Modelling single person and multi person event based synchronisation

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Elliott, Mark T., Chua, Wei Ling and Wing, Alan M. (2016) Modelling single-person and

multi-person event-based synchronisation. Current Opinion in Behavioral Science, 8 .

pp. 167-174.

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Modelling

single-person

and

multi-person

event-based

synchronisation

Mark

T

Elliott

,

Wei

Ling

Chua

and

Alan

M

Wing

A linearphase correctionmodelhasbeenshownto accurately reflectthecorrectiveprocessesinvolvedin synchronisingmotoractionstoanexternalrhythmiccue.The modeloriginated fromstudiesoffingertappingto an isochronousmetronomebeatandisbasedonthetimeseries ofasynchroniesbetweenthemetronomeandcorresponding fingertaponsets,alongwiththeirassociatedintervals.Over recentyears themodelhasevolved andbeenappliedto morecomplexscenarios,includingphaseperturbed cues, tempo variationsand,mostrecently,timing withingroups. Here, wereviewthestudiesthat havecontributedto the development ofthelinear phasecorrectionmodelandthe associated findingsrelatedto humantiming performance. Thereviewprovidesabackgroundtothestudiesexamining single-person timingto simplemetronomecues. Wethen furtherexpand onthe morecomplexanalysesofmotor timing tophase andtemposhiftedcues. Finally,recent studiesinvestigatinginter-personalsynchronisationbetween groupsoftwoormoreindividualsarediscussed,alongwitha briefoverviewontheimplicationsofthesestudiesforsocial interactions.Weconcludewithadiscussiononfutureareas of researchthat willbeimportantforunderstanding correctivetiming processesbetweenpeople.

Addresses

1_{InstituteofDigitalHealthcare,WMG,UniversityofWarwick,UK} 2_{SchoolofPsychology,UniversityofBirmingham,UK}

Correspondingauthor:Elliott,MarkT([email protected])

CurrentOpinioninBehavioralSciences2016,8:167–174

ThisreviewcomesfromathemedissueonTimingbehavior

EditedbyRichardBIvryandWarrenHMeck

http://dx.doi.org/10.1016/j.cobeha.2016.01.015

2352-1546/#2016TheAuthors.PublishedbyElsevierLtd.Thisisan openaccessarticleundertheCCBYlicense( http://creativecom-mons.org/licenses/by/4.0/).

Introduction:

variability

of

timing

Rhythmicactionwithperiodicmovementsthatare main-tained in synchronywithothersor withregulated phase across group members is a common feature of various humanactivities.Forexample,inarowingeight,atarate of30–40strokesperminute,therowersattempttobring thebladesoftheiroarsintothewateratthesametimeto achieveagood‘‘catch’’.Thisisfollowedbyaconcerted pull to drive the boat through the water [1]. In music

ensembles, at tempos ranging from50 to 200 beats per minute(bpm;largo—prestissimo),theplayersstrivefora commonpulsesothatnotesscoredassimultaneoussound togetheracrossthedifferentinstruments[2].Indance, theperformersnotonlymoveintimetothemusicbutmust also synchronise among themselves [3]. The question addressedinthisreviewis,howdoindividualparticipants engagedin suchactivitiesadjust theirrelative timingto achievesynchronywithotherindividualswithinthegroup?

Biological timingis inherentlyvariable and affectedby fluctuationsinproducedintervalswhich,forinstance,in simple tapping tasks, increase with duration [4,5].As a result,evenifthevariousmembersofanensemblestart exactly together and agree on the same target interval (tempo orrate), individual timingvariability meansthe membersoftheensemblewillinevitablyslipoutofphase with one another during the course of a performance. Tocompoundtheproblem,tempochangeisoftencalled forduringperformance(e.g.slowingattheendofapiece of music). Asa result, differences in the control of the rate of tempo change by each individual will further add to the tendency to develop differences in phase. Activeadjustmentoftimingisthereforerequiredtokeep theplayers’phasedifferencesclosetozero.Inthispaper, we review how adaptive feedback and predictive feed-forward mechanismsoperatein supportofinterpersonal timing.Westartbyconsideringonepersonsynchronising withafixedoranadaptivemetronome.Theevent-based timingmodelsthathavebeenusedtodescribecorrection mechanismsforanindividualtomaintainsynchronywith a metronome, are defined. We then turn to the case of groupsoftwo ormoreindividualssynchronisingwith one another. Tasks discussed in this review include finger tapping, arm movement, musical performance, androwing.

