),193{213. Naud,R.,Marcille,N.,Clopath,C.,&Gerstner,W.(2008).Firingpatternsintheadaptiveexponentialintegrate-and-remodel.BiologicalCybernetics,99(4),335{347. Rudin,W.(1964).Principlesofmathematicalanalysis.McGraw-HillNewYork. Squire,L.,Bloom,F.,Mcconnell,S.,Roberts,J.,Spitzer,N.,&Zigmond,M.(2003).FundamentalNeuroscience. Stanisz,T.(1969).FunctionswithSeparatedVariables.Master'sthesis,ZeszytyNaukoweUniwerstyetuJagiellonskiego. Stein,R.(1967).SomeModelsofNeuronalVariability.BiophysicalJournal,7(1),37. VandePanne,C.(1975).Methodsforlinearandquadraticprogramming.North-Holland. Vapnik,V.,&Kotz,S.(2006).Estimationofdependencesbasedonempiricaldata.Springer-VerlagNewYorkInc. 89

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BIOGRAPHICALSKETCH NicholasFisherwasborninTallahassee,FloridabutwasprimarilyraisedinGraniteFalls,NorthCarolinaandBrownsburg,Indiana.AftergraduatinghighschoolhereceivedhisBachelorofScienceinengineeringcomputersciencefromtheUniversityofMichiganinMayof2003.NicholasthencontinuedhisstudiesingraduateworkattheUniversityofFloridainGainesville.HewasawardedaMasterofEngineeringincomputerengineeringinDecemberof2009andwillreceiveaDoctorofPhilosophyincomputerengineeringinAugustof2010. 90