Constructing Numerical Scheme Using Python in Neural Field Model

Abstract

We analyze a two-population neuronal network model of the Wilson-Cowan type to inves- tigate the presence of localized stationary solutions called “bumps”. Specifically, we focus on a scenario where two distinct types of bump solutions exist: a narrow pair and a broad pair. To assess the stability of these bumps, we employ two different approaches: one that generalizes the Amari method and another based on a direct linearization procedure. These approaches yield consistent predictions, except for one significant difference. The narrow pair is generally unstable, while the broad pair remains stable for small and moderate values of the relative inhibition time. Interestingly, at a critical relative inhibition time, the broad pair typically undergoes a Hopf bifurcation, transitioning into stable breathers. Notably, in our numerical example, the broad pulse pair remains stable even when the inhibition time constant is three times longer than the excitation time constant. Consequently, our model findings contradict the assertion that slow excitation mediated by NMDA-receptors or similar mechanisms is nec- essary for the presence of stable bumps. In summary, our investigation of the two-population neuronal network model reveals the existence of unstable narrow bumps and stable broad bumps, except during a critical relative inhibition time when the broad bumps transform into stable breathers. These results challenge the idea that slow excitation mediated by NMDA-receptors is a prerequisite for stable bumps in our model