In this paper we analyze the global existence of classical solutions to the initial boundary value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary, which makes the equation nonlocal.
We show that by an appropriate change of variables it can be transformed to a non standard Stefan-like free boundary problem with a source term given by a delta function, and prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study the spectrum for the linear problem corresponding to uncoupled networks and its relation to Poincaré inequalities for studying their asymptotic behavior.
Copyright. All Rights Reserved.