Document Type

Honors Thesis

Major

Mathematics

Advisor(s)

Peter Stechlinski

Committee Members

David Hiebeler, Benjamin King, Jane Wang

Graduation Year

May 2024

Publication Date

Spring 5-2024

Abstract

Neural activity in the brain involves a series of action potentials that represent “all or nothing” impulses. This implies the action potential will only “fire” if the mem- brane potential is at or above a specific threshold. The Wilson-Cowan neural mass model [6, 28] is a popular mathematical model in neuroscience that groups excita- tory and inhibitory neural populations and models their communication. Within the model, the on/off behavior of the firing rate is typically modeled by a smooth sigmoid curve. However, a piecewise-linear (PWL) firing rate function has been considered in the Wilson-Cowan model in the literature (e.g., see [5]). This function, however, is non- smooth, and cannot be analyzed using standard mathematical theory. In this thesis, we considered the Wilson-Cowan neural mass model using a nonsmooth PWL firing rate function and analyze its behavior using techniques from generalized derivatives theory. To accomplish this, we calculated the sensitivities of the model parameters in order to determine the parameters that most impact the dynamics of the model across a set of parameter values. We also determined the stability of the model to better understand the long-term behavior of the model. We then compared the results of these analyses to that of the Wilson-Cowan model with a smooth firing rate function.

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