Optimizing and Modeling Phase-Locked Deep Brain ...ruthcfong.github.io/files/ruth_fong_msc_dissertation...While conventional DBS has some parameters, such as the strength and frequency

OptimizingandModelingPhase-LockedDeepBrainStimulationtoSuppressTremorRuthFongSupervisor:ProfessorRafalBogaczMScNeuroscienceDissertation,HilaryTerm13April2016WordCount:10355words

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AbstractDeepbrainstimulation(DBS)effectivelysuppressestremorsinessentialtremor,dystonictremor,andParkinson’sdiseasepatientswhenadministeredascontinuous,high-frequencystimulationinthethalamus.However,adverseeffectsofsuchnearlyconstantstimulation,whichincludespeechandmotorimpairmentandhighpowerconsumption,motivatethesearchforsmarter,moreenergy-efficientDBSstrategies.Thisdissertationaimstoinvestigateonesuchstrategybasedonrecentresearch,whichsuggeststhattremorsuppressioninessentialtremorpatientsmaydependonthephaseoftremoratwhichlow-frequencystimulationisdelivered(Cagnanetal.,2013).Theworkpresentedinthisdissertationattemptstodemonstrateanoptimal,phase-lockedstimulationstrategythatmaximallysuppressesthetremorofasingleessentialtremorpatient.ItalsousesandextendstheKuramotoneuraloscillatormodel(Kuramoto,1984;Tass,2003;WilsonandMoehlis,2014)toshowthelimitationssuchamathematicalmodelhasinexplaininganoptimal,phasicstimulationstrategythatisconsistentwiththeonedemonstratedintheclinicalsetting.Twoadditionalcontributionsareoutlinedinthisdissertation:First,buildingofftheRayleightest(Mardia,1975),anovel,parameterizedcircularstatistictotestweightedcircularuniformityisdescribedandanalysed.Second,awebsimulationoftheproposedoscillatormodelispresentedasbotharesearchtoolandapublicengagementresource.

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TableofContentsAbstract..........................................................................................................................................................2TableofContents.......................................................................................................................................3TableofFigures..........................................................................................................................................5TableofTables............................................................................................................................................6TableofEquations.....................................................................................................................................7Introduction.................................................................................................................................................8Methods......................................................................................................................................................10ExperimentalDataCollection.......................................................................................................10ExperimentalDataProcessing.....................................................................................................13ExperimentalDataAnalysis..........................................................................................................14CalculatingChangeinAmplitudeandinFrequency......................................................14CalculatingthePhaseResponseCurve(PRC)...................................................................15SimilaritybetweenChangeinFrequencyandChangeinPhase...............................17

SignificanceTestingusingCircularStatistics........................................................................18RayleighStatistic...........................................................................................................................18Moore-RayleighStatistic............................................................................................................19ScaledRayleighStatistic.............................................................................................................20ComparisonofWeighted,CircularStatistics.....................................................................21SignificanceTestinginChangeofAmplitude,Frequency,andPhase....................21

KuramotoModel.................................................................................................................................22StimulationFunctionExtensiononTass,2003...............................................................24

OptimizedModelParameters.......................................................................................................25SimulatingPhase-LockedDBS......................................................................................................26ExtendedKuramotoModelOnlineSimulationTool&Tutorial.....................................28

Results.........................................................................................................................................................29AnalysisofExperimentalData.....................................................................................................29ChangeinAmplitude,Frequency,andPhaseCurves....................................................29SignificanceTesting.....................................................................................................................31CorrelationamongChangeinAmplitude,Frequency,andPhase............................33

AnalysisofModelledData..............................................................................................................33PredictingaConsistentTremorSuppressiveModel.....................................................33DefaultParametersforExtendedKuramotoModel......................................................36ResultsusingDefaultParameters..........................................................................................36ResultsusingDifferentStimulationFunctions................................................................43Non-LinearTransformationfromIndividualPRCtoChangeinMeanPhaseofOscillatorPopulation...................................................................................................................47ResultsusingExperimentalChangeinFrequencyandChangeinPhaseCurvesasPRCFunction.............................................................................................................................48

Discussion..................................................................................................................................................54SummaryofResults..........................................................................................................................54FutureWork.........................................................................................................................................55QuantifySignificant,Phase-SpecificTremorModulation............................................55ModelNegativeCorrelationbetweenChangeinTremorAmplitudeandFrequency/Phase..........................................................................................................................55

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DescribePhase-dependentDBSStrategyforDifferentTremulousPathologiesandPatients.....................................................................................................................................57

Conclusion..................................................................................................................................................59References.................................................................................................................................................60Appendix1:AlternativeMetricsofEvaluatingEffectsofTremorSuppression..........63ChangeinAmplitude........................................................................................................................63ChangeinFrequency&Phase......................................................................................................63AlternativeMetrics............................................................................................................................64'subtract_last_block'.....................................................................................................................64'subtract_last_block_norm'........................................................................................................65'point_normalize'...........................................................................................................................65'simple_median'.............................................................................................................................65'normalize'.......................................................................................................................................65'subtract_prev'................................................................................................................................65'subtract_prev_norm'...................................................................................................................65'regression'......................................................................................................................................65'poly2_latent'...................................................................................................................................65'poly2_diff'........................................................................................................................................65'prc_point_normalize'..................................................................................................................66'prc_first_last_median'................................................................................................................66

Appendix2:SimulatedChangeinPhaseCurveswithDifferentNumbersofOscillators..................................................................................................................................................67Appendix3:DifferentStimulationFunctions.............................................................................70Appendix4:ExtendedKuramotoModelSimulationTool....................................................75

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TableofFiguresFigure1:AnExperimentalBlockwithPhasicStimulationatphase240°.....................11Figure2:ExperimentalDataforWholeTrial.............................................................................12Figure3:FirstsecondofanExperimentalBlockwithPhase-lockedStimulationat

240°.....................................................................................................................................................13Figure4:EpochsUsedtoCalculateMetricforChangeinAmplitudeandFrequency15Figure5:UnwrappedPhaseoftheTremorAmplitudeforanExperimentalBlock...17Figure6:ExamplePhasesPlottedAlongUnitCircle...............................................................18Figure7:ExamplePhaseswithDifferentRadiiorWeights.................................................19Figure8:VisualizationofNeuronalOscillators.........................................................................22Figure9:SimulatedBlockwithPhasicStimulationat240°usingDefaultParameters

...............................................................................................................................................................27Figure10:ChangeinAmplitude,Frequency,andPhaseofExperimentalData..........30Figure11:VisualizationsofMoore-RayleighTestforNegativeChangeinAmplitude,

ChangeinFrequency,andChangeinPhase......................................................................32Figure12:CosineCurveasIndividualNeuron’sPhaseResponseCurve(PRC)..........34Figure13:DiagramoftheOptimalStimulationStrategyfortheKuramotoModel...35Figure14:DefaultPRCfunction,𝒛𝜽𝒊 = 𝐜𝐨𝐬(𝜽𝒊 + 𝝅𝟔),forExtendedKuramotoModel

...............................................................................................................................................................35Figure15:ChangeinAmplitude,Frequency,andPhaseofSimulatedDatausing

DefaultParameters......................................................................................................................38Figure16:VisualizationsofMoore-RayleighTestforNegativeChangeinAmplitude,

ChangeinFrequency,andChangeinPhaseofSimulatedDatausingDefaultParameters.......................................................................................................................................40

Figure17:ChangeinAmplitudeCurvesforSimulatedDatawhenusing“Half”,“Random”,and“Mixture”Stimulationfunctions.............................................................47

Figure18:MeanSquaredErrorbetweenDefaultPRCFunctionandMedianChangeinPhaseinSimulatedDataasNumberofOscillatorsIncreases..............................47

Figure19:Scaled,ExperimentalChangeinFrequencyCurveasPRCFunction𝒛(𝜽𝒊)forExtendedKuramotoModel................................................................................................49

Figure20:ChangeinAmplitude,Frequency,andPhaseCurvesofSimulatedDatausingScaled,ExperimentalChangeinFrequencyCurve(Figure19)asPRCFunction............................................................................................................................................50

Figure21:Scaled,ExperimentalChangeinPhaseCurveasPRCFunction𝒛(𝜽𝒊)forExtendedKuramotoModel.......................................................................................................51

Figure22:ChangeinAmplitude,Frequency,andPhaseofSimulatedDatausingScaled,ExperimentalChangeinPhaseCurve(Figure21)asPRCFunction.......52

Figure23:ChangeinPhaseCurveforSimulatedDatawhenDifferentNumbersofOscillators(N=2,5,8)...............................................................................................................68

Figure24:ChangeinFrequencyCurvesforSimulatedDatawhenusing“Half”,“Random”,and“Mixture”Stimulationfunctions.............................................................71

Figure25:ChangeinPhaseCurvesforSimulatedDatawhenusing“Half”,“Random”,and“Mixture”Stimulationfunctions....................................................................................73

Figure26:ScreenshotofSimulationTool....................................................................................76

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TableofTablesTable1:p-valuesandResultPhasesfromMoore-RayleighandScaledRayleigh

SignificanceTestingofChangeinAmplitude,Frequency,andPhaseofExperimentalData........................................................................................................................31

Table2:CorrelationbetweenChangeinAmplitude,Frequency,andPhaseofIndividualExperimentalBlocks.............................................................................................33

Table3:p-valuesandResultPhasesfromMoore-RayleighandScaledRayleighSignificanceTestingofChangeinAmplitude,Frequency,andPhaseofSimulatedDatausingDefaultParameters...............................................................................................39

Table4:CorrelationsbetweenMedianChangeinAmplitude,Frequency,andPhaseofExperimentalDataandthoseofSimulatedDatausingDefaultParameters..41

Table5:CorrelationbetweenChangeinAmplitude,Frequency,andPhaseofIndividualExperimentalBlocksandIndividualSimulatedBlocksusingDefaultParameters.......................................................................................................................................41

Table6:CorrelationsamongMedianChangeinAmplitude,Frequency,andPhaseaswellasSelectDerivativeApproximationsofExperimentalandSimulatedBlocksusingDefaultParameters..........................................................................................................42

Table7:CorrelationsbetweenMedianChangeinAmplitude,Frequency,andPhaseofExperimentalDataandthoseofSimulatedDatausingDifferentStimulationFunctions..........................................................................................................................................43

Table8:CorrelationsamongMedianChangeinAmplitude,Frequency,andPhaseaswellasSelectDerivativeApproximationsofExperimentalandSimulatedBlocksusingDefaultParameters..........................................................................................................44

Table9:StatisticalSignificanceofAlternativeMetricsMeasuringChangeinAmplitudeinExperimentalData............................................................................................63

Table10:StatisticalSignificanceofAlternativeMetricsMeasuringChangeinFrequencyandPhaseinExperimentalData.....................................................................64

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TableofEquationsEquation1:AnalyticSignalofFilteredTremorSignal...........................................................13Equation2:TremorAmplitudefromAnalyticalSignal(Equation1)..............................14Equation3:InstantaneousFrequency...........................................................................................14Equation4:MetricforChangeinAmplitudeandinFrequency.........................................14Equation5:Actual,UnwrappedPhaseofExperimentalBlock...........................................15Equation6:Reference,UnwrappedPhaseofExperimentalBlock....................................15Equation7:1st-OrderTaylorApproximationofPhaseAroundt=0...............................16Equation8:kth-OrderTaylorApproximationofPhaseAroundt=0...............................16Equation9:1st-OrderTaylorApproximationofActual,UnwrappedPhaseof

ExperimentalBlock......................................................................................................................16Equation10:ApproximationofFrequencyasaResultof5-secondPhase-Lock

Stimulation.......................................................................................................................................16Equation11a-b:MetricforChangeinPhase...............................................................................16Equation12a-c:RayleighStatistic...................................................................................................18Equation13a-c:Moore-RayleighStatistic....................................................................................20Equation14a-c:ScaledRayleighStatistic....................................................................................20Equation15:Z-Score.............................................................................................................................21Equation16:UpdateRuleforExtendedKuramotoModel...................................................22Equation17a-c:OrderParametersforKuramotoModel......................................................23Equation18:Tass,2003’sStimulationFunction......................................................................23Equation19:ModelingTremorAmplitude.................................................................................24Equation20:IndicatorFunctionforStimulation......................................................................24Equation21:GeneralformforStimulationFunction.............................................................24Equation22:UniformStimulationEffectFunction..................................................................25Equation23a-b:RandomStimulationEffectFunction...........................................................25Equation24:HalfStimulationEffectFunction..........................................................................25Equation25:MixtureStimulationEffectFunction...................................................................25Equation26:ProposedExtensiontoKuramotoModel’sUpdateRule............................57

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IntroductionTremorsareacommon,debilitatingsymptomofseveralmotordisorders.Continuous,high-frequencydeepbrainstimulation(DBS)inthethalamushasbeenahighlyeffectivetreatmentforsuppressingtremorsinpatientswithParkinson’sdisease,essentialtremor,anddystonictremor(Benabidetal.,1991).However,DBSalsodisruptsphysiologicalmotorfunction(Chenetal.,2006;Rayetal.,2009)andcancauseadverseeffectslikespeechimpairmentsandgaitinstability(Zhangetal.,2010;Baizabal-Carvalloetal.,2014).Thecontinuous,energy-consumingnatureofconventionalDBShasanadditionaltollinrequiringthereplacementofDBSimplantbatterieseveryfewyears.Together,theseeffectsmotivateresearchforamoreefficientDBSparadigmthatreapsthesametremor-suppressingbenefitsoftypicalDBSyetstimulatethethalamussparinglyandsmartly.WhileconventionalDBShassomeparameters,suchasthestrengthandfrequencyofstimulation,thatcanbeoptimallytuned,typical,continuous,high-frequencystimulationdoesnotchangeonceparametersareset.Recently,aclosed-loopparadigmknownasadaptiveDBS(aDBS)onlyadministersstimulationwhenathresholdlevelofbetaoscillations–neuronsfiringwithinthebetafrequencyrangeof13-30Hz–inthebasal-gangliaisexceeded(LittleandBrown,2012;Littleetal.,2013,2015;Rosaetal.,2015).Highlevelsofbetaoscillationsinthebasal-gangliacorticalloophavebeenlinkedtoParkinsoniansymptomssuchaslimbrigidityandbradykinesia,aconditionofslownessofmovement,yettheyhavenotbeenassociatedwithtremor(Hammondetal.,2007).Thus,currentaDBSparadigmsarelimitedtotreatingnon-tremor,Parkinsonian-specificsymptoms.Recentworksuggeststhepotentialforanalternative,tremor-suppressingaDBSparadigmbytimingstimulationtoaspecificphaseoftremor,thatis,aparticularpointinatremoroscillation.Thisresearchshowsatremor-suppressingeffectwhenthephaseoflow-frequencyDBSbecomessynchronizedwiththephaseofhandtremorintremulousessentialtremor(Cagnanetal.,2013)andParkinson’sdiseasepatients(Azodi-AvvalandGharabaghi,2015).Insuchpatients,tremoroscillationsoccurataround4-6Hz,whichisalsothecharacteristicoscillationfrequencyincertainbasalgangliaregionsoftremulouspatients(Hirschmannetal.,2013).Ithasbeenshownthatthalamicneuralactivitysynchronizedtotremorfrequencydrivestheautomatic,repetitivemovement(Hua,2004).Thus,itmaybeplausiblethatstimulatingthethalamusaccordingtotremorphasemaydesynchronizetheneuralactivityresponsiblefortremulousbehavior(BeudelandBrown,2015).Acomputationalmodel(Tass,2003)describeshowDBScanbeincorporatedintoacommonneuraloscillatormodel,theKuramotomodel(Kuramoto,1984),tosimulatethesynchronyofthalamicneuronsandthedesynchronizingeffectofDBS.Arecentworkproposesanoptimal,close-loopedDBSstrategythatdependsonthephaseofanindividualneuronaloscillator(WilsonandMoehlis,2014).However,thiscomputationalresearchhasnotbeenusedtodetermineanoptimal,adaptiveDBSstrategybasedonthephaseoftremor.

