Howard Hughes Medical Institute, Department of Physiology, Neuroscience Graduate Program, and W. M. Keck Foundation Center for Integrative Neuroscience, University of California, San Francisco, California 94143
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ABSTRACT |
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Churchland, Mark M. and Stephen G. Lisberger. Experimental and Computational Analysis of Monkey Smooth Pursuit Eye Movements. J. Neurophysiol. 86: 741-759, 2001. Smooth pursuit eye movements are guided by visual feedback and are surprisingly accurate despite the time delay between visual input and motor output. Previous models have reproduced the accuracy of pursuit either by using elaborate visual signals or by adding sources of motor feedback. Our goal was to constrain what types of signals drive pursuit by obtaining data that would discriminate between these two modeling approaches, represented by the "image motion model" and the "tachometer feedback" model. Our first set of experiments probed the visual properties of pursuit with brief square-pulse and sine-wave perturbations of target velocity. Responses to pulse perturbations increased almost linearly with pulse amplitude, while responses to sine wave perturbations showed strong saturation with increasing stimulus amplitude. The response to sine wave perturbations was strongly dependent on the baseline image velocity at the time of the perturbation. Responses were much smaller if baseline image velocity was naturally large, or was artificially increased by superimposing sine waves on pulse perturbations. The image motion model, but not the tachometer feedback model, could reproduce these features of pursuit. We used a revision of the image motion model that was, like the original, sensitive to both image velocity and image acceleration. Due to a saturating nonlinearity, the sensitivity to image acceleration declined with increasing image velocity. Inclusion of this nonlinearity was motivated by our experimental results, was critical in accounting for the responses to perturbations, and provided an explanation for the unexpected stability of pursuit in the presence of perturbations near the resonant frequency. As an emergent property, the revised image motion model was able to reproduce the frequency and damping of oscillations recorded during artificial feedback delays. Our second set of experiments replicated prior recordings of pursuit responses to multiple-cycle sine wave perturbations, presented over a range of frequencies. The image motion model was able to reproduce the responses to sine wave perturbations across all frequencies, while the tachometer feedback model failed at high frequencies. These failures resulted from the absence of image acceleration signals in the tachometer model. We conclude that visual signals related to image acceleration are important in driving pursuit eye movements and that the nonlinearity of these signals provides stability. Smooth pursuit thus illustrates that a plausible neural strategy for combating natural delays in sensory feedback is to employ information about the derivative of the sensory input.
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INTRODUCTION |
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Motor systems rely on sensory feedback. A primary function of feedback is to tell the system how its output differs from the intended output, and to guide corrective movements. In such cases, sensory inputs provide the system with negative feedback. Negative feedback systems have the practical advantage that they are resistant to both noise and miscalibration of internal signals. They also have the drawback that they are prone to instabilities if sensory feedback is not immediate. Sensory processing delays of 100 ms or more are common in the nervous system. How do motor systems maintain good performance in the face of such feedback delays if they rely heavily on a negative feedback architecture?
Smooth pursuit eye movements provide an ideal model system in which to
approach this problem. Pursuit has an explicit negative feedback
architecture because the source of its sensory inputs, the retina, is
attached to the motor effector, the eyeball. As illustrated in Fig.
1, the primary input to the pursuit
system is retinal image motion, defined as target motion with respect to the potentially moving eye (
)
(Rashbass 1961
). The pursuit system is designed to
minimize image motion by matching eye velocity to target velocity. The
image velocity input to pursuit thus provides both a feed-forward
signal that drives changes in eye velocity, and a feedback signal
regarding the adequacy of those changes. Feedback regarding the effect
of motor commands is delayed due the 60- to 130-ms latency between
visual input and motor response. Pursuit performance is generally much
better than would be expected given this feedback delay. For example, Fig. 2 shows typical pursuit responses to
a step-ramp of target position (Rashbass 1961
), which
delivers a step of target velocity. After a delay of approximately 100 ms, eye velocity increases rapidly, overshoots target velocity little
or not at all, and then either oscillates near target velocity with a
period of about 200 ms (Fig. 2B), or tracks target velocity
almost perfectly (Fig. 2C).
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Models of the pursuit system have been used extensively to form and test hypotheses about how neural systems achieve the performance measured in human and monkey subjects. The simplest pursuit model uses only a single visual input, an image velocity signal, and instantiates the "visual system" box in Fig. 1 as a simple gain. Eye acceleration would then be proportional to image velocity with a delay of 60-130 ms. In the absence of delay, such a "velocity-servo" model performs well, as eye acceleration is always in the direction that reduces image motion. However, in the presence of delay, the velocity servo model performs poorly and fails to account for a number of the features exhibited by pursuit. Figure 2, B and C, shows typical examples of eye velocity during pursuit, while the performance of the velocity-servo model is illustrated in Fig. 2D. The model exhibits much more overshoot than is seen in the data and oscillates with much too low of a frequency.
These failings of the simple velocity-servo model are eliminated, in
different ways, by three classes of pursuit model that have been
published previously, each of which is able to account for pursuit
responses to step changes in target velocity like those shown in Fig.
2, B and C. However, the three models use different control strategies and make fundamentally different predictions about the organization of the neural circuits that mediate
pursuit. Two of these, the "target velocity" model of Robinson
(Dicke and Thier 1999; Huebner et al.
1990
, 1992
; Pola and Wyatt 2001
;
Robinson et al. 1986
) and the "tachometer feedback" model of Ringach (1995)
, assume that pursuit eye
movements are guided by motor feedback signals, and that the visual
pathways driving pursuit are sensitive only to image velocity. The
third, the "image motion" model of Krauzlis and Lisberger
(1989
, 1994b
) assumes that pursuit is driven not
only by image velocity, but also by image acceleration.
Goldreich et al. (1992) have shown that the target
velocity model cannot account for the changes in spontaneous
oscillation frequency produced by altering the visual feedback delay.
We therefore focus on experiments and simulations designed to
discriminate between the tachometer feedback and image motion models,
and to determine whether the accuracy of pursuit is due to motor
feedback or to elaborated visual inputs. Our experiments provide new
data that could be reproduced only by a modified version of the image motion model. We conclude that the remarkable accuracy of pursuit in
human and nonhuman primates is due to visual inputs related to image
velocity and image acceleration. We further conclude that the
unexpected stability of pursuit, remarked on under a variety of
circumstances (Goldreich et al. 1992
; Ringach
1995
; Robinson 1965
), results from a
nonlinearity implied by our data.
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METHODS |
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Experiments on monkeys
The majority of pursuit data shown was collected specifically
for this paper. The exception is the data from monkey Jo in Fig. 11, which is reprinted from Fig. 3 of Goldreich et al.
(1992) and shows responses under conditions of artificially
increased feedback delays. These data were included because
Ringach (1995)
has argued that the image motion model
cannot account for them.
Data were obtained from three rhesus monkeys using methods that had
been approved in advance by the Committee on Animal Research at UCSF.
Using sterile procedure under isoflurane anesthesia, monkeys were
implanted with a platform that allowed head restraint and a scleral
search coil for monitoring eye movements (Judge et al.
1980). They were given postoperative analgesic doses of Buprenorphine (0.01 mg/kg) every 12 h for 2-3 days.
Monkeys were trained to track visual targets for a juice reward. During
the experiment, monkeys sat in a primate chair with their head
restraint device fixed to the ceiling of the chair. Targets were
projected onto the back of a tangent screen that was 114 cm in front of the monkey. A red fixation spot was provided by direct projection of
the image from a red light-emitting diode (LED) onto the center of the
screen. A 0.5° white moveable tracking target was created by
reflecting the beam from an optical bench off a pair of orthogonal mirror galvanometers. The fixation and tracking targets had luminances of 0.2 and 3.5 cd/m2, respectively. The room was
otherwise dark.
Stimuli were presented in individual trials. Daily experiments lasted about 2 h, during which we collected eye movement responses for 1,600 to 3,000 trials. Trials began with the appearance of the red fixation spot, which the monkey was required to fixate. After a variable delay of 700-1,100 ms, the fixation spot was extinguished, and the white tracking target appeared 1.5-2.5° to the left or right and began to move immediately. The exact eccentricity was set to reduce the occurrence of saccades and varied depending on the monkey. The target always moved toward and then well past the extinguished fixation point. The target moved for a variable period of at least 1,600 ms. Each monkey was required to track the target with an accuracy of 3° until it was extinguished, at which time he received a reward. Usually, the target moved at a constant velocity of 15°/s. In some trials, perturbations were imposed on the constant target velocity. Perturbations were either sinusoidal variations of target velocity, brief pulses of target velocity, or combinations of pulses and sine waves. The position excursion of the perturbations ranged from 0.06 to 1.2° and was therefore small enough so that their presence did not affect the monkey's ability to keep eye position within the window required for reward. Different trial types were presented in random order, weighted so that perturbations were present in only 25% of the trials.
