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Receptive fields were mapped by hand using bar stimuli and were located 1.5-3° from the fovea in one monkey and 5-6° from the fovea in the other. Stimuli were always 3.5° on a side, and they covered the receptive field and part of the surround. The stimuli that were used (Fig. 1) included 32 oriented bars (A), 32 sine-wave gratings (B), 32 Walsh patterns (C), and 32 photographic images (D). Although it is still a small subset of all possible stimuli, this is, to our knowledge, the most extensive set of stimuli used to examine the mean-variance relation in monkey primary visual cortex. For each neuron, each stimulus was presented on a video monitor in randomized order approximately the same number of times; the median number of presentations per stimulus ranged from 8 to 52 (median 14) in different neurons. No significant differences were found between the results from the two monkeys, so we present data from both together.
The Journal of Neuroscience, October 15, 2001, 21(20):8210-8221
Consistency of Encoding in Monkey Visual Cortex
Matthew C. Wiener, Mike W. Oram, Zheng Liu, and Barry J. Richmond Laboratory of Neuropsychology, National Institute of Mental Health, National Institutes of Health, Bethesda, Maryland 20892-4415
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
Are different kinds of stimuli (for example, different classes of geometric images or naturalistic images) encoded differently by visual cortex, or are the principles of encoding the same for all stimuli? We examine two response properties: (1) the range of spike counts that can be elicited from a neuron in epochs representative of short periods of fixation (up to 400 msec), and (2) the relation between mean and variance of spike counts elicited by different stimuli, that together characterize the information processing capabilities of a neuron using the spike count code. In monkey primary visual cortex (V1) complex cells, we examine responses elicited by static stimuli of four kinds (photographic images, bars, gratings, and Walsh patterns); in area TE of inferior temporal cortex, we examine responses elicited by static stimuli in the sample, nonmatch, and match phases of a delayed match-to-sample task. In each area, the ranges of mean spike counts and the relation between mean and variance of spike counts elicited are sufficiently similar across experimental conditions that information transmission is unaffected by the differences across stimulus set or behavioral conditions [although in 10 of 27 (37%) of the V1 neurons there are statistically significant but small differences, the median difference in transmitted information for these neurons was 0.9%]. Encoding therefore appears to be consistent across experimental conditions for neurons in both V1 and TE, and downstream neurons could decode all incoming signals using a single set of rules.
Key words: coding; visual; cortex; V1; TE; inferior temporal cortex; monkey; natural images; mean-variance relation
INTRODUCTION
Many different kinds of visual stimuli are used in neurophysiological experiments. This raises the question of whether results obtained using one class of stimuli can be expected to hold for others. For example, photographic or naturalistic images might somehow be processed differently from geometric stimuli frequently used in experiments. Previously, we have shown that knowledge of the operating range of the responses of a neuron, along with the linear relation between log(mean) and log(variance) of spike counts elicited by different stimuli (Dean, 1981
; Tolhurst et al., 1981
We examine operating range and the relation between mean and variance in responses (here, spike counts) that are elicited from V1 by four kinds of stimuli: three kinds of geometric stimuli, i.e., bars, sine-wave gratings, and Walsh patterns, and photographic images, which are often used to study the statistics and processing of natural images (Field, 1987
MATERIALS AND METHODS
Data collection
V1. Responses were recorded using standard single-electrode techniques from complex cells in primary visual cortex of two awake rhesus monkeys. At the beginning of each trial, a fixation point appeared on the screen. One hundred milliseconds after the monkey fixated the point, a stimulus was flashed on the receptive field of the neuron for 300 msec and then replaced with the background. The monkey was rewarded for fixating within 0.5° of the fixation point from the appearance of the fixation point until the stimulus disappeared and was not required to react to the stimulus in any way. After a delay of 300 msec, the next trial began.
