Hil G.E. Meijer
Travelling waves in a neural field model with refractoriness
Meijer, Hil G.E.; Coombes, Stephen
Abstract
At one level of abstraction neural tissue can be regarded as a medium for turning local synaptic activity into output signals that propagate over large distances via axons to generate further synaptic activity that can cause reverberant activity in networks that possess a mixture of excitatory and inhibitory connections. This output is often taken to be a firing rate, and the mathematical form for the evolution equation of activity depends upon a spatial convolution of this rate with a fixed anatomical connectivity pattern. Such formulations often neglect the metabolic processes that would ultimately limit synaptic activity. Here we reinstate such a process, in the spirit of an original prescription by Wilson and Cowan (Biophys J 12:1-24, 1972), using a term that multiplies the usual spatial convolution with a moving time average of local activity over some refractory time-scale. This modulation can substantially affect network behaviour, and in particular give rise to periodic travelling waves in a purely excitatory network (with exponentially decaying anatomical connectivity), which in the absence of refractoriness would only support travelling fronts. We construct these solutions numerically as stationary periodic solutions in a co-moving frame (of both an equivalent delay differential model as well as the original delay integro-differential model). Continuation methods are used to obtain the dispersion curve for periodic travelling waves (speed as a function of period), and found to be reminiscent of those for spatially extended models of excitable tissue. A kinematic analysis (based on the dispersion curve) predicts the onset of wave instabilities, which are confirmed numerically. © 2013 The Author(s).
Citation
Meijer, H. G., & Coombes, S. (2014). Travelling waves in a neural field model with refractoriness. Journal of Mathematical Biology, 68(5), 1249-1268. https://doi.org/10.1007/s00285-013-0670-x
Journal Article Type | Article |
---|---|
Acceptance Date | Mar 7, 2013 |
Online Publication Date | Apr 2, 2013 |
Publication Date | 2014-04 |
Deposit Date | Mar 27, 2014 |
Publicly Available Date | Mar 28, 2024 |
Journal | Journal of Mathematical Biology |
Print ISSN | 0303-6812 |
Electronic ISSN | 1432-1416 |
Publisher | Springer Verlag |
Peer Reviewed | Peer Reviewed |
Volume | 68 |
Issue | 5 |
Pages | 1249-1268 |
DOI | https://doi.org/10.1007/s00285-013-0670-x |
Keywords | neural field models; travelling waves; refractoriness; delay differential equations |
Public URL | https://nottingham-repository.worktribe.com/output/723878 |
Publisher URL | http://link.springer.com/article/10.1007%2Fs00285-013-0670-x |
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https://creativecommons.org/licenses/by/4.0/
Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0
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