Eye and target position and eye velocity were sampled at 1 kHz on each
channel. The eye velocity voltage was obtained by using an analog
circuit that differentiated signals at frequencies up to 25 Hz and
attenuated signals at higher frequencies (20 dB per decade). The
target position voltages were obtained as feedback from sensors on the
mirror galvanometers. Actual target velocity lagged commanded target
velocity by 2 ms or 7.2° at 10 Hz. Analog differentiation of eye
position voltages to create the eye velocity signal introduced phase
lag that was negligible at low frequencies but corresponded to up to 7 ms of time delay in the frequency range of 8-11 Hz. These lags were
corrected in the data analysis.
Saccades were identified by eye, and the resulting deflections of eye
velocity were replaced with straight-line segments. For our purposes,
the practice of linear interpolation is practically and theoretically
preferable to other alternatives, such as treating the excised saccades
as missing data (see Churchland and Lisberger 2000), although under our stimulus conditions the two
methods produce very similar results. For the majority of our data,
saccades were both small and rare, occurring about once per second,
although there was day-to-day and monkey-to-monkey variability in the
number of saccades. Trials were rejected in the rare (1-5% depending on the experiment) instances when saccades were numerous, indicating that smooth tracking was poor. For experiments in which we examined the
response to 100-ms-long perturbations of target velocity, trials were
rejected from the analysis if a saccade obscured the response or
occurred during the target perturbation. Additionally, after
discovering the effect of baseline image velocity on the response to
perturbations, for some analyses (Figs. 4 and 5) we rejected trials in
which eye velocity was not close to target velocity when the
perturbation was imposed. For some experiments, the requirement both
that eye velocity be near target velocity at the time of the
perturbation, and that no saccade interrupt the target perturbation or
obscure the response, caused as many as 40% of trials to be excluded
from analysis. Although we considered it important to exclude these
trials on principle, their inclusion had only the minor (and expected)
effect of making the average responses slightly smaller.
For each trial type, we aligned individual trials on the onset of
target motion and computed the average eye velocity evoked by the
target. We isolated the pursuit response to perturbations of target
velocity (Fig. 3C) by
computing the millisecond-by-millisecond difference between the average
response to target steps that included (Fig. 3A) or did not
include (Fig. 3B) a perturbation. We presented perturbations
on top of ongoing target motion because the pursuit system responds
well to high-frequency perturbations only after it has already been
engaged by some other target motion (Goldreich et al.
1992; Schwartz and Lisberger 1994
).
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For sinusoidal perturbations, we used the Fourier transform to compute the gain and phase shift of both monkey and model responses. The gain was estimated as the amplitude of the relevant frequency component of the eye velocity response, divided by the amplitude of the target velocity perturbation. The phase shift was estimated as the difference between the phase of the response and that of the target. For perturbations that delivered pulses of image velocity, response amplitude was calculated as the difference between the maximum and minimum eye velocities over a period that began 40 ms before the response and ended at the time of the peak of the response. To remove the small contributions of noise, we subtracted the same measure for trials in which perturbations were not presented. The impact of this correction was minor (<10%).
Computer simulations
Models were simulated on a DEC Alpha workstation using our revision of the ASP software originally written by L. M. Optican and H. P. Goldstein. To allow for greater flexibility in the cost function used during optimizations, some optimizations were run using compiled Matlab functions with gradient descent provided by the "constr" function. Simplified versions of both models, employing an intuitive graphical interface, can be explored with common web browsers at http://keck.ucsf.edu/~sgl/top_pursuitmodel.htm.
Models were built by interconnecting time delays, nonlinearities, and
low-pass filters. Optimized parameters were the time delays, the
coefficients describing each of the nonlinearities, and the time
constants of the low-pass filters, which converted step inputs into
exponentially relaxing outputs with a single time constant.
Optimization was typically initiated by manually adjusting the model
parameters until the responses were reasonably close to those of the
monkey. For example, for the fits shown in Figs. 12 and 13, the
parameters of both models were initially set so that each provided
reasonable approximations to the 15°/s step responses and produced
responses to sinusoidal perturbations that were in the right amplitude
range. Optimization employed a gradient descent algorithm
("stepit," Chandler 1965). To speed the optimization
process, limits were initially set on the range of most parameters. If
a good fit was not achieved within this range, the limits were relaxed
or eliminated. When a good fit was achieved, the optimization algorithm
typically took little time in finding it. In cases where a good fit was
not achieved, we repeatedly restarted the optimization algorithm using
different initial parameters. Different initial parameters were
obtained by 1) "jiggling" the parameter values slightly
from their value at the error minimum, 2) setting the
parameters to new random values, and 3) setting the
parameters by hand to attempt to improve the fit. The cost function was
usually simply the sum of the millisecond by millisecond squared
difference between the model and the data. For the simulations in Fig.
10, we included in the cost function the error between measured aspects
of the pursuit and model responses (e.g., the response amplitude). When
some aspects of the data were fit better than others, we increased the
cost function for those traces that were not fit well (multiplying
their error by a constant), and continued optimization. This was useful
in fitting the model simultaneously to large magnitude responses (e.g.,
the response to a 15°/s step) and small magnitude responses (e.g., the response to a 10-Hz target). Thus in instances where fits were
consistently poor we think it unlikely that we missed the global error
minimum. Our confidence in this assertion is increased by the nature of
the failures observed in instances of poor fits, which are readily
explained by reference to the architecture of the models. Pursuit
initiation from fixation is typically 10-20 ms slower than the true
pursuit latency. Models were allowed to compensate by having the
optimization algorithm add an additional delay at initiation.
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RESULTS |
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Open-loop responses to brief perturbations of target velocity
We begin by presenting the results of experiments designed to test the feed-forward properties of pursuit. We recorded responses to 100-ms-long perturbations consisting of pulses (Fig. 4A) or sine wave modulations (Fig. 4B) of target velocity. Responses were recorded to various amplitudes of each perturbation type. Figure 4, A and B, shows averages of eye velocity. The pursuit response to a 12°/s velocity pulse was more than twice as large as the response to a 4°/s pulse, although not quite three times as large. In contrast, the response was nearly the same for a 12°/s sinusoidal perturbation as for a 4°/s perturbation. Note that, for graphical visibility, the eye velocity responses in Fig. 4, A and B, are plotted at twice the vertical scale as the target velocity traces.
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The differing nonlinearity of the responses to pulse and sinusoidal
perturbations is summarized for monkey Na in Fig.
4C, which plots response amplitude as a function of stimulus
amplitude. The response to velocity pulse perturbations () saturates
moderately with stimulus amplitude over the range of 1-12°/s. The
response to a 12°/s pulse is 60% as large as expected given a linear
extrapolation from the response to a 2°/s pulse. The dashed line
shows that a logarithmic fit captures well the small to moderate
response saturation. The responses to sinusoidal perturbations
saturated much more severely (
and
). The saturation was similar
whether we measured the response to the first cycle of a perturbation that was five cycles long (
) or the average response to all five cycles (
). The response to the first cycle of a 12°/s amplitude sine wave is only 33% as large as expected given a linear
extrapolation from the response to a 2°/s sine wave. The response to
all five cycles is 38% as large as expected given a linear
extrapolation. These data were well fit by a logarithmic relationship
(solid and finely dashed lines). Note that the responses to sinusoidal perturbations are smaller than the responses to pulses and have been
plotted on a different amplitude scale. Amplitudes for the two kinds of
perturbations have been scaled so that they appear at a similar
position on the y-axes for low stimulus amplitudes, to
emphasize the difference in the degree of saturation.
