A novel automated spike sorting algorithm with adaptable feature extraction

ContentslistsavailableatSciVerseScienceDirect

Journal

of

Neuroscience

Methods

j o u r n al ho m e p age :w w w . e l s e v i e r . c o m / l o c a t e / j n e u m e t h

Computational

Neuroscience

A

novel

automated

spike

sorting

algorithm

with

adaptable

feature

extraction

Robert

Bestel

∗

, Andreas

W.

Daus, Christiane

Thielemann

BioMEMSLab,UniversityofAppliedSciencesAschaffenburg,63743Aschaffenburg,Germany

h

i

g

h

l

i

g

h

t

s

 Newfeatureextractionapproachbasedonreviewofcommonspikesortingmethods.  Variablefeaturederivationbyevaluatingthefeature’sprobabilitydistribution.  Validationofthealgorithmwithneuronalmonolayerand3Dspheroidsignaldata.  Comparisonoftheresultswithcommonfeatureextractiontechniques.

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received8May2012

Receivedinrevisedform13August2012 Accepted15August2012 Keywords: Spikesorting Microelectrodearray Neuron Actionpotential Biosensor

a

b

s

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c

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Tostudytheelectrophysiologicalpropertiesofneuronalnetworks,invitrostudiesbasedon microelec-trodearrayshavebecomeaviabletoolforanalysis.Althoughinconstantprogress,achallengingtask stillremainsinthisarea:thedevelopmentofanefficientspikesortingalgorithmthatallowsanaccurate signalanalysisatthesingle-celllevel.Mostsortingalgorithmscurrentlyavailableonlyextractaspecific featuretype,suchastheprincipalcomponentsorWaveletcoefficientsofthemeasuredspikesignalsin ordertoseparatedifferentspikeshapesgeneratedbydifferentneurons.However,duetothegreatvariety intheobtainedspikeshapes,thederivationofanoptimalfeaturesetisstillaverycomplexissuethat currentalgorithmsstrugglewith.Toaddressthisproblem,weproposeanovelalgorithmthat(i)extracts avarietyofgeometric,Waveletandprincipalcomponent-basedfeaturesand(ii)automaticallyderivesa featuresubset,mostsuitableforsortinganindividualsetofspikesignals.Thus,thereisanewapproach thatevaluatestheprobabilitydistributionoftheobtainedspikefeaturesandconsequentlydetermines thecandidatesmostsuitablefortheactualspikesorting.Thesecandidatescanbeformedintoan individ-uallyadjustedsetofspikefeatures,allowingaseparationofthevariousshapespresentintheobtained neuronalsignalbyasubsequentexpectationmaximisationclusteringalgorithm.Testresultswith sim-ulateddatafilesanddataobtainedfromchickembryonicneuronsculturedonmicroelectrodearrays showedanexcellentclassificationresult,indicatingthesuperiorperformanceofthedescribedalgorithm approach.

© 2012 Elsevier B.V.

1. Introduction

Themostbasicgoalofneuroscienceisundoubtedlyto

under-standtheprocessesofthehumanbrain,notonlytocomprehend

thethoughtprocessitself,butalsotofindmethodsthattreat

neu-rologicaldiseases suchas epilepsy,depression and Alzheimer’s

disease.Onewaytostudythebehaviourofneuronalnetworksis

tomonitorelectricsignalsgeneratedbyvariousneuronsinsuch

anetwork. In vitroexperiments enabletheanalysisof cell–cell

interactionsincontrolledlaboratoryconditionsandfurtheroffer

improvedresearchopportunities becauseofa reduced network

complexity,aswellasbetteraccessibility.Thestandardtechnique

∗ Corresponding authorat: WürzburgerStraße 45, D-63743 Aschaffenburg, Germany.Tel.:+496022813612;fax:+496022813611.

E-mailaddress:[email protected](R.Bestel).

toderiveneuronalsignalsisthepatch-clampmethod(Sakmann

andNeher,2009).Inrecentyears,however,theusageof

micro-electrodearrays(MEAs)hasbecomeincreasinglypopular,asthis

techniqueallowsnon-invasiveandlabel-freerecordingsfrom

mul-tiplepositionsintheneuronalnetwork(Thomasetal.,1972).

However,fundamentalnetworkanalysisstillfacescertain

dif-ficulties,sinceitcannotbeguaranteedthattherecordedelectrode

signaloriginatesonlyfromonesinglecellratherthanfrom

mul-tipleneuronsintheproximityofthatspecificelectrode. Dueto

thisuncertainty,specialmethodsarerequiredtoseparate

differ-entcellsignalswithinthemeasuredelectrodesignal.Onepossible

approachtakesintoaccountthefactthatthederivedshapeofthe

signalchangeswithrespecttothepositionoftheneuronandthe

correspondingelectrode(seeFig.1).

However, none of the proposed spike sorting methods can

bebroadlyestablished,mainlyduetoaccuracyandperformance

issues.Inthefollowingreport,wepresentanoverviewofspike

0165-0270© 2012 Elsevier B.V.

http://dx.doi.org/10.1016/j.jneumeth.2012.08.015

Open access under CC BY-NC-ND license.

Fig.1.Schematicofthesignalcomponentsthatcanparticipateinthegenerationofameasuredelectrodesignal.Thesignalcancontainspikesgeneratedbydifferentneurons. Theshapesoftheseneuralspikesshouldvaryduetodifferentcouplingqualitiesbetweentheindividualneuronandtheelectrode.Inaddition,noise,originatingfromvarious sources,isusuallypickedupbytherespectiveelectrode.

sortingalgorithmsalreadydevelopedinthisarea.Wealsopresent

anewapproachtothespikesortingproblem,focussingon

possi-blefeaturesthatareusedtodescribethedifferentshapepatterns

ofneuronalspikes.Insteadoflimitingthepoolofpossiblefeatures

toacertaintype,suchasprincipalcomponentsorWavelet-based

features, the proposed algorithm calculates a variety of spike

parameters. Witha newlydeveloped evaluation method,based

onexpectationmaximisation(EM)approximation,thealgorithm

selectsfeaturesthatarethemostsuitableforthespikesortingofa

particulardataset.

2. Stateoftheart

Ananalysisofthecurrentliteratureonneuronalpattern

recog-nition reveals a wide variety of spike sorting approaches (see

Table1).Usually,mostspikesortingalgorithmscanbedividedinto

twomajorsteps:featureextractionfollowedbyclassificationor

clustering.Inaddition,Table1alsoincludesinformationaboutthe

usedspikedetectiontechnique,asthispre-processingstepcanalso

influencetheresultofthespikesortingsignificantly.Many

differ-entapproacheshavealreadybeentestedandintroducedinorder

toaddresstheproblemathand.

