| Summary: | 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.
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