Figure 4D shows the same general results from experiments on monkey Mo. Responses to sine waves showed more saturation (only 38% of the linear expectation) than did responses to pulses (59% of the linear expectation). For this experiment, sine wave perturbations were only one cycle long. Monkey Mo was used for this experiment only; all subsequent experiments use monkeys Na and Ka. The parameters of the logarithmic fits are given in the figure legend and, as expected, show a greater saturation for the fits to sine wave perturbation responses than for the fits to pulse perturbation responses. Linear regressions to the responses to sine waves showed y-intercepts significantly greater than zero (Mo, P < 0.05; Na, P < 0.05 for response to 1st cycle, P < 0.005 for response to all cycles), indicating that the response saturated. To determine whether the degree of saturation was significantly greater for sine waves than for pulses, we divided the response to the latter by the response to the former. This ratio was significantly larger for larger amplitude stimuli (Mo, P < 0.05; Na, P < 0.05 for response to 1st cycle, P < 0.005 for response to all cycles). While one expects the biphasic sinusoidal perturbations to evoke lower amplitude responses overall, relative to the pulse perturbations, some nonlinearity must be proposed to explain why they also evoke more response saturation.
Interaction of sinusoidal and pulse perturbations of target velocity
Our interpretation of the data in Fig. 4 is that the responses to
high-frequency sine wave perturbations are driven largely by a
fast-saturating sensitivity to image acceleration. While other
interpretations are possible, this interpretation guided the design of
the following experiment, which seeks to analyze the nature of the
proposed saturation further. We asked how the response to a sine wave
perturbation depended on the baseline image velocity at the time of the
perturbation. Our purpose was to determine whether the response to
image acceleration saturates with increasing image acceleration, or
with increasing image velocity. In other words, is the saturation best
approximated as S(d/dt) or as
d[S(
)]/dt, where
is image velocity and
S is a saturating function? If the first possibility holds,
then baseline image velocity should have no effect on the response to
sinusoidal perturbations. If the second holds, then the response
amplitude should decrease with increasing baseline image velocity.
Figure 5 illustrates that the response to
sinusoidal perturbations depended strongly on the size of a concurrent
pulse. Responses were evoked by perturbations that consisted of a
single cycle of a 10-Hz sine wave superimposed on different amplitude
100-ms velocity pulses, all presented during maintained pursuit of
target motion at 15°/s (methods shown in Fig. 3D). To
isolate the response to the sine wave and exclude the response to the
pulse itself, we computed the difference between the response to the
sine wave with pulse (stimulus shown in Fig. 3D, 3rd
trace) and the response to the pulse alone (Fig. 3D, 2nd
trace). The resulting difference traces in Fig. 5A show
that the sinusoidal perturbation of target velocity caused very little
modulation of eye velocity when the pulse of image velocity was either
4°/s in the direction of ramp target motion (trace labeled "4
deg/s") or 6°/s in the direction opposite target motion (trace
labeled "6 deg/s"). The amplitude of the response grew as the
pulse size was reduced and was largest when the pulse was 2°/s in the
direction opposite target motion (trace labeled "
2 deg/s").
Average eye velocity at the time of the perturbation was 13.8°/s,
which is 1.2°/s smaller than the original 15°/s target velocity.
Thus the image velocity offset caused by the pulse was smallest during
the
2°/s pulse, when the response to the sine wave was largest.
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Figure 5, B and D, summarizes, for monkeys
Na and Ka, the finding that the response amplitude
depended strongly on the mean image velocity during the perturbation.
Response amplitude is shown normalized by its maximum value. The
amplitude was largest when the mean image velocity was near zero, and
declined sharply with increasing image velocity in either the same or
opposite direction from target motion. This decrease was significant
for both monkeys (linear regression, P < 0.001 for
both). These results support the hypothesis that high-frequency sine
wave perturbations are driven in part by a sensitivity to image
acceleration, and that this sensitivity saturates with increasing image
velocity. That is, the saturation is best expressed as
d[S()]/dt. However, the results of Fig. 5, B
and D, could also have been obtained if sine wave
perturbations are driven purely by a strongly saturating sensitivity to
image velocity. To test this possibility, we assessed the linearity of
the pursuit response to pulses of target velocity presented alone.
Control trials presenting pure pulses were interleaved with
experimental trials.
The responses to pulse perturbations are summarized in Fig. 5, C and E. As in Fig. 4, the response to pulses alone saturated only moderately with increasing stimulus amplitude. For monkey Na (C), the response to an 8°/s pulse was 76% of the linear expectation given the response to a 2°/s pulse. This mild saturation is unlikely to account for the results of B, where sine wave response amplitudes fell to less than one-half when image velocity was between 4 and 8°/s. Monkey Ka (E) showed an asymmetry: considerable response saturation was observed only when the pulse increased target velocity (right-hand side for the open squares plotting rightward pursuit, left-hand side for the filled circles plotting leftward pursuit). The response to an 8°/s decrease in target velocity was 88% of the linear expectation, while the response to an 8°/s increase was only 58% of the linear expectation. This asymmetric response saturation is almost certainly due to eye velocity saturation and cannot account for the responses of monkey Ka in D, where response amplitude decreased similarly regardless of whether the pulse increased or decreased eye velocity.
Given that the response to sinusoidal motion is reduced by the addition of image velocity, one would expect that the natural fluctuations in image velocity during pursuit maintenance might have a similar influence. Perturbations presented when eye velocity is near the baseline target velocity would evoke larger responses than perturbations presented when eye velocity is farther from the baseline target velocity. To test this hypothesis, we pooled data from a number of experiments using 10-Hz perturbations (those shown in Figs. 4, 5, and 8, along with others not shown in this paper). For each experiment, we calculated the average absolute image velocity, across all trials, during the time the perturbation was presented. Within each experiment, we then divided individual trial responses into two bins: one in which baseline image velocity at the time of the perturbation was higher than average, and one in which baseline image velocity was lower than average. The trials in each bin were then averaged, and the response to the perturbation was calculated as described above. It was necessary to bin and average because responses were obscured by noise in the majority of individual trials, and were clear only in the averages. For monkey Na, image velocity averaged 2.4°/s for the first bin and 0.9°/s for the second (averages over 16 experiments). The response to a 10-Hz sinusoidal perturbation was on average 34% smaller in the first bin (2-tailed t-test, P < 0.0002). For monkey Ka, image velocity averaged 1.7°/s for the first bin and 0.7°/s for the second (averages over 11 experiments). The response was on average 16% smaller in the first bin (P < 0.05). Thus even the small naturally occurring departures from zero image velocity that occur during pursuit maintenance can reduce the pursuit response to sinusoidal perturbations. For comparison, consider only those responses in Fig. 5, B and D, where image velocity during the pulse was <2°/s (mean = 1.1°/s) or >4°/s (mean = 6.4°/s). Across both directions and monkeys, the response in the latter condition was 59% smaller (standard error = 6%).
Closed loop responses to steps and sinusoidal perturbations of target velocity
To provide a data set for testing the models under closed-loop conditions, we conducted modified versions of experiments already in the literature. We recorded the responses of monkeys Na and Ka to 15°/s steps of target velocity, and to multi-cycle sine wave perturbations over a range of frequencies from 1 to 10 Hz. Sine wave perturbations were imposed during maintained pursuit of 15°/s steps. Perturbations were presented in the minority (25%) of trials; target velocity was usually a pure 15°/s step.
Figure 6 shows 12 responses of
monkey Na to a 15°/s step of target velocity. For this
monkey, spontaneous oscillations at approximately 5 Hz were a typical,
if not universal (see examples 11 and 12) feature
of maintained pursuit. However, comparison along the vertical dashed
line reveals that the phase of these oscillations was not consistent
between trials. As a result, the average eye velocity response, when
aligned on the onset of target motion, showed no spontaneous
oscillations (e.g., Fig. 3B). For comparison of different
models, we did not wish to use average responses that were
unrepresentative of the majority of individual responses. We wished
particularly to preserve the spontaneous oscillations present in
individual trials, since fitting the oscillation period has been an
important challenge for models of pursuit (Goldreich et al.
1992; Ringach 1995
; Robinson et al.
1986
).
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Our solution to this problem, shown in Fig.
7, was to make two averages: one with
individual trials aligned on the initiation of pursuit, and a second
average aligned on the first trough of the spontaneous oscillations.
Most records exhibited at least one full oscillation cycle, making this
strategy feasible. For the few trials that did not show any clear
oscillations, we made an estimate based on the end of the initial eye
acceleration and the typical period of the oscillations. After
obtaining the two averages, we spliced them at the point were their
accelerations and velocities were both equal, just before the end of
the initial eye acceleration. The resulting "spliced average" (Fig.
7) was representative of both the initial eye acceleration and the
typical amplitude and period of the spontaneous oscillations, and was used as the goal for the models' responses. The spliced average is
related to the method used by Robinson et al.