Intermsofspikedetection,themostcommonmethodsuse

sim-plethresholdingtechniquestoextracttheactualspikesignalsfrom

thenoisymeasurementdata.Moreelaboratedalgorithmsfurther

addspecificsignaltransformations,e.g.usingthenonlinearor

Tea-gerenergyoperator(NEOorTEO)(KimandKim,2000;Choietal.,

2006)tothethresholdingprocessinordertodecreasethe

influ-enceofnoiseorlowfrequencysignalartefacts.Anotherapproach

presentedbyHulataetal.(2002)usesWavelet-based

decomposi-tiontodiscriminateactualspikeshapesfromnoise.Suchextended

detectionmethodsusuallyyieldamoreprecisedetectionresult,

whichfacilitatestheactualsortingprocess,butattheexpenseofa

highercomputationalcomplexity.

Themost commonmethodof feature extractionand

reduc-tionisstilltheprincipalcomponentanalysis(PCA)(Woodetal.,

2004; Wanget al., 2006;Biffi et al.,2008).However,

Wavelet-orWaveletpacket-basedfeatureshavealsobecomequite

popu-larastheresultingcoefficientsarealsocapableofdescribingthe

differencesofvariousspikesignals(OweissandAnderson,2002;

Hulataetal.,2002;Quirogaetal.,2004).Adrawbackofthis

com-plexmethod is thenecessity ofan additional selection step to

identifythecoefficientsthatmostpreciselydiscriminatebetween

thevariousspike shapes.In somecases, geometricfeaturesare

alsoobtainedfromthemeasuredspikeevents(Feeetal.,1996;

Vogelsteinetal.,2004)asanalternativeor,morecommonly,a

com-plementtostandardPCAfeatures.Thevarietyoftheusedspike

sortingfeaturesindicatesthatnooptimalfeaturesethasyetbeen

established.Thismightbeduetothefactthatitisquestionable

whetherafixedsetoffeaturesexiststhatcangenerallyguarantee

asuitablespikeshaperepresentation.

While ananalysis ofcommon spikesorting features already

showscertaindiversity,thisisevenmoreapparentintheareaof

possibleclassificationalgorithms.Manydifferentpattern

recogni-tionapproaches,rangingfromsimpleclusteringtechniquessuch

ask-means(Hulataetal.,2002;Satoetal.,2007),moreelaborate

expectationmaximisation(KimandKim,2003;Woodetal.,2004)

andsuperparamagneticclusteringmethods(Quirogaetal.,2004),

totrainableclassificationalgorithmssuchasneuralnetworks(Kim

andKim,2000)andsupportvectormachines(SVMs)(Vogelstein

etal.,2004),havealreadybeentestedforneuralspikesorting.

How-ever,noneoftheseapproacheshasbeenabletoprovidegenerally

optimalresultstobewidelyestablished.Whereastrainable

algo-rithmscertainlyprovideagreaterpotential,theirperformancestill

seemstosufferduetoalackoffundamentaltrainingdata.Unlikein

otherareas,forexamplevoicerecognitionorpersonidentification,

inwhichbasictrainingdatabasesalreadyexist,suchdatasetshave

notbeenestablishedforneuralpatterns,therebylimitingactual

algorithmtraining.

In addition, many spike sorting algorithms also include a

final template matching step to further refine the

Table1

Overviewofthepublishedspikesortingalgorithms.

Author/year Detection Features Classification

Hulataetal.(2002) Threshold;Wavelet reconstruction

Waveletcoefficientsbasedon Shannonentropy

k-meansclustering

Wangetal.(2006) NEO+threshold PCA Subtractive;k-meansclustering;templatematching Vargas-IrwinandDonoghue(2007) Threshold PCA Densitygridcontourclustering;templatematching Feeetal.(1996) Threshold Waveformalignmentbasedon

multi-electrodewaveformpair

Bisectionclustering;iterativefusionofclusterprocesses

Woodetal.(2004) Threshold PCA Spectralclustering;EMwithMoG

Satoetal.(2007) Peak–peakthreshold PCA Iterativek-meansusingDBVI;templatematching KimandKim(2003) – Projectionpursuit/neg.entropy

maximisation

EMwithMoG

KimandKim(2000) NEO+threshold Spikeshape Artificialneuralnetwork(MLPwithRBFN)

Shohametal.(2003) Threshold PCA EMwitht-distributions

Biffietal.(2008) Adaptivenoisefiltering; pos.+neg.thresholddetection

PCA Hierarchicalclustering

Zhangetal.(2004) Threshold PCA Subtractiveclustering;templatematching Hortonetal.(2007) Threshold PCA+spikeshapecurveindices Kohonennetwork

OweissandAnderson(2002) Threshold Waveletpacket decomposition;bestbasis evaluatedbySVD

–

Quirogaetal.(2004) Threshold Waveletpacketcoefficients (Lillieforstest)

Superparamagneticclustering

Choietal.(2006) MTEO PCA Fuzzyc-meansclustering

Vogelsteinetal.(2004) SVM Spikeshapes SVM

Chenetal.(2011) NEO+threshold Waveletcoefficientsevaluated byPCA

Watershedalgorithm Takegawaetal.(2010) Threshold Waveletcoefficientsevaluated

bydistribution

VariationalBayesclusteringusingt-distributions

2007). However, since the result depends on prior template

extractionmethods,whichusuallyincludethealready-mentioned

clustering techniques, inaccuracies cannot always be

compen-sated.

Inconclusion,therestillseemstobenooptimalmethodthat

isgenerallycapableofsortingneuralspikepatterns.Onereason

forthismightbethelackoffundamental testdatasets

contain-ingaprincipalspectrumoftheoreticallypossiblespikeshapesthat

couldbeusedtoevaluatethedifferentclassificationalgorithms.As

thisproblemalsopreventsathoroughevaluationofthe

suitabil-ityofdifferentspikefeatures,itisgenerallydifficulttoevaluate

thefeasibilityofspecifictypesoffeaturessuchasprincipal

compo-nentsorWaveletcoefficients.Althoughcertainstudieshaveshown

thatWavelet-basedfeatureextractionpotentiallyoutperformsthe

PCA,thisfactcannotbegeneralisedtoallpossibledatasets(Pavlov

etal.,2007).Therefore,anaprioriexclusionofPCAfeaturescan

limittheability todiscriminatebetweencertainspikeforms. In

fact,this isalsotrueforotherfeatures thatmightbeinferiorin

mostcasesbutnotingeneral.Hence,itmightnotbeusefultofocus

onaspecificfeatureextractiontechnique,asotherfeaturesmight

bemoresuitable onspecificoccasions.Therefore,incontrastto

mostmethods,thealgorithmpresentedinthispaperregards

var-iouspossiblefeatures,suchasprincipalcomponentsanddifferent

Waveletpacketcoefficients,aswellasgeometricfeatures.Assuch

anapproachrequiresauniquefeaturereductionstepinorderto

determinethefeaturesmostsuitablefortheparticularspikesof

thesignal,anewlydevelopedfeatureevaluationstepisproposed

inthispaper.Byanalysingtheprobabilitydistributionofthe

dif-ferentfeaturecoefficients,thecandidateswiththemostdistinctive

multimodalcharactercanbefound.Resemblingthedifferentspike

shapecharacteristics,thismethodallowsthederivationofspike

sortingfeaturesetsthatarecustomisedforeachindividualsetof

neuronaldata.

