End Points of Planar Reaching Movements Are Disrupted by Small Force Pulses: An Evaluation of the Hypothesis of Equifinality

Florin C. Popescu and W. Zev Rymer

Northwestern University, Evanston 60208; and Rehabilitation Institute of Chicago, Chicago, Illinois 60611


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Popescu, Florin C. and W. Zev Rymer. End Points of Planar Reaching Movements Are Disrupted by Small Force Pulses: An Evaluation of the Hypothesis of Equifinality. J. Neurophysiol. 84: 2670-2679, 2000. A single force pulse was applied unexpectedly to the arms of five normal human subjects during nonvisually guided planar reaching movements of 10-cm amplitude. The pulse was applied by a powered manipulandum in a direction perpendicular to the motion of the hand, which gripped the manipulandum via a handle at the beginning, at the middle, or toward the end the movement. It was small and brief (10 N, 10 ms), so that it was barely perceptible. We found that the end points of the perturbed motions were systematically different from those of the unperturbed movements. This difference, dubbed "terminal error," averaged 14.4 ± 9.8% (mean ± SD) of the movement distance. The terminal error was not necessarily in the direction of the perturbation, although it was affected by it, and it did not decrease significantly with practice. For example, while perturbations involving elbow extension resulted in a statistically significant shift in mean end-point and target-acquisition frequency, the flexion perturbations were not clearly affected. We argue that this error distribution is inconsistent with the "equilibrium point hypothesis" (EPH), which predicts minimal terminal error is determined primarily by the variance in the command signal itself, a property referred to as "equifinality." This property reputedly derives from the "spring-like" properties of muscle and is enhanced by reflexes. To ensure that terminal errors were not due to mid-course voluntary corrections, we only accepted trials in which the final position was already established before such a voluntary response to the perturbation could have begun, that is, in a time interval shorter than the minimum reaction time (RT) for that subject. This RT was estimated for each subject in supplementary experiments in which the subject was instructed to move to a new target if perturbed and to the old target if no perturbation was detected. These RT movements were found to either stop or slow greatly at the original target, then re-accelerate to the new one. The average latency of this second motion was used to estimate the voluntary RT for each subject (316 ms mean). Additionally, we found that the hand neither exerted target-oriented force against the handle nor drifted toward the desired end point just before coming to rest, making it unlikely that the mechanical properties of the manipulandum prevented the hand from reaching its intended target.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

From the earliest studies of motor control, special significance was granted to the elastic behavior of muscle. Experiments as early as the 19th century produced now-familiar force-length curves, which showed that muscle behaves like a nonlinear spring (Fick 1867). This led early theoreticians to assign a regulatory role to the "spring-like" behavior of muscle (Bernstein 1935). Other investigators have sought to integrate the control of posture and movement by emphasizing the elastic properties of active muscle (Asatryan and Feldman 1965; Bizzi et al. 1978; Feldman et al. 1990). In particular, these authors proposed a model for generating voluntary movements in which the central command specifying a movement simply specified a new postural state. The visco-elastic muscle and reflex properties produce physiologically realistic transitions from the initial to the desired state. In theory, this sequence can also be maintained in the face of a large class of external perturbations because of the conservative properties of the muscle and reflex "springs."

This manner of controlling limb movement by simply specifying desired end-point position, known as the equilibrium point hypothesis (EPH) provides an elegant explanation of postural regulation, which implicitly regulates joint position about some defined joint angle. Furthermore movement control, which requires systematic changes in the commanded joint angle, can be integrated with postural regulation and hence greatly simplified. The most basic prediction of the EPH is that for a given limb load at rest and a given central command, movement end points will be unaffected either by small, transient perturbations or by variations in the starting point of the motion. This is the property termed "equifinality" (Kelso and Holt 1980). Our present study aims to test the presence of equifinality rigorously by using a series of small, transient perturbations, building on previous studies addressing this behavior in human and nonhuman primate reaching movements.

There have been several pertinent studies indicating that muscle behavior may not be simply spring-like even under full reflexive control. There is evidence that active areflexive muscle is itself not spring-like because a viscous load imposed on muscle in this condition yields a large undershoot in the final position (Lin and Rymer 1998; Rothwell et al. 1982a; Sanes 1986), a finding that contradicts common beliefs about muscle spring-like properties. Furthermore muscle yield (Joyce et al. 1969), which is an abrupt change in stiffness recorded once active muscle is stretched more than a fraction of a millimeter, may provide a mechanism by which even small transient loads can change muscle state, inducing further deviations from spring-like behavior. Although stretch reflex action is able to provide excellent compensation for changes in intrinsic muscle properties, this compensation is often not complete as there are routinely systematic changes in stiffness as a function of perturbation amplitude and direction (Nichols and Houk 1976) and does not necessarily guarantee spring-like behavior (Lackner and Dizio 1992). Given that limb position is governed by both the intrinsic mechanical properties of muscle and by feedback regulation of reflexes, spring-like behavior and any resultant equifinality must also be a property of these two factors, presumably acting in concert with the descending central command.

