{"title": "A Computational Mechanism to Account for Averaged Modified Hand Trajectories", "book": "Advances in Neural Information Processing Systems", "page_first": 619, "page_last": 626, "abstract": null, "full_text": "A Computational Mechanism To Account For \n\nAveraged Modified Hand Trajectories \n\nEalan A. Henis*and Tamar Flash \n\nDepartment of Applied Mathematics and Computer Science \n\nThe Weizmann Institute of Science \n\nRehovot 76100, Israel \n\nAbstract \n\nUsing the double-step target displacement paradigm the mechanisms un(cid:173)\nderlying arm trajectory modification were investigated. Using short (10-\n110 msec) inter-stimulus intervals the resulting hand motions were initially \ndirected in between the first and second target locations. The kinematic \nfeatures of the modified motions were accounted for by the superposition \nscheme, which involves the vectorial addition of two independent point-to(cid:173)\npoint motion units: one for moving the hand toward an internally specified \nlocation and a second one for moving between that location and the final \ntarget location . The similarity between the inferred internally specified lo(cid:173)\ncations and previously reported measured end-points of the first saccades \nin double-step eye-movement studies may suggest similarities between per(cid:173)\nceived target locations in eye and hand motor control. \n\n1 \n\nINTRODUCTION \n\nThe generation of reaching movements toward unexpectedly displaced targets in(cid:173)\nvolves more complicated planning and control problems than in reaching toward \nstationary ones, since the planning of the trajectory modification must be per(cid:173)\nformed before the initial plan is entirely completed. One possible scheme to modify \na trajectory plan is to abort the rest of the original motion plan, and replace it with \na new one for moving toward the new target location. Another possible modifica-\n\n\u00b7Current address IRCS/GRASP, University of Pennsylvania. \n\n619 \n\n\f620 \n\nHenis and Flash \n\ntion scheme is to superimpose a second plan with the initial one, without aborting \nit. Both schemes are discussed below. \nEarlier studies of reaching movements toward static targets have shown that point(cid:173)\nto-point reaching hand motions follow a roughly straight path, having a typical bell(cid:173)\nshaped velocity profile. The kinematic features ofthese movements were successfully \naccounted for (Figure 1, left) by the minimum-jerk model (Flash & Hogan, 1985). In \nthat model the X -components of hand motions (and analogously the Y -components) \nwere represented by: \n\nVI \no. \n\n! \nB \n..... -+ \n\n-0.2 \n\n--\n\n, \n., \n,., \n\nc \n)\n\nA \n\n~B \nA \nIKlaft \n\nVy \n\no. \n\nb. \n\n(1) \n\nTime \n\n---\n\nFigure 1: The Minimum-jerk Model and The Non-averaged Superposition Scheme \n\nI Computer \n\no. \n\nc \n\u2022 \n\nB \u2022 \n\n100 \n\n.. \n2' \u00ab \n\no \n\n100 \n\nI \n\n\u2022 \n\n\u2022 \n\n.: . \n.-.. \n..... -\n.. \n. .. :: . . \n. . \n. \n.:. :. \n\no \n\no \n\nD msec \n\n500 \n\nFigure 2: The Experimental Setup and The Initial Movement Direction Vs. n \n\n\fA Computational Mechanism ro Accoum for Averaged Modified Hand Trajecrories \n\n621 \n\nwhere tf is the movement duration, and XB -XA is the X-component of movement \namplitude. In a previous study (Henis & Flash, 1989; Flash & Henis, 1991) we have \nused the double-step target displacement paradigm (see below) with inter-stimulus \nintervals (ISIs) of 50-400 msec. Many of the resulting motions were found to be \ninitially directed toward the first target location (non-averagerl) (for larger ISIs a \nlarger percentage of the motions were non-averaged). The kinematic features of \nthese modified motions were successfully accounted for (Figure 1 right) by a super(cid:173)\nposition modification scheme that involves the vectorial addition of two time-shifted \nindependent point-to-point motion units (Equation (1)) that have amplitudes that \ncorrespond to the two target displacements. \n\nIn the present study shorter ISIs of 10-110 msec were used, hence, all