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The Journal of Neuroscience, March 15, 1999, 19(6):2224-2246
Visual Motion Analysis for Pursuit Eye Movements in Area MT of
Macaque Monkeys
Stephen G.
Lisberger1 and
J. Anthony
Movshon2
1 Howard Hughes Medical Institute, Department of
Physiology, and W. M. Keck Foundation Center for Integrative
Neuroscience, University of California, San Francisco, San Francisco,
California 94143 and 2 Howard Hughes Medical Institute and
Center for Neural Science, New York University, New York, New York
10003
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ABSTRACT |
We asked whether the dynamics of target motion are represented in
visual area MT and how information about image velocity and
acceleration might be extracted from the population responses in area
MT for use in motor control. The time course of MT neuron responses was
recorded in anesthetized macaque monkeys during target motions that
covered the range of dynamics normally seen during smooth pursuit eye
movements. When the target motion provided steps of target speed, MT
neurons showed a continuum from purely tonic responses to those with
large transient pulses of firing at the onset of motion. Cells with
large transient responses for steps of target speed also had larger
responses for smooth accelerations than for decelerations through the
same range of target speeds. Condition-test experiments with pairs of
64 msec pulses of target speed revealed response attenuation at short
interpulse intervals in cells with large transient responses. For
sinusoidal modulation of target speed, MT neuron responses were
strongly modulated for frequencies up to, but not higher than, 8 Hz.
The phase of the responses was consistent with a 90 msec time delay
between target velocity and firing rate. We created a model that
reproduced the dynamic responses of MT cells using divisive gain
control, used the model to visualize the population response in MT to
individual stimuli, and devised weighted-averaging computations to
reconstruct target speed and acceleration from the population response.
Target speed could be reconstructed if each neuron's output was
weighted according to its preferred speed. Target acceleration could be reconstructed if each neuron's output was weighted according to the
product of preferred speed and a measure of the size of its transient response.
Key words:
smooth pursuit; eye movements; visual motion processing; temporal dynamics; gain control; models; MT; monkeys
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INTRODUCTION |
What we do is often guided by what
we see. Sensory-motor systems must therefore transform visual signals
into commands for accurate movement. Smooth pursuit eye movements
provide an excellent opportunity to investigate the neural circuits
that perform visual-motor transformations. The basic neuroanatomy of
pursuit is known, the sensory inputs and motor outputs are well
understood, and the behavior itself has been studied extensively.
Eckmiller (1987) , Lisberger et al. (1987), Tusa and Ungerleider (1988),
Leigh (1989), Kowler (1990), and Keller and Heinen (1991) have provided
reviews of these issues.
The particular visual signals needed to control pursuit are related to
the motion of visual targets (Rashbass, 1961), and previous work has
implicated extrastriate visual area MT as a major source of these
visual motion signals. Lesions of MT cause deficits in the initiation
of pursuit for targets moving in any direction across the part of the
visual field represented at the site of the lesion (Newsome et al.,
1985). Electrical stimulation of MT affects the initiation of pursuit
if the stimulation coincides with the motion of a tracking target (Groh
et al., 1997). Single neurons in MT are selective for the direction and
speed of the motion of small targets and prefer speeds in a range that
is relevant to pursuit (Maunsell and Van Essen, 1983; Albright,
1984).
Pursuit is configured as a negative feedback control system; its visual
input is target motion with respect to the (potentially moving) retina,
defined as "image motion." As a result, the visual input for
pursuit varies as a function of time according to the pattern
illustrated in Figure 1. For this
single-pursuit trial, the stimulus was a ramp of target position from
an eccentric starting point, which provides a step of target velocity.
After the onset of the step, the eye remained still for ~100 msec,
accelerated for ~150 msec, and then oscillated around target velocity
at a frequency of ~6 Hz. To estimate the image velocity, we
subtracted eye velocity from target velocity. This reveals that the
visual input during pursuit is, sequentially, a step increase in image velocity, constant image velocity for 100 msec, a ramp decrease in
image velocity toward zero, and small oscillations around zero at ~6
Hz.
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Figure 1.
Schematic diagram showing typical pursuit eye
movements and the visual image motion that drives them. From
top to bottom, the traces
are as follows: superimposed eye and target position, superimposed eye
and target velocity, image velocity computed as target velocity minus
eye velocity, and image acceleration computed as the low-pass filtered
derivative of image velocity. The image acceleration at the onset of
target motion is represented by a brief square pulse that has been
clipped as a reminder that it is an impulse of acceleration. The
short horizontal dashed line below the eye position
trace shows the position of a fixation point that went
out when the eccentric tracking target started to move.
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Several laboratories have made computer models that reproduce the eye
movements illustrated in Figure 1 on a millisecond time scale. An issue
that distinguishes many of these models is whether the dynamics of the
biological system are generated by direct sensory drive or by feedback
from motor computations. For example, the model of Krauzlis and
Lisberger (1994) requires that the discharge of visual neurons
providing input to pursuit represents image velocity and image
acceleration, both of which are shown in Figure 1. In contrast, the
models of Robinson et al. (1986) and Ringach (1996) assume that the
visual inputs to pursuit encode only image velocity and that the
dynamics of pursuit arise in motor circuits.
The present experiments were designed to record the dynamics of visual
motion signals in area MT, to determine whether those dynamics could
represent image acceleration, and to explore ways to extract
information about image velocity and acceleration from the population
responses in area MT. Thus, we used trajectories of stimulus speed that
varied in the same way as image velocity and acceleration vary during
pursuit. Our recordings showed that many individual MT neurons have
transient responses that can provide information about image
acceleration. Individual neurons, however, do not carry an invariant
acceleration-related signal across all image speeds, so the true value
of image acceleration can be represented only by the activity of a
population of MT cells. We developed a model of MT neuron responses
that simulates the responses of the population of MT neurons for a
variety of stimuli, and we used the model to visualize the distributed
representation of target motion in MT and to reconstruct target
velocity and acceleration from this distributed representation.
Parts of this paper have been published previously (Movshon et
al., 1990; Lisberger and Movshon, 1991, 1994; Lisberger et al.,
1995).
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MATERIALS AND METHODS |
Surgical preparation and maintenance. We recorded the
activity of single units in area MT in 10 hemispheres of eight macaque monkeys (six fascicularis and two nemestrina).
The monkeys were prepared for acute single-unit recording using methods
we have described in detail previously (Levitt et al., 1994; Kiorpes et al., 1996). They were premedicated with atropine (0.25 mg) and with
acepromazine (0.05 mg/kg) or diazepam (Valium, 0.5 mg/kg). After
induction of anesthesia with intramuscular injections of ketamine HCl
(Vetalar, 10-30 mg/kg), cannulae were inserted into the trachea and
the saphenous veins, the monkey's head was fixed in a stereotaxic
frame, and surgery was continued under intravenous anesthesia with the
opiate anesthetic sufentanil citrate (Sufenta, 4-8
gm · kg 1 · hr1).
Infusion of the surgical anesthetic continued throughout the recordings.
To minimize eye movements, paralysis was maintained with an infusion of
vecuronium bromide (Norcuron, 0.1 mg · kg1 · hr1) in
lactated Ringer's solution with dextrose (5-20 ml/hr). Monkeys were
artificially ventilated with room air or a mixture of 50-70% N2O in O2. Peak expired CO2 was
maintained near 4% by adjusting the tidal volume of the ventilator.
Rectal temperature was kept near 37°C with a thermostatically
controlled heating pad. Monkeys received daily injections of a
broad-spectrum antibiotic (Bicillin, 300,000 units) to prevent
infection, as well as dexamethasone (Decadron, 0.5 mg/kg) to prevent
cerebral edema. The electrocardiogram (EKG), EEG, autonomic
signs, and rectal temperature were monitored continuously to ensure the
adequacy of anesthesia and the soundness of the monkey's physiological condition.
Tungsten-in-glass microelectrodes (Merrill and Ainsworth, 1972) were
introduced by a hydraulic microdrive through a small guide needle. To
obtain the most consistent access into the portions of MT representing
the central visual fields, we used a vertical approach to MT through
the anterior bank of the superior temporal sulcus. After the electrode
was in place in the cortex, the exposed dura was covered with warm
agar. Action potentials were amplified conventionally, displayed, and
played over an audio monitor. The recording sessions lasted between 72 and 108 hr.
Physiological optics. The pupils were dilated, accommodation
was paralyzed with topical atropine, and the corneas were protected with +2D gas-permeable hard contact lenses. When necessary,
supplementary lenses were chosen by direct ophthalmoscopy to make the
retinas conjugate with the display screen. The power of the lenses was then adjusted as necessary to optimize the visual responses of recorded
units. Contact lenses were removed periodically for cleaning. At this
time, the eyes were rinsed with saline and infiltrated with a few drops
of ophthalmic antibiotic solution (gentamycin). At least once a day,
the locations of the foveae were recorded using a reversible ophthalmoscope.
Characterization of receptive fields and stimulus
presentation. We initially mapped the receptive fields of single
MT neurons by hand on a tangent screen using small black-and-white
geometric targets. For each neuron, we recorded the location and size
of the minimum response fields and determined its selectivity for the
direction of motion. With the exception of a small group clustered around 30° eccentric, our neurons had receptive field centers that
were fairly evenly distributed between 1 and 17° from the fovea.
After receptive fields had been determined, we positioned a mirror to
place the preferred eye's receptive field on the face of a display
oscilloscope that subtended 10-15° at the monkey's eye. For most
experiments, textures consisting of several hundred randomly placed
bright dots were generated and moved with 100% coherence under
computer control. The mean luminance of the random-dot displays was
between 5 and 10 cd/m2, and the frame rate was 128 or 250 Hz. In some experiments, we also used a separate display
generated by a raster frame buffer to measure responses to drifting
gratings or plaid patterns. Methods for generating these stimuli are
detailed elsewhere (Levitt et al., 1994). Gratings and plaids were
presented at a frame rate of 107 Hz. Time-sampled motion stimuli can
contain energy in the direction opposite to the motion, because of
spectral replicas created by the temporal sampling (Watson et al.,
1986; Britten et al., 1993). Because cells having high preferred
speeds tend also to prefer low spatial frequencies (Levitt et al.,
1988), the intrusion of energy from these spectral replicas was
almost certainly outside the cells' spatiotemporal sensitivity range for the speeds that we used.
Before starting quantitative analysis, we attempted to optimize the
stimulus to elicit strong and reliable responses to stimuli moving at
the optimal speed and direction. Many cells responded much better if
dots moved through a small window inside a surround of stationary dots;
this often meant presenting a surround of stationary dots around dots
that moved through a window over the classical receptive field. For the
remainder of the paper, the moving texture will be called a
"target," and we will refer to target speed and acceleration even
though these attributes actually belong to the individual dots within
the moving part of the texture stimulus. Because our recordings were
made in anesthetized monkeys, target and image motion are
interchangeable; this would not be the case during pursuit in awake monkeys.
Stimuli were presented as a series of trials with durations that ranged
from 1024 to 3072 msec with intertrial intervals of ~1 sec. Target
motions consisted of steps and ramps of target speed of different final
speeds and durations, double pulses of target speed at different
interpulse intervals, and sinusoidal variation of target speed around
baselines ranging from zero to several times the neuron's preferred
speed. Trials were blocked by stimulus type to generate coherent,
controlled experiments while keeping each block small enough so that we
would obtain useful data for at least part of our series if neuronal
isolation were lost. Each block of trials was repeated 8-32 times, and
the order of trials was shuffled between blocks. The seed used to initiate the pseudorandom sequence that placed the random dots was
changed, and the dots were replotted for each trial; we cannot reconstruct the locations of the dots because we did not record the
seed for each trial. Early experiments were controlled by a PDP11
computer, and later experiments were controlled by an 80486 personal
computer using a DSP board with 16-bit DACs to generate
random dots. Data were analyzed after the experiment by aligning the
responses to identical target motions and compiling average response
histograms. Details of the data analysis are provided in the
descriptions of the relevant figures.
