 |
INTRODUCTION |
Vergence eye movements function to align both
eyes on the same object and facilitate the binocular fusion of visual
images. An important cue in this process is the slight difference in
the locations of the two retinal images that arises from the slight difference in the viewpoints of the two eyes: binocular disparity. Most
studies of disparity-induced vergence have examined the transfer of
fixation between small targets presented at different distances and
have reported latencies ranging from 150 to 200 ms in humans (Jones 1980
; Mitchell 1970
;
Rashbass and Westheimer 1961
; Westheimer and
Mitchell 1956
) and from 135 to 177 ms in monkeys
(Cumming and Judge 1986
). However, it has been shown
that step changes in the horizontal disparity of a large pattern result
in machine-like vergence responses with very short latencies: <60 ms
in monkeys and <0.80 ms in humans (Busettini et al.
1996
; Masson et al. 1997
). It has been suggested
that these disparity vergence responses are important for the rapid
automatic correction of residual (i.e., small) vergence errors
(Busettini et al. 1996
). In line with this suggestion,
these short-latency vergence responses have disparity tuning curves
that resemble the derivative of a Gaussian, are well fit by odd (sine
phase) Gabor functions and have a roughly linear servo region that
extends only a degree or so on either side of zero disparity
(Masson et al. 1997
). It has recently been reported
(Masson et al. 1997
) that horizontal disparity steps applied to dense anticorrelated random-dot patterns (in which the
patterns seen by the two eyes have opposite contrast so that each black
dot in one eye is matched to a white dot in the other eye) also elicit
short-latency vergence responses that are very similar to those
observed with normal correlated patterns except that they are in the
opposite direction. In line with the earlier observations of
Cogan et al. (1993)
, Masson et al. also showed that all
of their human subjects were able to discriminate between 1.2°
crossed and uncrossed disparity steps applied to dense correlated patterns, but none was able to make these discriminations with dense
anticorrelated patterns, which evoked strong binocular rivalry. Cumming and Parker (1997)
have reported that monkeys too
can make such discriminations with correlated patterns but not with
anticorrelated patterns. These anticorrelated patterns therefore
provide an interesting disparity stimulus in that they can support
vergence eye movements but not depth perception.
It has often been suggested that disparity-induced vergence utilizes
disparity-selective neurons to sense vergence errors, and such neurons
have been recorded in various regions of the monkey cortex, including
striate and extrastriate visual areas (Burkhalter and Van Essen
1986
; Cumming and Parker 1999
, 2000
; Felleman and Van Essen 1987
; Hubel and
Livingstone 1987
; Hubel and Wiesel 1970
;
Poggio and Fischer 1977
; Poggio and Talbot
1981
; Poggio et al. 1988
; Prince et al.
2000
; Smith et al. 1997
; Trotter et al.
1996
), as well as the middle temporal area (MT) (Bradley and Andersen 1998
; Bradley et al. 1995
;
DeAngelis and Newsome 1999
; DeAngelis et al.
1998
; Maunsell and Van Essen 1983a
), MST (Eifuku and Wurtz 1999
; Roy et al. 1992
),
the posterior parietal area (Sakata et al. 1983
), the
lateral bank of the intraparietal sulcus (LIP) (Gnadt and Mays
1995
), and the frontal eye fields (FEF) (Ferraina et al.
2000
; Gamlin et al. 1996
). Most of the earlier
studies grouped cells according to the shapes of their disparity tuning
curves using the classification scheme of Poggio and Fischer
(1977)
, which recognized two general groupings of disparity-selective neurons, later termed "tuned" and
"reciprocal" by Poggio et al. (1988)
. Tuned neurons
had narrow tuning curves with either a peak ("tuned-excitatory"
neurons) or a trough ("tuned-inhibitory" neurons) centered on the
plane of fixation. Reciprocal neurons had asymmetric tuning curves and
responded to a broad range of disparities in front ("near" neurons)
or behind ("far" neurons) the plane of fixation. Subsequently,
Poggio et al. (1988)
recognized two additional tuned
groups with peaks in front ("tuned near" neurons) or behind
("tuned far" neurons) the plane of fixation and this led to the
renaming of the "tuned excitatory neurons" as "tuned zero
neurons." Cumming and Parker (1997)
recently showed that most disparity-selective neurons in cortical area V1 of monkeys also respond to the disparity of anticorrelated random-dot patterns, often with the opposite sign, e.g., neurons that had "tuned
excitatory" disparity tuning curves with correlated patterns had
"tuned inhibitory" tuning curves with anticorrelated patterns. Such
responses are in line with the suggestion that these neurons act as
purely local filters (Cumming and Parker 1997
, 2000
;
Nomura et al. 1990
; Ohzawa 1998
;
Ohzawa et al. 1990
).
Our purpose in the present study was to see if there are neurons in
area MST of the macaque monkey that have the necessary properties to
initiate the short-latency disparity-vergence responses described by
Busettini et al. (1996)
and Masson et al.
(1997)
. A strong motivating factor came from preliminary
evidence that bilateral lesions in MST cause a significant reduction in
these vergence responses (Takemura et al. 1999
, 2000
).
Our study was restricted to the initial (open-loop) neuronal responses
that occur before the associated vergence responses have had time to modify the central neuronal responses via the disparity feedback loop.
We here report that MST neurons can be activated by horizontal disparity steps applied to either correlated or anticorrelated random-dot patterns at latencies that are probably short enough for
these neurons to have a causal role in producing even the earliest
vergence responses. However, comparatively few of these neurons had
disparity tuning curves that resembled the tuning curves for vergence
and, when sorted using a fuzzy clustering algorithm, the curves fell
into four major groups, corresponding roughly to classes of
disparity-selective neurons described in the visual cortex
(Poggio et al. 1988
). Thus qualitatively at least, most
individual tuning curves resembled those at earlier, overtly sensory,
stages. Interestingly, when these curves were simply summed together,
they fitted the tuning curves for the vergence responses elicited by
both correlated and anticorrelated stimuli, indicating that the
associated motor responses are encoded at the population level.
Nonetheless, using a genetic algorithm, it was possible to identify
subsets of neurons whose tuning curves, when summed together, gave an
even better fit to the vergence tuning curves, though these subsets
invariably included cells from all four groups. Additional analyses of
the spike trains elicited by disparity steps revealed considerable
variation across cells in the latency, amplitude, and time course of
the changes in discharge rate. When all of the spike trains elicited by
a given disparity step were summed together to give an average
discharge profile for the whole population of cells, many were rather
noisy, but others that were less so matched the temporal profile of the motor response, vergence velocity, quite well. Based on these findings,
we hypothesize that the disparity-sensitive cells in MST each encode
only some aspect(s) of the sensory input and/or motor output, but that
the population of cells as a whole encodes the complete motor output
(vergence velocity).
