Insights into oscillatory neural dynamics using a phase-amplitude framework

Model reduction techniques can provide useful insight into the dynamics behaviour of high dimensional oscillatory systems such as networks of neurons or neural field models. However, the utility of the classical technique of phase reduction is limited by the assumption that the local dynamics for each node in a network remain on the stable limit cycle of the uncoupled system. Here we will discuss the use of a framework where each oscillator is described by an amplitude given by its slowest decaying isostable coordinate in addition to its phase on limit cycle. This allows for representation of trajectories away from (but near) the limit cycle.

We first discuss the application of this framework to discrete networks where the resulting phase-isostable network equations can be used compute existence and stability conditions for phase-locked states in networks of identical nodes, extending known results for phase-reduced equations beyond the weak coupling limit. We demonstrate the power of the general framework by considering small and large networks of Morris-Lecar neurons, capturing stability changes and quasiperiodic behaviour that are beyond the descriptive scope of classical first-order phase reduction, and are shown to be in good qualitative agreement with the dynamics of the original network through numerical simulations and bifurcation analysis. Delays in both the node dynamics and network interactions can strongly influence patterns of phase-locked states and their bifurcations. We will briefly discuss ongoing work incorporating delayed interactions as well as phase and amplitude response of delay induced node oscillations within a phase-amplitude network setting. Finally, we will show how phase-isostable reduction can provide useful information about the stability breakdown of phase waves in continuum neural field models, revealing parameter regimes where more exotic behaviour such as weak turbulence or chimera states may exist.

References:

[1] R Nicks, R Allen and S Coombes 2024 Insights into oscillator network dynamics using a phase-isostable framework, Chaos, Vol 34, 013141

[2]  R Nicks, R Allen and S Coombes 2024 Phase and amplitude responses for delay equations using harmonic balance, Physical Review E, Vol 110, L012202