Reconstruction of recording sites. During recording, small
electrolytic lesions were produced at locations of interest along the
electrode tracks by passing DC current (2 µA for 2-5 sec; tip
negative) through the electrode. At the end of the experiment, the
monkeys were killed with an overdose of Nembutal and perfused through
the heart with 2 l of 0.1 M PBS followed by
2 l of a solution containing 4% paraformaldehyde in 0.1 M PBS. Blocks containing the region of interest were stored
cold overnight in a post-fix solution of 4% paraformaldehyde and 30%
sucrose, after which 40 µm sections were cut on a freezing microtome.
Sections were stained for Nissl substance with cresyl violet or for
myelin using the methods of Gallyas (1979). Most recordings were
verified to lie within area MT, as defined by standard histological
criteria (Van Essen et al., 1981). In the few cases in which we were
unable to recover all the electrode tracks, we used the distinctive
concentration of direction-selective neurons and the relatively small
sizes of their receptive fields to identify recording sites as lying within MT (Desimone and Ungerleider, 1986).
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RESULTS |
Experimental results
Our results are presented in separate experimental and simulation
sections. In the experimental results, we show first that MT cells vary
substantially in the dynamics of their responses to motion. Some cells
encode only target velocity, whereas others have large transient
responses to changes in target motion. We will conclude that the
distribution of transient responses across the population of MT cells
has the potential to provide information about target acceleration,
although no individual cell does. Second, we use paired pulses of
target velocity at short intervals to characterize the mechanism that
controls the time course of transient responses. We will conclude that
a class of mechanisms that falls under the general term
"adaptation" is responsible for the transients, although our data
could be accounted for by any of several specific cellular or neural
mechanisms. Third, we describe the responses of MT cells for sinusoidal
oscillation of target speed at frequencies normally seen during pursuit
eye movements.
Responses to steps of target speed
It is well known that MT cells are tuned for the direction and
speed of target motion. Our first step was to look for dynamic features
in the responses to target motion by measuring the time course of
neuronal firing for target motions that had instantaneous onsets.
Figure 2 shows the responses of two cells
to steps of target speed for stimuli that moved in the neurons'
preferred directions. For these experiments, the target initially was
visible and stationary for 256 msec, then moved at one of 8-10 speeds for 512 msec, and finally stopped and remained stationary for 256 msec.
For the cell in Figure 2A, steps of target speed
evoked a large transient pulse of activity followed by a sustained
response. Both the transient and sustained responses were tuned for
speed, with a preferred speed of ~9°/sec for the transient response
and ~5°/sec for the sustained response. The offset of target motion caused only a slight transient response, in contrast to the large transients emitted by this cell for the onset of motion. For the cell
in Figure 2B, a step of stimulus speed caused little
or no transient response at the lower speeds, only a small transient response at speeds >19°/sec, and a clear sustained response with a
preferred speed of ~19°/sec. In our sample of ~100 cells, we always observed a speed-tuned sustained response, but we observed a
range of transient responses. The example in Figure
2A is near one end of the range, whereas that in
Figure 2B is in the middle of the population.
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Figure 2.
Representative responses of MT cells to steps of
target speed. A, Responses of a neuron
(321r10) with one of the largest transient responses we
recorded. B, Responses of a neuron
(324r4) with a more typical transient response.
Each histogram shows the accumulated response of the
neuron to steps of target speed from zero to the final speed indicated
at the left of A. The
histogram labeled stationary shows each
neuron's response to the appearance of a stationary dot field that did
not move. Bin width was 8 msec. The traces at the
bottom of the figure show the time course of target
speed, which always started at zero. Histograms were
accumulated from nine repeats of each stimulus in A and
eight repeats in B.
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For each MT cell, speed tuning and direction tuning were similar for
the transient and sustained responses. Figure
3A illustrates speed-tuning
curves for the transient (open symbols) and sustained (filled symbols) responses of the typical MT cell
whose responses to steps of target speed are shown in Figure
2B. For each speed, the size of the sustained
response was computed as the mean firing rate in the interval from 256 to 512 msec after the onset of target motion. The size of the transient
response was estimated as the largest mean firing rate in a 24 msec
window within 120 msec of the onset of target motion. The spontaneous
firing (Fig. 3A, horizontal dashed
line) was taken as the sustained firing rate for a target that remained stationary throughout the trial. We fitted each speed-tuning curve (with spontaneous firing subtracted) with the function:
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(1)
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where Rs is the response at each
speed, a is the amplitude, b is the preferred
speed, c is the tuning bandwidth, d is a
parameter controlling the skew of the curve, and s is target
speed. As shown in Figure 3B, there was a strong correlation
between the preferred speeds of the transient and sustained responses
(r = 0.92), with a tendency for the preferred speed for
the transient response to be slightly higher than that for the
sustained response.
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Figure 3.
Quantitative analysis of the responses of 104 MT
neurons to steps of target speed. A, Firing rate plotted
as a function of target speed for neuron 324r4, which also appears in
Figure 2B. Open and filled
symbols show the transient and sustained responses,
respectively. The horizontal dashed line shows the
spontaneous firing in the presence of stationary dots. The solid
curves show the best-fitting function based on Equation 1. The
vertical arrows labeled aT
and aS are positioned at the preferred
speeds of the transient and sustained responses, respectively, and show
the peak responses, which were used to compute the transient/sustained
ratio. B, Comparison of preferred speed for the
transient and sustained response. Each symbol summarizes
the response of one MT neuron. C, Representative
speed-tuning curves showing the range of preferred
speeds of the sustained responses. The curves were
normalized so that each is plotted on the scale shown by the
calibration bar on the lower right but were shifted
vertically to facilitate viewing. The horizontal
dashed line at the right of each
curve shows the baseline for that neuron, and the
number next to the curve gives the
amplitude of the sustained response at the preferred speed. The three
vertical dashed lines were drawn at speeds of 1, 10, and
100°/sec to facilitate comparison of the different tuning
curves.
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We computed the transient/sustained ratio as
aT/aS, where
aT and aS are
the fitted values of a for the transient and sustained responses, respectively. For the two cells illustrated in Figure 2, the
transient/sustained ratio was 7.8 and 1.9. Among 104 cells that were
studied during steps of target speed, the transient/sustained ratio
varied from 1.08 to 9.4 with a median of 1.77 and a mean of 2.1. The
size of the transient/sustained ratio was not correlated with any of
the other parameters we measured, including laminar location of the
recorded neurons, preferred speed, sustained response strength, spatial
or temporal tuning for sine wave gratings, or pattern versus component
characteristics during stimulation with moving "plaids" (Movshon et
al., 1985; Movshon and Newsome, 1996).
The stack of speed-tuning curves in Figure 3C illustrates
the range of observed speed-tuning characteristics for the cells' sustained responses, with preferred speeds varying over two orders of
magnitude from 0.6 to 65°/sec. Every cell we recorded showed speed
selectivity when the range of testing speeds went as low as
0.125°/sec. Note that the three cells plotted at the bottom of Figure
3C would have been classified as low-pass (Lagae et al.,
1993) had they been tested only at speeds >1°/sec.
Latency of response
We measured the latency of the neuronal response for steps of
target speed; to avoid contaminating our data with noise, we did not
make measurements if a stimulus evoked sustained firing that was <10%
of the sustained response at the preferred speed. Latencies were
measured manually from histograms with 4 msec bin widths by following
the rising phase of the averaged response back in time until the
average firing was within the fluctuations in spontaneous rate. Because
of the low or absent spontaneous rates and the brisk rises of the
initial responses to steps of target speed, this procedure almost
always yielded an estimate of latency that had an error of 8 msec or
less. In 80% of the responses, the manual analysis yielded latencies
in perfect agreement with those obtained by an objective procedure that
found the time after which the firing rate remained >1 SD above the
base rate. We elected to use the manual analysis throughout, however,
because it was clearly superior to the objective analysis in the
remaining 20% of responses, in which the objective analysis was
unreliable because of response variability.
In almost all cells, the latency was quite long for low target speeds,
decreased with increases in target speed, and became as short as 40 msec at the highest speeds. We have plotted the latency of response as
a function of the inverse of target speed (Fig.
4A) because straight
lines in these graphs have a clear functional interpretation if
response latency has two components a variable component that requires
the target to traverse a given distance before the response is
initiated and a fixed delay affecting all responses. The
y-intercept latency from the linear model estimates the
fixed minimum latency. The slope of the line has the units of degrees,
estimating a "space constant" that measures the distance the target
has to traverse before a response is initiated. We have not analyzed
the linear model rigorously, but linear fits were generally excellent
for speeds as low as 0.5°/sec (Fig. 4A). Latency
often failed to follow the linear model for speeds <0.5°/sec, which
we excluded from the linear regression for this reason.
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Figure 4.
Determinants of response latency in MT neurons.
A, Plots of latency versus 1/speed for three
representative MT cells. Symbols show the measurements,
and the straight lines show the result of regression
analysis with the equation: latency = b + a/speed, where a is the space constant
and b is the fixed, minimum latency. B,
Space constant (a converted to minutes of arc) plotted
as a function of preferred speed of the transient response.
C, Intercept latency (b) plotted
as a function of the transient/sustained ratio for steps of target
speed. D, Intercept latency (b)
plotted as a function of preferred speed of the transient response. In
B-D, each symbol shows
the response of one MT neuron.
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The regular relationship between latency and inverse speed appeared to
be robust despite wide variations in response strength across speed.
For example, note the difference in latency between the weak responses
of each of the example neurons to targets of low and high speeds. The
fixed component of latency showed no convincing (or statistically
significant) relationship to either the transient/sustained ratio (Fig.
4C) or the preferred speed (Fig. 4D),
suggesting that cells preferring different speeds did not differ in
their temporal properties. In contrast, the graph in Figure
4B shows that the space constant was larger for cells that had higher preferred speeds. This relationship can be understood by noting that neurons with different speed preferences can be constructed by variation in either their spatial or temporal
properties. In agreement with the previous finding that cells
responding well to high speeds tend to prefer lower spatial frequencies
(Levitt et al., 1988), our results suggest that spatial
variations are more important than temporal variations in
differentiating neurons with different preferred speeds.
Stimulus-dependent variations in latency provide information about the
neural mechanisms that lead to cellular responses, whereas the latency
of the population constrains how the neuronal responses in MT might be
decoded by downstream circuits. Figure 5
summarizes the distribution of latencies in all cells in which we
measured latencies at target speeds of 1, 8, and 64°/sec. Both the
distributions and the medians (Fig. 5, vertical
arrows) shifted toward shorter latencies as speed increased.
For speeds >1°/sec, we recorded some latencies as short as 40 msec;
the median latencies were 88, 72, and 65 msec at target speeds of 1, 8, and 64°/sec, respectively.
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Figure 5.
Distribution of response latency for MT neurons
with measurable responses at three selected speeds: 1, 8, and
64°/sec. Vertical arrows show the median latency for
each speed.
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Responses to ramps of target speed
In this section, we show that the firing rate of some MT neurons
is influenced by target acceleration. The data do not support a
conclusion that the firing of individual MT neurons encodes target
acceleration. Instead, these data provide the basis for the final
section of the paper, in which we demonstrate that target acceleration
can be reconstructed from the population response in MT.
