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A study of wave propagation in the FitzHugh Nagumo system Paton, Kelly Marie


An excitable medium has two key properties: a sufficiently large stimulus provokes an even bigger response (excitability), and immediately following a stimulus the medium cannot be excited (refractoriness). A large class of biological systems from cardiac tissue to slime mold are examples of excitable media. FitzHugh Nagumo (FHN) is the canonical model of excitable media. Its two variables are the state of excitation and refractoriness of the one- or two-dimensional medium. Although one of the simplest models, FHN exhibits complex dynamics that have not been fully explored. For example, it supports a stable traveling pulse solution. However, this pulse can be destabilized by large perturbations. In Chapter 2 I explore a one-dimensional example where the perturbation is an increasing refractory profile. This perturbation can lead to collapse of the pulse depending on the steepness of the profile, as conjectured by Keener [Keener, J. (2004) J Theo. Bio. 230(4):459-73]. In Chapter 3 I consider a perturbation in two dimensions which can cause the stable traveling pulse to wrap around the perturbation and generate self-sustaining spiral activity. The one-dimensional example for exponential refractory profiles is explored numerically for a piecewise linear FHN system. Steep profiles lead to collapse while milder profiles allow propagation. The exponential profiles are used as bounds for more general profiles to predict where collapse and propagation will occur. I also make use of a singular FHN system in the limit ε→ 0 to provide insight into the behaviours of the full FHN system for small ε and small diffusion. I conclude this chapter by showing analytically that, in contrast to the full system, a wave in the singular system will propagate for any exponential refractory profile. The two-dimensional case is explored numerically in a FHN system. The use of a temporarily refractory region as a perturbation is a novel mechanism for generating spiral activity. Moreover, it is shown to be robust for refractory regions of a large area. This situation models the appearance of abnormal electrical activity in the heart. In particular, it models the appearance of abnormal electricity activity in undamaged cardiac tissue.

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