Stability of rhythmic visuo-motor tracking does not ... - Olivier Oullier

Exp Brain Res (2008) 184:269–273DOI 10.1007/s00221-007-1180-0RESEARCH NOTEStability of rhythmic visuo-motor tracking does not dependon relative velocityAymar de Rugy Æ Olivier Oullier ÆJean-Jacques TempradoReceived: 14 March 2007 / Accepted: 15 October 2007 / Published online: 1 November 2007Ó Springer-Verlag 2007Abstract It is well established that the in-phase pattern ofbimanual coordination (i.e. a relative phase of 0°) is morestable than the antiphase pattern (i.e., a relative phase of180°), and that a spontaneous transition from antiphase toin-phase typically occurs as the movement frequency isgradually increased. On the basis of results from relativephase perception experiments, Bingham (Proceedings ofthe 23rd annual conference of the cognitive science society.Laurence Erlbaum Associates, Mahwah, pp 75–79,2001; Ecol Psychol 16:45–53, 2004; Advances in psychology135: time-to-contact. Elsevier, Amsterdam, pp421–442, 2004) proposed a dynamical model that consistsof two phase driven oscillators coupled via the perceivedrelative phase, the resolution of which is determined byrelative velocity. In the present study, we specifically testbehavioral predictions from this last assumption during aunimanual visuo-motor tracking task. Different conditionsof amplitudes and frequencies were designed to manipulateselectively relative phase and relative velocity. While theknown effect of phase and frequency were observed, relativephase variability was not affected by the differentconditions of relative velocity. As such, Bingham’s modelA. de Rugy (&)Perception and Motor Systems Laboratory,School of Human Movement Studies, Room 424,Building 26, University of Queensland,St Lucia, QLD 4072, Australiae-mail: aymar@hms.uq.edu.auA. de Rugy J.-J. TempradoUMR Mouvement et Perception (UMR 6152),CNRS and Université de la Méditerranée, Marseille, FranceO. OullierLaboratoire de Neurobiologie Humaine (UMR 6149),Université de Provence and CNRS, Marseille, Franceassumption that instability in relative phase coordination isbrought about by relative velocity that affects the resolutionof the perceived relative phase has been invalidated forthe case of rhythmic unimanual visuo-motor tracking.Although this does not rule out the view that relative phaseproduction is constrained by relative phase perception, themechanism that would be responsible for this phenomenonstill has to be established.IntroductionSince the seminal studies of Kelso (1981, 1984), the relativestability between different patterns of bimanualcoordination has been extensively studied to investigate theconstraints acting upon the coordination of interlimbmovement. The in-phase pattern of coordination (i.e. arelative phase of 0°) was typically found more stable thanthe antiphase pattern (i.e., a relative phase of 180°), andspontaneous transitions from antiphase to in-phase weretypically observed as the movement frequency was graduallyincreased.Although neuromuscular correlates of those behavioralprinciples have been clearly identified (e.g., Carson 2005;Carson and Kelso 2004; Swinnen 2002), similar principleswere also found under the sole influence of visual coupling,as in interpersonal coordination (Schmidt et al. 1990, 1998;Temprado and Laurent 2004; de Rugy et al. 2006) and inunimanual visuo-motor tracking of a movement on a screen(Wimmers et al. 1992; Stins and Michaels 2000; Buekerset al. 2000).This motivated Bingham and colleagues to conduct a setof studies showing that perception of the variability of therelative phase between two balls moving on a screen123