Preliminary results have been presented elsewhere (Takemura et
al. 1997
, 1998
, 1999
).
 |
METHODS |
We recorded single unit activity in the MST area of five
adolescent Japanese monkeys (Macaca fuscata, 6-8 kg) in
response to disparity steps applied to large random-dot patterns. Prior to any surgery, animals were trained to fixate small target spots on a
tangent screen for a liquid reward using the dimming task of
Wurtz (1969)
. After the completion of training, animals
were anesthetized with pentobarbital sodium for the surgical
implantation (under aseptic conditions) of 1) a
pedestal, secured to the skull to permit the head to be fixed in the
standard stereotaxic position during the experiments, 2) a
cylinder, secured to the skull over the superior temporal sulcus for
the chronic recording of single neuron activity, and 3)
scleral search coils, around both eyes for recording eye movements
(Judge et al. 1980
). All experimental procedures were
approved by the Electrotechnical Laboratory Animal Care and Use
Committee and have been fully described elsewhere (Kawano et al.
1994
).
Disparity stimuli
During recording sessions, which were each several hours long,
the animal sat in a primate chair with the head secured and faced a
translucent tangent screen onto which two identical random-dot patterns
were back-projected. The screen was 50 cm in front of the eyes and
subtended 90° along the vertical and horizontal meridia. Orthogonal
polarizing filters in the two projection paths and in front of each eye
ensured that each pattern was visible to only one eye. Mirror
galvanometers in the two light paths were used to control the
horizontal positions of the two images (binocular disparity). Using
fluid reinforcement, animals were rewarded for fixating stationary red
target spots, which were projected onto the patterns on the screen.
Binocular disparity stimuli were applied to random-dot patterns that
were either exactly matched at the two eyes (standard correlated
patterns) or of opposite contrast (anticorrelated patterns).
Standard paradigm using correlated patterns.
At the beginning of each trial, the patterns seen by the two eyes were
identical, overlapped exactly (zero binocular disparity), and filled
the screen. The patterns consisted of white dots on a black background
(dots had a luminance of 0.8 cd/m2, a diameter of
1.5° and covered 50% of the image space). Horizontal disparity steps
(crossed and uncrossed, ranging in amplitude from 0.5 to 6.0°) were
applied by displacing the two images equally in opposite directions.
Because previous studies had shown that the vergence responses were
subject to transient postsaccadic enhancement (Busettini et al.
1996
), these steps were applied 50 ms after 10° leftward
centering saccades guided by target spots projected onto the screen.
The experimental situation was the same as that used by
Busettini et al. (1996)
in all essentials. Because we
were interested only in the initial vergence responses, the disparity
steps lasted only 230 ms, and, if there were no saccades during this
time, then the data were stored on a hard disk and the animal was given
a drop of water; otherwise, the trial was aborted and fluid was
withheld. At this point, both images were blanked for 500 ms by
mechanical shutters and then reappeared once more for the start of the
next trial. Note that all experiments included control trials in which
no steps were applied (saccade-only trials). By applying the disparity
steps in the immediate wake of centering saccades, we ensured that the animal was alert during the steps, the stimulus pattern was always centered on the retina at the onset of the steps, and the vergence responses were subject to postsaccadic enhancement.
Paradigm using anticorrelated patterns.
For this paradigm, the right eye saw the same white dots on a black
background as when the binocular patterns were correlated, and the left
eye saw a matching negative image (black dots on a white background).
However, trials started with the screen a featureless gray (with the
same space-averaged luminance as for the patterns), until 50 ms after a
10° leftward centering saccade (again, guided by projected target
spots), at which time the anticorrelated random-dot patterns suddenly
appeared with a fixed horizontal disparity. Here too the disparity
stimuli were presented only briefly (230 ms) before the screen was
blanked, ending the trial. This procedure for applying disparate
anticorrelated stimuli was the same as that used by Masson et
al. (1997)
and was necessitated because disparity steps applied
directly to anticorrelated patterns at best elicit only weak vergence
eye movements (unpublished observations).
Data collection
The techniques for recording unit activity in MST were the same
as previously described (Kawano et al. 1992
,
1994
) and will only be given in brief here. A hydraulic
microdrive (Narishige Mo-9) was mounted on the recording cylinder, and
glass-coated tungsten microelectrodes were used for the initial
identification and mapping of the MT/MST region. Subsequently, a fixed
grid system (Crist et al. 1988
) was used to position a
guide tube through which a flexible tungsten microelectrode was
introduced into the MST area for single-cell recordings. The tip of the
guide tube was positioned 3-5 mm above the MST. Neuronal activity was
recorded using standard extracellular techniques. Spikes were detected with a time-amplitude window discriminator with a resolution of 1 ms.
We selectively isolated neurons whose discharge was modulated by
disparity steps applied to correlated patterns, and only after obtaining
40 samples of the responses to the complete set of disparity steps did we record responses to similar "steps" applied to anticorrelated patterns. Eye velocity signals were sampled at 500 Hz
and all other analog signals at 250 Hz. All data were transferred to a
work station (SunSparc) for quantitative analysis.
Data analysis
Horizontal vergence was computed by subtracting the horizontal
position signal for the right eye from that of the left eye. We used
the convention that rightward positions and velocities are positive,
hence, crossed disparities and convergence were also positive. The
latency of the vergence eye movements (and associated neuronal
responses) was taken to be the time when the mean vergence acceleration
(and the mean discharge rate) first exceeded the baseline noise by 2 SD. Vergence responses were quantified by measuring the change in
vergence position over the 60-ms time period beginning 50 ms after
stimulus onset, and disparity tuning curves were constructed by
plotting these measures against the amplitude of the disparity step. To
quantify the neuronal responses, we first constructed peristimulus time
histograms (binwidth, 1 ms) from the responses to multiple
presentations of each disparity step, computing a spike density
function by convolving each spike with a Gaussian pulse whose standard
sigma was 3 ms (MacPherson and Aldridge 1979
;
Richmond et al. 1987
). These histograms were then used
to compute the (average) instantaneous discharge frequency over time.
The mean discharge frequency over the 60-ms period starting 40 ms after
stimulus onset was computed for the responses to each disparity step,
and these values were then plotted against the amplitude of the step:
these plots will be referred to as disparity tuning curves. All data
shown in the figures, including both eye movements and neuronal
discharges, have had the responses on saccade-only trials (i.e., no
disparity step applied) subtracted to eliminate any postsaccadic
vergence drift and postsaccadic neuronal activity. This has the effect
of forcing all the disparity tuning curves through the origin.
Fuzzy clustering analysis.
It has been usual to group disparity-selective neurons according to the
shape of their disparity tuning curve using a classification scheme
described by Poggio and coworkers (Poggio and Fischer
1977
; Poggio et al. 1988
). This scheme has been
very successful but relies on a subjective assessment. In an effort to
classify the cells according to the shape of their tuning curve in an
objective manner, we turned to clustering methods. These methods
attempt to achieve an optimal partitioning of the data into groups.
