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The Journal of Neuroscience, September 15, 2002, 22(18):8334-8345
Signals in Macaque Striate Cortical Neurons that Support the Perception of Glass Patterns
Matthew A. Smith1, Wyeth Bair1, 2, and J. Anthony Movshon1, 2 1 Center for Neural Science and 2 Howard Hughes Medical Institute, New York University, New York, New York 10003
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
Glass patterns are texture stimuli made by pairing randomly placed dots with partners at specific offsets. The strong percept of global form that arises from the sparse local orientation cues has made these patterns the subject of psychophysical investigations, yet neuronal responses to Glass patterns have not been studied. We measured the responses of neurons in macaque striate cortex (V1) to dynamic, translational Glass patterns as a function of dot separation and dot-pair orientation. Responses were selective, but were on average more than an order of magnitude weaker than responses to sinusoidal gratings. Response and selectivity were greatest when the dot-pair orientation matched that of the preferred grating and when dot separation was between one-quarter and one-half of the spatial period of the optimal grating; changing the dot-pair separation or inverting the contrast of one of the dots radically changed the orientation selectivity. We computed the expected responses for a receptive field model to translational Glass patterns and found that the complexity of our V1 tuning curves could be understood in terms of the responses of linear filters to pairs of dots. This modeling connects our understanding of V1 receptive fields as rectified, quasi-linear filters with results from psychophysical studies of Glass patterns. Our results provide a basis for studying how subsequent visual areas integrate weak, local signals into global form percepts.
Key words: Glass patterns; macaque monkey; primary visual cortex; V1; random dots; orientation selectivity; linear filter; spatial frequency
INTRODUCTION
Glass patterns (Glass, 1969
; Glass and Perez, 1973
Consideration of the structure of Glass patterns suggests that they are processed in two stages. The first stage must identify local orientation cues in the otherwise random pattern, and the second stage must combine those local signals to extract larger-scale global structures. The local cues for orientation in Glass patterns are individually quite weak, because each dot pair is embedded in a random noisy background. The absence of strong local contours means that the first stage of orientation-selective cells in the cortex might provide sparse, irregular signals; knowledge of these signals is a prerequisite for studying their integration by neurons tuned for global form.
In this paper we report on the responses of neurons in striate cortex (V1), the earliest neurons in the visual pathway with the orientation selectivity needed to begin to parse Glass patterns. V1 receptive fields are quite small compared with the Glass patterns typically used in perceptual experiments and so would typically contain only a small part of the pattern, as demonstrated by the square apertures in Figure 1. A small aperture over any type of extended Glass pattern is approximated well by a translational pattern. We therefore reasoned that a first account of Glass pattern responses in V1 could be obtained by studying translational patterns like the one in the top panels of Figure 1.
We also simulated the response of V1 neurons to such Glass patterns and derived the response of an oriented filter to arbitrary translational Glass patterns. Simulations of rectified, linear spatial receptive fields (Movshon et al., 1978a
MATERIALS AND METHODS
Electrophysiology. We recorded extracellularly from single units in primary visual cortex of 12 Cynomolgus macaques (Macaca fascicularis) and 1 pig-tailed macaque (M. nemestrina), ranging in weight from 3.0 to 5.0 kg.
The techniques used in our laboratory for recording from the visual cortex of anesthetized, paralyzed primates have been reported in detail elsewhere (Carandini et al., 1997
1 · hr
In most experiments, we recorded with tungsten-in-glass microelectrodes (Merrill and Ainsworth, 1972
Visual stimulus generation. We displayed all visual stimuli at a resolution of 1024 × 731 pixels and a video frame rate of 100 Hz on either a Nanao T550i or an Eizo T550 monitor. We used look-up tables to correct for nonlinearities in the relation between input voltage and phosphorluminance in the monitors. We generated drifting sinusoidal grating stimuli with a Cambridge Research Systems VSG 2/2 board (Kent, UK) running on an Intel ×86-based host computer and random dot stimuli with a Silicon Graphics workstation. The mean luminance of the display was ~33 cd/m2 when gratings were displayed. All of the gratings were presented at 100% contrast in a circular aperture surrounded by a gray field of the average luminance.
