Some theoretical results for a class of neural mass equations

Pascal Chossat, Grégory Faye, Olivier Faugeras

Abstract: We study the neural field equations introduced by Chossat and Faugeras in their article to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincar'e disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, i.e. time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.

Mathematics Subject Classification ID

Hyperbolic and elliptic geometries (general) and generalizations (51M10) Convolution as an integral transform (44A35) Other nonlinear integral equations (45G10) Neural biology (92C20) Neural networks for/in biological studies, artificial life and related topics (92B20) Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations (34A12) Nonlinear differential equations in abstract spaces (34G20) Harmonic analysis on homogeneous spaces (43A85) Stability of solutions to ordinary differential equations (34D20) Conformal metrics (hyperbolic, Poincaré, distance functions) (30F45) Simulation of dynamical systems (37M05) Global stability of solutions to ordinary differential equations (34D23) Classical hypergeometric functions, ({}_2F_1) (33C05)

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