To determine whether the firing of MT cells was influenced by target
acceleration and how, we used stimuli in which the target either
accelerated or decelerated through a given range of speeds. Figure
6 shows the time course of firing rate
during these stimuli for the two cells whose responses to steps of
target speed appear in Figure 2. As shown by the target speed traces at
the bottom of the figure, the target was initially stationary and
visible for 256 msec. Target speed then increased at a constant rate
for 128 msec up to final speeds indicated by the numbers at the left of
each histogram in Figure 6A, moved at constant speed
for 512 msec, decelerated at a constant rate for 128 msec back to zero velocity, and finally remained stationary and visible for 256 msec. In
our later experiments, the entire sequence of constant acceleration,
constant speed, and constant deceleration was presented in single
trials like those illustrated in Figure 6. In our earlier experiments,
we presented the accelerations and decelerations in separate trials and
then spliced the averages together. The results were identical for the
two methods.
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Figure 6.
Representative averaged responses of MT neurons to
ramp increases and decreases in target speed. A, MT
neuron (also shown in Fig. 2A) that had one of
the largest transient responses and one of the largest asymmetries and
that was recorded between ramp increases and decreases in target speed.
B, MT neuron (also shown in Fig.
2B) that fell approximately in the middle of our
sample in terms of transient response and asymmetry between ramp
increases and decreases in target speed. For each neuron, the different
firing rate histograms show responses for ramp increases to and
decreases from different target speeds, given by the
numbers at the left of A.
Target speed always started at zero. The traces at the
bottom of A and B show the
time course of target speed. The vertical dashed lines
delimit two analysis intervals in the first 256 msec after the onsets
of ramp increases and decreases in target speed. Histograms were
accumulated from 9 repeats of each stimulus in A and 16 repeats in B. Bin width is 8 msec.
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In principle, the response of a cell to this stimulus could be
determined entirely by the sequence of speeds traversed by the target
during ramp accelerations and decelerations, or it might also be
influenced by features of motion other than target speed. If a ramp
acceleration took target speed from zero through the cell's preferred
speed to a final speed well above preferred speed, then we would expect
to see a large transient response because of the speed tuning of the
cell. However, we would expect to see approximately the same transient
as ramp deceleration took target speed from well above preferred speed
back through preferred speed down to zero. Thus, MT cells that give
symmetric responses to ramp increases and decreases in target speed
probably do not provide information that could be used to determine
target acceleration. On the other hand, an asymmetric response to ramp
increases and decreases in speed would show that a cell's response
could carry information about target acceleration as well as speed.
The cell whose responses appear in Figure 6A shows
one extreme of a continuum of neuronal behavior for ramp increases and decreases in target speed, whereas the cell in Figure
6B is near the median. For all six final ramp speeds
shown here, the cell in Figure 6A showed a pronounced
transient response during ramp increases in target speed and either
none or much less of a transient during ramp decreases in target speed.
The cell in Figure 6B showed a clear transient
response during ramp increases in target speed to 74°/sec as well as
to other speeds >18°/sec. However, this cell also emitted a large
transient for ramp decreases to zero target speed from these higher
speeds. Thus, inspection of the histograms demonstrates, at least for
ramps of target speed that started at zero, that the firing of the cell
in Figure 6A differentiates between target
acceleration or deceleration for all final speeds, whereas the firing
of the cell in Figure 6B did so only over a middle
range of speeds. Inspection of Figure 6, A and B,
reveals that the latency of the peak response during ramp increases in target speed becomes shorter as final target speed increases. This
suggests that the transients of firing during ramp increases in target
speed occur as the target passes through the cell's preferred speed
and leads to the question (addressed below) of why similar transients
are not always evident for ramp decreases in target speed.
We quantified the asymmetry in each cell's responses to ramps of
target speed by measuring the peak firing rates over a 24 msec window
in the two 256 msec intervals starting at the onset of the ramp
increases and decreases in target speed (Fig. 6, intervals marked by
vertical dashed lines). The analysis intervals
always included the peak of the responses without including other peaks that occasionally occurred later. The results of this analysis appear
in Figure 7 for the two example cells in
Figure 6. Figure 7, A and B, plots the peak
firing during ramp increases (open symbols) and
decreases (filled symbols) in target speed as a
function of the magnitude of target acceleration. As was clear in the
histograms, the cell in Figures 6A and 7A
showed a large asymmetry, whereas that in Figures 6B
and 7B shows a consistent but smaller asymmetry. To gain an
impression of the size of the asymmetry in relation to each cell's
sustained and transient responses to steps of target speed, we computed
the difference between the peak firing for target accelerations to and
decelerations from a given target speed and plotted the difference
firing rate as a function of target acceleration (Fig.
7C,D). For the cell in Figure 7C, the difference firing between ramp acceleration and deceleration increased consistently as a function of target acceleration up to an asymptote of
~320 impulses/sec. The asymptote was much larger than the peak of the
sustained response to steps of target speed (Fig. 7C,
bottom horizontal dashed line). For the
cell in Figure 7D, the difference firing rate was clearly
tuned and reached a peak that was less than the maximal sustained
response for steps of target speed. Comparison with the maximal
sustained response is useful because it reveals the size of the
asymmetry, which may provide information about target acceleration,
relative to the size of the same neuron's responses to sustained
speed. We conducted this analysis on all 89 cells that were studied
during steps and ramps of target speed, and for each cell, we fitted
the difference firing with Equation 1, with speed supplanted by
acceleration. Examples of the fits are shown by the smooth curves in
Figure 7, C and D.
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Figure 7.
Quantitative analysis of the asymmetry between
responses to ramp increases and decreases in target speed.
A, B, Open and
filled symbols plot the peak firing rate in the first
256 msec after the onset of ramp increases and decreases,
respectively, in target speed as a function of the magnitude of
target acceleration. The horizontal dashed lines show
the maximum sustained firing rate, computed as the maximum sustained
response plus the spontaneous firing rate. C,
D, Open symbols plot the difference
between the peak firing rates for ramp increases and decreases
in target speed as a function of target acceleration. The solid
curves plot the best fit obtained with Equation 1, with target
acceleration substituted for target velocity. The two horizontal
dashed lines show the largest transient and sustained responses
for steps of target speeds, with spontaneous firing subtracted.
A, C, Data are for cell
321r10, which also appears in Figures
2A and 6A.
B, D, Data are for cell
324r4, which also appears in Figures
2B, 3A, and
6B.
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The selection of curves in Figure
8A shows the diversity
of the asymmetries among the cells that had the largest responses to
ramp acceleration and deceleration. These examples were selected from
the 40 cells that had transient/sustained ratios of 1.8 or larger
during steps of target speed and that were studied during ramps of
target speed. Each curve plots the difference firing rate described in
the previous paragraph, normalized for the largest transient response
evoked by steps of target speed, as a function of the value of the ramp
acceleration. We chose to normalize for the largest transient response
during steps of target speed so that most of the normalized points
would have values between 0 and 1 and different cells could be compared
easily. The curve labeled e in Figure
8A is from the neuron that was used to
construct Figure 7, A and C, and shows an
asymmetry evident across the full range of accelerations we tested.
Curve g in Figure 8A is from the
neuron used to construct Figure 7, B and D, and
shows an asymmetry tuned for a middle range of accelerations. Other
cells showed tuning for a narrow range (Fig. 8A,
curve h) or a broad range (curve f) of
accelerations or had high-pass characteristics with positive values of
difference firing rate starting at low (curves c,
d), medium (curve b), or high (curve
a) accelerations. None of the 39 cells that were tested with
target accelerations as low as 0.9°/sec2 showed
low-pass characteristics. The presence of large differences in the
response to ramp accelerations and decelerations demonstrates that the
firing of some MT cells is influenced by target acceleration. The
diversity of the tuning across the population suggests that target
acceleration across different speed ranges is probably represented by
different MT cells.
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Figure 8.
Asymmetry between responses to ramp increases and
decreases in target speed. A, Family of
curves showing the relationship between the difference
firing rate (peak during acceleration minus peak during deceleration)
and the magnitude of target acceleration for eight MT neurons.
Difference firing rates are normalized to the peak transient response
for steps of target speed and are plotted according to the calibration
bar on the lower right of the graph. The
curves for different cells have been shifted
vertically to facilitate viewing. The horizontal
dashed lines show difference firing rates of zero.
Curves e and g are from the two neurons
that appear in Figure 7. B, Comparison of the
transient/sustained ratio for ramps and steps. C,
Comparison of individual neurons' preferred stimulus acceleration for
ramps and preferred speed for steps of target speed. The solid
line is the type 2 regression line for the neurons plotted with
open circles. In B and C,
open circles are for cells with transient/sustained
ratios for ramps >1.8, and small x symbols are for
cells with transient/sustained ratios <1.8. Because the ramp increases
in target speed always started from zero and were always 128 msec in
duration, the two y-axes in C are
proportional and differ only by the factor of
0.1281 used to convert the change in target speed
(left y-axis) into target acceleration (right
y-axis).
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Two facts argue against the possibility that the transient responses of
MT neurons to accelerating and decelerating targets might arise because
the apparent contrast of rapidly moving stimuli is lower than that of
stationary targets. First, this effect should be the same for both
increases and decreases in target speed, because MT cells usually
respond equally to increases and decreases in contrast (Maunsell and
Van Essen, 1983) (J. A. Movshon, unpublished observations). Thus,
it cannot explain the asymmetries we observed; if anything, it would
attenuate them by adding a response at the point of both increasing and
decreasing target speed. Second, an effect of apparent contrast on the
transient responses of MT neurons predicts that the transient responses
should be unselective for target speed. Figures 3A and
8A contradict this prediction.
Comparison of responses to ramps and steps of target speed
To summarize the ramp asymmetry for each cell, we computed the
transient/sustained ratio for ramps, defined as
(aR + aS)/aS, where
aR is the peak value of the curve fitted
to the difference firing rate from graphs like Figure 7, C
and D, and aS is the peak value
of the curve fitted to the sustained firing rate during steps of target
speed. The transient/sustained ratio for ramps of target speed was 4.9 for the cell in Figures 6A and 7, A and C, and was 1.6 for the cell in Figures 6B
and 7, B and D. Figure 8B shows
that the transient/sustained ratio for steps and ramps of target speed
agreed well for almost every cell we recorded (r = 0.87). The different symbols indicate cells with transient/sustained ratios for ramps greater than and less than 1.8 (compare Fig. 8C). Type 2 regression analysis on the log10 of
the data in Figure 8B, under the assumption that the
two values of transient/sustained ratio were equally well estimated
from the data, gave a slope of 0.74. As expected, there was also
excellent agreement between the peak sustained responses for steps and
ramps of target speed (r = 0.96; type 2 regression
slope = 1.19).
Figure 8C illustrates that there was also a correlation
between the preferred speed of the transient response to steps of target speed and the preferred acceleration from the fits to the difference curves for ramps of target speed. The values plotted along
the y-axis (Fig. 8C) were all obtained for 128 msec ramps from zero to final speed, creating a proportionality between
the final speed (left y-axis) and the acceleration of
the preferred ramp (right y-axis). The scatter plot
suggests a more consistent relationship for neurons with
transient/sustained ratios >1.8 (Fig. 8C, open
circles, r = 0.61) than for those with
smaller transient/sustained ratios (small x
symbols, r = 0.16). For the neurons with
transient/sustained ratios of 1.8 or greater, type 2 regression
analysis on the log10 of the data in Figure 8C
(solid line) revealed a regression slope of 1.0, indicating that the derived values on the y- and
x-axes are proportional. The constant of proportionality of
approximately four implies that most cells could contribute most
effectively to a distributed representation of target acceleration when
the target accelerates from zero to final speeds higher than the
cell's preferred speed.
