Manual Tracking in Two Dimensions

Kevin C. Engel and John F. Soechting

Department of Neuroscience, University of Minnesota, Minneapolis, Minnesota 55455


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

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.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

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.


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

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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Fig. 1. An example of tracking performance during a directional change in target motion. Top: results from a single trial. The thin line denotes the path of the target with the arrows indicating the direction of target motion. The open arrow (slightly below the start of the target's trajectory) marks the starting position of the finger. After the change in target direction, the position of the finger and position of the target are connected every 100 ms by a thin straight (isochronic) line. Inset: description of how angular changes, both of the target and of the finger, are defined with counterclockwise rotations from the vertical defined positive. Bottom: all 10 trials from this subject.

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.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

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 theta  = 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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Fig. 2. Tracking paths for 8 of 24 target directions. Dotted lines represent the path of the target and the thick lines represent the average trajectories of the finger. Inset: approximation to the behavior: the finger trajectory is represented by 2 straight lines. With this approximation, the finger appears to intercept the target at a constant time, represented by the circle centered on the point where the target changes direction. Thin lines emanate from the finger position at the end of a constant reaction time (250 ms) and intersect the target paths on the perimeter of the circle. Note that these lines generally parallel the straight line portion of the finger path following the reaction time. (The fit between the model and experimental data are better for target changes to the right, as the finger position on the touch screen deviates slightly for right-handed subjects moving toward the left hand of the screen.)

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 (alpha ) 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
<IT>d</IT><SUB><IT>i</IT></SUB><IT>=</IT><RAD><RCD><IT>d</IT><SUP><IT>2</IT></SUP><SUB><IT>r</IT></SUB><IT>+</IT><IT>d</IT><SUP><IT>2</IT></SUP><SUB><IT>t</IT></SUB><IT>−2</IT><IT>d</IT><SUB><IT>r</IT></SUB><IT>d</IT><SUB><IT>t</IT></SUB><IT> cos &thgr;</IT></RCD></RAD> (1)
Furthermore, the direction to intercept the target is given by
&agr;=arcsin <FENCE><FR><NU><IT>d</IT><SUB><IT>t</IT></SUB></NU><DE><IT>d</IT><SUB><IT>i</IT></SUB></DE></FR><IT> sin &thgr;</IT></FENCE> (2)
The straight lines in Fig. 2 represent the idealized finger paths of the model, from the point representing the reaction distance (dr) to the point where the path of the target intercepts the circle. The model fits the data reasonably well. In this example, the fit appears better for counterclockwise changes in target direction. The larger error for clockwise rotations may be due to a slight rotation of finger position on the touch screen. This typically occurred during leftward movements of the right-handed subjects.

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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Fig. 3. A comparison between the model and experimental performance. Experimental data (+) for 1 subject and the model's prediction are shown for the direction (A) and maximum velocity (B) of the finger's trajectory. In A, angular difference is defined as the difference between the angular change of the finger and the angular change of the target. Finger heading was determined by taking the inverse tangent of the ratio of horizontal and vertical velocities at 350 ms after the target changed direction. In B, the straight line represents the speed of the target.


                              
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Table 1. Ratio of target reacquisition distance to reaction distance

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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Fig. 4. The temporal profile of finger speed during target reacquisition. A: the finger speed for 12 of the 24 target directions for 1 subject. In each case, the baseline indicates the speed of the target. The horizontal and vertical scales are identical for each trace. Each trace begins at the point when the target changed direction. The variations in speed have been superimposed for 13 of the 24 total target directions (half of the circle) in B. Line thickness increases with increasing angular changes of the target.

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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Fig. 5. Tracking obliquely moving targets. Left: path of the target, which always started in the lower-left corner of the screen and the average path of the finger during tracking to 8 of the 24 directions tested. Right: comparison between the model and tracking performance of the obliquely moving targets. The conventions are the same as those used in Fig. 3.

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 (alpha ) 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 (alpha ) should not depend on target speed. However, if the goal is to keep distance (dt) constant, alpha  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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Fig. 6. The effect of target speed on tracking performance. Thin lines indicate the path of the target traveling at 4 speeds (5.4, 10.8, 16.2, and 21.6 cm/s), changing by 135° (left) and 60° (right). Bold lines demonstrate the average path of the finger for each of the 4 speeds. After the reaction time, finger paths are parallel to each other, intercepting the target at a constant time.

