Rupture de symétrie et formation de structures dans certaines équations de champs neuronaux

Grégory Faye 1
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : 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 modeling the visual cortex. Neural field 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 field equations are integro-differential equations set on domains particular to the modeled anatomical / functional properties. The first part of the Thesis is an in- troduction to mesoscopic modeling 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 geome- tries, 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 centered on the study of localized solutions of neural field equations set on unbounded domains.
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Grégory Faye. Rupture de symétrie et formation de structures dans certaines équations de champs neuronaux. Mathématiques générales [math.GM]. Université Nice Sophia Antipolis, 2012. Français. ⟨tel-00850269⟩

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