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Thisdissertationusestremordatacollectedfromanexperimentaltrialthattimestheadministrationofshort,35msburstsofhigh-frequencyDBStoaspecifictremorphaseofanessentialtremorpatientthroughoutadurationof5seconds(Cagnanetal.,2016).Usingthisexperimentaltremordata,thisdissertationdemonstratesthattremorphase-lockedDBSyieldsastatisticallysignificant,phase-dependentchangeintremoramplitudeandidentifiesanoptimalrangeoftremorphasesatwhichphase-lockedDBSbestsuppressestremorforthispatient.ThisworkalsopresentsseveralextensionstotheKuramotomodelandteststheextenttowhichthemodelcanexplaintheclinical,phase-dependenttremorsuppressionobservedintheessentialtremorpatient.Insummary,itishopedthatthisworkwillserveasasteppingstonetowardsfurtherresearchindevelopingnotonlyaclosed-loop,tremorphase-lockedDBSparadigmbutalsoabiologically-plausiblemodeltoexplainhowsuchstimulationeffectivelydampenstremor.

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MethodsExperimentaldatacollectionwasconductedbyDr.HayriyeCagnan(Cagnanetal.,2016),andtheoptimizationofmodelparameterswasdonebyDr.GihanWeerasinghe.Allothermethodsdescribedinthissectionweredesignedandcarriedoutbytheauthor.

ExperimentalDataCollectionTheexperimentaldataanalyzedinthisdissertationwascollectedwiththeapprovaloftheappropriate,localethicscommitteeandwiththeinformedconsentofallpatients(Cagnanetal.,2016).OneessentialtremorpatienthadfourDBSelectrodesinsertedintotheirleftventralateralthalamus;placementofelectrodeswasconfirmedusingCTandMRIimagingaftertheoperation.Anaccelerometerwiththreeaxeswasattachedtotheindexfingerofthepatient’srighthand.Minimal,onlineprocessingwasconductedontheacceleratoraxiswiththemosttremor.Withasamplingrateof10,417Hz,thisslightlyprocessedsignalfromthedominanttremoraxisisusedandanalyzedthroughoutthisdissertation.Thetremorsoftheessentialtremorpatientwererecordedwhenphase-locked,35msburstsof130HzDBSwereadministeredthroughouta5-secondblock(Figure1).Stimulationwasphase-lockedtotremor,whichwasmeasuredbythehand-mountedaccelerometer.Givenaphase𝜃forablock,wheneverthetremoroscillationpassedphase𝜃,a35msburstof130HzDBSwasadministered(Figure3).9blockswererecordedforeachofthefollowing12phases:0°,30°,60°,90°,120°,150°,180°,210°,240°,270°,300°,330°(withtheexceptionofthe180°phase,forwhichonly8blocksweresuccessfullyrecorded).Therewasatleast1secondofnostimulationbetweenblocks.

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Figure1:AnExperimentalBlockwithPhasicStimulationatphase240°

(1st row) The unfiltered tremor acceleration signal (m/s2)was recorded by a single hand-mountedaccelerometer.(2ndrow)Thetremorsignalfromthe1strowwasband-passfilteredbetween3Hzand6Hz;thetremoramplitude(m/s2)istracedinorangeandwasderivedfromtheanalyticalsignal.(3rdrow)Theinstantaneousfrequency(Hz)ofthetremoramplitudewasalso derived from the analytical signal. (4th row) 35 ms bursts of 130 Hz, phase-lockedstimulationwereappliedwheneverthetremoramplitudepassedthe240°phase.

Tremorswererecordedin9batches(Figure2).Ineachbatch,12blockswererecorded,oneforeachofthe12phases.Tomitigatetheeffectaparticularorderofphasicstimulationblocksmayhave,theorderofphaseblockswasrandomlypermutedforeachbatch.Inbetweenbatches,thepatientrestedtheirhandsfromtheextended,elevatedhandpositionmaintainedduringthebatches,resultinginnegligibletremoramplitudebeingrecorded(Figure2,seefilteredamplitudesignalduringt=1000-1200seconds).Theserestperiodsexplainthenoisyinstantaneousfrequencycalculatedinbetweenbatches.

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Figure2:ExperimentalDataforWholeTrial

(1st row) The unfiltered tremor signal (m/s2) was recorded by a single hand-mountedaccelerometer.(2ndrow)Theunfilteredtremorsignalwasband-passfilteredbetween3Hzand6Hz;thetremoramplitudeistracedinorange.(3rdrow)Theinstantaneousfrequency(Hz)ofthefilteredtremoramplitudewasderived.(4throw)Thephase(degrees)ofstimulationwassetaprioriviaarandompermutationforeachbatch.(5throw)35msburstsof130Hz,phase-lockedstimulationwereappliedwheneverthetremoramplitudepassedthephaseforagivenblock;phaseoftremorwascalculatedonlinethroughoutthetrialfromthefilteredamplitude.

Duringtheexperiment,tocalculatethephaseofthetremor,thesignalfromthedominanttremoraxisoftheunfilteredtremorwasband-passfilteredonlineto±2Hzaroundthetremorfrequency.Then,thetremorcycle’sphasewasestimatedfromthetremorfrequencyandthepreviouscycle’szerocrossingoftheonline,filteredsignal.Whenthetargetphasewasdetectedfromtheonline,filteredsignal,DBSisadministeredat130Hzfor35milliseconds;thiscorrespondstothedeliveryof6pulses(Figure3).Becausethefilteredsignalwascalculatedonline,itwasshiftedintimeandresultedinaphaseshift.Thisshiftwascorrectedforinthesubsequentdataprocessingandanalysisaswellasallpresentedfigures.

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Figure3:FirstsecondofanExperimentalBlockwithPhase-lockedStimulationat240°

(1st row) The unfiltered tremor signal (m/s2) was recorded by a single hand-mountedaccelerometer.(2ndrow)Theunfilteredtremorsignalwasband-passfilteredbetween3Hzand6Hz; the tremoramplitude(m/s2) is traced inorange. (3rdrow)Thephase(degrees)of thetremorwascalculatedonlinefromthefilteredtremorthroughoutthewholetrial.(4throw)A35msburstof130Hzstimulation,resultingin6pulses,wasappliedeverytimethetremoramplitudepassedphase240°.

ExperimentalDataProcessingAlldataprocessingandanalysiswasconductedusingcustomMATLABcode.Thetremorsignal,withanoriginalsamplingrate10,417Hz,wasresampledat1000Hz.Theresampledsignalwasthenband-passfilteredbetween3Hzand6Hzusingthesecond-orderButterworthfilter(Butterworth,1930).Theresampledandfilteredsignalisusedinallanalysesunlessotherwisementioned.Thetremoramplitude(Equation2)wascalculatedbytakingthecomplexmagnitudeoftheanalyticsignal(Equation1)ofthefilteredtremorsignal.

𝐴(𝑡) = 𝑓(𝑡) + 𝑖𝐻[𝑓 𝑡 ]Equation1:AnalyticSignalofFilteredTremorSignal

Theanalyticalsignalofthetremorconsistsoftheoriginalsignalofthefilteredtremor,𝑓(𝑡),andtheHilberttransformoftheoriginalsignal,𝐻[𝑓(𝑡)].

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𝑎(𝑡) = 𝑓(𝑡)9 + 𝐻[𝑓 𝑡 ]9

Equation2:TremorAmplitudefromAnalyticalSignal(Equation1)

Theinstantaneousfrequencywascalculatedbysmoothingthederivativeofthefilteredtremorsignal.First,thedifferencebetweeneach1millisecondtimestepofthetremoramplitudewascalculated;thisdifferencevectorwasthensmoothed.Finally,theinstantaneousfrequency,𝑓(𝑡),wascalculatedbyscalingthesmootheddifferencevector,𝑑(𝑡),bythesamplingrate,r=1000Hz,andby1/(2𝜋)toreflectthederivativeofthefilteredtremorsignal(Equation3).

𝑓(𝑡) =𝑟𝑑(𝑡)2𝜋

Equation3:InstantaneousFrequency

Thefrequencyofatimeperiodcanalsobecalculatedbycountingthenumberofzerocrossingsofthefilteredtremorsignal.Theresultsreportedinthisworkuseinstantaneousfrequency;however,someadditionalresultsincludedinAppendix1usedfrequencycalculatedviathezero-crossingsmethod.Qualitatively,resultsobtainedusingthetwodifferentmethodsforcalculatingfrequencyweresimilar.

ExperimentalDataAnalysisForeachblock,measuresofthechangeintremoramplitude,frequency,andphaserespectivelywerecalculatedinordertoanalyzetheeffectsofphase-lockedDBS.Themainmetricsaredescribedbelow;however,additionalmetricsweredevelopedandareincludedinAppendix1.

CalculatingChangeinAmplitudeandinFrequencyThesamemetricwasusedtocalculatechangeinamplitudeandchangeinfrequency(Equation4).Let𝑔> 𝑥, 𝑦 ,where𝑥, 𝑦 ∈ −1,5 seconds,denotethemedianofthetremoramplitudebetween𝑡 = 𝑥secondsand𝑡 = 𝑦secondsinablock.Correspondingly,let𝑔O 𝑥, 𝑦 denotethemedianoftheinstantaneousfrequency.Then,foragiven5secondblock,thedifferencebetweenthemedianoftheamplitude(orinstantaneousfrequency)ofthelastsecondoftheblock,𝑔P 4,5 ,andthatoftheprecedingonesecondperiodofnostimulation,𝑔P −1,0 ,wascalculated,where𝑚 ∈ 𝑎, 𝑓 denoteamplitudeorinstantaneousfrequency.

∆𝑚 =𝑔P 4,5 − 𝑔P(−1,0)Equation4:MetricforChangeinAmplitudeandinFrequency

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Figure4:EpochsUsedtoCalculateMetricforChangeinAmplitudeandFrequency

Themediansofthefilteredamplitude(1strow,inorange)andoftheinstantaneousfrequency(2ndrow)from4s-5sofablock(epochboundedinorange)andarecomparedtothatfrom-1sto 0s (epochbounded in blue), duringwhich there is no stimulation to compute change inamplitudeandinfrequency.

Forchangeinamplitudeandfrequency,thismetricwascalculatedforeachofthe107blocksandplottedinFigure10,1stand2ndrows,withthemedianofeachphaseplottedaswell.

CalculatingthePhaseResponseCurve(PRC)Let𝑔O 𝑥, 𝑦 bethemedianfunction𝑔definedintheprevioussectionforinstantaneousfrequency.Tocalculatethechangeinphase,areferencefrequency,𝑓U ,wascalculatedbytakingthemedianinstantaneousfrequencyoftheprecedingonesecondperiodofnostimulation,i.e.𝑓U = 𝑔O −1,0 .Then,for𝑡 = 0secondsto𝑡 = 5ofagivenexperimentalblock,theunwrappedphase,𝜃>(𝑡),inradians,ofthefilteredtremorsignal,𝑓(𝑡),wascalculated.

𝜃> 𝑡 = unwrap[atan2(𝐻[𝑓 𝑡 ]𝑓(𝑡) )]

Equation5:Actual,UnwrappedPhaseofExperimentalBlock

FromEquation1,𝐻[𝑓 𝑡 ]istheimaginarycomponentoftheanalyticsignal,computingusingthe Hilbert transform, of the filtered tremor signal,𝑓(𝑡) . The MATLAB unwrap functioncorrects forphase jumpsbyadding360°whenphase jumps from359° to0°.TheMATLABatan2functionwasusedinsteadofinversetanfunction,tan-1,inordertohandlediscontinuitiesat90°andat270°.

Then,accountingforthestartingphaseoftheblock,𝜃> 0 ,inradians,thereferenceunwrappedphasewascalculatedbyprojectingthereferencefrequency(Equation6).

𝜃U 𝑡 = 𝜃> 0 + 2𝜋𝑡𝑓U Equation6:Reference,UnwrappedPhaseofExperimentalBlock

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Theequationforreferencephase(Equation6)correspondstothefirst-orderTaylorapproximationaroundtheneighborhoodoft=0oftheactual,unwrappedphase(Equation7).Thisisbecausethederivativeofphaseisinstantaneousfrequencyscaledby2𝜋,asonecompleteoscillationcycleisequivalenttotravelling2𝜋or360°.Thus,𝜃>′ 0 isapproximatedby2𝜋𝑓U .

𝑃\ 𝑡 = 𝜃> 0 + 𝜃>′(0)𝑡Equation7:1st-OrderTaylorApproximationofPhaseAroundt=0

Thisfirst-orderapproximationsufficed,asthesecondderivativeofphasewasnearlyalways0;thus,furtherorderapproximations(Equation8)didnotsignificantlydifferfromthefirst-orderapproximation.

𝑃] 𝑡 = (𝜃>^ 0𝑖! 𝑡^)

]

^`a

Equation8:kth-OrderTaylorApproximationofPhaseAroundt=0

Thetime-evolutionoftheactualphasecanalsobeapproximatedusingafirst-orderTaylorexpansion,whichusesadifferentfrequency,𝑓> ,thatreflectstheactualfrequencyasaresultofthecurrentblock’sphase-lockedstimulation(Equation9).

𝜃> 𝑡 ≈ 𝜃> 0 + 2𝜋𝑡𝑓>Equation9:1st-OrderTaylorApproximationofActual,UnwrappedPhaseofExperimentalBlock

Then,aftersolvingfor𝑓>intheequationfortheapproximationof𝜃> 5 (Equation10),whichcorrespondstothefrequencyattheendofthe5-secondphase-lockedstimulationblock,themetricforchangeinphasecanbecalculatedasthedifferenceindegreesbetweentheactualfrequencyandreferencefrequency(Equation11a).Notethattheapproximationof𝑓>istheunwrappedphaseoftheblocknormalizedbytime,i.e.5seconds,andthelengthofacycle,i.e2𝜋(Equation10).

