Networks and Heterogeneous Media (NHM)

A spatialized model of visual texture perception using the structure tensor formalism

Pages: 211 - 260, Volume 8, Issue 1, March 2013      doi:10.3934/nhm.2013.8.211

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Grégory Faye - School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN 55455, United States (email)
Pascal Chossat - J-A Dieudonné Laboratory, CNRS and University of Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France (email)

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.

Keywords:  Neural field equations, symmetry, equivariant bifurcation, pattern formation, Poincaré disk.
Mathematics Subject Classification:  Primary: 37G40, 34C23; Secondary: 92B20.

Received: January 2012;      Revised: December 2012;      Available Online: April 2013.