Local/Global Analysis of the Stationary Solutions of Some Neural Field Equations
Résumé
Neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integro-differential equations. The solutions of these equations represent the state of activity of these populations when submitted to inputs from neighboring brain areas. Understanding the properties of these solutions is essential to advancing our understanding of the brain. In this paper we study the dependency of the stationary solutions of the neural fields equations with respect to the stiffness of the nonlinearity and the contrast of the external inputs. This is done by using degree theory and bifurcation theory in the context of functional, in particular, infinite dimensional, spaces. The joint use of these two theories allows us to make new detailed predictions about the global and local behaviors of the solutions. We also provide a generic finite dimensional approximation of these equations which allows us to study in great detail two models. The first model is a neural mass model of a cortical hypercolumn of orientation sensitive neurons, the ring model [O. Shriki, D. Hansel, and H. Sompolinsky, Neural Comput., 15 (2003), pp. 1809-1841]. The second model is a general neural field model where the spatial connectivity is described by heterogeneous Gaussian-like functions.