(1986) to solve the same problem. For these experiments, we had
many more repetitions of the responses to steps of target velocity than
we needed. Therefore we averaged only a randomly selected subset of
25-35 repetitions.
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Figure 8A shows the spliced
averages of eye velocity () for monkey Na to rightward and
leftward 15°/s steps of target velocity (- - - - -). We estimated
the feedback delay of this monkey's pursuit system to be about 80 ms,
based on the latency to respond to perturbations of target velocity
imposed during maintained tracking. The spontaneous oscillations were
lightly damped and persisted for several cycles with a period of 207 ms, a little over twice the feedback delay. In contrast to monkey
Na, monkey Ka exhibited spontaneous oscillations in only a small
subset of his responses (individual responses not shown). Thus aligning his responses on the onset of target motion produced average responses that resembled individual responses. However, we found that by averaging his responses time locked to the peak of the usual small overshoot of target velocity, we better preserved this aspect of
individual responses. Responses of monkey Ka are shown
alongside those of the models in Fig. 13A. On rare occasions
when spontaneous oscillations were present in the individual responses
of monkey Ka, their period was around 150 ms, just over
twice the estimated feedback delay of 65-70 ms.
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Figure 8B shows the responses of monkey Na to
sinusoidal perturbations over a range of frequencies, collected during
the same experiment as the step response in A, and isolated
as described in METHODS. As expected, eye velocity ()
always lagged target velocity (- - - - -), even for 500-ms period
perturbations (2 Hz). As the period of the sinusoidal perturbations
decreased, the phase lag increased until it exceeded 360° for
perturbations with a period of 100 ms. In agreement with previous data
(Goldreich et al. 1992
), the gain of the response was
close to one for 500-ms period perturbations (2 Hz), and declined to
somewhat less than 0.5 for 100-ms period perturbations (10 Hz). One
feature of these responses is particularly notable, even though it can
be predicted from the unexpected stability of pursuit in the face of
artificially increased feedback gains (Robinson 1965
).
Given the lightly damped spontaneous oscillations in response to steps
of target velocity, linear systems analysis predicts that the system
should resonate strongly when stimulated at the oscillation frequency.
The amplitude of the eye velocity response should increase with each
cycle of the perturbation, and the final gain should be quite high.
This did not occur. In Fig. 8B, the gain of the response to
a perturbation of period 190 ms is less than one and is smaller than
that to a perturbation of period 350 ms. However, the absence of the
resonance predicted by linear control models is perhaps not surprising, as we observed above that the pursuit response to sinusoidal target motion is far from linear. In particular, fluctuations of eye velocity
about target velocity (as in Fig. 6) increase baseline image velocity
and are thus expected, given the data in Fig. 5, to reduce the response
to sinusoidal perturbations.
The responses of monkey Ka to sinusoidal perturbations of target velocity were similar to those of monkey Na and are shown later (Fig. 13C). Again, there was no pronounced increase in the response gain when the perturbation frequency was near that of the (rarely observed) spontaneous oscillations.
Architecture of the models
Both the image motion model (Goldreich et al. 1992;
Krauzlis and Lisberger 1989
, 1994b
) and
the tachometer feedback model (Ringach 1995
) attempt to
explain why pursuit eye velocity follows trajectories like those in
Fig. 2, B and C, rather than that of a simple
velocity servo as in Fig. 2D. For the image motion model, the use of image acceleration (and deceleration) information limits the
overshoot of target velocity and produces the correct oscillation frequency of approximately twice the visual feedback delay.
Intuitively, performance of the simple velocity-servo model is poor
because what is needed to ensure perfect performance is current image velocity,
t).
The image motion class of models essentially makes the first-order
estimate:
t) + gÏ(t
t). For the estimate to be optimal, g should
be approximately
t. If the image acceleration signal is
too weak, large overshoots will result. If it is too strong,
spontaneous oscillations are produced with a period of twice the visual
delay. These spontaneous oscillations represent repeated
over-corrections. The period is twice the delay because one delay
interval expires between maximum image acceleration (maximum eye
deceleration), and the maximum of the subsequent overcorrecting
response: maximum eye acceleration. The response delay thus accounts
for one-half cycle, from maximum eye deceleration to maximum eye
acceleration. These explanations are complicated somewhat by the
addition of filters and of nonlinear gain elements, but remain a
fruitful way of understanding the behavior of the image motion model. A
formal explanation of these effects can be found in the Appendix of
Goldreich et al. (1992)
.
Figure 9A shows a block
diagram of a revised image motion model. Target velocity () and
eye velocity (
) are compared by the model at a summing junction
that represents the retina, yielding image velocity (
). The eye velocity command is then passed through a low-pass
filter (labeled "plant") to yield actual eye velocity (
).
The relatively simple dynamics of the plant are based on the widely
accepted idea that neural circuits not included in our model compensate
for the physical dynamics of the eye (Skavenski and Robinson
1973
).
|
The top visual pathway in Fig. 9A takes image velocity as its input, delays it, processes it with a nonlinear gain element, and subjects it to low-pass filtering with a single time constant. The middle pathway has the same three elements (with different parameters), plus a switch that is on only for 30 ms following the onset of target motion, to produce a "Motion onset transient." This pathway was included to account for certain nonlinearities present during the initiation of pursuit, but is silent during maintained pursuit, and is thus of minor relevance to the majority of our analysis. The bottom pathway receives an image velocity input that is first delayed and passed through a saturating nonlinearity. The derivative is then taken, and the resulting image acceleration signal then passes through a second nonlinear gain element and filter. An additional low-pass filter is associated with the differentiation.
The model in Fig. 9A has three departures from the image
motion model of Krauzlis and Lisberger (1994b). First,
it adds a nonlinearity to the acceleration pathway, situated before the derivative of image velocity is taken. This nonlinearity was crucial in
fitting much of the data presented in this paper and was motivated by
the data in Figs. 4 and 5. When set to saturate, the nonlinearity reduces the sensitivity to image acceleration when image velocity is
high; an acceleration from 0 to 5°/s produces a larger output than
does the same magnitude acceleration from 30 to 35°/s. Second, each
of the parallel pathways now has an independent time delay. These
delays were allowed to vary slightly from one another (see Table A1) to
achieve the best fit to the data. Third, the current model does not use
temporal filters with inherent resonance. Given that relatively little
is known about the filtering properties of the pursuit system, we made
the simplest possible assumption and used low-pass filters described by
a single exponential time constant.
We converted the image motion model into an equivalent tachometer
feedback model by replacing the image velocity input to the
acceleration pathway with a negative eye velocity input. The tachometer
feedback model, shown in Fig. 9B, shares the first two
pathways with the image motion model. However, the bottommost pathway
embodies a sensitivity to eye acceleration, rather than a sensitivity
to image acceleration. This version of the tachometer model uses
potentially nonlinear gain elements and is capable of showing all the
features of the original linear tachometer model and more. The
inclusion of the motion onset pathway does confer a limited type of
image acceleration sensitivity (at the onset of target motion) to the
tachometer feedback model. However, we felt this pathway should be
included given the ample evidence for a motion onset transient that
drives eye acceleration at the initiation of pursuit (Krauzlis
and Lisberger 1994a; Lisberger and Westbrook
1985
). To further facilitate comparison of the two models, we
designed the differentiator in the acceleration pathway of the image
motion model to ignore the brief pulse of image acceleration present
when the target first began to move. Any response to motion onset could
still be modeled by the motion onset pathway.
The two models are functionally nearly identical for target motion at a constant velocity. Eye deceleration is identical to image acceleration when target acceleration is zero, and the eye acceleration pathway of the tachometer feedback model thus has a similar effect on model performance as does the image acceleration pathway of the image motion model. The only differences arise due to the potentially nonlinear gain that precedes differentiation within the acceleration pathway. The nonlinearity acts on image velocity for the image motion model, and on eye velocity for the tachometer feedback model, and image velocity is typically not equal to eye velocity. In general, optimized versions of the tachometer feedback model tended to use a nearly linear transfer function in this position, while the image motion model used a saturating nonlinearity. Still, the performance of the two models was nearly identical for steps of target velocity. The two models produced dramatically different responses 1) when the target accelerated continuously and 2) when visual feedback was artificially delayed. The latter condition creates different predictions for the two models because it delays both types of feedback (image velocity and image acceleration) for the image motion model, but only the image velocity feedback for the tachometer feedback model.