3. Method

Aftertheshortoverviewofspikesortingalgorithmsgiveninthe

previouschapter,thefundamentaltheoryofthemethodpresented

inthispaperisdescribedinthefollowing.

3.1. Featureextraction

Theimplementationofageneralisedfeatureevaluationstepin

thespikesortingprocessdoesnotonlyprovidetheopportunityto

useamixtureofPCAandWaveletfeaturesfortheclassification

step,butalsoallowstheinclusionofotherfeaturesthatare,for

example,basedontheshapeofthemeasuredspikesthemselves.

Certain characteristicsof the spike,e.g. itsnegative or positive

amplitude,aswellasvariousgradientsoritssignalenergycanbe

usedasadditionalcandidates.Ontheonehand,such

geometry-basedfeaturesareusuallyvulnerabletonoisesignalsandvoltage

offsets.Ontheotherhand,theycanbeverypotentinparticular

situations,inwhichsuchafeaturecanspecificallyhighlighta

dif-ferencebetweentwospikeshapes.Furthermore,thesefeaturescan

bequicklyandeasilycalculatedandrepresentnoimminentthreat

tothealgorithm’scomputationalcomplexity(seeTable2).

3.1.1. Geometricfeatures

Thecalculationofthepositiveandnegativeamplitudeis

basi-callyself-explanatory.Theleftandrightspikeanglesrepresentthe

leftandtherightgradientofthecharacteristicspikeminimumto

thepointinthesignal,atwhichitcrossesthemarginof50%of

thatvalue,asshowninFig.2.Thechoiceofthisparticularmargin

canbeexplainedbylowliabilitytothenoisecomponentsofthe

signal.Theresultingvaluecaneasilybetransformedintoan

angu-larmeasure,usingbasictrigonometricfunctions.Inaddition,the

Table2

Overviewofneuronalsignalfeatures.

Feature Advantages Disadvantages

Negativeamplitude Easytointerpret Vulnerabletosignaloffsets

Positiveamplitude Stressesspecificshapecharacteristics Onlysuitableinparticularcases;vulnerabletosignal offset

Left/rightspikeangle Stressesspecificshapecharacteristics Onlysuitableinparticularcases;vulnerabletonoise distortions

Neg./pos.signalenergy Robustfeaturesintermsofnoisedistortionsorlow samplerates

Naturallycorrelateswithamplitudeandangle features;vulnerabletosignaloffsets

Corespikeduration Stressesspecificshapecharacteristics Onlysuitableinparticularcases;vulnerabletonoise distortions

NEOcoefficient Higherresolutionthannegativeamplitudein particularcases

Naturallycorrelateswithnegativeamplitude Principalcomponents Usuallyaccuratedescriptionofdatasetswithasetof

uncorrelatedparameters

Possibleinaccuraciesifmorethantwospikeshapesare presentinthedataset

Distribution/ShannonWaveletcoefficients PotentWaveletscalefeatureswithnosignificant correlation

Highcomputationalcomplexitycomparedtoother methods;multi-resolutionanalysislimitedbysample rate

intersectionpointsofbothgradientswiththe0Vlevel.The

posi-tiveandnegativesignalenergyofacontinuous-timesignals(t)can

beapproximatedbasedonthefollowingequation(OhmandLüke,

2010):

Es=



_∞

−∞

s2(t)dt (1)

Likewise,thesignalenergycanalsobecalculatedfordiscrete-time

signalsbysummingthesquareddiscretesignalvalues.Inthiscase,

thepositiveandnegativeenergycomponentsofthespikeare

calcu-latedinparticulartoseparatemonopolarfrombipolarspikeshapes

moreaccurately.

3.1.2. NEOcoefficients

ThevalueoftheNEOoperator(x[n])iscalculatedforthe

char-acteristicspikeminimumvaluewiththeequationbelow(Choietal.,

2006):

(x[n])=x2[n]−x[n−1]×x[n+1] (2)

Theresultingparameter(x[n])givesanestimateoftheenergy

contentofthesignalatthediscretetimestepnofthesignalx(n).

3.1.3. Principalcomponents

Thefirstfourprincipalcomponentsarederivedbythestandard

principalcomponentanalysis(PCA)algorithmbasedonthe

eigen-vectors and their corresponding eigenvalues of the covariance

res

ulting

wavelet

coe

fficien

ts

h[n] g[n] h[n] g[n] 2

1. iteration

2. iteration

3. iteration

ti

m

e re

so

lu

ti

o

n

fr

equenc

y r

esol

u

tion

discrete

time sign

al

digital hig

h

pass fil

ter

digital low

pass fil

ter

h[n] g[n]g[n] h[n] g[n]g[n] h[n] g[n] h[n] g[n]g[n] h[n] g[n]g[n] 2 2 2 2 2 2 2 2 2 2 2 2 2

Fig.3.SchematicoftheWPDmulti-resolutionanalysis.

matrix(Joliffe,2002).Thelatteriscomputedwithallthevoltage

valuesrepresentingtheshapeofeachspike,forallspikeevents

foundinthepreviousspikedetectionstep.

