Dimension Reduction: Establishing Proofs and Extending Applications to the Cowan-Wilson Model

Abstract

Models governed by systems of ordinary differential equations (ODEs) often produce complex and unpredictable behaviors. To address this, we can use dimension reduction techniques, which simplify these models, allowing for the retention of specific behaviors while greatly decreasing the cost of numerical solutions and, in some cases, enabling analytical derivations of sufficient conditions for the existence of nonzero fixed points of the model’s ODEs. This thesis reviews the state-of-the-art reduction theories and extends the established proofs by Wu et al., by providing necessary assumptions, lemmas, and a novel proof of an important proposition used in their work. We additionally verify and confirm Wu et al.’s findings and predictions for a cooperative version of the Cowan-Wilson model, which describes a population of neurons’ firing activity. We derived a one-dimensional reduction, inspired by Laurence et al., for a generalized Cowan-Wilson model, which we introduced in this thesis. Unlike the original, this generalized model can produce oscillatory behavior without external stimulus. A valuable finding is that the method of reduction does not depend on the specific form of the Cowan-Wilson function, allowing it to be applied to a broader class of nonlinear dynamics. Our reduction is able to predict system behavior effectively, given the network yields a unique reduction. However, the reduction parameter was not unique in approximately half of the networks studied, which saw a 66% increase in average error, suggesting these networks are inherently multi-dimensional. This opens the door for future research into the existence of a multi-dimensional reduction framework that could mitigate this discrepancy.