Analysis of bit-rate definitions for Brain-Compute

Analysisofbit-ratedefinitionsforBrain-ComputerInterfacesJulienKroneggSvyatoslavVoloshynovskiyThierryPunComputerVisionandMultimediaLaboratory,ComputerScienceDept.,UniversityofGeneva,SwitzerlandEmail:{firstname.surname}@cui.unige.ch,phone:[phone],fax:[phone]Abstract-Acomparisonofdifferentbit-ratedefinitionsusedintheBrain-ComputerInterface(BCI)communityisproposed;assumptionsunderlyingthosedefinitionsandtheirlimitationsarediscussed.CapacityestimatesusingWolpawandNykoppbit-ratesarecomputedforvariouspublishedBCIs.ItappearsthatonlyNykopp'sbit-rateiscoherentwithchannelcodingtheory.Wol-paw'sdefinitionmightleadtounderestimatetherealbit-rateandtoinferwrongconclusionsabouttheoptimalnumberofsymbols;itsuseshouldbeavoided.Theus-ageofaproperbit-rateassessmentismotivatedandadvocated.Finally,itisfoundthatthetypicalsignal-to-noiseratioofcurrentBCIsliesaround0dB.Keywords:brain-computerinterface,bit-rate,informa-tiontransferrate,numberofclasses,informationtheory.1IntroductionABrain-ComputerInterface(BCI)isaninputdevicethatallowsausertodriveaspecificapplication(e.g.virtualkeyboard[10],cursorcontrol[28],robotcontrol[24])usingEEGdatainducedbythinkingtoaspecificnotionormentalstate(e.g.mentalcalculation,imaginationofmovement,mentalrotationofobjects).Thismentalstateisthenrecognizedbythemachineusingaclassifier.ThefirstobjectiveBCIperformancemeasureisduetoWol-pawetalin1998[28],wherethebit-rate,orinformation-transferrate,wasdefinedbasedonShannonchanneltheorywithsomesimplifyingassumptions.Bit-ratescommonlyreportedrangefrom5toabout25bits/minute[29].Inthisarticle,wecomparethebit-ratedefinitionsusedintheBCIdomainandproposerecommendationsforoptimizingthenumberofmentalstatesinaBCI.Thearticleisorganizedasfollows:first,areviewofthenoisychanneltheoryispresented,asasupporttotheBCImodeldescribed.Existingbit-ratesdefinitionsusedintheBCIdomainarethenpresentedandanalyzed.2NoisyChannelTheoryAchannelisacommunicationmediumthatallowsthetransmissionofinformationfromasenderAtoare-ceiverB.Duetoimperfectionsinthatmedium,thetransmissionprocessissubjecttonoiseandBmightreceiveinformationdifferingfromtheoneemittedbyA.ThesimplestnoisychannelistheadditivenoisechannelwherethereceivedsignalYisthesumofanemittedsignalXandsomeindependentnoiseZhereassumedGaussian.Sincewedealwithreal,physicalinputsignals,theinputsignalenergyislimited(whichalsoimpliesthatXhaszeromeaninordertominimizeitsenergy,E[X2]≤σX).Theinformationchannelcapac-ityisthequantityofreliableinformationcarriedbyonesymboltransmittedthroughthechannel.Thechannelcapacitydependsontheinputsignaldistri-butionaswellasonthesignal-to-noiseratio(SNR)[5].ForcontinuousinputsignalandusingSNR=10⋅log10(σX2/σZ2),thecapacity(inbits/symbol)is:()210log1100.5SNRC=+⋅FordiscretePulseAmplitudeModulatedinputwithNsymbolsofaprioriprobabilityp(X=xi)=1/N(denotedp(xi)),thecapacityCNisdefinedbyEq.2.()()()()()()()()1222122()|||log1|ZNjjjkZNiNiiiykyxpypxpyxpyxCpyxpxdypypyxeσπσ=+∞==−∞−−===∑∑∫012345678910-20-100102030405060SNR[dB]Information[bits/symbol]N=256N=128N=64N=32N=16N=8N=4N=3N=2continuousgaussiansourceasymptoticalcapacityfordiscreteequiprob.signalN=512Figure1:ComparisonofthecapacityforGaussianinputCandfordiscreteequiprobableinputCNasafunctionofthenumberofsymbolsNandoftheSNR.(1)(2)2005Int.Conf.onHuman-computerInteraction(HCI'05),LasVegas,Nevada,USA,June20-23,2005.http://vision.unige.ch/Theprobabilityp(y|xi)istheprobabilitythatthecon-tinuoussymbolyisrecognizedwhenthesymbolxiissent.ThevarianceσZ2ofthenoiseZisgivenbyσZ2=σX2⋅10-SNR/10.ThecapacityCNhastobedeterminedbynumericalintegration.Figure1comparesthecontinuouscapacityC(Eq.1)andthediscreteequiprobablecapacityCN(Eq.2).Thereisanasymptoticdifferenceof1.53dBbetweenCandC∞(thediscretecapacityforN=∞),socalledshapingloss.Inallcases,thecontinuouscapac-ityisgreaterthanthediscreteequiprobablecapacity.3BCIModelTheBCIismodeledusinganadditivewhiteGaussiannoise(AWGN)channelasfollows[15],[25],[26],seeFigure2.Amentaltaskwi(e.g.mentalcalculation),selectedamongstNpossiblementaltasks,isencodedbythebrain,producingadiscretememorylessfeaturexi(e.g.powerspectraldensity,auto-regressivemodelcoefficients).ThisfeatureisperturbedbyanadditiveGaussiannoiseZ~N(0,σz2)inducedbythebackgroundbrainactivity,consideredasindependentofX.TheresultingfeatureY=X+ZisdecodedasˆWbyaclassi-fierabletorecognizeMsymbols(M=N+1forclassifi-erswithrejectioncapabilityandM=Nforclassifierswithoutrejectioncapability).Nopriorknowledgeisassumedabouttheoccurrenceprobabilityofclasses,thereforetheycanbeingeneralconsideredasnon-equiprobable(p(wi)≠1/N).ThecapacitycanthusbecomputedbynumericalintegrationusingEq.2.Figure2:ModeloftheBCIusinganAWGNchannel.Thetransitionmatrixˆ()jipww,alsocalledconfusionmatrix,iscomputedduringtheclassifiertrainingphase.Thismatrixdescribetheprobabilitythatamen-taltaskwiisrecognizedasamentaltaskˆiw.Thedi-agonalofthetransitionmatrixistheclassifieraccu-racy,thusaperfectclassifierhasanidentityconfusionmatrix[20].ThisBCImodelhoweverdoesnottrulycorrespondtoarealBCIapplication,butmoretoanidealBCI.First,thesourceisnotalway