Computational models with electrophysiological input predict behavior on a numerical cognition task
{ Draft Version 11.4.2016 }
RichardWPrather&SaraHeverly-Fitt
Abstract
Thedesignofcognitiveinterventionsislimitedbytheabilitytopredictparticipants’behaviorand
howitmaychangewithexperience.Wepresentanovelcomputationalmodeldesignthatusesbehavioral
andelectrophysiologicalinputtopredictparticipants’behavioronindividualtrialsofanumerical
comparisontask.Wemodelparticipants’behaviorusingindependentmodelinstantiationsthatare
optimizedforeachparticipantusinganevolutionaryalgorithm.Themodelisalsothenabletogeneralize
tonovelparticipantdatawithoutasignificantreductioninpredictionaccuracy.Wediscussboththe
potentialandlimitationsofthecurrentparadigmindevelopingtrainingregimensforchildrenwithearly
mathdifficulty.
Introduction
Thecurrentinterestininterventionsandtrainingparadigmsincludesareasofcognitionsuchas
languagelearning,memory,numericalcognition.Thegoalofsuchinterventionstypicallyistoimprove
learners’performanceonatargettask(Park,Bermudez,Roberts,&Brannon,2016;Park&Brannon,
2014;Prather&Alibali,2011),orinthecaseofolderadultspreventdecline(e.g.,Angueraetal.,2013).
Therehasbeenanincreaseintheautomaticization(e.g.,Iuculanoetal.,2015)andindividualizationof
interventions(e.g.,CohenKadosh,Dowker,Heine,Kaufmann,&Kucian,2013;Vanlehn,2011).Thedesign
ofeffectiveinterventionsrequiresbothaccurateassessmentofthelearner’sbehaviorandpredictionof
howthatbehaviormaychangewithexperience.
Thecurrentstudyisanevaluationofpredictivecomputationalmodeloflearners’behaviorona
numericaltask.Thelong-termgoalistocreateindividualizedtrainingproceduresthataredesignedby
thecomputationalmodel.Asignificantlimitationininterventiondesignisthatthereislargeindividual
variationinparticipantbehavior.Aparticulartrainingsequenceortaskthatiseffectiveforparticipants
ofaverageskillmaynotbeforparticipantswithlow-numeracy.Asaninitialstepwedevelopeda
computationalmethodforsingle-trialpredictionsofparticipants’behavior.Wefocusonevaluatingthe
canonicalnon-symbolicnumericalcomparisontask(Figure1).Participant’sskillinthistaskisusedto
indextheirnumeracy;aconstructthathasbeenshowntocorrelatewithimportantoutcomesrelatedto
mathperformance(e.g.,Libertus,Feigenson,&Halberda,2013;Libertus,Odic,&Halberda,2012;
Mazzocco,Feigenson,&Halberda,2011).Participants’skillinnon-symbolicnumericalcomparisonis
correlatedwiththeirskillonsymbolicarithmetic(e.g.,Chen&Li,2014;Fazio,Bailey,Thompson,&
Siegler,2014).Theuseofnon-symbolicnumericalcomparisonasatrainingtaskthatmaytransferto
symbolicarithmetichasmixedresults(Parketal.,2016;Park&Brannon,2013,2014).
Thecurrentstudycombinesthebehavioralandelectrophysiologicaldataasinputforapredictive
computationalmodel.Wefocusonthenon-symbolicnumericalcomparisontask,whichhaswell
cataloguedbehavioralandneuralcorrelates.Comparisonsinwhichthetwonumbersarerelativelycloser
invaluetendtohavemoreerrorsandlongerreactiontimes.Forexample,acomparisonbetween20and
22objectsproducesmoreerrorsthanacomparisonbetween20and50(foralternateseeRatcliff,
Thompson,&Mckoon,2015).Thisphenomenonistermedthedistanceeffectanddependsontheratio
differencebetweenthetwovaluesbeingcompared.Theneuralcorrelatesofthecomparisontask,asseen
withelectroencephalography,includeaneuraldistanceeffectwheretheamplitudeofeventrelated
potentialsvarybasedonthedistanceofthecomparison(e.g.,Ben-Shalom,Berger,&Henik,2013;Heine,
Tamm,Wissmann,&Jacobs,2011;Jiangetal.,2010).Ourapproachforthemodelpredictsthespecific
responsesforeachindividualparticipantoneachindividualtrial.Thisallowsthemodeltoaccountfor
individualdifferencesinperformance.
