Mathematical frameworks for oscillatory network dynamics in neuroscience

Ashwin, Peter; Coombes, Stephen; Nicks, Rachel

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 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.

Citation

Ashwin, P., Coombes, S., & Nicks, R. (2016). Mathematical frameworks for oscillatory network dynamics in neuroscience. Journal of Mathematical Neuroscience, 6, Article 2. https://doi.org/10.1186/s13408-015-0033-6

• 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
• DOI: https://doi.org/10.1186/s13408-015-0033-6
• 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
• Public URL: https://nottingham-repository.worktribe.com/output/773511
• Publisher URL: http://mathematical-neuroscience.springeropen.com/articles/10.1186/s13408-015-0033-6
• Contract Date: Mar 6, 2017