In this paper we focus on a mean-field model for a spatially extended FitzHugh-Nagumo neural network. In the regime where strong and local interactions dominate, we quantify how the probability density of voltage concentrates into a Dirac distribution. Previous work investigating this question have provided relative bounds in integrability spaces. Using a Hopf-Cole framework, we derive precise $L^\infty$ estimates using subtle explicit sub-and super-solutions which prove, with rates of convergence, that the blow up profile is Gaussian.