Spikes were counted in a 300 msec window starting at stimulus onset. We use this period because during normal primate vision, a new image appears on each receptive field one to three times per second because of saccadic eye movement, after which the image is kept nearly still on the retina (compared to saccade velocities). TE. Responses were recorded from neurons in visual area TE while a monkey performed a sequential delayed match-to-sample task using eight Walsh patterns (Fig. 1E). The monkey touched a contact lever to start each trial; a fixation point appeared at the center of a screen immediately. The monkey was required to fixate within ±5° of this spot for the entire trial. As soon as the monkey fixated the fixation point, the fixation point was replaced by a sample Walsh pattern 8.5° on a side, followed by up to two nonmatching patterns and a repeat of the original (now matching) pattern. When the original stimulus reappeared, the monkey was required to release the bar within 2 sec to receive a reward. Stimuli were displayed for 500-1000 msec, with 300-800 msec between stimuli. Further experimental details can be found in Liu and Richmond (2000)
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Figure 1. Stimuli used in the experiments. For the experiment in V1, the stimulus set consisted of 32 oriented bars (A), 32 sine-wave gratings (B), 32 Walsh patterns (C), and 32 photographic images (D). All stimuli except the bars were equiluminant with the background at 1.2 cd/m2. Bars appeared at luminances of 0.08, 0.63, 1.78, and 2.19 cd/m2. The brightest bars had a contrast [measured as (max min)/(max + min) of luminance] of 87.2%, the gratings had a contrast of 65.4%, the Walsh patterns had a contrast of 96.4%, and the contrasts of the pictures ranged from 56 to 96%. For the experiment in TE, eight Walsh patterns (E) were used. The contrast was the same as for the Walsh patterns in the V1 experiments.
Regression analysis
For each neuron, each stimulus produces a sample mean spike count, µi, and a sample variance of spike count,, where the subscript i labels stimulus. We fit the line log 2 = b + m log µ to the set of points (µi,
). Residuals were weighted by the estimated variance of the logarithm of the variance (which depends on the number of trials available for each stimulus; see below). The logarithmic transformation of means and variances makes the regression residuals more nearly uniform across the range of the mean responses (so that the data more closely conform to the assumptions underlying the regression analysis) and ensures that the model can never predict variances <0 (because the model is equivalent to , where is a chi-squared distribution with n , and for moderate values of n is approximately Gaussian. Thus, we can approximate symmetric points of the distribution by E[S2] ± k SD[S2], or , where k varies depending on the percentile of the distribution desired (for example, for the 5th and 95th percentiles, k 1.645). The logarithms of these points are:
Thus, the logarithm of the sample variance has mean approximately log
Information analysis
Information theory is a statistical approach that deals with the relation between inputs, or stimuli, and outputs, or responses (Shannon and Weaver, 1949xp(x) log2p(x), measured in bits, quantifies the uncertainty of the signal. The conditional entropy H(R|S) measures the uncertainty in a response if the stimulus s
S is known. The mutual, or transmitted, information between a stimulus and a response, I(R; S), is the reduction in uncertainty about which stimulus has been presented, provided by knowing the response, or vice versa: I(R; S) = H(S)
2 test) (Gershon et al., 1998
Analysis of scatter around the regression line
Scatter around a regression line represents variability not explained by the regression. In our regression of log(variance) versus log(mean) of spike count, we know of at least one source of such variability: both means and variances are estimated from samples. The amount of scatter resulting from this measurement problem is determined by the number of trials available for estimating each mean and variance; as the number of trials decreases, the scatter around the regression line increases. Assuming that the regression is valid, that is, that log(variance) is a linear function of log(mean), the mean residual sum of squares around the log(variance) versus log(mean) regression line is an estimate of the variance of log(variance) of the responses. Estimating means and variances using only a subset of the n points available will cause the sum of squared residuals to increase. Only neurons with a median of at least eight trials per stimulus in the subsampled data sets (so at least 16 trials per stimulus in the full sets) were included in the analysis of scatter around the regression line. We use simulated data to estimate how quickly the sum of squared residuals decreases with increasing numbers of trials per stimulus under the assumption that all of the scatter around the regression line is attributable to finite sample size. The artificial responses have the same number of trials per stimulus and are generated from distributions with the same mean spike count, as observed for each stimulus in the corresponding real neuron. However, in the artificial data, the variance of spike count for each stimulus is calculated from the observed mean of spike count using the regression line relating log(variance) and log(mean), and spike counts are generated by sampling from a truncated Gaussian distribution with the given mean and variance. Thus, in the artificial data, all scatter around the mean-variance regression line arises from sample size effects only. If the residual sum of squares in the real data increases less quickly than expected based on the artificial data, then some of the scatter around the line is not caused by sampling. (We do not know or speculate here on the source of this nonsampling variance.) The nonsample variance c can be obtained by solving k = (aRSSpart + c)/(aRSSfull + c), where aRSSfull and aRSSpart are the residual sums of squares from regressions from full and subsampled artificial data sets, respectively, and k is the ratio RSSpart/RSSfull measured from the actual data. The portion p of residual sums of squares attributable to sampling can then be calculated, and the total portion of variance explained is r2 + p(1Principal component analysis
Principal component analysis (Ahmed and Rao, 1975
RESULTS
Our data set included 27 complex cells from V1 (16 from one monkey, 11 from another) and 20 neurons from area TE. As explained in Materials and Methods, we performed the analyses using both standard regression methods and methods designed for situations in which both variables are measured with uncertainty. The results using the two regression methods were very similar. We present the results obtained using standard regression methods. At the end of this section, we compare results using the two regression methods.
