Hydrodynamic Limit for Spatially Structured Interacting Neurons

Aline Duarte, Guilherme Ost and Andrés A. Rodríguez

We study the hydrodynamic limit of a stochastic system of neurons whose interactions are not of mean-field type and are produced by chemical and electrical synapses, and leak currents. The system consists of ε−2 neurons embedded in [0, 1)2, each spiking randomly according to a point process with rate depending on both its membrane potential and position. When neuron i spikes, its membrane potential is reset to 0 while the membrane potential of j is increased by a positive value ε 2a(i, j), if i influences j. Furthermore, between consecutive spikes, the system follows a deterministic motion due both to electrical synapses and leak currents. The electrical synapses are involved in the synchronization of the membrane potentials of the neurons, while the leak currents inhibit the activity of all neurons, attracting simultaneously their membrane potentials to 0. We show that the empirical distribution of the membrane potentials converges, as ε vanishes, to a probability density ρt(u, r) which is proved to obey a non linear PDE of Hyperbolic type.

The whole paper is available here.

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