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Travelling waves in a neural field model with refractoriness

Meijer, Hil G.E.; Coombes, Stephen

Authors

Hil G.E. Meijer



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).

Journal Article Type Article
Publication Date 2014-04
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
APA6 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
DOI https://doi.org/10.1007/s00285-013-0670-x
Keywords neural field models; travelling waves; refractoriness; delay differential equations
Publisher URL http://link.springer.com/article/10.1007%2Fs00285-013-0670-x
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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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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