Our study examines this property of equifinality by applying a small force pulse to the hand during planar arm movements to a visual target and by assessing the resulting changes in movement end point. Our findings are that in the majority of perturbation trials, equifinality is not obeyed, and there are systematic errors in end point of the hand motion induced by the force perturbations. These end-point errors were not a result of failed voluntary corrective movements because the time from perturbation onset to movement end was too short for the RT corrections to be effective. Voluntary corrective movements occurred after a substantial delay, which was measured. We conclude that the EPH hypothesis is not supported by our experimental results and that upper regions of the CNS are involved in effective disturbance rejection in the moving limb in addition to muscle and reflex properties. Portions of this material were presented in abstract form (Popescu and Rymer 1997).


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

SUBJECTS. Five healthy subjects, naive to the aims and methods of the experiment, participated in this study. One subject was tested twice with a 5-mo interval in between the two experiments. A local institutional review board approved the methods of this study for human subjects, and all subjects provided informed consent.

APPARATUS. This study was performed on a 2 degree-of-freedom manipulandum operating in the horizontal plane. Two torque motors (PMJ JR24M4CH) were independently connected to each joint via a parallelogram configuration four-bar linkage (see Fig. 1) powered by two servo-amplifiers (Kollmorgen DCS/2500 100/18/20). Position and velocity measurements were made using optical encoders (Teledyne Gurley 25/045-NB17-IA-PPA-QAR1S) with 17-bit resolution and analog tachometers (PMI W6T), respectively, mounted on the mechanical joint axes. Forces were measured by a 6 df force transducer (ATI F/T Gamma 30/100) mounted on the manipulandum handle in two of the six experiments. The apparatus included a video display monitor mounted in front of the subject. This monitor displayed a screen cursor representing the position of the manipulandum's handle and the targets used for the reaching movements on a one-to-one scale. To avoid fatigue, the arm was supported by a low-friction planar device (Jaeco Orthopaedic Specialties Arm Positioner) at the forearm. The device consisted of a two-link arm made of 3/16-in-diam steel beams and ball bearings. It was attached to the chair. Some subjects elected not to have their arm supported. The apparatus has been used in previously published studies (Conditt et al. 1997).



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Fig. 1. Experimental setup. Subject's position viewed from above together with coordinate frame. Throughout this paper EXT (positive y direction) and FLX refer to the direction of the perturbation, not the direction of movement, which is always right to left (positive x direction). Each arrow represents a perturbation type, a force pulse of 10 N perpendicular to the direction of motion, lasting 10 ms.

Data were acquired by a dedicated Pentium PC (Dell XPS P90), with a high-speed A/D, D/A board (Computer Boards DAS 1602), and large-scale data storage capacity. The position and force data were sampled at 100 Hz.

PROCEDURE. All movements were made with the right hand, had the same starting point, and were directed to a target 10 cm away. There were two successive protocols: the "end-point shift" trials and the "RT" trials.

End-point shift trials

All subjects were given the same instructions: try to reach the target in one natural motion without stiffening your arms. Do not move before the target appears. Try to end the movement within the target within the allotted time. The end-point shift experiment was composed by a group of 768 movements, which were used to determine whether the end-point shifts following a perturbation. The first 768 were composed of a randomly shuffled sequence of the following conditions.

UNPERTURBED VISUALLY GUIDED MOVEMENTS. Movements (384 total) were from the same starting target to the same ending target 10 cm away, from a point in the sagittal plane of the right shoulder toward the left (see Fig. 1). Targets were represented by 1-cm squares on the screen with a smaller (0.2 × 0.2 cm) square as a position-indicating cursor with a 1-to-1 scaling factor. As a reward for speed and accuracy, the end target "exploded" only if the movement was completed within 350 ms, which encouraged subjects to produce movements consistent in duration and if the motion settled within the target, which encouraged accuracy. In all other cases, the targets were unmodified and return movements from ending to starting target were recorded but not used.

UNPERTURBED "BLINDED" MOVEMENTS. There were 192 total movements. The targets remained on the screen but the cursor disappeared at the beginning of the motion. The subjects did not look at the hand but focused on the screen.

PERTURBED BLINDED MOVEMENTS. These movements (192 total) featured a 10 N, 10-ms pulse, applied perpendicular to the direction of motion. Perturbed movements were equally divided into six different conditions. These included the two directions of the perturbation (flexion/extension) and three different "onsets" of the perturbing pulse, applied at 25, 50, and 75% of the way from the start point to the target. Vision of the cursor was not allowed in these trials (see Fig. 1 for a schematic of the protocol and Fig. 2 for typical movements), as in the unperturbed blinded condition described in the preceding text.