Experiment1
FirstweattemptedtoreplicatetheERPneuraldistanceeffectusinganon-symbolicnumerical
comparisontask.
Method
Participants.Participants(n=26)wereadultsrecruitedthroughauniversityparticipantpool.Protocols
wereapprovedbytheUniversityofMarylandInternalReviewBoard.
Procedure.Weevaluatedparticipants’performanceonthenon-symbolicnumericalcomparisontask
whilemeasuringneuralcorrelatesviaEEG.Participantswereinstructedthattheywouldcompletea
seriesofcomputer-basedtaskswhilesimultaneousEEGrecordingsweretaken.Afterabriefsetupto
initializetheEEGparticipantsbeganthebehavioraltasks.Instructionsfortaskweregivenorallyand
againwrittenonscreen.Participants’responseswereviakeypress.Theexperimentalsessionwas
approximately45minutes.
TaskandStimuli.Stimuliwere90visuallypresentedpairsofshapearrayswithamidlineseparator.
Shapearraysrangedinnumberfrom23to111(seeAppendixB)withamaximumratiodifferenceof2.6.
Participantswereinstructedtoindicatewhichsidecontainedmoreshapesviabuttonpress.Stimuliwere
displayedfor2secondsafterwhichthescreenwasblank.Therewasnoresponsetimelimit;participants
wereinstructedtorespondasquicklyaspossible.
ElectrophysiologicalRecordingsSpecifications.Recordingwasimplementedusinga32-channelEEGcap.
Twentyninetinelectrodeswereheldinplaceonthescalpbyanelasticcap1(Electro-Cap
International,Inc.,Eaton,OH)ina10-20configuration(O1,FP2,O2,P7,2P3,Pz,P4,P8,TP7,Cp3,CPz,
CP4,TP8,T7,C3,Cz,C4,T8,FT7,FC3,FCz,FC4,FT8,3F7,F3,Fz,F4,F8,FP1).Basedonpriorresearchwe
limitedanalysistochannelsintheleftparietalarea(P3,P7,CP3,TP7).Bipolarelectrodeswereplaced
aboveandbelowthelefteyeand4attheoutercanthusoftherightandlefteyestomonitorverticaland
horizontaleye5movements.Additionalelectrodeswereplacedovertheleftandrightmastoids.Scalp6
electrodeswerereferencedonlinetotheleftmastoidandre-referencedofflinetothe7averageofleft
andrightmastoids.Impedancesweremaintainedatlessthan5kΩforall8scalpelectrodesites,less
than2kΩformastoidsites,andlessthan10kΩforocular9electrodes.TheEEGsignalwasamplified
byaNeuroScanSynAmps®Model508310(NeuroScan,Inc.,Charlotte,NC)withabandpassof0.05-100
Hzandwascontinuously11sampledat500Hzbyananalog-to-digitalconverter.
Results
Behavioralresults.Participantperformanceonthenumericalcomparisontaskrangedfrom83%to97%
correct(Figure2).Medianreactiontimesrangedfrom352msto1604ms.Correlationbetweentrialratio
differenceandpercentcorrectwasnon-significant,r=0.30,t=1.54,likelyduetoaceilingeffect(Figure
2).Correlationbetweenreactiontimeandtrialratiodifferencewasr=-0.13,p<0.01
Electrophysiologicalresults.WeevaluatediftherewasanERPdistanceeffectinparietalareas.MeanERP
amplitudewithinthetimewindowof280-380mswasextracted.Trialswerebinnedintoequalsized
groupsoflarge,mediumorsmallcomparisondistances.WecomparedthemeanERPamplitudeforlarge
andsmallcomparisondistances.DatawereanalyzedusingrepeatedmeasuresANOVA,withtrial
comparisondistancebeingthewithin-subjectsvariable.Ouranalysisshowsthatwhenevaluatingdata
fromtwoparietalchannelsthereisasignificantdifferenceinERPamplitudebetweenlargeandsmall
comparisondifferences.WefoundasignificanteffectofratiodifferenceonmeanERPamplitudeforthe
P3andCP3,F(1,25)=6.33,p=.018andF(1,25)=4.58,p=.042respectively.Thisanalysisoftheneural
distanceeffectisconsistentwithpriorresultsintermsofEEGchannellocation,ERPtimewindow,and
trialcomparison.
