Intuition from the frequency domain

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 3-2.

The response of a Gabor function (Fig. 4-1A) to a stimulus can be determined by multiplying the FT of the Gabor function with the FT of the stimulus. The FT of the filter in Figure 4-1A 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 4-1B. 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. 4-1C) with the power spectrum of its FT (Fig. 4-1D). 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 panel C, a ridge of elevated power (panel 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 following 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 panel 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.

Figure 4-1: 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, while the spatial period of the Gabor is indicated by $\lambda$. B, The Fourier transform of a Gabor function is a pair of Gaussian blobs. The orientation, size, and separation of the blobs depend on the characteristics of the Gabor function. C, Our experiments used translational Glass patterns, composed of many dot pairs, like the one in panel C. D, Because the spatial pattern now contains many dot pairs with random positions, the Fourier representation is a noisy grating.