𝑓> ≈𝜃> 5 −𝜃> 0

2𝜋(5) Equation10:ApproximationofFrequencyasaResultof5-secondPhase-LockStimulation

∆𝜃 = 360°(𝑓> − 𝑓U)

∆𝜃 = 360°𝜃> 5 − 𝜃U 5

10𝜋

≈ (360°)𝜃> 0 + 2𝜋(5)𝑓> − (𝜃> 0 + 2𝜋 5 𝑓U)

10𝜋 = 360°(𝑓> − 𝑓U)Equation11a-b:MetricforChangeinPhase

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Alternatively,themetricforchangeinphasecanbecalculatedasthedifferenceindegreesbetweentheunwrappedactualphaseandunwrappedreferencephase(Equation11b,Figure5).

Figure5:UnwrappedPhaseoftheTremorAmplitudeforanExperimentalBlock

Theactualandrealunwrappedphases(θaandθrrespectively,inradians)areplottedforanexperimentalblockinwhich240°phase-lockedstimulationwasadministered.Thedifferencebetweentheactualandrealunwrappedphasesatt=5secondswasusedtocalculatethe

changeinphase(Equation11).Forchangeinphase,thismetricwascalculatedforeachofthe107blocksandplottedinFigure10,3rdrow,withthemedianofeachphaseplottedaswell.

SimilaritybetweenChangeinFrequencyandChangeinPhaseNotethatthechangeinphasemetricissimplyanotherwaytocalculatechangeinfrequency.Bothusethesamereferencefrequency,𝑓U = 𝑔O −1,0 ,thatisthemedian,instantaneousfrequencyofthesecondprecedingtheblockinwhichnoDBSwasadministered.Theonlydifferencesbetweenthemetricsaretheirunits–changeinfrequencyisgiveninHzwhilechangeinphaseisgivenindegrees–andthewaytheycomputethe“actual”frequencyasaresultofphase-lockedDBSforagivenblock.Thechangeinfrequencymetricapproximatesthe“actual”frequencyby

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Time (in sec)

-20

0

20

40

60

80

100

120

140

160

180

Unw

rapp

ed P

hase

Unwrapped Phase for Experimental Data Block #37 at phase 240 degrees

ReferenceActual

18

computingthemedian,instantaneousfrequencyofthelastsecondofagivenblock,i.e.𝑔O 4,5 (Equation4),whilethechangeinphasemetricapproximatesthe“actual”frequencyusingtheunwrappedphaseofagivenblock(Equation10).Table2showsthestrongcorrelationstrengthbetweenthechangeinfrequencyandchangeinphasemetrics.Thus,thetwometricsareapproximationsofthesamemeasure.

SignificanceTestingusingCircularStatisticsInordertotestwhethertherewasasignificanteffectinthechangeinamplitude,frequency,andphase,circularstatisticstestswereused.Inthissection,anexplanationoftherelevantstatistics–includingthepresentationofanovel,weighted,parametriccirculartest–willfirstbepresented,followedbyadescriptionofhowtheywereusedtotestsignificanceintheexperimentaldata.

RayleighStatistic

Figure6:ExamplePhasesPlottedAlongUnitCircle

TheRayleighcircularstatistic,𝑧U ,isanon-parametricmetricusedtotestwhetherasetofphasesissignificantlynon-uniform(Mardia,1975).TheRayleighstatisticcomputesthemeanvectorofallphasesplottedalongtheunitcircle(Figure6),whereRistheresultvector’smagnitudeand(X,Y)isitsCartesiancoordinate;thestatistic𝑧U isthesquareoftheradius,R,scaledbythenumberofphases,N(Equation12).Thisstatisticfailsonequallyspacedoutn-modaldata,suchasdiametricallybimodaldatawithclustersat0°and180°ortrimodaldatawithclustersat0°,120°,or240°,astheequally-spacedclusterswouldcanceleachotherout.

𝑋 =1𝑁 cos 𝜃i

j

i`\

, 𝑌 =1𝑁 sin 𝜃i

j

i`\

𝑅9 = 𝑋9 + 𝑌9𝑧U = 𝑁𝑅9

Equation12a-c:RayleighStatistic

𝜃

19

AcomputedRayleighstatisticcanthenbecomparedagainstap-valuetableindexedbythenumberofelements,N(Zar,2010).Alternatively,anulldistributioncanbegeneratedbyrepeatedlyuniformlysamplingNphasestoformasetofphasesandcomputingtheRayleighstatisticonsuchrandomlygeneratedsets.Then,anempiricalcumulativedistributionfunction(CDF)canbecomputedonthenulldistributionbysimplyorderingitsRayleighstatisticsandathresholdzvaluecanbefoundforaparticularp-value.Thisistheprocedurebywhichthestandardp-valuetablefortheRayleighstatisticwasgenerated(Zar,2010).

Moore-RayleighStatisticHowever,theRayleighstatisticfailstoweightphases.Becausethephasesareplottedalongtheunitcircle,theyallassumearadiusorweightof1.Inordertotestthesignificanceofthechangeinamplitude,frequency,andphaseofexperimentaldata,aweightedcircularstatisticisrequired,wheretheradiusofeachdatapointcorrespondstooneoftheabovemetricsanditsphaseisthatdatapoint’scorrespondingphase(Figure7).

Figure7:ExamplePhaseswithDifferentRadiiorWeights

Anon-parametricextensionoftheRayleighstatistic,theMoore-Rayleighstatistic,𝑧P,weightsphasesbytherankorderoftheirradii(Moore,1980).Givenasetofpolarcoordinates, 𝜃i, 𝑟i 𝑛 = 1…𝑁},thephasesarereorderedinascendingorderbytheirradii:(𝜃(\), 𝜃(9), … , 𝜃(j)).Then,theMoore-Rayleightestusesonlytheorderedphasestoweightthembytheirrank-orderwhencalculatingtheresultantvector.ThisresultsinthephaseofthelargestradiusgettingaweightofNwhilethephaseofthesmallestradiusgetsaweightof1.Finally,theresultantvector’sradiusisthenscaledbytheinverseofthe1.5powerofthesizeoftheset,N(Equation13).

𝑋 = 𝑛 cos 𝜃(i)

j

i`\

, 𝑌 = 𝑛 sin 𝜃(i)

j

i`\

𝑅9 = 𝑋9 + 𝑌9

(𝜃, 𝑟)

20

𝑍 =𝑅

𝑁r/9Equation13a-c:Moore-RayleighStatistic

SimilartotheRayleighstatistic,acomputedMoore-Rayleighstatisticcanthenbecomparedagainstastandardp-valuetableindexedbythenumberofelements,N(Moore,1980).Thestandardp-valuetablefortheMoore-RayleighstatisticwasgeneratedwithasimilarprocedureasthatfortheRayleighstatistic,byuniformlysamplingphasesandradiusestogenerateanulldistribution.Alternatively,apermutationtestcanbeusedtogeneratethenulldistributionbyrepeatedlyshufflingtheorderofthedataset’sphasesandcomputingtheMoore-Rayleighstatisticoneachpermutation.Then,anempiricalCDFofthenulldistributioncanbeusedtofindthresholdzvaluesforp-values.

ScaledRayleighStatisticIntheprocessofconductingtheresearchdescribedinthiswork,aparametric,scaledRayleighstatistic,𝑧t,wasderived.Tothebestoftheauthor’sknowledge,thisisthefirstparametric,weightedcirculartestbasedontheRayleighstatistic.ThescaledRayleightestweightseachphase𝜃ibyacorrespondingradius𝑟iwhencalculatingtheresultantvector;thesquareoftheresultvector’sradiusisthenscaledbytheinverseofthesizeoftheset,N(Equation14).

𝑋 = 𝑟i cos 𝜃i

j

i`\

, 𝑌 = 𝑟i sin 𝜃i

j

i`\

𝑅9 = 𝑋9 + 𝑌9

𝑧t =𝑅9

𝑁 Equation14a-c:ScaledRayleighStatistic

Becausethesetofphasesisparameterizedbycorrespondingradii,apermutationtestisthemostappropriatetesttocomputep-values.AnulldistributioncanbegeneratedbyrepeatedlyshufflingthesetofNphases,randomlypairingthemwithashuffledsetoftheNradii,andcomputingthescaledRayleighstatisticonshuffledsets.Then,thresholdzvaluescanbefoundforp-valuesfromtheempiricalCDFofthescaledRayleighstatisticsofthenulldistribution.Alternatively,anulldistributioncouldbegeneratedbysamplingfromwell-knownstatisticaldistributions,suchastheuniformdistributionforphasesandtheNormaldistributionforradii,butmomentparameterssuchasmeanandvariancewouldneedtobefittothedataset’sradiiinorderforthenulldistribution’ssampledradiitobereasonable.ThisdependencyonthedatasetmakesthescaledRayleighstatisticparametric.

21

ComparisonofWeighted,CircularStatisticsHistorically,non-parametrictestswerepreferablebecauseastandardp-valuetablecouldbecomputedandusedwithoutregardtothespecificdatasetbeingtested.Giventhatcomputingpowerwasmuchmorelimitedjustafewdecadesago,eliminatingtheneedtogenerateanulldistributionwashighlyappealing.Thiswasthemotivationbehindthedevelopmentofthenon-parametricMoore-Raleightest.However,generatinganulldistributiontodayisnowanegligibleconcernonmostmoderncomputers.Thus,morepreciseandpowerfultestsarenowmorepreferablethanonesthathistoricallyrequiredlesscomputationalpower.IncontrasttotheMoore-Rayleightest,thescaledRayleightestproportionallyweightsdatapointsbythemagnitudeoftheirradii.TheMoore-Rayleightestcaninaccuratelysuggestasignificanteffectinadataset,whenthedifferencesamongthedataset’sradiiarenegligiblysmallyetarenotcapturedinthenon-parametricstatistic.Inthiscase,thedataset’sphasesarescaledbytherankorderoftheircorrespondingradii,whichdonotaccuratelyreflectthemagnitudeoftheradiiandthusarguablyleadstoaninaccurateweighting.Dependingonthedataset,exactlywhatkindofeffectisbeingtested,andwhetherdiametricphasesarerelatedtoeachother,radiimayormaynotwanttoberestrictedtothesetofpositiverealnumbersforthescaledRayleightest.FurtheranalysisonthestrengthsandlimitationsoftheproposedscaledRayleightestwasoutsidethescopeofthisprojectbutmaybeaninterestingandpromisingdirectionforfuturework.TheprimaryMATLABresourceforcircularstatisticsdidnotincludeweightedcircularstatisticstests(Berens,2012).TheMATLABimplementationsoftheMoore-RayleighstatisticandthescaledRayleighstatisticusedinthisworkwereoptimizedandmadeavailableforpublicuse(Fong,2016a).

SignificanceTestinginChangeofAmplitude,Frequency,andPhaseFor𝑛 ∈ 1,… ,𝑁 ,whereN=107blocks,let𝜃idenotethephaseatwhichphase-lockedstimulationwasadministeredduringthen-thexperimentalblockandlet∆𝑎i, ∆𝑓i,and∆𝜃idenotetherespectivechangeinamplitude,infrequency,andinphasemetricsforthen-thblock.Then,foreachmetric,boththeMoore-RayleighstatisticandthescaledRayleighstatisticwerecomputed,wherez-scores(Equation15)ofthemetricvalues,i.e.{∆𝑚i|𝑛 = 1,… ,𝑁},where𝑚 ∈ {𝑎, 𝑓, 𝜃},wereusedasradiifortheweightedtests.Permutationtestsasdescribedabovewereusedtogeneratenulldistributionsagainstwhichthresholdzvaluesandp-valuescouldbeevaluated.

𝑧 =𝑥 − 𝜇𝜎

Equation15:Z-Score

22

ForthescaledRayleightest,bothradiithatincludenegativevaluesandradiishiftedsothatallradiiarepositivewereused.Therewasnopalpabledifferenceinresults,sothesignificancetestsreportedintheResultssection(Figure11,Table1)andinAppendix1directlyusemetricvalues,someofwhicharenegative,asradii.

KuramotoModelApopulationofneuronscanberepresentedasasetofoscillators.TheKuramotomodel(Kuramoto,1984)isonesuchneuronaloscillatormodelandhasbeenextendedtocapturetheeffectsofdeepbrainstimulationonhighlysynchronizedthalamicneuronalpopulations,likethoseofessentialtremorandParkinson’sdiseasepatients(Tass,2003).Inthissection,themodelwillfirstbeexplained;subsequently,theuseofthemodelinthisprojectwillbeoutlined.Tass’Kuramotomodelcapturesfourneuronalqualities–thattheyfire1.regularly,2.synchronously,3.noisily,and4.responsibly(toDBS).Let𝜃^ denotethephaseofthei-thoscillatorinasimulationofthemodel.Then,Equation16showstheupdateruleforthephaseofasingleoscillator,withthelastfourtermscorrespondingtothefouraforementionedqualities.

𝜃^ ← 𝜃^ + 𝑑𝑡 𝜔^ + 𝐾𝑟 sin 𝜓 − 𝜃^ + 𝛼𝒩 0, 𝑑𝑡 + 𝛽𝑧 𝜃^ 𝑢^ 𝑡

Equation16:UpdateRuleforExtendedKuramotoModel

Withsuchaframework,aneuronisassumedtospikeregularlyatagivenfrequency,𝜔^;aspikeisthenmodelledasthecompletionofacyclebyanoscillator,wherethetimetheoscillatortakestocompleteeachcycleisgivenbytheinverseofitsfrequency,i.e.itsperiod.Figure8visualizesapopulationof10oscillatorsorbitingclockwise;inthisvisualization,onecanassumethatanoscillatorspikeswhenitcrossesthe0°mark.

Figure8:VisualizationofNeuronalOscillators

AnotherfeatureoftheKuramotomodelisthatitmodelsthesynchronyofapopulationofneurons.Thestrengthofsynchronyismodulatedbythecoupling

23

factor,K.InEquation16,thecouplingterm,𝐾𝑟 sin 𝜓 − 𝜃^ ,includestwoorderparametersthatarecalculatedateachtimestep:𝜓,whichisthepopulation’smeanphase,and𝑟 ∈ [0,1],whichisthemagnitudeofthemeanvectorofthephasesplottedontheunitcircleandrepresentsthecoherenceofthepopulation’sphases.rcloseto1meansthepopulationishighlysynchronized(Equation17).

