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Laboratory of Sensorimotor Research, National Eye Institute, National Institutes of Health, Bethesda, Maryland 20892-4435
Submitted 27 November 2002; accepted in final form 3 April 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MAXIMUM LIKELIHOOD...DISCLOSURESACKNOWLEDGMENTSREFERENCES |
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0.1°. This error is large enough to explain the SD of
measured vergence in the absence of any real changes in vergence state. This,
and a variety of other arguments, indicate that the real variation in vergence
is much smaller than coil measurements suggest. These results suggest that
monkeys, like humans, maintain very stable vergence. The error has a slower
time course than fixational eye movements so that search coils report the
difference in eye position between two consecutive trials more accurately than
the eye position itself on either trial. Receptive field estimates are
unlikely to be improved by assuming the coil record is veridical and
correcting for eye position accordingly. However, receptive field parameters
can reliably be determined by a fitting technique that allows for eye
movements. It is possible that suturing coils to the globe reduces the
artifacts, but no method has been available to demonstrate this. These
receptive field measurements provide a general means by which the reliability
of eye-position measurements can be assessed.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MAXIMUM LIKELIHOOD...DISCLOSURESACKNOWLEDGMENTSREFERENCES |
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;
Gur et al. 1997). This raises a potentially serious problem for the study of receptive fields in awake animals. Even animals with extensive training on a fixation task make small eye movements during fixation, so that there is a variable relationship between the external stimulus location and the retinal stimulus location. Of course, the extent to which this disrupts RF measures will depend on the relative size of the eye movements and the RF. In monkey striate cortex, many RFs are sufficiently small that fixational eye movements pose a substantial problem. The structure of the RF may be blurred, and its size is likely to be overestimated. While for many applications an overestimate of RF size may be a relatively benign error, it nevertheless means that the sensitivity of V1 cannot be accurately assessed. Blurring of the RF by eye movements will result in an underestimate of the ability of individual neurons to discriminate changes in stimulus position.
This problem is most commonly ignored in the hope that the animals'
fixation is sufficiently precise that measures of RF size or structure are not
in fact disrupted. More recently, several studies have used measures of eye
position to correct for movements and estimate the retinal location of each
stimulus, either with scleral search coils
(Conway 2001;
Livingstone 1998;
Livingstone and Tsao 1999) or
a double Purkinje image eye tracker (Gur and Snodderly
1987,
1997;
Gur et al. 1997;
Kagan et al. 2002;
Snodderly and Gur 1995).
Gur, Snodderly, and colleagues have used the Purkinje eye tracker to
stabilize images on-line (that is, they add the recorded eye position to the
stimulus position to keep the retinal position constant)
(Gur and Snodderly 1987;
Gur et al. 1997;
Kagan et al. 2002;
Snodderly and Gur 1995). These
studies show convincingly that adjusting the image position with the measured
eye position is superior to making no adjustment. This is the clearest
evidence available that V1 receptive fields are fixed in retinal coordinates.
However, these data do not rule out any artifact in the eye-position measures.
This would require a quantitative comparison between measured eye-position
variation and RF size, both with and without correction. Furthermore, the data
demonstrating the effectiveness of stabilization were all collected over short
time periods. These have been used with care by these authors to establish the
relevant scientific points, but the long-term stability of the eye-position
measures remains unproven.
In contrast, other authors (Conway
2001; Livingstone
1998; Livingstone and Tsao
1999) have used scleral search coils to correct for eye movements
over extended recording sessions (hours). Clearly, such an approach places
heavy reliance on the long-term accuracy of eye-position recording. The
scleral search coil yields very precise measures. However, the very small
high-frequency noise does not guarantee accuracy—it is quite possible
that there are instrumental errors that change slowly enough not to compromise
the precision. Any adjustment of the instrument's offset during an experiment
would tend to conceal any such drift. We are unaware of any study that has
assessed the absolute accuracy of eye-position recordings with the scleral
search coil.
That scleral search coil signals contain significant inaccuracies is not
merely a logical possibility. When recording the position of both eyes with
scleral search coils, we have often noticed slow drifts in recorded vergence
angle, and others have noted the same phenomenon (F. Miles, personal
communications). Thus the raw data appear to show that the subjects are
maintaining a stable misconvergence, which seems unlikely given that a variety
of methods suggest that fixation disparities in humans are small and show very
little variation (SDs >2 min arc)
(Collewijn et al. 1988;
Duwaer 1983;
Enright 1991;
Jaschinski-Kruza and Schubert-Alshuth
1992; Ogle 1964;
Riggs and Neill 1960;
St Cyr and Fender 1969).
Because these observations are anecdotal, we attempt here to examine the reliability of these measures, exploiting the small RFs of V1 receptive fields. In one animal, we also implanted two coils in one eye. All of these measures suggested that there are substantial errors in the estimation of eye position from scleral search coils. This led us to develop a method for measuring RF size that does not depend on information about absolute eye position.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MAXIMUM LIKELIHOOD...DISCLOSURESACKNOWLEDGMENTSREFERENCES |
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); in particular, the
coil was not sutured to the sclera, and a strain-relief pouch was created. The
monkeys were trained to maintain fixation on a small white dot for fluid
reward. If fixation was maintained to within ±1° of the dot for 2
s, a reward was delivered. This relatively large window was chosen so as not
to suppress small eye movements during fixation. Coil offsets were set at the
beginning of each experiment and not adjusted thereafter. All protocols were
approved by the Institute Animal Care and Use Committee and complied with
Public Health Service policy on the humane care and use of laboratory
animals. Glass-coated platinum-iridium electrodes (Frederick Haer) were introduced transdurally into the operculum of striate cortex. Electrode position was controlled with a custom-made microdrive that used an ultra-light stepper motor mounted directly onto the recording chamber. The signal was amplified (Bak Electronics), band-pass filtered (100 Hz to 10 kHz), digitized at 32 kHz and stored to disk on a PC running the Datawave Discovery package. Single-unit isolation was always checked off-line. The vertical and horizontal positions of both eyes were sampled at 5.3 kHz and then averaged in groups of eight consecutive samples to give a sampling rate of 660 Hz.
Stimuli were generated on a Silicon Graphics Octane workstation and
presented on two Eizo Flexscan F980 monochrome monitors (mean luminance: 41.1
cd/m2, contrast: 99%, frame rate: 72 Hz) via a Wheatstone
stereoscope. That is, the monitors were placed on either side of the monkey,
who viewed the images via mirrors placed at an angle of 45° 2 cm in front
of each eye. At the viewing distance used (89 cm) each pixel in the 1,280
x 1,024 display subtended 1.1 min arc. Within each 2-s fixation period,
four stimuli were presented, each lasting 415 ms with an inter-stimulus
interval of 
100 ms.
Receptive fields were mapped with sinusoidal luminance gratings. Initially,
a circular patch of grating was manipulated manually to determine
approximately the preferred orientation and spatial frequency and the
boundaries of the minimum response field. Quantitative measurement of
orientation preference used circular grating patches (9 different
orientations spanning 180°), quantitative estimation of spatial and
temporal frequency tuning used larger rectangular grating patches.
Quantitative estimation of RF width and location then used a narrow strip of
grating at the preferred orientation, whose location varied from trial to
trial along an axis orthogonal to the preferred orientation. In many cases,
the width of the strip was smaller that the spatial period of the grating, so
the stimulus was similar to a stationary bar with a sinusoidal variation in
luminance over time. The temporal frequency was usually 4 Hz, but in a few
neurons (12), higher frequencies were necessary to elicit a brisk response.