3.1.4. Waveletcoefficients

TwodifferenttypesofWaveletcoefficientsarecalculatedusing

theWaveletpacketdecomposition(WPD),whichisanextension

ofthe standard discrete multi-resolutionWavelet analysis.The

Wavelettransformationofacontinuous-timesignalcanbewritten

assuch(Stark,2005): Lf(a,t)= 1



(|a|)



∞ −∞ 



_u−t a



f(u) du (3)

wherethefunctiona,t(u)denotestheresultingWaveletbasedon

aspecificmotherWaveletandmodifiedbythetwoindependent

variablesaandu.Thelatterisanadditionaltimevariabledescribing

thetemporalshiftoftheWaveletwithrespecttotheactualsignal,

whiletheactualfrequencyorscaledecompositionisvariedbythe

parametera.Inotherwords,differentscalingvaluesofagenerate

differentdaughterWavelets,amodificationofthemotherWavelet,

whichhighlightsspecificfrequencyareasinthedecomposed

sig-nal.Fordiscrete-timesignals,aniterativeWaveletdecomposition

process,alsocalledmulti-resolutionanalysis,isusuallydescribed

asafilteringprocessthatbreaksupthesubjacenttimesignalat

alow-passandaband-passscalewitheachiteration.Hence,the

resultingfrequencyscaleresolutionincreaseswitheverystepofthe

process,whereasthetimeresolutiondecreasesrespectivelywith

eachsplitting.Astheratiobetweenfrequencyandtimeresolution

ispredefinedbytheuncertaintyprinciple,thestandard discrete

Wavelettransform(DWT)multi-resolutionanalysisonlyfocuses

onthelow-passcomponentinsubsequentiterations,sincean

accu-ratefrequencydeterminationisespeciallycrucialinthisparticular

range.Incontrast,theWPDcontinuestodecomposeboththe

high-andlow-passcomponentwitheachstepoftheanalysis(seeFig.3).

Thisallowsamoreaccuratesignalanalysisattheexpenseof

additionalcomputationalcomplexity.Theresultisusuallyreferred

toastheWaveletpackettree(seeFig.3),withadifferentsetof

Waveletcoefficients,callednodes,foreachWaveletfilter.Eachof

thesecoefficientsdescribesthemagnitudeofacertainfrequency

rangeinaspecifictimeinterval,whichresultsinaconsiderable

amount of redundant information with every additional

itera-tion. As a consequence, the most crucial step with respect to

thespikesortingtaskis probablythederivationoftheWavelet

scalecoefficientsthatdescribethedifferencesofthespikeshapes

detectedintheactualtimesignal.

TheShannoninformationcriteriacanbeusedtodeterminethe

amountof information explained byeach node ofthe Wavelet

packettree(Hulataetal.,2002).Basedontheresultingvalues

cal-culatedforeverydecomposedspike,itispossibletoevaluatethe

divergenceofexplainedinformationforallnodesthroughoutall

spikeevents.Themostsuitablenodecanbeanalysedbasedonthe

distributionofthevaluesoftheShannoninformationcriteriaacross

everyspike.Undertheassumptionthattwodifferentspikeshapes

produceamultimodaldistributionatcertainnodes,a

determina-tionofthesecanprovideusefulfeaturetypeforthespikesorting

process.

Toobtainasecondtype ofWavelet-basedfeatures,thesame

process can be implemented for the resulting Wavelet packet

coefficientsthemselves.Again,thegoalistofindacoefficientthat

showsamultimodaldistributionconsideringitsvalueacrossall

detectedspikes.Therefore,thefundamentalextractionalgorithm

isidenticalandonlytheinputvariablesareactuallychanged.

Intheproposedalgorithm,threeWaveletcoefficientsforboth

typeswiththemostpromisingprobabilitydistributionwere

cho-senasspikesortingfeatures.Inotherwords,thebestthreeWavelet

coefficientscalculated withtheShannoninformation criteriaas

wellasthebestthreeWaveletcoefficientswithrespecttotheir

naturaldistributionwereused.Asonlytheresultingdistribution

and not the calculated values themselves wereof interest, the

Fig.4. Exampleofdifferentfeaturedistributionsextractedfromaneuronaldataset.The“negativeamplitude”andthe“firstprinciplecomponent”inAandB,respectively, seemtobesuitable,whereasthesecondnativeWaveletcoefficientinConlyoffersalimitedperspective.ThefirstShannonWaveletcoefficientinDmayalsobequitesuitable, butitsdistributionislessdistinctivecomparedtothatinB.

themulti-resolutionanalysis,sinceitprovidesthefastest

comput-ingtimewithoutsignificantlychangingtheoutcomecomparedto

othermotherWavelets.Toavoidahighlyredundantfeatureset,the

correlationbetweenthecandidateswasalsotakenintoaccount.

EvenapromisingWaveletcoefficientwasdiscardedifits

correla-tionvaluewithrespecttothesuperiorcoefficientsalreadychosen

crossedacertainthresholdvalue.

Inconclusion,thedistributionapproachanalysesthe

decom-position result in further detail; however, it also requires a

considerableamountof additional computation time compared

to the Shannon entropy-based method. Nevertheless, as each

approachcanidentifydistinctspikecharacteristics,itisbeneficial

toextractbothtypesofWaveletfeaturesforfurtherevaluation.

3.2. Featureevaluationandreduction

Witheveryspikesortingfeaturecalculated,thekeyistofind

thesetthat is most suitablefor the taskathand. Therefore, it

isimportant tokeep evaluatingthefeature’s distributions with

respecttotheirseparability.Inthefollowingsection,thecrucial

partofthepresentedalgorithmisdescribed,whichisusedto

eval-uateandcomparetheextractedfeatures.Forthesakeofclarity,this

isthesamealgorithmthatwasappliedtoidentifythemostsuitable

WaveletcoefficientsforbothWaveletfeaturetypes.

Tofurtherspecifytheactualgoalofthealgorithm,itishelpful

tolookatcertaindistributionfunctionsthatcanbefoundinthis

context(seeFig.4).

LookingatFig.4,itisquiteobviousthat thetwo featuresin

Fig.4AandBseemtobepromisingexampleswithaclearly

identifi-ablemultimodalcharacter,possiblyallowingtheseparationoftwo

spikeshapes.Thus,thedesiredfeaturedistributionshouldnotonly

showatleasttwodistinctmaxima,butalsooughttobeclearly

sep-aratedfromeachother.Therefore,asbothcriteriashouldbeideally

describedinasinglevalue,thisinevitablyleadstosomeblurriness

whenevaluatingthevariousfeaturecandidates.Asaconsequence,

itiscrucialthatbothcriteriaareratedasaccuratelyaspossiblein

ordertominimisetheresultinguncertainty.Hence,the

expecta-tionmaximisation(EM)methodwaschosenforthispurpose,asit

offerssuitableparametersforassessment.AnEMalgorithmusing

Gaussianbasisfunctionswasselected,assumingthatthesumof

thenoisecomponents,usuallypresentineverymeasurementdata,

leadstoaGaussiandistortionofgenerallystablespikeshapes.The

EMalgorithmisusedtoapproximatethecalculateddistributionof

eachfeaturewithaso-calledmixtureofGaussians(MoGs)thatcan

bemathematicallydescribedwiththefollowingequation(Polanski

andKimmel,2007): fmix(x,˛1,...,˛k,p1,...,pk)= K



k=1 ˛kfk(x,pk) (4)

where˛kspecifiestheweightofeachfunctionwithinthemixture

modelandpkdenotesitsactualparameters.Inturn,everyGaussian

componentisspecifiedwithitscharacteristicformula:

fk(x,pk)=fk(x,k,k)= 1 



(2n) exp



(x−)2 22 k



(5)