Distribution of mean responses
It is well known that different stimuli elicit different numbers of spikes, but this does not require that stimuli from different sets consistently elicit different numbers of spikes. Across the 27 V1 neurons, the stimulus set significantly affected the median of mean spike counts elicited (Fig. 2) (Friedman test; p < 0.05); the same was true in 26 of the individual neurons (Kruskal-Wallis test; p < 0.05). Stimulus set accounts for 11% [median; interquartile range (iqr), 5-18%] of the variability in spike counts. The least effective photographic images and Walsh patterns used in these experiments elicited mean spike counts larger than those elicited by the least effective bar and grating stimuli; that is, Walsh patterns and photographic images were less likely than bars and gratings to elicit mean responses near zero. We did not explicitly search for optimal bar or grating stimuli for the neurons we recorded. Figure 2 shows that the median of mean spike counts was between 20 and 27 spikes per second. Reich et al. (2001)
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Figure 2. Distribution across 27 V1 neurons of mean responses elicited by bars, gratings, Walsh patterns, and photographic images. The line in the middle of each shaded box shows the median response. The notch shows a 95% confidence interval around the median (if two notches do not overlap vertically, the corresponding medians are different at the 5% level). The bottom and top edges of the boxes show the 25th and 75th percentiles, and the extended whiskers show 1.5 times the interquartile range. Mean responses outside a range 1.5 times the width of the interquartile range from the median are shown as separate points. The fifth percentile and median of mean responses elicited by bars and gratings are lower than the fifth percentile and median of mean responses elicited by Walsh patterns and photographic images, although the 95th percentile of mean responses is not distinguishable across the four stimulus sets. Each shaded box is based on 896 measurements: the mean response of each of 27 neurons to each of the 32 stimuli in each set. We have no explanation for the greater number of outlier points for bars as opposed to the other stimulus sets.
Behavioral context did not significantly affect the median of mean spike counts across TE neurons (Fig. 3) (Friedman test; p > 0.05), or in any individual TE neuron (Kruskal-Wallis test; p > 0.05). However, behavioral context did affect the largest mean responses elicited by stimuli; the largest mean spike counts in the match condition were larger than the largest responses in the sample and nonmatch conditions (Friedman test on 95th percentile of mean responses; p < 0.05).
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Figure 3. Distribution across 20 TE neurons of mean responses elicited by stimuli in the sample, nonmatch, and match phases of a delayed match-to-sample task. For interpretation of boxplots, see Figure 2. The distributions of responses are not distinguishable across the three behavioral conditions. Each shaded box is based on 160 responses: the mean response of each of 20 neurons to each of the eight stimuli when viewed in the indicated phase of the task.
