We consider a spatially extended mesoscopic FitzHugh-Nagumo model with interactions and prove that in the regime where strong and local interactions dominate, the probability density of the potential throughout the network concentrates into a Dirac distribution whose center of mass solves the classical non-local reaction-diffusion FitzHugh-Nagumo system. In [2], we proved that the profile of concentration is Gaussian by providing a weak convergence result. Our main purpose here consists in strengthening this result by deriving two quantitative and strong convergence estimates: the first one in a L1 functional framework and the second in a weighted L2 functional setting. Our approach is based on relative entropy techniques in the first case and on propagation of regularity in the second.