RT trials

For tests of RT latency in each subject, we examined 192 additional movements under a slightly revised protocol, such that we could determine which of the end-point shift movements were so fast as to exclude the possibility of voluntary intervention. For these RT tests, we added a third target to the screen, 10 cm directly in the extension direction from the previous ending target (see Figs. 1 and 7 for relevant schematics). The subject was instructed to move to the new target as soon as he/she felt a perturbation pushing them toward the body (flexion), except the subject tested twice who was told in the second experiment to react to any perturbation. If no perturbation was sensed, then the subject was instructed to proceed to the original target. Of these RT trials, 96 were visually guided, 64 were blind (with target removed), and 32 were perturbed in a random sequence---so roughly the same proportion of blinded and perturbed trials were kept as in the end-point shift trials. For these perturbed RT trials, only the 50% (mid-motion) pulse was used (in both directions), with 16 trials perturbed in flexion and 16 in extension, except for the doubly tested subject's second experiment, which used 75% onset pulses.

Movements were accepted if the time between the perturbation onset and the end of the movement (ST) is less than the measured voluntary RT for that subject. The movement is considered to have ended when subsequent motion does not exceed 1% of the desired movement distance (i.e., is confined to a 1-mm square).

DATA FILTERING. Neither the position nor the force was filtered. The reason for this is that the encoder output is practically noiseless and force data used in analysis (forces during the stopping phase of the movement) had frequency content so low as to make 60 Hz or higher electrical noise a nonfactor. Data analysis was performed using MATLAB.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

End-point shift

We compared the end points of movement trajectories of perturbed to unperturbed planar voluntary movements. Figure 2 shows a typical perturbed trace (A), averaged blind unperturbed (B), and averaged blind perturbed (C) trajectories for one particular perturbation type in one subject. Note the similarity between the typical and average perturbed trajectories (A and C, respectively). In all trajectory averaging, unperturbed movements are aligned at movement onset time, while perturbed movements are aligned at perturbation onset time. The movements shown in Fig. 2, aside from a typical unperturbed trace, were perturbed in the extension direction at mid-motion, which was 50% of the movement distance. The trial rejection criteria based on RT estimation described in METHODS resulted in 21.3% of perturbed trials to be rejected from our analysis.



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Fig. 2. Effects of a perturbation: planar traces. Curve A is a randomly picked motion trace. The sharp feature in the middle of the trace is the result of a 50% onset extension pulse. Curve C is the average trace for the same perturbation condition and the same subject. Note the significant difference between the randomly picked trial and the average, suggesting high variability of motion. Curve B is the average unperturbed trajectory for that subject. Width in all traces is proportional to instantaneous speed. The ellipses at the end of the average traces represent 2 df SEs in end point. The shapes of movements are not used in analysis---the end-point ellipses are. Averages are accumulated per subject and perturbation condition with traces aligned at perturbation time and with only trials sufficiently fast to eliminate the possibility of a RT accepted.

The main results of this work depend on statistical analyses of perturbed versus nonperturbed movements' end points. Figure 3 shows the mean perturbed and unperturbed end points along with the estimation uncertainty of each, represented by standard error ellipses, for all five subjects and six experiments. These ellipses' orientations are determined by the major and minor axes (eigenvectors) of the variance matrix for the end-point position. We use standard error rather than deviation as it is more directly related to statistics of differences in mean.



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Fig. 3. Average nonvisually guided trajectories for each subject. Each perturbation condition is represented by a different color. The target (1 ×1 cm) is shown as a gray square (the x and y scaling factors are equal). The dotted vertical lines indicate the positions at which a perturbation is presented, i.e., 25, 50, 75% along the way from the start point to the target, as indicated. Width in all traces is proportional to instantaneous speed, scaled in a similar manner to Fig. 2. The ellipses at the end of the average traces represent 2 df SEs in end point. The shapes of movements are not used in analysis---the end-point ellipses are. Averages are accumulated per subject and perturbation condition with traces aligned at perturbation time and with only trials sufficiently fast to eliminate the possibility of a RT accepted. One subject was tested twice with a 5-mo interval (top left and bottom right), 1st with, then without a brace supporting the lower arm.

Each subject was given six different types of perturbations: two directions each at three different onsets (see METHODS). Figure 4 summarizes the main results of the statistical evaluation of end-point error for these different stimulus conditions. The top graphic illustrates the target acquisition frequency for each perturbation, showing that the effect of the flexor and extensor perturbations is asymmetric. To sum up, over all subjects and all onsets, 61.3% of unperturbed motions settle within the target as opposed to 60.8% of flexion perturbed motions but only 20.0% of extension perturbed motions (these numbers vary from subject to subject and are approximate due to within subject variability).