Consistency of the mean-variance relation across stimulus sets and behavioral conditions
To determine whether a single regression line adequately described responses under different conditions, we examined two models for each set of responses. One model used a single regression line to predict log(variance) of spike count from log(mean) of spike count for all stimuli presented to a particular neuron. The other model used a different regression line to predict log(variance) of spike count from log(mean) of spike count for each stimulus set in V1 or each behavioral condition in TE. If different conditions do give rise to different relations between log(mean) and log(variance), the model using several regressions should predict log(variance) significantly better than the model with a single regression. We test this by comparing the variance of the residuals from the two models. The variance of the residuals is the residual sum of squares divided by the residual degrees of freedom. A model using several lines has fewer residual degrees of freedom than a model using a single line, so the residual sum of squares must decrease more quickly than the residual degrees of freedom to justify using the additional parameters.
In V1, we asked whether a model using four regressions, one for each stimulus set, predicted log(variance) of spike count from log(mean) of spike count significantly better than a model using a single regression for all four data sets. In 17 of 27 neurons, the two models were statistically indistinguishable (Fig. 4, top), but in 10 of 27 neurons, the reduction in sum of squared errors did justify using the extra parameters (f test; p < 0.05) (Fig. 4, bottom). Even in the neurons in which the change was significant, however, the increase in percentage of variance explained was small (Fig. 5, Table 1). Across all stimuli from all V1 neurons, the model using a single regression explained about two-thirds of the variance (median r2, 0.65; iqr, 0.44-0.76), and the model using four regressions explained only slightly more (median r2, 0.65; iqr, 0.49-0.79). Across all neurons, the median increase in r2 was 0.03 (iqr, 0.01-0.05); for only those neurons in which the four-regression model predicted variance significantly better than the single-regression model, the median increase in r2 was 0.06 (iqr, 0.04-0.07). We will show below that these small changes in predicted variance do not affect the ability of the neuron to distinguish among different stimuli. Therefore, for each neuron only a single regression line is needed to describe the relation between log(mean) and log(variance) of spike count for stimuli of all four kinds. Across 27 V1 neurons, the median intercept of the single regression line is 0.6 (iqr, 0.4-0.8), and the median slope is 1.1 (iqr, 1.0-1.2).
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Figure 4. Relation between log(mean) and log(variance) in two V1 neurons: one in which the model using four regression lines is not significantly better than the model using a single regression line (top), and one in which it is better (bottom). x-, y-axes, Mean and variance of number of spikes elicited by each stimulus on a logarithmic scale. Note the different scales on the x- and y-axes in the two panels. Each panel shows log(mean) versus log(variance) for each of four stimulus sets in a single V1 neuron. Oriented bars, Red squares; oriented gratings, blue circles; Walsh patterns, green triangles; photographic images, purple diamonds. Stimulus-set specific regression lines are in the corresponding colors, extended to the edges of the plot for visibility. The regression line for all stimuli taken together is shown in black. Colored bars at the bottom show the range of means for each stimulus set. The neurons shown have the median p values for the f test comparing the model using a single regression line to the model using four regression lines among those neurons for which the f test is (bottom) and is not (top) significant.
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Figure 5. Portion of variance explained (r2) increases for a model using a separate log(variance) versus log(mean) regression for each stimulus set as compared to a model using a single log(variance) versus log(mean) regression for all stimuli, for 27 V1 neurons. The horizontal and vertical axes show the r2 values for the single-regression and multiple-regression models. Filled circles represent neurons for which the improvement in prediction of log(variance) from log(mean) is significant (f test, p < 0.05), and open circles represent those for which it is not.
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Table 1. Summary of r2 for regression of log(variance) versus log(mean) for V1 neurons Although in 10 of 27 neurons, using four regression lines predicts variance significantly better than using a single regression line, naturalistic (photographic) stimuli are not consistently treated differently from the geometric stimuli. Examining models using two regression lines, one for stimuli from one set and another for stimuli from the other three sets, we found that bars, gratings, Walsh patterns, and photographic images were distinguishable from all other stimuli in 4, 7, 8, and 5 of the 10 neurons, respectively. There was no clear pattern to which stimulus sets were distinguishable from others in individual neurons.
