We consider a coarse grained neural field model for synaptic activity in spatially extended cortical tissue that possesses an underlying periodicity in its microstructure. The model is written as an integro-differential equation with periodic modulation of a translationally-invariant spatial kernel. This modulation can have a strong effect on wave propagation through the tissue, including the creation of pulsating fronts with widely-varying speeds, and wave-propagation failure. Here we develop new analysis for the study of such phenomena, using two complementary techniques. The first uses linearized information from the leading edge of a traveling periodic wave to obtain wave speed estimates for pulsating fronts, and the second develops an interface description for waves in the full nonlinear model. For weak modulation and a Heaviside firing rate function the interface dynamics can be analyzed exactly, and gives predictions which are in excellent agreement with direct numerical simulations. Importantly, the interface dynamics description improves upon the standard homogenization calculation, which is restricted to modulation that is both fast and weak
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