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Fig. 4. Statistics of end-point shift. A: target attainment frequencies. The bar graph is arranged in 6 groups corresponding to 6 subjects. The height is proportional to the frequency (0-100%) and each condition in a group is in the position indicated schematically. B: t-test results (alpha  < 0.01), presence of bar indicates statistically significant shift (perturbed, no vision versus unperturbed, no vision). The arrangement is same as A, with the bar corresponding to unperturbed result of a similar t-test to measure whether unperturbed no-vision end points are different from those unperturbed with vision (yes in each subject).

Given the size of the error ellipse, it is possible that some perturbations might have the effect of shifting the mean end point from inside the target to outside the target but in a statistically insignificant manner substantially away from the target. To evaluate this possibility, a series of t-tests was performed comparing mean end points of each perturbed condition to mean end points of nonvisually guided movements of the same subject. The appropriate type of t-test for determining differences in means of different populations of more than one dimension (our data being 2 dimensional) is Hotteling's t-square test (Johnson and Wichern 1992)---it can be applied in the case in which each population has a different underlying variance (as opposed to MANOVA).

In Fig. 4B, it is evident that 18 of 18 extension-direction perturbed conditions yield a significant shift (with a minimal significance level alpha  of 0.01 for each test), while only 8 of 18 flexion perturbed end points shifted significantly. This is consistent with the observations of target acquisition frequency asymmetry. Overall 26 of 36 cases had significantly different end points. Using Bonferroni post hoc corrections for power of multiple t-tests, we can say that flexion directed perturbation shift the end point with a significance level of 0.001 (each of the t-tests affected is much more significant than our threshold of 0.01), whereas only for one subject can we say that any perturbation type shifted the end point (with alpha  of 0.05).

Figure 5 shows the distance between the mean end points of blind perturbed movements and the mean end points of blind unperturbed movements, classified by perturbation direction and onset for each subject. Although each subject responds differently, it appears that there is no large effect of perturbation onset on the location of the end point (8 of 12 end point populations were statistically different at 25%, 8 of 12 at 50%, and 10 of 12 at 75%). This failure was also evident in the target-acquisition frequency results. In all cases, the size of the end-point shift is quite large, accounting for between 9 and 34% (i.e., 9-34 mm) of the movement distance itself [14.4 ± 5.9 mm (SD)]. In other words, the observed end-point shift is not a minor deviation, but a substantial and physically discernible effect. A 14.4-mm mean error places an end point well outside of the target, whose "radius" is 5 mm.



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Fig. 5. Dependence of end-point shift on perturbation onset. Each subject is represented by a point symbol, SEs are indicated as uncertainty intervals. The movement distance was 100 mm (desired) so the end-point shift represents both the distance between perturbed end point and unperturbed end point in mm and a percentage of movement distance.

For unperturbed movements under visual guidance, the end points were not found to be statistically different from the target center. The standard deviation of the blind unperturbed movement end point in the two-dimensional direction of highest end-point variability for each subject averaged 13.2 mm (11.5, 12.6, 7.70, 15.7, 22.8, and 8.56 mm). This is comparable to a circle of high end-point probability (higher than 68%) of 6.1-mm radius centered on the mean unperturbed end point. A mean 14.4-mm shift clearly places perturbed end points outside this region.

In summary, we conclude that unexpected small perturbations can significantly affect the location of the end point when vision is eliminated, casting doubt on the existence of equifinality.

Reaction time

In a set of supplementary trials, we assessed the ability of each subject to detect a perturbation pulse, to determine its direction, and to react voluntarily to it (see METHODS). The subject tested twice was given slightly different instructions in the second experiment to check for effects of decision choice and perturbation onset on the all-important voluntary RT latency (none were observed).

If terminal errors are induced by "unsuccessful" RT movements, (which may be emitted in an effort to bring the perturbed limb closer to the desired target destination), then we would expect that the secondary or corrective movements would present a separate peak in the velocity profile with secondary movement onset arising at RT latency, referenced to the perturbing pulse. The presence of such a second, clearly identifiable peak is confirmed in Fig. 6, which shows a typical "double-bell" velocity profile trial arising when the subject is asked to interject an additional movement when a pulse is detected. The figure shows one motion, almost coming to rest---then another motion is initiated. The time interval over which such movements were performed made it likely that the original movement would either have come to rest or slowed greatly before the re-acceleration.



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Fig. 6. Velocity plots and RT estimation. Right: velocity profile for a typical reaction time (RT) estimation trial (gray trace). The vertical dotted line indicates the RT deduced from the particular trace shown, not the average RT for that subject. Dark trace is a similarly perturbed trial without a RT movement from the main body of trials. Left: velocities for a typical normal trial and a typical RT trial. Perturbation occurs at t = 0 (trials are synchronized by perturbation onset prior to averaging). Note average velocity profile for RT trials does not go to 0 because the resting point of individual traces arrives at different times---the typical perturbed trial goes to rest before an induced voluntary reaction occurs. The time of rest for the individual trial is denoted by a vertical dotted line.

