Multiple Timescales in the Fitzhugh-Nagumo ModeldissertationsBergland, ErikcreatorSandstede, BjornAdvisorMenon, GovindReaderCarter, PaulReaderBrown University. Department of Applied Mathematicssponsor202413, 137 p.born digitalThesis (Ph. D.)--Brown University, 2024thesesAbstract of Multiple Timescales in the Fitzhugh-Nagumo Model, by Erik Bergland, Ph.D., Brown University, October 2024. In addition to its importance to the field of biology, the Fitzhugh-Nagumo model of neuron activity is a well-established laboratory in the setting of multiple-timescale analysis. However, the model is often studied in a traveling-wave setting, in order to simplify the reaction-diffusion partial differential equations to a system of ordinary differential equations. In this work, we seek to understand whether solutions to the kinetic equations obtained by neglecting diffusion are attractors to the solutions of the full model in the case where initial data is nearly constant. Since these solutions are well understood via Fenichel theory and constant initial data implies that no diffusion occurs, we can obtain a useful picture of the behavior of the full model. We make use of pointwise estimates to establish boundedness of offsets from these solutions, or algebraic decay to the kinetic solutions when the initial data meets certain geometric constraints. A full analysis is possible while the solutions to the kinetic equations travel along fast jumps or near the locally invariant slow manifold. We also compute estimates of key quantities near points where normal hyperbolicity is lost and discuss their inadequacies for controlling the solutions to the PDE. In addition to the work on the Fitzhugh-Nagumo model, we also examine the behavior of weakly-coupled oscillators of Ginzburg-Landau type, where blowup analysis is crucial to proving the existence of snaking solutions.Differential equations, PartialGeometric Singular Perturbation TheoryOrdinary Differential EquationsMathematical analysisEnglishRPB20241015