COBEHA170X–X
Please cite this article in press as: Addyman C, et al.: Computational models of interval timing, Curr Opin Behav Sci (2016),http://dx.doi.org/10.1016/j.cobeha.2016.01.004
Computational
models
of
interval
timing
Caspar
Addyman
Q1
1
,
Robert
M
French
2and
Elizabeth
Thomas
3Inrecentyearsgreatprogresshasbeenmadeinthe computationalmodelingofintervaltiming.Awiderangeof modelscapturingdifferentaspectsofintervaltimingnowexist. Thesemodelscanbeseenasconstitutingfour,sometimes overlapping,generalclassesofmodels:pacemaker– accumulatormodels,multiple–oscillatormodels,memory– tracemodels,anddrift–diffusion(orrandom-process)models. Wesuggestthatcomputationalmodelsshouldbejudged basedontheirperformanceonanumberofcriteria—namely, thescalarproperty,theirabilitytoreproduceretrospectiveand prospectivetimingeffects,andtheirsensitivitytoattentional andneurochemicalmanipulations.Futurechallengeswill involvebuildingintegratedmodelsandsharingmodelcodeto allowdirectcomparisonsagainstabatteryofempiricaldata.
Addresses
1DepartmentofPsychology,GoldsmithsU.,London,UK 2LEAD-CNRSUMR5022,U.ofBurgundy,Dijon,France 3INSERMU1093,U.ofBurgundy,Dijon,France
Correspondingauthor:French,RobertM([email protected]) CurrentOpinioninBehavioralSciences2016,8:xx–yy
ThisreviewcomesfromathemedissueonTimingbehavior EditedbyRichardBIvryandWarrenHMeck
doi:10.1016/j.cobeha.2016.01.004 2352-1546/PublishedbyElsevierLtd.
Althoughtherearenumerouswaysinwhichcomputational modelsofintervaltimingcanbeclassified,wehavechosen togroupthesemodelsintofourmajor,althoughsometimes overlapping,classes:firstly,pacemaker–accumulator mod-els(PAmodels),secondly,multiple–oscillator-coincidence detectionmodels(alsosometimescalledtimestamp mod-els),thirdly,memoryorneuralprocessmodelsand,finally, fourthly random-process(or drift–diffusion) models.For alternativeclassificationschemes,see,forexample[1,2].
Inwhatfollowswewillsuggestthatcomputational mod-els of interval timing be judged on the basis of the followingcriteria:thescalarproperty,prospectiveand retro-spectivetiming,andtheeffectsofattentionand neuropharma-cological manipulations.
Extensive empirical evidence [3–6] suggests that time-estimation errorsin interval timing growapproximately
linearlywiththesizeoftheestimate.Knownasthescalar propertyoftimeestimation,thisfactsetsahardconstraint onthenatureoftheunderlyingprocessesinvolvedintime estimation[7].Thiseffecthasbeenwidelyreplicatedin humans,pigeons,androdents(see[8–10].Similar behav-ioralresponsestotimescalescanevenbefoundin rate-dependenthabituationinCaenorhabditiselegans[11].Even though the scalar property has not been found to hold underallconditions[12],modelingithasprovedtobea significantchallengeforanumberof existingmodelsof interval-time judgments[7,13].In arecent paper, Hass andHermann[7]useinformationtheoreticargumentsto show how the scalar property places several important restrictionsonthenatureof any intervaltiming mecha-nism.Crucially,theyarguethat,inordertodisplayscalar errorprofiles,theneuralprocessunderlyingtime percep-tionmustbebasedonameasureofgrowingvariancein thesystem.
Secondly, it has been established that the perceived passage of time by human adults differs according to whethertheyareforewarnedthattheywillneedtomake atimingjudgment,andarethereforeactivelyattendingto itspassage(prospectivetimeestimation),orwhetherthey arerequiredtomakeanunexpected,after-the-fact judg-mentofthepassageoftime(retrospectivetimeestimation). Models shouldbejudged onhowwellthey accountfor bothof theseregimes.
And thirdly, there are varioussystematiceffects onthe lengthsofestimatescausedbylevelsofattention[14]and neurochemistry, suchasendogenouslevelsofdopamine or theeffects ofdopaminergic drugs[15–17].
Weavoidedthecriterionof‘neurobiologicalplausibility’ becauseitisnotoriouslydifficulttopindownexactlywhat ismeantbythisexpression.So,forexample,howrealistic do computationalneurons haveto bebeforethe model thatusesthem canbesaidtobebiologicallyplausible?