Theexperiment1resultsdemonstrateareplicationoftheneuraldistanceeffect(e.g.,Ben-Shalom,
Berger,&Henik,2013;Heine,Tamm,Wissmann,&Jacobs,2011;Jiangetal.,2010).Wefindthatevent
relatedpotentialsfortwoparietalchannels(P3andCP3)significantlyvariedforclosenumerical
comparisonsversusfarcomparisons.
Experiment2
Forexperiment2weusedthebehavioralandneuraldatafromexperiment1asinputfora
computationalmodelofparticipantsbehavior.Weusedthefirsttwo-thirdsoftrials(60)astrainingdata
forthemodelandthelastthirdoftrials(30)asapredictiontest.Thekeycomparisonisifmodel
predictionsduringthetestphaseareasaccurateasmodelfitduringthetrainingphase.Thecomputational
modelisamultilayereddynamicfieldtheorymodel(e.g.,Erlhagen&Schöner,2002;Sandamirskaya,
2014)thatemploysneuraltuningcurvesrelatedtonumericalcognition(e.g.,Prather&Heverly-Fitt,
2016;Prather,2012,2014)alongwithanevolutionaryoptimizationalgorithm.Theevolutionary
algorithmadjustsspecificationsofthemodeldynamicstofiteachparticipant’strainingdata.Thus,each
participant’sdatawasmodeledbyaseparateandindependentinstantiationofthecomputationalmodel.
Theresultingmodelspecificationswerethenusedtopredicttheremainderofeachparticipant’sdata
withoutfurtheruseoftheevolutionaryalgorithm.Weevaluatethemodelfirstbyhowwellthemodelfits
participant’sdatainthetrainingphase,thenbyifthatmodelinstantiationcanextrapolatetothetest
phasedatawithoutfurtheradjustment.
Method
ModelSpecificationsandProcedure.ThemodelwasimplementedusingMATLAB(Mathworks).
Thearchitectureisthatofamultilayereddynamicsystemsmodel(e.g.,Simmering&Perone,2013;
Spencer,Smith,&Thelen,2001).Layersincludedtwoperceptualneuraltuningcurvesandadecision
layer.Layersmodeledneuraltuningcurvesassociatedwithneuralcodingofstimuli(e.g.,Prather,2012,
2014;Tudusciuc&Nieder,2007).Oneachtrialtheexternalinputsforthemodelwerethetwonumerical
valuestobecompared,takenfromthebehavioraltaskinexperiment1.Thetwovalueswererepresented
byproportionallyscaledGaussiancurvesthatreproducetheratiodependentdistanceeffect.Perceptual
layersofthemodelreproducedthestimuliwhileactivitywasforwardedtothedecisionlayer.The
internaldecisionlayerconnectionswerespecifiedtoproducecompetitionwithinthelayerthrough
lateralinhibitionandself-excitation.Thusthetwoperceptuallayersoutputcreatedacompetitionwithin
thedecisionlayer.Thisdynamiccorrespondedtothe“decision”whichwastheindexofthefirststable
activationpeakinthedecisionlayer.Eachtrialwascomprisedof500timesteps.Thedecisionlayer
producedareliabledecisionthroughasteadypeak(activitywithapeakvalueatthesamelayerindexfor
10straighttimesteps).Thetimestepofthedecisionwasconvertedtothepredictedreactiontimeofthe
decision.Thusontrialsinwhichthemodelpredictedafastdecisionthesteadypeakwasreacheda
relativelylowtimestep.Ontrialsinwhichthemodelpredictedaslowerdecisionthesteadypeakwas
reachonahighertimestep.