Synchronisation

with

a

fixed

metronome

Perhapstheearliestpublisheddemonstrationofthe vari-abilityinindividualperiodictimingisthatofStevens[6]. Participants tappedaMorsecodekey, firstin timeto a metronome then unpaced, at rates in the range of 60–

the nth movement implementation event (response), causing negative covariation between successive IRIs [7].Intermsofpacedtapping,itisthecentraltimekeeper interval, Tn and its variability, s2T, that determine

syn-chronisationaccuracywiththemetronome.Timekeeper variability tends to increase with longer interval dura-tions,whereasmotorvariability,s2_M,remainsata relative-lysmall value[7–9].

Theabilitytosynchronisewithametronome(forreviews see[10,11])despitethepresenceofvariabilityintimed periodicmovement,impliesfeedbackcorrection.Vorberg andcolleaguesproposedafirst-orderlinearphase correc-tionmodel,inwhichtheasynchronybetweenthefinger tap and related metronome pulse is used to effect a

proportional correction of the time to the next tap [12,13];seeEq.(1).

Anþ1¼ð1aÞAnþTnþMnþ1MnSn (1)

where ais the correction gain, Anis thecurrent event

asynchrony, Tn is the time interval generated by an

assumed internal timekeeper, Mn is the current motor

implementationdelay,andSnisthecurrentmetronome

interval(seeFigure1a).

Ifthe correction gain, a,lies between0 and 2, Eq. (1) resultsinstableperformanceinthesensethata synchro-nisation error at tap n is progressively reduced over successive taps, n+1, n+2, etc. Here, we focus the review on this linear phasecorrection approach, where

168 Timingbehavior

Figure1

A_n–1 A_n A_n+1 A_n+2

IRI_n–1 IRI_n IRI_n+1 Sn–1 Sn Sn+1

M_n–1 M_n M_n+1 M_n+2

T_n–1 Tn Tn+1

S_n–1 S_n S_n+1 S_n+2

Tn–1 Tn Tn+1 T*n T*n+1

T*n–1 T*n+2 T*n+3

T_n+2 (b)

(a)

(c)

T_n~ N(t_n,

σ

α

Tn~ N(tn,

σ

β

S_n–1 S_n S_n+1 S_n+2

Tn+3 S_n+3

Current Opinion in Behavioral Sciences

(a)Schematicofthetwoleveltimingmodel.Weassumethatwhenparticipantstapintimetoametronome(withinterval,S,showninbrown)the observedvarianceintheasynchronies(A)andinter-responseintervals(IRI,blue)isaresultofthevarianceinthetimekeeperintervals(T,red)and themotordelays(M).Becauseoftheresultingvariance,acorrectionmechanismmustbeimplementedtoadjustfortheerrormadeonthe precedingtap.Thiscorrectionisappliedtothetimekeeperintwoways,phaseandperiodcorrection.(b)Aphasecorrectionisappliedtothe timekeepertoadjusttherelativephasebetweenfingertapevents(notshown)andthemetronomebeats.Thecorrectionismadetothe timekeeperinterval,Tn,whichissampledfromanormaldistributionwithmeanintervaltnandstandarddeviation,sT.Theamountofcorrectionis basedonthelastasynchrony(An1)multipliedbyacorrectiongain,alpha(a).Afullcorrectionofthelastasynchronythereforeoccurswhen

movement correctionsare performedwithin thisstable range. Analternativemethodtoanalysing synchronisa-tion is the non-linear dynamical systems (NLDS) ap-proach.CharacteristicsoftheNLDSapproachincludea focusoninstabilityandanemphasisonthecontinuous nature ofbehaviour. Acomparison oflinearphase cor-rection and NLDS approaches may be found in Pressing[14].

