Synchrony in firing rate neural networks with multiple delays: A harmonic balance approach

Networks of neural mass nodes with delayed interactions are increasingly being used as models for large-scale brain activity. To complement the growing number of computational studies of such networks, it is timely to develop new mathematical studies of their solution structure and bifurcations. The analysis of steady states and their stability is relatively well developed, though that of time-periodic solutions is far less so. Here, we show how the method of harmonic balance is ideally suited to describing delay-induced and delay-modulated periodic oscillations at both the node and network level. This approach reduces the formally infinite dimensional setting of the delayed differential equation network to a finite dimensional one, opening the way for a practical combined analytical and numerical treatment. At the node level, we show how to construct periodic orbits and develop an associated linear stability analysis to determine the Floquet exponents and, thus, stability. At the network level, we further show that explicit progress for analysing the stability of the synchronous state can be made for networks with a circulant structure. This is achieved with the use of an adjacency lag operator that decouples the linearised network equations into a set of equations that can each be analysed using the techniques previously developed for analysing a single node. When combined with numerical continuation techniques this allows us to build the skeleton of a network bifurcation diagram, and highlight the role of distance-dependent delays in contributing to novel spatio-temporal patterns arising from the instability of a synchronous state, including travelling periodic waves, alternating anti-phase solutions (in which only next nearest neighbours are synchronised), cluster states, and more exotic behaviours.