Therefore,theEMalgorithmneedstoscalethevaluesk,kand

˛kforeachmixturecomponent.Tofindasuitableapproximation,

ititeratesbetweentwodistinctivesteps,theexpectation(E-step)

andthemaximisation(M-step).Atthestartoftheprocess,either

arandomoranapriorivalueispresumedforeachparameter.On

thisbasis,thedifferenceQ(p,pold_{)betweentheactualdistribution}

andthecurrentapproximationiscalculatedintheE-step,usingthe

log-likelihoodfunction(PolanskiandKimmel,2007):

Q (p,pold)= N



n=1 K



k=1 p(k|xn,pold)ln˛k + N



n=1 K



k=1 p(k|xn,pold)lnfk(xn,pk) (6)

TheparameterNequalsthenumberofdatapointsofthefeature

distribution,whileKstilldenotesthenumberofGaussiansinthe

mixture.InthesubsequentM-step,anewsetofparametersis

deter-minedtoimprovetheparitybetweenapproximationandfeature

distribution.Inthiscase,theparameterspk,new_k ,_knewand˛new_k

canbecalculatedwiththefollowingsetofequations(Polanskiand

Kimmel,2007): new k =

N n=1xn×p(k|xn,pold)

N n=1p(k|xn,pold) (7) (new k ) 2 =

N n=1(xn−newk ) 2 ×p(k|xn,pold)

N n=1p(k|xn,pold) (8)

˛new k =

N n=1(k|xn,p old₎ N (9)

Thesenewparameterscanbeevaluatedandrefinedwithanother

iterationoftheE-stepandM-step,respectively,eitheruntilQ(p,

pold_{)cannotbeimprovedanyfurtheroruntilacertainthreshold}

criterionismet.

Tofacilitatetheapproximationprocess,eachnativefeature

dis-tribution,derivedwithitshistogram,issmoothedinordertolimit

theeffectsofoutlyingdatapoints.Furthermore,thenumberand

correspondingpositions of existing maximaare determined by

simplecurvesketchingandserveasinitialvaluesfortheEM

pro-cess.

Oncethefinal approximationresultisacquired,theobtained

parametersprovideasuitablebasisfor evaluation.Whereasthe

distance between each Gaussian component in the mixture is

expressedbynew

k ,˛ new

k modelsthedistinctionoftheindividual

functions,whilenew

k canbeusedtodeterminetheoverlapping

oftheidentifiedGaussians.Asitishardtodetermineany

hierar-chyamongtheseparameters,theirvalueshavetobetransformed

intoanoverallratingfortheexaminedfeatures.Forthispurpose,

thefollowingequationhasbeenspecificallydesignedtoprovidea

solutiontotheresultingproblem:

v

(k)=

K

k/=1Co

v

(i,k)

((max)−(k))ײ(k) (10)

Thevalue

v

(k) ratesthe distinctivenessof an individual

Gauss-ianfunction,k,presentinthemixturecomparedtotheGaussian

surroundingitsglobalmaximum,denoted by(max).Onlythe

covariancemeasureputseverymixturecomponentintoaccount,

toaccuratelyexpresstheactualoverlap.AsdescribedinSection

3.3,thepresentedspikesortingalgorithmisspecificallydesigned

toseparatetwospecificclusters.Therefore,thefeaturesshowinga

bimodalcharacteristicareofparticularinterest.Asaconsequence,

inthefirststep

v

(k)iscalculatedindividuallyforeveryGaussian

componentofeachmixture.Afterwards,theminimumvaluefor

v

(k)overallcomponentsofthemixtureisdeterminedandserves

inasanoverallratingforthecorrespondingfeature.Inaddition,

featuresthatappeartohaveonlyaunimodaldistributionarealso

evaluatedbythevariancetheyexplain,butaretreatedassecondary

candidates.

It is this rating that allows thederivation of a unique

fea-turesetforeachindividualdataset.Theparticularcandidatesthat

minimise

v

(k)areselected,althoughagainwiththecorrelationas

anadditionalcriterion.Inthisalgorithm,themaximumsubsetsize

wasadjustedtosix,simplytolimitthecomputationalcomplexity

ofthesubsequentclusteringprocess.However,itispossiblethat

thisnumberisnotreached,asthecorrelationcriteriacanfurther

limitthesizeofthedeterminedsubset.

3.3. Clustering

Asthefinalstepinthespikesortingprocess,theresulting

fea-turesethastobeclassifiedintodistinctclustersthatcontainthe

particularspikes ofan individualneuron. For this purpose, the

derived subsetis firstused tospan a multidimensionalfeature

space.Inordertoidentifytheclustersinvolved,theEMalgorithm

isusedonceagain.Ontheonehand,thisclusteringmethodhasthe

advantageofprovidingasoftclassificationresult,whichallowsthe

exclusionofoutliers,unlike,forexample,thek-meansalgorithm

(Clarkeetal.,2009).Ontheotherhand,asaplainclustering

algo-rithm,itdoesnotrequireanytrainingprocessinadvance,whichis

especiallyadvantageousinthiscase.

Similarlytothefeatureevaluationstepdescribedinthe

previ-ouschapter,theEMalgorithmapproximatesthedatadistribution

ofagivenspace,usingGaussianbasisfunctions.Consequently,each

datapoint, resemblinga specificspike event,is sortedintothe

resultingmixturecomponentsbasedontheprobabilityassociated

witheachGaussian.Asmentionedbefore,thealgorithmisdesigned

tospecificallyseparatethedataintotwoGaussianclusters.Firstof

all,thiseliminatestheproblematictaskofdeterminingthenumber

ofclusters,i.e.thenumberofspikingcells.Secondly,thisgivesthe

opportunitytoderivefeaturesetsthatcanresemblethedifference

inshapeoftheremainingspikesignalsmoreaccurately,thereby

enhancingtheseparability.Thedescribedalgorithmisgenerally

abletoclusterthedataintomorethanjusttwoclasses.

Neverthe-less,intheproposedalgorithm,aniterativeapproachisfavoured,

asitgivestheopportunitytoseparateonespecificclassfromthe

restofthesignalineachiteration.Thus,thederivationofsuitable

featuresbecomessignificantly easier,improvingthesubsequent

clusteringresult.Asafinal consequencethisapproach basically

minimisesthechanceofoversorting,whichmeanstheseparation

ofthespikesof oneindividual cellintomultipleclusters, since

thefeatureevaluationprimarilycreditsthedistinctivenessofthe

Gaussiansinthemixtureofafeatureratherthanthenumberof

Gaussiansinit.ThoughMoG’swithmorethantwoGaussian

com-ponentsarenotexcludedingeneral,thecovariancecriteriaofEq.

