Department of Neuroscience, University of Minnesota, Minneapolis, Minnesota 55455
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ABSTRACT |
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Engel, Kevin C. and John F. Soechting. Manual Tracking in Two Dimensions. J. Neurophysiol. 83: 3483-3496, 2000. Manual tracking was studied by asking subjects to follow, with their finger, a target moving on a touch-sensitive video monitor. The target initially moved in a straight line at a constant speed and then, at a random point in time, made one abrupt change in direction. The results were approximated with a simple model according to which, after a reaction time, the hand moved in a straight line to intercept the target. Both the direction of hand motion and its peak speed could be predicted by assuming a constant time to intercept. This simple model was able to account for results obtained over a broad range of target speeds as well as the results of experiments in which both the speed and the direction of the target changed simultaneously. The results of an experiment in which the target acceleration was nonzero suggested that the error signals used during tracking are related to both speed and direction but poorly (if at all) to target acceleration. Finally, in an experiment in which target velocity remained constant along one axis but the perpendicular component underwent a step change, tracking along both axes was perturbed. This last finding demonstrates that tracking in two dimensions cannot be decomposed into its Cartesian components. However, an analytical model in a hand-centered frame of reference in which speed and direction are the controlled variables could account for much of the data.
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INTRODUCTION |
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In recent years, it has become evident that the
study of movement in two- and three-dimensional space introduces
questions that are not apparent in the study of one-dimensional
movement. With respect to arm movements, this has been demonstrated by
various investigators who have recorded from motor and premotor
cortical areas during movements to stationary targets and have shown
movement direction to be prominently represented in the
activity of these neurons (e.g., Fu et al. 1995;
Georgopoulos et al. 1986
; Kalaska et al.
1997
; Schwartz et al. 1988
). Much less is known
about the control of arm movements for tasks in which the target itself is moving in space, such as tracking or intercepting a target moving in
two dimensions (Johnson et al. 1999
; Port et al.
1997
; Viviani et al. 1987
).
Sensory reception, neural computation, and motor output all require a
finite amount of time. Therefore time delays are inherent in the task
of manually tracking a moving target. However, during the normal
tracking of a predictable target moving along one dimension, the
tracking error can be very small (Poulton 1974),
implying that predictive algorithms are employed by the nervous system. Studies of tracking in one dimension have shown that this predictive behavior is generated by a velocity error signal in combination with a
positional error signal (Poulton 1974
; Viviani et
al. 1987
; see also Lisberger et al. 1987
). The
question then can be posed: what is the form of the error signal for
tracking in two dimensions?
The current study was undertaken to determine how speed and directional error signals are used in two-dimensional tracking. To this end, we asked subjects to track a target that moved initially in a straight line and then changed direction (and sometimes speed) abruptly. We identified a rather unexpected strategy. A conceptual model based on a constant time to intercept could predict the new direction of the finger motion as well as its maximum speed. This conceptual model provided the basis for a formal quantitative model in which direction and speed are the controlled variables.
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METHODS |
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Motor task
The manual tracking of targets moving in two dimensions was assessed in six different experimental conditions. Before describing each of the experimental conditions in detail, we will describe those aspects which were common to all experiments.
Subjects sat in front of a touch-sensitive computer video monitor. Seat height was adjusted such that each subject was comfortable and could easily reach all areas of the video screen. Room lighting was dimmed to increase the contrast of the display. No restrictions were imposed on head or eye movements. Each experiment typically consisted of 300-360 trials and lasted from 45 min to 1 h. The subjects gave their informed consent to the experimental procedures, which were approved by the Institutional Review Board of the University of Minnesota.
All subjects were right handed and were asked to track, with their right index finger, the motion of a target presented on the video monitor. In most experiments, a box 1.6 cm per side initially appeared 1.6 cm from the top edge of the screen to indicate the starting position for the subject's finger. When the subject placed his or her finger in the box, a round target 1.6 cm in diameter appeared at the same edge of the screen and began to move at a constant downward velocity toward the box (see Fig. 1). Subjects were to begin tracking the target as soon as it entered the box. In all cases, the target initially moved at a constant speed and in a direction that was constant from trial to trial. After the target had traveled a random distance of from 10.9 to 17.4 cm, it made a single abrupt change in direction. On average, the target motion changed direction when the target was in the middle of the screen ~1-2 s after the start of the trial.
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Experiment 1: tracking a target moving at a constant speed
In this experiment, the target appeared at the middle of the top edge of the screen, and moved straight downward at a speed of 10.8 cm/s. Then after the target had traveled a random distance, it made an abrupt change in direction to 1 of 24 equally spaced directions. These directions were varied randomly from trial to trial. The speed of the target remained constant throughout the experiment. Four subjects participated in this experiment.
