Tuning of oriented filters to Glass patterns

Figure 3-2A shows a linear spatial filter that represents the spatial receptive field of a V1 simple cell as a Gabor function (Marcelja, 1980); we chose the parameters to match the shape, orientation and spatial frequency selectivity of typical simple cells in monkey or cat V1 (DeValois et al., 1985,1982; Jones and Palmer, 1987; Foster et al., 1985). Light and dark shading represents sensitivity to light increments and decrements, respectively. The essential aspects of the tuning of such a filter to Glass patterns as a function of orientation and dot separation can be grasped by considering the alignment of a single pair of dots with the receptive field. First, consider the case where the dot separation, $r$, is half of the optimal spatial period, $\lambda$ (Fig. 3-2A). When such a dot pair is orthogonal to the receptive field, the signals elicited by the dots will tend to cancel. For example, the unmarked dot and the dot marked ``-'' fall in opposite-signed regions of the receptive field (Fig. 3-2 A). When the pair is aligned to the receptive field, however, the dots will tend to reinforce because they fall in same-signed regions (Fig. 3-2A, plain and ``+'' dots). The cancellation and reinforcement as a function of $\theta$ determine the orientation tuning curve for the filter (Fig. 3-2C, polar plot, taken from Fig. 4-2B in Ch. 4 where we derive the response for all values of $r$ and $\theta$). Now, consider the case of $r=\lambda$, where the dot separation matches the spatial period of the receptive field (Fig. 3-2B). The parallel alignment of the dot pair again causes response reinforcement, but the orthogonal alignment now escapes cancellation because one dot falls beyond the inhibitory flank of the filter. Around 30$^\circ$ from parallel, there is response cancellation. Thus, for $r=\lambda$, the model predicts the four-lobed tuning curve shown in Figure 3-2D. If the sign of one dot in each pair is inverted so that the dots are of opposite contrast, the orientation tuning curves will be inverted (Fig. 3-2E and F) because dot pairs that previously canceled will now reinforce, and vice versa. In Chapter 4, we provide an alternate approach to visualizing the tuning of a Gabor function to Glass patterns by examining the Fourier representation of both the filter and stimulus in the frequency domain, where Glass patterns have a convenient representation (Dakin, 1997b; DeValois and Switkes, 1980).

Figure 3-2: A, The spatial profile of a V1 simple cell receptive field is modeled as a Gabor function (a Gaussian times a sinusoid) with an aspect ratio of 0.6, frequency of 2.16 c/deg and width (one standard deviation of Gaussian) of 0.16$^\circ$. These are typical values for a macaque V1 simple cell (DeValois et al., 1982; Parker and Hawken, 1988; Foster et al., 1985) and are also representative of simple cells in cat area 17, which have similar structure (DeValois et al., 1985). Black represents negative values, white positive, and background gray is zero. Pairs of dots with separation, $r$, equal to half the spatial period, $\lambda$, of the grating are superimposed on the receptive field at several angles. Responses to dots in a pair aligned parallel to the grating reinforce (+), whereas responses to dots in a pair orthogonal to the grating cancel (-). B, For $r=\lambda$, however, both parallel and orthogonal alignments reinforce (+) whereas some intermediate angles cancel (-). C and D, Orientation tuning curves (response versus $\theta$) are plotted in polar coordinates for the $r$ values in A and B. For $r=\lambda/2$ the tuning is bi-lobed, similar to the classical tuning to edges and sine waves. For $r=\lambda$, the tuning becomes four-lobed. Gray circles represent responses to random dots. E and F, Orientation tuning curves similar to A and B, but for opposite-polarity (i.e., one white and one black dot). In E, the tuning is shifted by 90$^\circ$ and is wider for $r=\lambda/2$. In F, for $r=\lambda$, the selectivity is nearly abolished. Gray circles represent responses to opposite-polarity random dots (half black and half white).

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 3-2 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). Such a rectified output provides a signal related to the variance of the combined local activations. The tuning curves (Fig. 3-2C-F) are based on rectified responses averaged across all possible positions of a particular dot pair; the baseline for the simulated tuning curves (gray circles) is the response of the model to randomly placed dots of the same density as the Glass patterns. This model applies equally well to a Glass pattern stimulus comprised of many dot pairs and to single dot pairs, randomly placed. Our simulations were specifically designed to predict the responses of simple cells, but they also capture the responses of complex cells that sum the rectified outputs of linear filter subunits (Movshon et al., 1978b), and of even- and odd-symmetric filters (see Ch. 4).

The model makes several interesting predictions. (1) The shape of the orientation tuning curve for Glass patterns should depend on $r/\lambda$, the ratio of the dot separation to the preferred spatial period of the cell. (2) The response to an optimally oriented Glass pattern will exceed that to random dots, whereas the response to the least effective orientation will fall below that baseline. (3) For typical receptive fields, the greatest degree of orientation selectivity should occur for $r$ ranging from $\lambda/4$ to $\lambda/2$, and the maximal response should occur when the dot-pair orientation matches the classical preferred orientation of the cell. (4) A second mode of orientation tuning with four principal lobes is possible for relatively large dot separations; as detailed in Chapter 4, the strength of this mode depends on the specific properties of the linear filter used. (5) For Glass patterns made with opposite-polarity (black-white) dot pairs, the tuning is inverted compared to that of a same-polarity pattern. In particular, the preferred orientation will rotate to be orthogonal to the receptive field orientation, and orientation tuning will be broader and less modulated than for a same-polarity pattern of the same $r$. We will now consider how well these predictions hold for the responses of V1 neurons.