The RTs during motion were measured from perturbation onset to the first velocity minimum. We averaged this time across the appropriate trials and subtracted 1 SE to provide a conservative estimate of RT. We also checked that the first movement came to rest at the same end point as without a voluntary reaction, demonstrating no clearly measurable difference between the initial part of the "voluntary reaction" trials and movements without voluntary reactions. This end-point similarity was confirmed by a t-test (alpha  = 0.05) in four of six subjects, a result somewhat confounded by the fact that the determination of intended end point of the initial motion of the double-bell trials is not accurate.

The RTs were, for each subject: 285 ± 112 (SD), 345 ± 67, 306 ± 110, 345 ± 89, 307 ± 87, and 312 ± 63 ms (the last 2 are for the same subject tested 5 mo apart). The first velocity minimum was chosen because it proved to be the most practical approach and because it provided a relatively unbiased estimator of RT. Although the velocity minimum did not quite reach zero in some trials, other trials displayed a prolonged period of very low velocity, with the minimum occurring in the middle of this interval (therefore we did not actually use the longer time estimate of 2nd motion onset). Given an average movement time of 350 ms and a time from perturbation to settling on the order of 100-200 ms (see Fig. 8), there is clearly minimal or no opportunity for RT corrections in most trials.

We measured RTs with the 50% onset pulse and applied this value to the analysis of movements perturbed at all three onsets. We could not use the 25% onset pulse because separation of the control and perturbed movements would became too difficult---in the cases were the control motion was slower, there would be more time for the RT motion to begin prior to the end of the former. We have mentioned exploratory data (the subject tested twice) showed no dependence of RT on choice of response (it was no greater when the subject was asked to react to just flexion pulses than to any pulse) or pulse onset (50 vs. 75%). Moreover three of the five subjects changed targets to the new target in the RT determination trials more than 80% of the time for either direction of perturbation. This suggests that they were not making a choice (correctly or incorrectly) in the RT-determination trials anyway.

Dependence of end point on perturbation direction and RT response accuracy

We have further examined the possibility that RT movements were involved in producing the terminal errors by evaluating the relation between the direction of the perturbation and the resulting error. When subjects are asked to react to an extension pulse only, by making a fast movement to a new target, the subject may err by moving to the new target when he/she shouldn't and not moving when he/she should. Whether the subject moved to the old or new target was easily determined from the movement path (see Fig. 7). Our results indicate that subjects did not discriminate well between extension and flexion pulses during the supplementary trials. (A t-test of response frequency shows no statistical difference in 4 of 5 subjects, alpha  = 0.05.) Yet in 15 of 18 cases during the main body of trials, the end points did depend on the pulse direction (t-tests comparing "extension" pulse and "flexion" end points, alpha  = 0.01). The lack of knowledge of pulse direction combined with a clear effect of the latter on end-point location reduces the significance of voluntary reaction on end-point locations observed (see DISCUSSION).



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Fig. 7. Sample RT traces. A: normal reactions to 50% FLX perturbations ("do not react" instruction). B: RT trials to the same perturbation type ("move to new target if a perturbation toward from the body is felt" instruction). Only trials with the correct response are shown---on some trials the subject stopped at the old target.

Adaptation

Given the extended length of the experiments, coupled with the requirement that subjects repeatedly and accurately acquire a target within a designated time interval, it is conceivable that experience and increasing manual skill could have had an impact on the results. Figure 8 shows the end-point accuracy and movement time as a function of practice for a typical subject. Apart from an initial acclimatization period of approximately five trials during which the subject became familiar with the task, there was no evidence of learning as judged by any progressive decline in end-point error. The most important facet of the learning curves was that the end-point variation stabilized to a random variable with constant mean (14.4 mm)



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Fig. 8. Learning curves for a typical subject. Top: settling time (ms) following perturbation vs. practice. Bottom: distance from end point (mm) to desired end point vs. practice. Each trace represents a different perturbation condition. The RT for this subject was 379 ± 3.5 ms, so all perturbed trajectories were accepted except for errant trials. Other subjects had faster RTs and subsequently some of their trials were rejected. Total desired movement time was 350 ms.

Role of manipulandum dynamics

Whatever small transient forces were due to the manipulandum dynamics (such as link inertia), only static forces (such as friction) can be responsible for the lack of equifinality. In the latter case, the drift and residual force directions should be related to the direction of the target or the closely situated average unperturbed end point, where the "force field" ought to be centered. To assess potentially confounding effects of mechanical properties of the manipulandum, we examined the relation between either force direction just prior to stopping or hand movement direction, and the direction of the target (see Fig. 9 for graphical explanation). These angles were obtained by averaging the direction of the movement and force vector across a 50-ms window placed immediately before movement termination, which was computed as the time after maximum velocity when position held constant for at least 30 ms.



