A dynamic theory of coordination of discrete movement - Free

265described by the pattern dynamics, (19). This questionwas briefly discussed for rhythmic movement in Sect. 2in the context of phase resetting analysis (Kay et al.1990). In the case of discrete movement a perturbationof the coordination system Eq. (19) leads to relaxationback towards the phase plane curve (x, v) given by thelimit cycle attractor solution in the presence of behavioralinformation and towards the target fixed point inthe absence of behavioral information. The positionaltrajectory, x(t), itself is not stable, however. This isevident from the simulations shown in Fig. 5, where aperturbed trajectory is compared to an unperturbed one.While the phase plane trajectories of the two solutionsconverge, the spatial trajectories do not converge (due toa shift in the phase of the underlying oscillator).In experiments with monkeys who performed discretemovements towards learned visual targets Bizzixa2.50.0-2.5v0.0-50-10050.0,-g,E0.0-50.0-100.0L9 //"".. _ .--1.0 2.0 time [s Iii/ ..'" "'J-150.0-5.0 ~2.5 0.0 '2.5 5.0ix[cm]Fig. 5a, b. Simulations of (19) at the standard parameter values (25).Here an unperturbed solution (solid lines) is shown together with asolution that was perturbed at t = 1.1 s to assist the movement(velocity advanced: v =-2.3--+-50crn/s). In panel a the spatialtrajectories are shown together with their velocities. Note that there isno tendency to relax back to the unperturbed trajectory. By contrast,the phase plane trajectories shown in panel b converge due torelaxation of the perturbed solutionand colleagues (Bizzi and Abend 1982; Bizzi et al. 1982)found that perturbations assisting the movement (similaras in the simulation of Fig. 5) can lead to relaxationback toward the typical unperturbed trajectory. (Theeffect was particularly clear in deafferented preparations.)In the present framework this means that theunderlying coordination system was not perturbed, sothat the end-effector relaxes to the trajectory defined bythe coordination system. To address such effects wemust explicitly model the relationship between endeffectorand coordination system (cf. Discussion).5 Coordinated Discrete Movement: Model and ResultsAn important experimental result on interlimb coordinationof discrete movement can be expressed as atendency to synchronize the movements of differentlimbs. Kelso et al. (1979a) studied a bimanual task, inwhich each hand moved from a home key to a target.By varying the distance and the precision requirementsfor each target, the index of difficulty in the sense ofFitts' law (Fitts' 1954) could be varied. As a result, themovement times in single hand movement could bevaried by almost a factor of two. If the movements hadto be performed with the two hands simultaneously,one hand to a difficult (far, small) target, the other toan easy (near, large) target, the movement times ofboth hands were almost identical. Essentially, the fasthand was slowed down to move synchronously with theslow hand. Further studies in similar paradigms confirmedthis tendency to synchronize discrete movementsof different limbs and showed the breakdown of synchronizationwhen the intrinsic movement times of thetwo limbs become too dissimilar (Kelso et al. 1983;Marteniuk et al. 1984; Corcos 1984).To model the coordination of two end-effectors, weintroduce a single effector description for each limb.Each structure's collective variables, (xi, vi) withi = l, 2, are assumed to be governed by dynamics of theform of (19):Yc =v iD = --f/, total(Xi,/)i) (27)= --{a 2 + co2i + •[q, ti+Ati](t)Cintent, i}Xi+ 2ajvi - 4bix2vi + 2aibix~ 2 5 -bixi+ x/~irWe introduce a different set of parameters for eachend-effector to express different movement conditionsof the two limbs.The coordination tendency is captured by couplingthe two intrinsic dynamics. A similar coupling structureas used in the earlier work on rhythmic movements(Haken et al. 1985; cf. Sect. 2, (2)) can be used. Unlikein this earlier work, however, we treat here cases inwhich the two limbs are not equivalent but may, forexample, differ in movement amplitude and limit cyclefrequency. In such cases, the coupling functions as usedin (2) are dominated by one component and do notlead to a synchronization tendency. Coupling functions