The strip was substantially longer than the minimum response field (MRF; mean
length, 5°), so that only variation in stimulus and eye position along one
axis influenced the stimulus within the MRF. Monocular stimuli, presented to
the dominant eye, were used, although the fixation marker was always visible
in both eyes. To be included in the study, neurons had to respond with at
least three spikes to a 415-ms presentation at the optimal position (as an
average over 3 repetitions of the stimulus), and the mean number of
repetitions at each stimulus position had to exceed 3. Fifty-seven neurons
satisfied these criteria, including six pairs recorded simultaneously.
In addition to eye movements, any blinks that occurred during the
presentation of a stimulus could be a source of error. All trials which
included any part of a blink were discarded. Both blinks and microsaccades
were detected by differentiating the conjugate eye-position signals,
and
being the magnitude of the horizontal
and vertical velocities, respectively. Whenever the speed of conjugate
movement
exceeded 10°/s, an event was deemed to have started. Two characteristics
distinguished blinks from true saccades. First, the displacement was transient
so that the excursion during the event was large relative to the size of any
net displacement. Second, the transient displacements occurred at slightly
different times in the two eyes, giving rise to an apparent transient vergence
movement (with both vertical and horizontal vergence changes). The latter was
sufficiently distinctive to allow reliable blink detection. If an event
identified by a conjugate eye velocity >10°/s was also associated with
a vergence velocity (vertical or horizontal) that exceeded 5°/s for
100 ms, it was invariably a blink. To confirm this, calibration data were
gathered by viewing the pupil with an infrared sensitive CCTV camera. A
photodiode was placed over the video image of the pupil, and its output
low-pass filtered at 30 Hz. Reflections from the lids then produced a readily
detectable change in this output when a blink occurred. A total of 2,072
saccades (mostly microsaccades) and 162 blinks were recorded. No saccades were
misclassified as blinks, and all 162 blinks were correctly identified.
The variation in spike count as a function of stimulus location was fit
with a Gaussian, first by a simple least-squares fit and later by a more
sophisticated maximum-likelihood approach. It is inappropriate to use
least-squares fitting on neuronal spike counts directly because their variance
is proportional to the mean (Dean
1981). However, at least for large firing rates, this implies that
the square-root of the spike count has approximately constant variance. We
therefore fit the square-root of a Gaussian to the square-root of spike counts
(Prince et al. 2002b). We
compared the effect of ignoring eye movements, and of correcting for eye
movements using the scleral search coil. We also developed a novel fitting
method designed to enable an accurate reconstruction of receptive field
parameters even in the presence of eye movements and inaccurate coil
measurements (see APPENDIX). It is based on the observation that
the artifact affecting the coil is of lower frequency than the signal and
hence that the coil accurately reports changes in eye position between two
consecutive trials even if it is unreliable over long periods of time. Knowing
the difference in eye position between two trials, as well as the spike counts
recorded on each trial, provides important additional constraints on the eye
position and RF parameters. All analysis code was written in MATLAB 6.1 (The
Mathworks), and the fitting employed the routine FMIN-SEARCH from MATLAB's
optimization toolbox.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MAXIMUM LIKELIHOOD...DISCLOSURESACKNOWLEDGMENTSREFERENCES |
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In a separate study of disparity-selective neurons, selectivity for disparity in random dot stereograms (RDS) was measured. In several instances, the tuning curve was measured in two separate blocks separated by several minutes.
Figure 1 shows two examples
(1 for each monkey) in which the recorded vergence angle changed substantially
in the period between the two blocks, yet it is clear that there has been no
such displacement of the tuning curves. This implies either that there has not
in fact been a change in vergence angle or that the neuronal responses somehow
compensate for such changes. The latter interpretation seems very unlikely
because when vergence is explicitly manipulated disparity tuning curves are
displaced accordingly (Cumming and Parker
1999). Two other approaches were used to distinguish these
possibilities.
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Comparing search coils in the same eye
First, in monkey Duf, a third coil was implanted so that there
were two coils in the right eye. (This was done because the drifts in that eye
seemed larger than usual, so it was necessary to implant a new coil for other
reasons.) The eye-position readings of all three coils were measured
simultaneously for a series of 6,284 415-ms stimulus presentations similar to
those used in recording experiments. Trials where the monkey blinked or made a
saccade were removed, leaving 3,121 trials. The mean horizontal position
was calculated for each 415-ms stimulus presentation. The SD was then
calculated for the three possible "vergence" measures:
,
and 
. The
difference between two coils in the same eye shows substantial variation
(0.157°), establishing beyond doubt that eye-position measures with
scleral search coils can contain significant inaccuracies.
One might argue that some mechanical interaction between the two coils in
one eye induced inaccuracies that were not present when only one coil was
implanted. If this was the case, then the measures of
taken before the
second coil was implanted should show a smaller SD of vergence. In fact, the
SD of vergence prior to implanting the second coil was slightly larger (mean
SD across 28 experiments was 0.216°) than the equivalent measure on the
same two coils subsequently.
Coil R1 was replaced because of its unusually large drifts. Vergence
measured with this coil has a larger SD (0.202°) than with the newer coil
(0.088°). Thus a possible interpretation is that coils L and R2 are
veridical, while R1 is subject to a large artifact. Under this interpretation,
(L – R2) represents the monkey's vergence state, (R1 – R2)
represents the error on coil R1 (SD 0.16), and (L – R1) represents
OCvergence plus the error on R1. Because the real changes in vergence should
have a different structure than artifactual drifts, examining the time course
of the three "vergence" measures should reveal these differences.
We therefore examined the Fourier amplitude spectra of all three measures. Of
course, the animal may make saccades during inter-trial intervals, which may
add a common signal to all three. To prevent this, our analysis uses only data
recorded during fixation trials. To look at low frequencies, we first divided
the 3,000 trials into 10 groups of 300 roughly consecutive trials, and
plotted mean vergence angle
(
,
averaged over 1 trial) over the 300 trials. For each group of 300 trials, we
calculated the Fourier spectrum of this between-trials vergence variation,
normalized to unit power. Figure
2A shows the average value of this spectrum, averaged
over the 10 groups. The spectra of the three vergence measures are remarkably
similar. To look at high frequencies, we looked at fluctuations within each
415-ms trial (L – R1, L – R2, R1 – R2).
Figure 2B shows the
Fourier spectrum of these instantaneous vergences, averaged over all 3,121
trials. The three spectra are indistinguishable.
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If coils L and R2 are assumed to be veridical, this means that the error on coil R1 just happens to have exactly the same frequency profile as the animal's vergence movements. This seems an improbable coincidence. We suggest, rather, that our monkeys, like human subjects, have vergence errors close to zero and that the "vergence" measure obtained from the coils is dominated by artifacts on the coil signals. Under this interpretation, the errors on the coils have different magnitude (coil R1 is subject to a larger error than R2), but the Fourier spectra are similar because they all reflect the same underlying process.
In fact, the Fourier spectra at high frequencies are strikingly close to
that of a simple one-dimensional random walk, whose value is either
incremented or decremented at every time step, with equal probability. This
random walk has too much power at lower frequencies, but simply by making the
probability of stepping back toward the origin increase with distance from the
origin, (see legend to Fig. 2)
a much better match is obtained. This rough model provides a surprisingly good
match to the experimental Fourier spectra over frequencies from 330 Hz down to
0.008 Hz. At frequencies >0.008 Hz, the Fourier spectrum of the
experimental vergence measures has more power than the model, continuing to
rise down to the lowest frequencies measured (0.0008 Hz, not shown). It
is nonetheless striking that such a simple model explains the observed Fourier
spectra over a very wide frequency range. Importantly, the same deviations
from the model are seen in all three coil difference signals. The failure of
the simple model at very low frequencies in no way undermines the conclusion
that all three difference signals are generated by a similar process. This in
turn suggests that none of them has much power that reflects real changes in
vergence, which implies that real vergence changes are small relative to the
size of the artifact.