target displace(cid:173)\nments occurred before movement initiation. Most of the resulting hand motions \nwere found to be initially directed in between the first and second target locations \n(averaged motions). For increasing values of D, where D = RTI - lSI (RTl is the \nreaction time), the initial motion direction gradually shifted from the direction of \nthe first toward the direction of the second stimulus (Figure 2 right). The averaging \nphenomenon has been previously reported for hand (Van Sonderen et al., 1988) and \neye (Aslin & Shea, 1987; Van Gisbergen et al., 1987) motions. In this work we \nwished to account for the kinematic features of averaged trajectories as well as for \nthe dependence of their initial direction on D. \n\nIt was observed (Van Sonderen et al., 1988) that when the first target displacement \nwas toward the left and the second one was obliquely downwards and to the right \nmost of the resulting motions were averaged. Averaged motions were also induced \nwhen the first target displacement was downwards and the second one was obliquely \nupwards and to the left. In this study we have used similar target displacements. \nFour naive subjects participated in the experiments. The motions were performed \nin the absence of visual feedback from the moving limb. In a typical trial, initially \nthe hand was at rest at a starting position A (Figure 2 left). At t = 0 a visual target \nwas presented at one of two equally probable positions B. It either remained lit \n(control condition, probability 0.4) or was shifted again, following an lSI, to one of \ntwo equally probable positions C (double-step condition, probability 0.3 each). In a \nblock of trials one target configuration was used. Each block consisted of five groups \nof 56 trials, and within each group one lSI pair was used. The five lSI pairs were: \n10 and 80, 20 and 110, 30 and 150, 40 and 200, and 50 and 300 msec. The target \npresentation sequence was randomized, and included appropriate control trials. \n\n2 MODELING RATIONALE AND ANALYSIS \n\n2.1 THE SUPERPOSITION SCHEME \n\nThe superposition scheme for averaged modified motions is based on the vectorial \naddition of two time-shifted independent elemental point-to-point hand motions \nthat obey Equation (1). The first elemental trajectory plan is for moving between \nthe initial hand location and an intermediate location B i , internally specified. This \nplan continues unmodified until its intended completion. The second elemental \ntrajectory plan is for moving between Bi and the final location of the target. The \ndurations of the elemental motions may vary among trials, and are therefore a \n\n\f622 \n\nHenis and Flash \n\npriori unknown. With short ISIs the elemental motion plans may be added (to give \nthe modified plan) preceding movement initiation. Several possibilities for Bi were \nexamined: a) the first location of the stimulus, b) an a priori unknown position, c) \nsame as (b) with Bi constrained to lie on the line connecting the first and second \nlocations of the stimulus, and d) same as (b) with Bi constrained to lie on the \nline of initial movement direction. Version (a) is equivalent to the superposition \nscheme that successfully accounted for non-averaged modified trajectories (Flash & \nHenis, 1991). In versions (b), (c) and (d) it was assumed that due to the quick \ndisplacement of the target, the specification of the end-point for the first motion \nplan may differ from the actual first location of the target. The first motion unit \nwas represented by: \n\nXl(t) = X A + (XB. - XA)(10T3 - 15T4 + 6T5 ), where T = -\n\nt \nTl \n\n. \n\n(2) \n\nIn (2), (XBi - XA) is the X -component of the first unit amplitude. The duration of \nthis unit is denoted by T i , a priori unknown. The expression for Yi (t) was analogous \nto Equation (2). The X-component of the second motion unit was taken to be: \n\nX2(t) = (Xc - XB.)