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Starting from a microscopic model for a system of neurons evolving in timewhich individually follow a stochastic integrate-and-fire type model, we studya mean-field limit of the system. Our model is described by a system of SDEswith discontinuous coefficients for the action potential of each neuron andtakes into account the (random) spatial configuration of neurons allowing theinteraction to depend on it. In the limit as the number of particles tends toinfinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, withderivatives only with respect to one variable and discontinuous coefficients.We also study strong well-posedness of the system of SDEs and prove theexistence and uniqueness of a weak measure-valued solution to the PDE, obtainedas the limit of the laws of the empirical measures for the system of particles.

A mean-field model with discontinuous coefficients for neurons with spatial interaction