E.x_'mental BranResearch

Exp Brain Res (1987) 67:253-269

9 Springer-Verlag 1987

Trajectory control in targeted force impulses III. Compensatory adjustments for initial errors J. Gordon and C. Ghez Center for Neurobiology and Behavior, New York State Psychiatric Institute, College of Physicians and Surgeons, Columbia University, 722 W. 168th Street, New York, NY 10032, USA

Summary. In the preceding study (Gordon and Ghez 1987), we showed that accurately targeted isometric force impulses produced by human subjects are governed by a pulse height control policy. Different peak forces were achieved by modulating the rate of rise of force while force rise time was maintained close to a constant value and independent of peak force. An early measure of the rate of rise of force, peak daF/dt2, was scaled to the required force (target amplitude) and highly predictive of the peak force achieved. In six subjects examined, peak d2F/dt 2 accounted for between 70% and 96% of the total variance in peak force. In the present study, we further examined these targeted responses to determine whether the residual variability not predicted by peak d2F/dt2 could be accounted for by adjustments to the force trajectories which compensated for initial errors in the scaling of the d2F/dt 2. A statistical model of the determinants of peak force was tested. This model included two paths by which the target amplitude could independently influence the peak force achieved. The first path was preprogrammed pulse height control. In this path, target amplitude determined the initial rate of rise of force (peak d2F/dt 2) which in turn determined the final peak force achieved. The second path was an independent influence of errors in the initial scaling of peak d2F/dt2 on peak force. Multiple regression analysis was performed on trajectory variables within the sets of responses by each subject in each condition to determine whether the second path contributed significantly to explaining the variance in peak force. In each subject and condition, there was a significant independent influence of error in d2F/dt2 on peak force, and the direction of this effect was to decrease the magnitudes of peak force errors. These compensatory adjustments accounted for between Offprint requests to: C. Ghez (address see above)

1% and 14% of the total variance in peak force. Further multiple regression analyses revealed that inappropriate scaling of the initial phase of the trajectories was compensated for by shortening or prolonging the force rise time. These trajectory adjustments were in turn implemented by modulation of the timing and magnitude of the contractions in the agonist and antagonist muscles that produced the force trajectories. Because these compensatory adjustments were evident in the EMG pattern at latencies too short to be accounted for by peripheral feedback, we assume that they depend on internal monitoring of the unfolding neural commands. These internal feedback processes act in parallel with the programmed commands, both determining the force trajectory.

Key words: Human subjects - Isometric - Trajectory control - Motor programs - Corrections - Accuracy Agonist-antagonist EMG pattern

Introduction In the preceding paper (Gordon and Ghez 1987), we showed that, under two distinct sets of instructions (Fast and Accurate), subjects adopt a pulse height control policy to match the amplitude of force impulses to different target forces: This control policy implements a general rule: different peak forces are achieved by varying the rate of rise of force while the rise time of force is regulated about a constant value which is dependent upon the instruction set. We have argued that this control policy reflects the operation of a motor program and is strategic, chosen by the nervous system to optimize performance, rather than arising from experimental or mechanical constraints. In the six subjects

254 examined, the operation of this pulse height control policy accounted for between 71% and 96% of the variability in the peak force of their responses. We now turn to the question of whether some portion of the residual variability can be accounted for by compensatory adjustments to the trajectory. Movements aimed to a target are usually considered to have two phases: a preplanned initial impulse followed by a series of corrective adjustments. However, since the early studies of Woodworth (1899), it has generally been considered that very rapid responses have only a single impulsive phase, because there is no time for corrections based on peripheral feedback. According to this view, the entire trajectory of a rapid movement is preprogrammed. Such responses have been termed "continuous" (Brooks 1974), since the trajectories show no overt deflections indicating the presence of corrections. It has also been suggested that, once released, the motor programs responsible for controlling rapid movements must be run off as a unit (Desmedt and Godaux 1978) and that trajectory corrections can only be implemented when the program is completed, after the passage of about a reaction time (Craik 1948; Taylor and Birmingham 1948; Welford 1952; Poulton 1981). In recent years, however, it has become clear that even rapid movements may show overt corrections during their course. Corrections can occur in response to changes in target (Megaw 1974; Georgopolous et al. 1981; Vicario and Ghez 1984) and in response to initial directional errors (Higgins and Angel 1970; Cooke and Diggles 1984). Initial trajectory errors not due to an external perturbation are presumed to be detected by internal monitoring of neural signals (Higgins and Angel 1970; Vicario and Ghez 1984) because the corrections occur too soon to be accounted for by peripheral feedback. Such results indicate that trajectory control can, to some extent, be governed by coincident, parallel processes. Corrections must be prepared and implemented while the effects of the motor program are unfolding. These studies, however, which depended upon manifest trajectory deflections to disclose later compensatory adjustments, did not determine whether corrections might occur in continuous responses. In order to investigate whether trajectory adjustments are present even in the absence of overt corrections, we have relied upon an analysis of the variability of targeted force trajectories. Our results (Gordon and Ghez 1987) showed that the peak force achieved is largely, but not completely, predicted by the initial dynamics of the trajectory, as measured by the peak of the second time derivative of force (d2F/dt2). Thus, 4% to 29% of the variance in peak