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Fig. 9. Close-up of average trajectories around end point for a subject. X, desired end point. Note arrangement and skewness of standard error ellipses. Top: the drift direction Theta  and target direction Omega  shown schematically were calculated for each individual trace (see text) by averaging over a small time window prior to stopping. Even without small hooks most average trajectories do not point toward the target.

As shown in Fig. 10, there was no correlation between drift direction and target direction or between handle force direction and target direction. Force was recorded for only two of the subjects, whereas drift direction is shown for all six subjects, as it is computed from position. Furthermore the magnitude of the force vector at rest was <0.2 N, which was also the magnitude of the recorded static friction value in the manipulandum. In other words, whatever the shape of any terminal corrective responses ("hooks") revealed by the average trajectory profiles, they did not point toward the target. Furthermore the force exerted on the handle did not show a push toward the intended target immediately after movement had stopped. It follows that the terminal error was not mediated by frictional or viscous forces limiting the restoring actions of residual elastic forces (as predicted by the EPH).



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Fig. 10. Scatter plots of drift, residual force, and target directions. Left: direction to target Omega  vs. drift direction Theta  prior to stopping. Right: direction to target vs. residual force direction after stopping. Each point type corresponds to a different perturbation type and subject. If the arm was drifting toward the target or pushing, the manipulandum toward the target, all points would be close to the identity line (shown). Note most residual forces are pointing in the general direction of motion.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

The principal finding of this study is that transient force perturbations applied to the arm of a subject reaching toward a target induce substantial errors in final position, indicating that spring like properties of arm muscles are not inherently able to guarantee acquisition of the final target position. It follows that the central attribute of equifinality, which requires that a limb achieve the designated final position regardless of changes in either initial conditions or of transient perturbing forces (Kelso and Holt 1980), is almost certainly incorrect.

Are the observed end-point shifts significant?

The protocol of this experiment is similar to that of earlier studies that were advanced in support of the EPH (Bizzi and Polit 1978; Bizzi et al. 1978; Kelso and Holt 1980). That is, a transient perturbation is applied to the limb during point-to-point movement and the effect on end point is recorded. Since all such systems are inherently noisy, it is important to establish that the errors in final position reported here are both "statistically" and functionally significant.

To review relevant earlier studies briefly, Bizzi and colleagues (1978) performed a broadly similar test on deafferented monkeys and reported that there was no end-point shift due to the perturbation. However, the monkeys in the Bizzi study received extensive training after the deafferentation, which resulted in a substantial level of co-contraction of arm muscles, to a level not recorded in the control animals. This may result in a much higher arm stiffness, quite unlike that seen for normal voluntary movements. Such co-contraction may have increased limb net stiffness to a disproportionate degree, and it is reasonable to think that higher stiffness increases the likelihood of observed equifinality. In another pertinent study (Kelso and Holt 1980), unexpected perturbations were applied to normal and ischemic fingers of humans during 1 df movements, again showing no end-point shift. In this study, the sample size was very small making interpretation of their results rather difficult, although their findings were consistent with the existence of equifinality.

Another potential issue in comparing our results to those of other studies is the magnitude of the perturbation. We have used "small" perturbations, so designated because they are not startling to the subject. Indeed, one subject did not even perceive the perturbation, while all had great difficulty judging its direction---toward or away from the body. By confining the study to the use of such small perturbations, we hoped to avoid or at least to minimize the causation of nonlinear behaviors (so that we may assume muscle properties and geometry can be locally linearized for later modeling purposes) or to elicit nonphysiological reactions. Our perturbations were routinely smaller than others have used to test the equifinality hypothesis.

Other studies of equifinality provide more ambiguous information about its potential existence. For instance, some investigators (Day and Marsden 1982) claimed equifinality in intact but not "anesthetized" thumbs. The perturbation used in their study was an unexpected viscoelastic load. These authors report that intact subjects did not show significant errors, although with the largest viscosity there was a small but statistically significant undershoot of 1%. This outcome was interpreted as indicating that muscle properties by themselves do not guarantee equifinality, that muscles are not fully spring-like, and that reflex circuitry is required to maintain spring properties and to guarantee equifinality. Later Sanes (1983, 1986) was able to show significant errors due to the imposition of unexpected viscous loads in normal subjects but only for small movements (3°). Together with our present results, we can infer from these studies that neither muscle nor reflex circuitry is sufficient to guarantee conservative elastic behavior, although reflexes serve to make muscle action more spring-like in character and to improve our ability to reject disturbances. We did not test movements of varying amplitudes, but we did conclude that movements exposed to flexion perturbations normal to the motion are more likely to exhibit equifinality than are extension perturbations. Perturbations given later in the movement are slightly less likely to exhibit equifinality than early perturbations due to less time available for reflex correction.