Experiment 2: tracking an obliquely moving target
To determine whether the results from experiment 1 could be generalized to any initial direction, the experiment was repeated, with one modification: the initial direction of finger tracking was rotated counter-clockwise through 135°. Therefore the start box for the subject appeared in the lower left-hand corner of the video screen rather than at the top middle. The initial target motion was upward and rightward, rather than straight downward. As in experiment 1, target motion changed unpredictably to 1 of 24 equally spaced directions. Four subjects participated in this experiment.
Experiment 3: the effect of target speed on tracking
In this experiment, the target again appeared at the top, middle section of the screen and initially moved straight downward. At a random point in time, the target then changed to one of six equally spaced directions. The speed of the target remained constant throughout any particular trial but was varied randomly across trials. Four speeds were used: 5.4, 10.8, 16.2, and 21.7 cm/s. Five subjects participated in this experiment.
Experiment 4: tracking during an abrupt change in speed
In experiments 1-3, during any particular trial, only the direction of the target's motion varied. Experiment 4 was conducted to determine whether the results from these three experiments generalize when target speed also changes unpredictably. In experiment 4, the target again appeared at the middle of the top edge of the screen, moved straight downward at a constant velocity of 10.8 cm/s, and then changed to 1 of 12 equally spaced directions. However, the target sometimes also changed speed (unpredictably) at the same time it changed direction. In one third of the trials, the target abruptly slowed from its initial speed of 10.8 cm/s to a speed of 5.4 cm/s. In another third, it abruptly increased speed to 16.2 cm/s. In the last third, it maintained its original speed of 10.8 cm/s. Six subjects participated in this experiment.
Experiment 5: constant vertical velocity/variable horizontal velocity
As in experiment 1, the target initially moved downward at a speed of 10.8 cm/s. The motion of the target then changed unpredictably to 1 of 12 directions. For 7 of these 12 directions, target speed also underwent a step change. In these instances, the vertical component of velocity was held constant throughout the entire trial and the horizontal velocity underwent a stepwise change from 0 to some new constant value. Seven values of horizontal velocity were used such that the resulting directional change of the target was 0, ±22.5, ±45, or ±67.5°. Consequently, the speed and direction of the target's motion changed simultaneously as in experiment 4. For the remaining five directions, the target's motion changed to one of five upward directions (180, ±150, and ±120° with respect to the downward direction); the target's speed being held constant throughout the trial (as in experiment 1). Four subjects participated in this experiment.
Experiment 6: the effect of target acceleration on tracking
The parameters for experiment 6 were nearly identical to those for experiment 4 except that acceleration was changed instead of speed. In experiment 6, the target either accelerated or decelerated at a rate of 5.4 cm/s2 or maintained its original speed of 10.8 cm/s, at the time when its direction of motion changed. Five subjects participated in this experiment.
Recording system
The experiments were performed using a touch screen (Elo Touch
Systems, TN) mounted over a standard 20-in computer monitor (Mitsubishi
Diamond Scan 20 M). The touch screen has a spatial resolution of 0.08 mm. The target motion and recording of finger position were controlled
by a laboratory computer using custom software. The position of the
finger was recorded at a rate of 100 Hz. Target location was updated at
a rate of 60 Hz, equal to the refresh rate of the video monitor. The
output of the touch screen was scaled and aligned with the video image
through the use of cubic polynomials and a rectangular reference grid
of target positions (see Flanders and Soechting 1992).
Data analysis
Data were averaged by aligning the trials on the point at which the target changed direction. All subsequent analysis was performed on averaged data. Velocity was calculated by numerically differentiating the position data and digitally smoothed using a two-sided exponential filter with a cutoff frequency of 12 Hz. As will be shown in RESULTS, after the target changed direction, the finger maintained the original target direction for a reaction time period, changed direction, initially headed in a nearly straight line to intercept the target, and then finally curved to merge with the new target direction. This heading to intercept the target was defined by computing the inverse tangent of the ratio of the horizontal and vertical velocities at 350 ms after the change in target direction. This point in time was chosen because it is generally in the middle of the straight interception period, at a time when tracking speed was increasing (see Figs. 4B, 8, and 11).
