Hydrodynamic limit for interacting neurons

This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with $N$ neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value $0$ and, simultaneously, the membrane potentials of the other neurons are increased by an amount of {\sl energy} $\frac{1}{N} $. This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system. We show that, as the system size $N$ diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.