Responses to spatially restricted targets
To establish the generality of the response characteristics that
we measured in MT cells stimulated with speed steps and ramps for
random-dot texture targets, we studied the responses of 38 of the cells
to the same motion trajectories of small spots and textures. These
experiments were complex to design and execute, because the use of
small targets mandated that we explore the effect of varying the
receptive field position at which steps and ramps of target speed
occurred. The analysis was similarly complex, because the sensitivity
profile of the neuron's receptive field had to be taken into account
in evaluating the responses. We do not present a detailed analysis of
these experiments here, but the results showed that the important
dynamic features of MT cells' responses to textures could also be
discerned in their responses to small targets. We are therefore
confident that our measurements with textures represent adequately the
response of MT cells for the small targets normally used to study
pursuit eye movements.
Effect of base velocity and ramp duration on the responses to
target acceleration
The results in Figures 7 and 8 imply that many MT cells could
provide information about the direction and possibly the magnitude of
target acceleration for targets that are initially stationary and
accelerate smoothly through the range of speeds to which the cell is
sensitive. However, individual MT cells cannot signal acceleration for
all initial target speeds. Figure
9A-D shows the responses of
an example MT neuron to different combinations of acceleration and
initial target speed. This cell had large transient responses during
128 msec ramps of target speed from 0 to 2.25, 4.5, or 9°/sec (Fig.
9A-C). The transient response was completely absent when
the ramp started at the preferred speed for the sustained response,
which was ~4.5°/sec for this cell, and increased to 9°/sec (Fig.
9D). Thus, a target acceleration of
35°/sec2 caused a large transient in firing when
the target speed started at 0°/sec (Fig. 9B) and no
transient when the target speed started at 4.5°/sec (Fig.
9D). We obtained the same result on all 15 cells that were
tested. The converse experiment, of varying ramp duration and therefore
target acceleration while keeping the initial and final target speed
the same, also emphasized the nature of the relationship between firing
rate and target acceleration. When the initial target speed was
0°/sec and the final target speed was 9°/sec, the response of the
cell in Figure 9 was essentially the same when the target acceleration
was 2240°/sec2 (Fig. 9H, ramp duration
4 msec), 560°/sec2 (Fig. 9G, ramp
duration 16 msec), 280°/sec2 (Fig. 9F,
ramp duration 32 msec), 140°/sec2 (Fig.
9E, ramp duration 64 msec), or 70°/sec2
(Fig. 9C, ramp duration 128 msec). These results show that
individual MT neurons can contribute to a distributed representation of
target acceleration only for a limited range of target speeds.
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Figure 9.
Examples showing that MT neurons are sensitive to
target acceleration only for some stimulus trajectories. Each
panel shows a histogram of firing rate
obtained from nine repeats of the target speed trajectory shown by the
lower trace. The column of
histograms shows the responses for different initial and
final target speeds. The row of
histograms shows responses to different accelerations
produced by different duration ramps between zero and twice the
preferred speed. A, 0-2.25°/sec in 128 msec.
B, 0-4.5°/sec in 128 msec. C,
0-9°/sec in 128 msec. D, 4.5-9°/sec in 128 msec.
E, 0-9°/sec in 64 msec. F, 0-9°/sec
in 32 msec. G, 0-9°/sec in 16 msec. H,
0-9°/sec in 4 msec.
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Responses to double pulses of target velocity at different
interpulse intervals
Our data on responses of MT cells to steps and ramps of target
speed (Figs. 2, 6) revealed that many neurons have large transient responses with complex dynamics. Because the size and time course of
these transients are dependent on stimulus speed and acceleration, such
simple transient-forming mechanisms as linear high-pass temporal filters (e.g., spike frequency adaptation) probably cannot provide a
complete account of the data. To account for these complex dynamics, we
guessed that neuronal excitability might be influenced by a time-dependent adaptation of the responses to a given stimulus. Adaptation might arise either from complex synaptic or cellular mechanisms or from neurons outside those providing excitatory input to
the cell. We conducted the following experiments to probe adaptation,
even though we realize that they do not distinguish possible mechanisms
rigorously and that adaptation could be implemented by division or
subtraction. Thus, we use the term adaptation to encompass a
number of possible neural and cellular mechanisms.
As an initial probe of adaptation, we used a series of condition-test
experiments in which we measured the response to a test pulse of target
speed as a function of time after an identical conditioning pulse. As
shown in Figure 10, targets were
stationary and visible for 256 msec before undergoing one or two 64 msec pulses of motion of the preferred speed and direction. In Figure 10, top, for example, the histogram (Firing
rate) shows the response of one cell to two 64 msec pulses
of target speed that were separated by 64 msec. As a control, the solid
firing rate trace shows the response of the same cell to a single pulse
of target speed. The response to the second pulse can then be estimated
by subtracting the response to the first pulse from the response to two
pulses (Fig. 10, middle, Difference firing
rate).
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Figure 10.
Example histograms showing the
experimental design and data analysis for experiments that presented
two pulses of target speed at different interpulse intervals.
Top, The histogram shows the accumulated
response to 10 repeats of two 64 msec pulses of target motion in the
neuron's preferred direction at its preferred speed with an interpulse
interval of 64 msec. Bin width is 8 msec. The solid trace
superimposed on the histogram is the average
response of the same neuron to 40 repeats of just the first pulse.
Middle, The difference firing rate in the
histogram shows the response to two pulses minus the
response to the first pulse. Bottom, The
solid and dashed traces show the time
course of target speed for the double- and single-pulse stimuli,
respectively.
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Figure 11 shows two examples that
relate the results of the two-pulse condition-test experiments and the
responses of MT cells to steps of target speed (traces
labeled Step). The data in Figure 11A came
from a cell that had a clear transient response to 512 msec steps of
target speed, whereas the data in Figure 11B came from a cell whose response was almost purely tonic. For each cell, the
traces labeled Pulse (Fig. 11) show the response to the test pulse alone, and the traces labeled with different times show the
responses when the onset of the test pulse followed the offset of the
conditioning pulse by the specified delay. For each histogram in the
left columns of Figure 11, A and B, the companion
histogram in the right columns shows the difference firing rate,
obtained by subtracting the response to the conditioning pulse from the response to two pulses and aligning the differences on the onset of the
test pulse. For the response to the test pulse alone (Fig. 11,
Pulse), the difference histograms were obtained by
subtracting the spontaneous firing recorded during presentation of a
stationary texture.
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Figure 11.
Comparison of transient behavior during steps of
target speed with response attenuation in two-pulse experiments for two
representative neurons. A, A neuron with a large
transient during steps of target speed and strong attenuation of the
response to the second pulse at short interpulse intervals.
B, A neuron with no transient and no attenuation of the
response to the second pulse. In A and B,
the histogram labeled Step shows the
accumulated firing for a 512 msec step of target speed to the same
speed used for the two-pulse experiments. For the
histogram labeled Pulse in
A and B, the left column
is the accumulated firing for a single pulse of target speed, and the
right column is the same response with the spontaneous
firing for a stationary dot pattern subtracted. For the lower
six rows of A and B, the
left column shows the response histogram
for two pulses of target speed at the interval indicated by the
number to the left of the
histogram. The right column shows the
difference firing rate obtained by subtracting the response to the
first pulse and aligning the resulting histograms on the
onset of the second pulse of target speed. The vertical dashed
line shows the approximate start of the response. Bin width is
16 msec for the response to steps of target speed and is 8 msec in all
of the other histograms. The traces below
each histogram show the time course of target speed. For
each neuron, we performed this experiment for pulses of target speed
that were near the preferred speed. The pulses and step of target speed
were 2°/sec in A and 4°/sec in B. In
A, the step and single-pulse responses were obtained
from 40 repeats of each stimulus, and the other
histograms were from 10 repeats. In B,
the step and single-pulse responses were obtained from 60 repeats of
each stimulus, and the other histograms were from 16 repeats.
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For the cell with a transient response to steps of target speed (Fig.
11A), there was a clear effect of the time between
the pulses on the response to the second pulse. The second response was
attenuated for interpulse intervals <64 msec and returned to control
values when the interpulse interval reached 256 msec. The gradual
return of the amplitude of the response to the second pulse can be seen
in the histograms of the responses to two pulses (Fig.
11A, left column of
histograms) but is clearer in the difference firing rates
obtained by subtracting the response to the first pulse (Fig.
11A, right column of
histograms). The difference firing rates are aligned on the
time of onset of the second pulse, revealing that the latency of the
response to the second pulse (Fig. 11A, vertical dashed line) did not vary consistently
as a function of the interpulse interval. Note also that these
difference rate histograms suggest that the response to the second
pulse at short interpulse intervals is a scaled-down replica of the
unconditioned response (Fig. 11A, top
traces); these scaled replicas are a characteristic signature of a mechanism that controls response gain and are quite different from the abbreviated ("iceberg") responses that would result from delivering double-pulse stimuli to a linear high-pass filter.
For the cell that lacked a transient response to steps of target speed
(Fig. 11B), the two-pulse experiment yielded a
different result. The response to the second pulse did not depend
strongly on the interpulse interval and was nearly the same as that of the control even for interpulse intervals as short as 32 msec.
To quantify the relationship between the attenuation of the response to
the second pulse at short interpulse intervals and the transient
responses of MT cells to steps of target speed, we determined the
latency of the response to a single pulse, defined the next 64 msec as
an analysis interval, and measured the mean difference firing rate in
the analysis interval for each of the six condition-test intervals. We
then computed a "response attenuation index" defined as the mean
difference firing rate for an interpulse interval of 32 msec divided by
the average of the firing rates for interpulse intervals of 128 and 256 msec. Figure 12 shows the relationship
between the transient behavior of firing rate for steps of target speed
and the amount of attenuation revealed in two-pulse experiments. Each
point in this plot shows results from one of the 22 cells studied using
two pulse stimuli. The x-axis plots the response attenuation
index derived above, and the y-axis plots the
transient/sustained ratio for steps from stationary to the target speed
used for the two-pulse experiment. In general, cells that had little or
no transient response for steps of target speed also had attenuation
indices near 1.0 for two pulse stimuli, indicating no attenuation.
Cells with large transient responses for steps of target speed had
attenuation indices as small as 0.1. Although we have elected not to
show the data here, we obtained very similar results from the same
cells when we tested them with a 64 msec pulse of target speed at
different intervals after the offset of a 512 msec step of target
speed. We interpret the correlation between the existence of a
transient response to steps of target speed and attenuation of
responses at short interpulse intervals as evidence that adaptation
shapes the transient responses of MT cells. The recovery of the
response to the test pulse at longer interpulse intervals reflects
recovery from adaptation. In a later section of the paper, we will
implement adaptation as divisive gain control to create transient
responses in a model that reproduces the dynamics of MT cell
responses.
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Figure 12.
Quantitative analysis of the relationship between
response attenuation in two-pulse experiments and the transient
behavior for steps of target speed. Each point shows the response of
one MT neuron. The x-axis plots the response attenuation
index for two-pulse experiments: the mean firing rate for an interpulse
interval of 32 msec divided by the average of the responses for
interpulse intervals of 128 and 256 msec. The y-axis
plots the transient response divided by the sustained response for a
step from zero to the target speed used for the two pulse stimuli. Note
that this is different from the transient/sustained ratio computed
earlier as the peak transient response divided by the peak sustained
response across target speeds.