To determine when two averages of either speed or direction began to differ from each other, we performed a t-test at each point in time. The averages were said to diverge once the 0.05% confidence level was reached and the two series remained separated by at least this level of confidence for the next 70 ms. To determine reaction time (defined as the interval between the time at which target motion changed direction and the first observable change in the finger's trajectory), a baseline period was defined by averaging both direction and speed data over the interval from 150 ms before the target changed direction to 100 ms after the target changed direction. The standard deviation of direction and speed was also computed for this same 250-ms interval. The subject's reaction time was then defined as the point in time beyond this baseline interval when either finger speed or direction exceeded the 2 standard-deviation limit and continued to exceed it for at least 30 ms.
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RESULTS |
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Response to a change in target direction (experiment 1)
To study manual tracking in two dimensions, we began with the
simplest experiment, that of a target moving downward at a constant speed and making a single change in direction. Figure 1,
top, shows the result for one trial of tracking. The path of
the target is shown by the thin line. The direction of target motion is
indicated by arrows. In this example, the target changed direction to
= 135°. The path of the finger is shown by the thick line.
(During the vertical segment of the target's motion, the path of the
finger obscures the path of the target.) To demonstrate how the motion of the finger with respect to the target evolved over time, the position of the finger and the position of the target were joined by a
thin (isochronic) line, every 100 ms, starting at the point where the
target changed direction. After this point in time, the finger
maintained its original downward trajectory for a period of time,
slowed in speed to change direction, then accelerated to reacquire the
target (see Fig. 4B). For this example, the reaction time
was found to be 230 ms using directional data and 210 ms using speed.
For this subject, for all directions, the average reaction time was
239 ± 25 (SD) ms for direction and 235 ± 26 ms for speed.
Averaging across all directions for all subjects, the reaction time was
229 ± 24 ms for direction and 227 ± 28 ms for speed. There
was a statistically significant effect of target direction on reaction
time (ANOVA, P < 0.05). However, a post hoc comparison
(Tukey HSD) showed that the reaction times did not depend significantly
on the amount by which the target changed direction for changes
exceeding 30°. For smaller changes in target direction, the estimated
reaction times were about 30 ms longer, but these estimates are not as
reliable because the signal to noise ratio is much smaller in these
cases (see Fig. 4). These reaction times are in general agreement with
previous findings (Hanneton et al. 1997
; Poulton
1974
).
Figure 1 illustrates a general aspect of our results: after the finger's motion changed direction, the hand initially headed in a nearly straight line before curving to merge with the path of the target. It is clear that this new heading is not directed toward the current location of the target. (The target's location can be noted by considering the isochronic line that most closely connects finger and target location at the time of the directional change.) Rather, the finger heads in a direction anticipating the future location of the target.
In Fig. 1, bottom, all trials from this subject for this direction are shown with the trials aligned on the point at which the target changed direction. The consistency in the handpaths illustrated was typical of the results we obtained in this and the other subjects. We computed the deviation of finger position (the square root of the sum of the variances in X and in Y). For this example, the average deviation of the finger position after the change in target direction was 0.56 cm. This deviation was not significantly related to target direction (slope not significantly different from 0, P > 0.36). Across all subjects and directions the average deviation was 0.63 ± 0.1 cm. Since this value was fairly small, we restricted our analysis to averaged data.
Figure 1 shows the results from one out of 24 target directions tested in this experiment. Figure 2 summarizes the results that were obtained from one subject for 8 of the 24 directions, the dotted lines denoting the paths of the target. The finger trajectories begin to diverge from a common point, and during the reacquisition of the target, the path of the finger appears to be reasonably straight for a considerable period of time. Finally, once most of the positional error has been eliminated, the path of the finger curves to merge with the path of the target. Therefore while the movement is continuous, for the purpose of discussion, we may consider it as occurring in four steps: a "reaction phase," in which the subject continued on the original target heading, a change in direction, a "reacquisition phase," in which the subject moved in a relatively straight, anticipatory path to reduce the error between the target and the finger, and a gradual "merging" of finger velocity with target velocity.
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Simple model for tracking directional changes in target motion
In an attempt to gain some understanding as to how the reacquisition of the target is controlled by the nervous system, we developed a simple geometrical model of the subjects' behavior. In particular, we approximated the finger path by two straight lines, one representing the movement of the finger during the reaction phase, the other representing the movement of the finger during the reacquisition phase (see Fig. 2, inset). We also assumed that the reaction time did not depend on the amount by which target motion changed direction (see preceding text). These are clearly oversimplifications because the finger does not change direction instantaneously, because there is some curvature in the path of the finger, and because there may be some slight variability in the reaction time. However, this conceptual model allowed us to test a hypothesis that was suggested by the data: the path of the finger merges with the path of the target at a constant time after the target changes in direction, independent of the amount of the directional change. (In this experiment, the target moved a constant distance in a constant time. Accordingly, the predicted point of interception can be represented by a circle centered on the time of the target's change in direction, as in Fig. 2.)
