Visual Neuroscience (1992), 8, 277-279. Printed in the USA. Copyright © 1992 Cambridge University Press 0952-5238/92 $5.00 + .00
SHORT COMMUNICATION
Variability of responses of cat retinal ganglion cells
MICHAEL W. LEVINE, 1 BRIAN G. CLELAND, 2 AND ROGER P. ZIMMERMAN 3 1
Department of Psychology and the Committee on Neuroscience, University of Illinois at Chicago, Chicago Department of Physiology, University of Sydney, New South Wales, Australia Departments of Neurological Sciences and Physiology and Division of Cell Biology, Rush University, Chicago
2 3
(RECEIVED March 1, 1991; ACCEPTED August 10, 1991)
Abstract Previous studies of the variability of firing of retinal ganglion cells have led to apparently contradictory conclusions. To a first approximation, the variance of rate of maintained discharges of ganglion cells in cat is independent of the mean firing rate. On the other hand, the variability of responses to abrupt changes in lighting of ganglion cells in goldfish increases with increasing firing rate. To examine whether the difference is due to differences between species, we examined the variability of responses of cat ganglion cells, and find it similar to that of goldfish ganglion cells. The variance of rate of ganglion cells is neither independent of mean rate, as might be expected from maintained discharges, nor directly proportional to the mean rate, as it is for cat cortical cells. Rather, there is a nonlinear relationship between variance of rate and mean rate. Keywords: Retinal ganglion cells, Cat retina, Variance of rate, Variability of responses
Introduction Results from two kinds of experiments on variability of retinal ganglion cell firing appear contradictory. Experiments on maintained discharges of ganglion cells in cat indicate that variability is added to the rate-setting signal (Robson & Troy, 1987). Their conclusion is supported by the finding that the coefficient of variation (CV) of the intervals declines approximately as the square root of mean firing rate when the maintained rate is changed (Levine, 1987; Robson & Troy, 1987). The inverse square-root relationship implies that the variance of rate is independent of mean rate (Levine, 1987). The other type of experiment compiles responses to various strength stimuli, each repeated a number of times. Variability is expressed as the variance of rate (mean square difference between the rate on each trial and the mean rate). Such experiments showed that in cortical neurons in cat (Dean, 1981; Tolhurst et al., 1981, 1983) and in monkey (Tolhurst et al., 1983) variance of rate increases in direct proportion to mean rate. To determine whether the difference is between retinal and cortical cells, or between maintained discharges and responses to temporal changes in the lighting, Levine et al. (1988) measured the variance of rate of goldfish ganglion cells in response to flashes of light. They obtained larger variance for larger responses, reminiscent of the results in cortical cells. Since gan-
Reprint requests to: Michael W. Levine, Department of Psychology, M/C 285, University of Illinois at Chicago, Box 4348, Chicago, IL 60680, USA.
277
glion cells also produce a moderate maintained discharge, they were able to show a decrease in variance when discharge rate was lowered by a suppressive stimulus. The effect is thus related to discharge rate, not stimulus strength. The data from goldfish ganglion cells also presented some peculiarities (Levine et al., 1988). Although variance increased for higher firing rates, it did not increase in direct proportion to the firing rate. In general, the logarithm of variance of firing rate vs. the logarithm of the mean firing rate was best fit with a straight line whose slope was less than unity, indicating a fractional power law. The slope was somewhat steeper if each response was sampled for a longer period (geometric mean slopes: 3-s periods, 0.66; ^-s periods, 0.78; 1-s periods, 0.93). These results suggested that the main discrepancy is related to the type of experiment, not the type of cell. Nevertheless, the differences could have been attributed to the species, with goldfish ganglion cells operating one way, and cat ganglion cells operating in another. We now report the variance of rate of the responses to light of cat ganglion cells. Methods Extracellular recordings were made from ganglion cells in the retinae of anesthetized cats using a microelectrode inserted through a scleral guide tube. The animal was paralyzed with gallamine triethiodide, and anesthesia maintained by ventilating with a mixture of nitrous oxide (70%) and halothane ( 1 4%) in oxygen. Heart rate and the pupil of the unoperated eye were monitored to maintain a level of anesthesia that ensured freedom from pain. Temperature, blood pressure, and end-tidal
M.W. Levine, B.G. Cleland, and R.P. Zimmerman
