Nonlinear analysis methods in neural field models

• Main Author: Veltz, Romain
• Language: English
• Published: Université Paris-Est 2011
• Subjects: [INFO:INFO_OH] Computer Science/Other; [INFO:INFO_OH] Informatique/Autre; Neural fields; Propagation delays; Center manifold; Bifurcation; Hypercolumn; Neuronal illusion
• Online Access: http://tel.archives-ouvertes.fr/tel-00686695; http://tel.archives-ouvertes.fr/docs/00/68/66/95/PDF/TH2011PEST1056_complete.pdf

This thesis deals with mesoscopic models of cortex called neural fields. The neural field equations describe the activity of neuronal populations, with common anatomical / functional properties. They were introduced in the 1950s and are called the equations of Wilson and Cowan. Mathematically, they consist of integro-differential equations with delays, the delays modeling the signal propagation and the passage of signals across synapses and the dendritic tree. In the first part, we recall the biology necessary to understand this thesis and derive the main equations. Then, we study these equations with the theory of dynamical systems by characterizing their equilibrium points and dynamics in the second part. In the third part, we study these delayed equations in general by giving formulas for the bifurcation diagrams, by proving a center manifold theorem, and by calculating the principal normal forms. We apply these results to one-dimensional neural fields which allows a detailed study of the dynamics. Finally, in the last part, we study three models of visual cortex. The first two models are from the literature and describe respectively a hypercolumn, i.e. the basic element of the first visual area (V1) and a network of such hypercolumns. The latest model is a new model of V1 which generalizes the two previous models while allowing a detailed study of specific effects of delays