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We study the stochastic system of interacting neurons introduced in De Masiet al. (2015) and in Fournier and L\"ocherbach (2016) in a diffusive scaling.The system consists of $N$ neurons, each spiking randomly with rate dependingon its membrane potential. At its spiking time, the potential of the spikingneuron is reset to $0$ and all other neurons receive an additional amount ofpotential which is a centred random variable of order $ 1 / \sqrt{N}.$ Inbetween successive spikes, each neuron's potential follows a deterministicflow. We prove the convergence of the system, as $N \to \infty$, to a limitnonlinear jumping stochastic differential equation driven by Poisson randommeasure and an additional Brownian motion $W$ which is created by the centrallimit theorem. This Brownian motion is underlying each particle's motion andinduces a common noise factor for all neurons in the limit system.Conditionally on $W,$ the different neurons are independent in the limitsystem. This is the {\it conditional propagation of chaos} property. We provethe well-posedness of the limit equation by adapting the ideas of Graham (1992)to our frame. To prove the convergence in distribution of the finite system tothe limit system, we introduce a new martingale problem that is well suited forour framework. The uniqueness of the limit is deduced from the exchangeabilityof the underlying system.

Conditional propagation of chaos for mean field systems of interacting neurons