𝜓 =1𝑁 𝜃^

j

^`\

𝑋 =1𝑁 cos 𝜃^

j

^`\

, 𝑌 =1𝑁 sin 𝜃^

j

^`\

𝑟 = 𝑋9 + 𝑌9Equation17a-c:OrderParametersforKuramotoModel

Intuitively,thecouplingterm“speedsup”oscillatorsthatare“behind”themeanphase,𝜓,and“slowsdown”thosethatareaheadofthemeanphase.Forinstance,ifthei-thoscillatoris“behind”themeanphase,then𝜓 − 𝜃^ > 0,makingthesintermpositiveandthusthewholecouplingtermcontributesanincreasetothephase𝜃^ intheupdaterule(Equation16).ThemodelalsocapturesthenoisinessofneuronalspikingbyaddingGaussiannoisewithameanof0andastandarddeviationof 𝑑𝑡,whichensuresthattheeffectofthenoisedoesnotdependonthesizeofthetimestep,andthenscalingtheGaussiannoisebythenoiseconstant,𝛼.Finally,theextendedmodelincorporatesDBSinthelasttermoftheupdaterule.Thefunction,𝑧 𝜃^ : 0,2𝜋 → [0,1],representsthephaseresponsefunctionofanindividualneuron,aspreviousworkhasdemonstratedthataneuron’sresponsetostimulationisphase-dependent(Best,1979;Guttmanetal.,1980).Typically,intheliterature,thephaseresponsefunctiontakesonasinusoidfunction,i.e.𝑧 𝜃^ =cos 𝜃^ (Tass,2003).Notethatthisissimilaryetnotthesameasthechangeinphaseofanaggregatepopulationofneuronsorthatofabehavioraloutputlikeahandtremor.Beingabletoderiveanappropriatephaseresponsefunctionofanindividualneuronfromaggregatepopulationsremainsanareaofactiveresearch(Netoffetal.,2012;WilsonandMoehlis,2015).Next,thestimulationfunction𝑢^(𝑡)denoteswhetherthei-thoscillatorisstimulatedattimet(Equation18).Theparameterizationof𝑢^ 𝑡 byoscillatorallowsforDBStoeffectonlyasub-populationofoscillators.Lastly,theDBSstrengthparameter,𝛽,scalesthephase-dependentandoscillator-specificeffectofDBS.

𝑢^ 𝑡 = 1, ifoscillator𝑖isstimulatedattime𝑡0, otherwise

Equation18:Tass,2003’sStimulationFunction

24

ThisextendedKuramotomodelcanbeusedtomodeltremoramplitudebyaveragingthecosinepartsofthephases(Equation19).

𝑎 𝑡 =1𝑁 cos[𝜃^(𝑡)]

j

^`\

Equation19:ModelingTremorAmplitude

Insummary,thefreemodelparametersareoscillatorfrequencies,{𝜔^|𝑖 = 1,… ,𝑁},thecouplingconstant,K,thenoiseconstant,𝛼,andtheDBSstrengthconstant,𝛽.Additionally,thenumberofoscillators,N,andthesizeofthetimestep,dt,mustbechosen.With𝛽 = 0,theupdateruleisreducedtothetypicalformulationoftheKuramotomodel;thus,theDBStermwastheprimarytheoreticalextensionfromTass’work.

StimulationFunctionExtensiononTass,2003First,forthisproject,theassumptionismadethatstimulationinstantaneouslyaffectsoscillatorsatthesametime;thebinaryindicatorfunction,𝑖 𝑡 ,indicatesifwhenstimulationisbeinggivenattimet(Equation20).Furtherworkcouldbedonetoexploreusingdifferenttimedelaysfordifferentoscillators,simulatingthetimeittakesforapulsetotraveltoaneuron.

𝑖 𝑡 = 1, ifstimulationisgivenattime𝑡0, otherwise

Equation20:IndicatorFunctionforStimulation

Forthisproject,fourdifferentstimulationfunctions𝑢^ 𝑡 weretestedtoinvestigatemorebiologicallyplausiblemodelsofDBS.Unlessotherwisespecified,throughouttherestofthedissertation,𝑢^ 𝑡 intheupdaterule(Equation16)willbedenoted𝑢∗ 𝑖, 𝑡 wheretheasteriskindicateswhichstimulationeffectfunctionisbeingused.Astimulationfunction𝑢∗ 𝑖, 𝑡 (Equation21)capturesifstimulationisbeinggivenattimetwiththeindicatorfunction𝑖 𝑡 aswellaswhatkindofeffectstimulationhasonapopulationofneuronswithastimulationeffectfunction𝑠∗(𝑖).

𝑢∗ 𝑖, 𝑡 = 𝑖 𝑡 ×𝑠∗(𝑖)Equation21:GeneralformforStimulationFunction

First,auniformstimulationeffectfunction,𝑠� 𝑖 ,thatdecoupledDBSstrengthfromthenumberofoscillators,wasdeveloped(Equation22).Using𝑠� 𝑖 ,theuniformstimulationfunction𝑢t 𝑖, 𝑡 ismostsimilartotheoneusedinTass,2003,exceptthatitassumesthatstimulationaffectsalloscillatorsinstantaneously.

𝑠� 𝑖 =1𝑁

25

Equation22:UniformStimulationEffectFunction

Next,arandomstimulationeffectfunction,𝑠U 𝑖 ,wasdevelopedinanattempttocapturetheeffectthatathalamicneuron’sresponsetoDBSislikelyproportionaltoitsdistancefromtheDBSelectrode.ThisfunctionfirstdrawsNsamplesfromthestandarduniformdistributionandthennormalizesthem(Equation23).TheweightsofDBSstimulationeffectarefixedafterthesimulationisinitialized,whentheNuniformsamplesaredrawnandnormalized.

𝑥^ ← 𝒰 0,1 𝑠U 𝑖 =

𝑥^𝑥�j

�`\

Equation23a-b:RandomStimulationEffectFunction

Additionally,a“half”stimulationeffectfunction,𝑠� 𝑖 ,ispresentedinordertonaivelymodelthephenomenonthattwoneuronalpopulations,oneinthemotorcortexandoneinthethalamus,mightcontributetoParkinsonian-liketremor;however,onlythethalamusisstimulated;thus,DBSshouldonlyeffectonepopulation(Equation24).

𝑠� 𝑖 = 2𝑁 , if𝑖 ≤

𝑁2

0, otherwise

Equation24:HalfStimulationEffectFunction

Lastly,a“mixture”stimulationfunction,𝑠� 𝑖 ,attemptstocapturebothqualitiesoftherandomand“half”stimulationfunctions(Equation25).

𝑦^ ← 𝒰 0,1 , if𝑖 ≤ 𝑁2

0, otherwise

𝑠� 𝑖 =𝑦^𝑦�j

�`\

Equation25:MixtureStimulationEffectFunction

Notethatforallfunctions,, 𝑠∗ 𝑖 = 1j

^`\ .ThisnormalizationwasaddedtomakeDBSstrengthindependentfromNandtomakethestimulationeffectfunctionscomparabletooneanother,asthesameamountofDBSisdistributedtoalloscillators.

OptimizedModelParametersIncollaborationwithlabmemberDr.GihanWeerasinghe,modelparametersparameterswerefoundviaanon-linearoptimizationthatminimizedtheleast

26

squareerrorbetweenqualitiesofthemodel-generateddataandthoseofonepatientwhosehandtremorwasrecordedintheabsenceofDBS(thepatientwasincludedintheCagnanetal.,2013study).Theoptimizationfittedthemodelbasedonthefollowingtwoqualities:1.thepowerspectrumofthedata,and2.theprobabilityofobtainingaparticulartremoramplitude.WhenusingN=10oscillators,theseoptimizedmodelparameterswerefound:𝜔^ ← 𝒩(𝜇 = 32.3319, 𝜎 = 2.5804),𝐾 =1.89753,and𝛼 = 2.66466.Giveninradianspersecond,themeanangularfrequencyfromwhichoscillatorfrequenciesweredrawncorrespondsroughlytoabitmorethan5Hz,whichisclosetothenatural4-6HzfrequencyatwhichParkinsonianandessentialtremorsoccur.Unlessotherwisenoted,theaboveparameterswereusedinsimulationsreportedintheResultssection.

SimulatingPhase-LockedDBSUnlessotherwisespecified,modelsimulationsexplainedintheResultssectionusedtheaforementionedoptimizedparametersaswellasthefollowingparameterschosenapriori:𝛽 = 0.5,𝑑𝑡 = 1/2048,𝑁 = 10.Additionally,unlessotherwisenoted,modelsimulationsweredesignedtomimictheexperimentalset-up,withasimulationblockconsistingof1secondofnostimulationfollowedby5-secondsofphase-locked130Hz,6-pulsestimulationat12phases,i.e. 0°, 30°, … ,330° .ThephaseatwhichDBSwaslockedtowassimplythemeanphaseoftheoscillatorpopulation,whichwascalculatedateachtimestep(Figure9,3rdrow).Atthestartofeachblock,phaseswereuniformlydrawnfromtheinterval 0,2𝜋 .Similartotheexperimentaldesign,9blocksweresimulatedforeachofthe12phases.Similarpreprocessingwasconductedonthesimulateddata:whilethesimulateddatawasnotresampledto1000Hzliketheexperimentaldata,itwasband-passfilteredbetween3-6Hzinthesameway(Figure9,2ndrow).Theamplitudeandinstantaneousfrequencywerealsocalculatedinthesamefashionandthesamechangesinamplitude,frequency,andphasemetricswerecalculatedonthesimulatedblocksaswellasthesamestatisticalanalysis.Theprincipalaimofthesimulationswastoreproducethesameeffectsobservedintheexperimentaldatabyprimarilyinvestigatingwhatphaseresponsecurveyieldedthemostexperimentally-consistentresults.AdditionalanalysiswasconductedtoexplorethelimitsofthemodelinexplainingtheexperimentaldataandisdescribedintheResultssection.

27

Figure9:SimulatedBlockwithPhasicStimulationat240°usingDefaultParameters

(1strow)Simulated,unfilteredtremoraccelerationsignal,calculatedusingEqn19.(2ndrow)Filtered tremorsignalandamplitude(orange line).The tremorsignal fromthe1st rowwasband-passed filtered between 3Hz and 6Hz; the tremor amplitude was derived via theanalyticalsignal. (3rd row).35msof130HzDBSwasadministeredwhenthe filtered tremorsignalpassedthe240°phase,astrackedinthe4throw.(4throw)Meanphaseoftheoscillatorpopulation. (5th row) Coherence metric. r connotes how synchronized the population ofoscillatorsis.

28

ExtendedKuramotoModelOnlineSimulationTool&TutorialAspartofapublicengagementeventwithlocalOxfordshirehighschoolstudents,anonlinesimulationofthemodelusedinthisprojectwasbuilt(Fong,2016b).Itwasalsousedforresearchtoqualitativelyunderstandthemodelandobservehowcertainparameterchangesaffectedthemodel.Additionally,anintuitiveonlinetutorialofthemodelwasdeveloped(Fong,2016c).Appendix4providesmoredetailsaboutthetool.

29

Results

AnalysisofExperimentalDataAnanalysisofthechangeinamplitude,frequency,andphaseforall107experimentalblocksdemonstratedastatisticallysignificantphase-dependentsuppressionoftremorwhenphase-lockedDBSisadministeredaround210°-240°.Thechangeinamplitude,frequency,andphasewerecalculatedaccordingtothemetricsdescribedintheMethodssection.

ChangeinAmplitude,Frequency,andPhaseCurvesWhenthechangeintremoramplitudeforallblocksareplotted(Figure10,1strow),thereappearstobeasuppressiveeffectat120°and240°andanamplifyingeffectat30°,whichdiametricallyopposes210°.Notethatallbuttwo240°blocksyieldedatremor-suppressiveeffect,andallbutthree210°blocksdidthesame.Whenthechangeininstantaneousfrequencyofthefilteredtremoramplitudeforallblocksareplotted(Figure10,2ndrow),thereappearstobeamaximumpeakat240°andaminimumpeakat60°,whichdiametricallyopposes240°.Whenthechangeinphaseforallblocksareplotted(Figure10,3rdrow),thereisaglobal,maximumpeak240°andglobal,minimumpeakat60°,whichdiametricallyopposes240°.Therealsoisalocal,maximumpeakat150°andalocal,minimumpeakat180°.Mostnotably,at240°,themosttremor-suppressingphaseatwhichtophase-lockDBSto,thetremor-suppressingeffectcapturedinthechangeinamplitudecurveiscorrelatedwithamaximalspeed-upinfrequencyandphase.

30

Figure10:ChangeinAmplitude,Frequency,andPhaseofExperimentalData

Thechangeinamplitude(inm/s2),∆𝑎(1strow),frequency(inHz),∆𝑓(2ndrow),andphase(indegrees),∆𝜃(3rdrow),werecalculatedforall107experimentalblocksandplottedabove(bluecircles).Linearinterpolationsofthemedian∆𝑎, ∆𝑓,and∆𝜃foreachphaseofstimulationwereplotted(orangelines)alongwithstandarderror.

31

SignificanceTestingTotestwhetherthepalpablesuppressiveeffectsinthechangeintremoramplitudeaswellastheseemingly-phasedependentchangeinfrequencyandphasemetricsarestatisticallysignificant,theweightedMoore-RayleighandScaledRayleighpermutationtestswereused.Foreachtestrun,100,000sampleswithasamplesizeof100–comparableinsizetothe107experimentalblocks–wereusedtogeneratethenulldistributionwithwhichtocomparethecriticalzscoreoftherelevanttestto. p-value Phaseofresultantvector Moore-

RayleighScaledRayleigh

Moore-Rayleigh

ScaledRayleigh

∆𝒂 0.0056 0.0067 31.7° 30.9°−∆𝒂 0.0190 0.0065 216.6° 210.9°∆𝒇 0.0470 0.0422 241.3° 240.6°∆𝜽 0.0104 0.0130 232.6° 222.7°

Table1:p-valuesandResultPhasesfromMoore-RayleighandScaledRayleighSignificanceTestingofChangeinAmplitude,Frequency,andPhaseofExperimentalData

Accordingtothesetests,therewassignificantnon-uniformity(p<0.05)forallmetrics(Table1).Testswereconductedforthenegativechangeinamplitude,−∆𝑎,totestsignificantdepression,asopposedtosignificantamplification(Figure11,topplot).Thetestforchangeinamplitude,∆𝑎,stillprovidessomeinformationastothedirectionofthenon-uniformity.Thediametricallyoppositephasesofthosetest’sphasesoftheresultantvector,31.7°and30.9°fortheMoore-RayleighandScaledRayleightests,are211.7°and210.9°respectively.Thephasesofthetests’resultantvectorsareallwithintherange210°-242°.Notethatthesephasesarenotthemselvesstatisticallysignificantmetricsbutsimplyshowthemeandirectionofweightednon-uniformity.Theyarelikelyskewedtobelessthan240°becausethereisasmalltremorsuppressingeffectfor120°phase-lockedDBS.Figure11visualizesthetremorsuppressiveeffect,aswellasthecorrelatedamplificationofchangeinfrequencyandphase,whenDBSisphase-lockedaround210°-240°,astherearemanymoredatapointswithgreatermagnitudesinthethirdquadrant(correspondingto180°-270°)thanintheotherquadrants.