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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Fig. 7. The effect of changing target direction and target speed simultaneously. The thin line in A represents the path of the target. Target speed was constant (10.8 cm/s) during the downward phase. When the target changed direction, target speed either decreased (5.4 cm/s), increased (16.2 cm/s), or remained constant. Bold lines show the average path of the finger during tracking. The scale along the target path demonstrates the position of the target at a constant time for the 3 speed levels. B: a comparison of the direction of finger motion with that predicted by the constant time to intercept model. The angular difference between target and finger heading is shown for each target speed. Smooth curves represent the angular difference predicted by the model assuming the same time to intercept for all 3 speeds.

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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Table 2. Target direction variance as accounted for in experiment 4 

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 (down-arrow ). 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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Fig. 8. Reaction time to a change in target speed. - - -, point in time at which target speed and direction were varied. The target either increased to 16.2 cm/s or decreased to 5.4 cm/s (top). Bottom traces: time profiles of finger speed and direction. down-arrow , time at which the 2 movements diverged from each other.

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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Fig. 9. Tracking a target with a constant downward velocity. Left: the average target and finger paths for 1 direction for trials in which the vertical velocity was constant. Right: horizontal and vertical motion of the finger and the target vs. time. - - -, time at which the target changed direction.

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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Fig. 10. The effect of target acceleration on tracking performance. A and B: 2 examples of tracking an accelerating target. Thin lines indicate the path of the target for 3 acceleration profiles. During the downward phase of target motion, the target maintained a constant speed of 10.8 cm/s. At the point at which the target changed direction, the target speed either accelerated or decelerated at a rate of 5.4 cm/s2 or remained constant. The thick lines represent the finger paths for each of these acceleration profiles. C-E: increasing and decreasing target accelerations. The dotted line indicates the time at which the target changed direction and accelerated. The arrow indicates the time at which the finger speed profiles for accelerating and decelerating targets diverged. For this subject, finger direction did not differ for the 2 target motions.

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 (&puml;F) is related to a positional (ep) and a velocity (ev) error signal
<B><A><AC>p</AC><AC>¨</AC></A></B><SUB><IT>F</IT></SUB>(<IT>t</IT>)<IT>=</IT><IT>a</IT><SUB><IT>1</IT></SUB><B>e</B><SUB><IT>p</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>p</SUB></IT>)<IT>+</IT><IT>a</IT><SUB><IT>2</IT></SUB><B>e</B><SUB><IT>v</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>v</SUB></IT>) (3)
where
<B>e</B><SUB><IT>p</IT></SUB>(<IT>t</IT>)<IT>=</IT><B>p</B><SUB><IT>T</IT></SUB>(<IT>t</IT>)<IT>−</IT><B>p</B><SUB><IT>F</IT></SUB>(<IT>t</IT>)

<B>e</B><SUB><IT>v</IT></SUB>(<IT>t</IT>)<IT>=</IT><B><A><AC>p</AC><AC>˙</AC></A></B><SUB><IT>T</IT></SUB>(<IT>t</IT>)<IT>−</IT><B><A><AC>p</AC><AC>˙</AC></A></B><SUB><IT>F</IT></SUB>(<IT>t</IT>)
The subscripts T and F denote the target and the finger, respectively and p is a vectorial representation of the position {px, py}. The coefficients a1 and a2 are constants, as are the two time delays tau p and tau v. This model leads to two uncoupled equations, one in x and one in y
<IT><A><AC>p</AC><AC>¨</AC></A></IT><SUB><IT>Fx</IT></SUB>(<IT>t</IT>)<IT>=</IT><IT>a</IT><SUB><IT>1</IT></SUB><IT>e</IT><SUB><IT>px</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>p</SUB></IT>)<IT>+</IT><IT>a</IT><SUB><IT>2</IT></SUB><IT>e</IT><SUB><IT>vx</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>v</SUB></IT>)

<IT><A><AC>p</AC><AC>¨</AC></A></IT><SUB><IT>Fy</IT></SUB>(<IT>t</IT>)<IT>=</IT><IT>a</IT><SUB><IT>1</IT></SUB><IT>e</IT><SUB><IT>py</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>p</SUB></IT>)<IT>+</IT><IT>a</IT><SUB><IT>2</IT></SUB><IT>e</IT><SUB><IT>vy</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>v</SUB></IT>) (4)
where epx and epy are the x and y components of the position error vector and &puml;Fx and &puml;Fy are the x and y components of the finger acceleration.

