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Mathematical frameworks for oscillatory network dynamics in neuroscience

Ashwin, Peter; Coombes, Stephen; Nicks, Rachel

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Peter Ashwin


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 mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience.

Journal Article Type Article
Acceptance Date Oct 30, 2015
Publication Date Jan 6, 2016
Deposit Date Mar 6, 2017
Publicly Available Date Mar 6, 2017
Journal Journal of Mathematical Neuroscience
Electronic ISSN 2190-8567
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 6
Article Number 2
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, S
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