(lOT3 - 15T4 + 6T5 ), where T = \n\nt - t, \ntl - t, \n\nt - t, \n\n= - - . (3) \n\nT2 \n\nIn (3), (Xc - XBJ is the X-component of the amplitude of the second trajectory \nunit. The start and end times of the second unit are denoted by t, and t I, respec(cid:173)\ntively. The duration of the second motion unit T2 = tl-t, is a priori unknown. The \nX -component of the entire modified motion (and similarly for the Y -component) \nwas represented by: \n\n(4) \nThe unknown parameters T 1 , T 2 , BiX and BiY that can best describe the entire \nmeasured trajectory were determined by using least-squares best-fit methods (Mar(cid:173)\nquardt, 1963). This procedure minimized the sum of the position errors between \nthe simulated and measured data points, taking into account (in versions (a), (c) \nand (d)) the assumed constraints on the location B i . \n\n2.2 THE ABORT-REPLAN SCHEME \n\nIn the abort-replan scheme it was assumed that initially a point-to-point trajectory \nplan is generated for moving toward an initial target (Equation (2\u00bb). The same four \ndifferent possibilities for the end-point of the initial motion plan were examined. It \nwas assumed that at some time-instant t, the initial plan is aborted and replaced \nby a new plan for moving between the expected hand position at t = t, and the \nfinal target location. The new motion plan was assumed to be represented by: \n\nXNEW(t) = L:ai(t)i. \n\n5 \n\ni=O \n\n(5) \n\nThe coefficients a3, a4 and a5 were derived by using the the measured values of \nposition, velocity and acceleration at t = t I. For versions (b), (c) and (d) the \nanalysis was performed simultaneously for the X and Y components of the tra(cid:173)\njectory. Choosing a trial Bi and Tl the initial motion plan (Equation (2\u00bb was \n\n\fA Computational Mechanism to Account for Averaged Modified Hand Trajectories \n\n623 \n\ncalculated. Choosing a trial t\" the remaining three unknown coefficients ao, al and \na2 of Equation (5) were calculated using the continuity conditions of the initial and \nnew position, velocity and acceleration at t = t, (method I). Alternatively, these \nthree coefficients were calculated using the corresponding measured values at t = t, \n(method II). To determine the best choice of the unknown parameters B iX , Biy , Tl \nand t, the same least squares methods (Marquardt, 1963) were used as described \nabove. For version (a), for each cartesian component, a point-to-point minimum(cid:173)\njerk trajectory AB was speed-scaled to coincide with the initial part of the measured \nvelocity profile. The time t, of its deviation from the measured speed profile was \nextracted. From t, on, the trajectory was represented by Equation (5). The values \nof ao, al and a2 were derived by using the same least squares methods (method I). \nAlternatively, these values were determined by using the measured position, velocity \nand acceleration at t = t, (method II). \n\n3 RESULTS \n\nThe motions recorded in the control trials were roughly straight with bell-shaped \nspeed profiles. The mean reaction time in these control trials was 367.1 \u00b1 94.6 \nmsec (N = 120). The mean movement time was 574.1 \u00b1 127.0 msec (N = 120). \nThe change in target location elicited a graded movement toward an intermediate \ndirection in between the two target locations, followed by a subsequent motion \ntoward the final target (Figure 3, middle). Occasionally the hand went directly \ntoward the final target location (Figure 3, right). For values of D less than 100 ms \nthe movements were found to be initially directed toward the first target (Figure 3, \nleft). As D increased, the initial direction of the motions gradually shifted (Figure \n2, right) from the direction of the first (non-averaged) toward the direction of the \nsecond (direct) target locations (The initial direction depended on D rather than \non lSI). The mean reaction time to the first stimulus (RTI) was 350.4 \u00b1 93.5 msec \n(N=192). The mean reaction time to the second stimulus (RT2) (inferred from the \nsuperposition version (b)) was 382.8 \u00b1 119.9 msec (N=192) . This value is much \nsmaller than that predicted by successive processing of information: RT2 = 2RTl -\nlSI (Poulton, 1981), and might be indicative