ItcanbeshownthatcorrectionsbasedonEq.(1)produce a characteristic asynchrony autocovariance function (AACF). The AACF exhibits a negative value at lag one damping towards zero with increasing lag if a lies between0and1(over-damped).Incontrast,itoscillates betweennegativeand positivevalueswhiledampingto zeroifaliesbetween1and2(under-damped).Thevalue of a that minimises asynchrony variance is considered optimalandresultsinanAACFthatiszerobeyondlag1 (criticallydamped).Thisoccurswhena=1,droppingto slightly less than 1 when there is appreciable motor implementation variability. Vorberg and Schulze [13] developed an estimationprocedurebased onnumerical minimisation tofittheAACFof theexperimental asyn-chroniesto themodel’spredictions.However,the accu-racy of this method is limited by estimation bias and parameterinterdependencewhentheestimatedavalues are closeto optimal.Jacoby andcolleagues[15]showed thattheseproblemscanbeavoidedbyassumingthatthe motorvarianceissmallerthanthetimekeepervariance.A companion paper [16] developed a bounded General LeastSquares(bGLS)methodforparameterestimation by reformulating the linear phase correction model in termsof matrixalgebra(seeBox1).

Correcting

for

a

shift

of

phase

in

the

metronome

So far, thephase correctionmodel hasbeen discussed in the context of corrections arising from the partici-pant’sowntimingerrorswithrespecttoafixedinterval (or isochronous) metronome. However, it is also inter-esting to consider the complementary case when a timing error is introduced by the metronome. When just one of the metronome intervals is lengthened or shortened, shifting thephase ofall subsequentpulses, participantsmustadjusttheir timingtoregain synchro-ny with the beat (see Figure 1b). This corrective response can be used to give a direct measure of the correctiongainintermsoftheproportionalreductionin asynchrony on the response following the phase shift event.Interestingly,thesamecorrectionoccurs regard-less of effector, be it upper limb (e.g. finger tapping, [17]) orlowerlimb (e.g.stepping, [18]). However, the correctiongainimmediatelyfollowing aperturbationis much largerthan thegain estimatedfrom steady state series[19],suggestingadifferentcorrectionmechanism may come into play for sudden perturbations to the metronome phase.

Correctingforaphaseshiftrequiresachangeinthetarget tapping interval untilsynchronyis regained.The ques-tion whicharisesiswhether thisadjustment to the tap-pingintervalinvolvesachangeintheperiodofthecentral timekeeper?A variantof thephaseshift taskaddressed thisquestion,wherebythemetronomewasswitchedoff immediately afterthephase shift andparticipants were instructedtocontinuetappingatthesamepace[20].The period change observed in the subsequent finger tap

Boxtopic1 ThebGLSmethod

ThebGLSmethodusestheinter-stimulusintervals(i.e.theintervals betweencueonsets;Sn)andtheasynchroniesbetweenthe participant’smovementonsetandthecueonset(An)toestimatethe linearphasecorrectionparameters(motorvariability,sM,timekeeper variability,sT,andcorrectiongain,a).Thecomponentintervals(see Eq.(1))arearrangedinmatrixformatasfollows(from[16_]):

y ¼BxþZ

wherey=[A1+S1E(A+S),A2+S2E(A+S),...,

An+SnE(A+S)]T;B=[A0E(A),A1E(A),...,An1E(A)]T;

x=1a;Z¼½H0;H1;...;Hn1;andHn=Tn+Mn+1MnE(T)

E()representsthemeanvalueacrossthetimeseries.

ZisconsideredasmultivariateGaussiannoise,withthecovariance matrixdefinedas:

S¼CovðZÞ¼

gð0Þ gð1Þ 0 0

gð1Þ } } 0 0 } } gð1Þ

0 0 gð1Þ gð0Þ 2

6 6 4

3 7 7 5

IfSwasknown,thentheestimateofxcouldbeachievedusingthe GeneralLeastSquaresapproach:

x ¼ðBTS1BÞ1ðBTS1Þy

Alternatively,ifxwasknown,thenScouldbeestimatedbythelag 0andlag1autocovariancesofyBx

S¼g₀Iþg₁D

whereIistheidentitymatrixandDisamatrixcontainingonesonthe diagonalseithersideofthecentraldiagonalandzeroselsewhere.

However,bothxandPareunknownvaluestobeestimated.The estimatesarethereforeachievedbyaniterativeapproach,first estimatingxbysettingS=I.Usingthefirstestimateofx,anupdated estimateofSiscalculated,andsoon,untilthexandSvalues convergetoastablevalue.TheconstraintdefinedinJacobyetal.

