Improved temporal stability in multi-effector movements.

Ivry et al., The multiple effector advantage 1

Improved temporal stability in multi-effector movements.

Richard B. Ivry, Thomas C. Richardson, and Laura L. HelmuthUniversity of California, Berkeley

Final Draft: April 24, 2001 In press, Journal of Experimental Psychology:

Human Perception & Performance.

Abstract

Four experiments compared the temporal stabilit y of actions involving either one or more effectors. Areduction in within-effector temporal variabilit y was observed during two-effector tapping compared towhen either moved alone. This phenomenon was observed for various limb combinations, regardless ofwhether the two effectors were on the same or different sides of the body (Experiment 1) and did notrequire that the timed movements be produced in a repetitive manner (Experiment 3). Moreover, anadditional reduction in variabilit y was found when tapping with three effectors (Experiment 2). Thismultiple effector advantage is multiplicative: The magnitude of the multiple effector advantage wasgreater for longer target intervals (Experiment 4). A process-based account of these findings is proposed,based on the idea that independent temporal representations are generated for each effector. Theserepresentations are integrated to produce coordinated motor commands, and the multiple effectoradvantage is hypothesized to be a statistical consequence of the integration process.

Improved temporal stability in multi-effectormovements

Bimanual coordination has proven animportant avenue for understanding the dynamics ofmotor control. A central focus of this work has beento identify the constraints that characterize thetemporal coordination of the two hands (e.g.,Schöner & Kelso, 1988). When producing rhythmicmovements, the two hands naturally adopt a commonfrequency, establishing an in-phase, symmetricrelationship, or an anti-phase, opposing relationship.With training, we may learn novel phase relations(Zanone & Kelso, 1997), and skill ed musicians arecapable of performing complex polyrhythms(Krampe, Kliegl, Mayr, Engbert, & Vorberg, 2000).But even in these contexts, the gestures of the twohands remain strongly coupled. Drummers are likelyto exploit the hierarchical relationship between thetwo required rhythms when tapping out patterns suchas three against two or four against three. Indeed,

temporal coupling would appear to be the mostfundamental constraint associated with multi -limbmovements.

Temporal coupling has proven to be thecornerstone for much theorizing in the motor controlliterature, especially in terms of the development ofdynamic accounts of coordination (Kelso, 1997;Kugler & Turvey, 1987). Coupling provides animportant way in which control requirements can bereduced. In bipedal locomotion, the motions of thetwo limbs can be described as non-linear coupledoscill ators. A phase parameter can characterizedifferent modes of locomotion; for example, a fastmoving biped can be running (anti-phase) or hopping(in-phase). The lack of stabilit y at other phaserelations offers a means for understanding theattraction to certain categorical forms of behavior.

The focus of most coupled oscill ator modelshas been on the relationship between the two limbs(for reviews, see Kelso, 1997; Schöner & Kelso,1988). In particular, these models provide an


account of the stability of certain movement patterns,and provide an analytic tool for understanding howstability may change as a function of controlparameters (i.e., frequency) or experience. Stabilityin this context is generally assessed in terms of thevariability of the phase differences between the twolimbs. The individual may be asked to maintain aparticular phase relationship and the deviation fromthis target phase will be measured in terms of bothconstant and variable error as frequency is varied.

An alternative way to describe stabilityduring repetitive movements, one that is focused onthe component rather than coordinative level, is tolook at the performance of each limb individually.The mean and variability of the movement periodsfor each limb can be measured to assess how well anindividual can maintain a target frequency. Helmuthand Ivry (1996) examined this question in a series ofstudies using a repetitive tapping task. In their firstexperiment, the participants were required to tap witheither the left hand alone, the right hand alone, orwith both hands, trying to maintain a target inter-tapinterval of 400 ms. The mean inter-tap interval wasunchanged between the uni- and bimanualconditions. However, as measured by the variabilityof the inter-tap intervals, the performance of eachhand became more stable during bimanual tapping.That is, the variability of the within-hand inter-tapintervals was lower for each hand in the bimanual

context (see also Yamanishi, Kawato, & Suzuki,1980; Semjen & Ivry, in press). A secondexperiment demonstrated that this multiple effectoradvantage did not require the movement ofhomologous muscles. A similar improvement wasobserved when finger and forearm movements werecombined.

To account for the multiple effectoradvantage, Helmuth and Ivry (1996) proposed theMultiple Timer Model, a process account of thecontrol processes involved in timing and temporalcoupling (Figure 1). The model rests on three criticalassumptions. First, it is assumed that there areindependent central timing signals associated withthe movements of each effector. During left handtapping, it is assumed that an internal timing signal isgenerated to control when each left-hand responseshould occur. Likewise, during right hand tapping,an internal timing signal associated with the righthand is generated. These signals are assumed tooriginate in a central control process that operates asan internal timing system. The recruitment of specificelements within this timing system is dependent onthe output effector, with independent representationsassociated with different effectors.

The second assumption is that these separatetiming mechanisms continue to operate duringbimanual movements. That is, the multiple timermodel posits that during such movements, there are

Figure 1. The multiple timer model. Left column: Separate temporalrepresentations (Timers 1 and 2) are generated for each hand duringbimanual tapping. These representations are depicted as samples from anormal distribution (circles). Middle column: A


two temporal control signals being generated, one forthe left hand and one for the right hand.

The third assumption centers on how thesetwo signals are translated into motor commands.Helmuth and Ivry (1996) proposed that, althoughindependent timing signals are generated for eacheffector, the internal timing system does not havedirect access to the motor system. Instead, centralcommands to the effectors are regulated through amotor implementation process that we will refer to asan output gate. The output gate is constrained toupdate central commands to different effectorssimultaneously. It is this constraint which underliestemporal coupling in the model. Due to this couplingconstraint, the independent timing signals for the twohands become integrated.

We will contrast the dynamics of this formof coupling with more traditional coupled oscillatormodels in the General Discussion. At this point, wesimply point out that temporal coupling in themultiple timer model does not reflect interactionsbetween the timing mechanisms per se, but rather aprocess receiving input from multiple timers. Thetiming mechanisms are also coupled in the sense thatthe triggering of the gate not only initiates theresponses but also serves as a signal for the nexttiming cycle to begin. Without coupling of this form,the two hands would quickly become out of phase.

An obvious question is how does the gateoperate? How do the independent timing signalsinteract? We have conducted a series of simulationsto explore different ways in which the two timingsignals could be integrated (Helmuth and Ivry, 1996).In these simulations, two independent samples weretaken from distributions representing two timingsystems, one for the left hand and one for the righthand. The means and variances for thesedistributions were based on the observedperformance during unimanual tapping inExperiment 1 of Helmuth and Ivry. Differentprocedures were simulated in terms of how thesamples could be used by an output gate and fromthis, runs of inter-tap intervals were generated. Themeans and variances of these runs were thencompared with the observed bimanual data.F1

The gate could perform an OR operation,firing whenever it receives an input from eithertiming process. Alternatively, the gate could performan AND operation, firing only after it receives inputfrom both timing processes. While simulations of theOR and AND gating models indicates that variabilitywould be reduced during bimanual tapping, eachmodel also predicts that there should be a change inmean tapping rate. For the OR model, mean tappingwould be faster; for the AND model, mean tapping

rate would be slower. Neither prediction isconsistent with the observed data.

The best fitting model was one in which thetwo independent timing signals were averaged. Thatis, the output gate is triggered at a time thatcorresponds to the average of the two timing signals.With this model, the predicted reduction in variabilitycan be analytically derived. It is the standarddeviation of a new distribution formed by the averageof two independent samples from the constituentdistributions. If the constituent distributions areidentical, then

SDbim = SDuni / sqrt(2) (1)

Using the observed unimanual data from Helmuthand Ivry, the averaging model predicted a standarddeviation of 9.4 ms during bimanual tapping. Thisclosely approximated the observed value of 9.7 ms(averaged over left and right hands).

Averaging in the strict sense is illogical inthe temporal domain. Consider a situation in whichthe target interval is 400 ms. Suppose that for aparticular interval, the right hand timer signal is alittle fast and sends its output at 380 ms, whereas theleft hand timing signal is slow and sends its output at440 ms. By averaging, the output gate would initiatethe response in both hands after 410 ms even thoughthe left hand timer is not going to provide its signalfor another 30 ms.

This problem ceases to exist when thetiming signals are conceptualized as continuousvariables rather than discrete events. Figure 2 depictsthe operation of the output gate as a threshold device:The response is triggered when the input activationreaches a threshold level. The left panel shows theoperation of the gate for two successive intervals, onein which the threshold is reached earlier than thetarget time and one in which the threshold is reachedafter the target time. This variability might underliethe production of a long interval following a shortinterval, at least in terms of the timing signals. Theright panel shows the operation of the gate when thetwo signals arrive simultaneously and are summedtogether. Assuming that activity is normalized, thesummed activity from two signals will provide acontinuous record of the average and the normalizedthreshold will be reached at the average of the twosamples.

The threshold mechanism is ourinstantiation of the simultaneity constraint. It ensuresthat bimanual movements are coupled. A statisticalconsequence of this implementation is that thewithin-hand variability for each hand is lower duringbimanual tapping than during unimanual tapping.


Models that assume a single timer would also, ofcourse, predict coupling, but they do not predict the

multiple effector advantage. By postulating separatemechanisms for timing and temporal coupling, weare able to account for both phenomena.

The multiple timer model provides aparsimonious account of the performance of normalsubjects in Helmuth and Ivry (1996). It also providesa novel account of the paradoxical improvementobserved in unilateral ataxia patients during bimanualmovements. Franz, Ivry, and Helmuth (1996) testedfour patients with unilateral cerebellar lesions on arepetitive tapping task, comparing uni- and bimanualperformance in both the affected and unaffectedhand. As predicted, all four patients exhibited lowertemporal variability on the affected side duringbimanual tapping, presumably because the effects ofaberrant timing signals were mitigated by the timingsignals controlling the unimpaired hand. Moreover,an analysis of the individual cases confirmed aprediction derived from the model regarding changesin the performance of the unimpaired side. If thedifference in variability between the impaired andunimpaired sides was large during unimanualtapping, then the movements of the unimpaired limbduring bimanual tapping became less consistent. Ifthe impaired-unimpaired difference duringunimanual tapping was small, then the movements ofthe unimpaired limb became more consistent duringbimanual tapping. The general form of Equation 1,in which non-identical distributions are associatedwith the two effectors, predicts these results.

In the current paper, we examine a new setof predictions derived from the multiple timer model.

We focus on the hypotheses that independenttemporal representations are generated for each

effector and that these representations are integratedin a manner that resembles an averaging operation. Inthe General Discussion, we return to a discussion ofthe psychological and neural implications of thisprocess model, and examine it within the context of amore general class of dynamic systems models.

Experiment 1The goals of Experiment 1 are two-fold.

First, we examine the generality of the multipleeffector advantage by using various pairs ofeffectors. The studies of Helmuth and Ivry (1996)were limited to upper limb movements, although theydid observe that the improved temporal variabilityduring bimanual movements was evident formovements that either involved homologous or non-homologous muscles. In the current experiment,participants will tap with two hands, two feet, orhand-foot combinations that involve effectors fromeither different or the same side of the body.