278 P C o 2 were monitored to sustain a proper physiological state. (For details, see Cleland & Lee, 1985.) To measure the variance of rate at different rates, we applied stimuli of various strengths, with each strength repeated several times. We used flashed or square-wave modulated stimuli, and took as the responses the mean firing rates in time windows the first of which began at the time when the luminance in the middle of the receptive field was increased. Two different stimulation protocols were used. In the first, a tangent screen illuminated with white light at a mesopic level (1.8 cd/m 2 ) was placed 1.14 m in front of the cat; it subtended 50 X 50 deg. Receptive fields were within 20 deg of the area centralis. A spot of approximately the diameter of the receptivefield center was present for ^-s of every second. Each spot was flashed 9-60 times consecutively before a new luminance was selected. Four to 18 different luminances were used. The second protocol used a white CRT monitor at a photopic level (mean luminance = 300 cd/m 2 ). A sinusoidal grating pattern of nearly optimal spatial frequency for the cell was square wave modulated in counterphase at 1 Hz. Modulation was repeated for 39 cycles at each of 15-23 different contrasts ranging from 1-60%. To compare data from cat with the alternative outcomes (direct proportionality or no relationship), we compare the group mean slopes of lines fit to the data from each cell. The tests used incorporate an assumption that the slopes are distributed normally. The slopes of the principal component lines that we fit to our data present skewed distributions; a logarithmic transform (presenting the logarithm of the slope, rather than the slope itself) normalized the data (Levine et al., 1988). Note that the antilogarithm of the mean (of the logarithms) of the slopes is the geometric mean slope.
temporally modulated in cosine phase, it was difficult to extract data for |-s periods.) Each function was fit with a principal component line by linear regression; an example is shown in Fig. 1. All of the slopes obtained were positive, indicating that, as for the goldfish, the variance of rate increases as the rate increases. The geometric mean slopes were significantly less than unity (|-s periods: mean slope = 0.60; t = - 4 . 9 8 , 6 d.f.; P < 0.005; |-s periods: mean slope = 0.71; t = - 3 . 0 4 , P < 0.05), indicating a different relationship than the direct proportionality reported for cat visual cortex. These mean values for the cat are quite similar to those reported for goldfish (Levine et al., 1988; see Introduction). We could detect no tendency for a systematic difference according to cell type (X or Y cell, ON or OFF center). Similarly, there was no noticeable difference (due either to the lighting level or the difference in protocol) between responses recorded in mesopic and those recorded in photopic conditions. The two cells recorded in both conditions showed little difference between conditions. The slope was steeper for ^-s than for |-s periods in all but two of the nine cat ganglion cells from which both could be obtained, and a paired /-test showed the difference between the slopes significant (f = 2.27, 8 d.f., P < 0.05). The distribution of the differences is not normal, but the nonparametric Wilcoxon test gives the same outcome {T = 3, N = 8, P < 0.05). Although the variance of rate was not linearly related to the
2 -
Results Data were obtained from 12 cat retinal ganglion cells; there were nine ON- and three OFF-center cells, seven X and five Y cells, with all possible combinations represented. Seven cells were tested only in the mesopic state, three only in the photopic state, and two under both conditions. For comparison to goldfish, we use only the data collected in the mesopic state (3 ON-X, 2 ON-Y, and 2 OFF-X cells). For each cell tested, the logarithm of the variance of rate in each j-s period synchronized to the modulation was plotted against the logarithm of the mean rate in that period. The rate was taken as the number of impulses produced during each period, divided by the length of the period. We rejected periods in which the mean count was less than two.* There were thus up to four data points (one for each |-s period) for each stimulus strength used. For cells tested in mesopic lighting, a second function was produced on the same axes by dividing each cycle into two |-s periods; each stimulus strength contributed up to two points to this function. (Since the photopic stimulus was
0
1 -
o
1 Log firing rat©
•Since the count in any period must be an integral number of impulses, the variance of count when the mean is small is defined by the statistics of small numbers. Periods of very low mean rate must necessarily have low variance; including such periods would force the conclusion that variance increases with rate. By excluding periods with a small count, we avoided this potential confounding of our results. Note that if the mean rate is zero (as was the case for many of our rejected periods) the variance is also zero, and the point would have been located at (—oo,—oo) on our logarithmic axes.
Fig. 1. Log variance of firing rate vs. log mean rate of a cat retinal ganglion cell in mesopic illumination. Data analyzed in | - s periods (solid squares) and 2-s periods (open circles). Firing rate is in impulses/s; and variance of rate is in (impulses/s) 2 . Stimulus was a flashed spot, 0.5 deg in diameter, repeated at 1 Hz for 34 cycles at each of five luminances. Principal component lines fit to each duration have slopes of 0.62 and 0.84, respectively. ON-X cell; mean maintained rate = 24.8 impulses/s; and CV = 0.62.
