In this paper we analyze the global existence of
classical solutions to the initial boundaryvalue 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 coeffcients depending
on the probability ux through the boundary. We show by an
appropriate change of variables that this parabolic equation
with nonlinear boundary conditions can be transformed into a
non standard Stefan-like free boundary problem with a source
term given by a delta function. We 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 ....Preprin