This model is incompatible with the results presented in Fig. 9. In experiment 5, there was no perturbation along the y axis and therefore the positional and velocity error terms on the left side of Eq. 4 (epy and evy) are zero. Accordingly, the y component of finger acceleration is predicted to be zero by this model. Nevertheless it is instructive to consider its other predictions in more detail.

Figure 11, left, shows the finger trajectory and the time course of speed and direction predicted by this model (model 1) for one subject for experiment 1. Equation 3 was solved using a Runge-Kutta scheme (Press et al. 1992) and an iterative error minimization algorithm to identify the four parameters (a1, a2, tau p and tau v) that gave the best fit to data. We used the 11 directions in which target motion deviated to the right and minimized the mean square difference between finger and model speeds and directions (directional error being weighted 1/10th as much as speed, to give comparable weight to both). We began the model using as initial conditions, finger speed and direction 800 ms after the onset of target motion (i.e., at about -300 ms in the examples in Fig. 11). As noted, the subjects' tracking speed (before the target changed direction) was generally less than the speed of the target. The analytical model attempted to bring tracking speed back up to the target's value and accordingly there is an initial acceleration in the model's response (at the time the dotted and solid curves diverge in Fig. 11).



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Fig. 11. Comparison of the behavior of 2 analytical models in capturing tracking behavior in trials in which the target changed direction (experiment 1). In model 1, the finger acceleration vector is related to vectorial error signals of position and velocity. In model 2, acceleration is defined by its normal and tangential component. The tangential component is related to a vectorial error signal in position and velocity (as in model 1), but the normal component is related to a directional error signal. Results of the modeling for 3 directions are shown for subject 2. For each direction, the paths of the finger and the target are shown to the left and the temporal variation in speed and direction is shown to the right. The dotted traces represent the performance of the models. Note that model 2 provides a better fit to the time course of the change in direction of finger motion as well as the time course of the variations in speed.

From Fig. 11, it is clear that this model gave a reasonable fit to the data. It does predict the heading of the finger after the target changed direction (see the left-most plots in Fig. 11 which show the finger paths). The discrepancy in finger paths between experimental data and the model arises because, as noted previously, subjects lagged behind the target prior to the time at which the target changed direction, whereas the model attempted to match finger speed to the target speed. Model 1 accounted for 96.8% of the variance in speed and direction for this subject, averaged over the 11 directions. (Parameter values and the goodness of fit of model 1 are reported in Table 3.)


                              
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Table 3. Best fit parameters for model 1 

Despite the apparent success of model 1, there are consistent discrepancies between this model and the experimental data for this subject as well as for the others. First, the model failed to predict the maximum speed of the finger; maximum speed of the model generally being substantially lower than the actual data. Second, the model failed to match the time course of the directional change of finger motion. It did provide a good match to data in the bottom-most example, but it anticipated the change in direction for the other two examples. Furthermore, in the top two cases, the rate of change in direction was much slower than the experimental data. The results shown in Fig. 11 are representative of the results obtained for the other subjects as well. In all cases, the time course of the change in direction was well matched when the directional change was large (as in Fig. 11, bottom), but the model's response was in advance of the experimental data when the directional change in target motion was more modest.

