Northwestern University, Evanston 60208; and Rehabilitation Institute of Chicago, Chicago, Illinois 60611
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ABSTRACT |
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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.
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INTRODUCTION |
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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).
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METHODS |
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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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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 sequenceso 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.
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RESULTS |
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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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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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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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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 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
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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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 restthen 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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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 ( = 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 difficultin 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, = 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,
= 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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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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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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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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DISCUSSION |
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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 directiontoward 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 < 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 stiffnessdamper
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.
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ACKNOWLEDGMENTS |
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This work was supported by National Institutes of Health Grants T32 HD-07418 and R01 NS-19331 to W. Z. Rymer.
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FOOTNOTES |
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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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REFERENCES |
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