32

Figure11:VisualizationsofMoore-RayleighTestforNegativeChangeinAmplitude,Changein

Frequency,andChangeinPhase

33

The z-score normalized values of the negative change in amplitude (top plot), change infrequency(middleplot),andchangeinphase(bottomplot)metricsforall107experimentalblocksareplottedbytheirphasesontheunitcircleaspolarcoordinates,i.e. 𝜃^, −∆𝑎^ , (𝜃^, ∆𝑓 ),and(𝜃^, ∆𝜃^)respectively.Themetricsforthe8-9blocksassociatedtoasamephasearecoloredby the same color. “+”markers denote a positivemetric valuewhile “o”markers denote anegativemetricvalue;negativevaluesareplottedinthedirectionofthediametrically-oppositephase. The p < 0.05 threshold from theMoore-Rayleigh permutation test is plotted as thedottedredcircle,andtheMoore-Rayleighcriticalz-scoreoftheexperimentaldataisplottedasthemagnitudeoftheblackvector,whosephasedenotesthephaseoftheresultantvector.Forspaceefficacy,thevisualizationisrestrictedtoplotpointswithmagnitudeslessthan1.5.

CorrelationamongChangeinAmplitude,Frequency,andPhaseChangeinamplitudeisnegativelyrelatedtochangeinfrequencyandchangeinphase;whilechangeinfrequencyandchangeinphasearestronglypositivelycorrelated.Thecorrelationcoefficientsandcorrespondingp-valueswerecomputedbetweenthe107changeinamplitudevaluesandthoseofthechangeinfrequencymetric;thesamecomparisonwasdonefortheotherpairingsofthethreemetrics(Table2).Thestrong,positivecorrelationbetweenchangeinfrequencyandchangeinphaseistobeexpected,asthechangeinphasemetricissimplyadifferentmethodforcalculatingfrequency(byunwrappingtheactualphaseofthefilteredtremorsignal). R(correlation

coefficient)p-value

∆𝒂vs.∆𝒇 -0.4077 1.3052×10�a�∆𝒂vs.∆𝜽 -0.4021 1.7618×10�a�∆𝒇vs.∆𝜽 0.6735 1.8956×10�\�

Table2:CorrelationbetweenChangeinAmplitude,Frequency,andPhaseofIndividualExperimentalBlocks

Thecorrelationcoefficientsandp-valuesofcorrelationwerecomputedonthe107valuesofthechangeinamplitude,frequency,andphasemetrics.

Fromtheanalysisofexperimentaldata,phase-lockedDBSaround240°bestsuppressestremor(Figure10,1strow).Furthermore,tremorsuppressionaround240°correspondstomaximalincreasesinfrequencyandphaseacrossallphases(Figure10,2ndand3rdrows).

AnalysisofModelledData

PredictingaConsistentTremorSuppressiveModelAssumethatallthalamicneuronsrespondtoDBSinexactlythesamephase-dependentfashion,thatis,thattheeffectofDBSononeneuronataparticulartimeisidenticaltothatonanotherneuron.Then,basedonTass,2003extensionoftheKuramotomodel,assumingthatDBSaffectsalloscillatorsindiscriminatelyandinstantaneously,themostoptimalphaseatwhichtostimulateshouldbethephase

34

atwhichthephaseresponsecurve(PRC)functionofanindividualneuron,𝑧 𝜃^ ,hasthesteepest,positiveslope(WilsonandMoehlis,2014).

Figure12:CosineCurveasIndividualNeuron’sPhaseResponseCurve(PRC)

SupposethatthePRCforalloscillatorswas𝑧 𝜃^ = cos(𝜃^)(Figure12).Then,theslopeofthePRCismoststeeplypositive,i.e.𝑧′ 𝜃^ ,thederivativeofthePRC,hasamaximumpoint,atphase270°.IfapopulationofKuramotooscillatorswiththecosinefunctionasitsPRCis“stimulated”whenthepopulation’smeanphaseis270°,theoscillatorswillbecomemoredesynchronized.Thisisbecausetheoscillatorswithphasesgreaterthan270°willspeedup,because𝑧 270° + 𝜀 > 0,whiletheoscillatorswithphaseslessthan270°willslow,because𝑧 270 − 𝜀 < 0(Figure13).

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Figure13:DiagramoftheOptimalStimulationStrategyfortheKuramotoModel

Kuramotooscillators(orangecircles)areoscillatingcounter-clockwise(indicatedbythegrayarrow).Iftheoscillators’PRCfunctionis𝑧 𝜃^ = cos(𝜃^),thentheoptimalphaseofstimulationiswhenthemeanphaseoftheoscillatorsisequalto270°.Thisisbecausetheoscillatorswhosephasesaregreaterthan270°willspeedupwhilethosewhosephasesarelessthan270°willslowdown,therebydesynchronizingthepopulation.

BasedonthisinterpretationoftheextendedKuramotomodel,tomodelmaximaltremorsuppressionat240°aswasobservedintheexperimentaldata,aPRCsuchas𝑧 𝜃^ = cos(𝜃^ +

��)shouldbeused,becausethecurveismostpositivelyslopedand

crossesthey-axisat240°(Figure14).

Figure14:DefaultPRCfunction,𝒛 𝜽𝒊 = 𝐜𝐨𝐬(𝜽𝒊 +

𝝅𝟔),forExtendedKuramotoModel

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DefaultParametersforExtendedKuramotoModelUnlessotherwisementioned,thefollowingdefaultparametersareusedtosimulatetheKuramotomodel(Equation16).TheseparameterswerefoundviaanoptimizationthatchosetheparametersthatbestsimulatedtremorsignalofanessentialtremorpatientintheabsenceofDBS:

• Initialphasesofoscillators=𝜃^ ← 𝒰( 0,2𝜋 )• Angularfrequencyofoscillators:𝜔^ ← 𝒩(𝜇 = 32.3319, 𝜎 = 2.5804),• Couplingconstant:𝐾 = 1.89753,• Noiseconstant:𝛼 = 2.66466,

Theseparameterswerefixedapriori:

• Numberofoscillators:𝑁 = 10,• Timestep:𝑑𝑡 = 1/2048,• DBSstrengthconstant:𝛽 = 0.5,• PRCfunction:𝑧 𝜃^ = cos(𝜃^ +

��),

• Stimulationfunction:𝑢� 𝑖, 𝑡 = 𝑖 𝑡 × \j

Tosimulatetheexperimentaltrial,6-secondblocksoftheKuramotomodelweresimulated,inwhich1secondofnostimulationwasfollowedby5secondsof130Hz,6-pulsestimulationlockedtoaspecificphase.108blocksweresimulated,9blocksforeachofthe12phases,i.e. 0°, 30°, … ,330° .9,6-secondreferenceblocks,inwhichnoDBSwasadministered,werealsosimulatedandusedasabaselinetocomparewithwhichtocomparethephase-locked-DBSsimulatedblocks.SeetheMethodssectionformoredetailsonhowparameterswerechosenandthesimulationswereexecuted.

ResultsusingDefaultParametersWhensimulatingtheexperimentalset-upusingtheextendedKuramotowiththeaforementioneddefaultparameters,statisticallysignificanttremorsuppressionwasobserved(Table3).However,themodelfailedtoprovideanexplanationforthesimulatedsuppressionthatwasconsistentwiththeexperimentaldatafortremorsuppression.ChangeinAmplitude,Frequency,andPhaseThesamechangeinamplitude,frequency,andphasemetricswerecomputedonthesimulatedaswasdoneontheexperimentaldata.Fromthechangeinamplitudecurve,tremorsuppressionappearedstrongestinsimulatedblocksinwhichDBSwasphase-lockedto180°-270°(Figure15,1strow).Thisbroadsuppressioneffectforphases180°-270°differsfromthatobservedinthe

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experimentaldata(Figure10,1strow),whichhadasmallsuppressiveeffectatphase150°andanotherat240°.Whiletherelationshipbetweenthechangeinamplitude(Figure15,1strow)andchangeinfrequencycurves(Figure15,2ndrow)isnotapparentwithoutfurtheranalysis,tremorsuppressionforphase-lockedstimulationat180°-270°(Figure15,1strow)appearscorrelatedtothepositiveslopeandzero-crossinginthechangeinphasecurveat180°-270°(Figure15,3rdrow).Quantitatively,Table6confirmsthisrelationshipandshowsthatthechangeinamplitude,∆𝑎,isnegativelycorrelatedtothederivativeapproximationofthechangeinfrequency, �

��(∆𝑓),(𝑅 = −0.64, 𝑝 =

0.02)aswellastothederivativeapproximationofthechangeinphase, ���(∆𝜃)

(𝑅 = −0.76, 𝑝 = 0.004).

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Figure15:ChangeinAmplitude,Frequency,andPhaseofSimulatedDatausingDefaultParameters

The change in amplitude (m/s2), ∆𝑎 (1st row), frequency (Hz), ∆𝑓 (2nd row), and phase(degrees),∆𝜃(3rdrow),werecalculatedforall108simulatedblocksandplottedabove(redcircles).Linearinterpolationsofthemedian∆𝑎, ∆𝑓,and∆𝜃foreachphaseofstimulationwereplotted(bluelines)alongwiththestandarderror.Themedianchangeinamplitude,frequency,andphaseforthe9referenceblocksofnostimulationwerealsoplotted(orangelines).

SignificantTestingThesameweightedRayleightestswereusedtodeterminewhethertherewassignificantnon-uniformityinthechangeinamplitude,frequency,andphasecurvesforthesimulateddata.Thesetestsreportquitehighconfidenceinanon-uniform

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effectforallcurves(Table3).Whilenotarigorousstatistic,thephasesoftheresultantvectorsfromthesetestsforchangeinamplitude(Table3)comparedtothoseoftheexperimentaldata(Table1)suggestthatthemodelmayroughlyhaveasimilartremorsuppressioneffectaround210°-240°.However,thephasesoftheresultantvectorsforchangeinfrequencyandphasetestsonthesimulateddata,i.e.320°-330°,arenearlyperpendiculartothoseforthecorrespondingexperimentaltests,i.e.220-240°.Thissuggeststhatthemodelmaynotexplainthephase-dependenttremorsuppressionofthisparticularpatient.VisualizationsoftheMoore-Rayleightestforthenegativechangeinamplitude,changeinfrequency,andchangeinphaseareprovidedinFigure16. p-value Phaseofresultantvector Moore-Rayleigh ScaledRayleigh Moore-Rayleigh ScaledRayleigh∆𝒂 < 10�� < 10�� 34.8° 36.4°−∆𝒂 < 10�� < 10�� 214.8° 216.4°∆𝒇 10�� 10�� 323.8° 328.5°∆𝜽 < 10�� < 10�� 323.0° 321.7°

Table3:p-valuesandResultPhasesfromMoore-RayleighandScaledRayleighSignificanceTestingofChangeinAmplitude,Frequency,andPhaseofSimulatedDatausingDefaultParameters

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Figure16:VisualizationsofMoore-RayleighTestforNegativeChangeinAmplitude,Changein

Frequency,andChangeinPhaseofSimulatedDatausingDefaultParameters

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The z-score normalized values of the negative change in amplitude (top plot), change infrequency(middleplot),andchangeinphase(bottomplot)metricsforall108simulatedblocksareplottedbytheirphasesontheunitcircleaspolarcoordinates,i.e. 𝜃^, −∆𝑎^ , (𝜃^, ∆𝑓 ),and(𝜃^, ∆𝜃^)respectively.Themetricsforthe8-9blocksassociatedtoasamephasearecoloredbythesamecolor.“+”markersdenoteapositivemetricvaluewhile“o”markersdenoteanegativemetricvalue;negativevaluesareplottedinthedirectionofthediametrically-oppositephase.Thep<0.05thresholdfromtheMoore-Rayleighpermutationtestisplottedasthedottedredcircle, and the Moore-Rayleigh critical z-score of the experimental data is plotted as themagnitudeoftheblackvector,whosephasedenotesthephaseoftheresultantvector.Forspaceefficacy,thevisualizationisrestrictedtoplotpointswithmagnitudeslessthan1.5.

ComparisontoExperimentalDataToquantifythedifferencebetweenthesimulatedandexperimentaldata,thePearson’scorrelationcoefficientswerecomputedbetweenthecorrespondingchangeinamplitude,frequency,andphasecurvesofthesimulatedandexperimentaldata(Table4).Thereisasignificantlinearcorrelationbetweenthechangeinamplitudeeffectsinthesimulatedandexperimentaldata,butnosuchrelationshipbetweenthechangeinfrequencyorphase.

R(correlationcoefficient) p-value∆𝒂𝒆𝒙𝒑vs.∆𝒂𝒔𝒊𝒎 0.6437 0.0239∆𝒇𝒆𝒙𝒑vs.∆𝒇𝒔𝒊𝒎 0.0226 0.9445∆𝜽𝒆𝒙𝒑vs.∆𝜽𝒔𝒊𝒎 -0.0093 0.9770

Table4:CorrelationsbetweenMedianChangeinAmplitude,Frequency,andPhaseofExperimentalDataandthoseofSimulatedDatausingDefaultParameters

Thecorrelationcoefficientsandp-valuesofcorrelationwerecomputedbetweenthe12medianvaluesofthechangeinamplitude,frequency,andphasemetricsoftheexperimentaldataandthe12medianvaluesofthesamemetricsofthesimulateddatausingdefaultparameters.

Withinboththesetofexperimentalblocksandthesetofsimulatedblocks,thecorrelationcoefficientsamongthedifferentmetricswerealsocomputed(Table5).Whilethereweresignificant,inversecorrelationsbetweenchangeinamplitudeandchangeinfrequency(∆𝑎vs.∆𝑓)aswellaschangeinamplitudeandchangeinphase(∆𝑎vs.∆𝜃)amongexperimentalblocks,therewereonlyweak,positivecorrelationsamongthesamepairsofmetricsamongsimulatedblocks. R(correlationcoefficient) p-value Experimental Simulated Experimental Simulated∆𝒂vs.∆𝒇 -0.4077 0.1567 1.31×10�a� 0.1053∆𝒂vs.∆𝜽 -0.4021 0.0265 1.76×10�a� 0.7857∆𝒇vs.∆𝜽 0.6735 0.6955 1.90×10�\� 6.69×10�\¦

Table5:CorrelationbetweenChangeinAmplitude,Frequency,andPhaseofIndividualExperimentalBlocksandIndividualSimulatedBlocksusingDefaultParameters

Thecorrelationcoefficientsandp-valuesofcorrelationwerecomputedonthe107valuesofthechangeinamplitude,frequency,andphasemetricsoftheexperimentaldata(Table2)andonthe108correspondingvaluesofthemodelleddatausingdefaultparameters.

