Symmetry breaking and pattern formation in some neural field equations
Grégory Faye, CHOSSAT, Pascal, FAUGERAS, Olivier · OpenGrey (Institut de l'Information Scientifique et Technique) · 2012
The aim of this Thesis is to give a deeper understanding of pattern formation in neural field equations with symmetry, and to understand the significance of these symmetries in modelling the visual cortex. Neural fields equations are mesoscopic models that describe the spatio-temporal activity of populations of neurons. They were introduced in the 1970s and are often called the Wilson-Cowan-Amari equations in reference to their authors. From a mathematical point of view, neural fields equations are integro-differential equations set on domains particular to the modelled anatomical / functional properties. The first part of the Thesis is an introduction to mesoscopic modelling of the visual cortex and presents a model of the processing of image edges and textures. The second part is dedicated to the study of spatially periodic solutions of neural field equations, in different geometries, with applications to visual hallucination patterns. The results developed are general enough to be applied to other pattern formation problems. Finally, the last part is centred on the study of localized solutions of neural field equations set on unbounded domains.