Insights into neural oscillator network dynamics using a phase-isostable framework

Networks of coupled nonlinear oscillators can display a wide range of emergent behaviours under variation of the strength of the coupling. Of particular interest in this talk are neuronal networks where nodes are single neuron or neural population models. In these cases, interactions may be significant in magnitude compared to the rate of decay to the underlying stable limit cycle. Since the standard technique of first-order phase reduction breaks down beyond the weak coupling regime it therefore fails to capture many important features of the dynamics of these neural networks. Recent work has shown isostable coordinates to be a useful concept to characterise the transient behaviour of oscillators in directions where decay to the limit cycle is slow. Reduced network equations for two coupled oscillators, where the dynamics of each node is described by the evolution of its phase and slowest decaying isostable coordinate, have been shown to capture bifurcations and dynamics of the network which cannot be explained through standard phase reduction. An alternative framework using isostable coordinates to obtain higher-order phase reductions has also demonstrated a similar descriptive ability for two oscillators.
In this talk we discuss the extension of phase-isostable network equations to an arbitrary but finite number of coupled oscillators, giving conditions required for stability of phase-locked states including synchrony. For examples where the dynamics of the full system are known, we compare the accuracy of the phase-isostable network equations and higher-order phase reductions in capturing bifurcations of phase-locked states. We find the former to be the more accurate and therefore we may employ this framework to investigate the dynamics of a number of globally coupled neuronal networks of varying size for planar node models.