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A master stability function approach to cardiac alternans (2019)
Journal Article
Lai, Y. M., Veasy, J., Coombes, S., & Thul, R. (2019). A master stability function approach to cardiac alternans. Applied Network Science, 4(1), https://doi.org/10.1007/s41109-019-0199-z

During a single heartbeat, muscle cells in the heart contract and relax. Under healthy conditions, the behaviour of these muscle cells is almost identical from one beat to the next. However, this regular rhythm can be disturbed giving rise to a varie... Read More about A master stability function approach to cardiac alternans.

Synchrony in networks of Franklin bells (2019)
Journal Article
Sayli, M., Lai, Y. M., Thul, R., & Coombes, S. (2019). Synchrony in networks of Franklin bells. IMA Journal of Applied Mathematics, 84(5), 1001-1021. https://doi.org/10.1093/imamat/hxz023

The Franklin bell is an electro-mechanical oscillator that can generate a repeating chime in the presence of an electric field. Benjamin Franklin famously used it as a lightning detector. The chime arises from the impact of a metal ball on a metal be... Read More about Synchrony in networks of Franklin bells.

Complex patterns of subcellular cardiac alternans (2019)
Journal Article
Veasy, J., Lai, Y. M., Coombes, S., & Thul, R. (2019). Complex patterns of subcellular cardiac alternans. Journal of Theoretical Biology, 478, 102-114. doi:10.1016/j.jtbi.2019.06.016

Cardiac alternans, in which the membrane potential and the intracellular calcium concentration exhibit alternating durations and peak amplitudes at consecutive beats, constitute a precursor to fatal cardiac arrhythmia such as sudden cardiac death. A... Read More about Complex patterns of subcellular cardiac alternans.

Next-generation neural field model: The evolution of synchrony within patterns and waves (2019)
Journal Article
Byrne, Á., Avitabile, D., & Coombes, S. (2019). Next-generation neural field model: The evolution of synchrony within patterns and waves. Physical Review E, 99(1), doi:10.1103/physreve.99.012313

Neural field models are commonly used to describe wave propagation and bump attractors at a tissue level in the brain. Although motivated by biology, these models are phenomenological in nature. They are built on the assumption that the neural tissue... Read More about Next-generation neural field model: The evolution of synchrony within patterns and waves.

How do spatially distinct frequency specific MEG networks emerge from one underlying structural connectome? The role of the structural eigenmodes (2018)
Journal Article
Tewarie, P., Abeysuriya, R., Byrne, Á., O'Neill, G. C., Sotioropoulos, S. N., Brookes, M. J., & Coombes, S. (2019). How do spatially distinct frequency specific MEG networks emerge from one underlying structural connectome? The role of the structural eigenmodes. NeuroImage, 186, 211-220. doi:10.1016/j.neuroimage.2018.10.079

Functional networks obtained from magnetoencephalography (MEG) from different frequency bands show distinct spatial patterns. It remains to be elucidated how distinct spatial patterns in MEG networks emerge given a single underlying structural networ... Read More about How do spatially distinct frequency specific MEG networks emerge from one underlying structural connectome? The role of the structural eigenmodes.

Analysis of networks where discontinuities and nonsmooth dynamics collide: understanding synchrony (2018)
Journal Article
Yi Ming, L., Thul, R., & Coombes, S. (2018). Analysis of networks where discontinuities and nonsmooth dynamics collide: understanding synchrony. European Physical Journal - Special Topics, 227(10-11), 1251-1265. doi:10.1140/epjst/e2018-800033-y

Integrate-and-fire networks have proven remarkably useful in modelling the dynamics of real world phenomena ranging from earthquakes, to synchrony in neural networks, to cascading activity in social networks. The reset process means that such models... Read More about Analysis of networks where discontinuities and nonsmooth dynamics collide: understanding synchrony.

Relationships Between Neuronal Oscillatory Amplitude and Dynamic Functional Connectivity (2018)
Journal Article
O’Neill, G. C., Hunt, B. A. E., Tewarie, P., Hunt, B. A. E., O'Neill, G. C., Byrne, A., …Brookes, M. J. (2018). Relationships Between Neuronal Oscillatory Amplitude and Dynamic Functional Connectivity. Cerebral Cortex, 29(6), 2668-2681. doi:10.1093/cercor/bhy136

Event related fluctuations of neural oscillatory amplitude are reported widely in the context of cognitive processing and are typically interpreted as a marker of brain ‘activity’. However, the precise nature of these effects remains unclear; in part... Read More about Relationships Between Neuronal Oscillatory Amplitude and Dynamic Functional Connectivity.

Three-dimensional spatio-temporal modelling of store operated Ca2+ entry: insights into ER refilling and the spatial signature of Ca2+ signals (2018)
Journal Article
McIvor, E., Coombes, S., & Thul, R. (2018). Three-dimensional spatio-temporal modelling of store operated Ca2+ entry: insights into ER refilling and the spatial signature of Ca2+ signals. Cell Calcium, 73, doi:10.1016/j.ceca.2018.03.006

The spatial organisation of Orai channels and SERCA pumps within ER-PM junctions is important for enhancing the versatility and specificity of subcellular Ca2+ signals generated during store operated Ca2+ entry (SOCE). In this paper we present a nove... Read More about Three-dimensional spatio-temporal modelling of store operated Ca2+ entry: insights into ER refilling and the spatial signature of Ca2+ signals.

