Abstract

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling. To gain insight into the behaviour of neural networks when phase-oscillator descriptions are not appropriate we turn instead to the study of tractable piece-wise linear (pwl) systems. There has been an appreciation for some time in the applied sciences, and particularly in electrical engineering, of the benefits of studying caricatures of complex systems built from pwl and possibly discontinuous dynamical systems. Although a beautifully simplistic modelling perspective the necessary loss of smoothness precludes the use of many results from the standard toolkit of smooth dynamical systems, and one must be careful to correctly determine conditions for existence, uniqueness and stability of solutions.

In this talk I will describe a variety of pwl neural oscillators and show how to analyse periodic orbits. Building on this approach I will show how to analyse network states, with a focus on synchrony. I will make use of an extension of the master stability function (MSF) approach utilising saltation matrices, and show how this framework is very amenable to explicit calculations when considering networks of pwl oscillators. These can include pwl integrate-and-fire (IF) systems with smooth synaptic interactions, for which synchrony is ubiquitous in the case of a balance between excitation and inhibition. Moreover, the MSF approach is readily generalised to treat other phase-locked states such as clusters. For the case of nonsmooth synaptic interactions there is a further mathematical challenge that requires a careful treatment of the order in which perturbations cross the IF threshold. A similar issue arises in networks of switch-like elements, as exemplified by Glass networks and neural mass models with a Heaviside nonlinearity. Finally, I will discuss the dynamics of the famous Wilson-Cowan model posed on a realistic large-scale brain atlas and, time permitting, the important role that axonal delays can have on emergent network brain states and rhythms.