Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex

• Published in: Philosophical transactions of the Royal Society of London. Series B. Biological sciences 2001-03, Vol.356 (1407), p.299-330
• Main Authors: Bressloff, Paul C., Cowan, Jack D., Golubitsky, Martin, Thomas, Peter J., Wiener, Matthew C.
• Format: Article
• Language: English
• Subjects: Animals; Contours; Coordinate systems; Eigenfunctions; Eigenvalues; Flicker Phosphenes; Hallucination; Hallucinations; Horizontal Connections; Humans; Linear Models; Mathematical lattices; Models, Biological; Neural Modelling; Planforms; Rotation; Symmetry; Visual cortex; Visual Cortex - physiology; Visual fields; Visual Imagery
• ISSN: 0962-8436; 1471-2970
• DOI: 10.1098/rstb.2000.0769

. 1.33–2 mm. We next introduce a mathematical description of the large–scale dynamics of V1 in terms of the continuum limit of a lattice of interconnected hypercolumns, each of which itself comprises a number of interconnected iso–orientation columns. We then show that the patterns of interconnection in V1 exhibit a very interesting symmetry, i.e. they are invariant under the action of the planar Euclidean group E(2)—the group of rigid motions in the plane—rotations, reflections and translations. What is novel is that the lateral connectivity of V1 is such that a new group action is needed to represent its properties: by virtue of its anisotropy it is invariant with respect to certain shifts and twists of the plane. It is this shift–twist invariance that generates new representations of E(2). Assuming that the strength of lateral connections is weak compared with that of local connections, we next calculate the eigenvalues and eigenfunctions of the cortical dynamics, using Rayleigh–Schrödinger perturbation theory. The result is that in the absence of lateral connections, the eigenfunctions are degenerate, comprising both even and odd combinations of sinusoids in ϕ, the cortical label for orientation preference, and plane waves in