On solutions of advanced-retarded travelling wave equations arising in a neural field theory

We consider a firing rate and a spike frequency adaptation (SFA) model of a one-dimensional neuronal network with axo-dendritic synaptic interactions. The neuronal network we focus on has a compactly supported connectivity pattern. The model equations are integro-differential equations with spatio-temporal delays arising from the finite axonal conduction velocity. We show that travelling wave solutions of these models are determined by mixed-type functional differential equations with both advanced and retarded delays. Especially, both delays depend on a unknown travelling wave speed. Based on the discussion of homogeneous steady states of model systems, we investigate the travelling front solutions of the firing rate model and the travelling pulse solutions of the SFA model. For the special case of a Heaviside firing rate function (the limit of infinite gain of a sigmoidal firing rate function), we obtain a special relationship between wave solutions, and then find analytic travelling wave solutions. For the models with a general sigmoidal firing rate function, we investigate the dynamic behavior in tails of travelling wave solutions, and perform the numerical calculation of travelling front solutions.Keywords. Neuronal network; firing rate function; mixed-type functional differential equations; delays; travelling wave solution