279
Variability of responses
histograms) accounts for much of the scatter of the variance. Nevertheless, the main effect of increased variance of rate with mean rate is robust. 2 -
Discussion The analyses of responses from cat retinal ganglion cells indicate a qualitative similarity in the behavior of ganglion cells in fish and cat. The ganglion cells in the two species both present a nonlinear relationship when similarly tested. The variance of rate increases with rate but less than proportionally, with the relationship often depending upon the duration of the period in which the responses are measured. We conclude that an increase in the variability of responses with response size is not peculiar to cortical cells or to ganglion cells in the fish. Previous studies of the maintained discharges of retinal ganglion cells in cat indicated that the variance of rate should be independent of mean rate (Robson & Troy, 1987; Levine, 1987). We have proposed a model that reconciles the apparent contradiction between the results presented in this communication with the expectation from maintained discharges (Levine & Zimmerman, 1992).
1 -
0
0 -
Log fir
rote
Acknowledgments
Fig. 2. Log variance of firing rate vs. log mean firing rate of another cat retinal ganglion cell in photopic illumination. Data analyzed in \-s periods only. Sinusoidal grating is 1 cycle/deg; 39 cycles of counterphase modulation (1 Hz) at each of 19 contrasts. ON-X cell; mean maintained rate = 47.2 impulses/s; and CV = 0.28. Conventions are as in Fig. 1.
Research was supported by the NH&MRC of Australia. M.W. Levine was supported in part by a Senior International Fellowship TWO1317 from the Fogarty International Center of NIH. The analyses were supported in part by NIH Grant EY6163, the Rush University Committee on Research, the Chicago Community Trust, and the Regenstein Foundation. We are also grateful to Dr. Laura Frishman for her suggestions and comments on an earlier version of this manuscript.
mean rate, it was not always best characterized by a fractional power function. Negatively accelerated curves such as that in Fig. 2 (data from another ganglion cell) were clearly present in four of the 12 cells, and were less distinctly evident in two others. The distribution of this feature across cell types was similar to the distribution of cell types in the sample, with all subtypes except OFF-Y showing at least one exemplar. The variability in the plots is considerable, making it difficult to characterize the exact relationship. Similar scatter is evident in the previously published work (Tolhurst et al., 1983). We have been unable to attribute the scatter to the order of presentation of the stimuli or to adaptation during the course of the experiment. There were no long-term drifts or nonstationarities evident in the data. Scatter due to sample size is expected to decline as the square root of number of cycles at each contrast, but we see a minimal decrease in scatter for fourfold increases in the number of cycles. There is apparently some other determinant of variance of rate in addition to the strong effect of mean rate upon variance of rate; preliminary analyses suggest that the temporal structure of the responses (the relative amplitudes of the peaks and plateaus of the peristimulus-time
References CLELAND, B.G. & LEE, B.B. (1985). A comparison of visual responses of cat lateral geniculate nucleus neurones with those of ganglion cells afferent to them. Journal of Physiology 369, 249-268. DEAN, A.F. (1981). The variability of discharge of simple cells in the cat striate cortex. Experimental Brain Research 44, 437-440. LEVINE, M.W. (1987). Variability in the maintained discharges of retinal ganglion cells. Journal of the Optical Society of America A 4, 2308-2320. LEVINE, M.W. & ZIMMERMAN, R.P. (1992). A model for the variability of maintained discharges and responses to flashes of light. Biological Cybernetics, June 1992 (in press). LEVINE, M.W., ZIMMERMAN, R.P. & CARRION-CARIRE, V. (1988). Vari-
ability in responses of retinal ganglion cells. Journal of the Optical Society of America A 5, 593-597. ROBSON, J.G. & TROY, J.B. (1987). The nature of the noise in cat retinal ganglion cells. Journal of the Optical Society of America A 4, 2301-2307. TOLHURST, D.J., MOVSHON, J.A. & DEAN, A.F. (1983). The statistical
reliability of signals in single neurons in cat and monkey visual cortex. Vision Research 23, 775-785. TOLHURST, D.J., MOVSHON, J.A. & THOMPSON, I.D. (1981). The depen-
dence of response amplitude and variance of cat visual cortical neurones on stimulus contrast. Experimental Brain Research 41, 414-419.