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Lastly,forbothexperimentalandsimulateddata,the12medianvaluesofeachcurve,correspondingtothe12phasesDBSwaslockedto,werecorrelatedwitheachother(Table6).Additionally,themedianvaluesofthechangeinamplitudecurvewascorrelatedwithapproximationsofthederivativesofthechangeinfrequencyandinphase.Thederivativeapproximationforamedianvalueatphase𝜃wascomputedbytakingthedifferencebetweenthemedianvalueatphase𝜃 + 30°andthatatphase𝜃 − 30°.Intheexperimentaldata,changeinamplitudeisstrongly,negativelycorrelatedwithchangeinfrequencyandphase.Incontrast,inthesimulationdata,changeinamplitudeisstrongly,negativelycorrelatedwiththederivativeapproximationsofchangeinfrequencyandphase.ThisisexpectedbyWilsonandMoehlis,2014,whichexplainsthattheoptimalphasicDBSstrategyshouldbetostimulateatthephaseatwhichthePRChasapositivezero-crossing,thatis,whereitsderivativeismostpositive.Whilelatersimulationsdemonstratethepopulation’schangeinphasecurveisanon-lineartransformationoftheindividualPRCfunction,thechangeinfrequencyandphasecurvesaresimplypopulationmetricsofthesameconceptthatthePRCfunctionisanindividualoscillator’smetricfor.

R(correlationcoefficient) p-value Experimental Simulated Experimental Simulated∆𝒂vs.∆𝒇 -0.7427 -0.0842 0.0057 0.7947

∆𝒂vs. 𝒅𝒅𝜽(∆𝒇) -0.4774 -0.6447 0.1166 0.0236

∆𝒂vs.∆𝜽 -0.6869 0.0364 0.0136 0.9106∆𝒂vs. 𝒅

𝒅𝜽(∆𝜽) -0.4367 -0.7602 0.1558 0.0041

∆𝒇vs.∆𝜽 0.8246 0.7752 0.000967 0.0031Table6:CorrelationsamongMedianChangeinAmplitude,Frequency,andPhaseaswellasSelectDerivativeApproximationsofExperimentalandSimulatedBlocksusingDefaultParameters

For both experimental and simulated data, the correlation coefficients and p-values ofcorrelationwere computedon the12medianvaluesof the followingpairings:1. change inamplitudeandchangeinfrequency,2.changeinamplitudeandthederivativeapproximationofchangeinfrequency,3.changeinamplitudeandchangeinphase,4.changeinamplitudeandthederivativeapproximationof change inphase,and5. change in frequencyandchange inphase.

Thesequantifiabledifferencesbetweentheexperimentalandsimulateddatasuggestthatthemodeldoesnotexplainphase-dependentsuppressionforthepatientfromwhichtheexperimentaldatawasrecorded.Whileitcapturesphase-dependentsuppression,theextendedKuramotomodelproposestheoptimalphaseatwhichtostimulateistheoneatwhichthereisapositive,zero-crossinginitschangeinfrequencyorphase.Incontrast,intheessentialtremorpatientwithwhomthephase-lockedDBStrialwasconducted,theoptimalstrategyappearstobestimulatingatthephaseatwhichchangeinfrequencyorphaseismostpositive.

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ResultsusingDifferentStimulationFunctionsFromtheexperimentaldata,tremorsuppressionseemscorrelatedtothegreatest,positiveshiftinfrequencyorphase.ItwashypothesizedthatthegreatestphaseshifteffectyieldedthemosttremorsuppressionbecauseDBSdesynchronizedtwosub-populations,speedinguponlyaportionofneurons.Underthishypothesis,tremorwouldbemostsuppressedwhenasub-populationwasmostspeedupbyDBS.TotestthishypothesisandextendtheKuramotomodeltobettercapturethisrelationshipbetweentremorsuppressionandchangeinphase,afewmorebiologically-plausiblestimulationfunctionswereproposed.The“half”stimulationfunctionlimitstheeffectofDBStohalfoftheoscillatorpopulation;thiscapturesthefactthatboththethalamusandcortexcoordinatemotormovementsyetDBSonlyaffectsapopulationofthalamicneurons.The“random”stimulationfunctionrandomlyweightedtheeffectofDBSonindividualoscillators,simulatingthevaryingdistancesneuronsarefromtheDBSprobe.The“mixture”stimulationfunctionscombinesbothfeaturesfromthe“half”and“random”functions.Inthe“random”setting,desynchronizationwashypothesizedtooccurbetweenweaklyandstronglyDBS-affectedoscillators,whileinthe“half”setting,desynchronizationwasthoughttooccurbetweenthehalfsub-populationrepresentingDBS-affectedthalamicneuronsandtheotherhalf,representingunaffected,corticalneurons.SeetheStimulationFunctionExtensiononTass,2003sectionformoredetailsaboutthedifferentstimulationfunctions.Simulationsmodellingtheexperimentalset-upwereconductedusingeachoftheabovestimulationfunctions.TheMoore-Rayleightestdemonstratedsignificantnon-uniformeffects(𝑝 < 10��)forthechangeinamplitude,frequency,andphasecurvesofallsimulations.Oneconfirmationthatthesimulationswerebetterdescribingtheexperimentaldatawouldbeifthephasesoftheresultantvectorsofthedifferentmetricswereroughlyaround240°.However,whilethephasesoftheMoore-Rayleighresultantvectorsforthenegativechangeinamplitudecurveswereintherangeof213°-227.5°,thephasesoftheresultantvectorsforthechangeinfrequencyandphasecurveswereintherangeof315°-330°,a70°-90°shiftfrom240°.Likethesimulationusingdefaultparametersandtheuniformstimulationfunction,therewassignificantcorrelationbetweentheexperimentalandsimulatedchangeinamplitudecurvesyetnonebetweenthechangeinfrequencyandphasecurves(Table7).

R p-valueExpvs.Sim Uni Half Rand Mix Uni Half Rand Mix∆𝒂𝒆vs.∆𝒂𝒔 0.644 0.770 0.811 0.706 0.024 0.003 0.001 0.010∆𝒇𝒆vs.∆𝒇𝒔 0.023 -0.073 -0.080 0.071 0.944 0.821 0.804 0.825∆𝜽𝒆vs.∆𝜽𝒔 -0.010 0.180 0.119 0.022 0.977 0.576 0.713 0.945

Table7:CorrelationsbetweenMedianChangeinAmplitude,Frequency,andPhaseofExperimentalDataandthoseofSimulatedDatausingDifferentStimulationFunctions

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Thecorrelationcoefficientsandp-valuesofcorrelationwerecomputedbetweenthe12medianvaluesofthechangeinamplitude,frequency,andphasemetricsoftheexperimentaldataandthe12medianvaluesofthesamemetricsofthesimulateddatausingtheuniform(Uni),half(Half),random(Rand),andmixture(Mix)stimulationfunctions.

Similarly,unliketheexperimentaldata,whosechangeinamplitudeandchangeinfrequencyandphasecurveswerestrongly,negativelycorrelated,theonlystrongcorrelationswithinthesimulateddataforeachstimulationfunctionwasbetweenchangeinamplitudeandthederivativeapproximationofchangeinfrequencyandphase(Table8).

R p-value Uni Half Rand Mix Uni Half Rand Mix

∆𝒂vs.∆𝒇 -0.084 0.477 0.197 0.341 0.795 0.117 0.539 0.279∆𝒂vs. 𝒅

𝒅𝜽(∆𝒇) -0.645 -0.767 -0.682 -0.596 0.024 0.004 0.015 0.041

∆𝒂vs.∆𝜽 0.036 0.293 0.047 0.022 0.911 0.355 0.884 0.946∆𝒂vs. 𝒅

𝒅𝜽(∆𝜽) -0.760 -0.670 -0.878 -0.700 0.004 0.017 < 10�r 0.011

∆𝒇vs.∆𝜽 0.775 0.915 0.883 0.794 0.003 < 10�� < 10�� 0.002Table8:CorrelationsamongMedianChangeinAmplitude,Frequency,andPhaseaswellasSelectDerivativeApproximationsofExperimentalandSimulatedBlocksusingDefaultParameters

For simulated data using the four stimulation functions, the correlation coefficients and p-values of correlationwere computed on the 12median values of the following pairings: 1.change in amplitude and change in frequency, 2. change in amplitude and the derivativeapproximationofchangeinfrequency,3.changeinamplitudeandchangeinphase,4.changeinamplitudeandthederivativeapproximationofchangeinphase,and5.changeinfrequencyandchangeinphase.

Fromtheselimitedsimulations,thesestimulationfunctionsmaynotbeabletocapturethephase-dependentcorrelationbetweentremorsuppressionandpositivephaseshiftoftheexperimentalpatient.ChangeinAmplitudeCurvesWhenusingthe“half”stimulationfunction,thereappearstobeabroadtremor-suppressingeffectat180°-270°(Figure17,1strow).Incontrast,whenusingthe“random”stimulationfunction,thereappearstobeasharper,narrower,tremor-suppressingeffectat240°-270°(Figure17,2ndrow).Thisisalsomoresimilartotheclear,narroweffectobservedintheexperimentaldataat240°(Figure10,1strow);thechangeinamplitudecurveforthe“random”simulationisalsothemostcorrelatedwiththatoftheexperimentaldata(Table7).Lastly,thereisaweak,broadtremor-suppressingeffectinthe“mixture”simulation(Figure17,3rdrow).Morerepeatedsimulationswouldneedtobeconductedtorigorouslyanalyzewhichstimulationfunctionproducesachangeinamplitudecurvemostsimilartothatoftheexperimentaldata.Interestingly,thethreecurvesbelow(Figure17)showasmall,localminimumaround60°-90°similartothatintheexperimentaldataat120°(Figure10,1strow);

45

however,unliketheexperimentaldata,thesedipsinthesimulateddataarestillweaktremor-amplifyingeffects,astheyareabovethereferencechangeinamplitudebaseline.

46

47

Figure17:ChangeinAmplitudeCurvesforSimulatedDatawhenusing“Half”,“Random”,and“Mixture”Stimulationfunctions

Thechange inamplitude,∆𝑎,wascalculated forall108simulatedblocksgenerated ineachsimulationusingthe“Half”(1strow),“Random”(2ndrow),and“Mixture”(3rdrow)stimulationfunctionandplottedabove(redcircles).Whenusingeachofthesestimulationfunctions,linearinterpolationsof themedian∆𝑎foreachphaseof stimulationareplotted (blue lines)alongwiththestandarderror.Themedianchangeinamplitudevaluesforthe9referenceblocksofnostimulationarealsoplotted(orangelines).

Non-LinearTransformationfromIndividualPRCtoChangeinMeanPhaseofOscillatorPopulationTherewasapalpabledifferencebetweentheindividualPRCfunction(Figure14)usedbydefault,𝑧 𝜃^ = cos(𝜃^ +

��),whichrepresentsthephase-dependentshiftin

phaseofasingleoscillatorinresponsetoDBS,andthechangeinphasecurve(Figure15,3rdrow),whichcapturesthemeanphaseshiftoftheoscillatorpopulation.Theweaklycoupledinteractionsamongoscillatorswashypothesizedasthereasonforthisnon-linearrelationshipbetweentheindividualneuron’sPRCandthepopulation’schangeinphasecurve.Totestthishypothesis,simulationswereconductedthatvariedthenumberofoscillatorsfrom2to20andotherwiseusedthedefaultmodelparameters.Foreachsimulation,themeansquarederror(MSE)wascomputedbetweentheindividualPRCandthepopulation’smedianchangeinphasecurve(Figure18).Thereisastrong,positivecorrelationbetweenthenumberofoscillatorsusedandtheoscillatorpopulation’sdivergencefromtheindividualPRC.

Figure18:MeanSquaredErrorbetweenDefaultPRCFunctionandMedianChangeinPhaseinSimulated

DataasNumberofOscillatorsIncreases

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Using the default model parameters and only varying the number of oscillators used, theexperimental set-upwas simulated once for each𝑁 ∈ {2, … , 20} . Themean squared error(MSE)wascalculatedbetweenthedefaultPRCfunction,𝑧 𝜃^ = cos(𝜃^ +

��),whichrepresents

anindividualneuron’sphase-dependentresponsetoDBS,andthechangeinphasecurve,likethatinFigure15,3rdrow,fromthatsimulation.Thecorrelationcoefficientandcorrespondingp-valuewerecomputedbetweenthenumberofoscillators,𝑁,andtheplotted,correspondingMSE.

MedianchangeinphasecurvesforsimulationsusingN=2,5,and8oscillatorsareincludedinAppendix2:SimulatedChangeinPhaseCurveswithDifferentNumbersofOscillators,wherethedivergencecanbevisuallyobserved.Itisreasonablethatweakcouplingamongoscillatorswouldintroduceanon-lineartransformationbetweentheindividualPRCandthepopulationchangeinphase(WilsonandMoehlis,2015).ThiseffectaddsanotherlayerofcomplexitytousingtheextendedKuramotomodel,asatbest,onlyalargepopulationofneuronscanberecordedinthethalamusorcortex.Notethattheexperimentaldataisrecordingtremoramplitudeandphaseshiftsthere,notinthethalamusorcortex;thus,therelikelyisanothertransformationfromthethalamicand/orcorticalpopulationphaseshiftandthatobservedintremor.

ResultsusingExperimentalChangeinFrequencyandChangeinPhaseCurvesasPRCFunctionDespitethenon-lineartransformationdiscoveredabove,itwashypothesizedthatthemodelmaybeabletomodelthechangeinfrequencyandphasecurvesoftheexperimentaldataiftheywereusedasthePRCfunction.However,thesimulationsdescribedbelowshowthatwhileusingthesecurvesasPRCfunctionsproducedsomewhatqualitativelysimilarchangeinfrequencyandphasecurvestothoseoftheexperimentaldata,theywereunabletoreproducechangeinamplitudecurvesthatweresimilartothatoftheexperimentaldata.ThislastsetofsimulationsdemonstratesthelimitationoftheKuramotomodeltocapturetheoptimalphasicstrategythatworkedintheessentialtremorpatient;theKuramotosimulationscanonlymatchthechangeinamplitudecurveorthechangeinfrequencyorphasecurvesoftheexperimentaldata,butnotallthecurves.DirectlyUsingScaled,ExperimentalChangeinFrequencyCurveUsingallotherdefaultparameters,alinearinterpolationoftheexperimentalchangeinfrequencycurve(Figure10,2ndrow)wasscaledtorangefrom[−1,+1]andusedinamodelsimulationasthePRCfunction,𝑧(𝜃^)(Figure19).

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Figure19:Scaled,ExperimentalChangeinFrequencyCurveasPRCFunction𝒛(𝜽𝒊)forExtended

KuramotoModel

Thelinearinterpolationofthechangeinfrequencycurvefromtheexperimentaldata(Figure10, 2nd row) was scaled to range from −1,+1 and used as the PRC function𝑧(𝜃^) for asimulationoftheextendedKuramotomodel.

