Response of Population of Cells

We here consider the response of a population of ganglion cells distributed in the 2D space. To simplify the issue, we ignore the temporal aspect of their responses, so that the RF structure, such as a center-surround structure, of a cell located at x=x₀, y=y₀ can be represented by f(x-x₀, y-y₀). The response of this cell to some spatial stimulus s(x,y) can be obtained by

$$r(x_0, y_0)=\int \int s(x,y) f(x-x_0, y-y_0) dx dy$$

In general, the response of a spatially distributed population of cells of similar receptive fields can be expressed as

$$r(x,y)=\int \int s(u,v) f(u-x, v-y) du dv$$

This is the correlation of the stimulus s(x,y) and the RF function f(x,y). Recall that this correlation, as discussed earlier, also represents the response of a single cell to the same stimulus at different locations. As the receptive field is central symmetric, i.e.,

f(x,y)=f(-x,-y)

the correlation is equivelant to the convolution:

$$r(x,y)=\int \int s(u,v) f(u-x, v-y) du dv$$

This correlation model can give a possible explanation to the Mach band effect, as shown in the figure. This result can be verified by a simplified version of the problem. The center-surround RF is modeled by [-1, 3, -1] and the stimulus by [1, 1, 1, 3, 3, 3]. The response as the correlation of these two discrete functions is obtained as [1, 1, -1, 5, 3, 3], clearly showing the Mach band effect. The Hermann grid effect can be similarly explained.

web demo on Mach band