Second, by using effectors from the sameside of the body, we sought to test an implicitassumption of the multiple timer model. A centralquestion in the study of internal timing has beenwhether a common timing system is exploited acrossvarious task domains. Based on correlations intemporal acuity across time production and timeperception tasks, Keele et al. (1985) argued for sucha common timing system (see also, Ivry & Hazeltine,1995). Neuropsychological evidence has also beencited in support of a common timing systemhypothesis. Proponents of a cerebellar timing locus

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Figure 2 Activation process within the output gate. The gating process is depicted as athreshold mechanism. Activation arises as a consequence of timer inputs.


(e.g., Ivry & Keele, 1989; Ivry, 1997) haveemphasized that patients with cerebellar damageperform poorly on a variety of tasks that requireprecise timing. Similarly, proponents of a basalganglia locus (e.g., Harrington, Haaland, &Hermanowicz, 1998) have shown that Parkinsonpatients perform poorly on both tapping and timeperception tasks.

Ignored in this work has been the questionof what is meant by a common internal timingsystem. At one extreme, one might suppose there is aunitary timing mechanism whose output is gated todifferent processing systems that require precisetemporal representations. However, our ownneuropsychological studies make clear a limitationwith this hypothesis. Patients with unilateralcerebellar lesions have been used as their own controlto show that coordination problems on the affectedside are related to a loss of temporal control duringrepetitive movements (Ivry, Keele, & Diener, 1988;Franz et al., 1996). This impaired performance iscompared to their normal performance on the sametask when using effectors on their unaffected side.At a minimum, such results would argue for at leasttwo internal timing systems, one that is disturbed andone that is intact.

However, an alternative hypothesis wouldbe that, while a specific psychological process maybe specialized for temporal processing, theinstantiation of temporal representations will involvethe recruitment of computational elements that aretask specific. At a neural level, this hypothesiswould propose that distinct populations of neuralcircuits would be engaged for different tasks, eventhough the computational characteristics of thesecircuits is similar (Ivry, 1996). In this view, the ideaof a common timing system is misleading: Ratherthan take this statement to imply a single timingmechanism, one would conceptualize a near-infinitearray of timing mechanisms, each linked to particularinput or output systems (and perhaps particularintervals). Thus, different elements are invoked fortiming movements with the right index fingercompared to tapping with the left index finger, oreven the right wrist.

The comparison of various effectorcombinations, especially crossed and uncrossedcombinations allow a first test of this hypothesis.From the work of Helmuth and Ivry (1996), onemight argue that there are two timing mechanisms,one associated with the right side of the body andanother associated with the left side of the body. Theoutputs of these two timers are integrated because ofthe need to coordinate the two limbs during bimanualcoordination. However, the hypothesis raised in the

preceding paragraph would suggest that the multipleeffector advantage would hold regardless of whetherthe two effectors are on the same or different sides ofthe body.Method

Participants. Eighty-two undergraduatestudents at the University of California at Berkeleyparticipated in this experiment in partial fulfillmentof psychology course requirements. All participantswere right handed, as assessed by self-report.

Apparatus. Responses were produced onperipheral response devices, linked to a desktopcomputer. The temporal resolution of the system was1 ms. All hand responses were made withflexion/extension movements of the right and leftindex fingers. For these responses, a 20 x 30 cmresponse board with two piano-type keys (2 x 10 cm)was used. Response boards in the shape of a wedgewere designed for the extension/flexion anklemovements to produce foot responses. The surfaceof each board (10 x 7 cm) was fixed at a 20 degreeangle from a rubber base that rested on the floor. Araised button measuring 1.2 cm per side was mountedon the board. Responses were recorded when thebutton was depressed 0.4 cm, bringing it level to thesurface of the board.

Procedure. After reading and signing aninformed consent form, participants were seated infront of a computer terminal in a quiet room. Theywere told that the experiment would measure howaccurately and how consistently they could tap at agiven speed, which would be signaled by a series oftones from the computer. They were allowed toposition the response board(s) to a comfortableposition. The participants were told to minimizemovements during the experiment except in theeffectors required for responding.

A message appearing on the computerscreen before each trial indicated to the participantswhich effector or effector combination was to beused in the upcoming trial. Each trial consisted of asynchronization and continuation phase. Theparticipant initiated the trial by pressing the"ENTER" key on the computer keyboard. After a 1 sdelay, a series of 50 ms, 500 Hz tones werepresented, separated by an inter-onset interval of 550ms. The participants began tapping with the tonesonce they had internally established the beat. Afterproducing twelve intervals during thesynchronization phase, the tones were terminated.They were then required to continue tapping,attempting to maintain the target interval in as


consistent a manner as possible. After 32 unpacedintervals were recorded, a low-pitch tone indicatedthe end of the trial. The 550 ms target interval islonger than the 400 ms pace used in our previousstudies (Helmuth & Ivry, 1996; Franz et al., 1996). Itwas selected on the basis of pilot work designed todetermine a comfortable speed for foot tapping.

Feedback was provided immediately afterthe trial was completed. The target interval (550 ms)appeared at the top of the screen. Listed below thiswere two lines, one showing the mean and standarddeviation of the participants' inter-tap intervalsduring the synchronization phase and the secondshowing these measures for the continuation phase.The experimenter encouraged the participants toexamine these measures after each trial in order tosee how accurately they had maintained the pacingspeed during the unpaced phase of the trial. Primaryemphasis was given to the standard deviationmeasure during the unpaced phase. Theexperimenter explained that this number reflected theparticipants' consistency and that they should try tomake this as small as possible.

Design. The participants were assigned toone of four groups based on the required effectorcombination. The four groups were finger-finger(1a), foot-foot (1b), finger-foot, crossed sides (1c),and finger-foot, uncrossed sides (1d). Within thefinger-foot crossed group, half of the participantsused the left index finger and right foot and half usedthe right index finger and left foot. Within thefinger-foot uncrossed group, all of the participantsused the right finger and right foot.

All of the participants were tested in threeconditions. For two of the conditions, tapping wasperformed with a single effector; in the thirdcondition, tapping was performed with botheffectors. For example, participants in Group 1btapped with the left foot alone, the right foot alone,and both feet together. Similarly, participants inGroup 1d tapped with the right (left) index fingeralone, the right (left) foot alone, and the finger andfoot together. Three blocks of tapping werecompleted for each of the three conditions. ForGroups 1a, 1c, and 1d, each block consisted of seventrials, yielding a data set of 21 trials for eachcondition, or a total of 63 trials per participant. ForGroup 1b, each block consisted of six trials and thus,18 trials for each condition. We targeted a smallerdata set for this group because pilot testing indicatingthat foot tapping would likely lead to more errors(see below) and we didn't want the participants tobecome fatigued. The first block for each conditionwas preceded by two practice trials for that condition.

The order of presentation of the three conditions wascounterbalanced across participants with theconstraint that each condition was tested within atriad of blocks. The experiment lasted approximatelyone hour.

Data Analysis.An initial analysis was conducted after each

trial to identify any intervals that were either shorterthan 200 ms or longer than 1000 ms. Almost all suchtrials occur when the participant failed to depress theresponse key fully, thus faili ng to activate themicroswitch and leading to an interval measurementapproximately twice as long as the surroundingtapping intervals. These trials were repeated withinthe same block up to a limit of seven repeated trials.If the participant produced seven trials containing aninterval outside the minimum and maximum durationcriteria, the block was terminated prematurely.

The analyses reported below are based onthe data from the final 30 intervals obtained duringthe unpaced phase of the trials in which all of theintervals fell within the minimum and maximumduration criteria. The mean and standard deviationwere computed for each trial. Our primary analysisof temporal consistency followed the proceduredescribed in Helmuth and Ivry (1996), focusing ontotal variabilit y and a decomposition of thisvariabilit y based on the two-process model of Wingand Kristofferson (1973; Vorberg & Wing, 1996).This model assumes that tapping variabilit y reflectsthe contribution of two independent processes:variabilit y associated with an internal clock thatdetermines when each response should be emittedand variabilit y associated with motor implementationprocesses required to translate this central commandinto an action. Ivry and Hazeltine (1995) haveargued that the former process is composed ofvarious control operations only one of which is theclock, and thus will refer here to the two componentsas central and motor delay, respectively.

Vorberg and Wing (1996) provided acomprehensive discussion and derivation of the two-process model. Empirical confirmation of theassumptions of the model has been obtained in manystudies involving healthy and neurologicallyimpaired populations (e.g., Wing, 1980; Ivry et al.,1988; Ivry & Hazeltine, 1995). Here we provide abrief summary of the procedure used to derive theestimates of the variabilit y associated with the centraland motor delay components.

The duration of each Interval j can beexpressed as

Ij = Cj + MDj - MDj-1 (2)


where I represents the durations of the observedinterval, C the central processing time, and MD themotor implementation delays. Given the assumptionthat the central and motor processes are independent,the variances of the components are additive:

I2σ = C

2σ + 2 MD2σ (3)

Successive intervals are assumed to result fromindependent samples of the random variablesassociated with the central and motor processes. Inother words, unpaced tapping at rates in the hundredsof millisecond range is assumed to be an open-loopprocess. However, neighboring intervals share onesample of the motor delay with each other and arethus negatively correlated with each other. Giventhis, an estimate of motor delay variability is givenby

MD2σ = − Ιautocovar (1) (4)

where autocov(1) is the covariance between Intervalsj and j+1 (Lag 1). An estimate of central variabilitycan then be obtained by subtracting the motorestimate from the total variability obtained from theraw data.

Prior to calculating the estimates of the twocomponents, we performed a transformation on theraw data to remove the effects of global changes intapping rate. A regression line was fit through the 30unpaced intervals and the covariance function forlags 0 through 5 was based on this transformation.The values were averaged across the 21 trials percondition and the standard deviation scores as well asestimates of central and motor variability were basedon these data. The covariance function provides acritical test of the two-process model: the lag 1covariance should be negative and the values for lagsgreater than one should be zero. The lineartransformation has the effect of reducing positive

correlations between successive intervals, and thusthe estimate of the motor variability is higher thanthat obtained from the raw data. It turns out that thischange is minimal, usually on the order of less than 2ms, and the results reported below would be similar ifthe raw data had been used instead of the transformeddata.

It is important to note that this detrendingprocedure, and indeed, the two-process model ingeneral, ignore potential sources of noise that mightoperate at different time scales during repetitivemovements (e.g., 1/f noise, see Chen, Ding, & Kelso,1997). However, given that the trials in the currentexperiment were limited to about 15 s, it is unlikelythat any non-linear drift would contributesubstantially to the observed variability (seeMadison, in press)

Results and DiscussionThe data for two participants were excluded

from the final analysis because their mean tappingrates (less than 470 ms. in at least one condition)were much faster than the target interval. The datafor eight other participants were excluded becausethey produced blocks that did not contain a sufficientnumber of trials in which all of the intervals weregreater than 200 ms and less than 1000 ms. Intervalsfalling outside these criteria almost always occurredon trials involving foot responses and likely resultedfrom the failure of the participant to depress theresponse key with sufficient force. Of the 72participants retained in the analysis, 17 were ingroups 1a and 1d, 18 were in group 1b, and 20 werein group 1c. For these participants, 9.6% of the trialswere repeated due to trials in which at least oneinterval failed to fall within the criterion window of200 - 1000 ms.