Analysis of variability
The second approach exploited the variability of neuronal spike counts. If
there are real changes in vergence during the fixation task, then this should
influence the firing rate of disparity-selective neurons. However, this will
only be the case for stimulus disparities at which the neuron is sensitive to
small changes. During presentation of an uncorrelated dynamic RDS, change in
vergence angle should have no effect. Similarly, at the flanks of the
disparity tuning curve, where small changes in disparity have no effect on
activity, the variability in firing rate will not be influenced by vergence
fluctuations. Figure 3 shows
the relationship between disparity tuning and spike count variability (the
variance:mean ratio, VMR) for two narrowly tuned neurons. It is clear that the
variability is highest where the rate of change of spike count with respect to
disparity is greatest. This indicates that there are changes in vergence, and
the neuronal activity is determined by the resultant disparity changes. It is
possible to use data like these to estimate the variability of vergence, if
one assumes that the ratio (spike count variance)/(spike count mean), measured
across repeated presentations of the same stimulus, is a constant for each
neuron. This constant, k, can be estimated from the VMR for those
points on the tuning curve that are insensitive to disparity changes (we used
a slope of >10 spikes/s per degree of disparity). At each disparity, the
total variance of the count, 
, is
approximated by the sum of a term related to the mean firing rate, and a term
caused by vergence fluctuations (with an SD 
v)
 | (1) |
v is clearly most reliable when slope is high.
Indeed, when the slope is low enough that
becomes
small relative to k.mean, sampling variation can give rise to points
where 
, in which case
the vergence SD cannot be estimated.
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Restricting this analysis to data points associated with a high rate of
change of firing rate with respect to disparity therefore yields the most
reliable estimates of v, but limits the size of the
dataset. If only slopes >400 spikes ·
s–1 · °–1
of disparity are included in the analysis, then 6/114 disparity tuned neurons
yielded an estimate of v. For each neuron,
v was estimated at each point exceeding this slope,
and the mean v calculated. The mean of these across
the 6 neurons was 0.0111°, and they were in close agreement with one
another, with an SD of only 0.00059°.
Two conclusions can be drawn from the preceding analysis. First, it appears
that vergence fluctuations during fixation can alter neuronal response rates,
in a few neurons that are exquisitely sensitive to small changes in disparity.
Second, the additional variance this adds suggests that the SD of the vergence
angle in these two monkeys is 
1 min arc, similar to the estimates in human
subjects (see INTRODUCTION). This is substantially smaller that the
vergence SD simultaneously measured with scleral search coils, suggesting that
most of the measured vergence variation is artifactual.
Monocular responses
The preceding section used binocular eye-position measures to produce
evidence for some slow drifts in eye coil signals that do not reflect real
changes in vergence. That analysis was somewhat simplified by the fact that
real fluctuations in vergence angle seemed to be very small. In this section,
we evaluate the consequences of these drifts for monocular measures of RF
size. Because it is clear that real conjugate eye movements occur during
fixation (many aspects of microsaccades suggest that they at least are not
artifacts of the eye coil), the utility of eye-position signals will depend on
the relative size of real and artifactual variation in eye position.
Figure 4 shows data used to
estimate the size of one V1 receptive field. As in
Fig. 3, it is clear that the
spike count variability is greatest at locations where the response is most
sensitive to position changes. However, it is difficult to estimate the
underlying variation in eye position from such data as there is no point on
the tuning curve where it is possible to estimate a baseline VMR—the
only flat portions of the curve have zero firing rate. To quantify the
phenomenon shown in Fig. 4 across the population, we examined the correlation between the VMR and the
slope of the least-squares fitted Gaussian function (rearranging Eq.
1 shows that slope2/mean should be correlated with VMR if the
variance of eye position is substantial). The term slope2/ mean is
poorly defined when the response rate is low, so this analysis was restricted
to points which produced a mean spike count >5, and at which there were
6 repetitions. For 38/57 neurons, there were 4 points meeting this
criterion, and the mean value of the product-moment correlation coefficient
was >r > = 0.392 ± 0.067 (SE) (significantly different
from 0, P > 10–6, t-test).
Applying the same analysis unselectively to all data still yields a
significantly positive correlation coefficient (0.241 ± 0.043,
P > 10–6).
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Thus across the population of neurons, there is a systematic relationship
between the variability of the spike discharge and sensitivity to stimulus
position. This analysis, which makes no use of eye-position signals, supports
the view that the RF is fixed in retinal coordinates. Because there is
significant variation in eye position, it might be possible to improve
estimates of RF size, using eye-position signals from the coil to correct for
eye movements. Figure 5 shows
the results of doing this for two neurons (least-squares fit). In one case
(ruf128), applying the correction has clearly improved the RF
estimate—the amplitude is larger, and the SD smaller, after correction.
However, in the other example (duf217) the opposite is the case. This
indicates that the "correction" has failed, since any jitter in
the monkey's eye position must always tend to smear out the observed receptive
field. If the SD of the eye jitter, e, is small compared
with the size of the RF, RF, then the eye jitter has
negligible effect and the observed value of the RF, obs,
fitted without correcting for eye position, will closely approximate
RF. In the other extreme, e >>
RF, the apparent extent of the RF is almost all an artifact
of eye movements: obs 
e. We expect
that in general the relationship between these three quantities is
approximated by
 | (2) |
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Thus we expect that
 | (3) |
The left-hand side of this equation is approximately the fractional change in RF SD caused by correcting for eye movements; the approximation becomes exact in the limit of small fractional changes. Thus the equation states that correcting for eye movements always produces a decrease in RF SD and that the decrease is larger when the eye movements are large compared with the true RF size.
This prediction is tested across the population of neurons in
Fig. 6. In one cell, applying
the coil correction resulted in a scatter of points that could not be
adequately fit by a Gaussian; this cell is therefore omitted from this
discussion. For the remaining neurons, let corr denote the
SD of the corrected RF. In Fig.
6A, the "fractional change"
is plotted against the ratio e:corr. If
the coil signal is veridical, then fitting to the corrected values should
yield the true SD of the underlying RF: corr =
RF; - - - plots the prediction (Eq. 3). The
agreement is very poor: there is little evidence that correction produces a
larger fractional change in RF width for smaller RFs, as one would have
expected. More seriously still, in half of the cells the
"correction" has actually yielded a larger SD (fractional change
positive, 30/56 cells), as for the example in
Fig. 5B. This should
be impossible if the coil signal were veridical. On average applying the
correction has no effect (geometric mean:
corr/obs = 1.00, P = 0.97,
t-test on log ratios). Figure
6B plots the analogous quantities for the fitted RF
amplitude. Once again, in half the cells (26/56), the correction has yielded a
lower amplitude, the opposite of what should happen if eye-position
compensation was effective.
