Stimulus-driven Traveling Solutions in Continuum Neuronal Models with a General Smooth Firing Rate

We examine the existence of traveling wave solutions for a continuum neuronal network modeled by integro-differential equations. First, we consider a scalar field model with a general smooth firing rate function and a spatiotemporally varying stimulus. We prove that a traveling front solution that is locked to the stimulus exists for a certain interval of stimulus speeds. Next, we include a slow adaptation equation and obtain a formula, which involves a certain adjoint solution, for the stimulus speeds that induce locked traveling pulse solutions. Further, we use singular perturbation analysis to characterize an approximation to the adjoint solution that we compare to a numerically computed adjoint. Numerical simulations are used to illustrate the traveling fronts and pulses that we study and to make comparisons with our analytically computed bounds for stimulus-locked wave behavior.