We consider a spatially extended kinetic model of a FitzHugh-Nagumo neural network, with a rescaled interaction kernel. Our main purpose is to prove that its diffusive limit in the regime of strong local interactions converges towards a FitzHugh-Nagumo reaction-diffusion system, taking account for the average quantities of the network. Our approach is based on a relative entropy argument, to compare the macroscopic quantities computed from the solution of the kinetic equation, and the solution of the limiting system. The main difficulty, compared to the literature, lies in the need of regularity in space of the solutions of the limiting system and a careful control of an internal nonlocal kinetic dissipation.