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), doi:10.1103/PhysRevE.97.032213

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 this approach has recently been e... Read More about Clusters in nonsmooth oscillator networks.

Neural Fields: Localised States with Piece-Wise Constant Interactions (2018)
Book Chapter
Gökçe, A., Coombes, S., & Avitabile, D. (2018). Neural Fields: Localised States with Piece-Wise Constant Interactions. In Mathematical and Theoretical Neuroscience: Cell, Network and Data AnalysisSpringer Nature. doi:10.1007/978-3-319-68297-6_7

Neural field models are typically cast as continuum integro-differential equations for describing the idealised coarse-grained activity of populations of interacting neurons. For smooth Mexican hat kernels, with short-range excitation and long-range... Read More about Neural Fields: Localised States with Piece-Wise Constant Interactions.

Networks of piecewise linear neural mass models (2018)
Journal Article
Coombes, S., Lai, Y. M., Sayli, M., & Thul, R. (2018). Networks of piecewise linear neural mass models. European Journal of Applied Mathematics, 29(Special issue 5), 869-890. doi:10.1017/S0956792518000050

Neural mass models are ubiquitous in large scale brain modelling. At the node level they are written in terms of a set of ordinary differential equations with a nonlinearity that is typically a sigmoidal shape. Using structural data from brain atlase... Read More about Networks of piecewise linear neural mass models.

The dynamics of neural fields on bounded domains: an interface approach for Dirichlet boundary conditions (2017)
Journal Article
Gökçe, A., Avitabile, D., & Coombes, S. (in press). The dynamics of neural fields on bounded domains: an interface approach for Dirichlet boundary conditions. Journal of Mathematical Neuroscience, 7, doi:10.1186/s13408-017-0054-4

Continuum neural field equations model the large scale spatio-temporal dynamics of interacting neurons on a cortical surface. They have been extensively studied, both analytically and numerically, on bounded as well as unbounded domains. Neural field... Read More about The dynamics of neural fields on bounded domains: an interface approach for Dirichlet boundary conditions.

An analysis of waves underlying grid cell firing in the medial enthorinal cortex (2017)
Journal Article
Bonilla-Quintana, M., Wedgwood, K. C., O'Dea, R. D., & Coombes, S. (in press). An analysis of waves underlying grid cell firing in the medial enthorinal cortex. Journal of Mathematical Neuroscience, 7(9), doi:10.1186/s13408-017-0051-7

Layer II stellate cells in the medial enthorinal cortex (MEC) express hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels that allow for rebound spiking via an I_h current in response to hyperpolarising synaptic input. A computational... Read More about An analysis of waves underlying grid cell firing in the medial enthorinal cortex.

A mean field model for movement induced changes in the beta rhythm (2017)
Journal Article
Byrne, Á., Brookes, M. J., & Coombes, S. (2017). A mean field model for movement induced changes in the beta rhythm. Journal of Computational Neuroscience, 43(2), doi:10.1007/s10827-017-0655-7

In electrophysiological recordings of the brain, the transition from high amplitude to low amplitude signals are most likely caused by a change in the synchrony of underlying neuronal population firing patterns. Classic examples of such modulations... Read More about A mean field model for movement induced changes in the beta rhythm.

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, doi: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.

Combining spatial and parametric working memory in a dynamic neural field model (2016)
Journal Article
Wojtak, W., Coombes, S., Bicho, E., & Erlhagen, W. (in press). Combining spatial and parametric working memory in a dynamic neural field model. Lecture Notes in Artificial Intelligence, 9886, doi:10.1007/978-3-319-44778-0_48

We present a novel dynamic neural field model consisting of two coupled fields of Amari-type which supports the existence of localized activity patterns or “bumps” with a continuum of amplitudes. Bump solutions have been used in the past to model spa... Read More about Combining spatial and parametric working memory in a dynamic neural field model.

Synchrony in networks of coupled nonsmooth dynamical systems: extending the master stability function (2016)
Journal Article
Coombes, S., & Thul, R. (2016). Synchrony in networks of coupled nonsmooth dynamical systems: extending the master stability function. European Journal of Applied Mathematics, 27(6), 904-922. doi:10.1017/S0956792516000115

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbi... Read More about Synchrony in networks of coupled nonsmooth dynamical systems: extending the master stability function.

Neural field models with threshold noise (2016)
Journal Article
Thul, R., Coombes, S., & Laing, C. R. (2016). Neural field models with threshold noise. Journal of Mathematical Neuroscience, 6, doi:10.1186/s13408-016-0035-z

The original neural field model of Wilson and Cowan is often interpreted as the averaged behaviour of a network of switch like neural elements with a distribution of switch thresholds, giving rise to the classic sigmoidal population firing-rate funct... Read More about Neural field models with threshold noise.

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, doi: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.

Mathematical neuroscience: from neurons to networks (2015)
Book Chapter
Coombes, S. (2015). Mathematical neuroscience: from neurons to networks. In C. Dogbe (Ed.), Actes du colloque "EDP-Normandie" : Le Havre 2015Fédération Normandie Mathématiques

The tools of dynamical systems theory are having an increasing impact on our understanding of patterns of neural activity. In this talk I will describe how to build tractable tissue level models that maintain a strong link with biophysical reality. T... Read More about Mathematical neuroscience: from neurons to networks.