[15],meansanadditionalstepisaddedtotheiteration,thatchecks theautocovariancevaluesmeetthecondition:

0<gnð1Þ<gnð0Þþ2gnð1Þ

Ifnot,anadjustmentismadetogn(1),toensuretheconstraintismet beforethenextiterationisexecuted.

intervals suggestedthat phase correction resulted in an adjustment to the central timekeeper interval. Differ-encesin theinvolvementofperiodchangeinthephase shift paradigm and the constant metronome paradigm mightinpartaccountforthecorrectiongaindifferences observedin[19].

Tempo

tracking

Changesintempoaresometimesencounteredinperiodic timing tasks. For example, musicians often vary their timingbyspeedinguporslowingdownasanexpressive interpretationofamusicalpiece.Itistherefore interest-ingtoconsidertheresultsoffingertappingstudieswhich called fortempochangesto examineunderlying timing controlmechanisms.Schulzeetal.[21]studiedtransitions between pairs of target metronome intervals selected from 300, 370, 440, and 510ms. Each trial began with between17and22intervalsatthebasedurationbefore the metronome transited with a sigmoidal trajectory towardsthenewintervalduration.Asynchronydatafrom fiveparticipantsshowedsystematicdeparturesfromthe metronomein theformof two cyclesof gettingbehind (whenthemetronomewas speedingup)or ahead (met-ronome slowing down) of the beat. The lag or lead responseto the transition resulted in positive (upward) or negative (downward) signed asynchronies, thus pro-ducingacharacteristicMorWpatternofdeviationsfrom theinitialbaselineasynchrony.Amodelbasedonphase correction,asinEq.(1),combinedwithcorrectionofthe timekeeperperiodin proportion to thepreceding asyn-chrony, was used to fit the observed asynchrony time seriesusingnumericalminimisation.Themodelprovided reasonablequalitativefitstotheasynchronydataduring tempochange,andperformedbetterthananalternative model which was based on adjusting the timekeeper accordingto aweightedaverageofprevioustimekeeper andmetronomeintervals[22,23].However,different pa-rameterswererequiredtoaccountfortheasynchronytime seriesinthesteadystatebeforeandaftertempochange, suggestingtheneedtoturnthetimekeeperperiod correc-tiononandoffasanadhoccomponentofthemodel.

InSchulzeetal.[21]thetempochangesvariedfromtrial to trial, and so were unpredictable. However, where tempo changes are predictable (perhaps because of a blockeddesign or throughrehearsal), performers might adjust their timing in anticipation of further change in temporatherthanwaitandreacttoincreasingasynchrony aftertempochangeisfirstdetected.VanderSteenetal. [24]studiedfingertappingtoa400msmetronomewhich wentthroughanumberofcyclesoftemposlowingdown (to600ms)andspeedingup(backto400ms)withratesof tempochangevarying(inseparateblocksoftrials)atupto 14,28,or44msforeachsuccessiveinterval(with1,2,and 3 slowing-speeding cycles respectively). The results showed that the standard deviation of asynchrony in-creased with number of cycles (tempo change rate).

UsingthebGLSmethod([16],seeBox1)to estimate phase and period correction, as defined in [21], phase correctionincreasedandperiodcorrectiondecreasedwith rateofchangeoftempo.Cross-correlationsbetweenthe metronomeintervalsandIRIs,whichwerestrongly posi-tiveatlagzeroandsomewhatloweratlagone,decreased withtempochangerate.Thispatternsuggestsastronger tendency to anticipate (lag zero) than to match the previous(lag one) metronomeinterval, withless antici-pationatfastertempochangerates.

The correlation results mentioned above indicate the contribution of an anticipation process. Therefore, it is interestingtoconsiderextensionstothephaseandperiod correctionmodel[21]bycombininganticipatoryand ad-aptation components. Three such models based on an earliersimulation study [25] were explored in [24]. In thefirstmodel,thenextresponsewasdeterminedasthe weightedsumofthepreviousintervalandthepredicted nextinterval(basedonlinearextrapolationoftheprevious twointervals)plusaphasecorrectionterm(asEq.(1)).In thesecondandthirdmodels,therewas,again,a combina-tionofanticipationandadaptationcomponents(withthe minordifferenceofphaseversusperiodcorrectioninthe two models) but this was used as input to produce an internal prediction of the expected asynchrony. Feed-forwardcorrectionwasthenusedtoreducetheasynchrony whentheresponse was made. The parameters of these models were estimated using the bGLS method [16] whichalsoprovidedgoodnessoffitmeasures.In compari-son withamodelusing onlybasic adaptation,the three modelswithanticipationprovidedreliablybetterfitstothe dataand,inthecaseofthetwohighertempochangerates, thetwo modelswhichincorporated feed-forward correc-tionout-performedtheonethatdidnot.Inconclusion,the results suggestthe importance of including anticipatory componentsinfutureworkinthisarea.