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To determine whether the influence of the conditioning pulse was
direction selective, we tested seven cells with a variant of the
double-pulse experiment in which the first 64 msec pulse of target
speed provided motion in the null direction and the second pulse was in
the preferred direction. Figure 13
shows one example of the responses, for the same cell whose responses
are also shown in Figure 11A. This cell gave a
brisk response to a 64 msec pulse of target motion in the preferred
direction (Fig. 13, top pair of traces
labeled On-direction) and was inhibited slightly when the
same pulse of target speed was delivered in the null direction
(trace labeled Null-direction). The histograms of
firing rate for the double pulse stimuli reveal that this cell responded well to the pulse of target speed in the preferred direction for the shortest condition-test intervals, even though the same cell
showed almost complete attenuation of the response to the second pulse
at short intervals when the two pulses were in the same direction (see
Fig. 11A). We again isolated the response to the
second pulse by computing the difference between the responses to two
pulses and to the test pulse alone and aligning the difference firing
rate on the onset of the second pulse.
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Figure 13.
Example of a two-pulse experiment showing the
effect of conditioning motion in the null direction on subsequent
responses to target motion in the preferred direction. The same
neuron's responses are also shown in Figure 11A.
For the histograms labeled
On-direction, the left
column is the response histogram for a single
pulse of target speed in the preferred direction, and the right
column is the same response with the spontaneous firing for a
stationary dot pattern subtracted. The histogram labeled
Null-direction shows the response to a
single pulse of preferred target speed in the null direction. For the
lower six rows, the left column shows the
accumulated firing for two pulses of target speed at the interpulse
interval indicated by the number to the
left of the histogram. The right
column shows the difference firing rate obtained by subtracting
the response to the first, null-direction pulse and aligning the
resulting histograms on the onset of the second,
preferred-direction pulse of target speed. The vertical dashed
line shows the approximate start of the response to a single
pulse. The traces below each histogram
show the trajectory of target speed, which was 2°/sec for this
experiment and was close to the neuron's preferred speed in all
experiments. In all histograms, the bin width is 8 msec.
The on-direction and null-direction histograms were
accumulated from 40 repeats of the same stimulus; the other
histograms were from 10 repeats.
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In each of the seven cells tested, conditioning motion in the null
direction affected responses to subsequent test motion in the preferred
direction, but in a different way than did conditioning motion in
the preferred direction. There was no response attenuation at short
latencies. Instead, the dynamics of the response were affected. In
Figure 13, for an interpulse interval of 0 msec, the response to the
preferred-direction pulse was delayed by almost 24 msec and had a
sharper time course and a larger amplitude than did the response to the
control pulse (histogram labeled
On-direction). The effect on the latency of the
response to the preferred direction was absent when the interpulse
interval increased to 96 msec. But inspection of Figure 13 hints that
null-direction motion may have had a rather long-lasting effect on the
shape and amplitude of this cell's responses to subsequent motion in
the preferred direction, even for an interpulse interval of 256 msec.
We have confirmed this effect with more detailed observations on a
larger sample of cells (Priebe et al., 1998). However, our
limited sample of seven cells provided enough data to show that the
adaptation in MT cells is direction selective.
Responses to sinusoidal modulation of target speed
Traditional approaches to understanding dynamics have often relied
on sinusoidal-forcing functions to obtain estimates of response gain
and phase as a function of frequency. In a linear system, such analysis
provides the same information as the time domain analysis we have
presented so far. In this instance, however, it seemed useful to
analyze the responses of MT cells to sinusoidal modulation of target
speed, partly because their responses are clearly nonlinear and partly
to obtain data relevant to the performance of the smooth pursuit
system, which has been analyzed frequently with sinusoidal target
motion (e.g., Fuchs, 1967; Lisberger et al., 1981; Goldreich et al.,
1992).
Because of the nonmonotonic relationship between firing rate and target
speed, we expected that the firing elicited by sinusoidal modulation of
target speed would depend critically on the base speed around which the
oscillations occurred. For example, if target speed during the
oscillation were confined entirely to speeds on the rising phase of the
cell's speed-tuning curve, then we would expect firing rate to be
modulated approximately sinusoidally with a peak response near peak
on-direction target speed. If, however, target speed were greater than
preferred speed throughout the full sinusoidal oscillation, then we
would expect firing rate to be modulated at the frequency of the
stimulus but with peak firing at minimum rather than maximum target
speed in the preferred direction. Finally, if the oscillation of target
speed were centered on the peak of the speed-tuning curve, then we
would expect firing rate to decrease for both increases and decreases
in target speed, and the modulation of firing rate would be at twice
the frequency of the sinusoidal stimulus.
With these expectations in mind, we customized the parameters of
sinusoidal modulation of target speed for each cell according to the
strategy summarized in Figure 14. The
graph on the right shows the speed tuning for a hypothetical cell by
plotting firing rate in the preferred and null directions as a function
of target speed. The four sine waves on the left show target speed, on
the same axis as the speed-tuning curve, as a function of time for the
four conditions. For DC = 0, the baseline speed was zero, and
target speed oscillated between preferred speed in the preferred and
null directions. This was the only sinusoidal modulation of target
speed that delivered motion in the null direction. For DC = 0.5, target speed oscillated between zero and the preferred speed. For
DC = 1, target speed was centered on the preferred speed and
oscillated approximately between the two speeds that caused
half-maximal responses. For DC = 2, target speed oscillated along
most of the arm of the relationship between firing rate and target
speed that was above preferred speed. In practice, we did not always
achieve the goals outlined in Figure 14, partly because the parameters
of target motion had to be estimated during recording for each neuron
and partly because skew in the speed-tuning curves of real MT cells
made it impossible to achieve the ideal reflected by Figure 14. When
our stimuli did approach the goal envisaged in Figure 14, however, the
general features of the neuronal responses in MT conformed to our
expectations.
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Figure 14.
Schematic diagram of the experimental design for
presenting sinusoidal modulation of target speed on different base
speeds. The graph on the right plots
target speed on the y-axis, with preferred-direction
motion upward, and sustained firing rate on the x-axis,
with increased firing rate plotted to the right. The
curve shows a schematic speed-tuning
curve for an MT neuron. The four sine waves on the
left of the figure show target speed as a function of
time. The speed calibration for the sine waves is the same as that for
the speed-tuning curve at the right. The
sine waves labeled DC = 0, DC = 0.5, DC = 1, and DC = 2 show
schematically the situation we tried to achieve when we selected the
baseline speeds and amplitudes of modulation of target speed used to
study the responses of each MT cell.
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Figure 15 shows data from a cell that
exemplifies all the basic features of the responses to sinusoidal
modulation of target speed. When the baseline speed was zero (Fig.
15A, DC = 0), firing rate was strongly
modulated at frequencies up to 8 Hz. Modulation of firing rate
increased somewhat as the frequency of target speed oscillation
increased from 1 to 8 Hz but was small at 16 Hz. The attenuation of
modulation at 16 Hz is verified by the cycle histograms on the upper
right of each long histogram. The cycle histograms were constructed
with 24 bins per cycle by averaging the responses across the last
second of sinusoidal modulation. For all frequencies, MT neurons
emitted a pulse of firing for the first cycle of target motion at
DC = 0. When target speed oscillated between zero and the
preferred speed (Fig. 15B, DC = 0.5), the
modulation of firing rate was considerably weaker than when target
speed oscillated around a baseline speed of zero. Again, there were
responses at 1, 2, 4, and 8 Hz, but it is difficult to see any
modulation of firing rate at 16 Hz in spite of an increase in the
sustained firing of this MT cell.
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Figure 15.
Responses of a representative MT neuron to
sinusoidal modulation of target speed. The figure consists of 15 pairs
of target velocity traces and histograms
accumulated from six repeats of the same target motion. The
inset on the right above each
histogram is a cycle histogram showing
the modulation of firing rate averaged over the last second of the
sinusoidal speed modulation; the baseline of these cycle
histograms has a duration equal to one period of the
sinusoid. Cycle histograms have 24 bins per cycle.
Full-stimulus histograms have a bin width of 10 msec.
A, The amplitude of modulation of target speed was
3°/sec, and the base speed was zero. B, The amplitude
of modulation was 1.5°/sec, and the base speed was 1.6°/sec.
C, The amplitude of modulation of target speed was
10.5°/sec, and the base speed was 13.5°/sec. The preferred speed of
this neuron was 3.2°/sec. From top to
bottom in A-C, the frequency of the sine
wave was 1, 2, 4, 8, and 16 Hz. Data are from neuron 405r09.
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When the baseline target speed was above the preferred speed so that
target speed oscillated on the descending limb of the relationship
between firing rate and target speed (Fig. 15C,
DC = 2), the response of the cell was again modulated.
As expected, the phase of the neuronal responses was now reversed
relative to DC = 0 and DC = 0.5. For example, at 1 Hz,
increased firing occurred during the upward deflection of the
sinusoidal component of target speed for DC = 0 and DC = 0.5 but during the downward deflection for DC = 2. The single-cycle
histograms illustrate the effects of oscillation frequency on phase
shift. At each frequency, the response phase is similar for DC = 0 and DC = 0.5 but reversed for DC = 2, just as described above
for oscillations at 1 Hz. For Figure 15, A-C, response
phase lag increases with frequency, as expected of a response with a
latency that is a substantial fraction of the period of the higher
frequency oscillations. Responses for DC = 1 were quite small and,
when present, were dominated by the second harmonic of the oscillation
frequency (data not shown).
For target speed oscillation around a base speed of zero, the
relationship between the modulation of firing rate and the frequency of
oscillation varied widely from neuron to neuron. Figure
16A plots the
modulation of neuronal response as a function of the frequency of
oscillation of target speed for 10 example cells. The examples were
selected by ordering the 31 cells tested with sine waves according to
the maximum modulation of firing rate at any frequency and plotting
every third cell. For each cell, we normalized response modulation at
all frequencies to the maximum. In Figure 16A, the
functions are plotted on the normalized scale defined by the
calibration bar on the bottom right of the graph but at arbitrary
positions on the y-axis to facilitate viewing. The short
horizontal dashed line on the right end of each curve shows zero
modulation for that curve and demonstrates that the normalized
modulation of neuronal firing at 16 Hz was always <0.1. Inspection of
Figure 16A shows that many cells had increases in response modulation as the frequency of target oscillation increased from 1 toward 8 Hz, while a few had low-pass characteristics. Peak
response modulation occurred at frequencies that ranged from 1 to 10 Hz
in different cells, and response modulation was always small at
frequencies >10 Hz.
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Figure 16.
Analysis of responses to sinusoidal target
motion. A, Response modulation as a function of the
frequency of sinusoidal target motion for base speeds of 0 (DC = 0) in 10 representative MT neurons. Each
curve plots data from a different MT neuron selected
according to the procedure outlined in the text. Each
curve has been normalized for the maximum modulation of
the individual neuron's firing at any frequency of modulation of
target speed and then shifted vertically to facilitate
viewing. The calibration bar on the lower right of the
graph shows the full scale of each normalized curve. The
horizontal dashed line on the right of
each curve shows the baseline for that
curve, and the number next to each
curve gives the maximum amplitude of response modulation
for that neuron at any frequency. B, Phase shift between
target speed and firing rate plotted as a function of the frequency of
sinusoidal target motion with base speeds of zero. The
curves summarize the responses from 16 neurons. The
large filled circles show the phase predicted by a fixed
time delay of 90 msec. C-F, Polar plots summarizing the
responses of all the MT neurons that were studied with sinusoidal
modulation of target speed at 1 Hz. Each line is a
vector representing the response of an individual MT neuron; the
small filled circles show the center of the plot, i.e.,
zero response; and the large outer circles show a
response modulation of 25 impulses/sec. Vectors pointing
to the right would be in-phase with on-direction target
speed, and phase lag is indicated by counterclockwise rotation of the
vectors.