When visual inspection of the data reveals the presence of relatively tight and separate clusters, classic clustering algorithms successfully assign each datum to one of several discrete groups. But when a human
observer is uncertain about the separation between groups, as was the
case with our data set, these algorithms may not find any structure in
the data. In this case, one can try fuzzy clustering algorithms, which
retain much more information about the distribution of the data being
clustered. We implemented the fuzzy c-means clustering algorithm of
Bezdek (1981)
, and the technical details are given in
APPENDIX A. Here, we provide only a brief outline of the
general operations performed.
As we were interested in grouping the cells' responses solely on the
basis of the shapes of their disparity tuning curves, we first
normalized the individual curves by adjusting the gain and bias so that
all had the same peak-to-peak amplitude and mean. It was then necessary
to decide how many groupings would be used by specifying the number of
clusters, n, and the rationale for the number of clusters
finally used (4) is given in RESULTS. The algorithm begins
by randomly selecting n "center" waveforms from within
the available data space, one for each cluster, and then computes
n "membership values" for each of the tuning curves, one
for each of the n clusters. These membership values are a measure of the "similarity" of a given curve to each of the
n "center" curves and are constrained such that each
membership value can range from 0 to 1, the more similar the curve to
the center curve of a given cluster the higher its membership in that cluster, and the sum of the n membership values for each
tuning curve is 1. Through an iterative process, the algorithm selects cluster centers to maximize the homogeneity within each cluster and the
heterogeneity between clusters. The algorithm generally converged after
10-20 iterations, and we then assigned each cell to one of
n groups based on the cluster in which the cell had its
highest membership: cells whose membership values peaked in cluster 1 were placed in group 1, cells whose
membership values peaked in cluster 2 were placed in
group 2, and so forth.
Genetic algorithm.
In examining the correlation between the single-unit activity and the
vergence eye movements elicited by disparity steps, we attempted to
identify the subset of cells whose disparity tuning curves when summed
together best matched the disparity tuning curve for vergence. However,
if the number of units recorded in a given animal is n, then
2n subsets are possible. In the present study,
for example, we obtained disparity tuning curves for 49 units from one
animal, and we estimate that our current computers would require
hundreds of years to examine all 249 subsets.
After reviewing the options, we decided to use a genetic algorithm
(GA). The rationale behind this choice is that for problems in which
there is a large number of binary variables (as here), GAs are known to
outperform all other methods (computation time being equal): unlike
other optimization algorithms, GAs sample several regions of the
solution space in parallel. The technical details are given in
APPENDIX B and only a brief outline of the general
operations performed is provided here.
The data from each animal were treated separately because of small
differences in the shapes of their disparity tuning curves for
vergence. In our implementation of the GA algorithm, each "chromosome" in effect represents one of the
2n subsets of tuning curves. We started with
5,000 chromosomes, each having the same complement of n
"genes," one for each of the n units recorded in the
animal under scrutiny. All genes were randomly assigned a value of
either 1, indicating that they made a contribution ("were
expressed"), or 0, indicating that they made no contribution ("were
not expressed"). For each chromosome, the disparity tuning curves of
the units/genes that had been assigned a value of 1 were summed
together and then fitted to the disparity tuning curve for the vergence
response of that monkey. The task of the GA was to then "evolve" a
chromosome (or chromosomes) whose subset of "expressed" tuning
curves when summed together best fit the vergence data. The mean
squared error (MSE) was used to assess the goodness of fit, and a
histogram showing the distribution of MSEs among the first generation
of 5,000 chromosomes invariably indicated a wide range of values. New
generations of chromosomes (each having 5,000 chromosomes) with
progressively smaller MSEs were created using a set of standard
evolutionary rules. The algorithm ran for 50 generations, during which
the minimum MSE for the population gradually diminished, usually
stabilizing after ~30 generations. When the last (50th) generation
was reached, the vast majority of the "chromosomes" had the same
string of "expressed genes," and further evolution was virtually
impossible. This surviving string of expressed genes represents the
algorithm's estimate of the subset of units whose summed disparity
tuning curves best correlate with the disparity tuning curve for vergence.
Histology
At the conclusion of recordings in a given monkey, that animal
was deeply anesthetized with pentobarbital and perfused through the
heart with saline followed by 10% Formalin. The animal's brain was
removed, and frozen sections were cut at 50 µm in the sagittal plane,
mounted on microscope slides, and stained with cresyl violet for cell
bodies and with a modified silver stain (Gallyas 1979
) for myelinated fibers. Electrolytic lesions facilitated histological reconstruction of the electrode tracks, and recording sites were verified using both Nissl and myelination. Sample electrode tracks were
verified to pass through the MST using X-rays, and neurons were assumed
to be in the MST based on their physiological characteristics (such as
preferred speed and receptive field size).
 |
RESULTS |
Initial vergence responses to disparity steps (correlated
patterns)
Horizontal disparity steps applied to large-field, correlated,
random-dot patterns elicited vergence eye movements at short latency
that were, in all essentials, like those described by Busettini
et al. (1996)
and Masson et al. (1997)
. Latency
estimates for the five monkeys, based on the responses to the optimal
disparity steps, were 73 ms (monkey H, +2°), 68 ms
(monkey L, +2°), 65 ms (monkey M,
2°), 58 ms (monkey N, +1°), and 59 ms (monkey Q,
+1°).1 Although
no attempt was made to obtain formal estimates of latencies to
nonoptimal stimuli, the vergence velocity temporal profiles suggested
that for a given animal, latency was largely independent of the
stimulus amplitude: see, for example, the sample profiles from
monkey H in Fig. 1, those in
A showing responses to crossed disparity steps, and those in
B responses to uncrossed disparity steps, the stimuli
ranging in amplitude from 0.5 to 6°. The profiles in Fig.
1A reach a peak and then decline, often before the closing of the disparity feedback loop (twice the response latency), whereas the profiles in Fig. 1B have a more varied time course and
some fail to reach a peak within the time window shown (150 ms after the disparity step). The initial responses to small (<2°) steps were
always compensatory in that they operated to reduce the seen disparity:
small crossed steps elicited increases in the vergence angle and small
uncrossed steps the converse. Responses were generally maximal with
steps of 1-3° and declined with larger steps, sometimes showing
reversal (Fig. 1B). These trends are evident from the disparity tuning curves in Fig. 1C, in which the change in
vergence position (measured over the time period of 50-110 ms) is
plotted against the amplitude of the disparity step. These plots
indicate that the system showed appropriate servo-like behavior only
for small disparity steps. That is, increases in the input resulted in
roughly proportional increases in the output (in the compensatory direction) only for steps of less than a degree or so. (Note that the
amplitude and direction of the responses of any given monkey to the
largest steps were generally independent of whether the steps were
crossed or uncrossed and showed considerable variation from one animal
to another.) The disparity tuning curves resembled the derivative of a
Gaussian and were well fit by a Gabor function: the parameters for the
least squares best fits are listed for all five monkeys in Table
1. To evaluate the goodness of the fit,
we computed the fraction of the disparity-induced variation in the data
accounted for by the fit, r2,
as previously done by others (e.g., Cumming and Parker
1999
). We found that r2
ranged from 0.88 to 0.99, indicating that 88-99% of the variation in
the data were captured by the fit. (Note that the Gabor functions for
monkeys N and Q are plotted as the continuous
lines in Fig. 6, A and B.)