force was not accounted for by the peak d2F/dt 2. Moreover, the peak d2F/dt 2 was not as faithful a representation of the target amplitude as the subsequent peak force. This led us to ask whether adjustments to the later phase of the trajectory might compensate for errors in the initial trajectory that were evident in the peak d2F/dt 2, We reasoned that deviations from linear scaling of the peak d2F/dt 2 to the target can be considered as errors. These deviations represent differences between the initial trajectory dynamics of individual responses and those required by a stereotyped pulse height control model to achieve the target force. The possible relationships between peak d2F/dt 2 and peak force are shown schematically in Fig. 1. On the left, three superimposed traces are shown representing the time course of d2F/dt2. Only the middle trace is shown completely (in black). The other traces, representing erroneously larger (e+, gray) and smaller (e-, hatched) dZF/dt2 peaks, are shown only up to their peaks. According to pulse height control (upper right), the peak forces achieved will be proportional to the peak d2F/dt 2. Thus, if the peak d2F/dt 2 on a given trial is greater than the peak dZF/dt2 predicted by pulse height control for that target amplitude, the peak force can be expected to overshoot the target force. However, ff compensatory adjustments occur, as shown in the bottom part of the figure, the correct peak force may still be achieved by changing the force "deceleration". As a consequence, the force rise time must be altered. Initially overshot responses (~+) must be truncated and undershot responses (e-) extended in time. Compensatory adjustments, therefore, will be apparent as deviations of the force rise time from the regulated value. If we find that these deviations occur as in the lower part of the figure, we can conclude that compensatory adjustments have occurred. Of course, the variance in peak force that is not explained by peak d2F/dt ~ is not necessarily due to compensatory adjustments to trajectory. It could arise, instead, from random errors, either measurement errors or noise in the implementation of the motor program. In order to determine the sources of the variance in peak force, we have subjected the response trajectories and the associated EMG patterns made by subjects in the previous experiments to a multiple regression analysis. Using this statistical method and our measure of the contribution of motor programming, we were able to partition the overall variance in peak force into separate components determined by two factors: preplanned pulse height trajectory control and compensatory adjustments. This approach allowed us to test whether a causal model that includes both of these factors accounts for significantly more of the total variance

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in peak force than a model that only includes a preplanned trajectory. The same statistical procedure was then used to partition the variance in other trajectory variables (i.e. force rise time) and in variables of the EMG pattern (i.e. burst magnitudes and durations) to determine how compensatory adjustments were implemented. These results formed part of a doctoral thesis (Gordon 1985) and some of the data have appeared as an abstract (Gordon and Ghez 1985).

The statistical model of the determinants of peak force that was tested in this study is illustrated in Fig. 2. According to this model, the target can influence the peak force through two paths. In the first path (black arrows), the target influences the peak force through its effect on the initial rate of rise of force (peak d2F/dtZ). The squared correlation coefficient describing the relationship between peak d2F/dt2 and peak force (r2v.i in Fig. 1) represents the proportion of the variance in peak force that can be explained by variation in peak dZF/dt2. Therefore, r2r.1 measures the degree to which the individual trajectories are linear multiples of a stereotyped form. The squared correlation coefficient, fll.2, indicates the extent to which the peak d2F/dt 2 itself is determined by the required peak force (target amplitude). Thus, the path from target through peak d2F/dt2 to peak force in Fig. 1 represents the implementation of targeted pulse height control. The second path (hatched arrow) represents the corrective influence of the target amplitude on peak force. A corrective effect is defined as an influence of target amplitude on the peak force that: a. is independent of the target's effect on the peak dZF/dt2, and b. acts to reduce the error in the peak force. The method for assessing whether a corrective effect was present involved a statistical procedure which will be presented in the next section. Briefly, the strength of the independent effect of the target on peak force is its additional contribution to the prediction of peak force after controlling for the indirect effect of the target on peak force through peak d2F/dt2 (R2r.I2 - r2v.l). The direction of the target's influence, that is, whether it acts to reduce or increase the error, is determined by the sign of its regression coefficient in a multiple regression in which peak force is predicted by both peak d~F/dt2 and target amplitude. If the regression coefficient is positive, the effect is compensatory because, for a given peak d~F/dt2, larger targets will be associated with appropriately higher peak forces. A negative regression coefficient would indicate an anticompensatory effect, that is, one which increases the error associated with a poorly scaled initial trajectory.

