tailieunhanh - Journal of Mathematical Neuroscience (2011) 1:4 DOI 10.1186/2190-8567-1-4 RESEARCH Open

Journal of Mathematical Neuroscience (2011) 1:4 DOI RESEARCH Open Access Analysis of a hyperbolic geometric model for visual texture perception Grégory Faye · Pascal Chossat · Olivier Faugeras Received: 7 January 2011 / Accepted: 6 June 2011 / Published online: 6 June 2011 © 2011 Faye et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License Abstract We study the neural field equations introduced by Chossat and Faugeras 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. | Journal of Mathematical Neuroscience 2011 1 4 DOI 2190-8567-1-4 0 The Journal of Mathematical Neuroscience a SpringerOpen Journal RESEARCH Open Access Analysis of a hyperbolic geometric model for visual texture perception Grégory Faye Pascal Chossat Olivier Faugeras Received 7 January 2011 Accepted 6 June 2011 Published online 6 June 2011 2011 Faye et al. licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License Abstract We study the neural field equations introduced by Chossat and Faugeras 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 Poincaré 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 that is 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. Keywords Neural fields nonlinear integro-differential equations functional analysis non-Euclidean analysis stability analysis hyperbolic geometry hypergeometric functions bumps Mathematics Subject Classification 30F45 33C05 34A12 34D20 34D23 34G20 37M05 43A85 44A35 45G10 51M10 92B20 92C20 G Faye S P Chossat O Faugeras NeuroMathComp Laboratory INRIA Sophia Antipolis CNRS ENS Paris Paris France e-mail P Chossat Dept. of Mathematics University of Nice Sophia-Antipolis JAD Laboratory and CNRS Parc Valrose 06108 Nice Cedex 02 France Springer Page 2of51 .