View larger version (29K):
[in this window]
[in a new window]
|
Fig. 1.
Vergence responses: dependence on the amplitude and direction of the
disparity step (correlated stimuli). A: traces show vergence
velocity over time in response to crossed disparity steps.
B: traces show the same for uncrossed steps. Upward
deflections denote increased vergence and the numbers on the traces
indicate the magnitudes of the steps in degrees. C: the mean
change in vergence position in degrees, measured over the 60-ms period
starting 50 ms after the disparity step (see the horizontal bars in
A and B), is plotted against the magnitude of the
step in degrees, for each of the 5 monkeys: disparity tuning curves.
Continuous lines are spline interpolations with values at +6°
disparity replicated at disparities of +9 and +12°, and values at
6° disparity replicated at disparities of 9 and 12°, and
external knots at values with disparities of 18, 15, +15, and
+18°. This method was preferred to classic spline interpolation (or
cubic interpolation) because it dampens oscillations between absolute
disparities of 3 and 6° (where there were no data). Error bars, 1 SD.
|
|
Initial neuronal responses to disparity steps (correlated
patterns)
We recorded the activity of 586 MST neurons in seven hemispheres
of five monkeys while horizontal disparity steps were applied to
correlated large-field random-dot patterns. About 20% of the neurons
(122) responded to disparity steps at short latency, and the changes in
discharge rate ranged from very transient
some lasting little more
than 20 ms
to tonic. This is evident from the sample responses in Fig.
2, which shows the discharge rate temporal profiles of 20 units recorded from monkey N in
response to 2° crossed-disparity steps. If a neuron has a response
latency of Ln ms, and the associated
vergence has a latency of Lv ms, the
earliest time at which the neuronal response could be affected by the
decrease in retinal disparity secondary to the vergence response would
be Ln + Lv ms (after the disparity step). The
estimated latency of the vergence response (see METHODS) in
Fig. 2 was 59 ms (indicated by the dashed vertical line), which
means that only neurons with a latency of <51 ms could have been
affected by the closing of the disparity-feedback loop within the time
window shown (110 ms after the disparity step). Seven of the neurons in
Fig. 2 had a short enough latency to have been so affected (see *), but
even in those cases, the loop wouldn't have closed until long after
the neurons' initial burst of activity had ended. This is apparent
from the estimated time of closure of the disparity feedback loop for
those seven cells (indicated by
on the relevant traces in Fig. 2).
A similar scrutiny of the entire population of cells indicated that the
discharges of ~60% had a phasic component, which in every case was
independent of the closure of the disparity feedback loop. The
histograms in Fig. 3 show the
distributions of the latency estimates for the neurons that met the
response-onset criterion (2 SD above the mean control level: see
METHODS). The estimates in Fig. 3A are given
with respect to the onset of the disparity steps (n = 71; median, 55 ms; range 43-86 ms), whereas those in Fig.
3B are given with respect to the (measured) onset of the
vergence responses. The onset of the neuronal responses preceded the
onset of the vergence eye movements in 43/50 units (86%) by
24 ms
(median lead, 9 ms). Note that the measures in Fig. 3 were all obtained
from the response profiles elicited by the stimulus that was optimal
for the cell (see METHODS).

View larger version (12K):
[in this window]
[in a new window]
|
Fig. 2.
Time course of the neuronal responses (correlated stimuli).
Top: the changes in mean discharge rate over time in
response to 2° crossed disparity steps for each of the 20 units that
modulated most with this stimulus (ranked in descending order of their
mean discharge rates over the period 30-110 ms after the step).
Bottom: the changes in vergence velocity over time; the
vertical dashed line shows the estimated latency of the vergence
response (59 ms). Seven of the units (*) had response latencies <51
ms; , the estimated times at which the closing of the disparity
feedback loop could first influence the discharges of these units (see
text). The response latencies of the remaining units were too long for
the disparity-feedback loop to close during the 110 ms time window
shown. Calibration bars: 500 imps/s, 5°/s. Data from monkey
N.
|
|

View larger version (35K):
[in this window]
[in a new window]
|
Fig. 3.
Latency of onset of discharges (correlated stimuli). Histograms show
the distribution of latencies with respect to the onset of the stimulus
(A) and the onset of the associated vergence eye
movements (B). For the latter, negative times indicate
that the discharge preceded the vergence response.
|
|
DISPARITY TUNING CURVES OF SINGLE CELLS.
Neuronal responses were quantified by measuring the mean discharge
frequency over the 60-ms period starting 40 ms after the disparity
step. Of the 122 MST neurons that showed sensitivity to horizontal
disparity steps at short latency, 102 yielded sufficient data to allow
complete disparity tuning curves to be plotted. These curves showed a
variety of forms and, after normalization, we used the fuzzy c-means
algorithm to organize them into groups based on their shapes (see
METHODS). This algorithm sorted the curves into four
clusters, and we then assigned each curve to one of four groups based
on the cluster in which each curve had its largest membership. We chose
to partition the curves into four groups because this was the largest
number to yield a consistent grouping when the algorithm was run
multiple times (
50).
Figure 4 shows the normalized disparity
tuning curves for all 102 units arranged in the four groups, together
with the average disparity tuning curves for the vergence eye movements
of the two monkeys (N and Q) that yielded most
(80/102, 78%) of the data (bottom). The general shapes of
the tuning curves within the four groups conform very roughly to four
of the six classes of disparity selective units described by
Poggio et al. (1988)
: 1) Cells in group 1 tend to show increased activity in response to a
somewhat restricted range of uncrossed disparities, their tuning curves having relatively narrow peaks that are distributed mostly between 0 and
1° disparity and skewed to the left (toward uncrossed
disparities), c.f., the "tuned far" cells of Poggio et al.
(1988)
. 2) Cells in group 2 tend to show
increased activity in response to crossed or uncrossed disparities,
their tuning curves having a trough centered close to 0° disparity,
cf., "tuned inhibitory" cells (though in the present case the
trough is not the result of inhibition). 3) Cells in
group 3 tend to show increased activity in response to a
rather wide range of crossed disparities, their tuning curves having
relatively broad peaks that are distributed between +1 and +3°
disparity and skewed to the right (toward crossed disparities), c.f.,
"near" cells. 4) Cells in group 4 tend to
show increased activity in response to a somewhat restricted range of
crossed disparities, their tuning curves having relatively narrow peaks that are distributed between 0 and +1° disparity and skewed to the
right (toward crossed disparities), cf., "tuned near" cells.

View larger version (62K):
[in this window]
[in a new window]
|
Fig. 4.
