Abstract
The primary visual cortex (V1) can be partitioned into fundamental domains or hypercolumns consisting of one set of orientation columns arranged around a singularity or ``pinwheel'' in the orientation preference map. A recent study on the specific problem of visual textures perception suggested that textures may be represented at the population level in the cortex as a second-order tensor, the structure tensor, within a hypercolumn. In this paper, we present a mathematical analysis of such interacting hypercolumns that takes into account the functional geometry of local and lateral connections. The geometry of the hypercolumn is identified with that of the Poincaré disk $\mathbb{D}$. Using the symmetry properties of the connections, we investigate the spontaneous formation of cortical activity patterns. These states are characterized by tuned responses in the feature space, which are doubly-periodically distributed across the cortex.
Highlights
The formation of steady state patterns through Turing mechanism is a well-known phenomenon [50, 33]
In this paper we have analyzed a spatialized network of interacting hypercolumns in the context of textures perception in the primary visual cortex
Such a network is described by Wilson-Cowan neural field equations set on an abstracted cortex R2 × SPD(2, R), where the feature space SPD(2, R) is the set of structure tensors
Summary
The formation of steady state patterns through Turing mechanism is a well-known phenomenon [50, 33]. It occurs when a homogeneous state of a system of reaction-diffusion equations defined on the Euclidean plane becomes neutrally stable when a bifurcation parameter reaches a critical value. Any Fourier mode whose wave vector has critical length is a neutral stable mode and a consequence of the rotational symmetry of the system is that the kernel of linearized problem at the bifurcation point is infinite dimensional. Symmetry, equivariant bifurcation, pattern formation, Poincare disk.
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