of the presence of parallel planning. \nThe mean durations Tl and T2 of the two trajectory units (of superposition version \n(b)) were: 373.0 \u00b1 112.2 and 592.1 \u00b1 98.1 msec (N = 192), respectively. \n\n3.1 MODIFICATION SCHEMES \n\nThe most statistically successful model (Table-I) in accounting for the measured \nmotions was the superposition version (b), which involves an internally specified \nlocation (a priori unknown) for the end-point of the first motion unit. In par(cid:173)\nticular, the averaged initial direction of the measured motions was accounted for. \nSuperposition version (d) was equivalent to version (b). The velocities simulated on \nthe basis the other tested schemes substantially deviated from the measured ones \n(Table 1 and Figure 4). It should be noted that in both the superposition and abort(cid:173)\nreplan versions (b), (c) and (d) there were 4, 3 and 3 unknown parameters. In the \nabort-replan versions (aI) there were 3 unknown parameters, compared to 2 in the \nsuperposition version (a). Hence the relative success of the superposition version \n(b) in accounting for the data was not due to a larger number of free parameters. \n\n\f624 \n\nHenis and Flash \n\nTable 1: Normalized Velocity Deviations and The t-score With SP(b\u00bb \n\nSP(a) \n18.60 \n\u00b1 50.16 \n(4 .711) \n\nSP(b) \n0.035 \n\u00b1 0.036 \n(0 .000) \n\nSP(c) \n0.126 \n\u00b1 0.132 \n(8.465) \n\nSped) \n0.042 \n\u00b1 0.045 \n(1 .546) \n\nAB(aI) \n0.083 \n\u00b1 0.093 \n(6.126) \n\nAB(aU) \n\n0.084 \n\u00b1 0.088 \n(6.559) \n\nAB(bI) AB(bU) \n0.081 \n0.078 \n\u00b1 0.101 \u00b1 0.102 \n(5 .460) (5.050) \n\nAB(d) \n0.084 \n\u00b1 0.108 \n(5.4 78) \n\nAB(clI) \n0.083 \n\u00b1 0.096 \n(5.959) \n\nAB(dI) \n0.082 \n\u00b1 0.097 \n(5.782) \n\nAB(dU) \n0.085 \n\u00b1 0.101 \n(5.935) \n\nNon- Averaqed \n\nAveraged \n\nDirect --\n\nc+~ \n\nc~ \n\n~ ~ \n\nISI\u00b7200 \n0'80 \n\nl~em \n\n1St- 40 \n0\u00b7250 \n\nlSI '50 \n0\u00b7400 \n\nB+ \n\nB+ \n\nB \n\nA \n\n\" ... :~ \n\n0'120 \n\nlSI\u00b7 50 \nO' 280 \n\no \n\nC \n\nb \n\nB \n+ \n\nISI \u00b7 ao \n0 - 450 \n\nB \n\nA \n\n+ ~ \n\nc \n\n+B \n\nA \n\nlc \n\nB+ \n\nSP(h) \n\n\\A \n\nc \n\no~ o. \nf \n\nT .... \n\n\u00b702 \n\n---\n\n100_ \n\nFigure 3: Types of Modified Trajectories \n\n8 \n+ \n\n(+A \n\nAB(hII) \\ \n\nC+~A \nSP(h) \n\ni'-'0\" \n\nAB(bII) \n\nC \n\n\\~tm \n\n+B \n\n0 \n\ni \u2022 \n:.0 \nI \n\n+B \n\nv. \n\nr~ \n1-0 \n\nr.tooP--+----\n\nFigure 4: Representative Simulated Vs . Measured Trajectories \n\n\fA Computational Mechanism to Accoum for Averaged Modified Hand Trajectories \n\n625 \n\n3.2 THE END-POINTS INFERRED FROM SUPERPOSITION (b) \n\nThe mean locations Bi resulting from different trials performed by the same subject \nwere computed by pooling together Bi of movements with the same D \u00b1 15 msec \n(Figure 5 left). For D < 100 msec, the measured motions were non-averaged and \nthe inferred Bi were in the vicinity of the first target. For increasing values of D, \nBi gradually shifted from the first toward the second target location, following a \ntypical path that curved toward the initial hand position. For D 2:: 400 msec, Bi \nwere in the vicinity of the second target location. Since initially the motions are \ndirected toward B i, this gradual shift of Bi as a function of D may account for \nthe observed dependence of the initial direction of motion on D . The locations Bi \nobtained on the basis of the other tested schemes did not show any regular behavior \nas functions of D. \n\n4 DISCUSSION \n\nThis paper presents explicit possible mechanisms to account for the kinematic fea(cid:173)\ntures of averaged modified trajectories. the most statistically successful scheme in \naccounting for the measured movements involves the vectorial addition of two in(cid:173)\ndependent point-to-point motion units: one for moving between the initial hand \nposition and an internally specified location, and a second one for moving between \nthat location and the final target location. Taken together with previous results for \nnon-averaged modified trajectories (Flash & Renis, 1991), it