270 Exp Brain Res (2008) 184:269–273matches behavioral results: the antiphase pattern wasjudged more variable than the in-phase pattern, and relativephase was judged more variable as the movement frequencyincreased (Bingham et al. 1999, 2001; Zaal et al.2000). On the basis of those results, Bingham (2001,2004a, 2004b) proposed a dynamical model that capturesthe results of both behavioral and perception studies.Central to this model is the coupling of two phase drivenoscillators via the perceived relative phase, the resolutionof which is determined by relative velocity. In the presentstudy, we specifically test behavioral predictions from thislast assumption of the model, i.e., that instability in relativephase coordination is brought about by relative velocitythat affects the resolution of the perceived relative phase.This was achieved by manipulating selectively relativephase and relative velocity during a unimanual visuo-motortracking task.Bingham’s model is as follows:€x i þ b _x i þ kx i ¼ c sinð/ j ÞP ij€x j þ b _x j þ kx j ¼ c sinð/ i ÞP ji ð1Þwhere x i and / i are the position and phase of oscillator i,respectively, and the dots denote the first and secondderivatives. Each oscillator is driven by the perceivedphase of the other oscillator multiplied by a term Prepresenting perceived relative phase:P ji ¼ sign sinð/ i Þ sinð/ j Þþað _x i _x j ÞN t ð2ÞP (i.e. ± 1) is the sign of the product of the two driversincremented by a Gaussian noise term with a variance thatis proportional to the velocity difference between theoscillators.In the absence of noise, P = +1 ensures an in-phasepattern of coordination, whereas P = -1 ensures theopposite (antiphase) pattern. While the velocity differenceis minimal during the production of the in-phase pattern ofcoordination between oscillatory movements of similaramplitude (Fig. 1b1), it is oscillating at a maximal amplitudefor the antiphase behavior (Fig. 1b2); a noise term thatis function of this difference (Fig. 1c2) will therefore affectP (Fig. 1d2) and the stability of the resulting behavior. Asthe velocity signals increase with frequency, so do thevelocity difference and the resulting noise, leading to afurther increased variability of the behavior. Note that theuse of non-normalized velocity in Eq. 2 is crucial here,since normalizing velocity would prevent the increase invariability the model was precisely designed to reproduce.These features are illustrated in Fig. 2a together with predictionsfrom Bingham’s model in response to our selectivemanipulation of relative phase and relative velocity. Thisselective manipulation was achieved via the modulation ofthe amplitude of one of the two oscillations only: Bymultiplying the amplitude of one movement by three, thevelocity difference obtained during an in-phase behavior(Fig. 1b3) is similar to that of the antiphase behavior performedwith similar amplitudes (Fig. 1b2). According toBingham’s model, those two conditions should thereforeproduce similar variability profiles (Fig. 2a). Furthermore,the velocity difference obtained during an antiphase coordinationperformed with different amplitudes (Fig. 1c4) ishigher than that obtained with similar amplitudes (Fig. 1c2). According to Bingham’s model, the variability of thebehavior should therefore be higher in the former conditionas compared with the latter (Fig. 2a). If the mechanismsinvolved in the studied coordination are independent of therelative velocity, however, our manipulation of movementamplitude should have little effect on the behavior; i.e., theknown effects of phase and frequency should be observed,but different amplitudes should not affect relative phasevariability (Fig. 2b).We tested those predictions during a unimanual visuomotortracking task that permitted to keep the amplitude ofthe participants’ movement constant while manipulatingsolely the amplitude and frequency of the target to track(either in-phase or antiphase).Materials and methodsSix participants (four males and two females, aged 21–28)volunteered for this experiment. All were right handed withnormal or corrected to normal vision. Participants werenaive to the purpose of the experiment. They all gaveinformed consent prior to experiment, which was approvedby the local ethics committee and conformed to the Declarationof Helsinki.Each participant sat in a chair 70 cm from a computerdisplay positioned at eye level. With their right hand,participants grasped a handle fixed to a rotating horizontalshaft (instrumented to record angular displacement at asampling rate of 100 Hz) adjacent to the ulna. Pronation–supination movement resulted in the higher part of thehandle to move right–left from the middle position definedby the vertical (forearm mid-prone).The angular displacement of the handle controlled byparticipants was displayed online on the screen as a linearhorizontal displacement of a white ball moving in the samedirection as the higher part of the handle. An amplitude A1of 3.4 cm (corresponding to an angular displacement of thehandle of 62°) was prescribed by two thin vertical linesbetween which the participants were required to oscillate.Four centimeters above the ‘‘handle’’ ball was a ‘‘target’’(also white) ball moving according to an imposed sinusoidalhorizontal displacement. The white balls (1.7 cmdiameter) were presented on a dark background. To ensure123

Exp Brain Res (2008) 184:269–273 271Fig. 1 a The positions (in arbitrary unit) of two oscillatory objectsfor the combinations of two conditions of relative phase [in-phase(IN) and antiphase (ANTI)] and amplitude (A1 same amplitudes; A2one oscillation has an amplitude of three times the other). Rows 1–4correspond to condition IN-A1, ANTI-A1, IN-A2, and ANTI-A2,respectively. b–d Corresponding components of the coupling functionthat only the stimulus displayed on the screen could beseen, the experiment was performed in the dark and participantswore soldering glasses. This visual display wasdesigned to be as close as possible to that reported in thestudies conducted by Bingham and colleagues on the perceptionof relative phase (Bingham et al. 1999, 2001; Zaalet al. 2000).Participants were instructed to synchronize their oscillation(handle ball) either in-phase (IN) or antiphase(ANTI) with the oscillation of the target dot. In the INcondition, the handle ball had to move in the same directionas the target ball, and in the ANTI condition, it had tomove in the opposite direction. While the amplitude of thehandle ball’s displacement was always prescribed at3.4 cm, the amplitude of the target’s displacement waseither 3.4 cm (condition A1) or three times 3.4 cm (i.e,10.2 cm, condition A2). Three frequencies were employed:in Bingham’s model (Eq. 2). b Relative velocity ð _x i _x j Þ; c product(prod) of the drivers incremented by a Gaussian noise with a varianceproportional to the velocity difference (i.e., prod ¼ sinð/ i Þsinð/ j Þþað _x i _x j ÞN t ; with a = 0.0015); d sign of prod (i.e., Pin Eq. 2)0.75, 1.25, and 1.75 Hz. After a few practice trials, theacquisition session consisted of five trials of 35 s performedin each of the 12 conditions [phase(2) 9 amplitude (2) 9 frequency (3)], presented in a fullyrandomized order.Continuous relative phase was determined in phasespace, i.e., the space spanned between position and velocityfor each oscillation. The angular velocity obtained by differentiationof angular position was normalized by dividingthe velocity signal by the mean frequency. Next, the phaseangles were computed for each sample of oscillatorymovement as the arctangent of the position and velocity.Mean circular relative phase w was then determined usingcircular statistics (Fisher 1993). To this end, the cosine andsine of the difference between the phase angle for thehandle and the target were averaged separately, and w wasobtained as the arctangent of their ratio (for more detail,123