Because the path of the hand merged gradually with the path of the
target, we could not measure this time with any confidence. However,
using the simple model we could test a corollary of the hypothesis: the
motion of the finger during the "reacquisition phase" is in a
direction such as to intercept the target at a constant time. For the
reacquisition phase of the finger, we predicted the direction of travel
() such that the points of interception were a constant distance
(dt) away from the point in space at which the target changed direction, assuming a constant distance of
travel (dr) during the reaction time.
Therefore by trigonometry, for each target direction the distance the
finger must travel to intercept the target is simply
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To evaluate the extent to which this model fit the experimental data,
we used an error minimization algorithm (Nelder and Mead
1964) to compute and compare the predicted heading of the finger (Eq. 2) with the heading measured experimentally.
Figure 3A shows how well the
model was able to predict the heading of the finger. The plot
illustrates the angular difference between the initial heading of the
finger and the heading of the target during the reacquisition phase
plotted as a function of target direction. The crosses represent the
average angular differences recorded for the 24 target directions for
one subject. The smooth curve represents the angular differences
calculated by the model. For this subject, a ratio of target
reacquisition distance (dt) to
reaction distance (dr) of 3.63 provided the best fit. [Assuming a reaction time of 250 ms, this would
imply a time to reacquire the target (reaction time + interception
time) of 900 ms.] The model accounted for 94% of the variance in
angular differences across directions for this subject. For the four
subjects, the ratio of
dt/dr
ranged from 2.51 to 3.63 (see Table 1),
and the variance accounted for by the model was above 0.85 in each
case.
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The simple conceptual model also makes clear predictions about the maximum speed of the finger during the reacquisition phase. If one is to intercept a target moving at a constant speed at a constant time, the average speed of the finger must scale proportionally to the distance traveled (di). If the speed profile is similar for each direction (e.g., is bell-shaped), then the maximum speed should be proportional to di. Therefore the variation in maximum finger speed with direction should be predicted by Eq. 1. Figure 3B displays the average peak speed reached by the finger for another subject as a function of target direction (crosses). The smooth curve represents the prediction of the model. Finally, the horizontal line indicates the target speed (10.8 cm/s). For this subject, the scaling factor relating interception distance to peak velocity was 2.73, while the variance in peak speed accounted for by the scaled distance was 0.93. For the four subjects, the scaling factor relating interception distance to peak velocity ranged from 2.70 to 4.54 (Table 1), and the variance in peak speed during the reacquisition period accounted for by the distance to the reacquisition circle ranged from 0.89 to 0.94.
To better demonstrate the modulation of the speed profile as a function of interception distance, finger speed was plotted in two different formats in Fig. 4. Figure 4A shows how speed varied with time for one subject for 12 of the 24 target directions. The traces all begin at the time the target changed direction; the baseline indicates the speed of the target. In each case, initially the speed of the finger was slightly slower than the target's speed, indicating that this subject was slowly starting to fall behind the target (perhaps anticipating the change in target direction). Then after a nearly uniform period of time (the subject's reaction time), the finger's speed changed, first slowing to allow for the change in direction of the finger, then accelerating to reacquire the target. (For 0°, there was no change in target direction and the finger maintained its original velocity throughout.) As the amount by which target motion changed direction became greater, the amount of deceleration as well as acceleration also increased such that the time to reacquire the target remained constant.
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Figure 4B, where the traces for the 13 directions ranging from 0 to 180° (the right hemisphere) have been superimposed, provides further support for our hypothesis. One can observe that there are only minor variations in the reaction time. Furthermore the duration of the reacquisition phase does not depend on target direction; after ~750 ms the finger has re-assumed the velocity of the target. In summary, since the distance to intercept the target is not constant, it appears that both the direction and speed of the finger's motion are coordinated in such a manner that the time to intercept, or possibly the distance the target travels before interception, is held constant.
Obliquely moving targets (experiment 2)
The initial path of the finger in the first experiment was always straight downward. It is possible that the tracking of a downward moving target is somehow unique or that gravity might have an effect on tracking performance. To determine whether the results from experiment 1 can be generalized to other tracking directions, the experiment was repeated with the target initially moving obliquely at an angle of 135°. Figure 5 shows the paths of the target and of the finger to 8 of the 24 directions (left). As was the case in experiment 1, the hand generally headed in a straight line to intercept the target. Also as was true for initial downward target motion, the simple model was able to account for the angular differences between finger heading and target heading as well as modulation in the finger's peak speed (right).
