A Center Manifold Result for Delayed Neural Fields Equations

A Center Manifold Result for Delayed Neural Fields Equations

We develop a framework for the study of delayed neural fields equations and prove a center manifold theorem for these equations. Specific properties of delayed neural fields equations make it impossible to apply existing methods from the literature concerning center manifold results for functional differential equations. Our approach for the proof of the center manifold theorem uses the origina...

Center Manifold Theory for Functional Differential Equations of Mixed Type

We study the behaviour of solutions to nonlinear autonomous functional differential equations of mixed type in the neighbourhood of an equilibrium. We show that all solutions that remain sufficiently close to an equilibrium can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. In addition, we provide a Hopf bifurcation theorem for s...

Kernel Reconstruction for Delayed Neural Field Equations

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of mod...

The Nehari Manifold for a Class of Indefinite Weight Semilinear Elliptic Equations

Center Manifold: a Case Study

Following Almgren’s construction of the center manifold in his Big regularity paper, we show the C regularity of area-minimizing currents in the neighborhood of points of density one, without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren’s construction in its full generality. 0. Introduction In this note we consider area-minimizing i...