Phase Equation for Patterns of Orientation Selectivity in a Neural Field Model of Visual Cortex

Abstract

In this paper we consider a neural field equation on L${\mathbb R}^2\times \mathbb{S}^1$, which models the activity of populations of spatially organized, orientation selective neurons. In particular, we show how spatially organized patterns of orientation tuning can emerge due to the presence of weak, long-range horizontal connections and how such patterns can be analyzed in terms of a reduced phase equation. The latter is formally identical to the phase equation obtained in the study of weakly coupled oscillators, except that now the spatially distributed phase represents the peak of an orientation tuning curve (stationary pulse or bump on $S^1$) of neural populations at different locations in ${\mathbb R}^2$. We then carry out a detailed analysis of the existence and stability of various solutions to the phase equation and show that the resulting spatially structured phase patterns are consistent with numerical simulations of the full neural field equations. In contrast to previous studies of neural fi...