Useofelectrophysiologicaldata.WeusedERPdataasreportedinexperiment1asmodelinput.Datawas
limitedtotheP3channelasdataindicatedasignificantcorrelationbetweenERPamplitudeandtrial
rationdifference(i.e.theneuraldistanceeffect).ForeachtrialwecalculatedthemeanERPamplitude
acrossthetimewindowusedinexperiment1.Trialamplitudeswereusedtopredictincorrectresponses,
whichweremostlyfortrialswithratiodifferencesinthelowerhalfoftherange.ThusERPamplitude
onlyaffectedmodelparametersontrialswitharatiodifferenceof1.7orless(44of90trials).
ModelFitnessCalculationandEvolutionaryAlgorithmProcedure.Modelingprocedureincludedtwoparts,
trainingphaseandtestphase.Duringtrainingthemodelwasfittothefirst60trialsoftheparticipants
data.Fitwasoptimizedusinganevolutionaryalgorithm.Afterdeterminingthespecificationsofthe
modelthatbestfitwethenusedthatmodeltopredicttheentiretyofparticipantsdatawithoutfurther
adjustmentsviathelearningalgorithmic.Themodelmustgeneralizewhatislearnedduringthetraining
phasetonoveltrialsduringthetestphase.Thisprocessisdoneindependentlyforeachparticipant.
Specificationsthatvariedforeachmodelinstantiationincludedparametersfornoise,tuningcurve
width,rateofactivationchange,inhibitioninconnections.Thefirstgenerationofthealgorithmincluded
10instantiationsofthemodelwithrandomlyselectedvaluesofthespecifications(withina
predeterminedrange).The10modelsarethenrankedbytheresultingfitness.Thebottom5performing
modelswerediscarded.Thebestperformingmodelwasduplicatedforthenextgeneration.Models
ranked2–5weremutated,suchthattheirspecificationswerealteredslightlybyarandomprocess.The
nextgenerationalsoincluded5newmodelswithrandomlyselectedspecifications.Thisprocesswas
repeatedover25generationstoproduceamodelinstantiationthatfittheparticipants’data.Sincethe
modelhasasmallamountofrandomnoiseperformancedoesnotreplicateexactly.Tobestdeterminethe
fitnesslevelofagivensetofmodelspecificationswethenranthemodelrepeatedlyandcalculatethe
meanfitnesslevel.Forthetestingphaseweusedthebest-fitspecificationstopredictthefinalhalftrials
oftheparticipantsdata,whichthemodelhadnotbeentrainingwith.
Modelfitnessiscalculatedusinganoverallerrortermsthatcombinesthenumberofcorrectly
predicteddecisions,thedeviationofthepredictedreactiontimesandcorrelationbetweenpredictedand
truereactiontimes.Theoverallfitofthemodelwasweightedtowardsminimizingresponseprediction
errorviathefollowingequation:
Results
Modeldataaddressestwoquestions.First,howwelldoesthemodelfittoparticipantdatausing
theevolutionalgorithm.Second,howwelldothemodelinstantiationsgeneralizetopredictnovel
participantdata?Analysisofmodelperformancewasmeasuredbydeviationbetweenempiricalhuman
dataandmodeldataforbothtrainingandtestphases.Wecalculatedtheproportionoftrialsinwhichthe
modelresponsematchedtheparticipantresponse.Thekeycomparisonisbetweentheproportionof
matchesthetrainingandtestphase.Thatis,ifthespecificationsofthemodellearnedfromthetraining
canbeusedtopredictfutureperformancewithnoreductioninaccuracy.
Forthetrainingphasedeviationsbetweenthemodelresponseandparticipantresponse,as
measuredbyproportionofmismatchesrangedfrom0.26to0.10withameanof0.16.(seeFigure3).For
thetestphase,deviationsbetweenthemodelresponseandparticipantresponserangedfrom0.26to0.05
withameanof0.159.Forthefinalthirdoftrialsspecificallythedeviationrangewas0.26to0witha
meanof0.19.(seeFigure4).