266that account for the tendency of synchronization also inasymmetric cases can be obtained by normalizing eachcontribution. Such normalization can be based on themovement amplitude, Ai, and peak velocity, Vi.p, asestimated in the harmonic balance approximation:(i = 1, 2)Ae = ~/ b, ' v,,, ~ Io~,lA, (28)or similar measures. The coupled system is then:.~] = v l~)l=--fl, total(Xl'Vl)--k(V-~,p I)V~,; )3C 2 = V 2-t Z Z//)2: --f2, total(X2'V2)--k ( UV~,2p UU~I;)(29)where k and l are coupling constants. In the rrgime ofstable limit cycle solutions, that is, when both oscillatorsare in part (e) of the phase diagram at moderatecoupling strengths, the same results as obtained in theearlier work (Haken et al. t985) are reproduced.To address the coordination of discrete movementwe study the case, where the two components havedifferent movement amplitudes and movement timesindividually. In the model this corresponds to choosingthe parameters of each component such that they correspondto different points within the parameterization ofFiRs' law (Fig. 4). For example, by selecting a small]coT[ value for x~ (with bl chosen correspondingly, cf.Sect. 4), we may endow this component with a smallmovement amplitude and a small movement time (seeFig. 6, top panel, solid line). The other component, x2,may make a large amplitude movement with largermovement time, if its og~-parameter is chosen corre,spondingly (Fig. 6, top panel, dashed line). When thetwo components are coupled, corresponding to thesimultaneous performance of the discrete movements,we observe the synchronization of the movements (Fig.6, bottom panel) although the movement amplitudesare unchanged and thus still different. This result is anaccount of Kelso et al. (1979a) observations. (We aretaking into account only one spatial component of themovement, e.g., the projection of the movement ontothe line connecting target and initial position. This islegitimate in the absence of any effect in the space curveof the movement.) We remark, that the tendency ofsynchronization persists when in the model the parametersof behavioral information, A t; and Cintent ' i are differentfor the two components (see Fig. 9).If we make the movement conditions more dissimilar,the tendency to synchronization is no longer strongenough to achieve complete simultaneity and a breakdownof synchronization is observed. In Fig. 7 the twox [cml10 ....... "'",0.0 ~-10 ',x [em]....................... --410 "'"---...,0.0 ~,-lo1.0 2.0 time [s]iii I1.0 2.0 time [s]IFig. 6. Top panel: Two components in discrete movement withdifferent amplitudes and movement times. The solid line is a solutionat the standard parameter set [(25), leading to movement amplitudeA = 6.85cm and (half-distance) movement time To=0.53 sl. Thedashed line is a solution with larger amplitude and movement time[m2= _5Hz 2 and b =0.0794Hz/cm 2, leading to A =25cm andT o = 0.77 s]. Both movements are initiated at t = 1.0. The two componentsare uncoupled in the top panel corresponding to individualmovements. In the bottom panel the same conditions prevail, but thecomponents are coupled (k =-40Hz2), corresponding to simultaneousperformance of the movements. Note the almost perfect synchronizationof the movementsx [ore20 ........................100.0 '~K_-10-20 1.0x Eoml20 ..........................10 "",0.0-i0-20 1.02 0 ." .... time Is]Fig. 7. Similar to Fig. 6, but the second component (dashed line) hasparameter values ~o 2 = -5.5 Hz (b 2 = 0.036 Hz/cm2), so that its amplitudeand movement time are even further removed from those ofthe first component in individual movement (top panel). When coupled(bottom panel) with the same strength as in Fig. 6, the twocomponents still tend to synchronize, but do not achieve the samedegree of simultaneity

267individual movements differ more strongly in movementamplitude and hence movement time (top panel)and evidently their synchronization is less complete(bottom panel) than in the case of Fig. 6. This accountsfor the gradual breakdown of synchronization reportedin Marteniuk et al. (1984) and Corcos (1984), as themovement conditions for the two components were••lO0.0-10100.0-10Xltcm]•100.0-10100.0-101.0 2.0 time [s]1.0 2.0 time [s]1.012.0 time [s]1.0 2.0 time [s]Fig. 8a, b. The coordinated movements (with coupling) shown in Fig.6 are reproduced as solid hnes: the small amplitude component in thetop, and the large amplitude component in the bottom panel. Thedashed lines show simulations at the same parameter values in whichone component was perturbed at t = 1.2 s. In panel a of the figure theintrinsically faster component x~ (top) is perturbed to assist themovement (xl ~ 3 cm ~ 2 cm, v ~ - 2 cm/s ~ - 4 cm/s), leading to aspeeding up of the unperturbed component to restore synchronization.In panel b the intrinsically slower component x2 (bottom) isperturbed to delay its movement (v I ~ -8 em/s--, +50 cm/s) leadingto a slowing down of the unperturbed component again to restoresynchronization3-made more dissimilar. A similar breakdown of synchronizationwas observed by Kelso et al. (1983), when theyplaced an obstacle in the movement path of one componentso that its space curve and hence movement timebecame much longer than that of the other hand.In the present theoretical account the tendency tosynchronize discrete movement is due to the formationof a stable coordination pattern between the two components.This leads to the prediction that a remotecompensatory reaction occurs in one component, if theother component is perturbed. (In