272 Exp Brain Res (2008) 184:269–273Fig. 2 Predicted (a, b) andeffective (c) variability of therelative phase presented as afunction of frequency for thedifferent experimentalconditions of phase andamplitude. a Qualitativeprediction based on a variabilitythat scales with relative velocity(i.e., Bingham’s model). bQualitative prediction based ona variability that does notdepend on relative velocity (i.e,the known effects of phase andfrequency only are displayed). cExperimental results (SDw).Plain lines represent the inphase(IN) and dashed lines theantiphase (ANTI) conditions,while dark lines representconditions of similar amplitudes(A1) and gray lines conditionsin which one oscillation has anamplitude of three times theother (A2). Error bars representthe standard deviationssee Russell and Sternad 2001). A measure of dispersion ofcircular relative phase, uniformity U, was calculatedaccording to Fisher (1993). As this measure is bounded by0 and 1 and is nonlinear with respect to the distributionaround the mean relative phase angle, it is converted into ameasure of dispersion SDw that varies approximately linearlybetween 0 and infinity according to:SDw ¼ ð 2 log e UÞ 1=2As in the linearly computed standard deviation measurehigh values of SDw denote high variability, and low valuesindicate low variability.Relative phase variability SDw was averaged over thefive trials performed under the same experimental condition,and analyzed using three-way repeated measuresANOVA (phase 9 amplitude 9 frequency). When a significanteffect was obtained, the proportion of totalvariability attributable to the factor concerned was reportedas the value of partial eta-squared (g 2 ) (see Pierce et al.2004, for information about partial eta-squared measures).ResultsFigure 2c shows that the variability of the relative phasewas higher for the antiphase than for the in-phase pattern ofcoordination; that this variability increased with the frequencyof movement; and that this increase was higher forthe antiphase than for the in-phase pattern of coordination.The three-way repeated measures ANOVA(phase 9 amplitude 9 frequency) conducted on SDwrevealed a strong effect for the phase (F(1, 5) = 30.24,P = 0.003, g 2 = 0.86), for the frequency (F(2, 5) = 23.06,P = 0.006, g 2 = 0.92), and a significant interaction betweenthem (F(1, 5) = 23.19, P = 0.006, g 2 = 0.92). No effect,however, was found for the amplitude (F(1, 5) = 0.05,P = 0.83) or for the other interactions (all F \ 0.65 andP [ 0.56). Figure 2c confirms that our manipulation ofamplitude had little effect on the behavior, and that theobtained variability profiles correspond to a coordinationmechanism that does not depend on relative velocity(Fig. 2b) rather than on the coordination mechanism proposedin Bingham’s model (Fig. 2a).DiscussionBingham (2001; 2004a, b) proposed a perceptually drivendynamical model of bimanual coordination that consistsof two phase driven oscillators coupled via the perceivedrelative phase, the resolution of which deteriorates as afunction of relative velocity. In the present study, we123

Exp Brain Res (2008) 184:269–273 273specifically test behavioral predictions from this lastassumption, which is implemented Eq. 2 with a noiseterm that is proportional to the relative velocity. Therationale for this was twofold: first the noise term inEq. 2 was supposed to reflect known sensitivities to thedirection of optical velocities (De Bruyn and Orban1988; Snowden and Braddick 1991). And second,because relative velocity is higher for antiphase than forin-phase coordination and increases with frequency, itsmagnitude already matches variability results of bothbehavioral and perception experiments. In the absence ofinfluence of neuromuscular coupling between hands forour unimanual case, and while the motor component ofthe task was carefully kept constant (because theamplitude of the movement to perform was constant, themotor command had to be virtually the same for thedifferent conditions of relative phase and target amplitudeperformed at the same frequency), our experimentwas designed to promote as much as it possibly couldthe influence of the a perceptual component as proposedin Bingham’s model. Yet, we failed to observe aninfluence of relative velocity on the variability of thebehavior, and must conclude that the coupling termproposed in Bingham’s model does not reflect the couplingthat operates in rhythmic visuo-motor tracking.Although this does not rule out the view that relativephase production is constrained by relative phase perception,the mechanism that would be responsible for thisphenomenon still has to be established.Acknowledgments This work was founded by The AustralianResearch Council, a University of Queensland Travel Award, andsupport from the UMR Mouvement et Perception, CNRS-Universitéde la Méditerranée, Marseille, France.ReferencesBingham GP (2001) A perceptually driven dynamical model ofrhythmic limb movement and bimanual coordination. In:Proceedings of the 23rd annual conference of the cognitivescience society. 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