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The phase shift between the sine wave of target speed and neuronal
firing showed a consistent relationship to the frequency and DC level
of the sinusoidal stimulus motion. For each cell, we measured the phase
difference between firing rate and target speed as the difference
between the phase shifts of the fundamental components obtained from
Fourier analysis. To correct for the fact that this difference was, by
definition, <360°, we assumed that phase lag would increase
monotonically and added 360° of phase shift whenever an increase in
frequency caused a large decrease in phase difference between the two
fundamental components.
Figure 16B plots the response phase for 16 cells as a
function of stimulus frequency during sinusoidal target motion around a
DC value of zero (DC = 0). The responses were extremely consistent across cells. Firing rate almost always lagged target speed. Phase lag
was small at 1 Hz (mean, 27°; n = 31) and increased
as a function of frequency. This behavior is expected if firing rate is
determined by the response to stimulus speed. If firing rate were
dominated by a response to image acceleration, then we would predict
some phase lead, which we observed in only a few neurons. Two factors probably contribute to the absence of phase lead. (1) Target
acceleration is small at the low frequencies in which phase lead would
be most evident. (2) For target speeds below preferred speed, neuronal responses are related more closely to stimulus speed than to
acceleration (compare Figs. 6, 8C).
It was possible to account for the increases in phase lag as a function
of frequency with a fixed time delay. The relationship between response
phase  and frequency ft is described by
the relation:
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(2)
|
where  t is a fixed time delay. We applied Equation 2
to the data plotted in Figure 16B and found that a
t of 90 msec provides an acceptable fit to the data for
DC = 0 (large filled circles). Similar analysis
revealed that the data for DC = 0.5 were fit by a t
of 85 msec (data not shown). There are a number of reasons why the time
delay that accounts for the phase shifts in Figure 16B is longer than the latency measured for steps of
target speed (Figs. 4, 5). (1) The use of Fourier analysis to compute
phases emphasizes the center-of-mass of the neuronal response rather than its early peak. (2) The phase shift was computed for steady-state stimuli and low target speeds, which gave longer latencies even for
steps of target speed. (3) The effect of null-direction motion on the
latency of responses to subsequent motion in the preferred direction
would lengthen the latency and add phase lag for DC = 0 (compare
Fig. 13).
Figure 16, C-F, shows that the responses to 1 Hz sinusoidal
modulation of target speed conformed with our expectations when the
sine wave was imposed on different baseline target speeds. These four
polar plots represent the responses of each cell at 1 Hz as a vector,
in which the length of the vector indicates the amplitude of modulation
of the fundamental frequency component of firing rate and the angle
indicates the response phase. For DC = 0 (Fig. 16C),
the response magnitudes were uniformly large, and all but one of the
vectors pointed to the right and slightly up, indicating that firing
slightly lagged preferred-direction target speed. The plot for DC = 0.5 (Fig. 16D) was similar to that for DC = 0, although the mean response amplitude was slightly smaller (19.7 impulses/sec for DC = 0.5 vs 22.3 impulses/sec for DC = 0).
For DC = 2 (Fig. 16F), many of the responses
were still quite large (mean, 13.7 impulses/sec), but most of the
vectors pointed to the left, indicating that firing lagged on-direction target speed by ~180° and was approximately in-phase with
off-direction target speed. Finally, for DC = 1 (Fig.
16E), the response amplitudes were small (mean, 7.5 impulses/sec), and the vectors pointed haphazardly in all directions,
indicating that the population as a whole showed no consistent response phase.
Simulation results
We began the experiments described above to determine whether the
image acceleration signals used in some models of smooth pursuit eye
movements were represented along with image velocity in the distributed
response of neurons in MT. The previous section suggests an affirmative
answer by showing that the firing of many MT neurons is influenced by
image acceleration. The goal of the present section is to test the
hypothesis that the population response in MT represents image velocity
and acceleration. To accomplish this goal, we (1) create models to
simulate the responses of all MT cells in our sample, (2) use the
models to simulate the population response to a given set of stimuli,
and (3) demonstrate computations that reconstruct image velocity and
acceleration from the simulated population response.
We have simulated the responses of MT neurons using the model structure
illustrated in Figure 17; the details
of the implementation of this model can be found in the APPENDIX. The
model (Fig. 17A) received an input that represented image
velocity as a function of time. Image velocity was processed by a time
delay that recreated the relationship between latency of MT responses
and target speed, three parallel elements that used divisive gain
control to model the responses to preferred-direction motion, a summing
junction to add the outputs of the three elements, and a directional
interaction to account for the effects of motion in the null direction
on the subsequent responses of MT cells to motion in the preferred direction. The elements in Figure 17A were constructed
according to the diagram in Figure 17B. Each consisted of a
numerator pathway that produced output A (Fig.
17B, top pathway), a denominator pathway that produced output B (bottom pathway),
and a division junction so that the output from each element was
A/(1 + B). Each numerator pathway consisted of a
low-pass filter with a single time constant and a speed-tuned
nonlinearity. Each denominator pathway consisted of a time delay, a
low-pass filter with a single time constant, and another speed-tuned
nonlinearity. The tuned nonlinearities were of the form given by
Equation 1, used above to describe the relationship between MT cell
firing rate and target speed. The same model architecture was used for
all cells, but different parameter values were needed for different
cells. For each cell, we adjusted the parameters of the model within
realistic limits [using the minimization program STEPIT (Chandler,
1965)] to optimize its fit to different subsets of the responses we
recorded. For most cells, models containing only a single element were
capable of fitting the data adequately, but the fit was almost always improved by using two or three elements in parallel.
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Figure 17.
Architecture of the model used to simulate the
responses of MT neurons to the stimuli used in this paper.
A, General structure of the model. Delay
refers to a time delay that depended on target speed. Elements
1, 2, and 3 each have the
structure defined by panel B. The circle
with a plus sign in it is a linear-summing junction.
Directional interaction is used to account for the
effects of null-direction motion on subsequent preferred-direction
motion. B, Structure of each element. The input is image
velocity after the delay in panel A. Low-pass filters
responded to a step input with an exponential increase in output. The
two tuning functions were implemented using Equation 1. The
circle with a division sign in it
produces A/(1 + B) as its output. The
four traces on the right show signals at
different points within an element during a ramp of target speed like
those used to study MT neurons. From top to
bottom, the traces are delayed image
velocity (I), the output of the numerator
pathway (A), the output of the denominator
pathway (B), and the output of the gain control
element. The traces have been scaled arbitrarily along
the amplitude axis to allow easy viewing. The vertical dashed
line is aligned on the onset of the ramp increase in delayed
image motion. The calibration bar on the right of
trace I shows the preferred speed of the nonlinearity in
the numerator pathway for this element. vel.,
Velocity.
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The denominator pathways were designed to produce two of the main
features of the response dynamics of MT neurons in our experiments: transients in the responses to steps of target speed and an asymmetry between the responses to ramp accelerations and decelerations. These
dynamics are present in neither the numerator nor the denominator pathways but instead arise from their interaction. For example, the
stack of traces at the right edge of Figure 17B shows
the time courses of the signals in a single element when delayed image velocity (I) follows the trapezoidal
trajectory shown in the top trace of the stack, with sustained target
speed twice the cell's preferred speed. Because the ramp increase and
decrease in target speed provide the same sequence of speeds in
opposite order, they evoke similar small pulses that overshoot a
substantial sustained response in the numerator pathway
(trace labeled A). Similarly, the denominator
pathway (trace labeled B) emits a response that is symmetrical for ramp increases and decreases in target speed. In
this example, the output of the denominator pathway lacks overshoot because the tuned nonlinearity has a higher preferred speed in the
denominator pathway than in the numerator pathway. Because of the extra
time delay in the denominator pathway, its response (B) is delayed relative to the numerator pathway's
response (A), and the initial, large output from the
model element is controlled entirely by the numerator pathway. When the
output of the denominator pathway starts to increase, the output of the
model declines toward its sustained response with a trajectory that is
determined by the delay and low-pass filter in the denominator pathway.
Because the nonlinearity in the denominator pathway need not have the same preferred speed, bandwidth, or skew as that in the numerator pathway, the effect of gain control on response dynamics in the model
can vary with target speed. This feature of our data would not emerge
from a model that used a single, linear high-pass filter to simulate
the dynamics of MT neuron responses.
Performance of the model for steps and ramps of target speed
Figures 18 and
19 illustrate the agreement between the
output of the optimized model (bold traces) and the
response histograms of two MT neurons with very different dynamics. For
a cell with a large transient (Fig. 18), for example, the response of
the model showed a realistic large transient at the onset of the
response to a step of target speed (Fig. 18A) for
targets at speeds below, near, and above preferred speed. The output
from the model also showed the same large asymmetry between the
responses to ramp increases and decreases in target speed as did the
neuronal response (Fig. 18B). For double pulses of
target speed (Fig. 18C), the output of the model showed
large attenuation of the response to the second pulse when the interval
between pulses was 32 msec (top trace). Attenuation
decreased as the interval between pulses was lengthened. Finally, for
sinusoidal modulation of target speed with DC = 0 (Fig.
18D), the model reproduced the pulse at the onset of
the response to each cycle of 1 Hz target motion, the slightly
asymmetric pulse of response to each cycle of 4 Hz target motion, and
the very small response to 16 Hz target motion.
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Figure 18.
Comparison of neuronal responses and model output
for selected target motions in a neuron with large transient responses.
A, Steps of target speed from 0 to 1, 4, or 16°/sec.
B, Ramps of target speed from 0 to 1, 4, or 16°/sec.
C, Double pulses of target speed. Each pulse was 64 msec
in duration, and the interpulse interval was 32, 64, and 128 msec from
top to bottom. D,
Sinusoidal modulation of target speed at 1, 4, and 16 Hz with DC = 0. Histograms give the neuronal responses, and
bold traces show the output from the optimized model.
Bin width was 8 msec for A-C and 10 msec
for D. Data are from unit 404r02.
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Figure 19.
Comparison of neuronal responses and model output
for selected target motions in a neuron with relatively small transient
responses. A, Steps of target speed from 0 to
0.25, 2, or 8°/sec. B, Ramps of target speed
from 0 to 0.25, 2, or 8°/sec. C, Double pulses of
target speed. Each pulse was 64 msec in duration, and the interpulse
interval was 32, 64, and 128 msec from top to
bottom. D, Sinusoidal modulation of
target speed at 1, 4, and 16 Hz with DC = 0. Histograms give the neuronal responses, and bold
traces show the output from the optimized model. Bin width was
8 msec for A-C and 10 msec for
D. Data are from unit 405r20.
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For an MT neuron with a much smaller transient response that depended
on target speed (Fig. 19), the model again reproduced the responses
accurately. The time course of the model output reproduced that of the
neuronal response for steps (Fig. 19A) and ramps (Fig.
19B) of target speed, double pulses of target motion (Fig.
19C), and sinusoidal modulation of target speed with DC = 0 (Fig. 19D). Especially noteworthy is the ability of the
model to produce quite different response time courses for steps or ramps of target speed to different sustained target speeds.
To quantify the performance of the model, we analyzed its output in the
same way as our neuronal data. This revealed excellent agreement
between the transient/sustained ratio of the model and the data for
steps (r = 0.97; type 2 regression slope = 0.88) and ramps (r = 0.95; type 2 regression slope = 0.98) of target speed, except that the model consistently produced
transient responses that were slightly too small. For two pulse
stimuli, the agreement between the model and the data was excellent for
interpulse intervals of 0 and 256 msec and less good but acceptable for
the other interpulse intervals. For sinusoidal modulation of target
speed, the performance of the model was excellent for frequencies of
1-8 Hz and DC = 0 and DC = 0.5. For DC = 2, however,
the modulation of model output fell short of the modulation of neuronal
firing by an average of 50%. For all DC values, the modulation of the
model output modestly exceeded the modulation of neuronal firing at the
highest frequencies (12 and 16 Hz). Thus, the model captured the basic features of each MT neuron's response dynamics, even if a few details
could not be simulated perfectly. Two features of the model were most
important for allowing it to fit the data. The denominator pathway
enabled it to emit transient responses for steps of target speed and to
reproduce the asymmetry in the transient responses to ramp increases
and decreases in target speed. The use of three parallel elements
allowed the model to reproduce the wide variation in dynamics in the
responses of individual MT neurons across target speeds.
