Analytical derivation of Glass pattern responses

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$^\circ$ 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 times a sinusoid) as follows:

$$f(x,y) = e^{-\frac{1}{2}\left(\frac{x^2}{\sigma_w^2} + \frac{y^2}{\sigma_h^2}\right)} \sin(x/\lambda),$$ (7)

where $\sigma_w$ and $\sigma_h$ set the receptive field width and height, $\lambda$ is the preferred spatial period, and the preferred orientation of the receptive field is vertical. One dot pair is represented as a pair of $\delta$-functions,

$$\Pi(x,y) = \delta(x+\Delta x,y+\Delta y) + \delta(x-\Delta x,y-\Delta y),$$ (8)

where

$$\Delta x = \frac{r}{2} \cos(\theta),$$

$$\Delta y = \frac{r}{2} \sin(\theta),$$

where $r$ is the dot separation and $\theta$ is the orientation of the line connecting the dots.

The response to a single dot pair falling at location $(x_0,y_0)$ on the receptive field is simply the integral of the product $f(x,y)\Pi(x-x_0,y-y_0)$; therefore, the response as a function of position for all possible dot-pair locations is given by the convolution:

$$S_1(x,y) = f(x,y) * \Pi(x,y).$$ (9)

From $S_1$, we can compute the probability density function (p.d.f.) for the response to a dot pair that falls randomly within a fixed region that contains the receptive field, $f$. In particular, given a uniform, random choice of the coordinates $(x_i,y_i)$ from within the stimulus aperture, let the probability that $S_1(x_i,y_i)=s$ be given by the p.d.f. $\rho^*_1(s)$. The p.d.f. for the response to $n$ dot pairs that fall independently and uniformly within the stimulus aperture is given by the $n$-fold auto-convolution of $\rho^*_1$, denoted $\rho^*_n$. For large numbers of dot pairs, application of the central limit theorem leads to an approximation for $\rho^*_n$:

$$\lim_{n \rightarrow \infty} \rho^*_n = {\rm G}(n\mu, \sqrt{n}\sigma),$$ (10)

where $\mu$ and $\sigma$ are the mean and standard deviation of $\rho^*_1$, and G is the Gaussian p.d.f. The rectification stage maps negative firing rates to zero, so the p.d.f. for the final neuronal response is

$$\rho_n(s) = \left\{\begin{array}{ll} 0 & \mbox{for } s < 0... ... = 0, \\ \rho^*_n(s) & \mbox{otherwise}. \end{array} \right.$$ (11)

The expected value, $M$, of the neuronal response for all $r$ and $\theta$ is then given by

$$M(r,\theta) = \int_0^\infty s \rho_n(s;r,\theta) \,ds,$$ (12)

where the notation $\rho_n(s;r,\theta)$ indicates $\rho_n$ computed for a particular value of $r$ and $\theta$. This function is simple to compute numerically for any spatial receptive field, $f$, that goes to zero beyond some finite region. Because a Gaussian extends indefinitely, we truncated our Gabor receptive fields beyond three standard deviations from the center.

Figure 4-2 shows $M$ plotted for three different model receptive fields (see legend for parameters). Receptive fields in macaque V1 show a wide range of selectivity for orientation and spatial frequency (DeValois et al., 1982). The three receptive fields were designed to represent the upper and lower extremes of this range (Fig. 4-2A and C) and its average (Fig. 4-2B). We varied the width of the receptive field envelope (one standard deviation of the Gaussian) to produce the model output in Figure 4-2A-C, while holding other parameters constant. The left column contains grayscale images which represent response strength - white shows high response and black shows low response. These plots can be interpreted as vertically stacked families of orientation tuning curves, parametric in dot separation. The right column shows polar plots of orientation tuning (similar to those in Fig. 3-2C-F) taken at different dot separations. These three model plots and the corresponding slices illustrate the variation in Glass pattern tuning across a plausible range of receptive field geometries. In cells with narrow orientation tuning, there is the potential for four-lobed tuning when $r\approx\lambda$ (Fig. 4-2A). Typical cells (Fig. 4-2B) show little or no tuning when $r=\lambda$, but have their strongest tuning when $r\approx\lambda/2$. Cells with broad tuning (Fig. 4-2C) show a similar lack of strong Glass pattern tuning.

Figure 4-2: The grayscale images in the left column show large responses in white and small responses in black for a range of $r$ and $\theta$. The polar plots in the right column show orientation tuning curves taken at different values of $r$ ($1/4$, $1/2$, $3/4$, $1$ and $3/2$ times $\lambda$, as indicated by the arrows). All three plots are generated from a receptive field with an aspect ratio of 0.6 and frequency of 2.16 c/deg. A, For a receptive field with narrow orientation tuning for gratings (width = 0.32$^\circ$), there is prominent four-lobed tuning when $r=\lambda$. B, Model response for a receptive field with typical orientation tuning for gratings (width = 0.16$^\circ$) show some four-lobed tuning, and the tuning with the highest selectivity and modulation is shifted to lower dot separations. This receptive field is the same as the one used for Figure 3-2. C, For a receptive field with broad orientation tuning for gratings (width = 0.09$^\circ$), there is a complete lack of tuning when $r=\lambda$. The region of highest selectivity and modulation is shifted even lower, near $r=\lambda/4$.