light of previousdiscussion it is clear that the perturbation must besufficiently strong to affect the nervous system level ofcoordination modelled here.) Moreover, the remotecompensatory responses are such as to restore thecoordination pattern, that is, to restore synchronization.We demonstrate this phenomenon for the modelin Fig. 8. The two components perform movementswith different movement amplitudes, but identicalmovement times due to coupling (solid lines of Fig. 8are the same trajectories as shown before in the lowerpanel of Fig. 6). In Fig. 8a we perturb component x~ toassist its movement (top panel, dashed line). We observean effect in the unperturbed component (bottompanel, dashed line): this component is also advanced inits movement, that means, the compensatory responsetries to restore synchronization. Figure 8b shows thesame effect when we perturb the slower component, x2,delaying its movement (lower panel, dashed line). Inthis case the effect in the other component is to movemore slowly (top panel, dashed line), again in thedirection of restoring synchronization.It would be very interesting to try to observe such aphenomenon in experiment, because this prediction isdue to the conceptual structure of the present theoryrather than to the detailed modelling assumptions. Experimentson perturbations of the movements of articulatorsin repetitive speech contain hints at such aneffect. For example, Kelso et al. (1984) perturbed thejaw during various utterances and observed fast compensatoryreactions in the upper lip if it was functionallynecessary to achieve final lip closure. Recently,Gracco and Abbs (1988) showed, that in similar situationsthe compensatory movement of the remote articulatoris such as to restore the normal relative timing ofdifferent articulators.Finally, we examine the consequences of the nonlinearcoupling measured by the coupling coefficient/, thatwas introduced to account for anti-phase locking inrhythmic movement (see Haken et al. 1985). Whatcould anti-phase locking mean in the case of discretemovement? The attraction to a relative timing correspondingto anti-phase locking leads to a tendency toperform two movements sequentially. Thus, if two discretemovements are initiated with sufficient delay (inthe model: to, ~ different from to. 2), the movement timeof the delayed movement increases to make the movementoccur with less temporal overlap. In Fig. 9 weshow two components moving with the same amplitude(standard parameter set, (25)), either individually (i.e.,uncoupled: solid lines) or together (i.e., coupled: dashed

268Xl[cml5coordination pattern so that the predictions about remotecompensatory responses arise analogously as forthe synchronization tendency.0.0-5~2[cm50.0-51.0i1.0J2.0itime [s]2.0 time [s]Fig. 9. Two identical components perform movements with the sameamplitude (standard parameter set, but At = 0.5 to allow for largerdelays). The solid lines represent the same solution with movementinitiation at to, L= 1.0 s (top) and to, 2 = 1.2 s (left trace in bottompanel). A second simulation is shown in the same panels withidentical first component and movement initiation of the secondcomponent at t~, 2 = 1.4 s (right trace in bottom panel). If the twocomponents are coupled (dashed lines) with the full nonlinear coupling(k = - 10 Hz 2, l = 10 Hz2), the tendencies to synchronization orto sequentialization can be observed depending on the delay betweenthe movement initiations of the two components. For small delay(=0.2 s: left dashed traces) the synchronization tendency prevailsleading to shorter movement time of the delayed component. Forlarge delay (=0.4 s: right dashed traces) the sequentialization tendencyprevails leading to larger movement time of the delayedcomponentlines). If the onset of behavioral information of thesecond component (bottom panel) is delayed by a smallamount compared to the first component (bottompanel, left solid line), the synchronization tendencyshortens the movement time of the second componentto restore synchronizity (bottom panel, left dashedline). If this delay becomes sufficiently large (bottompanel, right solid line), however, the tendency to sequentialmovement leads to a larger movement time ofthe second component, so that this movement overlapslittle with the movement of the first component.The numerical simulation suggests an experimentalparadigm to test whether such a coordination tendencyof sequencial movement exists: In bimanual, fast movement,"go" signals are given separately for each limb.The delay between these go signals is manipulatedsystematically and movement time is measured. Theprediction is that for small delay, the movements willtend to synchronize, while for larger delay, the movementtimes will adjust so as to produce more sequentialmovement. (Because movement times and reactiontimes can be measured separately, complications relatingto the information