Qualitatively,thereappearedtobeabroadtremor-suppressingeffectat120°-210°(Figure20,1strow),whichisroughlya60°phaseshiftfromthetremor-suppressingeffectobservedintheexperimentaldataat240°(Figure10,1strow).However,similartotheexperimentaldata,thereisabroadpeakinthechangeinfrequencyandphasecurvesthatincludes210°-270°(Figure20,2ndand3rdrows).TheMoore-Rayleightestwasusedonthechangeinamplitude,frequency,andphasecurvesandshowedsignificantnon-uniformityinallcurves(𝑝 < 10�r).ThephasesMoore-Rayleighresultantvectorsforthenegativechangeinamplitude,changeinfrequency,andchangeinphasecurveswas156.1°,258.1°,and255.5°respectively.Thisfurthercorroboratesthequalitativeobservationsthat,whilethechangeinfrequencyandphasecurves,whenusingthescaled,changeinfrequencycurveasthePRCfunction,aremoresimilartothatoftheexperimentaldata,thetremor-suppressingeffectisnownolongerconsistentbetweenexperimentalandsimulateddata.

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Figure20:ChangeinAmplitude,Frequency,andPhaseCurvesofSimulatedDatausingScaled,

ExperimentalChangeinFrequencyCurve(Figure19)asPRCFunction

Thechangeinamplitude(inm/s2),∆𝑎(1strow),frequency(inHz),∆𝑓(2ndrow),andphase(indegrees),∆𝜃(3rd row),werecalculated forall108simulatedblocksandplottedabove (redcircles).Linearinterpolationsofthemedian∆𝑎, ∆𝑓,and∆𝜃foreachphaseofstimulationwereplotted(bluelines)alongwiththestandarderror.Themedianchangeinamplitude,frequency,andphaseforthe9referenceblocksofnostimulationwerealsoplotted(orangelines).

Fromcorrelationcoefficientscomputedbetweenmedianchangeinamplitudeandthederivativeapproximationsofthemedianchangeinfrequencyandphaseaswellasthosecomputedamongallthemedianmetrics,highcorrelativeeffectsareonlyobservedbetweenchangeinamplitudeandthederivativeapproximations.Thus,themodelstillpreservestherelationshipsamongthethreecurvesasshownearlier.

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Lastly,whileusingthisPRCfunctionappearedtoalignthechangeinfrequencyandphasecurvesmorewiththoseoftheexperimentaldata,therewasnosignificantcorrelationbetweenanyoftheexperimentalandsimulatedcurvesusingthisPRCfunction.DirectlyUsingScaled,ExperimentalChangeinPhaseCurveThescaledlinearinterpolationoftheexperimentalchangeinphasecurve(Figure10,3rdrow)wasalsousedinasimulation(Figure21).

Figure21:Scaled,ExperimentalChangeinPhaseCurveasPRCFunction𝒛(𝜽𝒊)forExtendedKuramoto

Model

Thelinearinterpolationofthechangeinphasecurvefromtheexperimentaldata(Figure10,3rdrow)wasscaledtorangefrom −1,+1 andusedasthePRCfunction𝑧(𝜃^)forasimulationoftheextendedKuramotomodel.

Thissimulationinducedatremor-suppressingeffectat120°(Figure22,1strow),incontrasttothetremor-suppressiveeffectat240°observedintheexperimentaldata(Figure10,1strow).Furthermore,itschangeinfrequencyandphasecurves(Figure22,2ndand3rdrows)arequitenoisyandflat.Yet,thechangeinfrequencycurveat120°(Figure22,2ndrow)appearstoadheretothetraditionalKuramotomodelexplanationofoptimaltremorsuppressionatthepositive,zero-crossinginchangeinphase.

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Moore-Rayleightestsdemonstratesignificantnon-uniformityinallcurves(𝑝 <0.05),withthephasesoftheresultantvectorsfor−∆𝑎,∆𝑓,and∆𝜃are137.7°,226.2°,and215.9°respectively.Thelattertophasesarewithin10°-15°tothephasesoftheresultantvectorsforstatisticaltestsontheexperimentalchangeinfrequencyandphasecurves(Table1).

Figure22:ChangeinAmplitude,Frequency,andPhaseofSimulatedDatausingScaled,Experimental

ChangeinPhaseCurve(Figure21)asPRCFunction

Thechangeinamplitude(inm/s2),∆𝑎(1strow),frequency(inHz),∆𝑓(2ndrow),andphase(indegrees),∆𝜃(3rd row),werecalculated forall108simulatedblocksandplottedabove (redcircles).Linearinterpolationsofthemedian∆𝑎, ∆𝑓,and∆𝜃foreachphaseofstimulationwere

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plotted(bluelines)alongwiththestandarderror.Themedianchangeinamplitude,frequency,andphaseforthe9referenceblocksofnostimulationwerealsoplotted(orangelines).

Forthissimulation,therewerenosignificant(𝑝 < 0.05)correlationsbetweentheexperimentalandsimulateddata.Therewerealsonosignificantcorrelationsbetweenthemedianchangeinamplitudecurveandthemedianchangeinfrequencyandphasecurvesorwiththederivativeapproximationsofthosecurves.

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DiscussionThisresearchdemonstratesclinicallyhowtofindanoptimalphaseatwhichphase-lockedDBSwouldbestdampentremorandshowscomputationallyhowaclassicneuraloscillatormodelfailstosufficientlyexplaintheoptimalstrategyinanessentialtremorpatient.Asclinicalresearchprogressestodemonstratetheeffectivenessofaphase-dependentDBSstrategyinavarietyofpatientswithdifferenttremor-relatedmovementdisorders,acorrespondingtheorywouldideallybedevelopedinordertodescribewhyphase-specificDBSeffectivelydampenstremor.Thissectionfirstsummarizesthiswork’scontributionstosuchanunderstandingandthenproposesfurtherintegrated,experimentalandcomputationalworktobetterinvestigatetheunderlyingmechanismsofphasicstimulation.

SummaryofResultsInsummary,thereareafewkeyfindingsfromthiswork:

- Anovelparadigmfordetectingandanalyzingphase-dependenttremor-suppressioninbothexperimentalandsimulateddatawaspresented.

- Anew,weightedcirculartest,theScaledRayleighstatistic,wasoutlined.- Intheessentialtremorpatientinwhichtheexperimentaltrialwas

conducted,significantphase-dependenttremorsuppressionaround240°-270°significantlycorrelatedwithlargeincreasesinfrequencyandphase,whicharethesamemeasurecalculatedusingdifferentmethodsandunits.

- FromWilsonandMoehlis,2014,tremorsuppressionshouldcorrelatewiththederivativeofthechangeinfrequencyorphase.Inthisexplanation,optimalstimulationwoulddesynchronizetwooscillatorsub-populationsatthemeanpopulationphaseatwhichchangeinfrequencyorphasechangesfromnegativetopositive,therebycausingonesub-populationtoslowdownandtheothertospeedup.However,thisexplanationfailedtoexplainthedirect,correlationbetweenchangeintremoramplitudeandfrequency/phasewithintheessentialtremorpatient.

- Thereisevidenceofanon-lineartransformbetweentheindividualoscillatorPRCfunctionusedintheKuramotomodelandtheoscillatorpopulation’smeanchangeinfrequencyorphase.Thisislikelybecauseofweaklycoupledinteractionsamongtheoscillatorpopulation.

ThisresearchisthefirstknownworktoanalyzetheKuramotomodel’sabilitytosimulatephase-dependenttremorsuppressionincomparisonwithclinicaltremordata.

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FutureWork

QuantifySignificant,Phase-SpecificTremorModulationInthesingleessentialtremorpatientinwhichthephasicDBSexperimentwasconducted,thereappearedtobesmalldampeningeffectat120°,alargersuppressiveeffectat240°,andanamplificationat30°(Figure10,1strow).Yet,withoutreferenceblocksinwhichnoDBSand/orconstant130HzDBSwasadministered,thesignificanceoftheseobservedeffectscouldnotbetested.Providedsaidreferenceblocks,apairedt-testcanbeusedtoquantifythesignificanceofsucheffects,withBonferronicorrectiontocorrectformultiplecomparisonsamong12phases.TheMoore-RayleighandScaledRayleighstatisticstestednon-uniformityofexperimentalandsimulatedchangeinamplitude,frequency,andphasecurves.However,theycouldnotrigorouslyidentifythephasesatwhichtherewassignificanttremormodulationorchangeinfrequency/phase;thephaseoftheresultantvectorissimplyaweighted,meanphaseofthedatapoints.

ModelNegativeCorrelationbetweenChangeinTremorAmplitudeandFrequency/PhaseWhilethisworkprovidedevidencethattheKuramotomodelfailstoexplaintherelationshipbetweenchangeintremoramplitudeandfrequency/phaseforasingleessentialtremorpatient,afewfurthercomputationalexperimentsandextensionscouldpotentiallyshowthatthemodelcandescribethisphenomenon.SearchModelParameterSpaceWhilesomemodelparameterswereoptimizedtobestmatchtremorcharacteristicsintheabsenceofDBS,othersparameters,i.e.numberofoscillators(𝑁)andDBSstrength(𝛽),werechosenapriori.Usingthefourproposedstimulationfunctions,theseparameters,andpossiblythepreviouslyoptimizedparametersaswell,shouldbeoptimizedtobestmatchtremorcharacteristicsofphasicDBS.Thiscouldbedonebyfindingparametersthatyieldthebestfittotheexperimentalchangeinamplitude,frequency,andphasecurves.Suchanoptimizationmayfindasetofparametersthat,whencoupledwithoneormorebiologically-plausiblestimulationfunctions,candescribethedirect,negativecorrelationbetweentremorsuppressionandfrequencyamplification.ModelandUnderstandMultipleTremor-InducingNeuralPopulationsOneofthemotivationsbehindthepresented,biologically-plausiblestimulationfunctionswastoincorporatetheunderstandingthatboththesub-thalamicnucleus(STN)andtheglobalpallidusinterna(GPi),asub-corticalregion,contributetotremor.Typically,DBSelectrodesareonlyimplantedinoneoftheseregions.Yet,withrespecttophasicDBS,therelationshipbetweenthesetworegionsandtheireffectontremorhasnotbeenthoroughlyresearched.AphasicDBSprimatestudycanbeusedtoinvestigatetheirinteractions.DBSelectrodescanbeimplantedin

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eithertheSPNorGPiwhilerecordingelectrodesareinsertedintobothregions.Then,localfieldpotentials(LFPs)inbothregionscanberecordedasphasicDBSisadministeredinasimilarset-uptotheoneusedinthiswork(Cagnanetal.,2016).DBSwouldlikelyinterferewithLFPrecordings;thus,recordingsfromshortnon-stimulatedperiods(i.e.afterashortburstofphasicDBShasbeenadministered)canbeprocessedandused.FromLFPrecordings,themeanphaseofbothneuralpopulationscanbecalculatedandcomparedtoeachotheraswellasthetremorphaserecordedbyahand-mountedaccelerometer.Thesemulti-regionandmulti-modalphaserecordingscanalsobeusedtooptimizeandbetterunderstandtheKuramotomodel,byseekingtoreplicatethenon-lineartransformationfromthePRCfunction,whichcanbeestimatedbyLFPsfromtheregioninwhichDBSelectrodeswereimplanted,tothechangeintremorphasecurve.Specificresearchshouldbefocusedonhowchangeinphasecurvesbetweenaneuralpopulationandhandtremorrelate–aretheydirectlycorrelatedorisonecorrelatedtothederivativeapproximationofanother?–andhowtheKuramotomodelcansimulatetherelationshipbetweenaneuralpopulationandadownstreambehaviorliketremor.IdeallygroundedinclinicalresearchontheeffectofphasicDBSontherelationshipsamongSPN,GPi,andhandtremor,theKuramotomodelcanbeextendedinafewwaystobettercapturetheinteractionsbetweenSPNandGPi.First,thecouplingconstant,𝐾,canbeparameterizedandmadetovaryamongoscillatorsinanefforttomodelhowaneuralpopulation’sinternalcoupling(i.e.neuronswithinSPN)maydifferfromitsexternalcoupling(i.e.couplingbetweenSPNandGPineurons).Second,thedistributionfromwhichoscillatorfrequenciesaresampledcanbechangedtoabi-modaldistribution,toreflectthattheneuralactivityofdifferentregionsmayhavedifferentfrequencies.ThefrequencycharacteristicsofneuralactivityinSTNandGPicanbeextractedfortheirLFPrecordingsandeitheruseddirectlyinthemodelorusedtoinitializeaparametersearchoptimization.Finally,therelationshipsamongSTN,GPi,andtremorcanbeexplicitlydescribedbydesigningdifferentarchitecturesofKuramotooscillatorpopulationsinteractingwithoneanother.ExtendtheModeltoExplicitlyIncorporateChangeinFrequencyTheTass,2003formulationoftheKuramotomodelthatincorporatesDBShasfourtermsthatdescribeanoscillator’s1.meanfrequency,2.coupledrelationshipwithotheroscillators,3.noisiness,and4.phase-dependentresponsetoDBS(Equation16).Anotherterm,ℎ(𝜃^),couldbeaddedtoexplicitlycapturethedirectlyinverserelationshipbetweenchangeintremoramplitudeandfrequency/phase.Thistermmayneedtobeparameterizedbynotonlytheoscillator’scurrentphasebutalsoarecenthistoryofitspreviousphasesinordertoapproximateitschangeinphase.Then,aregularizing,mixtureparameter,𝛾,canalsobeadded(Equation26)toweighthowmuchthemodelexplainstremorsuppressionusingthe“slope”explanation–thattheoptimalphaseatwhichtostimulateistheoneinwhichthereisapositive-slope,zero-crossingchangeinanindividualoscillator’sPRC(Wilson

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andMoehlis,2014)–versusthe“peak”explanation–thattheoptimalphaseistheoneinwhichthereisamaximumpeakinanindividualoscillator’sPRC.Theseexplanationsassumethatachangeinhandtremorfrequencycurveissimilartoanindividualoscillator’sPRC,asimplificationthatthisworkshowsisproblematicandthatneedstobefurtherinvestigatedaswell(see“Non-LinearTransformationfromIndividualPRCtoChangeinMeanPhaseofOscillatorPopulationNon-LinearTransformationfromIndividualPRCtoChangeinMeanPhaseofOscillatorPopulation”). 𝜃^ ← 𝜃^ + 𝑑𝑡 𝜔^ + 𝐾𝑟 sin 𝜓 − 𝜃^ + 𝛼𝒩 0, 𝑑𝑡 + 𝛾𝛽𝑧 𝜃^ 𝑢^ 𝑡 + (1 − 𝛾)ℎ(𝜃^)

Equation26:ProposedExtensiontoKuramotoModel’sUpdateRule

DevelopAnotherModelthatIncorporatesExcitatory-InhibitoryDynamicsTheKuramotomodelmakesseveralstrongassumptionsthatmaycontributetoitsinabilitytodescribetheexperimentalresultsobserved.Forinstance,itassumesthatspikingisrelativelyconstantwithitsfixedangularfrequencyterm,therebyeliminatingthepossibilityofburstingneuronsorneuronsmodulatedbyinhibition.IftheaboveresearchdirectionsusingtheKuramotomodelprovefutile,another,morecomplexmodel,suchasonethatcaptureexcitatory-inhibitorydynamics,mayneedtobedeveloped.