Disparity tuning curves for the individual cells (correlated stimuli).
Top 4 graphs: mean change in discharge rate over the
60-ms period starting 40 ms after the disparity step is plotted against
the magnitude of the disparity step; curves are
normalized and arranged in 4 groups based on the outcome of the
fuzzy c-means clustering algorithm. Bottom: the
disparity tuning curves for the vergence responses of the two monkeys
that yielded most of the data (N and Q).
Traces are spline interpolations (see legend of Fig. 1 for details).
|
|
Although useful to distinguish between the groups, simple descriptors
like near, tuned far, and so forth do not capture the full complexity
of the shapes of the tuning curves within each group. For example,
cells in the near group 3 often show some increase in
discharge with large uncrossed steps, and cells in the tuned far
group 1 often show some increase in discharge with large
crossed steps.
Discrete groups or a continuum?
The extent to which the curves within a given group have significant
membership in only the "defining" cluster (that is, in cluster 1 for group 1, in cluster 2 for group 2, and so forth) provides some measure of the
extent to which that group constitutes a discrete entity. By this
token, the tuned far group 1, with an average membership in
cluster 1 of 0.75, was the most discrete and the near
group 3, with an average membership in cluster 3 of only 0.57, was the least discrete. All of the groups are to some
degree fuzzy, however, insofar as all have cells with significant membership in more than one cluster. (Note that a cell would be maximally fuzzy if all of its membership values were 0.25.) This leaves
open the possibility that we are dealing with a continuum, and there is
some coarse hint of this in Fig. 4. Thus in passing from one group to
another in the sequence, tuned near
near
tuned inhibitory
tuned far,
some gradual trends are evident: the responses to crossed steps start
with a peak at small disparities, then the peak flattens and shifts to
higher disparities before largely disappearing, while the responses to
uncrossed disparities show a similar trend but in the reverse order. If
we are dealing with a "smooth" continuum then these trends should
also be apparent within the groups.
Figure 5 (top) plots the
memberships in each of the four clusters for all 102 cells. The cells
are subdivided into the four groups and, within each group, are ranked
according to their memberships in the defining cluster of an adjacent
group.2 Figure 5
(bottom) shows all of the disparity tuning curves in a
color-coded form arranged along the abscissa in the order in which they
are plotted in Fig. 5, top, with the disparity step represented along the ordinate (crossed steps upward); increases in
activity are shown in red, zero activity in white, and decreases in
activity in blue. The general impression is that the tuned far
group 1 is distinct, with a sharp discontinuity at the
boundary with group 2 (see particularly the responses to
uncrossed disparities ranging from 0 to
2°) whereas the other three
groups lie along a continuum.

View larger version (81K):
[in this window]
[in a new window]
|
Fig. 5.
Rank ordering of disparity tuning curves based on their membership
values from the fuzzy clustering analysis (correlated stimuli).
Top: membership values; each point along
the abscissa represents an individual unit, whose four
membership values (1 for each of the 4 clusters) are plotted
along the ordinate; each unit is placed in the group
that corresponds to the cluster in which it has its highest membership
(group 1 have their highest memberships in
cluster 1, group 2 their highest in in cluster
2, etc) and, within each group, each unit is ranked according
to its membership in an adjacent cluster. Units in group
1 are ranked in ascending order of their memberships in
cluster 2, units in group 2 are ranked in
ascending order of their memberships in cluster 3, units
in group 3 are ranked in ascending order of their
memberships in cluster 4, and units in group
4 are ranked in descending order of their memberships in
cluster 3. Triangles, cluster 1; stars,
cluster 2; squares, cluster 3; circles,
cluster 4. Bottom: disparity tuning
curves; each point along the abscissa
corresponds to the unit whose membership value is plotted above, and
its disparity tuning curve is shown in a color-coded form in accordance
with the scale shown at the right (increases in discharge rate, in
imp/s, are shown in red and decreases are shown in blue) with the
amplitude of the disparity steps represented along the
ordinate.
|
|
We further examined the discreteness of the four groups using a
hierarchical clustering algorithm. This algorithm starts out with as
many clusters as there are data points and then progressively reduces
the number by combining clusters that fail to achieve a set criterion
for separation. This iterative process stops when the separation
between the remaining clusters exceeds the criterion level. When run on
our data, this algorithm did identify four clusters that were
essentially identical to the four groups that we identified using the
fuzzy c-means clustering algorithm. However, the algorithm did not stop
there but went on to first combine clusters 3 and
4 and then to combine this new cluster with cluster 2 before reaching the criterion level for separation. Thus this algorithm reinforced the view that group 1 is separate from
all others and that groups 2-4 have considerable overlap.
Even though this provides some support for the notion that groups
2-4 represent a continuum, we feel that, for descriptive
purposes, it is still useful to treat them separately: whether they are
samples from a continuum or from discrete entities, the three groups
show clear differences, especially between the extremes (groups
2 and 4). Note also that there were no major
differences in the latency distributions among the four groups of
neurons: see the different patterns of shading in Fig. 3.
DISPARITY TUNING CURVES FOR (SUMMED) POPULATIONS OF CELLS.
A characteristic feature of the disparity tuning curves for vergence in
Fig. 1C is that they are roughly odd functions, that is,
f(x) = 
f(
x),
with a linear segment passing smoothly through zero disparity defining
the critical servo range over which changes in the input (binocular
disparity) elicit roughly proportional changes in the output (vergence
eye movement). It is significant that very few cells had disparity
tuning curves with this exact same feature: The tuned inhibitory
group 2 cells have rather flat curves in the vicinity of
zero, conveying little information about disparity in this range, and
the tuning curves for most other cells undergo a substantial change in
slope near zero disparity. Exceptions are small numbers of cells in the
tuned near group 4 and tuned far group 1, with
positive and negative slopes, respectively, extending on either side of
zero disparity. (Given appropriate excitatory or inhibitory
connections, neurons with either positive or negative slopes could make
a positive contribution to the vergence responses. That the tuning
curves for vergence had a positive slope around zero disparity was
determined solely by our sign convention.) To obtain an estimate of how
well the vergence responses were encoded in the discharges elicited in
individual MST cells, we fitted the disparity tuning curves of the
individual cells to the tuning curves for vergence, the only free
parameters being gain and y offset. We assumed that all
neurons made a positive contribution to the vergence responses elicited
by disparity steps in the important servo range, ±1° and so reversed
the sign of those curves that had negative slopes over that range: this
involved 29/102 cells, including all 28 in the tuned-far group
1 and 1 in the tuned inhibitory group 2. The
least-squares best fits ranged widely
(r2: mean, 0.51; range,
0.009-0.92), but the very best of the fits were good enough to raise
the possibility that some cells might be more properly regarded as
"vergence-related" rather than "disparity-related." We will
revisit this issue later when we describe the responses of these cells
to anticorrelated stimuli.
Population coding?