processing, e.g., psychologicalrefractory period, may be kept separate from the issueof coordination.) In this theory, such a tendency tomove sequentially is also due to the formation of aL6 ConclusionWe have shown how a dynamic theory of discretemovement can be constructed on the basis of localdynamic descriptions of the initial and target posturalstates. The intention to move is captured as behavioralinformation, a part of the dynamics that stabilizes alimit cycle over a limited period of time. The timing ofthe movement is then largely determined by the intrinsicdynamics in the absence of an intention to move.We have shown that movement time-amplitude relationshipscan be captured in terms of relations amongthe parameters of the dynamics. We discussed the roleof temporal stability in such a theory. The coordinationof discrete movement was understood as the result ofcoupling different component dynamics in a fashionsimilar to the case of rhythmic movement coordination(Haken et al. 1985). Such coupling accounts for theexperimentally established tendency to synchronize discretemovements even when the movements of theindividual components have different intrinsic movementtimes. The stability of the coordination pattern inthis theory gives rise to the prediction that remotecompensatory responses occur when one component isperturbed. These responses will be such as to restore thestable relative timing of the two components. Furthermore,we proposed a discrete analogon to anti-phaselocked rhythmic movement: the tendency to move sequentiallyis manifest if movement initiation of onecomponent lags sufficiently that of the other component.We proposed an experiment to empirically identifysuch a tendency to movement sequentially. It maytranspire from these results, that a unified dynamictheory of discrete and rhythmic movement can not onlydescribe various relationships among movement parameters,but, through the important concept of temporalstability, enables prediction in suitable model systems.Comparison with experiment led us repeatedly todiscuss the nature of the coordination pattern that weobserve, and the nature of the collective variables onwhich the theory is founded. In particular, experimentson phase resetting in rhythmic movement (Kay et al.1990) and on stability of the trajectory in discretemovement (Bizzi and Abend 1982; Bizzi et al. 1982)show that it is necessary to distinguish the level ofend-effector position and velocity from the level ofnervous system control of these variables. The collectivevariables that we use in the present dynamic theory mustbe correctly interpreted as representing the coordinationactivity of the nervous system. As long as the relation ofthe nervous system activity to the end-effector kinematicsis unperturbed, these latter can be used to measurethe former. When mechanical perturbations are applied,however, the two levels appear as different: sufficientlyweak perturbations may not affect the nervous systemlevel of control at all, leading to a relaxation of the

269end-effectors to the centrally defined trajectory. Whenthe perturbation is sufficiently strong, however, thenervous system may be perturbed as well and we maysee its relaxation to its stable coordination patterns.With respect to such perturbations, therefore, a morecomplete theory which explicitly describes the two relevantlevels is necessary. Such a theory can be based, forexample, on the functional model (Feldman 1966, 1974)of muscle control, such that the trajectory as describedby pattern dynamics Eq. (19) is essentially a trajectoryof equilibrium points (cf. also Flash 1987).We have shown how dynamics and stability areimportant concepts to understand the coordination activityof the central nervous system in discrete movement.Maybe, this could be viewed as an element ofwhat Erich von Hoist called "die Mathematik der nerv6senOrdnungsleistung" (the mathematics of the orderinginfluence of the nervous system; see von Hoist1948).Acknowledgements. Research supported by NIMH (NeurosciencesResearch Branch) Grant MH 42900-01 and the U.S. Office of NavalResearch (Grant No. N00014-88-J-1191). I would like to thank PierZanone for a critical reading of the manuscript as well as AnatolFeldman and Scott Kelso for discussion.ReferencesAbend W, Bizzi E, Morasso P (1982) Human arm trajectory formation.Brain 105:331-348Beggs WDA, Howarth CI (1972) The movement of the hand towardsa target. Q J Exp Psychol 24:448-453Bizzi E, Abend W (1982) Posture control and trajectory formation insingle and multiple joint ann movements. In: Desmedt JE (ed)Brain and spinal mechanisms of movement control in man.Raven Press, New YorkBizzi E, Polit A, Morasso P (1976) Mechanisms underlying achievementof final head position. J Neurophysiol 39:435-444Bizzi E, Accornero N, Chapple N, Hogan N (1982) Arm trajectoryformation in monkeys. 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