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Oscillatory networks: insights from piecewise-linear modelling

Coombes, Stephen; Şayli, Mustafa; Thul, Rüdiger; Nicks, Rachel; Porter, Mason A.; Lai, Yi Ming

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Authors

Mustafa Şayli

Mason A. Porter

Yi Ming Lai



Abstract

There is enormous interest-both mathematically and in diverse applications-in understanding the dynamics of coupled-oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled-oscillator networks. We discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. We then focus, in particular, on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use saltation operators, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds.

Citation

Coombes, S., Şayli, M., Thul, R., Nicks, R., Porter, M. A., & Lai, Y. M. (2024). Oscillatory networks: insights from piecewise-linear modelling. SIAM Review, 66(4), 619-679. https://doi.org/10.1137/22M1534365

Journal Article Type Article
Acceptance Date Dec 8, 2023
Online Publication Date Nov 7, 2024
Publication Date Nov 7, 2024
Deposit Date Nov 7, 2024
Publicly Available Date Nov 7, 2024
Journal SIAM Review
Print ISSN 0036-1445
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 66
Issue 4
Pages 619-679
DOI https://doi.org/10.1137/22M1534365
Keywords Coupled oscillators; networks; phase reduction; phase-amplitude reduction; mas- ter stability function; network symmetries; piecewise-linear oscillator models; nonsmooth dynamics; saltation operators
Public URL https://nottingham-repository.worktribe.com/output/41552612
Publisher URL https://epubs.siam.org/doi/10.1137/22M1534365

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