Patterns on spatially structured domains

Schouten-Straatman, W.M. (2021)

Doctoral Thesis

We consider the propagation of electrical signals through nerve fibres. In these systems, it is well-known that the signal can only travel at appropriate speeds if the fibre is covered by a myelin coating. This coating admits regularly spaced gaps at the so-called nodes of Ranvier. Since the signal travels much faster through the coated regions, it appears to hop between the nodes of Ranvier. However, many mathematical models that describe this propagation do not take into account the discrete structure directly.
More recently, a discrete version of the famous FitzHugh-Nagumo model has been proposed to capture this discrete behaviour. In this thesis, we consider several extensions to and generalisations of this discrete FitzHugh-Nagumo model. In particular, we study infinite-range interactions, periodic behaviour and spatial-temporal discretization. Our general aim is to establish the existence and, sometimes, non-linear stability of travelling wave solutions. Our main tools in this analysis are the spectral convergence method and exponential dichotomies. In addition, we extend some general mathematical theory to systems with infinite-range interactions.

• Lattice differential equations
• FitzHugh-Nagumo system
• Infinite-range interactions
• Nonlinear stability
• Periodic coeffi-cients
• Singular perturbations
• Travelling waves
• Spatial-temporal discretisation
• Exponential dichotomies
• Nonlocal interactions

• All authors: Schouten-Straatman, W.M.
• Supervisor: Doelman, A.
• Co-supervisor: Hupkens, H.J.
• Committee: Bates, P.; Faye, G.; Van Vleck, E.; Duijn Schouten, F. van der; Merks, R.
• Qualification: Doctor (dr.)
• Awarding Institution: Mathematical Institute (MI), Faculty of Science, Leiden University
• Date: 2021-03-02
• Sponsorship: Netherlands Organisation for Scientific Research (NWO), grant 639.032.612