For each isolated neuron, we began by mapping its receptive field for each eye on a tangent screen by hand. We determined the dominant eye to be that which yielded the larger response and occluded the other eye. Using a front surface mirror, we brought the receptive field into register with the center of the video monitor placed between 80 and 180 cm from the animal's eye, where it subtended between 10 and 22° of visual angle. We then proceeded with experiments under computer control.
Experiments consisted of multiple blocks of stimuli, each composed of a randomly ordered group of all the stimuli in a set. All stimuli within a block were equal in duration and were separated by presentation of a uniform blank background (mean gray for sinusoidal stimuli and mean gray or black for Glass pattern stimuli, depending on the experiment) for ~1.5 sec.
We characterized the response properties of the cell to gratings in this order: (1) orientation and direction tuning; (2) spatial frequency tuning; (3) temporal frequency tuning; and (4) size tuning. We chose a small patch of optimized grating and adjusted the vertical and horizontal position by hand to obtain the maximal response. This patch was taken to be centered in the receptive field. We classified cells as simple or complex using the standard F1/DC ratio (Movshon et al., 1978a
Glass pattern characterization. Glass pattern stimuli consisted of randomly positioned dot pairs in which dot separation and pair orientation were constant across all pairs on a given trial (Fig. 1). On each video frame (every 10 msec), a new set of dot pairs was plotted that was independent of the previous frame. Thus, these patterns had local spatial structure within frames but no coherent spatial structure or motion between frames. We used these dynamic patterns to randomize the positions of the dots in the pattern over time and to minimize retinal adaptation at particular dot positions. All dot patterns were presented within a circular aperture.
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Figure 1. Examples of Glass pattern stimuli. A, A translational Glass pattern consists of a field of random dots shifted by a distance r in a direction and added to itself. B, A concentric, or rotational, Glass pattern is created by rotating a field of random dots about the center. C, A radial, or expansion, Glass pattern, is obtained by multiplying the radial component of each dot by a constant. The square apertures indicate hypothetical receptive fields of V1 neurons that would contain only a portion of the stimulus. The region within the aperture for the rotational (B) and radial (C) patterns can be approximated by a translational pattern like that in A.
All Glass pattern stimuli were presented for 1 sec, immediately preceded and followed by 500 msec periods of dynamic random dots (with the same mean luminance on each video frame). This allowed us to avoid contamination of the Glass pattern response by any response to the luminance change caused by the onset of the dots. An additional 2 sec stimulus of purely random dots and a blank screen were included in each block of trials, from which we measured baseline responses.
Dots were usually presented at maximum contrast (i.e., white dots on a black background). The maximum luminance was 68.4 cd/m2, and the minimum was near 0.0 cd/m2. The mean luminance of the display was ~0.2 cd/m2 when white dots were displayed on a black background. In some experiments, we presented maximum luminance ("white") or minimum luminance ("black") dots on a mid-gray background (34.2 cd/m2). All of these stimuli, in which the dots all have the same luminance, are called same-polarity patterns. We also used opposite-polarity patterns, in which half of the dots were white, the other half were black, and the background was mid-gray. For opposite-polarity Glass patterns, each pair consisted of one white and one black dot. The white dot in each pair was chosen at random. Dot size was typically 0.04° (range, 0.03°-0.12°) and density was typically 200 dots per degrees squared per second (range, 100-800). In this range, human observers readily perceive Glass patterns, and variations in dot density have no significant impact on perception (Alliston et al., 2001
For each neuron, we presented a matrix of Glass patterns with eight orientations (
/4 to 3
Quantitative measures. We collected responses for gratings, Glass patterns, and random dots as a function of stimulus size. We fit data from these size tuning curves with the integral of a difference of Gaussians (DeAngelis et al., 1994
To characterize orientation tuning curves, we determined the selectivity and preferred angle by calculating a tuning bias vector (Leventhal et al., 1995
To measure selectivity, we calculated the summed response vector:
and normalized its magnitude by the summed magnitude of all the response vectors:
The selectivity index is 0 for a cell responding equally at all orientations and 1 for a cell that responds only to a single orientation. To estimate the significance of each selectivity estimate, we used the permutation technique described in O'Keefe and Movshon (1998)