2-person

synchronisation

Wenowturntoconsider2-personsynchronisationwithan examplefrommusic. InastudyreportedbyGoebl and Palmer[26],pairsofparticipants,wearingheadphonesto controlauditoryfeedback,played upperandlowerparts of a simple duet at 133bpm on a MIDI piano. When players were able to hear each other, cross-correlation functionsshowedthattheintervalsbetweentoneonsets werepositivelycorrelatedatlagsplusandminusone(an interval of one player was related to the previous and followingintervaloftheother),andnegativelycorrelated at lag zero (concurrent intervals of the players were inverselyrelated).However,whenlisteningwas restrict-edso thatonly one player heardtheother, thelag one cross-correlationwaslimitedtodependenceofthe hear-ingplayerontheother.Similarresultswereobtainedfor finger tapping in [27]. These results suggest that, in normal 2-person synchronisation, each participant uses feedback from the other (bidirectional correction) to

correctthecurrentintervalformismatchoftheprevious interval orasynchrony intheprecedingnoteonsets.

A formal model of 2-personsynchronisation was devel-oped byVorberg[28]for thesituationin which parti-cipants synchronised with a computerised metronome. The metronome software was programmedto synchro-nise thebeat onsetswith the participant’sown tapping responses using the first order linear phase correction model(Eq.(1)).Thiseffectivelysimulatedthecorrection responsesofavirtualhumanparticipanttappingintime with the real participant. At the same time, the real participant was correcting his or her own responses to thoseofthevirtualparticipant.Theauthorsmanipulated the correction responses of the virtual participant by varyingthe correctiongain. Theyestimated the correc-tiongainsofthehumanparticipantstothevirtualpartner byfitting thelinearphase correctionmodel (Eq.(1))to theirtappingdata.Additionalmanipulationswere subse-quently investigated by Reppand Keller [29]. Given thetwocorrectiongains,aS,forthesimulatedresponses

andaH,forthemeasuredgainsofthehumanparticipant,

Vorbergshowed thatforcorrection betweenthehuman andvirtualpartnertoremainstable,thesumofthegains must be between zero and two (i.e. 0(aS+aH)2)

[28].Itshouldbenotedthatthisstablerangeisthesame as that for a single person synchronising with a fixed metronome(seeSection‘Synchronisationwithafixed met-ronome’andFigure1b),suggestingthatregardlessofthe numberof peoplesynchronising,thesumof the correc-tiongainsmustremainwithinthisrangetoachievestable synchrony. ThiswasfurthercorroboratedbyWingetal. [30]whoinvestigatedtimingcorrectionsbetweenstring quartet musicians (seeSection ‘Multi-person synchronisa-tion’).

A synchronisation model corresponding to that of Vor-berg’s [28] was also explored by Hayashi and Kondo [31]. They instructed six pairs of performers (pairings madeupfromfourindividualparticipants)totaptogether at differentfrequencies ranging from.5 to 1Hz.While tapping,participantssawavisualflashrepresentingeach tapoftheirpartner.Usingregressiontoestimate correc-tiongain,theyobserveddecreasesinthesumofthepairs’ correctiongainswithincreasingfrequency.Theyfurther foundthatthegainvaluesdifferedforthetwomembers ofeachpair.Theauthorssuggestedthiscouldbeseenasa division of roles (a smaller gain being consistent with being more of aleader).However,they also notedthat therewasnoconsistencyacrosspairings,withgainsfora certain individual fluctuating depending on partner. Therefore,leadershipmightbesaidtobedefinedrelative to eachpartnerratherthanin absoluteterms.