In area TE, we asked whether a model using three regression lines, one each for the sample, nonmatch, and match task conditions, predicted log(variance) from log(mean) significantly better than a model using a single regression line for all task conditions together (Fig. 6). In 19 of the 20 neurons, the improvement in prediction did not justify using the extra parameters (Table 2), and 1 of 20 neurons is expected to show an effect at the p = 0.05 level by chance. Thus, we conclude that for neurons in TE, only a single regression line is needed to describe the responses in all three behavioral contexts. Across 20 TE neurons, the median intercept of the regression line is 0.3 (iqr, 0.0-0.6), and the median slope is 1.3 (iqr, 1.2-1.5).
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Figure 6. Responses obey a single mean-variance relation across behavioral conditions in a TE neuron. x-, y-axes, Mean and variance of number of spikes elicited by each stimulus on a logarithmic scale. Log(mean) versus log(variance) for each of three task conditions in a single TE neuron. Sample, Red squares; nonmatch, blue circles; match, green triangles. Regression lines for individual conditions are shown in the corresponding colors, extended to the edges of the plot for visibility. The regression line for all conditions together is shown in black. Colored bars at the bottom show the range of means for each task condition (sample, nonmatch, and match). The neuron shown has the median p value for the f test comparing the model using a single regression line to the model using three regression lines.
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Table 2. Summary of r2 for regression of log(variance) versus log(mean) for TE neurons It has been reported that instability of eye position during presentation of an optimal moving bar increases response variability in V1 neurons (Gur et al., 1997
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Figure 7. Responses obey a single mean-variance relation for more and less stable fixation. x-, y-axes, Mean and variance of number of spikes elicited by each stimulus on a logarithmic scale. Log(mean) versus log(variance) for trials during which eye position remained within a square 6 min of arc on a side during the entire trial (+), and for trials during which eye position ranged over a larger region ( ) in a single V1 neuron. Regression lines for the separate conditions are shown, using a dashed line for the trials with more stable fixation, and a solid line for the trials with less stable fixation. The solid line for trials with more stable fixation merges with the regression line for both conditions taken together, which is shown with a thicker solid line. The greater scatter around the line for trials with more stable fixation (+) than for trials with less stable fixation (
Transmitted information and channel capacity
Transmitted information measures how well an observer can guess which stimulus elicited any particular observed response (here, spike count). For each V1 neuron, transmitted information was calculated for all stimuli together and for the four stimulus sets individually. The information for a particular stimulus set was calculated using the relation between mean and variance measured from stimuli from that set only; the information for all stimuli together was estimated twice, once using the model with a single regression for all stimuli and once using the model with a separate regression for each of the four stimulus sets. The information for all stimuli together was nearly identical no matter which model was used (difference of 0.6% median; iqr, 0.1-1.2%), even for the 10 of 27 neurons for which the model with four regressions predicted variance significantly better than the model with a single regression (difference of 0.9% median; iqr, 0.2%-1.1%). However, the information that was transmitted about the individual stimulus sets varied a great deal (Fig. 8) (Friedman test; p < 0.001). The differences in information are attributable not to differences in the relation between mean and variance for the stimulus sets (which make very little difference in information when all stimuli are considered), but rather to the different mean responses elicited by stimuli of different kinds (Fig. 2).
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Figure 8. Information transmitted by spike count about which of the stimuli was presented, for all stimuli together and for each of the stimulus sets separately, for neurons in V1. The vertical axis shows transmitted information measured in bits. Each boxplot shows the distribution across 27 V1 neurons of information for the set of stimuli indicated on the x axis: all stimuli together (using either a single regression for all four stimulus sets or using a separate regression for each of the four stimulus sets), bar stimuli alone, gratings alone, Walsh patterns alone, and photographic images (photo) alone. For interpretation of boxplots, see Figure 2. Responses become less informative as the range of distinct means becomes smaller and the distributions of responses become more variable.
The least effective Walsh patterns and photographic images do not elicit mean responses as small as those elicited by the least effective bar and grating stimuli, whereas the most effective stimuli from each group elicit similar responses. This means that mean responses to Walsh patterns and photographic images are, in effect, crowded into a smaller range than mean responses to bars and stimuli. Because response variance grows with response mean, responses to Walsh patterns and photographic images are on average also more variable. Thus, individual responses to photographic images and Walsh patterns are less informative about which stimulus was presented than individual responses to bars and gratings.
