@article { , title = {Travelling waves in a neural field model with refractoriness}, 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).}, doi = {10.1007/s00285-013-0670-x}, eissn = {1432-1416}, issn = {0303-6812}, issue = {5}, journal = {Journal of Mathematical Biology}, pages = {1249-1268}, publicationstatus = {Published}, publisher = {Springer Verlag}, url = {https://nottingham-repository.worktribe.com/output/723878}, volume = {68}, keyword = {neural field models, travelling waves, refractoriness, delay differential equations}, year = {2014}, author = {Meijer, Hil G.E. and Coombes, Stephen} }