Population coding under normalization

A common computation in visual cortex is the divisive normalization of responses by a pooled signal of the activity of cells within its neighborhood. From a geometrical point of view, normalization constraints the population response to high-contrast stimuli to lie on the surface of a high-dimensional sphere. Here we study the implications this constraint imposes on the representation of a circular variable, such as the orientation of a visual stimulus. New results are derived for the infinite dimensional case of a homogeneous populations of neurons with identical tuning curves but different orientation preferences. An important finding is that the ability of the population to discriminate between any two orientations depends exclusively on the Fourier amplitude spectrum of the orientation tuning curve. We also study the problem of encoding by a finite set of neurons. A central result is that, under normalization, optimal encoding can be achieved by a finite number of neurons with heterogeneous tuning curves. In other words, increasing the number of neurons in the population does not always allow for an improved population code. These results are used to estimate the number of neurons involved in the coding of orientation at one position in the visual field. If the cortex were to code orientation optimally, we find that a small number (∼4) of neurons should suffice.