Wednesday, August 5, 2020

4:00 p.m.

PhD Candidate: Afroditi Talidou

Co-Supervisors: Michael Sigal, Almut Burchard

Thesis title: Near-pulse solutions of the FitzHugh-Nagumo equations on cylindrical surfaces

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In 1961, FitzHugh [19] suggested a model to explain the basic properties of excitability, namely the ability to respond to stimuli, as exhibited by the more complex HodgkinHuxley equations [24]. The following year Nagumo et al. [42] introduced another version based on FitzHugh’s model. This is the model we consider in the thesis. It is called the FitzHugh-Nagumo model and describes the propagation of electrical signals in nerve axons. Many features of the system have been studied in great detail in the case where an axon is modelled as a one-dimensional object. Here we consider a more realistic geometric structure: the axons are modelled as warped cylinders and pulses propagate on their surface, as it happens in nature.

The main results in this thesis are the stability of pulses for standard cylinders of small constant radius, and existence and stability of near-pulse solutions for warped cylinders whose radii are small and vary slowly along their lengths. On the standard cylinder, we write a solution near a pulse as the superposition of a modulated pulse with a fluctuation and prove that the fluctuation decreases exponentially over time as the solution converges to a nearby translation of the pulse. On warped cylinders, we write a solution near a pulse in the same way as in standard cylinders and prove bounds on the fluctuation of near-pulse solutions.

A copy of the thesis can be found here: Talidou-thesis-draft

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