Pacemaker–accumulator
models
The pacemaker–accumulator models (PAM) [18,19] havehadagreatinfluenceonthewaythatexperiments on timing are conceived and interpreted. Many of the recent modelsof timingstill utilize thepacemakerand accumulator processes described by Treisman [20]. These models currently constitute the most popular computationalapproachestointervaltiming.Inthe pace-maker–accumulatormodel,thearrivalofastimulusstarts a clock which generates pulses that are counted by an accumulator.Timejudgmentsarethenmadebya com-parisonofwhatisstoredintheaccumulatorandwhatis
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2 Timing behavior COBEHA 170 X–X Please cite this article in press as: Addyman C, et al.: Computational models of interval timing, Curr Opin Behav Sci (2016), http://dx.doi.org/10 .1016/j.cobeha.2016.01.004 105 Table 1
Reference Model type/Name What keeps the time? What tells the time? Scalar property? Neurochemical
or attention effects? Prospective or retrospective? Comment [18] Pacemaker–accumulator: Scalar Q6 Expectancy Theory (SET) Poisson process pacemaker and error free accumulator
Comparing estimates to those retrieved from memory.
Via memory comparison not via clock
No Prospective The first Pacemaker model to
address the Scalar property.
[21] Pacemaker–accumulator Poisson pacemaker Unreliable/stochastic
multistage accumulator
Under special circumstances
No Prospective An unreliable counter
mechanism can give rise to scalar property under very narrow circumstances.
[19,62] Pacemaker–accumulator Pacemaker with
geometrically increasing tick length and Gaussian noise
Accumulator built into larger ACT-R model.
Via implausible pacemaker assumptions
Attention effects
Prospective A classical PAM embedded in
an ACT-R framework models attention effects as a result of resource competition.
[63] Pacemaker–accumulator Constant rate pacemaker ACT-R model with time
stored in working memory
No Some
attention effects
Prospective Simplistic PAM model built in
ACT-R.
[22] Pacemaker–accumulator Poisson pacemaker Accumulator and memory Via ad hoc
Gaussian error mechanism
No Prospective Notable for allowing direct
quantitative test of SET by implementing it in Framsticks simulation environment.
[23] Multiple–oscillator: beat
frequency
Set of cortical oscillators of different phase
Time measured by selecting subset that will be in phase at correct interval
No No Prospective Original multiple–oscillator
model.
[25] Multiple–oscillator: striatal
Beat Frequency (SBF)
Set of cortical oscillators of different phases
Coincidence detectors based on striatal spiny neurons Only under assumption of globally correlated phase variations Several neurochemical effects
Prospective A modern oscillator model
that takes good account of neuroscience evidence.
[26,27] Multiple–oscillator: SBF
with realistic noisy neurons
Set of cortical oscillators with different phases and uncorrelated noise Neural network ‘coincidence detector’ Yes Yes–numerous pharmacological effects.
Prospective A nice reinvention of SBF
where scalar property emerges naturally from network noise. [11,34] Memory decay: multiple
time scales (MTS)
Chain of decaying activations
Reading off absolute level of decay
By assuming fixed Gaussian error threshold
No Prospective First memory decay model
was actually model of habituation in C. elegans. Only models prospective timing because requires dedicated mechanism.
[36] Memory decay: Gaussian
Activation Model (GAMIT)
Spreading cortical activation from event to be timed and rate of change of activation.
Comparison of activation to learned reference curve
Yes Cognitive load
effects via attentional resource competition
Both Retrospective case a single
estimate is made at end of interval. In prospective case multiple estimates during interval contribute.
[37] Memory decay:
GAMIT-Net
Spreading cortical activation
Neural network learns to estimate time
Yes Attention effects
via resource competition
Both Neural network version of
GAMIT model. Current Opinion in Behaviora l Sciences 2016, 8 :x – x www.sci encedirect.com
Computati onal models of interval timing Addyman, French and Thomas 3 COBEHA 170 X–X Please cite this article in press as: Addyman C, et al.: Computational models of interval timing, Curr Opin Behav Sci (2016), http://dx.doi.org/10 .1016/j.cobeha.2016.01.004 105 106
[35,53] Memory decay: temporal
context model (TCM)
Set of leaky integrators that stores stimulus event plus ‘context’ from previous events Feedforward connections permit reconstruction of sequences of events Due to choice of reconstruction algorithm
No Both Adapts model of serial
memory performance to more general task of interval timing. Estimation method is relatively complex
approximate inverse Laplace Transform.
[64] Memory decay: coupled
leaky integrators
Decay in activation in a two neuron systems acts like a simple oscillator.
Network has wait or respond states.