We have diagrammed the architecture of the models to highlight their close formal relationship. In doing so, we have lumped into the total delay of each pathway a motor delay that in Ringach's original formulation affected the final common pathway. This formulation of the delays is formally identical to the original. As described, the models employed many free parameters, eight describing the delay and filtering properties of pursuit, with multiple additional parameters describing the four nonlinearities. The number of free parameters may seem excessive, and indeed it was designed to be. We wished to ensure that any failings of the models were due to limitations imposed by their architecture, and not to a lack of free parameters.
Simulation of open-loop responses
The simulations shown in Fig.
10A verify our earlier
conclusion that a saturating response to image acceleration is
sufficient to account for the discrepancy in response saturation
between pulses and sine waves. There is excellent quantitative
agreement between the output from the image motion model (Fig.
10A, , - - - - -, and - - -) and the responses of
monkey Na to pulse and sine wave perturbations of target
velocity (symbols, replotted from Fig. 4). For both the model and the
monkey, the response to pulses of target velocity (
,
) showed less
saturation than did responses either to the first cycle of a 10-Hz sine
wave (- - - - -,
) or to all five cycles (- - -,
). The
model and the monkey both showed slightly smaller response amplitudes
when measured over five cycles instead of one. Model responses were
obtained using target velocity inputs equal to the sine wave or pulse
perturbation of interest. Response gains were then calculated just as
they had been for the monkey. To fit these data, the image motion model relied on the combination of a nearly linear response to image velocity, and a response to image acceleration that saturated with
increasing image velocity. The parameters used to achieve these fits
are reported in Fig. A1 and Table A1.
|
The tachometer feedback model was less successful in reproducing the
responses to pulse and sine wave perturbations of target velocity. When
the parameters were set so that the responses to pulses were fit well
(Fig. 10B, ), the gain of the response to sine waves was
too low for all but the highest stimulus amplitudes (bottom 2 dashed
lines). This was especially true when all five cycles of the response
were considered (bottom dashed line). In addition, the response of the
model did not saturate quickly enough with increasing sine wave
perturbation amplitude. Scaling the responses of the model so that the
pulse and sine wave response gains overlap for low amplitude stimuli
(top dashed line) reveals that the response to sine waves saturated to
the same degree as the response to pulses. It was possible to configure
the tachometer feedback model to produce responses to sinusoidal
perturbations with the appropriate gain and saturation. However, the
model's responses to velocity steps were then too large and exhibited far too much saturation (simulation data not shown). As discussed later, these shortcomings are due not to the use of motor feedback, but
to the tachometer model's lack of sensitivity to image acceleration.
We also fit both models to the responses of monkey Na to
sine waves imposed on velocity pulses, and to his responses to the pulses alone, originally shown in Fig. 5. As in Fig. 10, A
and B, both models were able to reproduce accurately the
responses to different amplitude pulses (simulation data not shown, but fits were as good as in Fig. 10, A and B).
However, only the image motion model was also able to reproduce the
responses to sine waves presented on top of pulses. Figure
10C shows, for the two models (curves) and for the monkey
(symbols), the response gain for sine waves as a function of the mean
image velocity created by a concurrent pulse. The responses of the
models were isolated using the same method employed for the behavioral
data: the response to the pulse alone was subtracted from the response
to the pulse/sine wave combination. The image motion model ()
captures the steep decline in response gain with increasing image
velocity (
). The behavior of the image motion model is due to the
initial nonlinearity in the image acceleration pathway (bottom pathway
of Fig. 9A). When baseline image velocity is zero, the input
to the pathway engages the steep part of the nonlinearity, and the
signal to be differentiated has a relatively high gain. When baseline
image velocity is either above or below zero, the input to this pathway engages the shallower part of the nonlinearity, and the signal to be
differentiated has a relatively low gain. In effect, the gain of the
image acceleration pathway varies as a function of baseline image velocity.
The tachometer feedback model (- - - - -) showed only a modest decline in response gain with increasing baseline image velocity. The decline was due to the saturation of the response to image velocity. The tachometer model was able to produce a steeper decline, but only if its responses to pulses saturated much more than did the monkeys' (simulation data not shown). Because the data are expressed in normalized form, Fig. 10C obscures an additional failure of the tachometer feedback model; when image velocity was zero, the responses of the tachometer feedback model to sinusoidal perturbations of target velocity were only 50% as large as those of the monkey. The data from monkeys Mo and Ka were sufficiently similar to those of monkey Na that we did not repeat the simulations using their data as a goal.
Simulation of closed-loop responses under conditions of delayed visual feedback
We next ask whether the nonlinear visual properties of our
elaborated image motion model are sufficient to explain pursuit under
closed-loop conditions, or if sources of motor feedback must also be
assumed. Ringach has argued that the image motion model is unable to
account for some aspects of pursuit under conditions of delayed visual
feedback, and indeed this is true for a linear model. We now compare
the behavior of the nonlinear versions of both models with
the pursuit behavior that led Ringach to reject the linear
image motion model. The solid lines in Fig.
11 reproduce data from Goldreich
et al. (1992) and show responses of a monkey to a 15°/s step
of target velocity, with artificially imposed feedback delays indicated
by the numbers to the left of each trace. As reported in detail before,
and as expected from an image motion model, the period of the
spontaneous oscillations increased with feedback delay. The increase is
not linear, however. At short artificial delays the oscillation period
is approximately twice the total feedback delay, while at long
artificial delays the period is closer to four times the total feedback
delay. There is also an increase in the damping of the oscillations at
intermediate delays. These more subtle aspects of the data could not be
reproduced by a linear image motion model (Ringach
1995
), although it has previously been shown that a nonlinear
image motion model can produce a nonlinear increase in oscillation
period with increasing feedback delay (Krauzlis and Lisberger
1994b
).
|
The dashed traces in Fig. 11 show that the nonlinear versions of both
models provide good fits to the data of Goldreich et al.
(1992). Both models show realistic effects of increases in feedback delay with respect to the oscillation period and the damping.
Thus the primary criticism leveled by Ringach (1995)
against the linear image motion model is obviated by the nonlinear version of the model. The nonlinear image motion model succeeds where
the linear version failed primarily because of the addition of image
velocity saturation prior to the extraction of image acceleration (1st
nonlinearity in bottom pathway, Fig. 9A). The image
acceleration pathway dominates the model's dynamics when the visual
delay is short and the oscillations (and resulting image velocities)
are small. Dominance of the image acceleration pathway produces an
oscillation period roughly twice the feedback delay, as discussed
above. When the visual delay is long and the oscillations (and
resulting image velocities) become large, saturation weakens the image
acceleration pathway, and the image velocity pathway dominates the
dynamics, creating an oscillation period of roughly four times the
feedback delay. Between these extremes is a period where the two
pathways are more balanced, and damping is increased. The parameters
that produced these fits are shown in Table A1, and the nonlinearities
appear in Fig. A1.
The principle frequency of the spontaneous oscillations exhibited by
the tachometer feedback model increases appropriately with increasing
feedback delay, as described by Ringach (1995). However,
for moderate to long delays, higher frequency oscillations produced by
the eye acceleration feedback pathway, whose delay is unaltered, are
superimposed on the lower frequency oscillations produced by the image
velocity pathway. The model oscillates at two distinct frequencies. In
contrast, the average responses of the monkey did not show two
frequencies of oscillation. Although it is possible that high-frequency
oscillations were present in individual trials and were lost due to
averaging, this seems unlikely because for this monkey similar
high-frequency oscillations were visible in the average responses when
the visual feedback was not delayed. Unfortunately, the raw data from
Goldreich et al. (1992)
are no longer available to
investigate directly the presence or absence of simultaneous
oscillations at both high and low frequencies. For the responses of the
image motion model, there are also some small departures from
smoothness that appear as if they might represent a small higher
frequency component. In fact, these result from a small discontinuity
at zero in the derivative of the first nonlinearity in the image
acceleration pathway (curve labeled "Image velocity #2" in the
2nd row of Fig. A1). These small bumps would have been
eliminated if we had parameterized the curve to allow asymmetry without
a discontinuity in the first derivative.
Simulation of closed-loop responses to steps and sinusoidal modulations of target velocity
The image motion model was able to replicate the very different responses of monkeys Na and Ka to 15°/s steps of target velocity (Figs. 12A and 13A). Different parameters were used in fitting the responses of the two monkeys (see Table A1 and Fig. A1), a necessity given their very different profiles of pursuit. The tachometer feedback model was equally successful in fitting the responses to steps of target velocity (Figs. 12B and 13B). These results were expected: the two models are formally very similar under these conditions, and both had previously been shown to successfully emulate pursuit of target velocity steps.