Transmitted information describes the outcome of a particular experiment. Channel capacity, which depends on the mean-variance relation and the range of possible mean responses (Gershon et al., 1998
0.01) and are no longer statistically indistinguishable from one another (Friedman test; p < 0.05), the estimates using only the Walsh patterns having dropped more than the other estimates.
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Figure 9. Channel capacity depends on the lowest allowed mean response more than on stimulus set or highest allowed mean response in neurons in V1. The left (darkest) box in each set shows the distribution across 27 V1 neurons of channel capacity if the range of allowable mean responses extends from 0 (no spikes ever elicited) to 1.25 times the maximum observed mean response. The middle box shows channel capacity if the range extends from 0 to 1.1 times the maximum observed mean response. The right (lightest) box shows channel capacity if the range extends from the minimum observed mean response to 1.25 times the maximum observed mean response. Changing the minimum allowed mean response has a much greater effect than changing the maximum allowed mean response. For interpretation of boxplots, see Figure 2.
These results show the importance of the assumed minimum achievable mean response for estimates of channel capacity. Even when the same relation between mean and variance is used, a change in the assumed minimum achievable mean can change the estimate of channel capacity dramatically (Fig. 9, comparison between the two left columns and the right column in each set). The smaller the smallest observed mean response, the less dramatic the effect will be. Therefore, it is important in experiments seeking to examine the information-processing capabilities of a neuron to use a range of stimuli that elicit the largest possible range of mean responses from a stimulus and, in particular, to include stimuli that elicit few spikes as well as those that elicit many.
We cannot use these methods to calculate information or channel capacity for the TE neurons, because we do not have a good model for the spike count distributions from the TE neurons. However, the fact that both the range of mean responses and the relation between mean and variance are identical across the three behavioral conditions suggests that the information content of responses in the three conditions will be similar.
The mean-variance relation in different counting windows
Our focus in this paper is whether the mean-variance relation for spike counts in a particular counting window is consistent across different stimulus sets in V1 and across different behavioral conditions in TE. Above, we have come to the conclusion that for a particular counting window (0-300 msec after stimulus onset in V1, 70-470 msec after stimulus onset in TE), any differences in the mean-variance relation are sufficiently small as to not affect information transmission. Figure 10 shows the mean-variance relation over time in representative neurons from V1 (left) and TE (right). The horizontal axis shows mean response, the vertical axis shows the end of the counting window, and gray scale and contours show the variance predicted for each mean response by the mean-variance relation. In both V1 and TE, the variance associated with a particular mean spike count increases as the window expands. Variance increases more rapidly in V1 neurons than in TE neurons.
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Figure 10. Consistency of the mean-variance relation in different counting windows. The left and right panels show results for neurons from V1 and TE, respectively. The top panels show results for windows expanding from a fixed start time (stimulus onset in V1, 70 msec after stimulus onset in TE), and the bottom panels show results for sliding 50-msec-wide windows starting at different times. In each panel, the horizontal axis shows mean spike count (on a logarithmic scale), the vertical axis shows the end of the counting window at 25 msec intervals (windows start at stimulus onset in V1, and 70 msec after stimulus onset in TE), and gray scale represents the variance (lighter means higher variance). Contours representing variances of 2, 5, 10, 20, and 40 spikes per second squared are also shown. The mean-variance relation at a particular time corresponds to a horizontal slice through the plot. V1, The variance associated with a particular mean spike count is larger in longer counting windows (with fixed starting point). This increasing variance is also seen in sliding windows. TE, The variance associated with a particular mean spike count is larger in longer counting windows, but variance increases more slowly than in V1. Again, the increase is seen in sliding windows as well.
Although the relation between log(mean) and log(variance) changes over time (Fig. 10), the explanatory power of the relation does not change much; the r2 values for the regressions are similar for the full period analyzed and for the two half-periods (Tables 1, 2). In expanding windows, the number of neurons for which using multiple regressions is justified [that is, for which the model using multiple regressions predicts log(variance) from log(mean) significantly (f test; p < 0.05) better than the model using a single regression] is similar to that found in the full window: 9-11 of 27 neurons in V1, and 0, 1, or 2 of 20 neurons in TE. In sliding windows, up to 14 (in V1) or 4 (in TE) neurons show differences among the mean-variance relations. However, in both expanding and sliding windows in the V1 neurons, the small differences found among the relations between mean and variance of spike count in different windows did not affect information transmission; the amount of information found in the responses was nearly identical, whether a single regression or multiple regressions were used to estimate variance from mean. (As for the main counting window, we cannot explicitly calculate information for TE neurons using our model because the spike count distributions are not well-modeled by a truncated Gaussian distribution.)