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
<IT>a</IT><SUB><IT>1</IT></SUB><IT>=</IT><FENCE><AR><R><C><IT>a</IT><SUB><IT>xx</IT></SUB></C><C><IT>a</IT><SUB><IT>xy</IT></SUB></C></R><R><C><IT>a</IT><SUB><IT>yx</IT></SUB></C><C><IT>a</IT><SUB><IT>yy</IT></SUB></C></R></AR></FENCE> (5)
and similarly for a2. Such a model was proposed by Viviani and Monoud (1990), and it would be compatible with the x-y interaction presented in Fig. 9. However, we found in experiment 2 that the behavior was invariant under a rotation, i.e., that the response did not depend on the initial direction of the target motion (Fig. 5). Together with the mirror symmetry in the response (Fig. 2), these observations require that
<IT>a</IT><SUB><IT>xy</IT></SUB><IT>=</IT><IT>a</IT><SUB><IT>yx</IT></SUB><IT>=0 </IT><IT>a</IT><SUB><IT>xx</IT></SUB><IT>=</IT><IT>a</IT><SUB><IT>yy</IT></SUB>
Therefore there does not appear to be a simple way to improve the Cartesian model represented by Eq. 3.

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
<B><A><AC>p</AC><AC>¨</AC></A></B><SUB><IT>f</IT></SUB><IT>=<A><AC>&ugr;</AC><AC>˙</AC></A></IT><B><A><AC>t</AC><AC>ˆ</AC></A></B><IT>+&ugr;<A><AC>&thgr;</AC><AC>˙</AC></A></IT><B><A><AC>n</AC><AC>ˆ</AC></A></B> (6)
where upsilon  is the speed, theta  is the direction of motion, and t and n are unit vectors in the tangential and normal directions, respectively (see Fig. 12A). (The normal component of the acceleration can also be written as upsilon 2/R, where R is the radius of curvature.) We chose this frame of reference because it makes speed and direction explicit parameters in the model and because the experimental data suggested these two parameters were important controlled variables.



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Fig. 12. Schematic defining the error signals in model 2. A: the path of the finger, where t and n denote the tangential and normal directions and theta  denotes the direction of finger movement. B: how the directional target signal phi  is defined. pT and pF denote the position of the target and of the finger, and &pdot;T indicates the target velocity. The angle phi  is defined by a vector from the finger to target positions (pT - pF) plus the target velocity vector.

We began with a model in which the tangential and normal accelerations were proportional to the components of the positional (ep) and velocity (ev) error terms (Eq. 3) along these two directions
<A><AC>&ugr;</AC><AC>˙</AC></A>=<IT>a</IT><SUB><IT>1</IT></SUB><IT>e</IT><SUB><IT>pt</IT></SUB><IT>+</IT><IT>a</IT><SUB><IT>2</IT></SUB><IT>e</IT><SUB><IT>vt</IT></SUB>

&ugr;<A><AC>&thgr;</AC><AC>˙</AC></A>=<IT>a</IT><SUB><IT>3</IT></SUB><IT>e</IT><SUB><IT>pn</IT></SUB><IT>+</IT><IT>a</IT><SUB><IT>4</IT></SUB><IT>e</IT><SUB><IT>vn</IT></SUB> (7)
If a1 is equal to a3 and a2 is equal to a4, this model will give the same results as the Cartesian model in Eq. 3. (As a test of the simulations, we verified that this was the case.) We next let the four coefficients a1-a4 vary independently and found that this gave only a modest improvement in the fit. Furthermore, a1 and a3 differed by <15%, as did a2 and a4.

We then proceeded to test different error signals, beginning with the equation for the normal acceleration. We chose the difference between the present direction theta (t) and a desired direction phi (t) to be the directional error signal. We defined this desired direction based on the observation that the initial direction of motion appeared to be such as to intercept the target at a constant time. This could come about if the desired direction were defined by the vector sum of a positional error signal and the target velocity signal
&phgr;(<IT>t</IT>)<IT>=∡ </IT>{<B><A><AC>p</AC><AC>˙</AC></A></B><SUB><IT>T</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>dv</SUB></IT>)<IT>+</IT><IT>b</IT><SUB><IT>5</IT></SUB>[<B>p</B><SUB><IT>T</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>dp</SUB></IT>)<IT>−</IT><B>p</B><SUB><IT>F</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>dp</SUB></IT>)]} (8)
where the ∡ denotes the angle that the vector in brackets makes with a reference direction (see Fig. 12B). The coefficient b5 is a constant, as are the time delays tau dv and tau dp.