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|
We next ask how well the models reproduce the responses to sinusoidal perturbations. For each monkey, responses to sine wave perturbations were collected during the same experimental session as the responses to steps. We simulated responses by simply providing the target velocity perturbations themselves as the input to the models. This method ignores the fact that, for actual pursuit image, velocity was rarely zero at the time the perturbations were imposed, due to imperfections in pursuit maintenance and the presence of spontaneous oscillations. The method essentially assumes that pursuit maintenance was perfect at the time of the perturbation. This choice of method avoided the myriad complications of attempting to incorporate noise sources into the models, but it posed a different problem. For a linear model, the average perturbation response would not depend on the presence of random-phase fluctuations in baseline image velocity. The versions of the two models we employ are not linear, but considering the small (1-2°/s) size of fluctuations in baseline image velocity during pursuit, the departures from linearity are expected to be of little consequence, with one key exception. For the image motion model, even small fluctuations in baseline image velocity will reduce the effective gain of the image acceleration pathway. One would therefore predict that the image motion model, configured to reproduce the response to steps of target velocity, would successfully reproduce the responses to sinusoidal perturbations if and only if the gain of the image acceleration pathway is reduced. This is precisely what we found.
The responses of the image motion model to sinusoidal perturbations are shown in Figs. 12C and 13C. The simulated responses successfully capture both the amplitude and the phase of the monkeys' responses. To achieve these fits, the gain of the image acceleration pathway was reduced, respectively, by 34 and 38%, relative to the gain that reproduced the step responses shown in Figs. 12A and 13A. Fluctuations in baseline image velocity are expected to have little or no effect on the duration of the visual delay, or on the effective gain of the image velocity pathway (which is nearly linear in the relevant range). All other parameters were therefore restricted to be the same when simulating responses to target velocity steps and perturbations. The reductions in gain were achieved by simply reducing the output of the image acceleration pathway by the indicated percentage. This is equivalent to linearly scaling the second nonlinear gain function of that pathway (see Fig. A1).
The reduction in the gain of the image acceleration pathway (34 and
38% for monkeys Na and Ka) was chosen to produce
optimal fits. An independent estimate can be made of the decrease in
responsiveness expected given the fluctuations of image velocity. For
the data in Figs. 12 and 13, we measured image velocity during the time
the sine wave perturbations were presented, but for individual trials without perturbations (i.e., pure 15°/s steps of target
velocity). Image velocity during this interval fluctuated about target
velocity as in Fig. 6, and its absolute value averaged 1.4°/s for
monkey Na and 1.0°/s for monkey Ka. For
monkey Na, we reported above that increases in baseline
image velocity of 1.22°/s, 1.16°/s (both artificially produced),
and 1.5°/s (naturally occurring) reduced the response to sine waves
by 34, 33, and 34%, respectively. These values are very similar to the
value of 34% used by the model to fit the data of monkey
Na. For monkey Ka, increases in image velocity of
1.0°/s,
1.0°/s (artificially produced), and 1.0°/s (naturally
occurring) reduced the response by 15, 7, and 16%. These values are
smaller than that used by the model (38%). In summary, although the
decreases in responsiveness expected given the data are similar to the
decreases in responsiveness necessary to allow the image motion model
to achieve good fits, the two values do not agree exactly in both monkeys.
The responses of the tachometer feedback model were, like those of the image motion model, computed without regard to the fluctuations of image velocity during pursuit maintenance. The input to the model was simply the target velocity of the perturbation, as if maintenance were perfect. Unlike the image motion model, the presence of small fluctuations in image velocity do little to alter the behavior of the tachometer feedback model. Still, if we did not alter any parameters from the values that produced good fits to the responses to 15°/s steps, then the tachometer feedback model showed excessive resonance in its response to perturbations (these simulations are not those seen in Figs. 12D and 13D). We therefore allowed the optimization algorithm to use a lower gain for the eye acceleration pathway when fitting the tachometer feedback model to the perturbation responses to observe whether the tachometer feedback model exhibited any further failings, and to allow fair comparison with our simulations of the image motion model. In fitting the data of monkey Na, a 34% reduction in gain was used by the optimization algorithm. In fitting the data of monkey Ka, no change in gain was used. All other parameters were constrained to be the same for steps of target velocity and sinusoidal perturbations.
The responses of the tachometer feedback model to sinusoidal perturbations (Figs. 12D and 13D) differ from those of the monkeys in two ways. First, for all but the lowest frequencies, the response to sine wave perturbations showed excessive phase lag. Second, although the gain of the response was roughly appropriate for low frequencies, at high frequencies the responses were too small. Both of these failings can be traced to the tachometer feedback model's lack of image acceleration sensitivity, rather than to the presence of eye acceleration feedback. A hybrid model using both image acceleration and eye acceleration signals provided good fits (simulation data not shown), although not appreciably better than those provided by the pure image motion model.
Changes in the gain of the image velocity pathway also did little to improve the performance of the tachometer feedback model. If the response to high-frequency perturbations was made larger, then the response to low-frequency perturbations was overly large. The tachometer feedback model could reproduce the responses to sine waves for all frequencies, but only if the majority of the parameters were altered from the values that produced reasonable step responses. Excellent fits to sine waves were obtained by reducing the delay and filter time constant of the image velocity pathway, and by drastically reducing the delay of eye acceleration feedback, from 60-70 ms to 10-40 ms. However, such parameter changes made it impossible for the model to respond to 15°/s steps of target velocity with the appropriate dynamics, or to produce spontaneous oscillations of the appropriate period. For the tachometer feedback model, the oscillation period is governed almost entirely by the delay of the eye acceleration pathway, and is approximately twice that delay. Repeated optimization attempts failed to fit the tachometer feedback model simultaneously to the responses to 15°/s steps and to sinusoidal perturbations, even when the eye acceleration pathway was allowed to have widely different gains for the two conditions. The simulations shown in Figs. 12 and 13 represent the best fits we were able to obtain.
Figure 14 summarizes the data in Figs.
12 and 13 and plots response gain and phase as a function of frequency
for each monkey () and each configuration of the models. The best
simulation of the data was provided by the image motion model, using
the reduced image acceleration gain (
). If we did not reduce the gain of the image acceleration pathway, the image motion model (
for
monkey Na) showed excessive resonance near the frequency of
the spontaneous perturbations (vertical dashed line). For the tachometer feedback model (
) response gain was too low and phase lag
too large for all but the lowest frequencies. The simulation responses
analyzed in Fig. 14 were produced using the same parameters as in Figs.
12 and 13. Gain and phase were computed using the amplitude and phase
of the relevant Fourier component, just as was done for the responses
of the monkeys.
|
Model parameters
Because we desired accurate quantitative fits to large data sets,
both models were simulated using a large number of free parameters and
were allowed to use multiple nonlinear gain elements. The
APPENDIX describes the parameters and nonlinearities that
allowed the image motion model to reproduce the data in Figs.
10A, 11, 12, and 13. The use of multiple parameters and
nonlinearities was important in providing precise quantitative fits.
This is not surprising. The pursuit system itself exhibits a number of mild to profound nonlinearities. However, the exact shape of each nonlinearity should not be expected to capture exactly the
corresponding nonlinear response property of pursuit. Not only are the
nonlinearities somewhat under-constrained by our data, but changes in
one can often partially compensate for changes in another. Nonetheless, for the one nonlinearity that can be measured in a model-independent fashion, the sensitivity to image velocity, the nonlinearities we used
are very close to the measured functions (Krauzlis and Lisberger
1994a; Lisberger and Westbrook 1985
;
Morris and Lisberger 1987
; Robinson et al.
1986
).
Note also that the models are being asked to fit a number of responses, each of which is hundreds of milliseconds long. Past the open-loop interval, the response of the model depends on its previous behavior. Even very small failings will therefore tend to accumulate toward the end of the simulation. Multiple parameters and nonlinearities were necessary to achieve good quantitative fits. However, for the image motion model qualitatively appropriate behavior depended only on the saturation of image velocity prior to the extraction of image acceleration, and on reasonable choices for the visual delay. The version of the image motion model available on our website (http://keck.ucsf.edu/~sgl/top_pursuitmodel.htm) uses only a handful of free parameters and a single nonlinearity and can qualitatively reproduce pursuit behavior for all stimulus conditions described in this paper.