Wethencomparedmodelperformanceontrainingandtestphases.Forresponsepredictionof
trainedversusnoveldataweusedanon-parametriccomparison(Mann–WhitneyUtest)betweeninitial
trainingfitandtest.ThemodelerrorduringtrainingphasewasnotsignificantlydifferentthantestW=
312,p=0.626.Wethencomparedfitonthetrainingdatatoonlythenovelportionofthetestphase(i.e.
thefinal30trials).Themodelperformanceonthenoveltrialswassignificantlybetterthanthetraining
datafit,W=427,p<0.01.
GeneralDiscussion
Thisstudydemonstratesthatamultilayersdynamicfieldcomputationalmodelusingbehavioral
andneuraldatainputcanaccuratelypredictbehaviorforindividualparticipants.Specificallythemodel
instantiationsareabletogeneralizetofuturebehavior.Experiment1resultsincludeareplicationofthe
ERPnumericaldistanceeffect.Experiment2resultsdemonstratethatthecomputationalmodelisboth
abletoaccuratelyfitindividualparticipants’behavioraldataandextrapolatetopredictnoveldata.The
currentmodel,usingindependentinstantiationsandanoptimizationalgorithmwasabletofittoarange
ofparticipantbehavioraldata.Fittoparticipantdatawasaccuratewithinanerrorrateof0.10for
responses.Extrapolationtonoveldatawasofsimilaraccuracytotheinitialmodelfit.
LimitationsandFutureDirections.Evaluatingthecomputationalmodelresultsrequirescaution.
Thereareseveralquestionstoconsider.Therewasvariationinthemodelfittotrainingdata.Whymight
someparticipantshavedatathatisbetterfitbythecomputationalmodel?Itcouldbethecasethat
participantswithlowvariabilitydataarebetterfitbythemodel.Forexampleaparticipantthatcorrectly
respondedto100%oftrialswouldbetrivialtopredict.Participantsperformanceonthetaskmight
correlatewithmodelfit,wherebetterperformingparticipantshadbetterfitmodels.Acorrelation
betweenparticipantperformanceandmodelfitwassignificantR=-0.54,t(22)=3.009,p<0.01.Wethen
lookedmorespecificallyofmodelpredictionsofincorrectanswers.Modelperformanceinpredicting
incorrectresponsesofparticipantsdidnotcorrelatewithparticipantperformance,R=-0.22,t(22)=
1.05,p=0.30.Themodel’sperformancesomewhatdependsonthecharacteristicsoftheparticipantdata.
Theinitialmodelfitwasbestforparticipantswithahighdegreeofaccuracy.Howevermodelprediction
ofwhichtrialstheparticipantsansweredincorrectlywasnotcorrelatedwithparticipantperformance.
Additionallytherewasnosignificantcorrelationbetweenparticipantperformanceandhowwellthe
modelgeneralizedtothenovelasmeasuredbythedifferenceintrainingandtesterror,R=0.03,t(22)=
0.141,p=0.88.
Itisunclearifthereisalimitationtothenumberoftrialsthemodelcouldpredictbasedonthe
training.Theremaybeaninprincipallimitationtothenumberoftrialsthemodelcanpredict.Itmayalso
bethecasethatalargertrainingsetforthemodelresultsinmoreaccurateextrapolation.Perhapsa
multiplevisitexperimentwithamuchlargernumberoftrialswouldresultingreateraccuracyin
predictions.Thecurrentstudyonlyinvolvedabriefexperimentalsession.Longerandpotentially
multiplevisitexperimentalsessionscouldaddressthisquestion.
Weusethecanonicalnon-symbolicnumericaltaskhere.Thistaskiscentraltothestudyof
numericalcognitionandhasshowntocorrelatewithothertasks,whicharemorerelevanttoeducational
applications.Thequestionremainsissuchamodelingapproachcouldbeextendedtonon-symbolic
arithmetic,symboliccomparisonorevensymbolicarithmetic.Asimilarstyleneuraltuningcurve
computationalmodelhasbeenusedforothertaskssuchassymbolicnumberlineandsymbolic
arithmetic(Prather2012;2014;2016)
Conclusions.Inthisstudywepresentanovelcomputationalmethodforpredictinghuman
behavioronanumericalcomparisontask.Thecomputationalmethodusesbothbehavioralandneural
datatomakepredictionsofparticipants’futurebehavioronindividualtrials.Themethodmaypotentially
beexpandedtoincludeothernumerical,arithmeticorgeneralcognitivetasks.Themethodmayalsobe
adjustedtopredictaparticularstimulussetthatleadstoimprovedperformanceonthecurrenttaskor
othergeneralcognitivetasks.Suchexpandedandadapteduseofthecomputationalmethodcouldprove
tobeasignificantstepinbridgingresearchoncognitiveinterventiondesignandindividualizedtraining
regimens.