(10)ensuresthatsuchfeaturesarepicked,onlyifthesecomponents

canbeclearlyseparated.

4. Resultsanddiscussion

Thealgorithm described inthepreviouschapterwas

imple-mented in a self-made Matlab©-based software. To test the

feasibilityof thedeveloped algorithm,both simulated dataand

recordeddataderived from3Dneuronal cellcultures, so-called

spheroids(LayerandWillbold,1993;Layeretal.,2002),wereused.

4.1. Resultswithsimulateddata

Simulateddatasetswereprimarilyappliedtovalidatetheresults

ofthedescribedspikesortingalgorithm.Suchsimulateddata

sam-pleshavetheadvantageofbeinglabelledandbettersuitedforthe

testingofageneralalgorithm.Hence,simulateddatapublishedfor

suchpurposesbytheUniversityofLeicesterwasused(cf.Quiroga

etal.,2004).Itofferedvariousexamplesofneuronaldata,

simu-latedatasamplerateof24kHz.Thedifferentsetsalsovariedin

theirsignal-to-noiseratio(SNR),allowinganevaluationofthe

sor-tingresultsasafunctionofSNR.Asanexemplarydataset,thefile

cdifficult2noise01withaSNRof10wasapplied(seeFig.5).

Thedatasetcontainedthreebasicspikeshapesthatwere

ran-domlydistortedbyGaussiannoisecomponents.Asapreliminary

step,thespikeeventsweredetectedatasimplevoltagethreshold

thatwascalculatedbasedonthestandarddeviationofthenoise

containedinthesignal(Chiappaloneetal.,2005).Toexcludeany

unwantednoiseeventsthatweredetected,afirstsortingiteration

wasperformed.

Next,theactualspikesortingtaskwasmetwithasecond

iter-ationofthespikesortingalgorithm.Variousspikefeatureswere

calculatedforeachspikeevent,basedona1mswindow(24data

points)aroundthedetectedtime-stampatitsvoltageminimum.

Inthefirstattempt,thedatawereclusteredviatheEMalgorithm,

applyingWavelet-basedfeaturesexclusivelyandinanother

sor-tingattempt,principalcomponentsonly(seeFig.6).Inorderto

improveclarity,theclusteringresultwasprojected onlyonthe

firsttwo dimensions of themultidimensionalfeature space.To

excludeoutlyingdatapointsusuallyoriginatingfromhighly

dis-tortedoroverlappingspikes,aconfidenceprobabilityintervalof

5%wasused.Inotherwords,datapointsbelongingtoacertain

clusterwithaprobabilityoflessthan5%wereseparatedintoa

Fig.5. 150mssectionofthesimulateddatasampleofdatafromtheUniversityofLeicester.Thesignal,sampledatarateof24kHz,containsthreebasicspikeshapeswith anSNRvalueof10.Inordertoprocessthedatawiththeanalysisframework,thenormalisedsamplewasmultipliedwithanoffsetvalueof70␮V.Furthermore,thesignal wasinverted,asthespikedetectionalgorithmofourMatlab©interfacegenerallyexpectsneuronalspikeswithnegativeamplitudes.Thedifferentspikeshapesaremarked withtheirrespectiveclusternumber.

Fig.6. Clusteringresultofthedatasetcdifficult2noise01usingtheEMalgorithmonlywithWavelet-basedfeatures,indetailthethreeWaveletpacketcoefficientswiththe mostsuitableprobabilitydistribution(left)andonlywiththefirstthreeprincipalcomponents,describing90%ofthedata’svariance(right).Forimprovedclarity,theplots onlyshowaprojectionofthemultidimensionalfeaturespaceontothefirsttwofeatures.Neithercombinationcorrectlyseparatedallthreespikeshapes,asonlytwodistinct clusterscouldbedetermined.Outlyingdatapointswereseparatedintoaparticularclusterusingaconfidenceprobabilityintervalof5%.

algorithms,couldseparateallthreespikeshapesintheparticular

featurespacecorrectly.

Hence,thesubsequentspikesortingresultwasnotsatisfying

either.Inthenextstep,thenovelautomatedfeatureevaluation

algorithm was used to find a suitable mixture of spike shape

features.Thedeterminedsetoffeatureswasthenfedintothe

sub-sequentclusteringstep.Fig.7showstheresultoftheEMalgorithm

Fig.7. Clusteringresultofthedatasetcdifficult2noise01usingthe(1)Wavelet coef-ficient(distribution)andthe(1)principalcomponent.Thesortingalgorithmcould separatethethreedistinctclusters,whileoutlyingdatapointswereseparatedinto aparticularclusterusingaconfidenceprobabilityintervalof5%.

inthetwo-dimensionalfeaturespacespannedbythetwobest

fea-tures,namelythefirstprincipalcomponentandthefirstWavelet

coefficientevaluatedfromitsdistribution.

InthefeaturespaceseeninFig.7,threeclustersinsteadoftwo

wereidentified.Hence,thefeaturecombinationfoundbythe

algo-rithmclearlydescribedtheparticulardatasetmoreaccuratelythan

thesetsofthetwopreviousapproaches.

Therefore,themixtureofdifferenttypesoffeaturesprovidesa

betterclusteringresultcomparedtothecommonattemptswith

oneparticulartypeonlyusedinmostofthealgorithmsalready

established.To lookat the resultsin furtherdetail, thesorting

resultwasevaluatedforeachspikebycomparingthecalculated

classificationresultandtheoriginallabel(seeTable3).

Iftheeventsofoverlappingspikesareexcluded,thespike

sor-tingalgorithmalmostyieldsaperfectresult,withonlyninefalsely

clusteredevents.Thisnumber,however,issignificantlydecreased,

onceoverlappingeventsaretakenintoaccount.Inorderto

exam-inethealgorithm’sresultsinmoredetail,theerrorsmadebetween

missing and incorrectlyassigned spikes ina certain clusterare

summarisedinTable4.

Mostspikesclassifiedinoneofthethreeclustersweresorted

correctly,even when consideringoverlapping events. This

phe-nomenoncanbeexplainedbytwoseparateerrors.First,spikesthat

followtoocloselyafteroneanotherweredetectednotastwo

dif-ferentbutonesingleevent.Hence,oneofthetwocontainedspikes

wasbasicallyomittedandtherefore,lostbeforethesortingstep.

Second,ifbothspikesweredetectedseparately,butdistortedeach

Table3

Comparisonoforiginalandcalculatedspikesortingresults,usingthedatasetcdifficult2noise01withaSNRof10.

