title: Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience.
abstract: The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.
authors.affiliation: Centre for Systems Dynamics and Control, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, Exeter, EX4 4QF, UK. P.Ashwin@exeter.ac.uk.
authors.forename: Peter
authors.initials: P
authors.lastname: Ashwin
authors.affiliation: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. stephen.coombes@nottingham.ac.uk.
authors.forename: Stephen
authors.initials: S
authors.lastname: Coombes
authors.affiliation: School of Mathematics, University of Birmingham, Watson Building, Birmingham, B15 2TT, UK. r.nicks@bham.ac.uk.
authors.forename: Rachel
authors.initials: R
authors.lastname: Nicks
keywords: Central pattern generator; Chimera state; Coupled oscillator network; Groupoid formalism; Heteroclinic cycle; Isochrons; Master stability function; Network motif; Perceptual rivalry; Phase oscillator; Phase–amplitude coordinates; Stochastic oscillator; Strongly coupled integrate-and-fire network; Symmetric dynamics; Weakly coupled phase oscillator network; Winfree model