No No Prospective A very simple neural system
model animal learning data. Noise plays important role in stabilizing network behavior.
[38] Climbing activation Firing rate adaptation in
inhibitory neurons leads to increasing activity in excitatory neurons.
When active population crosses fixed threshold. Changes to adaptation rate change interval
Yes No Prospective Detailed neural model inspired
by recordings from macaque inferotemporal cortex.
[65] Climbing activation: Dual
klepsydra model
Leaky integrator Comparing one integrator
to another
No No Prospective Unclear why integrator values
cannot be accessed directly.
[42,43] Climbing activation:
evolved, embodied neural net model.
An evolved continuous time recurrent neural network
Networks seemed to work via climbing activation.
No No Prospective Evolved neural network with
standard leaky-integrator neurons tells time without clock-like control a robot in a simulated environment.
[45] Random process:
population of bistable units
Population of independent bistable units transitioning from off to on
When number of ON neurons crosses threshold
Yes No Prospective Different intervals measured
by different global transition probabilities. Not clear how this would be implemented. [46,47] Random process: drift–
diffusion model of interval timing & decision making
Random walk by competing random inhibitory and excitatory processes.
When total crosses particular threshold.
Yes No Prospective An probabilistic model than
accounts for decision making and interval production in same framework.
[66] Contextual change Estimates derived from
amount of activity, number of actions and ACT-R system time.
ACT-R model No Some
attention effects
Retrospective Underspecified mechanism
but embedding model in ACT-R framework allowed testing of attention effects. www.scien cedirect.com Current Opinion in Behavioral Sciences 2016, 8 :x – x
stored in memory. Gibbon’s Scalar Expectancy Theory (SET)modelemphasizedtheimportanceofreproducing the property of scale invariance observed in interval timing[3,18].Scalar errorin this modelarises notfrom theclockitselfbut rather fromnoisein thecomparison process. Several variants on this original pacemaker– accumulator design have been produced. For example, KilleenandTaylor[21] use adifferentapproach tothe scalar property by using a noisy accumulator process ratherthananoisycomparator
Q2 (Table1).
Recentmodels havetaken thepacemaker–accumulator processandincorporateditintoalargercognitivesystem. For example, Taatgen et al. [19] place a timekeeping moduleinthecontextofageneralACT-Rarchitectureto capturetheeffectsofattentionandresourcecompetition onintervaltiming.Thismodelincorporatesanattentional gate which modulates the rate of pulse accumulation henceleadingtochangesin theperception ofintervals. AnotherexampleisKomosinskiandKups[22]whobuild a classical PAM in a neural simulator environment to model time-judgment errors in successively presented timeintervals.
Onedifficultywiththesemodelsisthaterrorsin sequen-tialprocessesgrowtooslowly(asthesquarerootoflength oftheinterval).Anytimerbasedondirectaccumulation oftickswouldbetooaccurate.Inordertoaccountforthe scalarpropertyof time,pacemaker–accumulatormodels have to introduce a secondary source of multiplicative errorinthecomparisonprocess [7].
Multiple–oscillator
models
Multiple–oscillator models [23,24] refer to models of interval timing in which intervals are represented as a setofactivitiesofseveraloscillators.Anearlyformofthe model was developed byMiall [23]. In this model, re-ferred to as the beat frequency (BF) model, timing is carriedoutbytheactivationofseveraloscillators,eachof which oscillates at its own particular frequency. The arrival of a stimulus resets the oscillators so that they begintofiretogether.Thetimeelapsedsincethearrival of the stimulus would then depend on the oscillatory phases of the entire set of oscillators. However the distributionoffiringwasnotnormallydistributed,having asharppeakatthetargettimeandsmallerpeaksatthe majorharmonicsofthefundamentalinterval.Inaddition, thewidthofthepeakwasnotproportionaltothelengthof theinterval.Forthisreason,andbecausethemodeldid notcontain any noise, it was unable to account for the propertyofscalarinvariance.
The Striatal Beat-Frequency (SBF) model tried to ad-dress these problems [25]. They modified the BF to induce the scalar property. The SBF model took into account experimental findings that interval timing was notexclusivelytheresultofactivityinthebasalganglia
butalsoofactivityinathalamo-cortico-striatalcircuit.In thismodel,oscillationsaregeneratedbycorticalneurons andtimingisindicatedbythecoincidentalactivationof spinyneuronsinthestriatumofthebasalgangliabythe corticaloscillators. Oscillatorspeedsand neuronalfiring thresholdswereadjustedonatrialbytrailbasisinorderto reproducetheGaussianshapedresponseprofilesseenin timingexperimentsthatuse thepeakprocedure experi-mental method and thereby produce scalar invariance. However,theseadjustmentshadtobegloballycoherent, otherwisethecoincidence-detectionsmechanismswould notoperate appropriately.Thistendsto make theSBF modeloversensitivetosmallamountsof noise.
