NEWWAVE: New methods for analysing travelling waves in discrete systems with applications to neuroscience

Patterns of animal colouration, biological tissues, vegetation patterns and the formation of crystals, as well as spatio-temporalpatterns such as travelling (water) waves, (laser) pulses and (electrical) spikes have captivated researchers in various fields of science for many decades. Project NEWWAVE puts forward a unique and unprecedented approach to studying such patterns in discrete, spatially-extended and delay-coupled systems of excitable and bistable units, which play an important role for neural signal propagation. Such systems can be obtained, for example, from discretisation of an underlying partial differential equation model, from the discreteness of the measured data, or from the microstructure of the medium. They can also be obtained bottom-up, from studying the response of single active unit to its neighbourhood, and large systems of coupled units are gaining more and more attention through the emergence of complex systems science and network science. The specific proposed research involves overcoming considerable mathematical and numerical challenges associated with time delays of mixed-type that appear when either (A) taking into account the finite speed of communication between discrete loci, or (B) imposing a TW ansatz (co-moving coordinates) on a discrete system.

The projected results promise significant potential for innovating the bifurcation (qualitative changes) analysis of TWs in discrete systems and will advance our understanding of their dynamic repertoire through new analytical and numerical means, and by making the developed methods available as an open source library as part of the project. The techniques will be used to elucidate the functional role of travelling waves in the brain and, more specifically, the role of TWs in Parkinson's disease together with project collaborators in Neuroscience.