Multi-person

synchronisation

AnearlystudyintogrouptimingwasundertakenbyWing andWoodburn[1]whousedrowingeightstoinvestigate

betweenrowertimingcorrections.Inaracingeight,oars alongtheboatalternatetotheleftandright.Successful rowers must attempt to maintain the same tempowith synchronouscatchesastheoarsenterthewateratthestart ofeachstroke.Itisparticularlyimportantthatoarsonthe samesideshouldmaintaintheirphysicalseparationsoas nottomakecontactwitheachother.Thisledtheauthors tohypothesisethatbetween-oartimingcorrections, mea-suredusinginter-catchintervalcross-correlations,should bestrongerbetweenpairsofoarsonthesameside,than foroarsonoppositesidesoftheboat.However,the cross-correlationfunctionsfailedtoconformtothisprediction, suggesting that oars onthe two sides of theboat use a commontimingcue to maintainsynchrony, suchas the acceleration–deceleration cycle of the boat motion (or associatedsounds)during rowing.

Usingchamberquartetsastheirstudygroup,Wingetal. [30]extendedthelinearphasecorrectionmodeltoallow pairwise analysisofsynchronybetweenmembersof the quartets (Eq.(2)).

ti;n¼ti;n1þTi;n1 X4

j¼1;j6¼1aijðti;n1tj;n1Þþei;n

i¼1;...;4 (2)

whereti,nandti,n1arecurrentandpreviousobservedtone onset event times for Player i, Ti,n1 represents the timekeeper interval, aij refers to the correction gain

appliedbyPlayerifortheasynchrony(ti,n1tj,n1)with Playerjandei,nisarandomnoisetermidentifiedwiththe

assumedinternaltimekeeper(seeFigure2).

Whatvaluesof correctiongainwouldbeappropriatefor fourperformersplayinginanensemble?For stable per-formance when tapping with an adaptive metronome (‘duetperformance’),thesumofgainsshouldbebounded between0and2 [28;seeSection‘2-person synchronisa-tion’].Wingetal.[30]furthershowedthatthecondition for stable synchronisation, stated asthesumof thetwo correction gains in the dyadicstudies [28] extendsto larger groups, N>2. They showed thatstability ofthe linear phase correction model of ensemble timing requires each player to maintain a correction gain of between 0 and2/N,assuming allgains are equal.More specifically,theauthorsshowedthatagainof1/N mini-mises asynchrony variance, that is to say, as group size increases, theoptimal gain for eachmember decreases. They also showed that the form of the AACF is over-damped,criticallydampedorunder-dampedwhengainis respectivelylessthan,equaltoor greaterthan1/N.

eachtrial.Theasynchroniesbetweenplayersineachofthe quartetswereusedtoestimatecorrectiongainswithinthe phase correction model using an iterative least squares proceduretofittheobservedasynchronytimeseries.The resultsrevealedastablesetof between-playergain esti-mates.TheoverallaveragegainforquartetAwas0.19,and 0.23forquartetB.However,itwasexpectedthatthegain profileswoulddifferbetweenplayers,reflectingthelead traditionallytakenbythefirstviolinist.Wingetal.found thisto betruefor quartet A.The gain for Violin 1 was consistentlylowerthanotherplayersindicatingthatViolin 2,Viola,andCelloadjustedmoretoViolin1thanviceversa. Incontrast,allplayersinquartetBhadsimilargainvalues, suggestingthatallplayerscorrectedmoreorlessequallyto eachother.

This music study revealed two contrasting patterns of gain,withonequartetshowingasymmetriesincorrection gains(theleaderemployedreducedgains),whereasinthe other quartet the gains were symmetric (more or less equalbetweenallplayers).Thespatialarrangementofa quartet affords both visual and auditory cues between playerssoequalityofgainsisperhapstobeexpectedby default, with Quartet B playing in a more democratic, leader-lessstyle.Incontrast,QuartetAappearedtoadopt thetraditionalapproachofViolin 1takingaleadrole.