SHORT COMMUNICATION
Variability of responses of cat retinal ganglion cells
MICHAEL W. LEVINE, 1 BRIAN G. CLELAND, 2 AND ROGER P. ZIMMERMAN 3 1
Department of Psychology and the Committee on Neuroscience, University of Illinois at Chicago, Chicago Department of Physiology, University of Sydney, New South Wales, Australia Departments of Neurological Sciences and Physiology and Division of Cell Biology, Rush University, Chicago
2 3
(RECEIVED March 1, 1991; ACCEPTED August 10, 1991)
Abstract Previous studies of the variability of firing of retinal ganglion cells have led to apparently contradictory conclusions. To a first approximation, the variance of rate of maintained discharges of ganglion cells in cat is independent of the mean firing rate. On the other hand, the variability of responses to abrupt changes in lighting of ganglion cells in goldfish increases with increasing firing rate. To examine whether the difference is due to differences between species, we examined the variability of responses of cat ganglion cells, and find it similar to that of goldfish ganglion cells. The variance of rate of ganglion cells is neither independent of mean rate, as might be expected from maintained discharges, nor directly proportional to the mean rate, as it is for cat cortical cells. Rather, there is a nonlinear relationship between variance of rate and mean rate. Keywords: Retinal ganglion cells, Cat retina, Variance of rate, Variability of responses
Introduction Results from two kinds of experiments on variability of retinal ganglion cell firing appear contradictory. Experiments on maintained discharges of ganglion cells in cat indicate that variability is added to the rate-setting signal (Robson & Troy, 1987). Their conclusion is supported by the finding that the coefficient of variation (CV) of the intervals declines approximately as the square root of mean firing rate when the maintained rate is changed (Levine, 1987; Robson & Troy, 1987). The inverse square-root relationship implies that the variance of rate is independent of mean rate (Levine, 1987). The other type of experiment compiles responses to various strength stimuli, each repeated a number of times. Variability is expressed as the variance of rate (mean square difference between the rate on each trial and the mean rate). Such experiments showed that in cortical neurons in cat (Dean, 1981; Tolhurst et al., 1981, 1983) and in monkey (Tolhurst et al., 1983) variance of rate increases in direct proportion to mean rate. To determine whether the difference is between retinal and cortical cells, or between maintained discharges and responses to temporal changes in the lighting, Levine et al. (1988) measured the variance of rate of goldfish ganglion cells in response to flashes of light. They obtained larger variance for larger responses, reminiscent of the results in cortical cells. Since gan-
Reprint requests to: Michael W. Levine, Department of Psychology, M/C 285, University of Illinois at Chicago, Box 4348, Chicago, IL 60680, USA.
277
glion cells also produce a moderate maintained discharge, they were able to show a decrease in variance when discharge rate was lowered by a suppressive stimulus. The effect is thus related to discharge rate, not stimulus strength. The data from goldfish ganglion cells also presented some peculiarities (Levine et al., 1988). Although variance increased for higher firing rates, it did not increase in direct proportion to the firing rate. In general, the logarithm of variance of firing rate vs. the logarithm of the mean firing rate was best fit with a straight line whose slope was less than unity, indicating a fractional power law. The slope was somewhat steeper if each response was sampled for a longer period (geometric mean slopes: 3-s periods, 0.66; ^-s periods, 0.78; 1-s periods, 0.93). These results suggested that the main discrepancy is related to the type of experiment, not the type of cell. Nevertheless, the differences could have been attributed to the species, with goldfish ganglion cells operating one way, and cat ganglion cells operating in another. We now report the variance of rate of the responses to light of cat ganglion cells. Methods Extracellular recordings were made from ganglion cells in the retinae of anesthetized cats using a microelectrode inserted through a scleral guide tube. The animal was paralyzed with gallamine triethiodide, and anesthesia maintained by ventilating with a mixture of nitrous oxide (70%) and halothane ( 1 4%) in oxygen. Heart rate and the pupil of the unoperated eye were monitored to maintain a level of anesthesia that ensured freedom from pain. Temperature, blood pressure, and end-tidal
M.W. Levine, B.G. Cleland, and R.P. Zimmerman