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For the four subjects, the ratio of target interception distance (dt) to reaction distance (dr) varied from 2.84 to 3.77. In this experiment, the variance accounted for by the model ranged from 0.92 to 0.97. There was one subject who participated in this experiment as well as in experiment 1. The results for this subject were almost identical, with ratios of 3.71 and 3.63 respectively. Again, in this second experiment, the peak speed for all subjects was well correlated to the target interception distance. For the four subjects in this experiment, the factor relating distance to peak speed ranged from 3.17 to 4.44 with the variance in peak speed accounted for by the distance to reacquisition ranging from 0.86 to 0.95.
Effect of target speed on tracking performance (experiment 3)
In the first two experiments, the speed of the target was the same
for all trials. We observed that the initial heading of the hand ()
was such as to reacquire the target at a constant time. Since the
target moved at a constant speed, a constant distance to intercept
(dt, Eqs. 1 and 2) is
equally consistent with the data. By examining manual tracking at
different target speeds, it was possible to differentiate these two
possibilities. To accomplish this, we repeated experiment 1 varying the speed of the target randomly from trial to trial over a
fourfold range. As will be demonstrated in the following
text, if the goal is to keep time to intercept constant, the
initial heading of the finger (
) should not depend on target speed.
However, if the goal is to keep distance (dt) constant,
should depend in a
predictable manner on target speed, becoming larger for faster speeds
(to make up for the extra distance traveled during the reaction time).
Figure 6, A and B, shows the results from one subject for two different target directions. The thin line represents the path of the target, which was the same independent of target speed. The bold lines represent the average finger paths for each of the different target speeds. There was a small but statistically significant (ANOVA, averaged data for all subjects and all directions, P < 0.01) effect of speed on reaction time. A post hoc pairwise comparison showed that the reaction time for the slowest speed was longer (13%) than for the other three speeds. This difference in time was small (33 ms) compared with the mean reaction time of ~250 ms. Since the reaction time is approximately constant, the distance traveled during the reaction time is approximately a linear function of the target speed.
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After the finger changed direction, the four traces of the finger position during the reacquisition phase were parallel with each other. An ANOVA showed that for four of the five subjects, there was no effect of speed on reacquisition direction (P > 0.05). For the one remaining subject, linear regression showed that for only one of six target directions was the relation between speed and reacquisition direction significantly different from zero (with a slope of 0.44°/cm/s, corresponding to a 7° difference in heading between the slowest and fastest target speeds). In conclusion, the distance traveled by the finger during the reaction time scaled with target speed and the directions of the finger paths during the reacquisition phase were parallel to each other at the four speeds. Therefore from the law of similar triangles, the finger intercepted the path of the target at a distance that also scaled with target speed. Accordingly, since the target traveled at a constant speed throughout each trial, the time not the distance to reacquisition remained constant, irrespective of the speed of the target.
Response to an abrupt change in target speed (experiment 4)
In all of the experiments described so far the speed of the target was constant and thus predictable throughout any given trial. The question can then be posed, do the results generalize when speed as well as direction changes unpredictably during the trial? We explored this question by introducing, in some trials, a step change in the speed of the target at the same time its direction changed. Since the speed during the initial downward tracking segment was constant, the reaction distance remained relatively constant as well. This is evident in Fig. 7A where the traces for all three averages overshoot the change in target direction by nearly the same amount.
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The direction of the finger motion during the reacquisition phase depended on the new speed of the target. Figure 7B shows the results of fitting Eq. 2 to these data. The three curves of the model were calculated such that the time to intercept was the same for all three speeds (i.e., dt was proportional to target speed). For this subject, the model explained 89% of the variance for those trials where the target abruptly slowed, 95% of the variance for those trials where speed did not change, and 99% of the variance for those trials where the target abruptly increased speed.
Therefore it appears that the subject was able to detect the change in speed of the target and scale the speed of the interception trajectory such that time to contact remained constant. This can also be appreciated in Fig. 7A. The three tick marks are equally spaced and represent the position of the target at the modeled interception time for the three speeds. The location of these ticks correspond well to where the extrapolated linear motion of the finger intercepts the path of the target. The results in Fig. 7 are representative of the results for the five subjects who participated in this experiment. For all subjects, the constant time to intercept model was able to fit the data at all three speeds with a considerable amount of fidelity (Table 2).