Although the stimuli presented in these experiments were monocular, the positions of both eyes were recorded. The eye-position correction applied in the preceding text used the recorded position of the eye to which the stimulus was presented. Comparing this result with the effects of applying a correction based on the recorded position of the nonstimulated eye affords another way of detecting vergence fluctuations. Such fluctuations should make the recorded position of the nonstimulated eye a less reliable measure of stimulus location in the stimulated eye. Conversely, if vergence fluctuations are small, applying a correction based on the nonstimulated eye should yield fits very similar to those based on the position of the stimulated eye. This second pattern was observed—the geometric mean ratios of the two corrections (nonstimulated to stimulated) were 1.04 for SD and 0.96 for amplitude, neither significantly different from unity (t-test on log ratios). This further strengthens the conclusion that real fluctuations in vergence angle are much smaller than the scleral search coil measures indicate.
Thus correcting for eye position has little overall influence on the
estimated RF parameters. This might be taken to suggest that the RF is fixed
in spatial coordinates rather than retinal coordinates. However, if this was
the case, applying a correction based on eye position should systematically
make the RF appear larger, which is not observed. This, combined with the
analysis of spike count variability, can best be explained by supposing that
there is both real variation in eye position and artifactual fluctuations in
the signals, which are of approximately equal magnitude. To quantify this, let
us assume that the coil signal is affected by an artifact whose SD is
n. Then
 | (4) |
coil is the SD of the coil record. Correcting for eye
position on the basis of this inaccurate signal might then be expected to
yield
 | (5) |
Combining Eqs. 2, 4, and 5, we can deduce the SDs of eye
position, of the coil artifact, and of the RF
 | (6) |
Of course, the assumptions leading to these expressions may not be
satisfied exactly. In 15/56 cells, one or more of these variances comes out
negative, indicating a failure of the method. In the remaining 41, the means
of n and e came out to be 0.111
and 0.109 respectively, confirming the preceding indications that the artifact
on the coil is approximately equal to the real variation of eye position
during fixation.
Equation 6 is based on a rather informal argument, and this method
fails for a quarter of cells, suggesting that the assumptions (Eqs. 2,
4 and 5) are not accurate. Clearly, a more reliable fitting
technique would be desirable. Several lines of evidence suggest that
inaccuracies in the coil take the form of a slow drift (discussed in the
following text). The coil may then reliably report the difference in
eye position between two successive trials, 
c =
cj –
cj–1, even though
across several trials enough errors accumulate that the coil cannot be used to
track eye position throughout an experiment. In the following text, we shall
describe how these difference signals can be used to constrain the fitted RF
parameters. First, we discuss the evidence which leads us to conclude that
these differences are veridical.
Estimating receptive fields in the presence of eye movements
If the actual eye position and the error on the coil really were both
independent identically distributed random variables, then the difference
c would have an SD 
times larger than that of
the coil record c itself. Thus on average we would expect
. In fact, this
ratio is >1 in every one of our 51 recording sessions (geometric mean =
0.684, P > 10-6, t-test on log ratios). Thus
the differences between successive coil positions change more slowly than
expected for a white-noise process.
A similar result is found when we compare the signals from left and right
coils, cL, cR. The mean correlation
coefficient between cL and cR is
r
= 0.730 ± 0.037 (SE) (n = 51 recording
sessions). When we consider the differences between the coil signal
on successive trials, cL and
cR, the correlation improves in almost every case
(rdiff > r for 45/51 sessions, P >
10–6, binomial), and the mean value goes up to
rdiff = 0.838 (significantly different from
r, P = 0.035, 2-sample t-test).
These observations indicate that either the real eye position or the coil
artifact is correlated between successive trials. We can determine which by
comparing results from the two coils implanted in the right eye of monkey
Duf. The discrepancy between these coil signals, v =
cR2 – cR1, is entirely
artifactual, enabling us to focus on the correlations in the artifact. Once
again, the SD of the difference between consecutive trials,
(cR2 – cR1), is
smaller than expected on the basis of independent identically distributed
normal random variables. We compared the differences in the horizontal
component of the vergence (v) between consecutive trials
(v) with the differences between pairs of trials picked at
random (vrnd). If v were a white-noise
process, these would be the same. In fact, there was a highly significant
difference between the two. SD(vrnd) = 0.195, while
SD(v) is only 0.052 (1,457 pairs of trials, P >
0.001, resampling). This demonstrates that the coil artifact on successive
trials is correlated.
The Fourier analysis of the coil vergence measures, which we suggested
represent mainly the coil artifact, supports this conclusion. The Fourier
power spectrum has most of its power at low frequencies:
Fig. 2B shows that 65%
of the power is at less than the stimulus presentation rate of 2 Hz. Thus
we expect significant correlations between the coil error on two consecutive
trials.
All these lines of evidence lead us to conclude that the coil reports the
difference in eye position between two consecutive trials much more
accurately than it reports the position itself on either trial. So, even if we
cannot trust the coil over long periods of time, we can use the information it
provides over short time scales to provide a powerful new constraint on the
possible receptive fields. Figure
7 provides an intuitive picture of why this is so. The three
panels each show a hypothetical pair of trials: the 
represent the
response of the neuron to a stimulus at a given screen location. Two putative
Gaussian fits are shown (one plotted with — and one with ·
· ·). Given only pairs 1 and 2, it is possible to rule out
· · · because it is incompatible with the data in pair 1.
Given only pairs 2 and 3, both fits appear satisfactory. However, reconciling
these data with · · · requires us to postulate a large
change in eye position (hence the shift in the peak of the dotted Gaussian
between pair 2 and pair 3). Considering pairs 2 and 3 simultaneously, combined
with the assumption that large eye movements are less likely than small ones,
we can conclude that — is a more likely explanation of the data than
· · ·. This argument, repeated for scores of pairs of
data for each cell, underpins our fitting technique (described in detail in
the APPENDIX).
|
The preceding explanation ignored the possibility that eye movements might
occur between the two members of a single pair. This is where the coil is
critical. Having demonstrated that the coil probably reliably tells us
differences in eye position between two trials, we can use the
measured coil difference to correct for this. The mean eye position across the
pair of trials is unknown, but in assessing the likelihood of the data given
the currently postulated RF parameters, we integrate over all possible mean
eye positions weighted by their a priori probability. The coil record suggests
that mean eye position over a trial is normally distributed about the fixation
point, so we modeled this probability as a Gaussian, with SD e. The
value of e, plus the RF parameters, are free parameters in
our fit: we select the values which lead to the highest likelihood of the
observed data (maximum likelihood estimation, MLE). Simulations suggest that
this approach can recover the correct parameters to within 5% (the
precise value depending on the number of trials, etc.).
Figure 8 shows the result of this new fitting procedure ("MLE-pairs", —) for one cell, duf218. The results of the least-squares fit used so far in this paper (- - -) were shown, ignoring the problem of eye movements. The MLE-pairs method suggests a narrower RF of larger amplitude, exactly as one would expect if it has correctly taken into account the effects of eye movements. However, by itself this does not demonstrate that the method has correctly identified the underlying RF. Both methods give a good account of the mean firing rate as a function of position in the visual field, and many combinations of RF size and assumed eye-position variation could explain these data. Importantly, the combinations differ greatly in the variability in neuronal firing that they predict. According to the MLE-pairs fit, variation in eye position should add considerable variability to the spike counts in a way that depends on the stimulus location. Figure 8C compares the predicted and observed VMR. The close agreement is particularly striking because the MLE-pairs method does not explicitly fit these VMRs. Rather, the assumed combination of RF and eye movement that best explains the changes in firing rate between consecutive trials independently explains the observed VMR for each stimulus considered individually.
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Figure 9 examines how well
the MLE-pairs fit explains the observed pattern of VMR across the population.