To evaluate Sanes' claim that a critical variable for single degree-of-freedom studies is the total movement amplitude, or distance, we sought to compare the movement amplitudes used to generate our results with those of other studies. Some of these studies (Day and Marsden 1982; Kelso and Holt 1980) used 20 and 50° movements. Our study relied on planar movements of 10-cm amplitude and used perturbations orthogonal to the movement path. These movements were associated with a shoulder movement of about 15°. Therefore since the previous studies detected equifinality and we did not, we can substantiate Sanes' observation that smaller amplitude movements are more likely to have their end points shifted by small perturbations, although we did not test larger movements and have no physiological basis to expect equifinality in large movements, other than perhaps larger overall movement time and therefore more time for voluntary corrections (RT movements).

Finally, a rather different and recent group of studies (Coello et al. 1991, 1996; Lackner and Dizio 1992, 1994, 1996) also appears to reject the existence of equifinality. Subjects moving their free limbs to a target while sitting at the center of a spinning darkened room showed consistent end-point shifts, even when movement velocity was zero and therefore the applied load was minimal, in situations where residual elastic forces should readily have established the correct final position. Although there are potentially disruptive psychophysical attributes of this experiment and a potential confounding role for centrifugal force (since the centrifugal force due to the spinning room does not fall to 0 when the arm goes back to rest), the reported end-point shifts do not seem to be explained by this force either in their direction or magnitude. Furthermore Lackner has recently repeated the experiment on patients with severe labyrinthine dysfunction who did not perceive that the room was spinning, even sub-consciously (Lackner and Dizio 1996), and were therefore not aware of the perturbation. Yet these patients showed similar terminal errors. It follows that RT corrections, which depend on perception of error, could not have been responsible for the reported end-point errors. The Lackner subjects' end-point errors decreased to insignificant levels as a result of exposure to the spinning room "force field," without any visual feedback or knowledge of results---in contrast to our results. This effect shows that the "position sense" inferred a century ago by Woodworth (1899) and resulting from kinesthetic inputs from various receptors is strong enough to detect an end-point error and serve as a basis for adjusting the descending command. The reasons why the gradual correction was not seen under our current paradigm are (presumably) that our perturbations are infrequent, randomized and unexpected, thereby offering little incentive for the subject to adapt. The effect of the Coriolis forces was also quite large (approximately 5% of a longer planar motion).

Implications of the lack of equifinality

Our study casts doubt on the idea of equifinality in human reaching movements. However, one potential objection to our findings is that while we may have observed statistically significant end-point shifts, these are not necessarily functionally significant. In response, since the shifts average 14.4% of the total movement distance (1.44 cm, compared with a 0.5-cm target radius) and since they result in a target acquisition frequency reduction from 61 to 40% they are almost certainly functionally significant as well.

An alternative explanation for the failure to acquire a target, consistent with the EPH, is related to the spatial profile of the elastic stiffness that exists about the end point and that is proposed to regulate movement termination. For the range of limb positions examined, the stiffness profile at the hand for static postures is typically elliptical (Hogan 1985; Won and Hogan 1995), with the highest stiffnesses oriented away from the body, and the lowest values lying orthogonal to this axis (approximately). Let us imagine that these stiffnesses are low compared with viscosity, and if we had waited longer, the viscoelastic forces would have brought us back to the intended target. Not only is this discounted by the lack of target-oriented force and motion at the end of movement (the creeping approach to the target would have to take awfully complicated paths inconsistent with a parabolic "well"), but one may wonder what the physiological benefit of muscle and reflex properties is when they are only effective for time periods much longer than that required for visual based voluntary intervention.

Significance and role of voluntary reactions

The central argument underlying our analysis is that in our protocol, the end point is predetermined before a voluntary reaction can even begin. To validate this argument, we show that the available time after perturbation was usually insufficient to allow corrective RT movements to intercede (as per our RT estimates).

There are relatively few studies in which the voluntary RTs have been estimated during voluntary movements of a type comparable to that used in our study. Most published reports set a minimum bound of approximately 200 ms for a no-choice RT movement (Crago et al. 1976). In our case, the perturbations alternated in both direction and onset so that voluntary reactions (should they occur) would be expected to be significantly slower than this 200-ms boundary. The RTs measured in our study lay between 240 and 420 ms and are consistent with previous estimates of 230-350 ms in the finger (Day and Marsden 1982) and lower than an estimated 450 ms in similar arm movements (Won and Hogan 1995). Slower movements may allow further correction and processing of sensory data (such as vision, when available), making it more likely that high latency RT movements help rather than hinder target acquisition, thereby placing the onus of equifinality on higher levels of the CNS.

