Waves and Patterns in Discrete Media: Bridging the Gaps
What happens to electrical waves that have to cross gaps in insulation material around nerves in the human body
Differential equations posed on discrete lattices (LDEs) have by now become a popular modelling tool used in a wide variety of scientific disciplines. Such equations allow the inclusion of non-local interactions into models and lead to interesting dynamical and pattern-forming behaviour. The discrete FitzHugh-Nagumo equation is an important example of an LDE. It can be used to model the propagation of signals through myelinated nerve fibres. The coating of such a fibre admits gaps at regular intervals, which forms a strong incentive to use this discrete model instead of its more traditional continuous counterpart, the FitzHugh-Nagumo PDE. Nevertheless, the FitzHugh-Nagumo PDE has attracted far more mathematical attention than the corresponding LDE. In fact, due to the distinct time-scales involved, this PDE has served as a prototype system for the development of modern geometric singular perturbation theory in the 70s and 80s. One of the hallmark achievements of this theory is the construction of a branch of stable fast travelling pulses that bifurcates from a certain singular orbit and that connects to a branch of unstable slow travelling pulses. In recent work, we managed to extend part of this benchmark result to the FitzHugh-Nagumo LDE. However, the discreteness of the model causes an energy barrier that travelling pulses must overcome. In particular, waves that propagate in the continuous model may fail to do so in the discrete model. As a consequence, it is unclear at present what a full generalization of the result mentioned above should look like. One of the projects proposed here aims to fill this gap by building on recent mathematical results that were obtained for a class of functional differential equations that involve both retarded and advanced arguments. In addition, we intend to advance our understanding of discrete systems by studying several aspects of travelling wave solutions to LDEs.