Improvements to the SBF model have been made by [26,27].Thismodelretained theseparationof cortical andstriatalrolesusedintheSBFmodels.Theneuronsin the new models however, were far more realistic. The simplerneuronalmodelswerereplacedbymoredetailed Morris–Lecar neuronsand neural activity was now the result of the dynamics in several ionic channels. This model succeeded in replicating several experimental findings on the effects of dopamine and cholinergic agents on timekeeping. In a more generalized version ofthemodelinwhichaperceptronreplacedthestriatum anditscoincidencedetection,scalarerrorswerean emer-gentpropertyofthenetworkwithouttheneedforglobal coherence[26].TheSBFmodelhasalsobeenextended toincludeaunifiedaccountofduration-basedand beat-basedtimingmechanisms[28,29].
Memory-based
models
Athirdclassofmodelsreliesonmemorydecayandfalling (or rising)neuralactivation. Theseneural processesare relatively well understood and provide evidence that timing and memory use the same cognitive resources [30], recruiting neurons in the dorso-lateral prefrontal cortex [31–33]. Once again, the scalar property does notalwaysarise fromthese modelsin astraightforward manner.Forexample, theMultipleTimeScalesmodel (MTS,[11,34])reliesonaseriesofleakyintegratorswith powerlawdecayandtheseintegratorsmustbecarefully linked to approximate the required logarithmic decay function. The Temporal Context Model (TCM, [35]) relies onmany leaky integrators and farmore complex dynamicsthan theMTS model.
Computational memory models have been introduced whichtakeintoaccountnotonlytheamountofactivation decay of a memory trace but also the rate at which activation decays (GAMIT: [36,37]). In this model, thereisamechanismofattentional-resourcesharingthat allowsGAMITtomodelbothretrospectiveand prospec-tivetiming.
By contrast with these falling activation-trace models, Reutimannetal.[38]useasingleclimbingneuronaltrace
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that attains a threshold at the expected end of an interval.Thismodel[38]isbuiltonasinglemechanism using well-understood principles of synaptic plasticity and the decision rule is built into the model itself. Single cell recordings in the inferotemporal cortex of monkeys have, infact, found neurons withthe appro-priate time-dependent firing rates [39,40]. This inter-pretation of climbing activation remains controversial, however, see [41].
An interestingrecentadditiontothisclassis[42,43],in whichneuralnetworkswithstandardleaky-integrator neu-ronswereevolvedtocontrolarobotinasimulated envi-ronmentinordertoperformatemporalcomparisontask. When network activity was examined timing appeared tobeduetoaclimbingactivationmechanism.
Random
process
models
Modelsdiscussedsofarhavebeenbroadlydeterministic orbasedonprobabilisticprocesses(e.g.countingrandom ticks) that produce time estimates that have less than scalar error. The models in this section are based on probabilistic processeswith linearor greaterthan linear error.Thesimplestapproach[44]replacesasingle Pois-son process with a group of 100 independent Poisson process and a leaky integrate-and-fireneuron that fires andresetseverytimeitcrossesathreshold.Withafixed threshold this model underestimates intervals but improves withtheincorporationof adynamicthreshold thatisinhibitedbyrecentfirings.However,theactualfit to empirical data remains poor. A better fit to data is obtainedby[45]inwhichatimerstartsbysetting50 bis-tableunitsto‘off’.Thereafter,eachbistableunit transi-tionsto ‘on’independentlywith probabilityp (adjusted bylearning)andthetimerstopswhenatotalof40units are active.
Ifexcitatoryandinhibitoryprocessesbothcontributeto the sameintegratorthen, unlessthe processesare pre-ciselybalanced,theresultingrandomwalkwilldriftin onedirection.Adjustingthebalanceadjuststherateof driftallowingdifferentintervalstobelearned[46,47]. Thelearningprocess issimpler thanin[45]becauseit doesnotrelyonfinetuningagroupofprobabilities.The approachhasadditionaladvantagesthatthesame frame-work can model decision making and that it makes severalquiteprecisepredictionsaboutskewand coeffi-cientsofvariationofresponsesintemporalreproduction tasks.
