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.Electronic supplementary materialThe online version of this article (doi:10.1007/s00285-013-0670-x) contains supplementary material, which is available to authorized users.
Highlights
The continuum approximation of neural activity can be traced back to work of Beurle (1956), who built a model describing the proportion of active neurons per unit time in a given volume of randomly connected nervous tissue
We have considered periodic travelling waves in a one dimensional neural field model describing a single spatially extended population with purely excitatory interactions
Despite the long history and extensive study of this type of model, to the best of our knowledge this is the first analysis of moving N -pulses in a neural field model with refractoriness
Summary
The continuum approximation of neural activity can be traced back to work of Beurle (1956), who built a model describing the proportion of active neurons per unit time in a given volume of randomly connected nervous tissue. In this paper we reinstate the original refractory term of Wilson and Cowan in a minimal neural field model describing a single population in one spatial dimension This model is briefly reviewed in Sect. A novel numerical continuation scheme is developed to track solution properties in a co-moving frame (speed, period, and profile shape) as a function of physiologically important system parameters (such as refractory time-scale, strength of anatomical connectivity, and firing threshold). These are obtained after recognising that the original model can be reformulated as a delay-differential equation for an exponentially decaying choice of anatomical weight distribution.
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