DescribePhase-dependentDBSStrategyforDifferentTremulousPathologiesandPatientsAmajorlimitationofthisworkisthesamplesizeofoneessentialtremorpatient.Theexperimentaltrialwasconductedinseveralessentialanddystonictremorpatients(Cagnanetal.,2016).ApreliminaryanalysisoftheexperimentaldatafromotherpatientssuggeststhatdifferentpatientsmayhavedifferentoptimalphasicDBSstrategies:thechangeinfrequency/phasecurvesofafewpatientsseemtofitthe“slope”explanationwhilethoseofotherpatientsseemtobemoreconsistentwiththe“peak”explanationobservedinthiswork’sanalysisoftremorsfromasingleessentialtremorpatient.Duetotimeconstraints,multi-patientanalysiswasnotabletobeadequatelyconductedandincludedinthiswork.Furtherclinicalresearchshouldincludemultiplepatientswithavarietyoftremor-inducingdisorders,whilefuturecomputationalworkshouldbedoneinclosecollaborationwithexperimentalworksoastobeabletomodeldifferentkindsofphase-dependent,tremor-suppressingeffectsthatmayvarybypatientand/orpathology.Ifsuchabiologically-consistentmodelweretobedeveloped,ifevenforonespecificmovementdisorder,itcouldthenbeusedtoidentifynewoptimalphase-dependentDBSstrategies.Forexample,aparametersearchcouldbeconductedtofindthesmallestDBSstrengthparameter,lowestDBSfrequency,and/ortheshortestdurationofphasicDBSadministrationthatcouldstillyieldeffectivetremorsuppression.Inthiswork,theseparametersweresetaprioriasfollows:DBS

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strength(𝛽 = 0.05),DBSfrequency(130Hz),durationofBDS(35ms–thiscorrespondedto6pulsesat130Hz).

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ConclusionThisresearchwasthefirstknownworkofitskindinitsdual,interdisciplinarymissiontoidentifyanexperimentallyoptimaltremorphase-lockedDBSstrategyinasingleessentialtremorpatientaswellastoexplaincomputationallyhowphasicDBSmightdesynchronizeaneuralpopulationandthusdampentremor.Whileacoherent,underlyingmechanismthatcancomprehensivelyexplainthetremor-suppressingeffectsofphasicDBSinthetestedessentialtremorpatientisstillunknown,thisworknonethelessmadecontributionstothisgoalaswellastothefieldsofstatisticsandcomputationalneuroscience,bypresentinganew,weightedcircularstatistic,thescaledRayleightest,implementingmorebiologically-plausiblestimulationfunctionstobeusedintheKuramotomodel,andhighlightingseveralshortcomingsofthemodel.Finally,thisdissertationproposesfuturedirectionsofresearchthatcontributetothegrowingbodyofworktodevelopenergy-efficientyeteffectiveclosed-loopDBStherapies.ThisworkwouldnothavebeenpossiblewithoutthesupportandcollaborationofProfessorRafalBogacz,Dr.HayriyeCagnan,andDr.GihanWeerasinghe.

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Appendix1:AlternativeMetricsofEvaluatingEffectsofTremorSuppressionAlternativemetricsweredevelopedforcalculatingchangeinamplitude,frequency,andphase.Asummaryofthemwillbepresentedfollowedbyanexplanationofeachmetric.

ChangeinAmplitudeTheresultsreportedinthisdissertationusethechangeinamplitudemetricgivenbyEquation4.However,10alternativemetricsweredeveloped.Thep-valuesandphasesoftheirresultvectorsfromtheMoore-RayleighandScaledRayleightestsareprovidedinTable9.Themeanphaseoftheresultvectorforalltestsforwhichthep<0.05thresholdwasmet(boldedinTable9)was69.8°,withastandarddeviationof8.7°.Thisisthediametricoppositeof249.8°. p-value Phaseofresultant

vector Moore-

RayleighScaledRayleigh

Moore-Rayleigh

ScaledRayleigh

'subtract_last_block' 0.0137 0.0277 75.1° 82.8°'subtract_last_block_norm' 0.0165 0.9239 73.4° 147.5°'point_normalize' 0.205 0.7449 49.0° 13.7°'simple_median' 0.4424 0.4246 103.7° 117.1°'normalize' 0.2393 0.0521 64.1° 359.6°'subtract_prev' 0.0002 𝟏𝟎�𝟒 61.5° 61.4°'subtract_prev_norm' 𝟏𝟎�𝟒 0.2002 64.4° 24.8°'regression' 0.6202 0.5376 143.9° 148.4°'poly2_latent' 0.079 0.0586 109.9° 111.7°'poly2_diff' 0.6846 0.814 101.8° 126.7°Table9:StatisticalSignificanceofAlternativeMetricsMeasuringChangeinAmplitudeinExperimental

Data

Tests of metrics for which the p < 0.05 threshold was met are bolded. Using Bonferronicorrection (𝛼 = 10), the adjusted threshold to meet is p < 0.005; the tests that meet themultiplecomparisonsthresholdareunderlined.

ChangeinFrequency&PhaseTheresultsreportedinthisdissertationusethechangeinfrequencyandphasemetricsgivenbyEquation4andEquation11respectively.However,8alternativemetrics–2forchangeinphaseand6forchangeinfrequency–weredeveloped.Furthermore,therewere2methodsforcalculatingthefrequencyforaperiod:1.usingthemedianoftheinstantaneousfrequency,and2.estimatingfrequencyfrom

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thenumberofzero-crossingsintremoramplitude.Thep-valuesfromtheMoore-RayleighandScaledRayleightestsareprovidedinTable10.Themeanphaseoftheresultvectorforalltestsforwhichthep<0.05thresholdwasmet(boldedinTable9)was267.2°,withastandarddeviationof16.2°. p-value Moore-Rayleigh ScaledRayleigh'prc_point_normalize'(inst) 0.004 0.0026'prc_point_normalize'(zc) 0.0039 0.0146'prc_first_last_median'(inst) 0.0034 0.0016'prc_first_last_median'(zc) 0.0026 0.0111'subtract_last_block'(inst) 0.0174 0.1019'subtract_last_block'(zc) 0.0002 𝟏𝟎�𝟒'subtract_last_block_norm'(inst) 0.0164 0.1112'subtract_last_block_norm'(zc) 0.0002 0.0002'simple_median'(inst) 0.0992 0.0396'simple_median'(zc) 0.0386 0.0319'normalize'(inst) 0.3488 0.4537'normalize'(zc) 0.2842 0.2606'subtract_prev'(inst) 0.0019 0.0014'subtract_prev'(zc) 0.004 0.0132'subtract_prev_norm'(inst) 0.0018 0.0019'subtract_prev_norm'(zc) 0.0036 0.018Table10:StatisticalSignificanceofAlternativeMetricsMeasuringChangeinFrequencyandPhasein

ExperimentalData

Tests of metrics for which the p < 0.05 threshold was met are bolded. Using Bonferronicorrection (𝛼 = 16), the adjusted threshold tomeet is p < 0.0031; the tests thatmeet themultiplecomparisonsthresholdareunderlined.“inst”denotesmetricsthatusedinstantaneousfrequency;“zc”denotesthosethatusedthezero-crossingmethodtocalculatefrequency.

AlternativeMetrics

'subtract_last_block'Usedforchangeinamplitudeandfrequency,thismetrictookthedifferencebetweenthemedianamplitudeorinstantaneousfrequency(orzero-crossingestimateofthefrequency)overthewhole5-secondperiodofthecurrentblockandthatofthepreviousblock.Iftherewasnopriorblock,themedianamplitudeorfrequency(orzero-crossingestimate)wascalculatedforthe1-secondno-stimulationperiodimmediatelyprecedingthecurrent5-secondblock.

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'subtract_last_block_norm'Usedforchangeinamplitudeandfrequency,thismetricnormalizedthe‘subtract_last_phase’metricbydividingitbythemedianamplitudeorfrequency(orzero-crossingestimate)ofthepreviousblock.

'point_normalize'Usedforchangeinamplitude,thismetrictookthedifferencebetweentheamplitudeattheendofthecurrentblock(t=5s)andtheamplitudeatthebeginningofthecurrentblock(t=0s)andnormalizedthisdifferencebythelatterterm.

'simple_median'Usedforchangeinamplitudeandfrequency,thismetricwassimplythemedianoftheamplitudeorinstantaneousfrequency(orzero-crossingestimateoffrequency)overthewhole5-secondperiodofthecurrentblock.

'normalize'Usedforchangeinamplitudeandfrequency,thismetrictookthedifferencebetweenthemedianamplitudeorinstantaneousfrequency(orzero-crossingestimateoffrequency)ofthelastsecondofthecurrentblock(t=4-5s)andthatofthefirstsecondofthecurrentblock(t=0-1s)andnormalizedthisdifferencebythelatterterm.

'subtract_prev'Usedforchangeinamplitudeandfrequency,thismetrictookthedifferencebetweenthemedianamplitudeorinstantaneousfrequency(orzero-crossingestimateoffrequency)ofthewhole5-secondperiodofthecurrentblockandthatofthe1-second,no-stimulationperiodimmediatelyprecedingit.

'subtract_prev_norm'Usedforchangeinamplitudeandfrequency,thismetricnormalizedthe‘subtract_prev’metricbydividingitbythemedianamplitudeorinstantaneousfrequency(orzero-crossingestimateoffrequency)ofthe1-second,no-stimulationperiodimmediatelyprecedingit.

'regression'Usedforchangeinamplitude,thismetricwastheslopeofalinefittedtothetremoramplitudevaluesforthewhole5-secondperiodofthecurrentblock.

'poly2_latent'Usedforchangeinamplitude,thismetricwasthetimestepatwhichaquadraticlinefittothetremoramplitudevaluesforthewhole5-secondperiodofthecurrentblockwasaminimumormaximumpoint.

'poly2_diff'Usedforchangeinamplitude,thismetricwasthedifferencebetweentheminimumormaximumpointfromafittedquadraticlinetothetremoramplitudevaluesforthewhole5-secondperiodofthecurrentblockandthemidpointvaluebetweenthetremoramplitudevaluesatthebeginning(t=0s)andend(t=5s)oftheblock.

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'prc_point_normalize'Usedforchangeinphase,thismetrictookthedifferencebetweentheunwrappedphaseattheendofthe5-secondperiodofthecurrentblock(t=5s)andtheprojected,unwrappedphasebasedonthemedianfrequency(orzero-crossingestimate)ofthepreviousblockandnormalizeditbytime(5seconds)andbythelengthofacycle(2𝜋).Iftherewasnopreviousblock,the1-second,no-stimulationperiodimmediatelyprecedingthecurrentblockwasusedasthereferencesignalinstead.Theonlydifferencebetweenthismetricandtheoneusedinthisdissertationwasthereferencefrequency;forthedefaultchangeinphasemetric,thereferencefrequencywasthemedianinstantaneousfrequencyofthe1-second,no-stimulationperiodimmediatelyprecedingthecurrentblock.

'prc_first_last_median'Usedforchangeinphase,thismetricissimilartothepreviousmetric,exceptitusedthemedianunwrappedphaseofthelast100mssecondofthecurrentblock(t=4.9-5s)andtheprojected,unwrappedphaseofthesame100msperiodusingthepreviousblock’sfrequencyasthereferencefrequency.

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Appendix2:SimulatedChangeinPhaseCurveswithDifferentNumbersofOscillatorsFigure23showshowtheoscillatorpopulation’schangeinphasecurvedivergesmorefromtheindividualPRCfunction,𝑧(𝜃^) = cos(𝜃^ +

��),asthenumberof

oscillatorsincreases.

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Figure23:ChangeinPhaseCurveforSimulatedDatawhenDifferentNumbersofOscillators(N=2,5,8)

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The change in phase,∆𝜃 , was calculated for all 108 simulated blocks generated in eachsimulationusingthe2oscillators(1strow),5oscillators(2ndrow),and8oscillators(3rdrow)respectively and plotted above (red circles).When usingN= 2, 5, and 8 oscillators, linearinterpolationsof themedian∆𝜃foreachphaseof stimulationareplotted (blue lines)alongwiththestandarderror.Themedianchangeinphasevaluesforthe9referenceblocksofnostimulationarealsoplotted(orangelines).

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Appendix3:DifferentStimulationFunctionsThechangeinfrequency(Figure24)andphasecurves(Figure25)forsimulationsusingthe“half”,“random”,and“uniform”stimulationfunctionsareincludedhere.

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Figure24:ChangeinFrequencyCurvesforSimulatedDatawhenusing“Half”,“Random”,and“Mixture”

Stimulationfunctions

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Thechange in frequency,∆𝑓,wascalculated forall108simulatedblocksgenerated ineachsimulationusingthe“Half”(1strow),“Random”(2ndrow),and“Mixture”(3rdrow)stimulationfunctionandplottedabove(redcircles).Whenusingeachofthesestimulationfunctions,linearinterpolationsof themedian∆𝑓foreachphaseof stimulationareplotted (blue lines)alongwiththestandarderror.Themedianchangeinfrequencyvaluesforthe9referenceblocksofnostimulationarealsoplotted(orangelines).

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Figure25:ChangeinPhaseCurvesforSimulatedDatawhenusing“Half”,“Random”,and“Mixture”

Stimulationfunctions

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The change in phase,∆𝜃 , was calculated for all 108 simulated blocks generated in eachsimulationusingthe“Half”(1strow),“Random”(2ndrow),and“Mixture”(3rdrow)stimulationfunctionandplottedabove(redcircles).Whenusingeachofthesestimulationfunctions,linearinterpolationsof themedian∆𝜃foreachphaseof stimulationareplotted (blue lines)alongwiththestandarderror.Themedianchangeinphasevaluesforthe9referenceblocksofnostimulationarealsoplotted(orangelines).

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Appendix4:ExtendedKuramotoModelSimulationToolAwebsimulationoftheKuramotoneuraloscillatorswasdevelopedtobeusedinapublicengagementeventwithlocalOxfordshirehighschoolstudents.Itcanbeusedasagame,inwhichusersearnpointsbydesynchronizingapopulationofKuramotoneuraloscillators,tointuitivelyteachtheoptimalphase-dependentDBSstrategythattheKuramotomodelsuggests(i.e.stimulatingatthepositivezero-crossinginthePRCfunction).Itcanalsobeusedasaresearchandvisualizationtool,wherenearlyalltheKuramotosimulationparameterscanbesetbytheuser,includingaphasicstimulationstrategy.Theusercanthenobservethestateoftheneuraloscillatorpopulation,itssynchrony,andthetremorsignalasthesimulationrunswiththeuser-specifiedmodelparameters.Thesimulationtoolcanbeaccessedatthislink:http://ruthcfong.github.io/kuramoto.

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Figure26:ScreenshotofSimulationTool