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 has 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.
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