To see how well the vergence responses were encoded in the discharges
of the entire population of MST cells, we summed the raw (that is,
nonnormalized) tuning curves for all units and then determined how well
this population average fitted the tuning curve for vergence when gain
and y offset were once more the only free parameters (so
that the contributions of all neurons were given equal weight). Because
the shapes of the disparity tuning curves for vergence differed
slightly from one animal to another, it was necessary to treat the data
from each animal separately, and we had adequate numbers of units to
permit this for only 2/5 animals: monkey N (49 units) and
monkey Q (31 units). We again reversed the sign of those
curves that had negative slopes over the disparity range ±1°; this
involved 17/49 cells in monkey N and 7/31 cells in
monkey Q, all in the tuned far group 1. Given our
sign convention (convergence is positive, divergence is negative), this
meant that cells with positive slopes (e.g., tuned near) contributed to
convergence and cells with negative slopes (tuned far) contributed to
divergence: push-pull configuration. The resulting summed activity
showed a very good fit to the vergence data for both animals: compare
the
(summed activity) and * (vergence) plotted in Fig.
6A (monkey N,
r2 = 0.98) and Fig. 6B
(monkey Q, r2 = 0.93). Inverting the
sign of the curves with negative slopes around zero disparity was
critical for achieving such good fits. Thus failure to invert those
curves (before fitting) decreased the
r2 to 0.12 for monkey
N and to 0.21 for monkey Q, and the best fits now
showed little resemblance to the tuning curves for vergence: see Fig.
6, C and D. None of the single units had a
tuning curve that matched the vergence as well as these population
responses did: even with sign inversion, the highest
r2 for any individual unit was
0.92 for monkey N (mean
r2: 0.49) and 0.86 for
monkey Q (mean r2:
0.48). It is of interest that the disparity tuning curves for the
summed neuronal activity, like those for the vergence eye movements,
were well fit by Gabor functions,
r2 being 0.95 and 0.94 for
monkeys N and Q, respectively: see - - -
plotted in Fig. 6. The parameters for these fits are included in Table
1.

View larger version (26K):
[in this window]
[in a new window]
|
Fig. 6.
Spatial coding of vergence by populations of cells (correlated
stimuli). Disparity tuning data for vergence (*) and for the
least-squares, best-fit summed activity ( ) for
monkey N (A and C) and
monkey Q (B and D). For
the top plots (A and B),
the individual unit curves with negative slopes around zero disparity
were inverted before summing and for the bottom plots
(C and D), no curves were inverted.
Curves are the least-square, best-fit Gabor functions for the vergence
data ( ) and the summed activity data (- - -).
|
|
These data indicate that the disparity tuning curve for the vergence
eye movements was effectively encoded in the aggregate activity of all
the disparity-sensitive cells recorded in MST. The question arises as
to the relative contributions of the four groups identified by the
fuzzy cluster analysis. We addressed this issue by examining the effect
of excluding each of the groups of units on the goodness of the fit
between the disparity tuning curves for the summed activity and for the
vergence eye movements when, as usual, gain and y offset
were free parameters. Removing all of the cells in any one of the four
groups always increased the MSE (and thus reduced
r2) for monkey N. We
examined the statistical significance of this finding using a bootstrap
algorithm (Efron and Tibshirani 1991
), which involved
computing the MSE after randomly excluding equivalent numbers of units,
using 10,000 random combinations of cells. This indicated that the
probability that excluding a random sample of cells would have an
impact equal to that when any one of the four groups of cells was
removed was always <0.008. This implies that units in all four of the
groups made a significant positive contribution to the population
response of monkey N. However, the data for monkey
Q were less consistent and only the removal of the tuned far
group 1 resulted in a significant worsening of the best fit
(P < 0.001, bootstrap algorithm).
Discrete coding?
To further examine the possibility of the discrete coding of vergence
by a subpopulation of MST cells, we used a genetic algorithm to
determine if there was a subset of cells whose activity when summed
together resulted in a disparity tuning curve that fitted the disparity
tuning curve for vergence as well as, or better than, that from summing
the entire population. We again treated the data from each animal
separately and inverted the sign of those disparity tuning curves with
negative slopes in the disparity range ±1°. We ran the algorithm 100 times on each data set, and the number of cells that survived to the
last generation each time varied from 19 to 28 (of 49) for monkey
N and from 7 to 14 (of 31) for monkey Q. The MSEs for
the summed activity of the last generations of cells were smaller than
those when the whole population was summed, by a factor of 58-107 for
monkey N (r2 = 0.9995 for the subset giving the very best fit) and by a factor of
16-24 for monkey Q (highest
r2 = 0.9973). Clearly, all of
the subsets of cells selected by the genetic algorithm gave much better
fits to the vergence data than did the whole populations, and the very
best fits (i.e., the best of each 100) were almost indistinguishable
from the targeted vergence data: see Fig.
7, A and B. Every
one of the four groups contributed to these very best fits, though
group 2 always contributed fewest: for monkey N,
each group contributed 7, 2, 5, and 5 cells (in order from group
1 through to group 4), and for monkey Q,
these numbers were 4, 1, 2, 2. The special importance of groups
1 and 4 was evident when the genetic algorithm was run
after excluding all of the curves selected from either of those groups:
the very lowest MSE (i.e., the lowest when the algorithm was run 100 times) increased by a factor of 124 and 26, respectively, for the data from monkey N, and by a factor of 9 and 2, respectively, for
the data from monkey Q. In contrast, excluding the cells
selected from group 2 had no significant impact on the very
lowest MSE in either monkey, and excluding those selected from
group 3 had a significant impact only for monkey
N (increasing the very lowest MSE by a factor of 12).

View larger version (13K):
[in this window]
[in a new window]
|
Fig. 7.
Spatial coding of vergence by the subsets of cells selected by the
genetic algorithm (correlated stimuli). Disparity tuning data for
vergence (*) and for the summed activity of the subsets of cells
(selected by the genetic algorithm) that gave the very best fits to the
vergence data ( ) for monkey N
(A) and monkey Q (B).
Curves are the least-squares best-fit Gabor functions for the vergence
data ( ) and for the summed unit data (- - -).
|
|
Temporal coding?
In view of the fact that the disparity tuning curves for the summed
activity of the whole population of cells showed a good correlation
with the disparity tuning curves for vergence (spatial coding), we also
attempted to determine if the summed activity elicited by a given
disparity stimulus conveyed information about the time course of the
vergence responses (temporal coding). As before, we first inverted the
contributions of those cells whose disparity tuning curves had negative
slopes in the disparity range ±1° and treated the data from
monkeys N and Q separately. It was soon apparent
that noise was a major limiting factor here, but the summed discharge
profiles elicited by some disparity stimuli strongly resembled
the associated vergence velocity profile, and we obtained least-squares
best fits, allowing gain, x offset (time delay), and
y offset to be free parameters. Samples of two such fits can
be seen in Fig. 8, A and
B, which shows the time course of the average vergence
velocity response elicited by 1° crossed-disparity steps for
monkeys N and Q together with the best-fit
discharge-rate profiles using the summed activity of all cells from
each animal. The fit was clearly better for monkey N
(r2 = 0.98; n = 49) than for monkey Q
(r2 = 0.87; n = 31), possibly in part because of the difference in the numbers of
cells. The x offset that yielded these best fits was
18 ms
for monkey N and
23 ms for monkey Q. In
both cases, the peak in the relationship between
r2 and the time delay was
sharply convex (Fig. 8, C and D), indicating that
these time delays provide a reliable estimate of the time interval by
which the neuronal population response preceded the vergence response.

