All Outputs (14)

Oscillatory networks: insights from piecewise-linear modelling (2024)
Journal Article
Coombes, S., Șayli, 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

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... Read More about Oscillatory networks: insights from piecewise-linear modelling.

Phase and amplitude responses for delay equations using harmonic balance (2024)
Journal Article
Nicks, R., Allen, R., & Coombes, S. (2024). Phase and amplitude responses for delay equations using harmonic balance. Physical Review E, 110(1), Article L012202. https://doi.org/10.1103/PhysRevE.110.L012202

Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations (DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equations with limit cycle oscillations, there is now a gr... Read More about Phase and amplitude responses for delay equations using harmonic balance.

Insights into oscillator network dynamics using a phase-isostable framework (2024)
Journal Article
Nicks, R., Allen, R., & Coombes, S. (2024). Insights into oscillator network dynamics using a phase-isostable framework. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(1), Article 013141. https://doi.org/10.1063/5.0179430

Networks of coupled nonlinear oscillators can display a wide range of emergent behaviors under the variation of the strength of the coupling. Network equations for pairs of coupled oscillators where the dynamics of each node is described by the evolu... Read More about Insights into oscillator network dynamics using a phase-isostable framework.

The two-process model for sleep–wake regulation: A nonsmooth dynamics perspective (2022)
Journal Article
Şaylı, M., Skeldon, A. C., Thul, R., Nicks, R., & Coombes, S. (2023). The two-process model for sleep–wake regulation: A nonsmooth dynamics perspective. Physica D: Nonlinear Phenomena, 444, Article 133595. https://doi.org/10.1016/j.physd.2022.133595

Since its inception four decades ago the two-process model introduced by Borbély has provided the conceptual framework to explain sleep–wake regulation across many species, including humans. At its core, high level notions of circadian and homeostati... Read More about The two-process model for sleep–wake regulation: A nonsmooth dynamics perspective.

Mean-Field Models for EEG/MEG: From Oscillations to Waves (2021)
Journal Article
Byrne, Á., Ross, J., Nicks, R., & Coombes, S. (2022). Mean-Field Models for EEG/MEG: From Oscillations to Waves. Brain Topography, 35, 36–53. https://doi.org/10.1007/s10548-021-00842-4

Neural mass models have been used since the 1970s to model the coarse-grained activity of large populations of neurons. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological consideration... Read More about Mean-Field Models for EEG/MEG: From Oscillations to Waves.

Understanding sensory induced hallucinations: From neural fields to amplitude equations (2021)
Journal Article
Nicks, R., Cocks, A., Avitabile, D., Johnston, A., & Coombes, S. (2021). Understanding sensory induced hallucinations: From neural fields to amplitude equations. SIAM Journal on Applied Dynamical Systems, 20(4), 1683-1714. https://doi.org/10.1137/20M1366885

Explorations of visual hallucinations, and in particular those of Billock and Tsou [V. A. Billock and B. H. Tsou, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 8490-8495], show that annular rings with a background flicker can induce visual hallucinatio... Read More about Understanding sensory induced hallucinations: From neural fields to amplitude equations.

Brain-wave equation incorporating axodendritic connectivity (2020)
Journal Article
Ross, J., Margetts, M., Bojak, I., Nicks, R., Avitabile, D., & Coombes, S. (2020). Brain-wave equation incorporating axodendritic connectivity. Physical Review E, 101(2), Article 022411. https://doi.org/10.1103/PhysRevE.101.022411

©2020 American Physical Society. We introduce an integral model of a two-dimensional neural field that includes a third dimension representing space along a dendritic tree that can incorporate realistic patterns of axodendritic connectivity. For natu... Read More about Brain-wave equation incorporating axodendritic connectivity.

Clusters in nonsmooth oscillator networks (2018)
Journal Article
Nicks, R., Chambon, L., & Coombes, S. (2018). Clusters in nonsmooth oscillator networks. Physical Review E, 97(3), Article 032213. https://doi.org/10.1103/PhysRevE.97.032213

© 2018 American Physical Society. For coupled oscillator networks with Laplacian coupling, the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory... Read More about Clusters in nonsmooth oscillator networks.

Standing and travelling waves in a spherical brain model: the Nunez model revisited (2017)
Journal Article
Visser, S., Nicks, R., Faugeras, O., & Coombes, S. (2017). Standing and travelling waves in a spherical brain model: the Nunez model revisited. Physica D: Nonlinear Phenomena, 349, https://doi.org/10.1016/j.physd.2017.02.017

The Nunez model for the generation of electroencephalogram (EEG) signals is naturally described as a neural field model on a sphere with space-dependent delays. For simplicity, dynamical realisations of this model either as a damped wave equation or... Read More about Standing and travelling waves in a spherical brain model: the Nunez model revisited.

Mathematical frameworks for oscillatory network dynamics in neuroscience (2016)
Journal Article
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

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 nov... Read More about Mathematical frameworks for oscillatory network dynamics in neuroscience.

A classification of the symmetries of uniform discrete defective crystals (2014)
Journal Article
Nicks, R. (in press). A classification of the symmetries of uniform discrete defective crystals. Journal of Elasticity, 117(2), https://doi.org/10.1007/s10659-014-9470-9

Crystals which have a uniform distribution of defects are endowed with a Lie group description which allows one to construct an associated discrete structure. These structures are in fact the discrete subgroups of the ambient Lie group. The geometric... Read More about A classification of the symmetries of uniform discrete defective crystals.

Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals (2013)
Journal Article
Nicks, R., & Parry, G. P. (2014). Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals. Journal of Elasticity, 115(2), 131-156. https://doi.org/10.1007/s10659-013-9450-5

The Lie group structure of crystals which have uniform continuous distributions of dislocations allows one to construct associated discrete structures—these are discrete subgroups of the corresponding Lie group, just as the perfect lattices of crysta... Read More about Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals.

On symmetries of crystals with defects related to a class of solvable groups (S1) (2011)
Journal Article
Nicks, R., & Parry, G. P. (2012). On symmetries of crystals with defects related to a class of solvable groups (S1). Mathematics and Mechanics of Solids, 17(6), 631-651. https://doi.org/10.1177/1081286511427485

We consider distributions of dislocations in continuum models of crystals which are such that the corresponding dislocation density tensor relates to a particular class of solvable Lie group, and discrete structures which are embedded in these crysta... Read More about On symmetries of crystals with defects related to a class of solvable groups (S1).

Geometrical issues in the continuum mechanics of solid crystals
Journal Article
Nicks, R., & Parry, G. P. Geometrical issues in the continuum mechanics of solid crystals. Miskolc Mathematical Notes,

We shall outline geometrical and algebraic ideas which appear to lie at the foundation of the theory of defective crystals that was introduced by Davini [5] in 1986. The focus of the paper will be on the connection between continuous and discrete mod... Read More about Geometrical issues in the continuum mechanics of solid crystals.