7. Oscillations and Synchrony in a Network of Delayed Neural Masses

In this chapter, we study the dynamics of a network of delayed Wilson–Cowan (WC) masses. We begin by analysing a single WC mass without delay, using numerical simulations and tools from numerical bifurcation theory to interrogate its dynamical behaviour under variation of important system parameters. We then briefly review the necessary prerequisites in network science before considering some of the ways in which delay differential equations differ from ordinary differential equations. Next we extend our analysis of the undelayed system to the case of a WC mass with two distinct discrete delays before reviewing some recent results from the literature. The chapter then moves on to study the effect that network structure has on the WC dynamics in the presence of both intra- and internodal delays. Deploying both artificial and experimental network structures, we find that network structure, coupled with delay times, can have both a regularising and deregularising effect, depending upon the overall strength of network coupling. Importantly, these results suggest that the presence of delays in a weakly coupled system provides a mechanism for avoiding undesirable (disordered) states, whilst still allowing the system to retain a sufficiently rich repertoire of network behaviours.