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Figure 8 demonstrates that the reaction
time for a change in target speed is similar to the reaction time to a
change in target direction. The plots provide a comparison of the two
instances in which the target either increased or decreased in speed at the time of the directional change. The dashed line indicates the time
at which the target changed both direction and speed. The top
panel describes the speed of the target, while the bottom panels show the speed and direction of the finger for both
conditions. We performed a statistical comparison between the finger
data for the two target speeds to determine the time at which the two curves first diverged (P < 0.05, see
METHODS). In Fig. 8, this time is denoted (). For this
subject, the speed traces were found to diverge 250 ms after the target
changed speed. Note that finger speed decreases at about the same time
(~175 ms) for both target speeds, but that the finger begins to
accelerate earlier when the target speed is increased. Across all
subjects and directions the average time at which the speed of the
finger first differed significantly (P < 0.05) for
slow and fast target speeds was 270 ± 40 ms after the change in
target speed, slightly larger than the reaction time found for finger
speed to change in response to a change in target direction.
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Direction of finger motion also depended on target speed, with a reaction time that was again comparable to that found for a change in target direction. In this instance (Fig. 8), the direction of the finger diverged 180 ms after the target changed speed. Across all subjects and directions, finger direction for the two target speeds diverged on average 280 ± 80 ms after the target changed speed. This value is comparable with the reaction time for the finger direction to change in response to a change in target direction.
Tracking a constant downward velocity (experiment 5)
There has been considerable work, both in terms of experimentation
and modeling, attempting to understand manual tracking performance in
one dimension (cf. Poulton 1974; Viviani et al. 1987
). One might conjecture that tracking in two dimension
(2-D) is equivalent to two simultaneous cases of one dimensional
tracking occurring in the X and Y directions. To
ascertain whether this viewpoint was viable, we conducted the following
experiment. As in previous experiments, target direction changed
randomly through angles encompassing all 360°. However, for those
cases in which the change in direction was <90°, (maintaining a
downward directional component), the speed of the target was modified
such that the vertical (Y) velocity remained constant, and
the horizontal (X) velocity underwent a step change. If
manual tracking can be decomposed into two cases of independent
tracking along orthogonal axes, then one would expect that the addition
of an X component should not perturb the tracking in the
vertical dimension.
Figure 9, left, shows the path of the target as well as the average path of the finger for one of these downward trajectories. Figure 9, right, displays the X and Y components of the target and finger velocity. - - - indicates the time at which the target changed direction. For the X component of finger motion, after ~200 ms the finger accelerated, exceeded, and eventually matched the velocity of the target. At a comparable latency (~200 ms), the Y velocity of the finger first decreased, then increased to reacquire the target, even though the Y velocity of the target did not change. The same general pattern was seen in all four subjects for all angles. It is therefore clear that tracking behavior in two dimensions cannot adequately be expressed as two independent cases of one-dimensional tracking occurring simultaneously along orthogonal axes.
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Response to target acceleration (experiment 6)
In the preceding (experiment 4), we showed that target speed influenced the direction of finger motion. Can target acceleration also influence the direction of the reacquisition movement? To address this question, in a final experiment, the target either maintained its original speed or accelerated or decelerated, coincident with the change in target direction. For the two examples shown in Fig. 10, A and B, there is no discernable effect of target acceleration on the direction of finger motion. Figure 10, C-E, shows how the movement described in the left-hand panels evolved over time for the cases of accelerating and decelerating targets. The speed of the finger for the two movements did not begin to diverge until 590 ms after the target changed direction (arrow in Fig. 10D). For all subjects and all directions, target acceleration did not begin to have an effect on finger speed until 490 ± 80 ms after the target changed direction. The direction of finger motion was nearly identical for both the accelerating and decelerating targets (Fig. 10E). For this subject, target acceleration had no significant effect on the direction of finger motion. In only 11 of 50 cases was there a statistically significant divergence in finger direction at any point in the movement, at an average time of 450 ± 120 ms.
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Quantitative modeling of manual tracking in two dimensions
A simple conceptual model in which the direction of finger motion
changes abruptly in response to a change in the direction of target
motion so as to intercept the target at a constant time was able to
account for a large body of experimental data. This model is compatible
with the idea of intermittent control during tracking (Miall et
al. 1986; Young and Stark 1963
). Specifically, our results might be interpreted to imply that there is one major correction in the finger's trajectory, secondary corrections perhaps occurring as the finger's trajectory eventually merges with that of
the target. However, tracking behavior is more commonly modeled to be
under continuous control (Krauzlis and Lisberger 1994
;
Lisberger et al. 1987
; Viviani et al.