The results of Experiment 1 are summarizedin Table 1. The table lists the mean and standard

Mean ITISD Central MDOne Two One Two One Two One Two

Exp 1aR Fing 520 517 23.4 20.1 18.3 13.8 10.3 10.3L Fing 519 517 24.5 21.3 18.0 13.2 11.8 11.8

Exp 1bR Foot 523 522 32.2 26.0 21.3 16.5 17.1 14.2L Foot 520 522 32.6 28.0 20.5 16.1 17.9 16.2

Exp 1cFing, S1 534 526 25.1 26.8 20.4 17.5 10.3 14.3Foot, S2 531 526 28.0 28.4 20.8 17.3 13.2 15.9

Exp 1dFing, S1 529 529 25.3 29.5 19.9 18.0 11.1 16.6Foot, S1 525 530 30.4 30.4 23.1 18.5 14.0 17.0

Table 1 Mean interval produced, standard deviation of the inter-tap intervals, and estimates of the centraland motor delay components for Experiment 1. All values are in ms


deviation of the inter-tap intervals, and the estimatesof the central and motor delay component sources ofvariability. Within each group, there are two rows,one for each effector. The pairs of columns show thevalues for the one-effector and two-effectorconditions. All of the data are within-effectormeasures. While the two effectors were tightlycoupled in all conditions, we will not report anybetween-effector analyses here (see Experiment 4).

In all of the conditions, the participantstended to tap more quickly than the target interval of550 ms. The speed-up of about 25 ms was generallycontinuous, with the produced intervals close to thetarget rate at the end of the synchronization periodfollowed by a tendency to speed-up over the courseof the unpaced phase. We have observed a similarhastening in a previous study with college students(Ivry & Keele, 1989). While we suspect thephenomenon reflects a mild degree of impatience onthe part of our participants, the effect is not large andappears to be similar in the one-effector and two-effector conditions.

Turning to the standard deviation scores, aninteresting difference is apparent between the twogroups who performed homologous movements(Groups 1a and 1b) and the two groups whoperformed non-homologous movements (Groups 1cand 1c). Variability was lower during the two-effector condition for participants who tapped withtwo fingers (Group 1a) and the participants whotapped with the two feet (Group 1b). In contrast, thestandard deviation values tend to be higher for theparticipants who tapped with one finger and one foot,either on opposite sides of the body (Group 1c) or thesame side of the body (Group 1d). The standarddeviation data for each group were analyzed in aseries of 2 x 2 ANOVAs, with one factor referring tothe effector (e.g., right finger or left finger for Group1a, finger or foot for Group 1c) and the other factorreferring to the condition (single-effector or two-effector). Separate ANOVAs were conducted foreach group since our main interest here is on whetherthe multiple effector advantage is observed across arange of conditions.

The reduction during two-effector tappingwas highly reliable for the two finger participants inGroup 1a, F(1,16)=33.8, p<.001, and the two feetparticipants in Group 1b, F(1,17)=34.8, p<.001. Theopposite pattern was observed for the two groups inwhich finger and foot tapping were combined. Forthese groups, tapping with two effectors tended to bemore variable than tapping with a single effector. Forthe participants in Groups 1c performance wassignificantly more variable when tapping with afinger and foot on opposite sides of the body

compared to when tapping with either effector alone,F(1,19)=6.3, p<.05. Similarly, the standard deviationwas larger during right finger and right foot tappingfor Group 1d, F(1, 16)=7.2, p<.05, although thisincrease was only reliable for the finger as reflectedin the significant interaction, F(1,16)=7.9, p<.05.These results are similar to the pattern reported byHelmuth and Ivry (1996). In that study, the standarddeviation of the inter-tap intervals was lower duringbimanual tapping than in unimanual tapping, but didnot change when the two-effector conditioncombined finger and forearm movements.

We next turn to the decomposition of thetotal variability into estimates of the variabilityassociated with central and motor implementationprocesses. It is important to first verify that thecurrent data are consistent with the assumptions ofthe two-process model. The covariance functionprovides three such tests (Vorberg & Wing, 1996;Wing & Kristofferson, 1973). First, the lag 1covariance should be negative. Second, this valuemultiplied by negative two should be less than the lag0 covariance (since values outside this boundarywould imply a negative value for the estimate ofcentral variability). Across the 284 covariancefunctions (72 participants x 2 effectors x 2 modes,single- and two-effector), the lag 1 covariance valuewas positive four times and greater than the boundaryset by the lag 0 covariance value three times. Weincluded these seven scores in the subsequentanalyses, assuming they reflected noise in the data.

Third, the covariance function should bezero for lags greater than one. Figure 3 shows thecovariance function for representative conditions. Ineach panel, the data are from right hand tapping,either alone or paired with the left hand (Group 1a,top panel) or the right foot (Group 1d, bottom panel).For all four covariance functions, the values for lags2-5 are close to zero. Most important, the functionsare quite similar for the single- and two-effectorconditions, indicating that using two limbs does notintroduce gross changes in the time series. There area few data points that are significantly different thanzero (all negative). However, when we appliedvariants of the two-process model that can accountfor such deviations (Wing, 1977), we found littlechange in the component estimates (see also,Helmuth & Ivry, 1996). Thus, we restrict thediscussion to the estimates obtained from the basictwo-process model.

Turning first to the estimates of centralvariability, a multiple effector advantage wasobserved for all four groups (see Table 1). Thestandard deviation associated with central processeswas lower in the two-effector conditions compared to


the one-effector conditions. These effects wereconfirmed in a series of ANOVAs identical to that

described above. The number of effectors wassignificant for all four groups (1a: F(1,16)=52.1,p<.001; 1b: F(1, 17)=16.3, p<.001; 1c: F(1,19)=8.6,p<.01; 1d: F(1,17)=13.0, p<.01). No differenceswere observed between the two effectors within eachgroup nor were any of the interactions reliable.

The results for the estimate of motorimplementation variability are slightly morecomplicated. None of the main effects nor theinteractions were significant for the two homologousmovement conditions. For both of these groups, themotor delay estimate was larger for the non-dominant, left limb, although the effect onlyapproached significance for the hand, F(1,16)=3.8,p=.07, and the foot, F(1,17)=3.3, p=.09. In thegroups for which finger and foot movements werecombined, there was a significant increase in themotor delay estimate during the two-effectorcondition (Group 1c: F(1,19)=9.0, p<.01; Group 1d:F(1,16)=28.2, p<.001). For the crossed side group(1c), there was also a main effect for the limb factorwith the motor variability associated with the footgreater than that associated with the finger,F(1,19)=10.2, p<.01. Thus, the estimates of motordelay were unchanged when the two-effectormovements involve homologous movements. Whenthe two movements were non-homologous, anincrease in the estimate of motor delay variabilitywas observed.

A final analysis concerns the magnitude ofthe multiple effector advantage. By the multipletimer model, the improved temporal consistencyduring multiple effector tapping is a statisticalconsequence of sampling: The coupling constraintimposed by an output gate effectively acts to averagethe independent timing signals that have beengenerated for each effector. The observedimprovement in the standard deviation should followthe square root of n rule, where n is the number ofsamples if the distributions associated with theeffector-specific timing elements are identical.However, not all of the sources of variability willbenefit from averaging by the process model outlinedin Figure 1. For example, variability associated withmotor implementation processes is imposed after theoperation of the gate. Given this, we focused on theestimates of central variability, using the valuesobserved during single-effector tapping to predictcentral variability during two-effector tapping. Weaveraged the two single-effector conditions, althoughsimulations using the observed values yieldedessentially identical results.

Figure 4 shows the comparison between thepredicted and observed estimates of centralvariability. The two values are quite comparable for


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Figure 3. Covariance functions for Experiment 1.A: Finger-finger. Data are for right hand,unimanual and bimanual. B: Finger-foot. Dataare for right foot, tapping alone and with righthand.


the bimanual group, similar to that reported byHelmuth and Ivry (1996). For the other three groups,the observed values are higher than the predictedvalues, and the difference is largest for the twogroups in which upper and lower limbs werecombined. These results raise the possibility that theprocesses associated with temporal coupling of thelower limbs may be different than those associatedwith the upper limbs. However, previous work on thedynamics of multi-limb coordination have assumedthat similar principles apply for upper and lower limbcoordination (Carson, Goodman, Kelso, & Elliott,1995; Schmidt, Carello, & Turvey, 1990) and, asdemonstrated in Figure 3, the covariance functionsseem quite similar for the different limbcombinations.

An alternative hypothesis is that, while thebenefits of temporal averaging are similar in allconditions, new costs arise with multi-limbmovements that involve combined movements ofupper and lower limbs. Jeka and Kelso (1995) haveshown that stability during repetitive movementsinvolving the arm and leg is influenced bydifferences in the intrinsic frequencies of the twolimbs, and that these frequencies are related to mass.From the perspective of the two-process model, themotor delay estimates point to one way in whichthese mass differences may influence variability.The motor delay estimates increased whenever thetwo-effector conditions involved limbs of unequalmass. This increase is not only found in the finger-foot conditions in the current experiment, but hasalso been observed for finger-forearm tapping(Helmuth & Ivry, 1996) and for bimanual studies inwhich external masses are added to produce anasymmetry between the two arms (Turvey,Rosenblum, Schmidt, & Kugler, 1986). It may bethat additional central sources of variability are alsointroduced in such conditions and these attenuate themagnitude of the multiple effector advantage. Theadded peripheral noise may result from the fact thatdifferent forces are required for the two movements,and this requirement could also affect centralprocesses. An evaluation of this hypothesis wouldrequire unconfounding the effects of homology andmass, as well as measurements of kinetic variables.

In summary, the results of Experiment 1demonstrate the robustness of the multiple effectoradvantage. The temporal consistency with which asingle limb produces repetitive movements improveswhen another limb is moved in a synchronousfashion. Thus, the stability of multi-effectormovement patterns is not only manifest in terms ofthe coordination between the limbs, but is alsoapparent within the series of movements produced by

each limb. The current study shows that the multipleeffector advantage can be replicated at a new interval(550 ms compared to 400 ms in previous studies) andgeneralizes to leg movements. The phenomenon doesnot require movements with homologous effectors,although under these conditions, the temporalimprovement is only observed in the estimates ofcentral variability.

Experiment 1 provides new support for themultiple timer model outlined in the Introduction. Atthe heart of this model is the idea that the reducedvariability during multi-limb tapping reflects aninteraction between independent timing signalsassociated with the two limbs. The improvedtemporal performance was found for movements

restricted to either upper or lower limbs, as well asfor upper- and lower-limb combinations. Moreover,the pattern of results was quite similar for thoseparticipants using a finger and foot on the same sideof the body as for those using a finger and foot fromopposite sides of the body. These findings are


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Figure 4. Estimates of central variability inExperiment 1 and predicted estimate based onthe multiple timer model. Unimanual andbimanual values are averaged over left and righthands. A: finger-finger. B: foot-foot. C: finger-foot crossed. C: finger-foot uncrossed.


consistent with the central hypothesis of the multipletimer model that effector-specific elements arerecruited for controlli ng movement timing. At amore general level, an internal timing system wouldbe conceptualized as an array of dedicated timingelements that are linked to specific input and outputsystems.

Experiment 2To this point, the generality of the multiple

effector advantage has been established inexperiments comparing single- and two-effectortapping. In Experiment 2, we added a condition inwhich the participants tapped with three effectors atthe same time. Assuming independent timing signalsare generated for each effector, we expected toobserve an additional reduction in temporalvariabilit y in the three-effector condition comparedto the two-effector condition. This prediction wouldseem to be at odds with expectations based onattentional considerations. Although all of theeffectors are required to produce simultaneousmovements, the addition of extra effectors would beexpected to entail a cost, rather than a benefit.