For each cell, we calculated the correlation coefficient between the observed
and predicted VMRs. This correlation was positive in 51/57 cells (P
> 10–9, binomial distribution); the mean was
r = 0.347 (significantly different from 0, P
> 10–6, t-test). The cell shown in
Fig. 8C is one of
15/57 in which this correlation was significant at the 5% level (it has
r = 0.73). Given the noise associated with estimating variance, it is
not surprising that the correlations are often small. The strong bias toward
positive values is striking given that the model used in the fitting assumes
that the variance of neuronal firing is proportional to the mean, so that, if
it were not for eye movements, the expected correlation coefficient would be
zero.
|
Figure 10 summarizes the
MLE-pair estimates of RF size across the population, in the same way as
Fig. 6 summarized the results
of correcting for eye position using the coil. Once again, - - - shows the
prediction from Eq. 3, where now e and
RF are obtained from the MLE-pair fit. The agreement is much
improved compared with Fig. 6.
Whereas the "fractional changes" in
Fig. 6 were scattered equally
on either side of zero, now the change in SD is almost always negative, while
that in amplitude is almost always positive (RF >
obs for 51/57 cells, ARF >
Aobs for 51/57 cells, P >
10–6, binomial distribution; geometric mean
RF:obs = 0.797,
ARF:Aobs =1.35, both P >
10–6, t-test on log ratios). There is
also a much clearer tendency for the changes to be larger where the jitter in
eye position is large relative to the RF. Thus our MLE method incorporating
eye jitter does result in systematically narrower SDs and larger amplitudes as
expected if eye movements have been correctly accounted for and in contrast to
the results of treating the coil as veridical. In addition, the agreement with
the observed VMRs (Fig. 9) is
powerful independent evidence that the fits are correctly estimating the
extent of eye movements and the underlying RF size. However, before concluding
that these estimates are correct, a number of properties are examined below to
check their reliability.
|
Checks and validation
FITTED SD OF EYE MOVEMENTS IS CONSISTENT WITH THE COIL. In the
fitting procedure, we optimize to find the most likely SD of eye movements,
e. But if our assumptions are correct, and the coil
accurately reports the difference in eye position between two consecutive
trials, then we should be able to find e by considering the
distribution of those differences, c: we expect that
.
Figure 11 compares this
estimate of e to the value obtained by the MLE fit, for each
of our 57 cells. The two estimates are highly correlated (r = 0.89).
This is further evidence supporting both our assumptions about the nature of
the coil artifact and the success of the fitting procedure.
|
It is noticeable that the MLE estimate of e tends to be
slightly larger than expected from the coil differences. This is readily
explained by assuming that the real value of eye position tend to be
correlated from one 415-ms trial to the next. Under these circumstances,
will underestimate the true
variation in eye position. Note that the suggestion that eye position is
correlated from one trial to the next does not undermine our arguments that
the drift artifact is also correlated from trial to trial. It is quite
possible that both are correlated (although the correlation appears to be
stronger for the artifact). RESULTS ARE NOT SENSITIVE TO SMALL EYE
MOVEMENTS DURING A TRIAL. The analysis presented in the
APPENDIX implicitly assumes that the eye position remains constant
throughout a trial. However, although the monkey was fixating, there may be
microsaccades or significant drift in eye position during a trial. Our
analysis is still valid, providing the mean spike count reflects the mean eye
position during the trial. As a check that our results are not sensitive to
eye movements that occur during a trial, we repeated our analysis, this time
excluding all trials on which the SD of coil-measured eye position in the
stimulated eye exceeded 0.05° over the course of the trial. The consequent
reduction in the number of data points is exacerbated because, in our MLE-pair
method, we restrict ourselves to considering consecutive pairs of trials. Thus
for example, removing trials 30 and 32 due to excessive eye movement also
results in the loss of trial 31. Five cells could not be analyzed because
fewer than 10 pairs survived this culling. For the remaining 52 cells, the RF
parameters were not significantly changed (for fitted RF SD,
RF, the correlation between the 2 sets of results is 0.953;
geometric mean ratio is 0.967, no significant difference, t-test on
log ratios (P = 0.09). For fitted RF amplitude,
ARF, these figures are 0.97, 1.01, P = 0.7).
However, there was a small but significant reduction in the fitted SD of eye
jitter, e, by an average of 0.035° (correlation:
0.81).
RESULTS ARE NOT SENSITIVE TO THE EXACT MODEL OF SPIKE COUNTS.
The results presented so far were based on a Poisson model of spike counts.
This captures some aspects of our spike count statistics, such as the constant
VMR, but is clearly oversimplistic. It is possible that the statistics we have
assumed influenced our fit. The VMR of a Poisson process is exactly 1, whereas
the VMRs observed in our spike counts are often higher by an order of
magnitude. Our fits reproduce these large VMRs by postulating eye movements
(Figs. 8 and
9). However, if our spike count
model had larger intrinsic VMR, the MLE fit might produce smaller estimates of
eye jitter. To find out whether our results were sensitive to the precise
spike count model, we reran our fits using a bursty-Poisson model with VMR = 2
(see APPENDIX for details), a figure which is larger than most
estimates for the VMR in cortical neurons
(Britten et al. 1993;
Dean 1981;
Geisler and Albrecht 1997).
This analysis used only trials where the SD of the coil measure over a single
trial was >0.05°, as in the previous section. The results were
extremely similar. The correlation between fitted RF SD for the 52 cells was
0.986; for fitted amplitude, 0.995; for e, 0.969; the
gradient of fitted regression lines did not differ significantly from 1 for
any of these. There was some suggestion that the bursty Poisson model resulted
in a marginally larger fitted RF SD, a smaller RF amplitude and a smaller
e, as expected from a model allowing more variance in the
neuronal firing. However, these changes are extremely small (1
spike/trial for RF amplitude, 0.01° for the SDs), demonstrating that our
results are not unduly sensitive to the precise assumptions concerning
neuronal firing. Similarly, simulations indicate that the results are not
unduly sensitive to the exact shape of the RF. Even if the RF is not a
Gaussian, if we define the best-fitting Gaussian to be that which would have
been obtained by a least-squares fit if the eyes had been still, the MLE
technique returns a good approximation to this best fit even when the
simulated data have been contaminated by eye movements.
RESULTS ARE NOT SENSITIVE TO ADAPTATION. Neurons often show some response adaptation, which is a potential problem for our MLE-pair analysis. If the first stimulus in the pair elicited a particularly large response, the response to the second stimulus might be smaller than average, and this is not taken into account in our MLE fitting. However, it is not clear that response adaptation would cause any systematic errors as stimuli were presented in a random order. On any particular trial, the response is as likely to have been enhanced as suppressed by such a mechanism. The effect of adaptation is therefore primarily to increase the variance in neuronal firing, and we showed in the previous section that our results are not excessively sensitive to the precise assumptions about the variance of neuronal firing. Nevertheless, we verified explicitly that our fitting procedure is still valid given realistic amounts of adaptation.
One advantage of our short stimulus presentations (415 ms, with a gap of
100 ms between stimuli) is that strong adaptation is avoided. We examined
the extent of adaptation in these data with a median-split analysis. We
grouped pairs of trials according to the stimulus position for the second
member of the pair and calculated the median value of the spikes elicited by
the first member of the pair. We then calculated the mean of the square-root
spike-count elicited by the second member, given that the first member of the
pair had been greater/less than this median (m> and
m, respectively). Across the population of cells,
there was no tendency for the mean sqrt spike count to be smaller where the
preceding trial had evoked a larger than median spike count, suggesting that
little adaptation occurs (means could be evaluated for 562 stimulus positions
in 57 neurons; m> > m in
288/562 cases, P = 0.6, binomial; population average
>m> – m> =
–0.072 ± 0.040 SE, not significantly different from 0, P
= 0.07, t-test).