1987
). The question then arises: can the observed behavior be a
consequence of the workings of continuous feedback control? We will
show that this is indeed the case.
A CARTESIAN MODEL.
We begin with analytical models of tracking similar to models for
tracking in one dimension that have been used previously to account for
two-dimensional manual tracking (Viviani et al. 1987).
In this form of model, the acceleration of the finger
(
F) is related to a positional
(ep) and a velocity
(ev) error signal
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EXTENSIONS TO THE CARTESIAN MODEL. The errors in matching finger speed suggest that a nonlinear model might give an improved fit, i.e., that finger acceleration would depend on the square and/or the cube of the error terms (ep and ev) in Eq. 3. We can rule out a quadratic nonlinearity since it would violate the mirror symmetry that was obtained for targets moving to the right or to the left (Fig. 2). We did try a model including a cubic nonlinearity (which does not violate mirror symmetry) for the data for one subject. This nonlinear model gave a negligible improvement in the fit (<1% reduction in error).
Another modification to the model in Eq. 3 would be to replace the scalar coefficients a1 and a2 with matrices
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MODELING TRACKING BEHAVIOR IN CURVILINEAR COORDINATES.
As an alternative to the Cartesian model, we investigated models in
which the error signals are described in a curvilinear coordinate
system fixed to the hand (Flanders et al. 1992).
Specifically, we defined feedback error signals in directions
tangential and perpendicular to the finger's trajectory at each point
in time. In such a description, acceleration is defined by the rate of change of speed and direction
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DISCUSSION |
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When targets made an abrupt change in direction, after a reaction time, the hand changed direction and moved in a straight line so as to intercept the target at a fixed latency. This strategy was followed under a wide range of experimental conditions. It held true for all target angles when the target's direction was varied randomly, and it did not depend on the initial direction of the target's motion. Furthermore this held true across a broad range of speeds whether or not the speed was predictable. As we showed in the last section of RESULTS, this observation may be an emergent property of a control system in which speed and direction of motion are the controlled variables.
We did not anticipate this result. It is entirely possible that the time to intercept could have been minimized. At least, one could naively assume that a control scheme that minimized time to contact would be far more effective. If time to contact were minimized rather than held constant, the peak velocity during the reinterception phase would not scale with direction (Figs. 3-5). Our results clearly rule out this hypothesis.
The idea that time to intercept or () is a controlled variable has
been proposed by several investigators (Lee and Reddish 1981
; Port et al. 1997
; Savelsbergh et
al. 1993
; Tyldesley and Whiting 1975
). Moreover,
it has been suggested that
could be derived from optic flow
information. More generally, movement times to stationary targets tend
to remain constant (cf. Ghez and Vicario 1978
;
Martin et al. 1995
, Viviani and Flash
1995
). The fact that the time to intercept remains constant in
the current experiments (in which optic flow signals do not play an
important role) might indicate that
is indeed a controlled
variable. It could also be related to a control scheme that operates in
an intermittent fashion at a fixed frequency (Miall et al.
1986
; Neilson et al. 1988
; Young and
Stark 1963
). Our model, in which tracking behavior was modeled
in a local coordinate system fixed to the hand, was also able to
account for the experimental data. Thus the observations are consistent
with either intermittent or continuous control of tracking. Our
experiments were not designed to distinguish between these
possibilities. Imposing successive changes in the direction of target
motion at short intervals (Georgopoulos et al. 1981
;
Soechting and Lacquaniti 1983
) may be a means of testing
whether or not feedback control during tracking is intermittent.
Our second model suggests that a constant time to intercept in the
current experiments may be a consequence of the underlying control
algorithm. As was mentioned in the INTRODUCTION, it has been proposed that the error signals underlying manual tracking include
both position and velocity (Hanneton et al. 1997;
Viviani et al. 1987
). As demonstrated in
RESULTS, a control scheme whereby the direction
of finger movement is defined by the vector sum of target velocity and
positional error (Fig. 12B) can lead to a constant time to
intercept. For a target moving at a constant speed, at the reaction
time, the distance traveled by both the finger and the target would be
the same, irrespective of direction (arrows in Fig. 1). Therefore by
simple trigonometry, the vector sum of the positional error signal and
the target velocity signal at the end of the reaction time would result
in a directional error signal compatible with our results. The second
model, which incorporated this control scheme (see Fig. 12), was able
to reproduce the experimentally observed directional changes in finger motion.