In designing the study, two considerationswere taken into account. First, if as is assumed in themultiple timer model, the improvement is the resultof averaging independent samples, the effect ofadding a third effector will be relatively small . If thestandard deviation of each timing element was 20 ms,averaging two samples would result in almost a 6 msbenefit (14.1 ms) whereas averaging three sampleswould only confer an additional 2.6 ms advantage(11.5 ms). Since the expected effect size is small , wedoubled the targeted number of participants.

Second, and more important, it was diff icultto determine the appropriate combination ofeffectors. Using all possible combinations of threeeffectors would require seven conditions (3 singleconditions, 3 pairs, and 1 triad). Moreover, as shownin Experiment 1, there are differences betweencombining homologous effectors and non-homologous effectors. Motor implementationestimates consistently increase when limbs ofunequal mass are combined, and this might make itdiff icult to interpret a comparison between bimanualtapping and a three-effector condition consisting oftwo hands and one foot. Given these considerations,we elected to combine the index finger and foot inthe two-effector condition and focus on what happenswhen the other index finger is added in the three-effector condition. We expected that the addedmotor noise would be present in both the two andthree-effector conditions, thus allowing a cleaner

assay of changes in total variabilit y and estimates ofcentral variabilit y.

Method

Participants. Thirty-nine right-handedundergraduates at UC, Berkeley participated in theexperiment in partial fulfill ment of psychologycourse requirements.

Procedure and Design. Each participant wastested in three conditions: single effector (right fingeralone), two effector (right finger and right foot), andthree effector (right finger, right foot, and left finger).At the beginning of each block, a message wasdisplayed on the computer screen indicating theeffector(s) for the forthcoming set of trials. Theparticipants were instructed to restrict movements tothe designated effectors. All other aspects of thedesign and procedure were identical to Experiment 1with the exception that only six trials were includedin each block. The last 30 intervals during theunpaced phase of each trial were included in theanalyses. Trials in which an interval was shorter than200 ms or longer than 1000 ms were repeated. Anerror in the data acquisition program led to someinconsistency in terms of the number of trialscollected per condition. For some participants, thetrials with aberrant intervals were not repeated; forothers, seven trials were collected per condition.Thus, the actual number of trials per condition variedfrom 15 to 21.

Results and DiscussionAll 39 participants were included in the

analysis. Overall , approximately 9% of the trialswere repeated because they contained at least oneaberrant interval. As in Experiment 1, the longintervals appeared to result from instances in whichinsuff icient force was used to depress the footresponse key.

There was littl e variation in the meanproduced interval across the three conditions. Whentapping with only the right finger, the mean intervalduring the unpaced phase was 536 ms. In both thetwo- and three-effector conditions, the means for allof the effectors was 531 ms. Observation of theparticipants' performance as well as an informalexamination of the time series at the level ofindividual trials indicated that the movements werealways tightly coupled (see Experiment 4 for a moreformal analysis). The data in all conditionsconformed with the basic predictions of the two-process model. The lag 1 covariances were withinthe boundary conditions for all conditions except for


one participant in one condition (right foot duringtwo-effector tapping). Moreover, the covariancefunctions were similar in all three conditions.

The within-effector variabilityscores (calculated as deviations from the regressionline), as well as the component estimates of centraland motor variability are shown in Table 2. For thestatistical analysis, we focused on the data for theright index finger and the right foot during the two-and three-effector conditions. In terms of thestandard deviation scores, variability was lowerduring three-effector tapping compared to two-effector tapping, F(1,38)=38.9, p<.001. However,the effect of the number of effectors differed for theright finger and right foot as reflected in thesignificant interaction, F(1,38)=38.3, p<.001. Whentapping these two effectors together, only theintervals produced by the right index finger becamemore consistent when the left index finger wasengaged. Thus, at least for the finger, the data areconsistent with the prediction that temporal

variability will be inversely related to the number ofactivated effectors. Note that the central estimate forthe finger was lower for the two- and three-effectorconditions compared to when the finger tapped alone.

Clearer support for the prediction of themultiple timer model comes from the analysis of thecentral variability component. Here, only the numberof effectors variable proved reliable, F(1,38)=4.4,p<.05. Averaging over the right finger and right foot,the estimate is 16.6 ms during two-effector tappingand 15.0 ms during three-effector tapping. While themagnitude of the effect is greater for the foot, theinteraction term did not approach significance,F(1,38)=1.3, p>.25.

Unexpectedly, there was also a reduction inmotor variability during three-effector tapping,F(1,38)=4.5, p<.05, although this main effect wasqualified by the significant interaction, F(1,38)=25.3,p<.001. For the right foot, motor variability

increased when the left index finger was added in thethree-effector condition; for the right index finger,motor variability decreased when the left finger wasadded. It is possible that adding an effector stabilizesperipheral noise factors associated with similareffectors (e.g., left finger and right finger), an ideathat could be tested by making the third effector theleft foot. As in Experiment 1, it is also not possible todetermine if the effects here are related to thehomology of the two index fingers or their similarityin mass (see Jeka & Kelso, 1995).

The above analyses of the overall standarddeviation scores and the estimates of centralvariability provide a qualitative confirmation of thepredictions of the multiple timer model. From theindependent sampling assumption of the model, wecan also examine this issue quantitatively. The moststraightforward test would be to use the estimate ofcentral variability for the right index finger in thesingle-effector condition, and use that to predict theestimates in the two- and three-effector conditions.

The observed value during unimanual tapping was17.8 ms. The predicted values for the two- and three-effector conditions would then be 12.6 ms and 10.3ms. Both are considerably lower than the observedvalues (see Table 2). This is similar to what wasfound for the finger-foot conditions in Experiment 1,and there we suggested that there may be newcontributions to central variability when combiningeffectors of unequal mass.

An alternative way to derive quantitativepredictions based on the independent samplinghypothesis is to use the central estimates from thetwo-effector condition. For this, the average of theobserved estimates is multiplied by the square root of2 to estimate the variability of the underlyingsampling distribution. This value is then divided bythe square root of three, reflecting the number ofsamples presumed to occur in the three-effectorcondition. From this procedure, the predicted value

ITISD Central Motor Delay

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R Finger 23.9 27.6 24.1 17.8 16.0 15.1 11.3 15.9 13.3

R Foot 29.4 29.4 17.1 15.0 16.9 17.9

L Finger 25.7 14.7 14.9

Table 2 Standard Deviation of the inter-tap intervals and component estimates for Experiment 2.All values are in ms.


of the central estimate during three effector tapping is13.5 ms. Although closer to the observed value of15.1 ms, the results again show that the improvementin temporal performance is less than would beexpected by the strict version of the multiple timermodel. To date, we have only obtained the squareroot of n reduction during bimanual tapping.

Nonetheless, the results of Experiment 2show that temporal variabilit y is reduced when threeeffectors are used compared to two effectors. By thelogic of the multiple timer model, we would expectthat further reductions would be found if moreeffectors were added to the mix. Of course, it wouldbecome quite diff icult to observe such improvementsif we are correct in attributing these effects to theexploitation of independent samples of temporalrepresentations. The multiple effector advantage wehave elicited in these experiments may well be alaboratory demonstration of a phenomenon longappreciated by musical performers. Most musicianstap their feet or let their body sway when performing.Even a drummer who is using a snare to maintain thebeat for a group will t ap his or her feet, even whennot using a foot pedal. These actions are intended tostabili ze temporal performance. The multiple timermodel provides a mechanistic account of how this isachieved.

Experiment 3Studies on temporal variabilit y in motor

control have generally involved repetitivemovements. Performance is observed over cycles ofcontinuous behavior to ask questions about thestabilit y of different phase relations (Schöner &Kelso, 1988) or to examine whether people arecapable of producing complex polyrhythms (e.g.,Jagacinski, Marshburn, Klapp, & Jones, 1988;Krampe et al., 2000). Similarly, in our work to dateon the multiple effector advantage (Franz et al.,1996; Helmuth & Ivry, 1996; Ivry & Hazeltine,1999), as well as the first two experiments of thispaper, we have always required the participants toproduce a series of paced and unpaced intervals.This has allowed us to apply the two-process modelin order to partition the total variabilit y into centraland motor implementation components.

However, a strong prediction of the multipletimer model is that temporal variabilit y duringbimanual movements should be reduced even whenparticipants are producing single intervals inisolation. We tested this prediction in Experiment 3.The participants were trained to produce singleintervals by pressing the response key twice, once tomark the beginning of the interval and once to markthe end of the interval. After an initial phase in

which computer-generated tones were presented toprovide a reference for the target interval, a set ofsingle intervals was produced with a variable delaybetween each production. In this way, we obtaineddata sets comparable to that obtained in the earlierstudies, but now each interval was produced inisolation rather than as a series of rhythmicmovements.

We assume that the control processes, atleast for timing the intervals, are comparable for thesingle interval task as in the standard repetitivetapping task. Independent signals must be generatedfor each hand, indicating the target delay between thetwo taps. Assuming that the implementation of thesesignals is again constrained by the output gate, weexpected the observed variabilit y during bimanualmovements to be lower than that found duringunimanual movements. With this method, we did notexpect to observe the square root of two reductionsince it is not possible to isolate central sources ofvariabilit y from those associated with motorimplementation. Nonetheless, we tested the weakerprediction that the multiple effector advantage is notdependent on the production of repetitivemovements.

This experiment also allows us to explore analternative hypothesis for the multiple effectoradvantage. We have attributed this effect to thegeneration of multiple timing signals, one for eacheffector. As such, our model emphasizes an open-loop aspect of the task, the central signals thatrepresent the target intervals. An alternative idea isthat when people use more than one limb, there arenew sources of feedback that could confer stabilit yon the movements of each limb. The movements ofeach limb could serve as a reference for the otherlimb. For example, the time at which one limbactivates the response key might be used to modifythe movement of the other limb. A model of thisform emphasizes a closed-loop aspect of the task.While it is possible that feedback can be usefulduring the production of single intervals, we mightexpect that this sort of process would be most viableduring a repetitive movement task. Observing themultiple effector advantage during single-intervaltapping would be problematic for a feedback-basedaccount.

Method

Participants.Twelve right-handedundergraduates at UC, Berkeley participated inExperiment 3.


Procedure and Design. The participantswere only tested in finger tapping conditions, eitherusing their right hand alone, their left hand alone, orboth hands together. As in the previous experiments,a trial was composed of 8 paced and 21 unpacedintervals. However, each interval was produced as aseparate entity. During the paced phase, twocomputer tones were presented with a tone-onsetasynchrony of 400 ms. The 400 ms rate was chosensince only finger movements were used in this studyand we expected the potential to use feedback wouldbe further reduced as the interval becomes shorter.After a delay of 550 ms, 700 ms, or 850 ms., theword "TAP" was displayed in the center of thescreen. The participants were then required to maketwo keypresses, attempting to separate the two tapsby the target interval. The variable delays werechosen so that the participants could not adopt arhythmic mode of responding. The presentation ofthe tone pair was repeated 1 s after the second tap,and this procedure was repeated until 8 pacedintervals had been produced. Following this phase,the word "TAP was presented another 21 timeswithout the tones. The participants produced 21 pairsof responses each time to produce the set of unpacedintervals.F2 At the end of the trial, feedback wasprovided as in the preceding experiments. Themeans and standard deviations for the paced andunpaced phases were presented on the screen. Theinstructions emphasized that the primary task was toachieve the lowest possible scores on the standarddeviation score during the unpaced phase. The final20 intervals during the unpaced phase were used inthe analyses reported below.