View larger version (19K):
[in this window]
[in a new window]
|
Fig. 8.
Temporal coding of vergence velocity by populations of cells
(correlated stimuli). A and B: plots of
average vergence velocity ( ) and the least-squares, best-fit summed
activity (- - -) over time for monkey N
(A) and monkey Q (B) in
response to 1° crossed-disparity steps; gain and offset were free
parameters for these fits, which were obtained for a range of
x shifts (time shifts, at 1-ms intervals), and the data
shown are for the x shifts that gave the very best fits.
C and D: the dependence of the goodness
of these best fits (r2) on the time
shift is plotted for monkey N (C) and for
monkey Q (D); a negative time shift
indicates that the summed activity has been moved forward in time with
respect to the vergence response; both peak at negative values ( 18 ms
in A, 23 ms in B), indicating that the
summed activity preceded the vergence response and had to be moved
forward to get the very best fit.
|
|
However, r2 for the least-squares best
fits exceeded 0.9 for only 8/20 data sets, most of these (7) involving
responses to crossed disparity steps, and the x shifts that
gave these fits ranged from
15 to
22 ms (mean ± SD, 17.6 ± 2.3 ms): see Table 2 (values listed
under "all cells"). The fits were even worse when we summed only
the activity of the cells that had been selected with the genetic
algorithm: see Table 2 (values listed under "GA"). Visual
inspection revealed that the summed discharge profiles giving poor fits
were generally noisy, often before the onset of the disparity-driven
response, indicating the existence of appreciable noise unrelated to
the disparity stimulus. To our surprise, the summed discharge profiles
that gave good fits always included individual profiles that varied
widely. For example, the summed discharge profile that included the
data in Fig. 2 accounted for >93% of the stimulus-induced variation
in the associated vergence velocity profile
(r2 = 0.933), despite the
obvious variability among the individual cells in the latency and time
course of the (averaged) discharge profiles. Thus we suggest that the
paucity of good fits was due in large part to the inadequacy of our
data samples: on the one hand, the relatively small number of responses
averaged for each stimulus condition and, on the other, the relatively
small numbers of cells recorded from any given monkey. Another
important factor here was that not all units were active for all
stimuli, further limiting the number of discharges contributing to a
given temporal profile. In summary, although noise problems restricted
the amount of useful data, the temporal profile of the summed activity
associated with some disparity steps showed a reasonably good fit to
the vergence velocity profile.
Initial vergence responses to disparity steps (anticorrelated
patterns)
The vergence responses elicited by disparity steps applied to
anticorrelated patterns were recorded from two animals (monkeys N and Q). As previously reported by Masson et
al. (1997)
, these vergence responses were comparable in latency
with those produced by the same steps applied to correlated patterns
but were often in the opposite direction. This is apparent from the
sample data from monkey N shown in Fig.
9, which has a layout identical to Fig.
1. Thus the disparity tuning curves showed a negative slope in the
immediate vicinity of zero disparity, though it is clear that the curve
for monkey Q is shifted appreciably to the left of the curve
for monkey N. The peak-to-peak amplitudes of the (interpolated) disparity tuning curves for the anticorrelated data were
smaller than those for the correlated data: by 33% for monkey N and by 44% for monkey Q. The tuning
curve data were well fit by a Gabor function
(r2 was 0.99 and 0.98 for
monkeys N and Q, respectively), and the parameters of the least-squares best fits are listed in Table 1: see
also the continuous lines plotted in Fig. 12, A and
B. Of particular interest among the parameters of the Gabor
functions is the difference in the phase of the cosine terms for the
correlated and anticorrelated data: 177° for monkey N and
169° for monkey Q. This reinforces the impression that, to
a first approximation, the disparity tuning curves obtained with
anticorrelated stimuli were inverted versions of those obtained with
correlated stimuli. The above-mentioned shifts, however, are evident in
the x shifts of the best-fit Gabor functions, which differed
for the correlated and anticorrelated data, especially for monkey
Q: this parameter was always zero for the correlated data and
negative for the anticorrelated data (Table 1). In fact, the tuning
curves obtained with anticorrelated stimuli were quite well fitted by
the tuning curves obtained with correlated stimuli when the latter were
inverted provided that the x shift was a free parameter (in
addition to gain and y offset): r2 values for the least-squares
best fits for monkeys N and Q were 0.86 and 0.97, respectively, and the corresponding x shifts were 0.05 and
0.61°.

View larger version (26K):
[in this window]
[in a new window]
|
Fig. 9.
Vergence responses: dependence on the amplitude and direction of the
disparity stimulus (anticorrelated stimuli). Traces in A
show vergence velocity over time in response to crossed disparity
steps, and traces in B show the same for uncrossed
steps. C: the mean change in vergence position in
degrees, measured over the 60-ms period starting 50 ms after the
disparity step), is plotted against the magnitude of the step in
degrees, for each of the 2 monkeys: disparity tuning curves. All
conventions as for Fig. 1.
|
|
Initial neuronal responses to disparity steps (anticorrelated
patterns)
Neurons that were still well isolated after we had finished
recording their responses to disparity steps applied to correlated patterns were then recorded while the same steps were applied to
anticorrelated patterns. The activity of 56 MST neurons in three
hemispheres of two monkeys was so recorded (25/49 units from
monkey N and 31/31 units from monkey Q), and all
gave significant responses to the anticorrelated stimuli
(P < 0.005, 1-way ANOVA). Neuronal response latencies
to anticorrelated stimuli were roughly comparable to those obtained
with correlated stimuli.
DISPARITY TUNING CURVES OF SINGLE CELLS.
Figure 10 shows the normalized
disparity tuning curves for all 56 units whose responses to
anticorrelated stimuli were recorded, and is organized like Fig. 4,
with cells placed in the same four groups. That is, cells that were in
group 1 in Fig. 4 were also placed in group 1 in
Fig. 10, and so forth. Again, we fitted the disparity tuning curves of
the individual cells to the tuning curves for vergence (gain and
y offset, free parameters) and assumed that the contribution
of any given cell to the vergence response would always have the same
sign regardless of the stimulus used to drive it. Accordingly, cells
whose contributions had been inverted for the earlier analysis of the
correlated data were again inverted here. (For monkey N,
7/25 curves were inverted and for monkey Q, 7/31 curves were
inverted, all in group 1.) As for the correlated data, the
least-squares best fits for the anticorrelated data ranged widely
(r2: mean,
0.39; range, 0-0.97) and some were clearly good enough to be
considered vergence-related rather than disparity-related. However, no
single cell had responses that fitted the vergence responses obtained
with both correlated and anticorrelated stimuli with an
r2 value >0.67: Fig.