Representations of speed, direction and acceleration in the error signals for tracking
In the present study we found similar reaction times for a change
in target speed and a change in target direction when these were
computed as the times at which speed or direction deviated significantly from the control values. The model gave different results
(Table 4) with a reaction time for changes in speed that were much
lower than the reaction time to changes in direction. While the model
may have underestimated the reaction time for speed (see for example
Figs. 11 and 14), we suspect that the estimates provided by model
2 are more accurate than the statistical estimates derived from
the averaged data. Given the natural variability in finger speed, this
value must change by a substantial amount for the change to attain
significance and consequently the statistical method most likely
overestimated the reaction time for speed. This would be especially
true for instances in which speed changed very little (Fig. 4). The
values for the reaction times for directional changes obtained by
measurement and by modeling were much more similar especially for
dv. This may be due to the fact that the direction of finger motion during the control period was much less
variable than was the speed of the finger.
Recent studies have shown that speed and direction, the two parameters
that are explicit in our model, are represented in the firing of motor
cortical neurons as well as neurons in the cerebellum during tracking
tasks (Coltz et al. 1999; Johnson et al.
1999
). In experiment 6, the reaction time to target
acceleration was considerably longer than the reaction time to a step
change in speed or direction. Note that the amount of
acceleration/deceleration used was substantial, bringing the
decelerating target to a halt in 2 s. Therefore higher derivatives
such as acceleration do not appear to be strongly represented in the
error signal and our model, which did not include an acceleration error
signal, was able to account for the results of experiment 6 as well as it did for the other experiments. In fact, the reaction time
to target acceleration measured in the current study may reflect the
reaction time to the change in target speed brought about by the
acceleration. These conclusions agree with recent observations by
Port et al. (1997)
concerning a manual interception task
and psychophysical studies by Werkhoven et al. (1992)
.
Electrophysiological studies of the medial temporal cortical area (MT,
an area known to be important in the processing of visual motion) have
shown that direction and speed are coded by the firing frequency of
individual neurons but that partial information about acceleration is
present only in the population response of these neurons
(Lisberger and Movshon 1999
).
Oculomotor and manual tracking in two dimensions
Eye-hand coordination underlies the execution of many motor tasks
(Herman et al. 1981; Mather and Lackner
1980
; Vercher and Gauthier 1988
), and it has
been suggested that the tight coupling between eye and arm is evidence
that both these systems utilize the same error signal (Herman et
al. 1981
). Eye movements are subserved by two anatomically
distinct systems for the horizontal and vertical control of eye
movement (Leigh and Zee 1991
). Nevertheless, analogous
to our present results, the control of smooth pursuit eye movements
also cannot be decomposed into independent horizontal and vertical
components. Leung and Kettner (1997)
recently studied the oculomotor tracking of predictable targets moving in two dimensions by monkeys. They found that a perturbation of either the horizontal or
the vertical components of the target's trajectory affected both components of the monkey's response.
We recently completed a study of oculomotor tracking using essentially
the same paradigm described in the current study (Engel et al.
1999). In an analogous manner to manual tracking, oculomotor tracking has been considered to be driven primarily by a positional error signal as well as a velocity signal (Lisberger et al.
1987
). In contrast to manual tracking, oculomotor tracking is
characterized by two unique and fairly independent systems, the smooth
pursuit system and the saccadic system, which work in tandem during
tracking tasks. We found that during tracking in two dimensions, the
saccadic system was primarily driven by the positional error between
the target and the gaze position of the eye. In contrast, the smooth pursuit system reduced the velocity mismatch between the target and the
eye, gradually aligning the direction of eye movement to the direction
of target motion.
Two-dimensional tracking reveals numerous similarities which exist
between eye and manual tracking. Neither system can be adequately
described by two independent cases of tracking in one dimension. In
addition, both systems appear to use error signals which at some level
incorporate both positional and directional error as well as speed
mismatch. Furthermore both systems appear to be much less responsive to
errors in acceleration (Lisberger et al. 1987).
Therefore it is possible that both systems use the same error signals
derived from the original retinal error, and may in fact share some of
the same trajectory planning apparatus, varying at some point due to
the obvious differences in the end effectors.
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ACKNOWLEDGMENTS |
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We thank Dr. Martha Flanders for helpful suggestions and comments.
This work was supported by a grant from the Human Frontiers Science Program, by National Institute of Neurological Disorders and Stroke Grant NS-15018, and by National Science Foundation Training Fellowship GER9454163.
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FOOTNOTES |
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Address for reprint requests: K. C. Engel, Dept. of Neuroscience, University of Minnesota, 6-145 Jackson Hall, 321 Church St. SE, Minneapolis, MN 55455.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 18 October 1999; accepted in final form 2 March 2000.
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REFERENCES |
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