Even so, to evaluate the effect of any adaptation, we ran our MLE-fitting
method on simulated data incorporating a simple model of adaptation. Whereas
our MLE fit assumes that the spike count simply reflects the retinal position
of the stimulus, our simulation reduced the elicited spikes by a
"gain" factor depending on the number of spikes produced on the
previous trial, Nprev: the gain was 1 if no spikes had
been fired on the previous trial, and decreased linearly to zero with
Nprev. We set this gain so as to obtain adaptation clearly
stronger than that in most real cells (m> –
m = –0.28). Our fitting procedure still
extracted the correct parameters to within the same accuracy (5%) as when
no adaptation was included in the simulation.
SIMILAR FITS ARE PRODUCED WITHOUT USING COIL DATA AT ALL. We developed a second MLE method that makes no use of the coil at all (APPENDIX, Eq. A6). It simply looks at each individual spike count and considers how likely that spike count was given the postulated RF parameters and the postulated SD of eye jitter. Obviously, because this approach uses less information, it is not so well constrained. However, it has the advantage of being free of any assumptions concerning the coil.
The results are, again, extremely similar. The fitted amplitudes were marginally larger with the second MLE method (geometric mean ratio = 1.03, P = 0.02, t-test on log ratios), but the correlation between them was 0.99. There was no significant difference in either the SD of the RF or the SD of eye movements (correlations: 0.96 and 0.56, respectively, both highly significant, P > 10–5).
Estimates of eye position
An extension of our MLE-pair fitting also allows us to obtain an estimate
of the eye position on individual trials (details in APPENDIX).
After fitting the cell's entire dataset to obtain the RF parameters and the SD
of eye jitter, e, we then run individual MLEs for each
stimulus presentation in the cell's dataset, in which we estimate the most
likely eye position on that trial, given the RF parameters and eye jitter
previously obtained.
These estimates of eye position are considerably more noisy than the
estimates of RF parameters and eye jitter. There, five parameters are fitted,
using hundreds of observations (on average 270 per cell). Here, a single
parameter (the mean eye position across a pair of trials) is fitted to a pair
of observations. When the observations do not tightly constrain the mean eye
position, the a priori assumption that eye position is normally distributed
about zero makes the MLE fit choose an eye position close to zero. Thus the
MLE fit for eye position shows a slight bias toward zero, which emerges when
we consider the population of fitted eye positions for a particular
experiment. This is expected to be normally distributed with mean 0 and SD
e, where e for this
experiment has been obtained, along with the RF parameters, by the initial
MLE-pair fit. In fact, over all 57 cells, the SD of the fitted eye positions
is systematically about 0.014° less than e, reflecting
the bias toward zero.
Despite this, the fitted eye positions turn out to be well correlated with the values reported by the coil, even though these values were not used during the fit. Figure 12A shows this correlation for the cell examined in Fig. 8 (duf218). The shallow gradient, which was reflected across the population, again probably reflects the bias of the MLE fit toward small eye positions. The population mean of the correlation between the coil record and the fitted eye position over our 57 cells was 0.661, ranging from 0.28 to 0.90. This correlation was significant at the 5% level in all 57 cells, and at the 0.1% level in 54/57. This level of agreement, despite the bias in fitted eye positions and the artifact on the coil, is further evidence of the success of our MLE approach.
|
Duf218 is a particularly interesting cell because spikes from a second cell, duf218 2, were recorded at the same time. Figure 12B compares the results of the fitted eye positions for the two cells. There is a significant correlation. This certainly shows that the estimate of eye position is not simply noise, although of course because estimates on both cells are subject to the same bias, it does not allow us to assess the absolute accuracy of the fit. We had six such pairs of simultaneously recorded cells. The mean correlation coefficient between the two estimates of eye position was 0.36, and the correlation was significant at the 1% level in 5/6 cell pairs.
Coil artifact
By a heuristic argument in the preceding text (Eq. 6), we arrived
at estimates for the SD of the coil artifact and of eye movements
e, which were both 0.11°. We can now compare these
with the estimates from the MLE fits. The mean fitted e is
0.128° (ranging from 0.041 to 0.246° with an SD of 0.045°).
Figure 13 shows the SD of the
coil record, SD(c), against the fitted SD of eye movements,
e. The coil record has larger SD for all but 6/57 cases. For
51 cells, therefore we can estimate the coil drift n by
assuming that var(c) = e2 +
n2. The mean is 0.0952°, ranging from 0.0220
to 0.220° with an SD of 0.0467°.
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As an alternative estimate, if we assume that our MLE technique has
successfully recovered the eye positions on each trial, then subtracting the
fitted eye position from the coil record gives us a trial-by-trial estimate of
the drift on the coil, which works for all 57 cells. The population average SD
of the deduced coil drift is 0.116° (ranging from 0.052 to 0.217° with
an SD of 0.042°). All three different estimates are in good agreement.
They suggest that the drift on scleral search coils is of approximately the
same magnitude as fixational eye movements, with an SD of 0.1°.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MAXIMUM LIKELIHOOD...DISCLOSURESACKNOWLEDGMENTSREFERENCES |
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), a
series of studies by Gur and Snodderly
(Gur and Snodderly 1997;
Gur et al. 1997) provided
compelling evidence to the contrary. Our demonstration of a relationship
between spike count variability and stimulus location further reinforces the
view that RFs are fixed in retinal coordinates. When the stimulus was
presented at a location where the RF was most sensitive to small
displacements, spike count variability was systematically larger than at less
sensitive locations. Importantly, this demonstrates the effect of retinally
fixed RFs without relying on measures of eye position at all.
Because RFs are fixed on the retina, changes in eye position will interfere
with the estimation of RF size and shape. Some studies have attempted to
estimate the retinal location of each stimulus simply by combining knowledge
of the spatial location with measures of eye position
(Conway 2001;
Livingstone 1998;
Livingstone and Tsao 1999;
Livingstone et al. 1996). This
places high reliance on the accuracy of these records over extended periods,
yet no one to our knowledge has demonstrated such accuracy with any system.
Furthermore, none of the studies that have applied eye-position correction to
stimulus locations have evaluated the effectiveness of the procedure (e.g., by
comparing data with and without correction). We find that applying
eye-position correction did not in general produce smaller RF estimates. Taken
together, these observations suggest that a substantial fraction of the
measured variation in eye position is artifactual. This raises serious
problems of interpretation for those studies that have applied eye-position
correction without evidence of its effectiveness.
It is unclear what the source of these artifacts is. It does not appear to be a property of the electronics—we found that signals from calibration coils were stable to within very narrow limits over many hours. This was true even when a calibration coil was placed next to a working animal, so it is unlikely that the artifact is due to alterations in the field induced by postural changes. One important factor may be that coils implanted in animals are not rigidly mounted. Mechanical distortion of the coil or of the connecting wire, due to movements of the eyelids, brow, or temporal muscles, could give rise to changes in recorded position in the absence of an eye movement. Indeed, the fact that blinks are associated with substantial transients in the signal from search coils seems clear evidence that changes in lid tension can introduce artifacts in the position signal. To what extent these artifacts are reduced if the coils are sutured to the globe will require further investigation.
The fact that scleral search coil measures contain substantial artifactual
variation is particularly problematic for binocular measures. A number of
human studies have reported that the vergence angle between the eyes is much
less variable than conjugate eye position during fixation. If the same holds
for monkeys, it would mean that the real variation in vergence would be
substantially smaller than this artifact and hence very hard to estimate.