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DISCUSSION |
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Nonlinearities of visual inputs and stability of pursuit
Pursuit relies on negative feedback, yet is remarkably stable
given the long feedback delay. Pursuit does not break into uncontrolled resonance when driven at its natural frequency. Nor does it do so when
the feedback delay is artificially increased. This stability is
particularly surprising given that pursuit often appears to be on the
verge of instability under more benign circumstances. Fuchs
(1967) first noted that pursuit often breaks into spontaneous oscillations during tracking of a constant velocity target. In the
current paper, such spontaneous oscillations were frequently present in
the pursuit of monkey Na, and occasionally present in the
pursuit of Ka. The responses of monkey Jo,
reprinted from Goldreich et al. (1992)
exhibit nearly
undamped oscillations. It now seems fair to conclude that these
oscillations are driven primarily by a sensitivity to image
acceleration. The oscillation period (approximately twice the feedback
delay) is consistent with this explanation, and nearly identical
spontaneous oscillations are produced by the image motion model, due to
its sensitivity to image acceleration.
As we have noted, the presence of spontaneous oscillations gives way to
an unfulfilled expectation of resonance for stimulus frequencies near
that of the oscillations. The nonlinear response to image acceleration
we document provides a potential explanation for the surprising
stability of pursuit: the decline of image acceleration sensitivity
with increasing image velocity acts to limit resonance. If
oscillations, either spontaneous or driven, become too large, then the
resulting image velocities reduce the tendency toward resonance. Even
the initial response to perturbations is expected to be reduced by the
presence of fluctuations in image velocity. The existence of this
nonlinearity also has practical implications from the point of view of
experimental design and interpretation. Many pursuit experiments employ
changes in target velocity that are intended to probe the
responsiveness of pursuit (Das et al. 1998;
Krauzlis and Lisberger 1994a
; Schwartz and
Lisberger 1994
). Our data make it clear that such experiments
are difficult to interpret if the image velocity offset varies between
the conditions being compared. For example, in Schwartz and
Lisberger (1994)
the conclusion drawn from Fig. 10, that the
on-line gain of pursuit builds over the course of pursuit, is probably incorrect.
If fluctuations in image velocity during steady-state tracking reduce
the response to sinusoidal perturbations, then the converse should also
hold: creating fluctuations in image velocity prior to
imposing a step of constant target velocity should reduce the presence
of spontaneous oscillations in response to the step. This experiment
was performed, although with different motivations, by Goldreich
et al. (1992). They created ongoing sinusoidal target motion of
a random phase, and then imposed a step of target velocity. By
averaging across phases, they isolated the response to the step. For a
linear system, this response should have been identical to the response
recorded to the step alone, in the absence of sinusoidal target motion.
Instead, as we predict, the tendency toward spontaneous oscillation was
greatly reduced.
Summary of the comparison between models
We have found that an improved version of the image motion model
is able to account for open-loop pursuit responses to velocity pulses,
sine wave perturbations, and combinations of the two over a range of
stimulus amplitudes. For steps of target velocity, the image motion
model was able to reproduce the very different closed-loop responses of
the two monkeys we used. The model was also able to fit the closed-loop
responses of the same two monkeys to sine wave perturbations over the
frequency range of 2-10 Hz. Importantly, our nonlinear image motion
model was able to reproduce responses to target velocity steps under
conditions of delayed visual feedback, precisely the data that led
Ringach (1995) to reject the image motion class of models.
Because the image motion model and the tachometer feedback model are formally very similar when target velocity is constant, the tachometer feedback model was able to reproduce the step responses of both monkeys. However, for accelerating targets, the tachometer feedback model performed poorly. It was unable to capture the differing degrees of saturation seen in open-loop pursuit responses to brief pulses and sinusoidal perturbations. Furthermore, when using reasonable values for the visual delay, the tachometer feedback model produced closed-loop responses to sinusoidal perturbations that were too small and showed too much phase lag at all but the lowest frequencies. These failures were in spite of both repeated optimizations (see METHODS) and the large number of free parameters available to the model. The shortcomings of the tachometer feedback model are understandable given its architecture. Because the model lacks sensitivity to image acceleration, it produces responses to accelerating targets that are smaller and more phase lagged than those of pursuit, and that show less saturation with increasing target acceleration.
We do not think that the shortcomings of the tachometer feedback model
can be repaired by elaborating the details of the motor feedback (for
example by adding further derivatives or more complicated filters). The
motor feedback was included by Ringach (1995) so that
the tachometer feedback model could emulate the image motion model when
target acceleration was small or absent (e.g., for steps to a constant
target velocity). The latency of the motor feedback is thus necessarily
of the same order as the visuo-motor delay. However, the tachometer
feedback model's failings are evident even during the open-loop
period, before the proposed motor feedback could have an effect. Thus
elaborating the motor feedback signal would not relieve the model's
failings. We also do not think that elaborating the image velocity
pathways of the tachometer feedback model would resolve its failings.
For example, one might consider adding a second feed-forward image
velocity signal to the tachometer feedback model, with a different
filter or nonlinearity. However, if the response gain of this pathway
were high enough to produce accurate responses to high-frequency sine
waves, then responses to steps, pulses, and low-frequency sine waves
would be overly large. Using multiple band-pass filters might achieve
the desired amplitude responses for both high- and low-frequency
stimuli. The phase would still lag, however, unless the visual delay
were shortened considerably, which would then result in step and pulse responses that were too early. We do not see a way around the necessity
for a visual signal, such as image acceleration, that creates phase
lead without decreasing absolute latency. We therefore conclude that
the pursuit system uses image acceleration information in guiding eye
movements. The strength of this conclusion is, of course, tempered by
the possibility that an as yet unexplored model might be able to
account for the observed responses without using image acceleration information.
Although none of our experiments provides evidence for the use of eye
acceleration-based motor feedback, none of our experiments or
simulations rules out the possibility of such a signal. They show
simply that such a motor signal cannot substitute for an image
acceleration signal. In general, we interpret our results as evidence
against all current pursuit models that do not use image acceleration
information, or that assume the spontaneous oscillations are internally
driven (e.g. Dicke and Thier 1999; Pola and Wyatt 2001
). In particular,
the original target velocity model of Robinson et al.
(1986)
lacks such a signal and is expected to exhibit failings
similar to those of the tachometer-feedback model when presented with
accelerating targets. In fact, the model of Robinson et al.
(1986)
has already been shown to produce overly phase-lagged
responses for one frequency of sine wave perturbation (Goldreich
et al. 1992
).
Interpretation of model parameters
In the APPENDIX, we report the parameters and
nonlinearities that allowed the image motion model to reproduce the
pursuit responses. These features of the model are meant to describe
aspects of the behavioral response and may not have distinct
physiological correlates. For example, the delay in the image velocity
pathway reflects the cumulative effects of delays in the retina, the
visual pathways of the brain, and brain-stem motoneurons. The
nonlinearity in the same pathway is meant to capture the behavioral
response to image velocity and is indeed a reasonable approximation to
the measured response function (Lisberger and Westbrook
1985). The "plant" filter is meant to incorporate both the
mechanical properties of the plant and the properties of neural
circuits that compensate for those mechanical properties
(Skavenski and Robinson 1973
). Thus, although the
filtering properties of pursuit can be reasonably approximated with
simple single-exponential filters, the underlying physiology is
considerably more complex. In general, we wish to stress that our model
is a model of behavior. While it has implications for the neural
signals guiding pursuit, it is not a model of those neural signals. The
goal of the model is to test the hypothesis that pursuit responses can
be accounted for by assuming that eye acceleration is driven by a
combination of signals related to image velocity and image acceleration.
Possible sources of visual information about image acceleration
Krauzlis and Lisberger (1991) demonstrated
that the discharge of Purkinje cells in the floccular complex of the
cerebellum could be accounted for as the sum of signals related to eye
velocity and the three visual signals used by the image motion model:
image velocity, motion onset transient, and image acceleration. The apparent use of an image acceleration signal by the pursuit system, and
the presence of this signal in the cerebellum, raise the question of
how image acceleration could be extracted from the discharge of
motion-sensitive visual neurons. Motion-sensitive neurons in cortical
area MT of rhesus monkeys are known to drive pursuit (Dursteler
and Wurtz 1988
; Groh et al. 1997
;
Komatsu and Wurtz 1989
; Newsome et al.