To estimate analogous quantities for tuning curves with four lobes (rather than two), which we term "quadropoles," we modified the first two equations simply by substituting 4
RESULTS
We made extracellular recordings from 113 neurons (38 simple, 75 complex) in the primary visual cortex of 13 macaque monkeys. We characterized each cell with drifting sine wave gratings before testing with dynamic, translational Glass patterns. We did not test neurons that were not selective for the orientation of gratings, but did not otherwise exclude neurons from study with Glass patterns. Not all of the experiments described herein were performed on every cell. In our population of cells, the distribution of orientation bandwidths (mean = 64.5°, SD = 25.9°) and spatial frequency peak (mean = 2.1 c/deg, SD = 1.2 c/deg) and bandwidth (mean = 2.2 octaves, SD = 0.7) were similar to those found by other investigators at the eccentricities of our recorded cells (DeValois et al., 1982
To provide a framework for interpreting the neuronal responses to Glass patterns, we begin by describing theoretical responses of oriented filters, designed to represent V1 receptive fields, to these stimuli. The intuition gained from this exercise guides our data analysis.
Tuning of oriented filters to Glass patterns
Figure 2A shows a linear spatial filter that represents the spatial receptive field of a V1 simple cell as a Gabor function (Mar
elja, 1980
" fall in opposite-signed regions of the receptive field (Fig. 2A). When the pair is aligned to the receptive field, however, the dots will tend to reinforce because they fall in same-signed regions (Fig. 2A, plain and + dots). The cancellation and reinforcement as a function of
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Figure 2. Modeling the response of a V1 cell to Glass patterns. A, The spatial profile of a V1 simple cell receptive field is modeled as a Gabor function (a Gaussian × a sinusoid) with an aspect ratio of 0.6, frequency of 2.16 c/deg, and width (1 SD of Gaussian) of 0.16°. These are typical values for a macaque V1 simple cell (DeValois et al., 1982
We have so far considered only responses to a pair of dots at a particular location in the receptive field, but dot pairs are placed randomly in Glass patterns. Local responses to randomly positioned dot pairs will tend to cancel for a purely linear filter like that in Figure 2 because dots are as likely to fall in the inhibitory region as in the excitatory region. Cortical cells, however, produce rectified responses, which we simulated by applying a threshold to the model output (Movshon et al., 1978a
The model makes several interesting predictions. (1) The shape of the orientation tuning curve for Glass patterns should depend on r/
Orientation tuning of V1 cells to Glass patterns
Figure 3 shows the orientation tuning curves of a V1 cell plotted in polar coordinates. For grating stimuli, 180° opposite points indicate opposite directions of drift for the same orientation. However, for Glass pattern stimuli, which have no coherent motion, we reflected the orientation data to the 180° opposite point (which has the same orientation). For two example cells, Figure 3, A and B, we show tuning for the direction of a drifting sinusoidal grating (dotted lines). The Glass pattern orientation tuning curve for r
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Figure 3. A comparison of orientation tuning for gratings and Glass patterns for two example cells. A, Polar plots of orientation tuning curves are shown for a simple cell responding to drifting sinusoidal gratings (dotted line) and Glass patterns (r
We collected orientation tuning curves for Glass patterns at multiple values of r. The distribution of the values of r/
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Figure 4. Preferred Glass pattern configuration across the population. A, The frequency histogram of r/
To compare the shape as well as the peak of orientation tuning for Glass patterns across our population of cells, we scaled each tuning curve to have a maximum value of 1. The averages of all such normalized curves for r
To further explore the prevalence of four-lobed orientation tuning and its relationship to r, we modified the orientation selectivity index (see Materials and Methods). The usual index is designed to be sensitive to peaks and troughs differing by 180°, so we performed an analogous calculation sensitive to peaks and troughs differing by 90° to find values for the optimal "quadropole angle" and "quadropole selectivity" for r
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Table 1. Selectivity of orientation tuning for Glass patterns Our model predicts that cells with narrow tuning for orientation and spatial frequency should be the most likely to show four-lobed tuning (see Appendix). We used these two measures of the selectivity of the response of a cell to gratings and correlated them with the quadropole selectivity index measured when r