Several of our observations indicate that this is indeed the case. 1)
When monocular RF sizes were estimated with eye-position correction, the
results were very similar whether the eye-position measure used was that of
the stimulated eye or the nonstimulated eye. 2) The analysis of spike
count variability in disparity selective neurons showed that the variability
increases only in regions of extremely steep disparity tuning, suggesting a
vergence SD of 1 min arc. This is in good agreement with values from human
studies. 3) The measured SD of vergence is similar in magnitude to
our estimates of the artifactual coil drift. And 4) the differences
between two coils implanted in one eye were similar to those for vergence,
both in absolute magnitude and in their temporal structure (they had nearly
identical Fourier amplitude spectra). We conclude that binocular coils
overestimate vergence variability. In fact, very few studies of binocular
neurons have included quantitative analysis of vergence variability
(Cumming 2002;
Prince et al. 2002a). The
results presented here suggest that vergence variability poses even less of a
problem than such measures indicated.
Note that the coil artifact does not render the measurement of vergence
altogether invalid. Reliable measures of vergence velocity are still possible
(Busettini et al. 1996,
2001). Furthermore, if two
conditions are interleaved, vergence measures can still reliably detect any
systematic change between them (Cumming and
Parker 1999; Thomas et al.
2002). Any systematic changes in vergence with the stimulus
condition [e.g., changes in fixation distance (Trotter et al.
1992,
1996)] might influence the
response of binocular neurons. Without vergence measures, these changes might
be interpreted as a property of the neuron itself.
That there is both real and artifactual variation in the eye-position records from fixating monkeys poses a significant problem for estimating RF size at least for studies of foveal V1. This in turn makes it difficult to be confident that any stimulus is confined to within a single RF or that a stimulus that elicits some contextual modulation really remains outside the receptive field. However, our data suggest that the artifactual component is correlated across successive trials, and the influence on RF measures can be substantially reduced by considering the differences in reported eye position between consecutive trials. We therefore developed a new method of analysis, which examines pairs of stimuli and uses the two firing rates, in conjunction with the change in stimulus location, to estimate RF size in a way which is not disrupted by changes in eye position over time. A number of checks suggest that this approach can successfully estimate the RF parameters and amount of eye jitter, that it is not disrupted by small eye movements which occur during a trial or by errors remaining in the coil difference signal, and that it is not unduly sensitive to the precise model assumptions. This technique yielded systematically smaller RF estimates than methods that either ignore eye movements or assume the coil is veridical, it explained the observed variability of neuronal discharge, and it was able to match estimates of eye-position variability derived from the coil in a way that was not sensitive to slow drifts. Thus in every way we have examined, it behaves as if it has correctly separated the underlying RF and the influence of eye movements on the neuronal response.
This is important not only because it allows estimation of RF size in V1 of the awake monkey, but also because it provides a method by which the absolute accuracy of an eye-position recording technique can be assessed. If, for example, suturing the coil to the globe reduces these artifactual drifts, this can now be demonstrated using our MLE-pair technique. Additionally, we have argued in the preceding text that actual variations in vergence are very small so that measurements of vergence effectively give a direct estimate of the artifact. Thus if some eye-position recording technique produced vergence SDs of a few minutes of arc and the MLE-pair method indicated a small artifact, this would show beyond reasonable doubt that the measure of eye position was accurate.
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APPENDIX: MAXIMUM LIKELIHOOD ESTIMATION OF RECEPTIVE FIELD PARAMETERS AND EYE MOVEMENTS |
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 | (A1) |
|
In talking of the position of the eye during a trial, we are ignoring the possibility that eye movements occur within a trial. More realistically, e represents the average eye position over a trial, and we are assuming that the expected spike count, n, reflects the average retinal position r. (In the text, we present evidence that this assumption is sufficiently accurate.)
The RF is assumed to be Gaussian, i.e.
 | (A2) |
is the RF SD, c is
the center of the receptive field in retinal coordinates, and B is
the baseline spike count. Note that the only assumption we make about the
response of these neurons is that it is a Gaussian function of position. Our
analysis holds irrespective of any nonlinearities which may contribute to this
response.
Usually, the spike counts were assumed to be Poisson, so that the
probability of recording N spikes during a trial is
 | (A3) |
We also investigated the effects of modeling the spike counts as a bursty
Poisson. Here, bursts are generated as a Poisson process, and each burst
consists of m spikes, where m in turn has a Poisson
distribution. The spike count model thus has two free parameters: the mean
number of bursts per trial, n, and the mean number of spikes per
burst, m. The mean spike count per trial is 
and the
variance:mean ratio is 1 + m. We set m = 1 to obtain a VMR
of 2. In this case, the probability of recording N spikes during a
trial was obtained by numerical inversion of the generating function.
If retinal position was known
If either the scleral search coil or the animal's fixation were known to be
flawless, we would know the retinal position r for each stimulus.
Then for a particular set of RF parameters we can deduce the expected spike
count n(r) (Eq. A2), and hence the probability of observing
a particular number of spikes N. The likelihood L of the
entire data set is therefore, assuming results from different trials
j are independent
 | (A4) |
. We would estimate these parameters by adjusting
them so as to maximize this likelihood. Using information from the coil (MLE-pair)
In practice, fixation is not perfect and scleral search coils appear to be subject to an artifact. However, we can still obtain a well-constrained fit if we assume that the scleral search coil accurately reports changes in eye position over a sub-second timescale, although the actual values may be subject to a slow drift which means they cannot be relied on over the course of an experiment. (Evidence supporting this assumption is discussed in the text.)
Rather than using data-points individually, we therefore use them in pairs.
We assess the probability of recording N1 counts for a
stimulus at screen position s1, followed by
N2 counts for a stimulus at s2. From
the coil, we know the change in eye position, e =
e2 – e1, so we simply need to
integrate over all possibilities for the mean eye position averaged over both
trials, e = (e1 + e2)/2. This
is distributed normally with SD 
, so
 |
Re-expressing our data as a set of consecutive pairs of trials, the
likelihood of the dataset is
 | (A5) |
We fit the whole data-set for the RF parameters and for
e.
It is also of interest to obtain an estimate of the eye position on each
trial. In practice, we do this by obtaining an estimate of the average eye
position e in each pair of trials (because the difference
e is known, this tells us the eye position for each trial).
For the jth pair of trials, we seek ej
that maximizes
 |
We then obtain the eye position on the first and second members of the pair
 |
In expressing our data as a set of consecutive pairs of trials, the same trial often occurs as a member of two pairs. Strictly, this invalidates Eq. A5, because this assumes that all the pairs are independent. We ignored this complication. For the duplicated trials, we took the final fitted eye position to be the mean of the two estimates e1j and e2,j–1.
To derive the expected spike count for comparison with experimental
observations, we have to incorporate the differences e that
were used in fitting. The easiest way to do this is by simulation. For each
pair, we pick e randomly from a normal distribution with SD
. Given e from the
coil, this specifies e1 and e2 and
hence the position of the stimuli on the retina. The RF function specifies the
expected number of spikes on each trial, and the actual number was drawn from
a Poisson distribution with this mean. Repeating this many thousands of times
for every pair used in fitting allows us to determine the expected spike count
distribution, and the VMR, for each stimulus screen position used.