1985
). MT neurons are tuned for image velocity, but many also
respond to changes in image velocity: a large percentage of MT cells
exhibit a transient increase in firing when stimulus speed is
accelerated over 128 ms to the preferred speed velocity (Lisberger and Movshon 1999
). While these transient
responses could form the basis of an image acceleration
signal for increases in image velocity, recordings in anesthetized
monkeys indicate that transients provide little information about image
deceleration (Lisberger and Movshon 1999
).
The image acceleration signal used by our model provides both positive
image acceleration and image deceleration information. Image
deceleration information is crucial in preventing overshoots of target
velocity and is therefore more important to the performance of most
pursuit tasks than is positive acceleration information. Thus it does
not appear that a sufficient representation of image acceleration is
explicitly coded by the firing of MT cells. However, that MT
cells are tuned for image speed implies that an adequate image
acceleration signal could be extracted from their firing, provided that
the nervous system can approximate mathematical differentiation.
If the MT population code of image speed is converted to an explicit
scalar code of image speed, an image acceleration signal could be
extracted through a number of methods. The more appealing of these
methods use the well-known method of delayed normalization. If is
the input to delayed normalization and a is the output, then
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While humans can clearly discriminate an accelerating target from a
decelerating one, it does not appear that humans directly perceive
image acceleration. Whereas perception of velocity is not equivalent to
the perception of changes in position, the perception of acceleration
may be equivalent to the perception of changes in velocity, at least
for near-threshold judgements (Werkhoven et al.
1992). This finding is not in conflict with our
conclusion that pursuit is driven in part by image acceleration. We are
not claiming that image acceleration information is coded explicitly within visual cortex or is provided to the perceptual system.
Role of extraretinal signals in extrastriate cortex
Neurons in the medial superior temporal sulcus (area MST) are
known to carry extraretinal signals; they respond when the eye moves,
even when there is no visual motion (Newsome et al.
1988; Sakata et al. 1983
). Area MST is part of
the pursuit pathway, and prior physiology and modeling studies have
suggested that the extraretinal signals therein could provide a source
of motor feedback (often termed "corollary discharge") regarding
eye velocity (Dicke and Thier 1999
; Newsome et
al. 1988
). Such motor feedback is exactly what is proposed by
the target velocity model of Robinson, and could also form the basis of
the eye acceleration signal used by the tachometer feedback model.
However, this and previous modeling work (Goldreich et al.
1992
; Krauzlis and Lisberger 1994b
) demonstrates that the model of Robinson is incorrect and that motor feedback is
unnecessary to explain pursuit behavior. Why then are there motor
signals in MST? It should be noted that most models of pursuit, including the image motion model, implicitly use eye velocity feedback
to accomplish the integration of eye acceleration to eye velocity. MST
may be part of the pathway that accomplishes this integration.
Alternately, the extraretinal signals in MST may be involved in
estimating target velocity for the purpose of engaging and disengaging
pursuit (Huebner et al. 1992
; Krauzlis and Miles
1996
; Luebke and Robinson 1988
;
Mohrmann and Their 1995
; Pola and Wyatt
2001
; Robinson et al. 1986
; Scheuerer et
al. 1998
). We conclude that, although their true role is still
a mystery, the extraretinal signals in MST are probably not providing
motor feedback of the type proposed by the models of Robinson et
al. (1986)
or Ringach (1995)
.
Implications for the control of motor output
One of the major control problems faced by the pursuit system is
overcoming the negative impacts of feedback delays. The target velocity
model of Robinson et al. (1986), the tachometer model of
Ringach (1995)
, and the image motion model of
Krauzlis and Lisberger (1989
, 1994b
)
adopt three different strategies to overcome delays. The target
velocity model negates the feedback, sacrificing the advantages of a
negative feedback system to eliminate the disadvantages. The image
motion model drives eye acceleration using a combination of image
velocity and image acceleration, essentially using the outdated visual
inputs to make an estimate of the current image velocity. The
tachometer feedback model uses a similar strategy but employs the first
derivative of the system's output, rather than of its input. Each of
these models has fundamentally different implications for the basic
organization of the neural circuits that generate pursuit eye
movements, and each represents a strategy that could be applied to a
variety of other motor behaviors. Many motor systems other than pursuit
use visually derived error signals (e.g., reaching movements) and
suffer from feedback delays as long or longer than that of pursuit. To
overcome feedback instabilities, these systems may also use one or more
derivatives of the error signal. The presence of such signals could be
evaluated empirically. As in pursuit, the frequency of visually driven
spontaneous oscillations would be diagnostic of the presence of higher
derivatives, as would the response to visually introduced perturbations.
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APPENDIX |
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We report the parameters and nonlinearities that allowed the image motion model to reproduce the data in Figs. 10A, 11, 12, and 13. Table A1 shows four sets of values, one for each figure, for the pure delays and filter time constants. There was some variation in the parameters across the four simulations, which was necessary to account for the different responses of different monkeys, and even of the same monkey on different days. For example, the visual delays used to fit the data of monkey Na shown in Fig. 10A were slightly shorter than those used to fit data from the same monkey shown in Fig. 12. This is expected because the data in Fig. 10A were collected after the data in Fig. 12, and intervening training on other pursuit tasks had in fact reduced the monkey's pursuit latency. The greatest variability in parameters was found in the filter time constants. In part this was due to a tradeoff between the latency provided by the pure delay and that provided by the filter. But with the exception of the "plant" filter, the time constants of the filters appeared under-constrained by the data sets we used, and good fits to the same data could often be obtained using rather different filter time constants (data not shown).
|
The four rows of Fig. A1 show the four
sets of nonlinearities used to fit the image motion model to the data
from Figs. 10A, 11, 12, and 13. Each column shows the
nonlinearity for a particular gain element. The first column
shows the image velocity pathway nonlinearity, which was constructed
from three linear elements, joined at 4 and +4°/s. The image
velocity nonlinearities used by the four fits were similar, each
showing a small to moderate decrease in gain with increasing image
velocity. The second column shows the motion onset pathway
nonlinearity. As described previously, this pathway was not used in
fitting the responses to target velocity perturbations imposed during
ongoing pursuit and therefore is not shown for the fits to the data in
Figs. 10A. The motion onset pathway was available to the
model only when fitting the responses to 15°/s steps of target
velocity in Figs. 11-13. However, for the data of monkey Na
in Fig. 12, the pathway was not used by the optimization algorithm
(i.e., its gain was zero) and therefore is not shown in Fig. A1. We
implemented the motion onset pathway using an equation that saturated
quickly to a horizontal asymptote with an exponential relaxation. This
description provides continuity with the model of Krauzlis and
Lisberger (1994b)
, and with behavioral analyses of the
nonlinearities present at pursuit initiation (e.g., Lisberger and Westbrook 1985
). Note, however, that because this pathway was used only when the models responded to the initial step of target
motion (the pathway was not used for responses to perturbations during
ongoing pursuit), and because these steps were always 15°/s in
amplitude, only the value of the nonlinearity at 15°/s was ever used,
and the shape of the nonlinearity was not constrained by our
experiments. We acknowledge that the behaviorally observed motion onset
transient probably is not mediated by a separate visual pathway and
instead represents part of the response to image acceleration. We
nonetheless included it as a separate pathway, as an aid to the
tachometer feedback model, for the reasons previously discussed.
|
The third column of Fig. A1 shows the saturation of image
velocity prior to the extraction of image acceleration within the image
acceleration pathway. This nonlinearity was parameterized separately
for positive and negative values using an equation that allows for
nearly linear behavior, or for bounded or unbounded saturation
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The fourth column of Fig. A1 shows the second nonlinear gain
element used in the image acceleration pathway. The nonlinearity was
parameterized separately for positive and negative values using the
equation
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ACKNOWLEDGMENTS |
---|
We thank P. Churchland, A. Churchland, and the members of the Lisberger laboratory for helpful comments on an earlier draft of the manuscript. In particular we thank F. Crawford who created the java-based version of the model available on-line. S. G. Lisberger is an Investigator of the Howard Hughes Medical Institute.
M. M. Churchland was supported by a fellowship from the Department of Defense. Research was supported partly by National Eye Institute Grant EY-03878.
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FOOTNOTES |
---|
Address for reprint requests: M. M. Churchland, Dept. of Physiology, UCSF, Box 0444, 513 Parnassus Ave., San Francisco, CA 94143-0444 (E-mail: mchurchl{at}phy.ucsf.edu).
Received 13 October 2000; accepted in final form 3 May 2001.
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REFERENCES |
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