References
Anguera,J.a,Boccanfuso,J.,Rintoul,J.L.,Al-Hashimi,O.,Faraji,F.,Janowich,J.,…Gazzaley,a.(2013).
Videogametrainingenhancescognitivecontrolinolderadults.Nature,501(7465),97–101.
http://doi.org/10.1038/nature12486
Ben-Shalom,T.,Berger,A.,&Henik,A.(2013).Mybrainknowsnumbers!-anERPstudyofpreschoolers’
numericalknowledge.FrontiersinPsychology,4(October),716.
http://doi.org/10.3389/fpsyg.2013.00716
Chen,Q.,&Li,J.(2014).Associationbetweenindividualdifferencesinnon-symbolicnumberacuityand
mathperformance:ameta-analysis.ActaPsychologica,148,163–72.
http://doi.org/10.1016/j.actpsy.2014.01.016
CohenKadosh,R.,Dowker,A.,Heine,A.,Kaufmann,L.,&Kucian,K.(2013).Interventionsforimproving
numericalabilities:Presentandfuture.TrendsinNeuroscienceandEducation,2(2),85–93.
http://doi.org/10.1016/j.tine.2013.04.001
Erlhagen,W.,&Schöner,G.(2002).Dynamicfieldtheoryofmovementpreparation.PsychologicalReview,
109(3),545–572.http://doi.org/10.1037//0033-295X.109.3.545
Fazio,L.K.,Bailey,D.H.,Thompson,C.a,&Siegler,R.S.(2014).Relationsofdifferenttypesofnumerical
magnituderepresentationstoeachotherandtomathematicsachievement.JournalofExperimental
ChildPsychology,123,53–72.http://doi.org/10.1016/j.jecp.2014.01.013
Heine,A.,Tamm,S.,Wissmann,J.,&Jacobs,A.M.(2011).Electrophysiologicalcorrelatesofnon-symbolic
numericalmagnitudeprocessinginchildren:joiningthedots.Neuropsychologia,49(12),3238–46.
http://doi.org/10.1016/j.neuropsychologia.2011.07.028
Iuculano,T.,Rosenberg-Lee,M.,Richardson,J.,Tenison,C.,Fuchs,L.,Supekar,K.,&Menon,V.(2015).
Cognitivetutoringinduceswidespreadneuroplasticityandremediatesbrainfunctioninchildren
withmathematicallearningdisabilities.NatureCommunications,6,8453.
http://doi.org/10.1038/ncomms9453
Jiang,T.,Qiao,S.,Li,J.,Cao,Z.,Gao,X.,Song,Y.,…Chen,C.(2010).Effectsofsymboltypeandnumerical
distanceonthehumanevent-relatedpotential.Neuropsychologia,48(1),201–10.
http://doi.org/10.1016/j.neuropsychologia.2009.09.005
Libertus,M.E.,Feigenson,L.,&Halberda,J.(2013).IsApproximateNumberPrecisionaStablePredictor
ofMathAbility?LearningandIndividualDifferences,25,126–133.
http://doi.org/10.1016/j.lindif.2013.02.001
Libertus,M.E.,Odic,D.,&Halberda,J.(2012).Intuitivesenseofnumbercorrelateswithmathscoreson
college-entranceexamination.ActaPsychologica,141(3),373–379.
http://doi.org/10.1016/j.actpsy.2012.09.009
Mazzocco,M.M.,Feigenson,L.,&Halberda,J.(2011).ImpairedAcuityoftheApproximateNumber
SystemUnderliesMathematicalLearningDisability(Dyscalculia),82(4),1224–1237.
http://doi.org/10.1111/j.1467-8624.2011.01608.x
Park,J.,Bermudez,V.,Roberts,R.C.,&Brannon,E.M.(2016).JournalofExperimentalChildNon-symbolic
approximatearithmetictrainingimprovesmathperformanceinpreschoolers.Journalof
ExperimentalChildPsychology,152,278–293.http://doi.org/10.1016/j.jecp.2016.07.011
Park,J.,&Brannon,E.M.(2013).Trainingtheapproximatenumbersystemimprovesmathproficiency.