In summary, significant four-lobed tuning was present in roughly one-quarter of our cells and was weakly correlated with tuning bandwidth. As we show in the Appendix, the expected strength of four-lobed tuning is relatively modest, even for narrowly tuned cells. We would therefore expect four-lobed tuning to be prominent only in a fraction of V1 cells. The reliability of this effect across the population is confirmed by the average tuning curve for r
Response strength
V1 cells showed clear preferences for Glass pattern orientation, but their responses to these patterns were generally much weaker than to drifting gratings. Because Glass pattern tuning involves modulation about a mean rate of response to random dots, we took the difference between the maximum and minimum responses in a Glass pattern tuning curve as the modulation of response. We compared this with the modulation of response in the grating tuning curve. For each cell, Figure 5 plots the modulation of response (peak-trough) to Glass patterns against that to gratings. All points fell well below the unity line, indicating that the neurons responded more vigorously, often by more than an order of magnitude, to gratings (complex cells, geometric mean = 52.1 vs 8.6 spikes/sec; simple cells, geometric mean = 28.1 vs 2.8 spikes/sec). The marginal distribution of the ratios of responses for each cell is shown in the oblique histograms at the top right. The geometric mean of the response ratio of gratings to Glass patterns was 10.1 for simple cells and 6.0 for complex cells.
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Figure 5. Responses to Glass patterns were weaker than those to gratings. For both complex cells (filled circles) and simple cells (open circles), we defined the response modulation to be the peak minus the trough of the orientation tuning curve. For Glass patterns, we used the orientation tuning curve taken at the optimal dot separation. All points fall below the diagonal line of equality, indicating that all cells responded better, i.e., had more modulated tuning curves, to gratings than to Glass patterns. The response ratio (oblique histograms) was ~10 on average. The absolute peak firing rates for Glass patterns were also much lower than those for gratings (complex cells, mean = 28.5 spikes/sec; simple cells, mean = 8.6 spikes/sec).
From a linear spatial filter we can predict not only the orientation tuning to Glass patterns but also the response ratio. The ratio in response that we observed was roughly half as large as our simulations predicted for a range of dot size and density values that spanned the range used experimentally. Thus the cells we studied responded more strongly to Glass patterns than the linear model predicts. This could be caused by response saturation to the high-contrast grating stimuli or because the cells operate in a state of high gain when confronted with the dot stimuli, which have low-contrast energy in any given spatial frequency band (Carandini et al., 1997
Opposite-polarity dot pairs
A feature of Glass pattern perception that suggests a limitation of local orientation sensors is the relative weakness of percepts elicited by patterns in which the two members of a dot pair are of opposite contrast polarity, i.e., one black and one white dot on a gray background (Glass and Switkes, 1976
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Figure 6. A comparison of the responses to same- and opposite-polarity Glass patterns. A, The same-polarity Glass pattern shown here is displayed on a mean luminance background, as opposed to the usual black background, for comparison with opposite-polarity Glass patterns. B, This opposite-polarity Glass pattern has the same dot separation, orientation, and dot placement as in A, but it does not elicit the same percept of oriented structure. C, D, For two example cells, orientation tuning curves for opposite-polarity Glass patterns (thick solid line) are plotted with tuning curves to same-polarity Glass patterns displayed on a black background (dashed lines). Response amplitude for opposite-polarity patterns is less because of the reduced contrast, and there is an ~90° change in the orientation preference for opposite-polarity patterns. As always, orientation is plotted relative to the preferred orientation for gratings, which is plotted as upward. E, F, A direct comparison of orientation tuning curves for opposite- and same-polarity Glass patterns displayed on mean gray backgrounds is shown here for two example cells. Responses for same-polarity patterns (gray lines for white dots, thin black lines for black dots) differ little in amplitude but significantly in orientation tuning when compared with responses to opposite-polarity patterns (thick black line). An error bar (±1 SEM) is shown at the peak of the opposite-polarity dot tuning curve. G, Across a population of 31 cells tested with opposite-polarity patterns, the absolute value of the difference between the preferred grating orientation and the preferred Glass pattern orientation is shown in a frequency histogram. The distribution of differences in optimal orientation is concentrated around 90° orthogonal to the similar distribution for same-polarity dots (Fig. 4B).