Reconstruction of target velocity and acceleration from the
population response in MT
After we had obtained excellent simulations of the responses of
all MT cells in our sample, we could predict the population response to
a single target motion and ask how to reconstruct the dynamics of the
target motion from the population response. Figure
20 shows the population response in
area MT for single target motion that consisted of a 128 msec ramp of
target speed from 0 to 8°/sec, 512 msec of motion at a constant speed
of 8°/sec, and another 128 msec ramp back to 0°/sec. The 50 traces
in Figure 20 show the responses of 50 different neurons, simulated by
our model of MT cell responses, under the assumption that each cell had
the same preferred direction. Each trace is plotted at x and y coordinates determined by the neuron's preferred speed
(x-axis) and transient/sustained ratio for steps of target
speed (y-axis). This stimulus elicited measurable
responses from almost every cell, but the pattern of response varied in
a characteristic way according to each cell's speed preference and
response dynamics. Cells with preferred speeds near the maintained
target speed (Fig. 20, vertical dashed line) had
the largest sustained responses. Cells with higher values of
transient/sustained ratio had larger transient responses and more
pronounced asymmetries between the transients produced by ramp
accelerations and decelerations. For neurons with the same
transient/sustained ratio but different preferred speeds, the largest
asymmetries appeared in neurons whose preferred speeds were well below
sustained target speed.
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Figure 20.
Distributed response in MT to a particular target
motion. The graph plots 50 neurons' responses to a
standard stimulus as a function of time. Each trace is
plotted at a location determined by the preferred speed
(x-axis) and the transient/sustained ratio for steps of
target speed (y-axis) of the neuron that gave
rise to the trace. Each trace was
obtained by using the model fitted to the given cell to estimate how
the cell would have responded to the standard stimulus and then by
truncating the response to the stationary target at the start and end
of each stimulus. Each trace is normalized to the
cell's maximum sustained response to steps of target speed. The
trace at the top left labeled
Target speed shows the time course of the standard
stimulus; it consisted of a 128 msec ramp from 0 to 8°/sec, 512 msec
of motion at constant speed, and a 128 msec ramp back to 0°/sec. The
vertical dashed line is drawn at the sustained stimulus
speed on the x-axis. The responses of 50 cells are
plotted here. The other 39 cells have been omitted because their
traces plotted on top of one of those on the
graph. A number of the traces have been
displaced slightly to avoid crossed traces.
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Salinas and Abbott (1994) showed that a center-of-mass computation is a
useful method for pooling the responses of neurons in a population
whose members are dispersed over a parameter space, as in Figure 20. We
formulated their suggestion to calculate a pooled motion response as
follows:
![m[t]=<FR><NU><UP>∑</UP>w<SUB>i</SUB>MT<SUB>i</SUB>[t]</NU><DE><UP>ϵ + ∑</UP>MT<SUB>i</SUB>[t]</DE></FR>](/content/vol19/issue6/fulltext/2224/img003.gif)
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(3)
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where m[t] is the pooled response at time
t, MTi[t] is the
response at time t of the model fitted to the ith
cell, wi is the weight afforded the
outputs from the ith cell, and  has a small positive
value to make the computation less sensitive to noise. The
denominator of Equation 3 implements vector averaging of MT unit
responses, consistent with evidence that the command for pursuit eye
movements is created by vector averaging the outputs of the active MT
neurons (Ferrera and Lisberger, 1997; Groh et al., 1997).
Figure 21 illustrates how the pooled
response from Equation 3 depends on the way that the vector of weights
w is related to the response properties of each cell. For
these computations, we first simulated the distributed representation
in MT for trajectories of target speed consisting of a 128 msec ramp
increase in target speed, motion at constant speed for 512 msec, and a
128 msec ramp decrease in target speed. We used four final target
speeds of 4, 8, 16, and 32°/sec to create four different distributed
representations for the same set of 89 MT neurons. We then explored
different rules for setting the output weight of each neuron
(wi). If we set
wi to be proportional to the preferred
speed (PSi) of the ith cell (Fig. 21,
traces labeled Preferred speed only), then the
pooled responses for the four final target speeds provide reasonable
reconstructions of the full trajectory of target speed. If we set
wi to be proportional to the
transient/sustained ratio (TSRi) of the
ith cell minus 2.1 (Fig. 21, traces labeled TSR only, the choice of 2.1 to be explained below), then the
pooled responses reconstruct a transient in association with target
acceleration but do not differentiate the different rates of target
acceleration produced by the ramps for different final speeds. If we
set wi to be proportional to both
TSRi  2.1 and PSi (Fig.
21, traces labeled TSR and preferred
speed), then the pooled responses reconstruct a transient
during ramp increases in target speed; moreover, the transient scales
with target acceleration. We conclude that a center-of-mass computation
can reconstruct target acceleration from the distributed response in
area MT, although imperfectly; inspection of these traces reveals an
undesired, small sustained response for a final target speed of
32°/sec and, perhaps more importantly, shows that this approach fails
to reconstruct target deceleration.
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Figure 21.
Results of pooling the distributed representation
of target motion in MT under different assumptions about the weighting
of the output from each neuron. The bottom set of
traces shows the profiles of target speed. The
top three sets of traces show the pooled
responses. From top to bottom, unit
outputs were weighted in proportion to the preferred speed (PS) of the
unit, the transient/sustained ratio (TSR) minus 2.1, or PS × (TSR 2.1). Line weights indicate sustained
target speed: light dashed, 4°/sec; light
solid, 8°/sec; bold solid, 16°/sec; and
bold dashed, 32°/sec.
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Why do these different choices of wi
provide the pooled responses they do? First, subtracting 2.1 from the
transient/sustained ratio assigns negative weights to the outputs of MT
neurons with a low transient/sustained ratio. The computation then
rejects the sustained component of firing rate related to target
velocity, which is approximately equal in all neurons with the same
preferred speed, but retains the transient component, which is related
to TSRi. Second, weighting by the product of the
transient/sustained ratio and preferred speed capitalizes on a feature
of the data that we mentioned above in reference to Figures
8C and 20. Consider Figure 20 as a set of vertical arrays
of cells, in which the cells in a given array have similar
preferred speeds but a wide variety of transient/sustained ratios. For
a ramp increase in target speed, the response profile within each array
will depend on the relationship between the final target speed and the
preferred speed of cells within that array. Within arrays of cells that
have preferred speeds below the final target speed (Fig. 20, left
side), there will be a range of transient responses in
relation to the value of the transient/sustained ratio; weighting by
TSRi 2.1 will reveal transients. Within
arrays of cells that have preferred speed above the final speed, the
transient responses are quite attenuated; weighting by
TSRi 2.1 will not reveal large transients. It
follows that information about the rate of target acceleration can be
obtained by knowing the preferred speed of the array of cells that has
a wide range of transient responses. Thus, reconstruction of target
acceleration depends on weighting by TSRi to
detect the transient responses and by PSi to
make a reconstruction based on which vertical array of cells has large transients.
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DISCUSSION |
The representation of target motion by MT neurons
It has been known for many years that neurons in MT are tuned for
the direction and speed of target motion and that any given target
motion is represented by the distributed response of a large number of
neurons in MT. We have now shown that the same distributed
representation of target speed and direction may also use transient
signals to provide information about target acceleration. The
transients are seen at the onset of target motion at constant speed and
as an asymmetry in the responses to smooth increases and decreases in
target speed. Two important features of this asymmetry can be
appreciated from the data alone. First, the firing of a given MT neuron
is influenced by target acceleration only for targets that start from
close to zero speed or at least well below preferred speed. Thus, it
would be incorrect to state that MT neuron firing is "related" to
target acceleration. Second, most MT neurons show the largest
transient/sustained ratio for ramps to target speeds at least twice
their preferred speed. As a result, the best information about target
acceleration is available in the firing of neurons with preferred
speeds that are traversed by the accelerating target.
A model that reproduces the dynamic responses of MT cells
We constructed a model of MT cell responses with the initial goal
of simulating the distributed representation of image motion in MT.
Only after completing the model were we able to show that target
acceleration could be reconstructed from the population response of MT
neurons. The same models will be useful for the next obvious step,
which is to create a model of pursuit eye movements based on visual
inputs that simulate the population responses of MT. For these two
purposes, it did not matter whether the equations used to simulate cell
responses were biologically plausible. Our goal in simulating MT cell
responses was simply to obtain a good fit to the data that would
generalize well to image motions that had not been presented to the
cell, including the image motions normally seen during pursuit. The
model presented here accomplished this initial goal satisfactorily, in
spite of minor failings on some of the more complicated target motions
we used when recording from MT neurons.
The main feature of our model that allows it to reproduce the transient
responses of MT neurons is a nonlinearity configured to perform
divisive gain control. The gain control provided by the denominator
pathways in our model is similar to, indeed inspired by, the suggestion
that gain control may be a general mechanism of cortical function
(Bonds, 1989; Heeger, 1992; Heeger et al., 1996; Carandini et al.,
1997). Of course, our modeling neither proves that gain control is used
to create transient responses in MT nor constrains the mechanisms of
gain control. Plausible mechanisms include the divisive normalization
proposed for MT cells by Simoncelli and Heeger (1998), forms of
short-term synaptic depression like those revealed by Varela et al.
(1997), a nonlinear implementation of spike frequency adaptation
[after Wang (1998)], or any other mechanism that acts like a suitably
nonlinear high-pass filter.
After we realized that gain control made the model work well, we
explored the time course of the adaptation mechanisms we had modeled as
gain control by measuring neural responses to two pulse stimuli. This
approach used the conditioning pulse to invoke adaptation and test
pulses at different intervals to probe the recovery from adaptation. We
found a good correlation between attenuated responses to the test pulse
at short intervals and large transient responses to steps of target
speed. This finding confirms the existence of adaptation that can be
modeled by gain control to create transient responses. That adaptation
was directional in our recordings makes it likely that its mechanism
resides in MT, rather than in the primary visual cortex where contrast
gain seems to be set by a signal that is insensitive to the orientation and direction of the stimulus (Carandini et al., 1997). The gain control we postulate for MT has a different outcomethe creation of
transient responsesthan that proposed by many other authors. Perhaps
gain control has many functions in the cortex. These functions could
range from normalization to render steady-state neuronal responses
invariant with the contrast of a stimulus to sculpting the temporal
response profiles in a way that creates transient responses capable of
conveying information about the dynamics of the stimulus.
Image motion inputs to models of pursuit
It has been understood for some time that reconstruction of the
direction and speed of target motion requires comparison of the
responses of multiple neurons. The responses of any individual neuron
are ambiguous; the neuron might fire at less than its maximum rate
because target motion is below preferred speed, above preferred speed,
or in a nonpreferred direction. The same is true of attempts to
reconstruct target acceleration. Even with post hoc
inspection of the responses to a given ramp increase and decrease in
target speed, individual neurons provide information primarily about the direction of acceleration rather than about the rate of
acceleration. Thus reconstruction of target acceleration also requires
neural computations based on the responses of multiple MT neurons.