Classes No.oforiginalspikes(with overlappingspikes)

No.ofcorrectlysortedspikes (withoverlappingspikes)

%ofcorrectlysortedspikes (withoverlappingspikes)

Overall 2742(3462) 2733(3351) 99.67(96.79)

Cluster1 957(1187) 953(1146) 99.58(96.55)

Cluster2 898(1136) 898(1113) 100(97.98)

Cluster3 887(1139) 882(1092) 99.44(95.87)

Table4

Classificationoftheresultingclusteringerrorsofthealgorithm,usingthedatasetc difficult2noise01withaSNRof10. Classes No.ofcorrectlysortedspikes

(withoverlappingspikes)

No.ofmissedspikes(with overlappingspikes)

No.offalsepositives(with overlappingspikes)

Overall 2733(3351) 9(111) 9(14)

Cluster1 953(1146) 4(41) 5(5)

Cluster2 897(1112) 0(23) 1(1)

Cluster3 883(1093) 5(47) 3(8)

specialremnantcluster(seeredclusterinFig.7).Inotherwords,

onlyoverlappingspikesthatstillshowedacertainamountofits

distinctiveshapecouldactuallybeprocessedcorrectly.

Onewayof facingthechallengeof overlapping spikesis by

implementinganadditionaltemplatematchingstep,whichrefines

thediscardedspikeevents.Byfindingacombinationofthe

discov-eredspikeshapesinthespikesortingstepthatmatchestheshape

ofaneventinsidetheremnantcluster,thespikesinvolvedcanbe

identified(cf.Zhangetal.,2004).

Inordertoevaluatetheinfluenceofdifferentnoiselevels,the

algorithmwasalsotestedwithanotherdatasetwithaSNRof6.67.

Theused setcdifficult2noise015 basicallyconsists ofthe same

threespikeshapesasthedataabove,yetthedistributionofthe

differentspikesis changed slightly. Due totheincreased noise

distortionitismoredifficulttofindsuitablespikefeatures that

candiscriminateall threeshapes in just onefeatureextraction

step.Hence,usingthefirsttwoWaveletcoefficients(distribution)

andthefirsttwoprincipalcomponents,thealgorithmcouldonly

separateoneofthecellsignalsfromtheother,butnotallsignal

shapesintheinitialprocess.Inthiscasetheadvantageofthe

iter-ativestructureoftheproposedalgorithmcomesintoeffect.Asit

ismucheasiertofindfeaturesthatspecificallydescribethe

dif-ferencesofthetworemainingspikeshapes,thesortingresultis

greatlyimprovedonce thefirstcellsignalisexcludedfromthe

dataset.UsingtheadjustedfirstandsecondWaveletcoefficients

(distribution)aswellastheadaptedfirstprincipalcomponentthe

tworemainingcellsignalscanbeseparatedfromeachother.Ina

finaliterationtheremainingspikeshapecanbeexcludedfromthe

obligatoryremnantcluster.Theresultofthisprocessissummarised

inTables5and6,whicharestructuredsimilartoTables3and4of

thefirstdataset.

Asexpectedtheaccuracyofthespikesortingprocess

deterio-rateswithincreasingsignal-to-noiseratios.Inthiscaseboththe

resultdiscardingoverlappingspikesaswellasresultthatincludes

theseeventsareeffectedbyaboutthesamemargin.Eventhough

theincreasedsignal-to-noiseratioposesanadditionalchallenge,

themajorityofthespikescontainedinthedataweresuccessfully

sortedinthecorrectclusters.Thusthenovelfeatureextractionand

evaluationstepsofthedescribedalgorithmshowsignificant

advan-tagescomparedtothemethodscommoninmostestablishedspike

sortingalgorithms,especiallywhenusedinaniterativeprocess.

4.2. Resultswithspheroiddata

Tofurtherassessthepresentedspikesortingalgorithm,data

obtainedfromchickembryonicneuronscoupledtomicroelectrode

arrayswereused.

Forthispurposechickneuroepithelialtissue(stages28and29,

HamburgerandHamilton(1951))waschoppedandenzymatically

dissociatedwithTrypsin/EDTA(all chemicals: CCPro,Oberdorla,

Germany, unless stated otherwise), similarly as described for

cardiactissue inEgertand Meyer(2005).Approximately2

mil-lioncellsweretransferredto35mmPetridishescontaining2ml

of cell culture medium (DMEM with 10% FCS, 2% CS (Sigma,

Munich, Germany), 0.5mM glutamine, 50U/ml penicillin, and

50␮g/mlstreptomycin).Thecellswerere-aggregatedinto

three-dimensionalsphericalinvitronetworks,socalledspheroids,by

Table5

Comparisonoforiginalandcalculatedspikesortingresults,usingthedatasetcdifficult2noise015withanSNRof6.67.

Classes No.oforiginalspikes(with overlappedspikes)

No.ofcorrectlysortedspikes (withoverlappedspikes)

%ofcorrectlysortedspikes (withoverlappedspikes)

Overall 2631(3440) 2577(3243) 97.95(94.27)

Cluster1 858(1142) 839(1065) 97.78(93.28)

Cluster2 851(1113) 824(1040) 96.83(93.44)

Cluster3 922(1185) 914(1138) 99.13(96.03)

Table6

Classificationoftheresultingclusteringerrorsofthealgorithm,usingthedatasetcdifficult2noise015withanSNRof6.67. Classes No.ofcorrectlysortedspikes

(withoverlappedspikes)

No.ofmissedspikes(with overlappedspikes)

No.offalsepositives(with overlappedspikes)

Overall 2577(3243) 54(197) 53(92)

Cluster1 858(1065) 19(77) 34(49)

Cluster2 851(1040) 27(73) 7(26)

Fig.8.100mssectionoftherecordedspheroiddatasample,recordedwith50kHz.Thespikeswerelabelledaccordingtotheclusteringresultofthesubsequentclassification (cf.Fig.9).

placingthePetridishesonagyratoryshaker(72rpm)withinan

incubator(37◦C,5%CO2)(Dausetal.,2011).

The resulting spheroids were then plated on

microelec-trodearrays(MEA;MultichannelSystems,Reutlingen,Germany)

for extracellular recording. The MEA chips consisted of 60

substrate-embeddedtitaniumnitrideelectrodes(30_␮min

diam-eter) arranged in an 8×8 matrix. Extracellular field potentials,

so-called spikes, wererecorded (25–50kHz sampling rate) and

saved for offline analysis. The timestamps of the spikes were

determinedbyathresholdseparatingneuronalsignalsfromnoise

(Chiappaloneetal.,2005).Thesedataposeanadditionaldifficulty,

asmultipleneuronscontributetotheelectrodesignal.However,

mostoftheseneuronshadaninsufficientcouplingwiththe

elec-trode,which significantly hamperedthe ability toseparatethe

particularspikeshapesoftheemittingneurons(seeFig.8).