278 P C o 2 were monitored to sustain a proper physiological state. (For details, see Cleland & Lee, 1985.) To measure the variance of rate at different rates, we applied stimuli of various strengths, with each strength repeated several times. We used flashed or square-wave modulated stimuli, and took as the responses the mean firing rates in time windows the first of which began at the time when the luminance in the middle of the receptive field was increased. Two different stimulation protocols were used. In the first, a tangent screen illuminated with white light at a mesopic level (1.8 cd/m 2 ) was placed 1.14 m in front of the cat; it subtended 50 X 50 deg. Receptive fields were within 20 deg of the area centralis. A spot of approximately the diameter of the receptivefield center was present for ^-s of every second. Each spot was flashed 9-60 times consecutively before a new luminance was selected. Four to 18 different luminances were used. The second protocol used a white CRT monitor at a photopic level (mean luminance = 300 cd/m 2 ). A sinusoidal grating pattern of nearly optimal spatial frequency for the cell was square wave modulated in counterphase at 1 Hz. Modulation was repeated for 39 cycles at each of 15-23 different contrasts ranging from 1-60%. To compare data from cat with the alternative outcomes (direct proportionality or no relationship), we compare the group mean slopes of lines fit to the data from each cell. The tests used incorporate an assumption that the slopes are distributed normally. The slopes of the principal component lines that we fit to our data present skewed distributions; a logarithmic transform (presenting the logarithm of the slope, rather than the slope itself) normalized the data (Levine et al., 1988). Note that the antilogarithm of the mean (of the logarithms) of the slopes is the geometric mean slope.
temporally modulated in cosine phase, it was difficult to extract data for |-s periods.) Each function was fit with a principal component line by linear regression; an example is shown in Fig. 1. All of the slopes obtained were positive, indicating that, as for the goldfish, the variance of rate increases as the rate increases. The geometric mean slopes were significantly less than unity (|-s periods: mean slope = 0.60; t = - 4 . 9 8 , 6 d.f.; P < 0.005; |-s periods: mean slope = 0.71; t = - 3 . 0 4 , P < 0.05), indicating a different relationship than the direct proportionality reported for cat visual cortex. These mean values for the cat are quite similar to those reported for goldfish (Levine et al., 1988; see Introduction). We could detect no tendency for a systematic difference according to cell type (X or Y cell, ON or OFF center). Similarly, there was no noticeable difference (due either to the lighting level or the difference in protocol) between responses recorded in mesopic and those recorded in photopic conditions. The two cells recorded in both conditions showed little difference between conditions. The slope was steeper for ^-s than for |-s periods in all but two of the nine cat ganglion cells from which both could be obtained, and a paired /-test showed the difference between the slopes significant (f = 2.27, 8 d.f., P < 0.05). The distribution of the differences is not normal, but the nonparametric Wilcoxon test gives the same outcome {T = 3, N = 8, P < 0.05). Although the variance of rate was not linearly related to the
2 -
Results Data were obtained from 12 cat retinal ganglion cells; there were nine ON- and three OFF-center cells, seven X and five Y cells, with all possible combinations represented. Seven cells were tested only in the mesopic state, three only in the photopic state, and two under both conditions. For comparison to goldfish, we use only the data collected in the mesopic state (3 ON-X, 2 ON-Y, and 2 OFF-X cells). For each cell tested, the logarithm of the variance of rate in each j-s period synchronized to the modulation was plotted against the logarithm of the mean rate in that period. The rate was taken as the number of impulses produced during each period, divided by the length of the period. We rejected periods in which the mean count was less than two.* There were thus up to four data points (one for each |-s period) for each stimulus strength used. For cells tested in mesopic lighting, a second function was produced on the same axes by dividing each cycle into two |-s periods; each stimulus strength contributed up to two points to this function. (Since the photopic stimulus was
0
1 -
o
1 Log firing rat©
•Since the count in any period must be an integral number of impulses, the variance of count when the mean is small is defined by the statistics of small numbers. Periods of very low mean rate must necessarily have low variance; including such periods would force the conclusion that variance increases with rate. By excluding periods with a small count, we avoided this potential confounding of our results. Note that if the mean rate is zero (as was the case for many of our rejected periods) the variance is also zero, and the point would have been located at (—oo,—oo) on our logarithmic axes.
Fig. 1. Log variance of firing rate vs. log mean rate of a cat retinal ganglion cell in mesopic illumination. Data analyzed in | - s periods (solid squares) and 2-s periods (open circles). Firing rate is in impulses/s; and variance of rate is in (impulses/s) 2 . Stimulus was a flashed spot, 0.5 deg in diameter, repeated at 1 Hz for 34 cycles at each of five luminances. Principal component lines fit to each duration have slopes of 0.62 and 0.84, respectively. ON-X cell; mean maintained rate = 24.8 impulses/s; and CV = 0.62.