was shown that the \nsame superposition principle may account for both modified trajectory types. The \ndifferences between the observed types stem from differences in the time available \nto modify the end-point of the first unit . Our simulations have enabled us to infer \nthe locations of the intermediate target locations, which were found to be similar \nto previously reported (Aslin & Shea, 1987) experimentally measured end-points of \nthe first saccades, obtained by using the double-step paradigm (Figure 5 right!). \nThis result may suggests underlying similarities between internally perceived tar(cid:173)\nget locations in eye and hand motor control and may suggest a common \"where\" \ncommand (Gielen et al., 1984; 1990) for both systems. \n\nc ~ + \n\n410\u00b7 \n\n.... -~=--..... \n\nISc\", \n\n.1210 \n\n+ \nB \n\nB \n\nC~A A \n\n~ C~A \n\nB \n\nB \n\nFigure 5: Inferred First Unit End-points and Measured Eye Positions \n\n1 Reprinted with permission from Vision Res., Vol. 27, No. 11, 1925-1942, Aslin, R.N. \n\nand Shea S.L.: The Amplitude And Angle of Saccades to Double-Step Target Displace(cid:173)\nments, Copyright [1987], Pergamon Press pic. \n\n\f626 \n\nHenis and Flash \n\nWhy is the internally specified location dependent on D, which is a parameter \nassociated with both sensory information and motor execution? One possible ex(cid:173)\nplanation is that following the target displacement the effect of the first stimulus on \nthe motion planning decays, and that of the second stimulus becomes larger. These \nchanges may occur in the transformations from the visual to the motor system. A \npurely sensory change in the perceived target location was also proposed (Van Son(cid:173)\nderen et aI., 1988; Becker & Jurgens 1979). Another possibility is that the direction \nof hand motion is internally coded in the motor system (Georgopoulos et al., 1986), \nand it gradually rotates (within the motor system) from the direction of the first \nto the direction of the second target. It is not clear which of these possibilities \nprovides a better explanation for the observations. \nIn the superposition scheme there is no need to keep track of the actual or planned \nkinematic state of the hand. Hence, in contrast to the abort-replan scheme, an \nefference copy of the planned motion is not required. The successful use of motion \nplans expressed in extrapersonal coordinates provides support to the idea that arm \nmovements are internally represented in terms of hand motion through external \nspace. The construction of complex movements from simpler elementary building \nblocks is consistent with a hierarchical organization of the motor system. The \nindependence of the elemental trajectories allows to plan them in parallel. \n\nAcknowledgements \n\nThis research was supported by a grant no. 8800141 from the United-States Israel \nBinational Science Foundation (BSF), Jerusalem, Israel. Tamar Flash is incumbent \nof the Corinne S. Koshland career development chair. \n\nReferences \n\nAslin, R.N. and Shea S.L. (1987). The Amplitude And Angle of Saccades to Double-Step \nTarget Displacements. Vision Res., Vol. 27, No. 11, 1925-1942. \nBecker W. and Jurgens R. (1979). An Analysis of The Saccadic System By Means of \nDouble-Step Stimuli. 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An algorithm for least-squares estimation of non-linear param(cid:173)\neters. J. SIAM, 11, 431-441. \nVan Gisbergen, J.A.M., Van Opstal, A.J. & Roebroek, J.G.H. {1987}. Stimulus-induced \nmidflight modification of saccade trajectories. In J.K. O'Regan & A. Levy-Schoen (Eds.), \nEye Movements: From Physiology to Cognition, Amsterdam: Elsevier, 27-36. \nVan S~n~eren, J.F., D.eniex: Van Der Gon, J.J. & Gielen, C.C.A.M. (1988). Conditions \ndetenmrung early modificatlon of motor programmes in response to change in target loca(cid:173)\ntion. Exp. Brain Res., 71, 320-328. \n\n\f", "award": [], "sourceid": 486, "authors": [{"given_name": "Ealan", "family_name": "Henis", "institution": null}, {"given_name": "Tamar", "family_name": "Flash", "institution": null}]}