Parenthetically, the voluntary RTs quoted here refer to the delay between stimulus onset and position change. It seems clear from conduction delay estimates and RT estimates in isometric conditions (Rothwell et al. 1982a) that the RT is longer during movement and is longest at the beginning of the movement, during the psychophysically described "refractory period" (Desmedt and Godaux 1978).

Further observations from data

We have reported that although subjects did not discriminate well at all between extension and flexion pulses, the mean end points of extension and flexion pulses at each onset were significantly different. If the reactions to the pulse were somehow alterations of the central command, this alteration would have to be the same for extension as well as flexion because of the lack of discrimination between stimuli. If the EPH were correct, this alteration would consist in specifying an end point, so the end points of the flexion perturbed movements would have to be similar to those perturbed in the extension direction. This is not the case. If one believes this argument, the argument that our results are inconsistent with the EPH is even stronger.

A further point of interest is that in five of six subjects, there was a significant (t-test of cross-product of starting and ending y coordinates) and positive correlation between starting point error (deviation from average starting point) and end point error for blind movements (significance level alpha  < 0.10). This correlation was sometimes maintained despite the perturbation and was always compromised by a second voluntary movement like those that occurred in the supplementary experiment. Evidently it points to a stronger regulation of movement extent than movement end point, hinted at by properties of somatosensory feedback (Woodworth 1899), which offers a better sense of movement distance than limb position (Miall et al. 1990) and is consistent with theoretical formulations that place emphasis on amplitude (Gottlieb et al. 1996). This dependence on starting position is inconsistent with the equilibrium point formulation in its alpha formulation (Bizzi et al. 1978), which is exclusively a position control framework but is consistent with the lambda model (Asatryan and Feldman 1965). This dependence on starting position is inconsistent with the equilibrium point formulation, which is exclusively a position control framework. A position servo should not care about where it starts.

Final observations

The demonstrated breach of equifinality means that so called linear "K-B-I" models (linear stiffness---damper---inertia) cannot fit the responses seen in our studies, even if K and B vary with time. Many types of impedance models show equifinality (K-B-I models are a subset of this class of models) by having the end point specified by a position-dependent force in parallel with elements that produce forces that are functions of derivatives of motion and are not producing force once the movement comes to rest (like damping and inertia). These types of impedance models will not work.

What type of nonlinear effect must we include in a model of limb "impedance" to account for the observed responses? One possible modification is some type of history-dependent "plasticity" such as muscle yielding behavior, which falls under the dissipative type of mechanical element, like damping. This may explain the results of Rothwell (Rothwell et al. 1982a,b) in which an unexpected viscous load caused a nonreflexive wrist flexion to undershoot with faster movements exhibiting more undershoot for a similar applied damping ratio. Yet in our present study, small pulses could induce overshoots or could even result in end points lying closer to the push direction. A change to muscle properties alone is unlikely to explain the trajectory perturbations seen.

Alternatively, reflex mechanisms could contribute to lack of equifinality. It is well known that intrinsic muscle stiffness changes during movement, and reflex "gain" is also modulated during movement. At each perturbation onset, we would expect that there were different values for muscle stiffness and reflex gain. Yet the magnitude of the end-point shift remains relatively unaltered by perturbation onset. It is conceivable that both types of disturbance rejection (namely intrinsic muscle stiffness and reflex action) complement each other, such that a drop in muscle stiffness is associated with some compensatory increase in reflex gain, as previously reported for isometric conditions (Nichols and Houk 1976). Of course, the magnitude of the reflex gain is limited by stability considerations (Rack 1981), unless the higher latency reflexes do not act like automatic servos (like the stretch reflex). If we assume the inherent muscle stiffness during movement is low, there being evidence to that effect (Bennett et al. 1982), and the stretch reflex is limited in gain, we should not be surprised by the lack of ability of the human arm to be a good position servo (i.e., lack of equifinality). Therefore a model that consists of inertia, a force generator (muscle), some damping and gain-limited stretch reflex activity and longer latency reaction of nonautomatic type could account for the behavior seen. What the strategy of disturbance rejection in the moving limb really is, involving other types of reactions, remains to be discovered.


    ACKNOWLEDGMENTS

This work was supported by National Institutes of Health Grants T32 HD-07418 and R01 NS-19331 to W. Z. Rymer.


    FOOTNOTES

Address for reprint requests: F. C. Popescu, Sensory Motor Performance Program, Rehabilitation Institute of Chicago, 345 E. Superior St., Suite 1406, Chicago, IL 60611.

Received 10 November 1999; accepted in final form 22 June 2000.


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ABSTRACT
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