Finally, it should be noted that in subsecond timing most successful models are random-process models, basedonstochasticallyconnectedchainsofnoisyneurons [48,49,50].However,mostauthorsdonotthinkthatthese models can be extended to the multi-second domain of interval timing [51]. This inability to scale up to multi-secondtimingappliesonlytotheserandom-process
models.Itremainsanopenquestionastowhetherother classes of models can account for both subsecond and multi-secondtiming.
Difficulties
with
the
models
As currently implementedpacemaker–accumulator and multiple–oscillator models rely on a dedicated timing mechanism whichneedstobestartedwhenaparticular eventoccurs.Thisisproblematicforretrospectivetiming becauseallperceivedeventsarepotentialcandidatesfor retrospectivetimejudgmentsand,therefore,eachevent wouldrequireaseparatetimer.
Staddon[52]suggestedthatmemory–tracemodelscould overcomethisresetproblembecauseallperceivedevents encodedbythecognitive systemautomaticallyresultin representations that are governed by the same trace dynamics. However, most activation-tracemodels posit aspecialisttimingmechanismthatisonlyrecruitedwhen timingisrequired(e.g. [34,38])and modelsof thistype can only address prospective timing. The Temporal Context Model (TCM) [35] developed from a model of episodicmemory,canpotentiallyperformboth retro-spective and prospective timing. To the best of our knowledge, TCMis thefirstattemptto use featuresof memory directly as a mechanism for interval timing. GAMIT[36]hassimilar motivationsbut ismuch sim-plerthanTCM.
Our estimates of time passing can alsobe affected by whetherornotweareactivelyattendingtothepassageof timeandbycognitiveload.Blocketal.[14]foundthat highcognitiveloadincreasesretrospectivetimeestimates and decreasesprospectivetimeestimates.Modelingthis surprisingeffectisachallengeforallexistingmodelsof intervaltiming.Frenchetal.[36]suggestanattentional resource-sharingmechanismthatallowsprospectiveand retrospective timing to be accounted for in a single model.Moreover,thismodel,GAMIT[36],iscurrently theonlycomputationalmodel toaccountforthis inter-action.
Mostmodelssimplydonotconsiderattentionaleffectson intervaltimeperception[34,38,53].Onesimpleproposal is that attention might modulate clock speed directly [25].Ifdecreasedattentiontotimingcausesthe organ-ism’s internalclock to beat slower, then it will tend to underestimatethelengthofintervals.Thisideais devel-oped further in the time-sharing model [54]. Working memory,timingandattentionalldependon dopaminer-gicpathways[32,55,56].Thechangesobservedininterval timingestimatesfollowingpharmacologicalinterventions thatmodulateclockspeed[16,57]havebeenmodeledby letting dopamine levelsaffect oscillatorfrequency (e.g. [26,27,58]). Nevertheless, none of these models can account fortheincreasein retrospectiveestimatesunder highcognitive load.
ComputationalmodelsofintervaltimingAddyman,FrenchandThomas 5 COBEHA170X–X
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Farfewermodelsattempttoexplainretrospectivetiming, in part because retrospective timing does not have an equivalentinanimalbehavior.Acommonthemebehind allapproachestoretrospectivetimingisthatintervalsare estimatedbyreconstructingasequenceof remembered events.Cognitiveloadcouldaffectthisbychangingthe memorabilityornumerosity ofevents[59,60].
Future
challenges
In conclusion, computational models of interval timing have come a long way but are still faced with many challenges. Besides the difficulties already discussed, a genuinelymaturemodelneedsto:
fitindividualnotjustgroup data
give a coherent account of relationship between retrospectiveandprospectivetiming,
apply to the full range of timing tasks and their associated attentional and pharmacological modula-tions,
explaincommonalitiesanddifferencesbetweenanimal andhumantimeperception.
We have argued elsewhere [61] that modelers need to maketheircodeavailableanduseraccessiblesothattheir models can be directly compared and developed. The current variety of modeling approaches is a strength. Bringing the successes of these varied models into a comprehensive frameworkis thelong termgoalfor the field.
Conflicts
of
interest
Nothingdeclared.Acknowledgments
Thisworkwassupportedinpartbyagrantfromthe
Q3 FrenchAgence
NationaledelaRecherche(ANR-14-CE28-0017)tothesecondauthor,bya jointgrantfromtheANR(ANR-10-056GETPIMA)tothesecondandthird authors,andtheUKESRC(RES-062-23-0819)tothefirstauthor,within theframeworkoftheOpenResearchArea(ORA)France–UKfunding initiative.
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Please cite this article in press as: Addyman C, et al.: Computational models of interval timing, Curr Opin Behav Sci (2016),http://dx.doi.org/10.1016/j.cobeha.2016.01.004
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