Our same-polarity patterns consisted of white dots on a black background. Opposite-polarity patterns must have a gray background, so the contrast in the stimulus was inevitably reduced. Because of the decreased contrast present, we expected responses to be weaker to opposite-polarity patterns than those we collected with same-polarity patterns, and in fact this was the case. Many cells did not respond at all to these patterns when we used small dots (0.04°), so we increased the dot size by a factor of 2-4 to increase responses. In some cells, we compared the responses to opposite-polarity patterns with those to same-polarity patterns with all white dots or all black dots on a mean gray background and found them to be similarly weak (Fig. 6E,F). Responses were approximately an order of magnitude less than those to same-polarity patterns, even with dot size increased. In addition, both of the same-polarity patterns showed tuning consistent with that for white dots on a black background (i.e., orthogonal to that for opposite-polarity patterns). Therefore, the reduction in response to opposite-polarity patterns (Fig. 6C,D) is caused by contrast reduction in the stimulus.
In Figure 6G we show the distribution of differences in optimal orientations for opposite-polarity Glass patterns relative to gratings when r
Effects of stimulus size
In Figure 1 we showed that within a small aperture, such as that likely to be viewed by V1 cells, any extended Glass pattern can be approximated by a translational pattern. However, responses to small patches of a Glass pattern might not be representative of those to large stimuli. For example, form-sensitive cells in higher cortical areas could affect V1 through feedback connections, perhaps enhancing their response. Alternatively, large translational Glass patterns could engage the orientation-selective mechanisms that mediate surround suppression, thereby causing a reduction in response. It is well established that increasing the size of a grating or bar stimulus beyond the classical receptive field engages a nonclassical surround that suppresses the response of V1 cells (Hubel and Wiesel, 1968
To test the effect of stimulus size on Glass pattern tuning, we performed two additional experiments. These data were collected in 4 of the 13 animals. In the first experiment (in 34 neurons), we collected size tuning curves to optimally configured Glass patterns, random (i.e., unpaired) dots, and optimally configured sinusoidal gratings (see Materials and Methods). Examples of the two main patterns of response that we observed are shown in Figure 7A,B. In Figure 7A, the response of the cell was suppressed by grating stimuli that were more than ~1° in diameter. The tuning curves for both types of dot stimuli showed no sign of suppression, i.e., response increased up to the maximum size tested. We observed this behavior in a minority of our cells. The example cell in Figure 7B shows a different behavior. Here, the size tuning curves for dots showed signs of suppression for the same range of sizes that produced suppression for gratings (e.g., >2°). At around the same size, we observed suppression for both types of dot stimuli.
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Figure 7. Responses to stimuli of different size. A, B, These two cells show suppression for large grating stimuli but differ in their response to dot stimuli of increasing size. The area tuning curves were collected at the optimal orientation and spatial and temporal frequency for gratings. For Glass patterns, the orientation and dot separation were set to be optimal. The cell in A shows an increased response as the dot patterns are increased in size up to 5°, whereas its response peaks to a grating of ~1°. The cell in B is suppressed beyond ~1° for both dot patterns and gratings. C, For most cells, the optimal size for Glass patterns is similar to that for gratings (most points fall near the line of equality). Some cells, like the one in A, prefer larger Glass pattern stimuli. On average, the optimal size for Glass pattern stimuli is larger than for gratings. D, On average, cells show more suppression for gratings than for Glass patterns. We calculated a suppression ratio, the peak response minus the response at the maximum size, divided by the response at the maximum size. It ranges from zero (for no suppression) to 1 (for complete suppression). Most cells fall near or below the equality line, indicating that suppression for gratings is usually similar to or higher than for Glass patterns. E, F, We collected orientation tuning curves for Glass patterns of two sizes (the optimal size for gratings and an extended pattern) in the same two cells. The thick and thin lines indicate response to the large and small dot patterns, respectively. The black lines indicate Glass pattern response, whereas the gray lines indicate the response to random dots. The curves are rotated so that the vector fit of the orientation tuning to gratings is set to zero. An error bar (±1 SEM) is shown at the peak of the small Glass pattern tuning curve.