Each block consisted of 6 trials. Eachparticipant completed two blocks of tapping with theright index finger alone, the left index finger alone,and with both hands. The order of blocks wascounterbalanced with the constraint that each effectorcondition was presented once every three blocks.

Results and DiscussionOnly intervals that were greater than 200 ms

and less than 600 ms in duration were included in theanalysis. Overall, about 2% of all of the intervalsfailed to fall within this boundary. Most of theseoccurred when the participant made his or her firsttap prior to the onset of the imperative signal. Sincethe single interval method is not amenable to the two-component analysis, we did not repeat the entire trialwhen violations occurred, but rather simply excludedthe violations from the analysis.

The mean produced interval during theunpaced phase was 403.9 ms in the left-handcondition and 404.4 ms in the right-hand condition.

For the bimanual condition, the means were 410.5and 408.4 ms for the left and right hands,respectively. No significant differences wereobserved between these values.

We used the same detrending process as inExperiments 1 and 2 prior to analyzing the variabilitydata. A regression line was calculated with the seriesof 20 unpaced intervals and calculated the variabilityfrom this regression line. This procedure minimizedthe effects of any linear trend across the unpacedphase that would result from the participants eitherspeeding up or slowing down. However, thedetrending procedure had only a slight change on thevariability measures and the results for the raw dataessentially mirror that observed with the transformeddata.

The mean standard deviation scores arepresented in Figure 5. As can be seen in the figure,the multiple effector advantage was observed forboth the right and left hands. Averaging over the leftand right hands, the standard deviation was 23.0 msduring unimanual tapping. During bimanual tapping,this value fell to 20.5 ms, F(1,11)=6.64, p<.03.Neither the effect of hand, F(1,11)=1.341, p>.271,nor the hand by number of effectors interaction,F(1,11)=0.602, p>.454 were significant.

The multiple effector reduction isconsiderably less than would be expected if twoindependent signals were being averaged. However,it is not reasonable to expect this prediction to hold inthe current experiment. Because the intervals arebeing produced in isolation rather than as acontinuous series, we are not able to apply the two-

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Figure 5. Standard deviation of intervals producedindividually during uni- and bimanual conditions ofExperiment 3


process model to decompose the total variabilit y intocentral and implementation components. Asindicated previously, the multiple timer modelpostulates that only the former would benefit fromthe averaging operation. Nonetheless, the absolutesize of the reduction is less than was found for thecentral estimates in the two-hand condition ofExperiment 1 (see also, Helmuth & Ivry, Experiment1).

In summary, the results of Experiment 3demonstrate that the multiple effector advantage doesnot require that the temporal intervals be produced asa continuous series. The reduction in within-effectorwas observed even when each interval was producedas a separate entity. This finding accords with thepredictions of the multiple timer model. As inrepetitive tapping, we assume that central temporalcontrol signals are generated for each hand, but thatthe implementation of these commands is subject tothe operation of an output gate. The gate is assumedto instantiate a form of averaging as it integrates thetwo timing signals.

We do not claim that a feedback-basedhypothesis is ruled out by the current results. Twosuccessive taps produced the intervals in the currentexperiment. It is possible that afferent informationfrom the two effectors during these taps may stillprovide reference signals that improve the temporalstabilit y of the single interval. However, the currentresults constrain how such an account could accountfor the multiple effector advantage. First, the utilit yof salient sources of feedback such as the asynchronybetween when the two hands tap would be ofminimal help in the single interval condition sincethis information could not be used to adjustsubsequent responses. Second, the current designeliminates any benefit that might come about fromrhythmic entrainment between the two hands over thecourse of a series of continuous movements. Thebenefit of feedback, if relevant, would have to berestricted to that obtained during the course of asingle interval.

Experiment 4Numerous studies have shown that temporal

variabilit y on motor and perceptual tasks is afunction of the target interval. As the interval to betapped or judged becomes longer, variabilit yincreases. The nature of this relationship appears tofollow a form of Weber's law in the temporal domainsuch that the standard deviation divided by the meanequals a constant value over a range of intervals (e.g.,Getty, 1975; Ivry & Hazeltine, 1995). Thisphenomenon has been the focus of much theoreticalinterest, with the scalar property indicating that at

least one major source of variabilit y on such tasks ismultiplicative, growing in a proportional mannerwith the interval being represented (see Gibbon,Malapani, Dale, & Galli stel, 1997; Kill een & Weiss,1987)

In Experiment 4, we exploited this propertyto test a strong prediction of the multiple timermodel. Specifically, the hypothesis that independenttiming signals are averaged during bimanual tappingpredicts that the magnitude of the multiple effectoradvantage should become larger as variabilit yincreases. At a qualitative level, the prediction isthat, in terms of standard deviation scores, thereshould be an interaction between the number ofeffectors and the target duration. The reductionduring bimanual tapping should become greater asthe target interval is lengthened. At a quantitativelevel, the prediction is that the slope relating theincrease in variabilit y as a function of the targetinterval during bimanual tapping should be lower bythe square root of two than that observed duringunimanual tapping.

Participants in Experiment 4 were tested onthe repetitive tapping task at four different rates, 325ms, 400 ms, 475 ms, and 550 ms. This procedureallowed us to test two key predictions with the two-process model of Wing and Kristofferson (1973).First, the logic of the model suggests that only theestimate of central variabilit y should increase as thetarget interval (see Wing, 1980). Second and moreimportant for the present purposes, we expected thatthe improvement during bimanual tapping would berestricted to the estimate of central variabilit y andthat the multiple effector advantage would becomegreater as the target interval increased. Thesepredictions follow from the assumption thatimplementation variabilit y arises from processesdownstream of the internal timing system and thegating process. That is, implementation variabilit y isassumed to be duration independent.

The idea that temporal variabilit y duringtapping is composed of duration-independent andduration-dependent sources of variabilit y also affordsa second, independent method for partitioning totalvariabilit y into component sources. In a series ofexperiments, Ivry and Hazeltine (1995) applied aprocedure called slope analysis to show that acommon internal timing system was invoked in bothmotor and perceptual tasks that require precisetiming. The essence of this procedure is that theslope of the function relating the standard deviationas a function of the produced or perceived intervalprovides a direct estimate of the variabilit y associatedwith the internal timing system. One advantage ofthis procedure over the Wing-Kristofferson model is


that the slope analysis provides a more directestimate of timing variability. The Wing-Kristofferson model estimates implementationvariability from the lag one covariance values and,via subtraction, generates an estimate of centralvariability. The latter, as a residual, actually containsall sources of non-motor variability, only onecomponent of which is associated with an internaltimer. In contrast, the slope analysis method isolatesduration-dependent variability and uses the interceptto estimate all sources of duration-independentvariability, be they central or peripheral. In thecurrent study, we predicted that the slope valueswould be lower during bimanual tapping than duringunimanual tapping. Changes in the intercept valueswould indicate that the bimanual conditions alter thecontribution of other sources of variability.

We also used the richer data sets ofExperiment 4 to explore between-hand measures oftemporal performance. At all four durations, themovements in the bimanual condition should exhibitstrong temporal coupling given the task instructionsto move the hands in a synchronous fashion. A pointestimate of the phase relationship between the twohands can be made from the time difference at whichthe two microswitches are activated. From Figure 1,it can be seen that the multiple timer model wouldattribute these asynchronies to variability in motorimplementation: While the commands to initiate thetwo responses are issued simultaneously, peripheralvariability will influence the two handsindependently. From this perspective, twopredictions can be tested. First, it is expected that themean asynchrony will be invariant across the fourdurations. Second, the standard deviation of thedistributions of the asynchronies should also remainunchanged as tapping rate varies.

Finally, the phase differences can also beused to obtain a third estimate of central variability.Vorberg and Hambuch (1984) have proposed amodel for analyzing bimanual tapping data that issimilar to the general structure of the multiple timermodel in terms of the division of central andperipheral sources of variability. In their model, asingle timer is used to generate the target intervals foreach hand, and thus operates similar to the gatingoperation we propose in Figure 1. Central variabilitycan thus be estimated by the between-handcovariance. While this model cannot be applied tosingle hand data, it does offer another method forevaluating the change in temporal variability as afunction of tapping speed. We expect that the slopeobtained with this method will be comparable to thatderived from the slope analysis.

Method

Participants. Ten subjects from theUniversity of California, Berkeley participated in theexperiment and were reimbursed for theirparticipation. Each subject was tested on fourdifferent days and was paid $7/day for theirparticipation.

Procedure. Each experimental session wasdevoted to repetitive finger tapping at one targetduration. A Latin Square design was used todetermine the order for the four test durations acrosssessions. Within an experimental session, theparticipant completed three blocks of tapping withthe right index finger alone, the left index fingeralone, or with both fingers together. A trial wascomposed of 12 paced and 21 unpaced intervals, andthe participants produced six trials for each block, or18 trials per condition. Trials in which any intervalwas less than or greater than 50% of the targetinterval were repeated.

Data Analysis.Three methods were used to estimate

component sources of variability from the time seriesdata. First, similar to Experiments 1 and 2, we usedthe Wing-Kristofferson model, a method that focuseson the within-hand covariance function. Second, weapplied the slope analysis, a method that estimatesvariability directly from the observed variancemeasures (Ivry & Hazeltine, 1995). Third, we usedthe between-hand covariance function to estimatecentral variability (Vorberg & Hambuch, 1984). Thislatter method can only be used in the bimanualconditions.

Slope Analysis: The starting premise for theslope analysis is that total variability can bepartitioned into duration dependent (DD) andduration independent (DI) components (Ivry &Hazeltine, 1995):

VarianceTotal = VarianceDD + VarianceDI (5)

Duration dependent variability is assumed to reflectthe operation of an internal mechanism that providesthe timing signals needed to accurately initiate eachmovement. Duration independent variability isassociated with the implementation of the responses.By definition, duration dependent variability willincrease as a function of the interval being timedwhile the estimate of duration independent variabilitywill remain constant.

The relationship between temporalvariability and duration has been the subject of


considerable study (see Kill een & Weiss, 1987). Ingeneral, the literature indicates that a generalizedform of Weber's law holds in the temporal domainwhere the standard deviation is a linear function ofthe base duration (Getty, 1975; Ivry & Hazeltine,1995). This can be formally expressed as

VarianceDD = k2D2 (6)

where k is the Weber constant and D is the meaninter-tap interval produced. Substituting Equation 6into Equation 5 and replacing the duration

independent component with a constant, c, we obtain:

VarianceTotal = k2D2 + c (7)

A complete discussion as well empirical validation ofthe slope analysis can be found in Ivry and Hazeltine(1995). This equation provides an excellent accountof the data in both time production and timeperception studies. Moreover, alternativeformulations (e.g., where the linearity is assumedbetween duration and the variance rather thanduration and the standard deviation) provide a poorerfit with consistent negative intercepts.

For each participant, a regression analysisbased on Equation 7 was performed for the fourfunctions, left and right hands during unimanual andbimanual tapping. The primary analysis focused onthe slope and intercept values obtained from theseanalyses. The square root of the slope term yields k,the Weber constant.

