  We introduce and study a new model of interacting neural networks,
incorporating the spatial dimension (e.g. position of neurons across the
cortex) and some learning processes. The dynamic of each neural network is
described via the elapsed time model, that is, the neurons are described by the
elapsed time since their last discharge and the chosen learning processes are
essentially inspired from the Hebbian rule. We then obtain a system of
integro-differential equations, from which we analyze the convergence to
stationary states by the means of entropy method and Doeblin's theory in the
case of weak interconnections. We also consider the situation where neural
activity is faster than the learning process and give conditions where one can
approximate the dynamics by a solution with a similar profile of a steady
state. For stronger interconnections, we present some numerical simulations to
observe how the parameters of the system can give different behaviors and
pattern formations.