For each size tuning curve, we found the best response to Glass patterns and compared it with the response to a random dot field of the same size. If a cell did not respond more to Glass patterns than to random dots by at least 1 SEM (indicating that the cell was not tuned for size), we removed it from further consideration.
We omitted from further analysis 10 of 34 cells, for which the peak response to Glass patterns was <1 SEM larger than the response to random dots. For the 24 remaining cells, we plotted the optimal size (Fig. 7C) and amount of suppression (Fig. 7D) for Glass patterns and gratings. In this population of cells, the mean optimal size for Glass patterns (1.7°) was larger than that for gratings (1.0°), and this difference was statistically significant (t test; p = 0.004). Responses to Glass patterns showed significantly less suppression on average (0.14) than those to gratings (0.28; t test; p = 0.007). Furthermore, Glass pattern and random dot tuning for size were highly correlated (Pearson's r = 0.71; p = 0.0001). Even as the cell showed suppression or summation for an increase in size, the firing rate difference between Glass pattern and random dot stimuli was maintained at a constant rate (Fig. 7A,B).
In the second experiment (in 43 neurons), we studied orientation selectivity for Glass pattern stimuli of two sizes, one that was confined to the CRF and one that included both the CRF and surround (as measured with gratings). The results from this experiment on two example cells (the same as in Fig. 7A,B) are shown in Figure 7E,F. Although the response increased with stimulus size in Figure 7E, the orientation preference and selectivity remained similar. In Figure 7F, the same is true, although the cell showed suppression with increasing stimulus size. On average across all cells, the peak firing rate for the smaller stimulus (24.8 ips) was significantly higher than for the larger pattern (21.8 ips; t test; p = 0.006). This was also true for unpaired random dots (mean = 20.0 ips vs 16.6 ips; t test; p = 0.002). In addition, there was no significant difference in either the optimal orientation (mean difference =
In summary, in no case did stimulation of the surround importantly modify the selectivity of the cell for Glass pattern orientation. Some cells had similar optimal sizes for Glass patterns and for gratings (Fig. 7B), whereas others had larger optimal sizes for Glass patterns because suppression for large gratings was weaker or absent for large Glass patterns (Fig. 7A). The behavior in this latter group of cells is consistent with the findings of Casanova (1993)
DISCUSSION
Both simple and complex cells in V1 typically gave weak but reliable orientation-selective responses to dynamic, translational Glass patterns. Neuronal selectivity qualitatively matched our predictions from receptive field models on the basis of oriented, linear filters. Orientation selectivity was generally best when the separation of the dots was approximately one-quarter to one-half the optimal spatial period of the receptive field. For larger dot separations (r
If, in fact, the local structure of Glass patterns is detected initially by V1 cells and then pooled in downstream areas, limitations imposed in V1 might be reflected in psychophysical data. Two types of data indicate that this is the case. Psychophysical measurements in humans (Wilson et al., 1997
A second psychophysical finding that has been related to the responses of V1 cells is the relative ineffectiveness of Glass patterns in which the two dots in a pair are of opposite contrast. Glass and Switkes (1976)
Other aspects of Glass pattern perception do not have obvious correlates in our V1 data. DeValois and Switkes (1980)
The links between psychophysical studies of Glass pattern perception and V1 receptive field structure help us to understand the limits of the first stage of form perception, but they cannot provide specific information about downstream stages of analysis that integrate information in Glass patterns to provide signals about global form. Yet, some recent psychophysical and physiological studies do provide clues about this second stage. Wilson and his colleagues (1997)
FOOTNOTES Received May 10, 2002; revised July 10, 2002; accepted July 11, 2002.