PsychologicalScience,24(10),2013–9.http://doi.org/10.1177/0956797613482944
Park,J.,&Brannon,E.M.(2014).Improvingarithmeticperformancewithnumbersensetraining:An
investigationofunderlyingmechanism.Cognition,133(1),188–200.
http://doi.org/10.1016/j.cognition.2014.06.011
Prather,R.W.(2012).Connectingneuralcodingtonumbercognition:acomputationalaccount.
DevelopmentalScience,15(4),589–600.http://doi.org/10.1111/j.1467-7687.2012.01156.x
Prather,R.W.(2014).Numericaldiscriminationismediatedbyneuralcodingvariation.Cognition,
133(3),601–610.http://doi.org/10.1016/j.cognition.2014.08.003
Prather,R.W.,&Alibali,M.W.(2011).ChildrensAcquisitionofArithmeticPrinciples:TheRoleof
Experience.JournalofCognitionandDevelopment,12(3),332–354.
http://doi.org/10.1080/15248372.2010.542214
Prather,R.W.,&Heverly-Fitt,S.(2016).Neuralcodingpartiallyaccountsfortherelationshipbetween
children’snumber-lineestimationandnumbercomparisonperformance.FrontiersinDevelopmental
Psychology.
Ratcliff,R.,Thompson,C.A.,&Mckoon,G.(2015).Modelingindividualdifferencesinresponsetimeand
accuracyinnumeracy.Cognition,137,115–136.http://doi.org/10.1016/j.cognition.2014.12.004
Sandamirskaya,Y.(2014).Dynamicneuralfieldsasasteptowardcognitiveneuromorphicarchitectures.
FrontiersinNeuroscience,7(January),1–13.http://doi.org/10.3389/fnins.2013.00276
Simmering,V.R.,&Perone,S.(2012).Workingmemorycapacityasadynamicprocess.Frontiersin
Psychology,3(January),567.http://doi.org/10.3389/fpsyg.2012.00567
Spencer,J.P.,Smith,L.B.,&Thelen,E.(2001).TestsofaDynamicSystemsAccountoftheA-not-BError :
TheInfluenceofPriorExperienceontheSpatialMemoryAbilitiesofTwo-Year-Olds,72(5),1327–
1346.
Tudusciuc,O.,&Nieder,A.(2007).Neuronalpopulationcodingofcontinuousanddiscretequantityinthe
primateposteriorparietalcortex.Neuron,104(36).
Vanlehn,K.(2011).TheRelativeEffectivenessofHumanTutoring,IntelligentTutoringSystems,and
OtherTutoringSystemsTheRelativeEffectivenessofHumanTutoring,IntelligentTutoringSystems
,andOtherTutoringSystems,(August2014),37–41.
http://doi.org/10.1080/00461520.2011.611369
Figure1.Example stimuli for a non-symbolic numerical comparison task. Participants are instructed
to indicate which side of the mid-line has a larger number of objects. The stimulus is displayed for a
limited amount of time to discourage counting.
0.90
0.80
0.85
Proportion Correct
0.95
1.00
Participant Performance
Figure2.Participantperformanceonthenumericalcomparisontaskasproportionoftrialscorrect.
0.15
0.00
0.05
0.10
Error
0.20
0.25
0.30
Training Phase Error
Figure3.ModelErrorforthetrainingphaseintermsofproportionofmismatches.
0.15
0.00
0.05
0.10
Error
0.20
0.25
0.30
Test Phase Error
Figure4.Themeancorrectresponseandreactiontimedeviationforthemodelforeachparticipant.
Thesearethemodelpredictionsfromthetestingphase.
AppendixA
StimuliList
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