Discrete Dynamics of Dynamic Neural Fields

Large and small cortexes of the brain are known to contain vast amounts of neurons that interact with one another. They thus form a continuum of active neural networks whose dynamics are yet to be fully understood. One way to model these activities is to use dynamic neural fields which are mathematical models that approximately describe the behavior of these congregations of neurons. These models have been used in neuroinformatics, neuroscience, robotics, and network analysis to understand not only brain functions or brain diseases, but also learning and brain plasticity. In their theoretical forms, they are given as ordinary or partial differential equations with or without diffusion. Many of their mathematical properties are still under-studied. In this paper, we propose to analyze discrete versions dynamic neural fields based on nearly exact discretization schemes techniques. In particular, we will discuss conditions for the stability of nontrivial solutions of these models, based on various types of kernels and corresponding parameters. Monte Carlo simulations are given for illustration.