Then, the simplest model for the finger acceleration in the normal direction is
&ugr;<A><AC>&thgr;</AC><AC>˙</AC></A>(<IT>t</IT>)<IT>=</IT><IT>b</IT><SUB><IT>4</IT></SUB>[<IT>&thgr;</IT>(<IT>t</IT>)<IT>−&phgr;</IT>(<IT>t</IT>)] (9)
According to Eq. 9, the rate of change in direction of finger motion (<A><AC>&thgr;</AC><AC>˙</AC></A>) will be zero when theta  is equal to phi , i.e., the finger will move in a straight trajectory in the direction given by phi . Thus qualitatively, it appears that Eq. 9 could account for the experimental data.

For the acceleration in the tangential direction, we used an error term that was similar to the one we used in the Cartesian model (see Eq. 7). This choice was motivated by the observation that model 1 gave a reasonable fit to the speed of the finger, with a suggestion of a nonlinearity. We defined the error signal for speed to be
<IT>e</IT><SUB><IT>s</IT></SUB>(<IT>t</IT>)<IT>=</IT>[<IT>b</IT><SUB><IT>1</IT></SUB><B>e</B><SUB><IT>p</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>v</SUB></IT>)<IT>+</IT><IT>b</IT><SUB><IT>2</IT></SUB><B>e</B><SUB><IT>v</IT></SUB>(<IT>t</IT><IT>−&tgr;<SUB>v</SUB></IT>)]<IT>·</IT><B><A><AC>t</AC><AC>ˆ</AC></A></B>(<IT>t</IT>) (10)
where es represents the components of the positional and velocity errors terms, defined as before, in the tangential direction. The tangential acceleration of the finger is then given by
<A><AC>&ugr;</AC><AC>˙</AC></A>(<IT>t</IT>)<IT>=</IT><IT>e</IT><SUB><IT>s</IT></SUB>(<IT>t</IT>)<IT>+</IT><IT>b</IT><SUB><IT>3</IT></SUB><IT>e</IT><SUP><IT>2</IT></SUP><SUB><IT>s</IT></SUB>(<IT>t</IT>) (11)
As was the case for the first model, Eqs. 9 and 11 were integrated, using an iterative search over the five coefficients b1-b5, and the three time delays tau v, tau dv, and tau dp to obtain the best fit to the data.

The results of this procedure, for one subject for experiment 1, are shown in the right column of Fig. 11. Note that this second model gave a much improved fit, matching the time course of the direction of finger motion and the variations in finger speed much better than did the first model. For this subject, the second model gave a 47% decrease in the error of the fit. This was typical for all subjects, as can be appreciated in Table 4. Furthermore the model was also able to fit the results of experiments 4 and 6, in which target speed underwent a step change or accelerated at a constant rate, as can be seen in Figs. 13 and 14. For all three experiments, model 2 consistently gave an improved fit, with an average decrease of 46% in the error over model 1. 


                              
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Table 4. Best fit parameter values for model 2 



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Fig. 13. Performance of model 2 in matching tracking behavior in trials in which target speed underwent a step change. The data are the same as in Fig. 8 and are from subject 7.



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Fig. 14. Performance of model 2 in matching tracking behavior in trials in which target speed accelerated or decelerated. The data are the same as in Fig. 10 and are from subject 10.

It might be argued that such an improvement in fit should not be unexpected since the second model has eight parameters, whereas the first model only has four. However, even a simple version of the model, in which the quadratic nonlinearity was omitted and tau dv was set equal to tau v, gave a significant improvement in fit (by 21% for subject 1 in experiment 1) in the instance in which this model was tested. To the contrary, adding more free parameters to the first model (by including nonlinear terms) did not improve the fit. As we have noted, extending the first model to a matrix formulation is precluded by our experimental results.

As can be appreciated in Table 4, the parameter values that gave the best fit to the data were highly consistent from subject to subject and for all three experiments. This is especially true for the three time delays. The time delay for the error signal for speed (tau v) was consistently the lowest, with an average value of 115 ms, whereas the time delay for the velocity component of the directional error signal (tau dv) was consistently the largest, with an average value of 258 ms.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

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 (tau ) 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 tau  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 tau  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 tau 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.


    ACKNOWLEDGMENTS

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.


    FOOTNOTES

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.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

0022-3077/00 $5.00 Copyright © 2000 The American Physiological Society