Phase-isostable descriptions of coupled oscillator networks

"Phase-reduced models, where oscillator dynamics are reduced to the dynamics of their phase on limit cycle, have been extremely successful in describing dynamical behaviours of networks of coupled oscillators in the case where individual oscillators possess a strongly attracting limit cycle and coupling between oscillators in the network is weak [1]. However, for many biological (and particularly neural) oscillator networks these modelling assumptions are not appropriate; numerical simulations and results from other modelling techniques have revealed dynamics that cannot be captured in a framework where only phases of oscillators are considered [2].

The failure of phase reduced models to describe the known dynamics begs the question of whether an alternative framework that also tracks some notion of distance from the limit cycle, along with the phase on limit cycle, can more accurately describe observed dynamics. Such a framework, using phase-isostable coordinates has recently been proposed [3,4], but its ability to describe a range of observed network behaviours has yet to have been thoroughly investigated. The aims of this project will be to first investigate conditions under which the phase-isostable framework can accurately characterise the response of single oscillators to external forcing. After making any required refinements to the framework, network equations in phase-isostable coordinates will be developed and used to investigate network dynamics beyond the weak coupling limit. The network analysis will utilise any symmetries of the network dynamical system allowing for prediction of when certain cluster and chimera states may exist. It is anticipated that the phase-isostable paradigm will also be able to reveal a rich variety of more complex dynamical network behaviour."