11 shows a plot of the individual
r2 values obtained with
correlated stimuli against those obtained with anticorrelated stimuli,
and there is a relative paucity of cells in the upper right quadrant,
which is where pure vergence-encoding cells would be expected.

View larger version (65K):
[in this window]
[in a new window]
|
Fig. 10.
Disparity tuning curves for the individual cells (anticorrelated
stimuli). Top 4 graphs: mean change in discharge rate
over the 60-ms period starting 40 ms after the disparity step plotted
against the magnitude of the disparity step for the data from 2 monkeys
(N, data in red; Q, data in blue); curves
are normalized and each has been assigned to 1 of 4 groups (the curve
for a given cell being assigned to the group to which that cell's
curve was assigned in Fig. 4); thick gray
traces in each of the 4 graphs show the (inverted) median
tuning curves obtained for the same groups of cells with correlated
stimuli. Bottom graph: thick colored
traces, the disparity tuning curves for the associated mean
vergence responses of the 2 monkeys; thin colored
traces, the (inverted) vergence tuning curves
obtained from the same 2 monkeys with correlated stimuli. Traces are
spline interpolations (see legend of Fig. 1 for
details).
|
|

View larger version (22K):
[in this window]
[in a new window]
|
Fig. 11.
Coding of vergence by individual cells (correlated and anticorrelated
stimuli). The disparity tuning curves of the individual cells were
fitted to the disparity tuning curves of the associated vergence
responses and r2 values for the
least-squares, best fits were computed. This graph plots the
r2 values for the data obtained with
correlated patterns against those obtained with anticorrelated
patterns. No cell had r2 values that
exceeded 0.67 for both stimuli (indicated by the - - -). Also shown
(indicated by their identifying letters), are the
r2 values for the fits between the
summed activity and the vergence responses for the 2 monkeys,
N and Q (from which all the unit data
plotted here were obtained).
|
|
Comparing Fig. 10 with Fig. 4 indicates that, within each group, the
anticorrelated data have much more scatter than the correlated data:
cells that had similar tuning curves with correlated stimuli (and so
were assigned to the same group) often had very different tuning curves
with anticorrelated stimuli. Some of this scatter for disparities
between 3 and 6° (where there are no data points) comes from the
spline interpolation: see particularly the group 1 curves
with uncrossed disparities. Further, some of the scatter in Fig. 10
might have resulted from differences in the responses of the two
monkeys: for example, in group 4, the peaks for monkey Q (shown in blue) are often to the left of those for monkey
N (shown in red), and a similar (weaker) trend is evident in
group 3. As we saw earlier, the tuning curve for the
vergence eye movements of monkey Q is shifted to the left of
that for monkey N (see also the thick colored traces in the
bottom plot of Fig. 10), and we sought to investigate these
x offsets further.
We saw in the preceding text that, to a first approximation, the
vergence responses to correlated and anticorrelated stimuli had
opposite signs, so that the disparity tuning curve for the vergence
data obtained with one stimulus resembled the inverse of the other,
except for an x offset that was greater in monkey Q than monkey N. We attempted to determine if this also
applied to the single-unit responses, i.e., whether, for each cell, the response to anticorrelated patterns closely resembled the inverse of
the response to correlated patterns. There is some hint of this in Fig.
10, which includes, in addition to the individual disparity tuning
curves for the responses to anticorrelated patterns, plots of the
inverse of the group median responses to correlated patterns: see the
thick gray lines in each
group.3 Thus the
curves in groups 2-4, which show more consistency than group 1, often have a general form that resembles the
inverted median but shifted horizontally. When the individual tuning
curves obtained with anticorrelated stimuli were fitted to the inverted median responses obtained with correlated stimuli (with gain, y offset and x offset as free parameters), many
of the fits were quite good: for monkeys N and Q,
mean r2 values were 0.61 and
0.71 and mean x shifts were 0.31 and 0.73°, respectively.
This difference in the x shifts of the data from the two
monkeys (0.42°) was statistically significant (P = 0.02, 1-way ANOVA) and corresponded roughly with the difference in the x shifts reported in the preceding text when the inverted
tuning curve for the vergence responses to correlated stimuli was
fitted to the tuning curve for the vergence responses to anticorrelated stimuli (0.56°). The need for this latter shift is clear from the
bottom plot in Fig. 10, which includes the inverted
disparity tuning curves for the vergence responses to correlated
stimuli (thin colored traces). Note that we did not attempt to fit the response of each cell to anticorrelated stimuli with the inverse of the
response of the same cell to correlated stimuli because of the
x offsets, which would require the pairing of actual data points from one tuning curve with interpolated data points from the
other. As the use of interpolated data points makes the result sensitive to the quality of the fit, we used the median because, being
computed over a number of cells, it smoothes out the oscillations (compare Fig. 10 with Fig. 4).
DISPARITY TUNING CURVES FOR (SUMMED) POPULATIONS OF CELLS.
We saw earlier that, when correlated patterns were used, the disparity
tuning curves for the summed activity of the population of cells
matched the tuning curves for the associated vergence responses quite
well. We now sought to determine if the same was true for the data
obtained with anticorrelated stimuli. We again summed all of the
raw
that is, nonnormalized
tuning curves for each of the two monkeys
separately and then determined how well these population responses
fitted their respective tuning curves for vergence when gain and
y offset were the only free parameters. We assumed that the
contribution of any given cell to the vergence response would always
have the same sign regardless of the stimulus used, and cells whose
contributions had been inverted for the earlier analysis of the
correlated data were again inverted here. Figure
12, A and B,
shows that the disparity tuning data for the summed activity (
)
again matched those for the associated vergence responses (*) quite
well (r2: 0.96 and 0.95 for
monkeys N and Q, respectively). Once again, failure to invert the contributions of the relevant cells before fitting led to significantly worse fits
(r2: 0.77 and 0.57 for
monkeys N and Q, respectively), though the effects on the fits were not as dramatic as reported above for the data
obtained with correlated patterns. In the case of monkey N,
this is perhaps in part because a somewhat smaller proportion of the
curves obtained with anticorrelated stimuli were inverted: 28% (7/25),
compared with 35% (17/49) of those obtained with correlated stimuli.
There was one cell from monkey N whose tuning curve matched the vergence as well as the curve for the summed activity did (r2 = 0.97) but the mean
r2 for all cells from this
monkey was only 0.47. None of the single units from monkey
Q had tuning curves matching its vergence responses as well as its
summed activity did and the highest
r2 for any individual unit from
this monkey was 0.88 (mean r2:
0.32).