Without using the coil
We also developed a method of estimating RF parameters in the presence of
eye movements that makes no use of data from the search coil. Here, we simply
seek to maximize the likelihood of obtaining the observed spike counts as a
function of position without including any constraints from the coil. To
estimate the probability of observing a particular spike count, N,
given a stimulus screen position s, we must integrate over all
possible eye positions e, weighted by their probability of occurring.
As before, we assume that the distribution of eye positions, along the axis
that stimulus position is varying, is Gaussian with SD e,
where e is an additional parameter to be fit. Then
 | (A6) |
,e) is
given by the product over all trials j. |
DISCLOSURES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MAXIMUM LIKELIHOOD...DISCLOSURESACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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A recent study of V1 receptive fields in awake monkeys, using sutured
search coils, has also found that correcting for measured eye position does
not improve RF maps (Tsao and Livingstone,
Neuron 38: 103–114, 2003; data very similar to our
Figure 6). This suggests that
suturing coils to the globe does not, in fact, reduce the artifacts.
Address for reprint requests: J.C.A. Read, Laboratory of Sensorimotor Research, National Eye Institute, National Institutes of Health, Bldg. 49/Room 2A50, 49 Convent Dr., Bethesda, MD 20892-4435 (E-mail: jcr{at}lsr.nei.nih.gov).
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MAXIMUM LIKELIHOOD...DISCLOSURESACKNOWLEDGMENTSREFERENCES |
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Busettini C,
Fitzgibbon EJ, and Miles FA. Short-latency disparity vergence in humans.
J Neurophysiol 85:
1129–1152, 2001.
Busettini C,
Miles FA, and Krauzlis RJ. Short-latency disparity vergence responses and
their dependence on a prior saccadic eye movement. J
Neurophysiol 75:
1392–1410, 1996.
Collewijn H,
Erkelens CJ, and Steinman RM. Binocular co-ordination of human horizontal
saccadic eye movements. J Physiol
404: 157–182,
1988.
Conway BR.
Spatial structure of cone inputs to color cells in alert macaque primary
visual cortex (V-1). J Neurosci
21: 2768–2783,
2001.
Cumming BG. An unexpected specialization for horizontal disparity in primate primary visual cortex. Nature 418: 633–636, 2002.[Medline]
Cumming BG and
Parker AJ. Binocular neurons in V1 of awake monkeys are selective for
absolute, not relative, disparity. J Neurosci
19: 5602–5618,
1999.
Dean AF. The variability of discharge of simple cells in the cat striate cortex. Exp Brain Res 44: 437–440, 1981.[Web of Science][Medline]
Duwaer AL. New measures of fixation disparty in the diagnosis of binocular oculomotor deficiences. Am J Optom Physiol Opt 60: 586–597, 1983.[Web of Science][Medline]
Enright JT. Exploring the third dimension with eye movements: better than stereopsis. Vision Res 31: 1549–1562, 1991.[Web of Science][Medline]
Geisler WS and Albrecht DG. Visual cortex neurons in monkeys and cats: detection, discrimination, and identification. Vis Neurosci 14: 897–919, 1997.[Web of Science][Medline]
Gur M, Beylin
A, and Snodderly DM. Response variability of neurons in primary visual
cortex (V1) of alert monkeys. J Neurosci
17: 2914–2920,
1997.
Gur M and Snodderly DM. Studying striate cortex neurons in behaving monkeys: benefits of image stabilization. Vision Res 27: 2081–2087, 1987.[Web of Science][Medline]
Gur M and Snodderly DM. Visual receptive fields of neurons in primary visual cortex (V1) move in space with the eye movements of fixation. Vision Res 37: 257–265, 1997.[Web of Science][Medline]
Jaschinski-Kruza W and Schubert-Alshuth E. Variability of fixation disparity and accommodation when viewing a CRT visual display unit. Ophthalmic Physiol Opt 12: 411–419, 1992.[Web of Science][Medline]
Judge SJ, Richmond BJ, and Chu FC. Implantation of magnetic search coils for measurement of eye position: an improved method. Vision Res 20: 535–538, 1980.[Web of Science][Medline]
Kagan I, Gur M,
and Snodderly DM. Spatial organization of receptive fields of V1 neurons
of alert monkeys: comparison with responses to gratings. J
Neurophysiol 88:
2557–2574, 2002.
Livingstone MS. Mechanisms of direction selectivity in macaque V1. Neuron 20: 509–526, 1998.[Web of Science][Medline]
Livingstone M,
Freeman D, and Hubel D. Visual responses in V1 of freely viewing monkeys.
Cold Spring Harb Symp Quant Biol
61: 27–37,
1996.
Livingstone MS and Tsao DY. Receptive fields of disparity-selective neurons in macaque striate cortex. Nat Neurosci 2: 825–832, 1999.[Web of Science][Medline]
Motter BC and Poggio GF. Dynamic stabilization of receptive fields of cortical neurons (VI) during fixation of gaze in the macaque. Exp Brain Res 83: 37–43, 1990.[Web of Science][Medline]
Ogle K. Researches in Binocular Vision. New York: Hafner, 1964.
Prince SJ,
Cumming BG, and Parker AJ. Range and mechanism of encoding of horizontal
disparity in macaque V1. J Neurophysiol
87: 209–221,
2002a.
Prince SJ,
Pointon AD, Cumming BG, and Parker AJ. Quantitative analysis of the
responses of V1 neurons to horizontal disparity in dynamic random-dot
stereograms. J Neurophysiol 87:
191–208, 2002b.
Riggs L and Neill E. Eye movements recorded during convergence and divergence. J Opt Soc Am A 50: 913–920, 1960.
Snodderly DM and Gur M. Organization of striate cortex of alert, trained monkeys
(Macaca fascicularis): ongoing activity, stimulus selectivity, and
widths of receptive field activating regions. J
Neurophysiol 74:
2100–2125, 1995.
St Cyr GJ and Fender DH. The interplay of drifts and flicks in binocular fixation. Vision Res 9: 245–265, 1969.[Web of Science][Medline]
Thomas OM, Cumming BG, and Parker AJ. A specialization for relative disparity in V2. Nat Neurosci 5: 472–478, 2002.[Web of Science][Medline]
Trotter Y,
Celebrini S, Stricanne B, Thorpe S, and Imbert M. Modulation of neural
stereoscopic processing in primate area V1 by the viewing distance.
Science 257:
1279–1281, 1992.
Trotter Y,
Celebrini S, Stricanne B, Thorpe S, and Imbert M. Neural processing of
stereopsis as a function of viewing distance in primate visual cortical area
V1. J Neurophysiol 76:
2872–2885, 1996.
Tsao DY and Livingstone MS. Receptive fields of disparity-tuned simple cell in macaque V1. Neuron 38: 103–114, 2003.[Web of Science][Medline]
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, monkey
Ruf.
erfc(x/2n2), where erfc is the
complementary error function, x the current distance from the origin,
and n was set to be 30 units. For a sufficiently long walk, the
distribution of all the x values visited during the walk is
approximately normal with SD = n. The simulation was repeated 100
times with different random walks; the plots show the mean results (which is
why they are smoother than the experimental data). All spectra have been
normalized to contain the same total power.
and - - -, the variance:mean ratio (VMR)
of the spike count distributions (scale on the right axis). It is clear that
this ratio is high at disparities where the slope is high, exactly as expected
if the spike count depends only on the absolute retinal disparity of the
stimulus and there is some fluctuation in vergence. Note that very high VMRs
are observed here because the neurons are very sensitive to disparity. Only
very small vergence fluctuations are required to produce these high VMRs (an
SD of 0.012° is sufficient for duf069, 0.011° for
ruf066).