Vorberg-Hambuch model: As describedabove, the between-hand covariance functionprovides an estimate of shared variabilit y betweenthe two hands during bimanual tapping. The keyassumption here is that the shared component reflectsvariabilit y in the operation of a common centralsignal, a signal that Vorberg and Hambuch associatewith an internal clock. The between-hand covarianceat lag 0 (i.e., for simultaneous intervals) will be lessthan the within-hand variance because of noise inmotor implementation processes. That is, the two

hands will produce non-identical intervals because ofvariabilit y in implementing the right and leftresponses. Thus, by the Vorberg-Hambuch model,an estimate of the variabilit y of the centralcomponent is obtained by:

SDCentral = sqrt(Covar(Lag 0)) (8)

Note that because this calculation is based on abetween-hand measure, a single estimate of temporalvariabilit y is derived. The method does not provideseparate estimates for the two hands.

Results and DiscussionLess than 2% of the trials contained an

interval that was outside the 50% criterion window.The low number here compared to Experiments 1and 2 likely reflects the fact that the participants weretested over multiple sessions.


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Figure 6. Component estimates for four target durations in Experiment 4. The dataare plotted as a function of the mean produced interval rather than the target values.


Wing-Kristofferson analysis: We beginwith the two-process model of Wing andKristofferson (1973). As before, the effects of globaldrift in the mean produced interval were minimized

by a transformation on the time series that removedany linear components. The resulting within-handcovariance functions were similar to those obtainedin the earlier experiments. The lag 1 covariance wasnegative for all 160 conditions (10 participants x 2hands x 2 tapping modes x 4 durations), and thevalues for lags 2-5 were close to zero. There were,however, four conditions in which the lag 1correlation was less than the theoretical limit of -.50.For these conditions, central variability is estimatedto be zero. We assume these violations reflect noisein the estimation process.

Figure 6 presents the estimates of centraland implementation variability at each of the fourdurations. The data have been averaged over the leftand right hands to simplify the figure. As can beseen, the estimates of implementation variabilityremain essentially constant, although the mean valuein both the uni- and bimanual conditions for the 325ms condition are slightly lower than for the otherthree target durations. In contrast, the estimates ofcentral variability increase across the range ofdurations. While there are no systematic differencesbetween the uni- and bimanual conditions on theimplementation scores, a consistent bimanualadvantage is observed with the estimates of centralvariability. The magnitude of this advantage appearsto increase for the longer target durations.

These data were analyzed in a four-wayANOVA with the variables hand (left or right),tapping mode (unimanual or bimanual), duration(325, 400, 475, and 550ms), and component (central

or motor delay). Our initial focus is on the basicquestion of whether temporal variability increases asa function of duration. As expected, a main effect ofduration was observed, F(3,27=20.2, p<.001). Thetwo-process model makes a more specific prediction:The estimate of the central component shouldincrease with duration whereas the estimate of themotor delay component should remain invariant.Thus, there should be a Component x Durationinteraction. Indeed, this interaction was reliable,F(3,27)=3.3, p<.05. While the central and motordelay estimates both increase over the four durations,the interaction reflects the fact that the increase issignificantly greater for the estimate of the clockcomponent. We did not find a reliable differencebetween the two hands, F(1,9)<1, nor was there aninteraction between the hand and componentvariables. Thus, unlike in Experiment 1 and Helmuthand Ivry (1996), we did not find a right-handadvantage on the estimate of motor variability withthese more experienced participants.

Turning to the comparison of uni- andbimanual tapping, a highly significant effect wasobserved for tapping mode, F(1,9)=42.4, p<.001,indicating that overall, performance was consistentlyless variable during bimanual tapping. This effect isqualified by the interaction between tapping modeand the component variable, F(1,9)=17.9, p<.01.There was no difference between the motor delay

0.001305

0.000723

0.001447

0.000823

0.000904

150.62

155.38

97.64

181.10

-0.55

0.61

0.60

0.83

0.52

0.77

0.0337

0.0272

0.0388

0.0271

0.0295

Left

Unimanual

Bimanual

Right

Unimanual

Bimanual

V-H Analysis

Bimanual

Slope Intercept R 2 Weber Fraction Tapping Mode

Table 3

Regression analysis results for experiment 4


estimates in the unimanual (9.6 ms) and bimanualconditions (9.8 ms). In contrast, the estimates of thecentral component for the uni- and bimanualconditions were 12.5 ms and 9.5 ms, respectively.Thus, the multiple effector advantage was restrictedto the estimate of central variability. The TappingMode x Duration interaction was marginallysignificant, F(3,27)=2.6, p<.08. As can be seen inFigure 6, the increase in variability tended to begreater in the unimanual condition compared to thebimanual condition.

Based on the multiple timer model, we hadpredicted a three-way interaction of Tapping Mode xComponent Estimate x Duration. Specifically, weexpected that the increase in variability as a functionof duration would be greater in the unimanualcondition, but only for the estimate of the centralcomponent. This interaction, however, did notapproach significance, F(3,27)<1. One reason for thefailure of this interaction to hold is the drop in themotor estimate for the shortest target interval.

Slope analysis: The slope analysis providesan alternative to the two-process model that is notdependent on indirect estimates of componentsources of variability. The slope analysis isperformed on the observed data, thus avoidingproblems that may arise from error in the estimationprocess. Using Equation 7, regression analyses wereperformed on the four functions produced by eachparticipant (left and right hands during uni- andbimanual tapping). The results of these analyses arepresented in Table 3. While the percentage ofvariability accounted for by a linear component wasreasonably high, there were a number of individualcases in which the values were quite low. It is likelythat the low values reflect the fact that the data setsare not extensive (18 trials/condition) and the orderwith which individuals were tested on the fourdurations varied. Note that the R2 values in Table 3are the average of the individual values. If aregression was performed on the averaged data, theR2 values would be greater than .96 for three of theconditions and .87 for one condition (bimanual righthand).

The slope and intercept data were analyzedin separate 2 (hand) x 2 (tapping mode) ANOVAs.F3

For the slope values, there was a significant effect oftapping mode, F(1,9)=5.7, p<.05. As predicted bythe multiple timer model, the slope was reliably

lower in the bimanual condition, indicating that theadvantage became greater as the target durationincreased. Thus, the multiple effector advantage ismanifest as a multiplicative reduction in temporalvariability rather than as a constant (additive)improvement. The effect of hand, F(1,9)=1.1, p>.3and the Mode x Hand interaction, F(1,9)<1.0, werenot significant. None of the effects were significantfor the intercept terms.

The Weber fractions are calculated as thesquare root of the slope values. These fractionsindicate the magnitude of temporal variability as afunction of the target interval. In the unimanualconditions, the Weber fractions were between 3-4%,values that are similar to those reported in theliterature (e.g., Ivry & Hazeltine, 1995). In thebimanual conditions, the Weber fractions dropped tounder 3%. Theoretically, the multiple timer modelpredicts that the Weber fraction during bimanualtapping should be equal to the Weber fraction duringunimanual tapping divided by the square root of two.The predicted and observed values for the left handduring bimanual tapping are .024 and .027,respectively. Thus, the improvement duringbimanual tapping is slightly less than predicted. Thepredicted and observed values for the right hand areidentical, .027.

Vorberg-Hambuch analysis: Estimates ofcentral variability based on the between-handcovariance were calculated according to Equation 7.As would be expected of a measure of timingvariability, these estimates increase with duration,and the increase is generally linear. The meanregression values for these data are included in thebottom row of Table 3. Note that the mean R2 valueover individuals here is quite high. While thismethod can not be used to compare uni- andbimanual tapping performance, it does provide anindependent method for calculating the change invariability across durations during bimanual tapping.The Weber fraction calculated with the between-handcovariances is .029 (square root of the slope,calculated on an individual basis). Thus, we findexcellent agreement between the Weber fractionswhen measured using the within-hand variance datain the slope analysis and the between-handcovariance data in the Vorberg-Hambuch analysis.This results lends strong support for the assumptionthat these analytic tools are estimating a commonconstruct.


Tapping asynchrony analysis: Theasynchrony between the left hand and right handresponses was calculated for each interval duringbimanual tapping. From these data sets (18 trials x30 intervals per trial), the distribution of theasynchronies was tabulated. These distrubutionswere approximately normal and their means andstandard deviations are shown in Figure 7.

Based on the multiple timer model, wewould expect the mean and standard deviation valuesto remain invariant over the four durations. Thisprediction is based on the assumption that theasynchronies result from motor implementationprocesses. There may be a consistent lead in onehand over the other; for example, lowerimplementation noise in the dominant hand mightresult in right hand responses being initiated prior toleft hand responses. However, we would expect thisasymmetry to remain constant over the range of

target intervals. Similarly, the variability of theasynchronies should be independent of duration.Contrary to these predictions, an effect of durationwas observed for both the mean phase difference,F(3,27)=5.6, p<.01, and the standard deviation of thephase differences, F(3,27)=6.6, p<.01. There is noconsistent effect of one hand leading the other for thetarget durations of 400 ms, 475 ms, and 550 ms.However, for the fastest duration of 325 ms, the righthand led the left by over 8 ms on average. Note thatvariability is largest for the fastest interval, a resultopposite that found in the variability of the tappingintervals themselves. As with the mean asynchronies,the standard deviation of the asynchrony distributionsremains relatively constant across the three longerintervals.

Taken together, these data provide mixedsupport for the predictions derived from the multipletimer model. For the three slower target durations,


BBB

B

J

J

J

J

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

22

250 325 400 475 550

Mil

lise

cond

s

Target Interval (ms)

B Std. Dev. of Phase Difference

J Mean Phase Difference

Figure 7

Figure 7. Asynchrony data for experiment 4. Negative values for the mean phase difference dataindicate left hand leading. Positive values are for right hand leading. The data are plotted as afunction of the mean produced interval rather than the target values.


the mean and standard deviation of the asynchronydistributions were constant. However, differenceswere observed at the fastest rate of 325 ms. Itremains to be seen why the asynchrony measureschanged at this fastest rate. One possibilit y is that theparticipants increased the stiffness of their finger inthe 325 ms condition. Such an increase might makeit easier to tap at this relatively fast rate, perhapsbecause the finger movement can be triggered by asmaller descending volley. Assuming such a changehad a more pronounced effect on the dominant handcould account for both the lead in right hand tappingand an increase in the asynchrony variabilit y. Thishypothesis is, admittedly, speculative. Anotherpossibilit y is that some of the participants adopted adifferent tapping strategy for the fastest condition.The mean phase lead for the right hand was over 10ms for four of the participants; for the other seven, itwas less than 5 ms.

An alternative way to examine theasynchrony data is in terms of relative phasedifferences, that is, by dividing the asynchrony value,the point estimate of relative phase, by the producedinterval. Expressed this way, the standard deviationsof the phase differences for the target durations of325 ms, 400 ms, 475 ms, and 550 ms are 15.3°,11.0°, 9.0°, and 7.5°. This change would beconsistent with the hypothesis that the couplingstrength between the two limbs becomes stronger asthe tapping rate slows down. It is not clear why onewould find a concomitant change in the mean phasedifference at the fastest frequency. Moreover, thisdecrease is also what one would expect given that thestandard deviation of the asynchronies remainsrelatively constant across the four target durations.