Analysis of scatter around the regression line
Although a single regression line can be used to predict log(variance) from log(mean) across stimulus sets in each V1 neuron, the prediction is not perfect; substantial scatter around the regression lines remains (Fig. 4). In the V1 neurons examined here, the regression explained 65% of the variability of measured variance (median r2; iqr, 0.44-0.76). Thus 35% (median; iqr, 24-56%) of the variability in the V1 neurons is seen as scatter around the regression lines. As explained below, we estimate that about two-thirds of this scatter can be attributed to sample size effects.
The amount of scatter around the regression line relating log(mean) and log(variance) depends in part on the number of trials per stimulus that are used to estimate the means and variances; the more trials per stimulus, the less scatter. As explained in Materials and Methods, we estimated the measurement effect of sample size (number of trials per stimulus) on residual sum of squares around the regression line using artificial data in which log(mean) and log(variance) are exactly linearly related. In such artificial data, any change in residual sum of squares can be attributed only to the measurement effect of sample size. If the residual sums of squares do not increase as rapidly in regressions subsampling the real data, we can conclude that some of the scatter around the regression line has other sources. In the V1 neurons, 70% (median; iqr, 58-78%) of the residual sums of squares remaining after regression can be attributed to the measurement effects of sampling. (Using the formulas rather than simulation, which gives an upper bound, the median percent of scatter attributable to sampling is 75%, with iqr 63-89%.) Thus, the regression relating log(mean) and log(variance) is not only consistent across stimulus sets in V1 neurons; it is actually better than it looks, because about two-thirds of the scatter around the regression line is attributable to limited sample size. As expected, the more trials available for estimating the variance, the smaller the percent of scatter attributable to the measurement effect of sample size (Fig. 11). When both the predictive power of the mean response and the measurement effect of sample size are taken into account, only 13% (median; iqr, 7-15%) of response variance in V1 neurons remains to be explained by other factors.
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Figure 11. The amount of scatter around the regression line that can be attributed to the measurement effect of sample size decreases with increasing numbers of trials per stimulus. Each point shows the result for one V1 neuron. x-axis, Median number of trials per stimulus; y-axis, percent of scatter around the regression line attributable to sampling.
We have shown that in some of the V1 neurons a model using a separate regression line for each stimulus set predicts log(variance) from log(mean) significantly better than a model using a single line for all four stimulus sets, although the improvement is small. In the neurons for which the lines differed most, slightly more scatter could be attributed to sampling when residuals were calculated as deviation from the four lines individually rather than from a single line. In this group-by-group analysis, the percent of scatter attributable to sampling was 73% (median; iqr, 70-80%), which, when combined with the predictive power of the mean response, left only 7% (median; iqr, 5-12%) of the scatter to be explained by other factors. In five V1 neurons, there were sufficient trials to analyze scatter around the regression line separately for trials during which fixation was very stable and trials during which fixation was less stable (see Materials and Methods). Analyzing scatter separately for these two subsets resulted in very little additional variance explained.
Because spike count has been shown to influence spike timing (Oram et al., 1999
Different regression methods give similar results
When regression methods taking into account uncertainty in both variables are used, the sums of squared residuals in the x direction (around the logarithms of the means) are much smaller than the sums of squared residuals in the y direction (around the logarithms of the variances), by a factor of 33 (median; iqr, 14-110) in the V1 neurons and by a factor of 13 (median; iqr, 7-24) in the TE neurons. This suggests that taking into account uncertainty in the mean should have a relatively small effect and differences between results using the two regression methods will be small.