Methods

Statistical procedures

The data used for this study were derived from the experiments described in the preceding paper (Gordon and Ghez 1987).

The magnitude of the influence of target amplitude on peak force that is independent of pulse height control was computed in the

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Fig, 1. Schematic diagram of the possible relationships between peak dZF/dt2 and peak force. Left side of figure shows three superimposed d2F/dt2 traces. The middle trace (in black) shows the peak d2F/dt2 which will produce an appropriately targeted peak force, according to the overall relationship between peak dZF/dtz and peak force. The other traces are either overshot (e+, gray) or undershot e-, hatched). The right side of the figure shows two possible results. On top, marked "Pulse Height Control", the overshot and undershot dZF/dt2 peaks lead to correspondingly overshot (in gray) and undershot (hatched) peak forces. The appropriately targeted peak d2F/dt 2 (in black) leads to a correctly targeted peak force. On the bottom, marked "Compensation for Errors (e) in d2F/dt2'', all three traces accurately achieve the targeted peak force. The initially overshot trace (gray) has been cut off sooner and has a shorter rise time, while the initially undershot trace (hatched) has a prolonged rise time

256 following manner. First, peak d2F/dt2 was used to predict the peak force, according to the following linear transformation:

Y = a + bXl

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where Y = peak force, X1 = peak dZF/dt2, and a and b are constants (a is the Y-intercept, and b is the regression coefficient expressing the change in Y for a given change in X1). The squared correlation coefficient (r2y.l) was computed for this relationship. Next, target amplitude was added to the regression, yielding the following equation:

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where Y = peak force, )(1 = peak d2F/dt 2, X2 = target amplitude, a is the new intercept, and b~ and b2 are the new regression coefficients. R2y.12is the squared multiple correlation coefficient which represents the proportion of variance in peak force that can be explained by a linear combination of peak d2F/dt2 and target amplitude. The additional variance accounted for by target amplitude is equal to the difference between that proportion of the variance accounted for by the combination of both variables and that accounted for by the peak d2F/dt 2 alone (R2y.i2 - PY.I). The statistical test for corrective components in the response was whether the target amplitude significantly increased the proportion of variance in peak force explained when it was added to the regression equation. In other words, did the direct path from target to peak force account for a significant increment (R2y.12 - r2y.i) in the proportion of the variance in peak force? The null hypothesis was that the target amplitude did not significantly add to the prediction of peak force when added to the equation. If the null hypothesis could not be rejected, deviations from pulse height control would have to be considered random with respect to the target and would therefore not serve to compensate for initial errors. Separate multiple regression analyses were carried out for sets of responses made by individual subjects under the Fast and Accurate conditions. An F statistic was computed for each subject by condition. This statistic was used to test the significance of the increment in variance accounted for by the addition of the target variable to the regression equation (Pedhazur 1982). When target amplitude is added to the regression equation, as described above, an increase in the proportion of peak force variance accounted for reflects the influence of the target amplitude on peak force after controlling for the variance that target amplitude shares with peak d2F/dt2. The remaining variance, that is, the variance in target amplitude not shared with peak daF/dt 2 (1.00 - r21.2) can be taken as a measure of the variable error in the initial planned output, since it reflects deviations from linear scaling of peak d2F/dt2 to the target. Therefore, the amount of additional variance in peak force explained when target amplitude is added to the regression equation indicates the degree to which initial trajectory errors are correlated with the actual peak forces achieved. If there is a significant added contribution of target amplitude, then it can be concluded that the error in the peak d2F/dt2 affects the peak force independently of the effect of peak d2F/dt2 itself. If the null hypothesis is rejected, then the target amplitude contributes to the peak force independently of its indirect effect on peak force through the peak d2F/dt z. This means that deviations from pulse height control are not random. Such non-random deviations could either act to compensate for initial errors or could further degrade accuracy. Regression coefficients in a multiple regression express the relation between the particular predictor variable and the dependent variable when all other predictor variables are held constant. (Standardized regression coefficients, termed [3, express this relation in terms of the standard deviation of each variable.) If the independent effect of target does compensate for initial errors then, for any given peak d2F/dt2, the