Summary: Experiment 4 was designed toprovide a strong test of the multiple timer model. Wehave proposed that the output gate performs a formof temporal averaging when provided withindependent temporal control signals duringbimanual movements. Based on this prediction, wewould expect the multiple effector advantage to bemultiplicative rather than additive. We tested thisprediction by having the participants perform therepetitive tapping task under uni- and bimanualconditions over a range of target intervals. In accordwith our predictions, an interaction was observedbetween tapping mode and duration. The multipleeffector advantage became larger as the targetinterval increased. This interaction was observedwith two analytic techniques designed to isolatecentral variabilit y, the Wing-Kristofferson model andthe slope analysis of Ivry and Hazeltine (1995).Moreover, the magnitude of the improvement asmeasured by the change in slope from the unimanual

to bimanual conditions was close to what would beexpected based on an averaging hypothesis. Finally,the results of Experiment 4 are in accord withprevious studies showing that variabilit y associatedwith motor implementation processes remainsconstant over a range of tapping rates.

General DiscussionAs described by Helmuth and Ivry (1996),

people become more consistent in producing a seriesof isochronous intervals when the movements areproduced by more than one effector. In their initialstudies, the multiple effector advantage was observedfor bimanual movements requiring homologousmovements (i.e., bimanual finger tapping) and non-homologous movements (i.e., tapping with one fingerand one forearm). The current experiments examinedthe generality of this phenomenon. Moreover, theexperiments were designed to test the multiple timermodel proposed by Helmuth and Ivry to account forthe multiple effector advantage.

Evaluating the assumptions of the multiple timermodel

In Experiment 1, we observed that themultiple effector advantage was quite robust, holdingover various movement combinations involving thefinger and foot. There was no apparent differencebetween movements that involved effectors ondifferent sides of the body (e.g., right finger and leftfoot) compared to movement that involved effectorson the same side of the body (e.g., right finger andright foot). Using the two-process model of Wingand Kristofferson (1973), the improved temporalvariabilit y was attributed to a reduction in theestimate of central variabilit y. Indeed, overallvariabilit y tended to become larger when theparticipants were asked to tap with limbs of unequalmass (see also, Helmuth and Ivry, 1996). Thisincrease is assumed to result from instabiliti es ingenerating differential forces to activate theasymmetric limbs (Jeka & Kelso, 1995).Nonetheless, the variabilit y associated with theoperation of an internal timing system was alwaysreduced for all effector combinations.

The multiple timer model proposes that theimproved temporal performance is the statisticalconsequence of the control operations required in theproduction of multiple effector rhythmic movements.We assume that these operations include timingmechanisms that regulate the timing of each cycle.At this point, we do not make specific claims abouthow this regulation is achieved. It may be that acentral command initiates each cycle, triggering theonset of the downstroke of the movement during


finger or foot tapping. Or it may be that the centralcommand is in terms of a representation of thedesired temporal pattern to be formed by the contactof the effector with the response board (Billon,Semjen, & Stelmach, 1996). What is essential to ourmodel is that a representation of the target interval isgenerated for each cycle and it is this representationthat provides the primary control of the timing of themovements. Most critical, we assume that theserepresentations are effector-specific. For eacheffector that is engaged in the task, an independentrepresentation of the target interval is generated tocontrol the movements of that effector.

However, as outlined in Figure 1, weassume these central representations do not havedirect access to their associated effectors. Rather, theimplementation of the commands is constrained by agating operation, allowing the movements to beproduced in a synchronized fashion. We propose thatthe manner in which the gating operation integratesthe effector-specific signals effectively acts as anaveraging device, and it is this averaging process thatunderlies the multiple effector advantage (Helmuthand Ivry, 1996). In essence, the advantage ishypothesized to be a manifestation of the central limittheorem. Variability is reduced as the sample sizebecomes larger. The constraint imposed by the gatingoperation may result from the task demands. In ourstudies, we require that the movements be producedin a synchronized fashion. However, across a rangeof bimanual tasks, people have great difficulty inachieving temporal independence, even in situationsdesigned to promote such independence (Franz,Eliassen, Ivry, & Gazzaniga, 1996; Kelso, Southard,& Goodman, 1979; Krampe et al., 2000; Zanone &Kelso, 1997). These observations suggest that thegating constraint may reflect a fundamental limitationin the motor system, perhaps providing a means forreducing control requirements by ensuring thatselected actions are implemented in a coordinatedfashion (Ivry & Hazeltine, 1999; Ivry & Richardson,in press).

Experiments 2, 3, and 4 tested specificpredictions of the multiple timer model. InExperiment 2 we observed a further reduction inwithin-effector temporal variability when a third limbwas added to the mix. We attribute this reduction tothe activation of a third representation of the targetinterval and the added benefits obtained when thegating operation is now provided with three inputs.We assume that temporal stability would continue toimprove as more effectors were engaged, althoughour ability to empirically observe this benefit wouldbecome difficult given that the magnitude of theimprovement decreases with each additional effector.

Experiment 3 demonstrated that the multipleeffector advantage was not dependent on theproduction of repetitive movements. A significantreduction in within-effector variability was foundeven when the participants produced each interval inisolation. Thus, the effect does not depend on somesort of entrainment process.

Experiment 4 used a different approach totest a quantitative prediction of the multiple timermodel. In this study, the target duration was varied.The multiple effector advantage was expected to holdacross all tapping rates. More critical, given that thestandard deviation is proportional to the intervalbeing timed, we would expect the magnitude of thereduction to increase as the inter-tap interval islengthened. This prediction was confirmed. Notonly was the tapping mode by duration interactionsignificant, but the observed slope during bimanualtapping was close to that predicted by the model.This study provided a novel demonstration of theadvantage of the slope method (Ivry & Hazeltine,1995). This procedure offers an alternative tool foridentifying component sources of variability, one thatattempts to directly measure central variability ratherthan use the indirect, subtractive approach of the two-process Wing-Kristofferson model.

While the hypothesis of multiple timers maynot seem parsimonious, there are a number ofappealing features of this sort of model. First, themodel does not require that a single clock beaccessed by different tasks. Although outside thescope of this paper, we assume that distinct neuralelements are not only linked to specific effectors, butare also tuned to represent specific intervals, an ideapromoted in a number of recent papers (e.g., Ivry,1996; Meegan, Aslin, & Jacobs, 2000; Rosenbaum,1998; Wright, Buonomano, Mahncke, & Merzenich,1997). Thus, we assume that there exist a set oftiming elements to regulate tapping at different rateswith one effector, and that this organization isrepeated for other effectors (Ivry, 1996). Byassuming that the exact circuits required forrepresenting temporal information will vary fromtask to task, the anatomical prerequisites would seemto be simplified. The circuitry for an amodal, singlemechanism would have to be quite complex, havingthe capability to broadcast a signal to all outputsystems. Note that in the multiple timer model,correlations across different temporal tasks (e.g.,Keele et al., 1985) do not reflect the operation of asingle clock, but rather the fact that the timing systemas a whole is associated with common noiseproperties. Thus, the model is consistent with thehypothesis that temporal representations may dependon the operation of a common timing system (e.g.,


the cerebellum), but within this system, elements willbe recruited in a task-specific manner.

Second, the gating constraint may helpensure that all movements generated at any one pointin time are coordinated or, at least, not mutuallyexclusive of one another. For example, if competingactions, one calling for moving the right handforward and the other for moving the right handbackward were simultaneously active, inhibitoryconnections between these gestures would make itunlikely that either gesture would achieve sufficientactivation to cross threshold. In this sense, the gatecaptures the idea of a winner-take-all process (Berns& Sejnowski, 1996).

At present, we have focused exclusively ontasks in which the instructions emphasize that the twolimbs should move in a synchronized fashion. Ourmotivation for this approach comes from the fact thatpredictions derived from the multiple timer model arestraightforward when the gating process is assumedto occur simultaneously for both limbs. However,movements can be coupled, even when they are notsynchronized. For example, during paced, anti-phasetapping, only one limb is synchronized with thepacing signal if we define synchronization by eventssuch as the time of contact with the response key andthe pacing signal. Nevertheless, the stability of anti-phase tapping suggests the persistence of strongtemporal coupling. It remains to be seen how themultiple timer model can be extended to such tasks.One possibility is that under such conditions, thetiming signals for each limb are not integrated.Indeed, Ivry and Richardson (in press) suggest thatthe instability that emerges when frequency isincreased during anti-phase tapping may reflectunwanted interactions between the separate timingsignals. A second hypothesis is that hierarchicaltemporal representations are generated to ensure thatsuccessive actions continue to exploit thesimultaneous operation of the gating process. Forexample, during 2:1 tapping the gate might operate atthe fastest beat, but only initiate movement for theslower hand on every other cycle (Krampe et al.,2000; Semjen & Ivry, in press; Vorberg & Wing,1986). These are obviously important questions forfuture study.

Can the multiple effector advantage result fromfeedback between the two limbs?

While we have focused on the multipletimer model in our account of the multiple effectoradvantage, it is important to consider alternativemodels. One alternative is based on the idea thatfeedback signals generated during multi-effectormovements can lead to reduced temporal variability.

During unimanual tapping, feedback is, of course,available from multiple sources including the clicksgenerated when the response keys reach theirmaximal excursion as well as from thesomatosensory input from the moving effector.When tapping with two hands, these sources offeedback are now available from both hands, perhapsresulting in more salient feedback signals. Moreover,an additional source of feedback can be obtained bycomparing the movements of the two hands. Forexample, a discrete feedback process could monitorthe asynchrony between the two hands at the startand finish of each tap, or a continuous feedbackprocess could monitor the phase relationship betweenthe two hands throughout the movement cycle. Suchfeedback signals during bimanual tapping wouldseem to offer an opportunity to make adjustments fordeviations in performance that were not possibleduring unimanual tapping.

At present, our preference for the multipletimer model is based on a set of indirect argumentsagainst the feedback hypothesis. First, the multipletimer model is a relatively straightforward extensionof the Wing and Kristofferson (1973) model. In theirtwo-process model, the estimation of central andmotor estimates of variability assumes that the clockand motor implementation processes operateindependently of one another and that successiveoutputs from each process are independent. Studieswith neurologically healthy (e.g., Wing, 1980; Ivry &Hazeltine, 1995; see Pressing, 1999) and impairedpopulations (e.g., Ivry & Keele, 1989; Ivry et al.,1988) have, in general, provided strong support forthese basic tenets, at least when the inter-tap intervalis less than 1 s. In the multiple timer model, the sameprocesses operate during multiple effectormovements; by a feedback model, we would have toassume that new processes come into play duringsuch movements. Of course, as noted in thepreceding paragraph, the multiple effector conditionaffords new sources of information.

We have conducted simulations of feedbackmodels to evaluate the viability of such an approach.These simulations have taken various forms. Forexample, in one simulation, we assumed that a single,central timing command was projected to alleffectors. Any asynchrony between the two hands isattributed to independent variability in motorimplementation processes (e.g., Vorberg &Hambuch, 1984). This asynchrony could then beused to make adjustments in the timing of theresponses. In another simulation, we retained theidea of multiple clocks and independentimplementation processes, but again used theresultant asynchrony to adjust the timing of the


responses. For the adjustment, we opted for thesimple method. If the left hand led the right by Xms, we delayed the next tap of the left hand by X/2ms and increased the tap of the right hand by X/2 ms(after obtaining the next clock and motorimplementation samples for each hand). It turns outthat models in which the asynchrony on Response Nis used to adjust the timing of Response N+1 end upleading to an increase in overall variability.