To assess the practical effect of the uncertainty of estimates of mean response, we used both standard regression methods and methods designed for data with uncertainty in both variables (Fuller, 1987
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Figure 12. Regression lines calculated with and without uncertainty in the mean are similar. The x- and y-axes show, on a logarithmic scale, the means and variances of spike counts elicited from a single neuron by different stimuli. The regression lines calculated using standard regression methods (solid) and methods taking into account uncertainty in estimates of both variables (dashed) are shown. Ignoring the uncertainty in estimates of the sample mean biases the slope toward 0, but the effect is small. The cell shown has the median change in slope among V1 neurons.
The main result of this paper, that the relation between log(mean) and log(variance) is consistent across multiple stimulus sets in V1 neurons and across behavioral conditions in TE neurons, still holds when regression methods accounting for uncertainty in both variables are used. Furthermore, the amount of scatter around the regression line that can be attributed to the measurement effect of sample size is quantitatively similar whether the regression methods account for the uncertainty in estimates of the logarithm of the mean or ignore it. In these V1 data, the measurement effect of sample size accounts for 70% (median; iqr, 58-78%) of the sum of squared residuals around the regression line when standard regression methods are used (as reported above), and 67% (median; iqr, 61-80%) when uncertainty in the sample mean is taken into account.
DISCUSSION
We have examined spike-count coding in single neurons in monkey primary visual cortex. We find that the previously observed linear relation between log(mean) and log(variance) is sufficiently consistent across a wide range of stationary black-and-white images (including photographic images); for practical purposes, there is no reason to use more than a single relation. In particular, the relation between mean and variance is not systematically different for photographic images than for simple geometric stimuli. In area TE, the relation between mean and variance of spike count and the distributions of mean responses to stimuli presented in the sample, nonmatch, and match phases of a delayed match-to-sample task are statistically indistinguishable (Figs. 3, 6), consistent with the results of McAdams and Maunsell (1999)
The relation between mean and variance is not, however, the only factor affecting the ability of a neuron to transmit information. In our experiments in V1 neurons, photographic images and Walsh patterns elicited larger mean responses than bar and grating stimuli (Fig. 2). As a consequence, estimates of transmitted information depend on which stimuli are presented (Fig. 8). If four different researchers had conducted four different experiments using our four stimulus sets, there might be controversy over how much information neurons in V1 "really" transmit. Estimates of channel capacity based on results from the different stimulus sets are more consistent than estimates of transmitted information (Fig. 9), because they depend only on two fundamental statistical properties of the responses: the relation between mean and variance and the range of allowed mean responses. To characterize the information processing capacity of a neuron, it is important to elicit the largest possible range of mean responses, particularly low mean responses. Thus, it is important to use as large and varied a stimulus set as possible in neurophysiological experiments.
We have also shown for V1 neurons that the relation between mean and variance is better than it looks; approximately two-thirds of the scatter around the log(variance) versus log(mean) regression line is attributable to the measurement effect of sample size. Although we cannot perform such a quantitative analysis for the TE neurons (because the spike count distributions in TE are not well modeled by truncated Gaussian distributions, as are those in V1), the variances of spike count distributions from TE neurons are measured with uncertainty, and this must contribute to the scatter around the regression lines in TE.
Comparison with other studies
Previous studies of whether naturalistic stimuli are encoded with less variability than other stimuli reached contradictory conclusions. Rieke et al. (1995)
In another study, Warzecha et al. (2000)
Earlier, we presented an analysis of the mean-variance relation in TE neurons from a monkey performing a delayed match-to-sample task similar to the one in this paper (Gershon et al., 1998
In the experiments described here, we used only stationary black-and-white stimuli. Stimuli involving color or motion might give rise to a different mean-variance relation, as suggested by Croner and Albright (1999)
Implications for neural coding
We have shown that assuming the existence of stimuli that elicit on average very small numbers of spikes, or elicit no spikes at all, results in significantly larger estimates of channel capacity than does assuming that the experiment has revealed the smallest achievable mean response. The large effect of very small responses on channel capacity suggests a link to theories of sparse coding in which few neurons should respond to any particular stimulus (Rolls and Tovee, 1995
Stimulus features can be encoded not only by spike count but also by spike timing, although the nature and time scale of that encoding remain the subject of debate (Heller et al., 1995