This work was supported by the Howard Hughes Medical Institute and by a research grant to J.A.M. from the National Institutes of Health (EY02017). M.A.S. was supported in part by a National Eye Institute Institutional Training Grant (T32-7136), and W.B. was supported in part by a grant from the Alfred P. Sloan Foundation. Adam Kohn provided helpful comments on this manuscript and, along with James R. Cavanaugh and Najib Majaj, assisted with some of the data collection. Suzanne Fenstemaker provided assistance with histology.
Correspondence should be addressed to J. Anthony Movshon, Center for Neural Science, New York University, 4 Washington Place, Room 809, New York, NY 10003. E-mail: movshon{at}nyu.edu.
APPENDIX
Before deriving an exact expression for the response of our receptive field model to translational Glass patterns, we will briefly describe a way to intuit the shape of the model's Glass pattern orientation tuning curve in the frequency domain. This approach makes use of the power spectrum of a field of dot pairs, thereby avoiding the position dependence of an individual dot pair, which limited the generality of the demonstration in Figure 2.
The response of a Gabor function (Fig. 8A) to a stimulus can be determined by multiplying the Fourier transform (FT) of the Gabor function with the FT of the stimulus. The FT of the filter in Figure 8A is a pair of Gaussians in the complex frequency domain, but we will consider only the power spectrum of the filter (the square of the modulus of the FT), which is a real-valued, symmetrical function shown in Figure 8B. The profile of the power spectrum reveals which spatial frequencies will have the strongest influence on the response of the filter. Next, we visualize the distribution of spatial frequencies in the Glass pattern stimulus (Fig. 8C) with the power spectrum of its FT (Fig. 8D). The power spectrum of a Glass pattern is a noisy sinusoidal grating in which the orientation of the grating depends on the orientation of dot pairs, the wavelength of the grating depends on the dot separation, and the noise is random white-noise determined by the random locations of the dot pairs (here, we do not model the circular aperture of the Glass pattern). For the vertically oriented pattern in C, a ridge of elevated power (D) runs horizontally through the regions of sensitivity of the Gabor filter shown in B. At this orientation, the stimulus will cause large fluctuations around a mean of zero in the output of the filter, but after rectification these fluctuations will lead to a large positive response. The strength of the response to any Glass pattern stimulus can therefore be estimated by observing how well its power spectrum aligns with that of the Gabor function. If the pattern in C is rotated, its spectrum will rotate, and the bands of high and low power will pass through the sensitive regions of the filter spectrum, yielding a bi-lobed tuning curve. For larger dot separation, the noisy grating in D will have more bands, causing the orientation tuning curve at larger dot separation to have more lobes. Finally, changing the contrast polarity of one dot in the pair shifts the noisy grating in the frequency domain by a quarter cycle. This accounts for the change in orientation tuning with opposite-polarity Glass patterns.
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Figure 8. Intuition of Glass pattern tuning from the Fourier domain. The column at left contains representations in the space domain, and the column at right contains the corresponding frequency domain representations (shown as power spectra). In the right column, white indicates an area with higher energy and black indicates lower energy. A, A Gabor function is typically used to model the spatial profile of a V1 simple cell receptive field. The vertical and horizontal axes represent space in the x and y directions, whereas the spatial period of the Gabor is indicated by
We will now derive an analytical expression for the response of a Gabor patch V1 receptive field to our Glass pattern stimulus as a function of orientation and dot separation. For a simple cell, the neuronal response was modeled as the half-wave rectified output of a linear filter. For a complex cell, the output of two linear filters with a 90° phase shift were rectified and summed. The receptive field model had an explicit spatial structure, whereas temporal integration was handled implicitly by setting the number of dot pairs that were integrated. Even- and odd-symmetric Gabor patch models, as well as complex and simple cell models, performed in qualitatively the same manner. Below we outline the derivation of the mean simple cell response for any number of dot pairs falling on an arbitrary receptive field profile.
The neuronal receptive field, f, was modeled by a Gabor function (a Gaussian × a sinusoid) as follows:
where
(1) w and
-functions:
