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.

我们引入并研究了一种新的交互神经网络模型，其中包括空间维度（例如，整个皮层中神经元的位置）和一些学习过程。每个神经网络的动态都是通过经过的时间模型来描述的，也就是说，神经元是通过自上次放电以来经过的时间来描述的，所选的学习过程实质上是从赫比规则中得到启发的。然后，我们获得了一个积分微分方程组，在弱互连的情况下，我们可以通过熵方法和Doeblin理论，从中分析出稳态的收敛性。我们还考虑了神经活动比学习过程快的情况，并给出了可以通过具有近似稳态的解来近似动力学的条件。为了加强互连，