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 27, 2014
• 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
• Contract Date: Mar 27, 2014