larger the target amplitude the greater will be the peak force achieved. A corrective effect will therefore be reflected in a positive regression coefficient for the target variable (b2) when both peak d2F/dt2 and target amplitude are used to predict peak force as in (2). If the independent effect of target is anticompensatory (i.e., if later errors act in the same direction as initial trajectory errors), then the regression coefficient (b2) will be negative. The test for significance of the target's regression coefficient is equivalent to the previously described F test of the increment in variance accounted for by the target. To summarize, if target amplitude adds significantly to the prediction of peak force, and if its regression coefficient in the multiple regression is positive, there is a compensatory, or corrective, effect. To further analyze variability in peak force, we also calculated partial correlation coefficients. A partial correlation coefficient expresses the relationship between two variables after removing from both of them the effects of their linear relationships to another variable (Pedhazur 1982). For example, the partial correlation coefficient between peak force and target amplitude with peak d2F/dt2 partialled out expresses the degree of correlation between the variance in peak force not linearly related to peak d2F/ dt z (deviation from pulse height control) and the variance in target amplitude not linearly related to peak d2F/dt2 (errors of initial trajectory). Thus, this particular partial correlation coefficient provides a measure of the degree to which deviations from pulse height control are linearly related to initial trajectory errors. In order to understand how corrections were implemented, the same statistical methods were used to analyze the variance in the rise time of force as well as in the magnitude and durations of the agonist and antagonist EMG bursts. Each of these variables was used as the dependent variable in a multiple regression analysis in which peak d2F/dt2 was entered first followed by target amplitude. As explained above, the increment in R 2 obtained by this procedure estimates the degree to which initial errors in scaling of the d2F/dt~ are correlated with the particular dependent variable.

Results Compensatory adjustments: analysis o f trajectory variables I n o r d e r to i l l u s t r a t e h o w f o r c e t r a j e c t o r i e s a r e i n f l u e n c e d b y b o t h t h e initial p l a n n e d t r a j e c t o r y ( p e a k d2F/dt 2) a n d t h e i n t e n d e d e n d p o i n t ( t a r g e t a m p l i t u d e ) , w e will first p r e s e n t r e s u l t s f r o m a single s u b j e c t a n d t h e n discuss d i f f e r e n c e s b e t w e e n s u b j e c t s and conditions.

Independent influences o f the target on p e a k force: single subject F i g u r e 3 A is a p l o t o f p e a k f o r c e as a f u n c t i o n o f p e a k d2F/dt z f o r all t h e r e s p o n s e s o f s u b j e c t $3 in t h e F a s t c o n d i t i o n . W h e n all t h e r e s p o n s e s a r e c o n s i d e r e d t o g e t h e r , t h e p e a k d2F/dt 2 is a s t r o n g p r e d i c t o r o f t h e p e a k f o r c e a c h i e v e d (r = 0.91); t h e s l o p e o f this r e l a t i o n s h i p is i n d i c a t e d by t h e d o t t e d l i n e p a s s i n g t h r o u g h all t h e d a t a p o i n t s . H o w e v e r , w h e n t h e

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responses to targets of each amplitude are considered separately, the slopes of the sets of relationships between peak d2F/dt 2 and peak force within each target amplitude (solid regression lines) are lower than that of the entire ensemble of trials. The result is that the errors in peak force around the three targets (horizontal dashed lines in Fig. 3A) are less than they would have been if the peak force had been deter-

mined solely by the peak d2F/dt 2. Figure 3A also shows that, for any given peak d2F/dtz, the peak force achieved is greater on average when the target amplitude is higher. For the data illustrated in Fig. 3A, the percentage of variance in peak force that could be explained was significantly increased (from 83% to 93%) when target amplitude was included along with peak dZF/dt2 in the regression equation predicting peak force. The standardized regression coefficient (~) for target amplitude was 0.52. This coefficient can be interpreted as a ratio which gives a measure of the independent effect of target amplitude. In this case, when peak d2F/dt 2 is held constant, an increase of one standard deviation in target amplitude is associated with an increase in peak force of approximately onehalf of its standard deviation. This independent effect of target amplitude is compensatory, because, for each peak dZF/dt2, greater peak forces will be achieved for larger target amplitudes. A related statistical measure of the independent influence of the target amplitude on peak force when peak d2F/dt 2 is held constant is the partial correlation coefficient between target amplitude and peak force with the effect of peak d2F/dt 2 removed from both variables. This is equivalent to a correlation between the error in the initial trajectory and the variance in peak force that is not explained by the peak d2F/dt 2. In this case, the partial correlation coefficient is 0.76, indicating that 58% of the variance in peak force not explained by pulse height control can be accounted for by the direct effect of the target amplitude on the peak force (that is, the effect of target independent of its influence on peak dZF/dtZ). Both these measures (~ and partial correlation coefficient) indicate that the later phase of the force trajectory manifests a significant compensation for early trajectory errors.

Compensatory adjustment of force r&e time: single subject For a given peak d2F/dt 2, the longer the rise time of a force impulse, the greater the peak force should be. Therefore, one reason that the peak dZF/dt2 is not a perfect predictor of the peak force achieved is that the actual force rise time varies independently of peak force. For the responses shown in Fig. 3A, when rise time was added to the regression equation predicting peak force after peak d2F/dt 2 had first been entered, the percentage of variance in peak force explained by the linear combination of the two variables increased significantly (from 83% to 91%). The percentage of variance in peak force explained by rise time when peak dZF/dt2 is held constant was

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