At first glance, it might seem counter-intuitive that feedback would impair performance.However, this phenomenon has been observed inother conditions. For example, variability duringboth unimanual and bimanual tapping is significantlylarger if the pacing signals are preserved over theentire trial (unpublished observations). This cost isobserved even though the multiple effector advantagecontinues to be manifest. Similarly, the visualfeedback available when two individuals tap with asingle finger leads to an increase in the within-subjecttemporal variability (Helmuth & Ivry, 1996).Pressing (1999; see also Vorberg & Wing, 1996) hasformally analyzed the feedback situation, arguing inaddition to the clock and motor implementationsources of variability, paced tapping introduces anovel source of variability associated with theutilization of the error signal generated by themismatch between the pacing signals and the taps.Similarly, we have all experienced the problem ofover-correction when calibrating a motor skill suchas dart throwing. We tend to assume that all of theerror is central in origin; we fail to recognize that aproportion of the error is more peripheral in nature(e.g., Schmidt, 1975).

Our simulations have all been based on theidea that feedback signals are used in a relativelydiscrete manner. Asynchronies between the twohands on one response are used to adjust the timingof the next response. An alternative procedure wouldbe to use feedback in a continuous manner. Forexample, a proprioceptive-based feedback processcould continuously monitor the phase relationshipbetween the two hands and make adjustments to keepthis difference near zero. Such a process wouldsurely reduce the variability of each hand (see belowfor a qualitative description of this idea). Whethersuch processes are viable during tasks such asrepetitive tapping remain unclear. Studies involvingmulti-joint movements such as throwing, however,have shown that the timing of the finger release isunaffected by perturbations during elbow extensionor wrist flexion (Hore, Ritchie, & Watts, 1999).

In addition to our theoretical explorations,we have also tried to empirically evaluate thefeedback idea. In the present paper, we found that

the multiple effector advantage was evident evenwhen the intervals were produced in isolation. Theseresults argue against the idea that the improvedtemporal performance results from some sort ofextended entrainment between the two limbs duringcyclic movements. However, the interpretation ofthese data provides, at best, a weak test of a feedbackhypothesis. First, the participants did produce taps tomark both ends of the interval in this study and thuscould have used feedback from the first tap to adjustthe timing of the second tap. Second, the magnitudeof the effect appeared to be considerably less thanwhat would be predicted by the averaging model.

A second line of evidence against afeedback model rests on the finding that the multipleeffector advantage was obtained in a split-brainpatient, even when this patient tapped with herfingers (Ivry & Hazeltine, 1999). While there isample opportunity for cross-talk of afferents fromproximal muscles, including bilateral projections tosomatosensory cortex, the ascending pathways fromtheir fingers are thought to project exclusively to thecontralateral hemisphere (e.g., Guillemot, Richer,Prevost, Ptito, & Lepore, 1987; Iwamura, 2000;Shanks, Pearson, & Powell, 1985). Nonetheless, thebimanual finger movements of the patient remainedtightly coupled (see also, Tuller, & Kelso, 1989) andthe within-effector variability was significantlyreduced for each hand during bimanual tapping. Inthis study care was taken to eliminate auditory andvisual sources of feedback. Thus, it is unclear howafferent information from each hand would be able toinfluence the movements of the other hand.

Despite these arguments, definitiveevaluation of the feedback hypothesis remains a goalof future research. One approach would be tointroduce perturbations during the movement cyclefor one hand and evaluate the effects on the otherhand. However, we do not doubt that people can(and will) use feedback. The question is whether thisinformation can lead to reduced temporal variability.A more dramatic approach would be to test patientswho suffer peripheral neuopathies that render themfunctionally deafferented. Such patients, especiallythose with intact output pathways, are rare.

The multiple timer model considered within thedynamic systems framework

A second alternative is to consider themultiple effector advantage from the perspective ofthe dynamic systems framework. This approach hasbeen extremely prominent in the field of motorcontrol. Indeed, the influential work of Kelso,Turvey, and their colleagues was initially developedfrom the experimental analysis of bimanual


movements (Kelso, 1997; Kugler & Turvey, 1987),although the approach has now been applied to awide variety of task domains. The focus of this workhas been on inter-limb coordination. For example,the coupled oscill ator model provides an elegantdescription of the stabilit y of certain phaserelationships and the transitions observed as variouscontrol parameters are varied (Schöner & Kelso,1988).

With the exception of only a couple ofstudies (e.g., Yamanishi et al., 1980; see Semjen &Ivry, in press), the temporal stabilit y within each limbhas been of secondary concern within the dynamicsystems approach. However, at a descriptive level,an account of the multiple effector advantage can beconceptualized within the framework of a coupledoscill ator model. Consider the limit cycle, thedynamic state that describes stable conditions duringrepetitive movements under conditions to produce in-phase movements (Figure 8). Noise can have twoeffects on the position of a single oscill ator movingalong the limit cycle. First, it can perturb the

oscill ator to a position off of the limit cycle. Suchperturbations would be corrected due to the attractiveforces of the limit cycle. Second, it could perturb theoscill ator along the limit cycle. During unimanualmovements, such perturbations would gouncompensated: all positions along the limit cycle arestable (Figure 8a). However, during bimanualmovements, each oscill ator also can beconceptualized as point attractors and thus provide ameans for adjustment (Figure 8b).F4 This descriptioncan be seen as one instantiation of a continuousfeedback model. We would assume that the controlparameters (or output signals corresponding to thecurrent phase) are equivalent for the two oscill ators,reflecting the effects of coupling and the taskrequirements to tap in phase. Perturbations thatimpose phase deviations are assumed to reflect noiseand the adjustment to such noise would in essenceconstitute a feedback process. Alternatively, acomparison could be made of the state of the outputsignals and an adjustment made if these signals wereout of phase with one another.

There are points of similarity and differencebetween the multiple timer and coupled oscill atormodels. Both models posit separable timingmechanisms for each limb, as well as a form ofcoupling between the outputs of these mechanisms.In a sense, the multiple timer model entails a specifictype of coupled oscill ators. The model is, of course,dynamic, in that it attempts to account for the time-varying interactions that occur between the processesassociated with the movements of each effector(Schöner, 2000). However, the coupling is of a verydifferent form than that articulated in current formsof the coupled oscill ator model. Rather thanconceptualizing coupling as a continuous process, thegating operation operates as a threshold mechanism,introducing a level of discreteness in the interactionsbetween central control processes and movementimplementation processes. This threshold processprovides coupling in two ways. First, there are theinteractions between the activation functions of thetiming mechanisms, an interaction that culminates inthe common gating of the output signals and ensuresthat the movements are generated in a synchronousfashion. Second, there is the mutual resetting of eachtimer for the next cycle following the triggering ofthe gate. These properties of the hypothesized gatingoperation impose a discontinuity on the dynamics. Aconsequence of this discontinuity is that the formalapproaches developed for coupled oscill ator modelsare diff icult to adapt to the multiple timer model(Schöner, 2000).

Empirically, we have sought to identifyplaces where the models diverge. Schmidt et al.

A

B

Velocity

C

Figure 8

Figure 8. Reduced timing variability fromcoupled oscillators. Repetitive movement isdepicted as a limit cycle in which velocityvaries in a continuous manner with position.Panel A: Perturbations of a single oscillator offof the limit cycle will be corrected due to thestable nature of the limit cycle. Panel B:Perturbations along the limit cycle are notcorrected since performance is stable at allpoints on the limit cycle. Panel C: In bimanualtapping, perturbations along the limit cycle arecorrected due to the coupling between theoscillators.


(1990) looked at the dynamics during repetitivemovements when the movements were produced bydifferent individuals. In this experiment, twoindividuals faced each other and each moved one legin time with a metronome. Similar to what had beenobserved in traditional within-individualexperiments, the movements of the two individualswere tightly coupled, and when the frequencyincreased under anti-phase conditions, a phasetransition was observed. This led the authors toargue that the coupled oscill ators operate at anabstract level; a common framework can be used toaccount for dynamical interactions that arise withinan individual and between individuals.

Helmuth and Ivry (1996) examined thissame issue in a finger tapping study, but with thefocus on within-effector variabilit y. Contrary to theresults of Schmidt et al. (1990), this dependentvariable showed a striking difference between thewithin- and between-individual conditions. Themultiple effector advantage was only found in thewithin-individual condition; for the between-individual condition, total variabilit y and the estimateof central variabilit y increased. This increase islikely similar to that observed when tapping with apacing signal with the pacing signal now being theother person's finger movements. The lack ofreduced temporal variabilit y in the between-individual condition is in accord with the multipletimer model. As sketched in Figure 1, the gatingprocess would not be expected to receive input fromthe timing mechanism of another individual!

We have also sought to identify predictionsthat are specific to the multiple timer model. Theprediction that central variabilit y will be reduced bythe square root of two and the reduced slopedescribing temporal variabilit y as a function ofduration are two such examples. The results ofExperiments 1 and 4 in the current study as well asthe findings of Helmuth and Ivry (1996) providereasonable support for these predictions, at leastwhen the movements are produced by effectors ofsimilar mass. Thus, the data are consistent with thequantitative predictions of the multiple timer model.Of course, tests that confirm a hypothesis offer aweaker form of argument than tests that disconfirman alternative hypothesis. At present, however,quantitative predictions based on the coupledoscill ator model are not as constrained as those basedon the multiple timer model.

The relationship between the multiple timerand coupled oscill ator models remains an issue fordebate. At one level, the two seem quite disparate.Certainly the issue of discrete versus continuouscoupling should be ripe for investigation (see

Schöner, 1990 for a theoretical analysis of therelationship between discrete and continuousmovements from a dynamic systems perspective).On the other hand, the two approaches may becompatible, offering different levels of description.The coupled oscill ator model offers a rich, abstractdescription of the dynamics across a wide range ofmovement conditions. The multiple timer model isnarrower, specifying component processes involvedin the control and coordination of timed movements.In its favor, the multiple timer model embodiesspecific hypotheses concerning the control processesinvolved in the temporal representations for suchmovements and the dynamics that allow thesemovements to be coupled. Whether the basic ideascan be extended to provide more general principlesof coordination, for example those observed withdifferent coordination modes as well as thetransitions observed between coordination modes,remains to be seen.

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Notes

F1. In the simulations, the means and variances forthe interval distributions were based on estimates ofthe central variance in Helmuth and Ivry (1996),derived according to the two-process model of Wingand Kristofferson (1973). We also added anadditional delay to represent the time required toimplement a motor command. The delaydistributions were independently sampled for the leftand right hands, using a fixed mean for the two handsand distribution variances based on the observedunimanual data.

F2. To minimize feedback even further, we wouldhave preferred to have the participants produce asingle response for each interval. We piloted a studyin which the onset of the interval was indicated by atone and the participants were instructed to produceone tap, marking the end of the interval. However,the participants were considerably more variable,both in terms of the mean interval produced andvariability of the produced intervals with thismethod. Another alternative would have been to askthe participants to make one response by pressingand holding the key for the requisite interval.However, this type of movement would also likelyinvolve two sub-movements, one related to the holdphase and the other related to the lift.

F3. One participant produced a negative slope in twoof the conditions, indicating that she was morevariable when tapping at the faster rates. Thestatistics were run twice, once with her data included,and once with her data excluded. No differences


were seen in the two analyses so we only report thestatistics involving the complete data set.

F4. We thank Gregor Schöner for his helpfuldiscussions in developing these ideas.

Author Notes

We are grateful to Andras Semjen, Jörn Diedrichsen,Eliot Hazeltine, and Brent Stansfield for theircomments on the ideas expressed in this paper. Thiswork was supported by grants from the NationalInstitute of Health (NS30256, NS2778) and theNational Science Foundation (ECS-987

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