We have demonstrated that it is possible to reconstruct both image
velocity and image acceleration from the distributed response we
recorded in MT. However, this does not achieve our long-term goal of
using the population code in MT to drive a model of pursuit that
reproduces the features shown in Figure 1: short latency, brisk smooth
eye acceleration in the first 100 msec of tracking, lack of a large
overshoot in the transition from initial eye acceleration to maintained
eye velocity, and a relatively high frequency of eye velocity
oscillations during sustained pursuit of target motion at constant
speed. We have not yet evaluated this question fully because we are
aware of two problems that must be solved before such a model is likely
to succeed. (1) Although it is clear that Equation 3 can extract
information about target acceleration from the distributed response in
MT, it does not provide information about image deceleration, such as
occurs during the initial, brisk eye acceleration toward target
velocity. In the model of pursuit eye movements proposed by Krauzlis
and Lisberger (1994), image deceleration is used to prevent eye
velocity from overshooting target velocity excessively. (2) From what
we know now, it is not clear exactly how to extract information about
image acceleration from the population response in MT during sinusoidal
modulation of target speed. Image acceleration information should cause
MT neuronal discharge to lead sinusoidal target velocity. Phase lag, not phase lead, was revealed by our recordings. Furthermore, when eye
velocity is oscillating sinusoidally around target velocity, the
maximum image acceleration occurs at zero image speed, when MT cells
are mostly silent. It will be difficult to extract meaningful information from silent neurons. In the model proposed by Krauzlis and
Lisberger (1994), sinusoidal image motion provided an image acceleration signal that was essential for modeling the propensity of
monkey pursuit to oscillate at ~6 Hz during pursuit of target motion
at constant velocity. To a first approximation, a negative feedback
system like pursuit will oscillate at the frequency at which the total
phase lag around the feedback loop is 180°. Some of the phase lag is
caused by the 80 msec latency, some by 90° of phase lag inserted by a
neural integrator, and some by the dynamics in visual processing. If
visual processing introduces no lead or lag, then the system will
oscillate at the frequency at which the latency introduces 90° of
lag: 3.125 Hz. If visual processing introduces 90° of phase lead,
because of image acceleration signals, then the system will oscillate
at the frequency at which the latency introduces 180° of phase lag:
6.25 Hz.
Several approaches might be used to extract more information from the
population code in MT. Image deceleration information may be
extractable by a number of neurally plausible computations such as
dividing along the transient/sustained ratio axis or weighting the
outputs of each cell differently from the modified center-of-mass computation we have performed. The problem of silent MT neurons at
maximum acceleration during sinusoidal image motion may be resolved by
creating a true opponent representation in which baseline firing is
increased by image motion in the preferred direction and decreased by
image motion in the nonpreferred direction. It also is possible that
the dynamics of MT neuron responses in awake monkeys will be richer
than those in anesthetized monkeys and will support reconstruction of
image acceleration and deceleration for the full range of image motions
normally seen during pursuit eye movements.
General principles for reading a distributed neural code
Our success in reading image velocity and acceleration from a
single distributed code suggests a principle that might provide a
neurally plausible solution to this general problem. Suppose that
multiple variables are represented in a single population response and
that it is possible to parameterize the distributions of neural
responses along multiple eigen-axes as we did in Figure 20 for the
population response in area MT. Then, it should be possible to extract
the different variables by center-of-mass computations along the
eigen-axis of the variable that is being reconstructed. For the
population response in MT, this approach reconstructed target velocity
or acceleration by center-of-mass computations along the eigen-axes of
preferred speed or the product of preferred speed and transience.
Salinas and Abbott (1994) have shown for a single variable that this
approach is close to optimal under reasonable assumptions about the
statistics of the distributed responses in the population.
There are two major strengths of the center-of-mass approach to
reconstructing individual variables from a distributed representation of multiple stimulus features. First, the center-of-mass computation is
biologically plausible. In neural terms, Equation 3 suggests that the
population response is pooled by summing each neuron's output weighted
according to some feature of the neuron's responsei.e., preferred
speed or the product of preferred speed and transienceand then
normalizing according to the overall activity of the population. These
features correspond to the age-old ideas of "labeled lines" or
"The Law of Specific Nerve Energies" (Müller, 1840) and the more modern concepts of normalization, adaptation, and gain control (e.g., Heeger, 1992; Chance et al., 1998). Second, the existence of a
plausible method for extracting different features from the distributed
code means that a given sensory feature such as visual motion needs to
be represented only once in the brain. Different functions, such as
perception and motor control, need only to choose the weights used to
pool the distributed code to reconstruct different features of motion
according to their individual needs.
| |
FOOTNOTES |
Received Oct. 28, 1998; revised Dec. 23, 1998; accepted Dec. 29, 1998.
This work was supported by the Howard Hughes Medical Institute, the
McDonnell-Pew Program in Cognitive Sciences, National Institutes of
Health Grants EY02017 and EY03878, and ONR contract N00014-94-1-0269. We are grateful to Maninder Kahlon, Henry Mahncke, Larry O'Keefe, and Nicholas Port for their assistance with the recording experiments. Suzanne Fenstemaker and Christine Brown did the
histological processing and track reconstructions. We are grateful to
Wyeth Bair and Jennifer Groh for their comments on an earlier version
of this manuscript.
Correspondence should be addressed to Dr. Stephen G. Lisberger,
Department of Physiology, Box 0444, University of California, San
Francisco, 513 Parnassus Avenue, Room HSE-802, San Francisco, CA 94143.
| |
APPENDIX |
The basic structure of the model is defined by Equations A1 and
A2, the structure of each element (Ej) is
defined in Equations A3-A7, and the terms of the "directional
interaction" used to convert R(t) into
MT(t) are defined in Equations A8 and A9:
|
(A1)
|
|
(A2)
|
MT is the output of the model for a given MT neuron,
R is an intermediate output, Ej
is the output of the jth element,
 v is the visual time delay, and t
is time.
When there has not been any null-direction target motion for a long
time, MT(t) is equal to R(t). As a
result, the three elements (E1,
E2, and E3)
contained all the parameters that had to be adjusted to fit the data
that included only motion in the preferred direction. The components of
these elements are defined in Equations A3-A7.
The latency of the visual input is computed dynamically for each time
point in the simulation as:
|
(A3)
|
The gain control within each element is defined by:
![E<SUB>j</SUB>(t)=<FR><NU>g<SUB>nj</SUB>(M<SUB>j</SUB>(<A><AC>I</AC><AC>˙</AC></A>[t]))</NU><DE>1+g<SUB>dj</SUB>(L<SUB>j</SUB>(<A><AC>I</AC><AC>˙</AC></A>[t−&tgr;<SUB>dj</SUB>]))</DE></FR>](/content/vol19/issue6/fulltext/2224/img007.gif)
|
(A4)
|
In equation A4, gnj and
gdj are nonlinear gain elements,
Mj and Lj are
filters, and dj is a time delay for the
denominator pathway. All except the time delay are defined below. The
low-pass filter Lj(x) has a
single time constant Tj and was simulated
by a numerical solution to the differential equation:
|
(A5)
|
The low-pass filter Mj(x) has
different time constants for increases and decreases in input and was
simulated by a numerical solution to the differential equation:
|
(A6)
|
The nonlinearity in the numerator and denominator pathways of each
element is:
|
(A7)
|
The extra time delay in the directional interaction, caused by previous
motion in the nonpreferred direction, is computed as:
![&tgr;<SUB>di</SUB>=<FENCE><AR><R><C>0</C></R><R><C>g<SUB>l</SUB>(<UP>−</UP>L<SUB>l</SUB>(<A><AC>I</AC><AC>˙</AC></A>[t−&tgr;<SUB>l</SUB>]))</C></R></AR> <AR><R><C><UP> if </UP><IT><A><AC>I</AC><AC>˙</AC></A>≥0</IT></C></R><R><C><UP> if </UP><IT><A><AC>I</AC><AC>˙</AC></A><0</IT></C></R></AR></FENCE>](/content/vol19/issue6/fulltext/2224/img011.gif)
|
(A8)
|
The gain enhancement in the denominator of Equation A1 is defined by
setting Gdi equal to the lesser of 0.9 and
the quantity computed in Equation A8:
![G<SUB>di</SUB>=<FENCE><AR><R><C>0</C></R><R><C>g<SUB>G</SUB>(L<SUB>G</SUB>(<A><AC>I</AC><AC>˙</AC></A>[t−&tgr;<SUB>G</SUB>]))</C></R></AR> <AR><R><C><UP> if </UP><IT><A><AC>I</AC><AC>˙</AC></A>≥0</IT></C></R><R><C><UP> if </UP><IT><A><AC>I</AC><AC>˙</AC></A><0</IT></C></R></AR></FENCE>](/content/vol19/issue6/fulltext/2224/img012.gif)
|
(A9)
|
The full model, including directional interaction and the three
elements, has 51 adjustable parameters, of which 20 are used to adjust
the parameters of the five nonlinearities. There are 3 parameters in
the velocity-dependent time delay (Eq. 3), 12 parameters in each gain
control element, and 12 parameters in the directional interaction.
Although this is a large number of parameters, it is small compared
with the ~5000 time points of image velocity and MT neuron firing
rate that were used as the basis for adjusting the parameters.
Simulations were conducted on a DEC Alpha computer with a modification
of A Simulation Program (ASP), first written by L. Optican
and H. Goldstein. The model was conceived as a series of differential
equations, time delays, and nonlinear gain elements. The equations for
these elements were then described in the BOMOL language and converted
into C code by the ASP software. The C program was compiled and linked
into a simulation shell that could operate in either manual mode under
operator control or automatic mode under the control of a gradient
descent optimization algorithm [STEPIT (Chandler, 1965)]. In manual
mode, the operator was able to adjust the parameters one at a time to
try to get the model started in the correct direction. In automatic
mode, we selected a subset of parameters for adjustment, specified the
order of priority for adjusting them, indicated upper and lower bounds on parameter values, and provided a set of target velocities and neuronal firing rates as a desired output for the optimization algorithm. Error was defined as the sum over all time points of the
mean squared difference between the output of the model and the firing
rate of the neuron being fitted. We will be happy to make our
simulation software available to anyone who wishes it.
Many iterations of manual and automatic adjustment were often needed to
get the model to fit the neuronal responses well. Sometimes it was
possible to simply feed the model the full set of data described in
this paper. More frequently, however, it was necessary to add specific
classes of target motions one at a time, allowing the optimization
algorithm to adjust only the parameters that were intended to fit those
target motions. On the one hand, we doubt that we have demonstrated
either the optimal parameter set or the optimal model architecture for
fitting the responses of MT cells to the set of target motions we used
in our experiments. Thus, it would almost certainly be possible to reduce the computed error by further simulations. However, we were
satisfied with the performance of our model because it succeeded in
capturing all the basic response properties we had measured in MT
neurons and because it could simulate closely a heterogeneous set of
temporal responses merely by changing parameters.
| |
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Y. Gu, P. V. Watkins, D. E. Angelaki, and G. C. DeAngelis
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N.S.C. Price, S. Ono, M. J. Mustari, and M. R. Ibbotson
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M. R. Carey and S. G. Lisberger
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M. M. Churchland and S. G. Lisberger
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N. J. Priebe, M. M. Churchland, and S. G. Lisberger
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M. Tanaka and S. G. Lisberger
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M. Suh, H.-C. Leung, and R. E. Kettner
Cerebellar Flocculus and Ventral Paraflocculus Purkinje Cell Activity During Predictive and Visually Driven Pursuit in Monkey
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K. C. Engel and J. F. Soechting
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M. Kahlon and S. G. Lisberger
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S. E. Raiguel, D.-K. Xiao, V. L. Marcar, and G. A. Orban
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Top
Abstract
Introduction