Honischetal.[32]usedexplicitleader-follower relation-shipstoinvestigatecorrectiongainsinasynchronisation

task involving passing of visual timing cues along two separate‘chains’.Thetaskin[32]requiredsix individu-als,seatedfacing outward in acircle,to movetheirleft andright armsup and downtogether ata rateset bya metronomewhichonlyoneperson,designatedthe Lead-er,couldhear.TwopairsofFollowersoneithersideofthe Leader formed chains whose timing was linked to the Leader’sleftandrighthands. Ineachchain Follower-1 (F1)wasrequiredtowatchthehandoftheLeader,while Follower-2(F2)watched thehandof F1to pick upthe tempo.Thisarrangementwasintendedtoencouragethe passing of timing cues from the Leader around the circumference of the circle in a way that would be reflectedinthecorrectiongains.Largercorrection gains wereexpectedfortheFollowerrelativetothetargethand providingthetimingcue onthesidenearertheLeader than the correction gain computed with respect to any otherparticipant. To completethe circle,an individual designated theIntegrator, sat betweenthe ends of the twochainsandobservedmovementsfromthetwo parti-cipantsontheirleftandrightsides.Aprimaryfindingwas that participants in the Leader and Follower positions usedastrategyof minimisingtheirasynchronyvariance whereas,intheIntegratorposition,participantsswitched to astrategy that minimised theirown movement vari-ability.However,from thepoint ofthepresent review, particularlyinterestingwas thefindingofanasymmetry in the correction gains that reflected access to visual information.ThusLeader-F1 and F1 —F2 gains were

172 Timingbehavior

Figure2

t1,n–1 t1,n

t2,n–1

t3,n–1 t3,n

A41,n–1

A42,n–1

A43,n–1

1

2

3

4

1

2

3

4

α

α

α

t_4,n–1 t_4,n t_4,n+1

t1,n+1

t2,n t2,n+1

t3,n+1 (b)

(a)

Current Opinion in Behavioral Sciences

consistentlylargerthangainsestimatedwithrespecttoall other pairingsof group members.Thus,correction gain was influencedbythespatialconstraintsof thetask.

Conclusions

In summary,we have reviewedhow event-based linear modelsforasingleperformertappingwithafixed metro-nome cangeneralise into testable accounts of interper-sonaltiming.Wehavetakenalinearsystemsapproachin whichbehaviour,andunderlyinggenerativemechanisms, aretreatedasstableovertime.However,instabilityover repetition is often observed in cyclic movement tasks. Non-lineardynamicalsystemsapproachestotimingfocus specifically onsuch instabilities and there is a growing body of work taking this perspective focused on two-person timing interactions (for review see Schmidt and Richardson [33]).Within thelinear approach described here,anumberoffurtherquestionsmightbeexploredin futureresearch,someofwhichwehighlighthere.Inthe above,wehavenotexaminedmeanasynchrony,butitis interestingtoask,forinstance,whethermeanasynchrony oftheleaderinagroupincreaseswithgroupasynchrony variance, which would be useful in aiding a listener to detect the timing lead. In thinking about asynchrony variance, itis pertinenttoinvestigate ifvariance reduc-tion, for example with practice,is more determined by sensory(listening),cognitive(timekeeper),ormotor con-straints. More specifically,are reductions in asynchrony variance relatedto familiaritybetweenplayersorto the musical materialbeing performed? We might also con-sider if correction gain values are consciously adjusted higherorlowerbythegrouptoinfluencetheplayers’(or listeners’) experienceofgroup timing.

Wenoteonefurther potentialareafor researchandthat involves brain mechanisms underlying synchronisation processes. For example, group timing effects on brain activationmightbeinvestigatedinneuroimagingstudies followingseminalworkbyKellerandcolleaguesonbrain activations whensynchronising with anadaptive metro-nome [34,35]. Taking all these questions together, we therefore feelconfidentin claimingthathereisafertile areaforquantitativemodelsofsocialinteraction[seeBox2] inareasofactivitythatimpactonmanyaspectsofcultural life and, as such, this is an area which merits further research blending empiricalstudy with theoreticaltools ofthekinddescribedinthisreview.

Funding

ThisworkwassupportedbygrantsfromtheEngineering and Physical Sciences Research Council, UK (EP/ I031030/1)andtheEconomicandSocialResearch Coun-cil,UK (ES/K002678/1).Thefunder hadnorolein:the studydesign;thecollection,analysisandinterpretationof data; inthewritingofthereport;andin thedecisionto submitthearticlefor publication.

Conflict

of

interest

statement

Nothingdeclared.

References

and

recommended

reading

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