279
Variability of responses
histograms) accounts for much of the scatter of the variance. Nevertheless, the main effect of increased variance of rate with mean rate is robust. 2 -
Discussion The analyses of responses from cat retinal ganglion cells indicate a qualitative similarity in the behavior of ganglion cells in fish and cat. The ganglion cells in the two species both present a nonlinear relationship when similarly tested. The variance of rate increases with rate but less than proportionally, with the relationship often depending upon the duration of the period in which the responses are measured. We conclude that an increase in the variability of responses with response size is not peculiar to cortical cells or to ganglion cells in the fish. Previous studies of the maintained discharges of retinal ganglion cells in cat indicated that the variance of rate should be independent of mean rate (Robson & Troy, 1987; Levine, 1987). We have proposed a model that reconciles the apparent contradiction between the results presented in this communication with the expectation from maintained discharges (Levine & Zimmerman, 1992).
1 -
0
0 -
Log fir
rote
Acknowledgments
Fig. 2. Log variance of firing rate vs. log mean firing rate of another cat retinal ganglion cell in photopic illumination. Data analyzed in \-s periods only. Sinusoidal grating is 1 cycle/deg; 39 cycles of counterphase modulation (1 Hz) at each of 19 contrasts. ON-X cell; mean maintained rate = 47.2 impulses/s; and CV = 0.28. Conventions are as in Fig. 1.
Research was supported by the NH&MRC of Australia. M.W. Levine was supported in part by a Senior International Fellowship TWO1317 from the Fogarty International Center of NIH. The analyses were supported in part by NIH Grant EY6163, the Rush University Committee on Research, the Chicago Community Trust, and the Regenstein Foundation. We are also grateful to Dr. Laura Frishman for her suggestions and comments on an earlier version of this manuscript.
mean rate, it was not always best characterized by a fractional power function. Negatively accelerated curves such as that in Fig. 2 (data from another ganglion cell) were clearly present in four of the 12 cells, and were less distinctly evident in two others. The distribution of this feature across cell types was similar to the distribution of cell types in the sample, with all subtypes except OFF-Y showing at least one exemplar. The variability in the plots is considerable, making it difficult to characterize the exact relationship. Similar scatter is evident in the previously published work (Tolhurst et al., 1983). We have been unable to attribute the scatter to the order of presentation of the stimuli or to adaptation during the course of the experiment. There were no long-term drifts or nonstationarities evident in the data. Scatter due to sample size is expected to decline as the square root of number of cycles at each contrast, but we see a minimal decrease in scatter for fourfold increases in the number of cycles. There is apparently some other determinant of variance of rate in addition to the strong effect of mean rate upon variance of rate; preliminary analyses suggest that the temporal structure of the responses (the relative amplitudes of the peaks and plateaus of the peristimulus-time
References CLELAND, B.G. & LEE, B.B. (1985). A comparison of visual responses of cat lateral geniculate nucleus neurones with those of ganglion cells afferent to them. Journal of Physiology 369, 249-268. DEAN, A.F. (1981). The variability of discharge of simple cells in the cat striate cortex. Experimental Brain Research 44, 437-440. LEVINE, M.W. (1987). Variability in the maintained discharges of retinal ganglion cells. Journal of the Optical Society of America A 4, 2308-2320. LEVINE, M.W. & ZIMMERMAN, R.P. (1992). A model for the variability of maintained discharges and responses to flashes of light. Biological Cybernetics, June 1992 (in press). LEVINE, M.W., ZIMMERMAN, R.P. & CARRION-CARIRE, V. (1988). Vari-
ability in responses of retinal ganglion cells. Journal of the Optical Society of America A 5, 593-597. ROBSON, J.G. & TROY, J.B. (1987). The nature of the noise in cat retinal ganglion cells. Journal of the Optical Society of America A 4, 2301-2307. TOLHURST, D.J., MOVSHON, J.A. & DEAN, A.F. (1983). The statistical
reliability of signals in single neurons in cat and monkey visual cortex. Vision Research 23, 775-785. TOLHURST, D.J., MOVSHON, J.A. & THOMPSON, I.D. (1981). The depen-
dence of response amplitude and variance of cat visual cortical neurones on stimulus contrast. Experimental Brain Research 41, 414-419.
