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We consider a mesoscopic model for a spatially extended FitzHugh-Nagumoneural network and prove that in the regime where short-range interactionsdominate, the probability density of the potential throughout the networkconcentrates into a Dirac distribution whose center of mass solves theclassical non-local reaction-diffusion FitzHugh-Nagumo system. In order torefine our comprehension of this regime, we focus on the blow-up profile ofthis concentration phenomenon. Our main purpose here consists in deriving twoquantitative and strong convergence estimates proving that the profile isGaussian: the first one in a L1 functional framework and the second in aweighted L2 functional setting. We develop original relative entropy techniquesto prove the first result whereas our second result relies on propagation ofregularity.

Large coupling in a FitzHugh-Nagumo neural network: quantitative and strong convergence results