Thespikedetectionprocessforthistypeofdatawasasdescribed

above.However,duetothesignal’scomplexity,asupplementary

sortingstepwasnecessarytoexcludeanyshapesoriginatingfrom

distantneuronsthatcouldnotbesatisfyinglyprocessedandonly

actedasadditionalnoisyartefacts.

Similartotheevaluationprocessusingthesimulateddata,the

spikesofthespheroid signalweresortedusingeither

Wavelet-based features, principal components or the feature mixture

derived byournovelapproach.Allmethods couldidentifytwo

differentspikeshapesinthespheroidsignal,whichmeansthat

therecordedspheroidsignalcontainedatleastthesignalsoftwo

neurons,withatightcouplingtotheelectrode(cf.Fig.9).Yetthe

differentfeatureextractionapproachesshowavaryingabilityto

discriminatebetweenthetwoshapes.Theresultingclustersofthe

Wavelet-basedapproachshowedasignificantoverlap,unlikethe

othertwomethods.Theresultusingprincipalcomponents

how-everwasquitesimilartotheclusteringofourproposedmixture

approach,onlywithslightdifferencesattheborderbetweenthe

twoidentifiedclusters.Eventhoughtheresultusingthefeature

mixtureoftheproposedalgorithmmightbeconsideredmarginally

moreaccurate,suchimprovementwouldbequitedifficulttoassess,

duetothelackoflabellingfortherecordedspheroiddata.

In general,eventhough both Waveletand principal

compo-nentbasedfeatureextractiontechniquesarecapableofproducing

acceptablespike sorting results, theyusually fail toachieve an

Fig.9.Scatterplotofthespheroiddatainthefeaturespacespannedbythenegative amplitudeandthe(2)Waveletcoefficient(distribution).Againthescatterplotshows aprojectionoftheclusteringresultsontothetwodimensionalfeaturespaceoffirst andsecondfeature.Thedatacanbesortedintotwospikeshapeclustersusingthe (2)principalcomponentasadditionalfeature.Thedatapointsinredagainhighlight theremnantclustercontainingoutliersinthespheroiddata.(Forinterpretationof thereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversion ofthearticle.)

optimaldiscriminationbetweendifferentcellsignals,simplydue

totheirlack ofvariability. Whereasouralgorithm includes the

strengthofprincipalcomponentswithoutneglectingthepotency

ofotherfeatures,themethodsusingonlyasinglefeaturetype

suf-ferfromtheirrespectiveweaknesses.Hence,itismostfeasibleto

includebothfeaturetypesinordertopreservetheirstrengthsand

tocompensatetheirweaknessesrespectively.

5. Conclusionandoutlook

Insummary,thedescribedalgorithmyieldsaccurateresultsfor

simulatedand recordedneuronaldatasets.Theconcept of

eval-uatingabroadsetofextractedfeaturestofindasuitablesubset

thatdescribesaparticulardatasetmostaccuratelyseemstohave

featuretype, thedifferentcapabilitiesofvariousfeature

extrac-tiontechniquesbecomeavailableforthesubsequentsortingstages

andtherefore,significantlyfacilitatetheclusteringtask.Therefore,

thisenablesevenacomparablysimpleEMclassificationmethodto

produceapreciseclusteringresult.

Basedonthisfeatureevaluationtechnique,itisalsopossible

toidentifythefeaturesthatexplicitlydiscriminatebetweentwo

differentspikeshapes,whichisadvantageousinneuralrecordings

thatincludemorethanjusttwosignalshapes.Thisabilityisused

bythepresentedalgorithm,giventhataniterativesubtraction

pro-cessofparticularspikeshapesfromtherestofthesignalcanbe

performedifnecessary.Thisallowstheidentificationofanew

fea-turesubsetmostsuitablefortheremainingspikesinthedataset

afterexcludingacertainshape.

Theonly issuethatcouldbeidentifiedisthelackof an

ade-quatehandlingofoverlappingspikeevents.Nevertheless,thiscan

befixedquiteeasilywithanadditionaltemplate matchingstep

basedonthealgorithm’sclusteringresult.

Theremightstillbesomeroomforusefuladaptations.One

pos-sibleapproach,forexample,istheuseoft-distributionsinsteadof

Gaussiansasabasisfunction(Shohametal.,2003).Withan

addi-tionaladjustmentparameter,thedegreesoffreedom(DOF),these

functionsmightbemorerobusttoanynon-Gaussiannoise

distort-ions.However,asadrawback,thisadditionalparametercouldalso

resultinasignificantlyhighercomputationalcomplexity.

Onthe otherhand,the EMalgorithm itselfcouldbe

substi-tutedwiththeVariationalBayes(VB)algorithm(Takegawaetal.,

2010).BeingamoregeneralapproachthanEM,itoffersthesame

advantages,butalsodisadvantages,astheswitchfromGaussianto

t-distributions.

Anotherissuenecessarytoaddressisthecomputational

com-plexityofthealgorithmitself.Thetimerequiredtoapproximatethe

distributionofacertainfeaturesignificantlydependedonthe

par-ticulardistributioncharacteristics.Therefore,itishardtogenerate

ageneralisedstatementregardingtheprocessingtimeneededby

thealgorithm.However,thecurrentversionofthealgorithmhas

tobemodifiedinordertofunctioninanonlinesignalanalysis

pro-cess.Forthispurpose,itispossibletocomputeexplicitoperations

inparallel,usingmultipleCPUorGPUcores(cf.Chenetal.,2011).

Thus,theapproximationofdifferentfeaturesviatheEMalgorithm

canbecalculatedsimultaneously,effectivelyspeedingupthespike

sortingalgorithmaltogether.

Althoughtheremightbesomepossibleadaptationsofthe

algo-rithm, itsgeneral feasibility and advantages compared tomost

common spike sorting algorithms are quite obvious. The

rea-son for this is the improved feature extraction approach that

enablesanadaptivedeterminationofsuitable spikesorting

fea-tures.Thisallowstheinclusionofavarietyofdifferentfeaturetypes,

whichincreasesthechanceofidentifyingparticularkeydifferences

betweenasetofuniquespikeshapes.

In conclusion, the disadvantage of only beingable to use a

limitedsubsetoffeaturetypesthatmostestablishedspikesorting

methodsfacecan beovercomewiththenovelapproaches

pre-sentedinthispaper.Hence,itisalsopossibletoenhancealready

existingalgorithmswiththeseideas,improvingtheirclusteringor

classificationsignificantly.

Acknowledgement

ThisworkwassupportedbytheBMBFAN2208PT-AIF.

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