An Understanding of the Physical Solutions and the Blow-up Phenomenon for Nonlinear Noisy Leaky Integrate and Fire Neuronal Models (Communications in Computational Physics, 2021)
The Nonlinear Noisy Leaky Integrate and Fire neuronal models are mathematical models that describe the activity of neural networks. These models havebeen studied at a microscopic level, using Stochastic Differential Equations, and ata mesoscopic/macroscopic level, through the mean field limits using Fokker-Plancktype equations. The aim of this paper is to improve their understanding, using anumerical study of their particle systems. This allows us to go beyond the mesoscopic/macroscopic description. We answer one of the most important open questionsabout these models: what happens after all the neurons in the network fire at the sametime? We find that the neural network converges towards its unique steady state, ifthe system is weakly connected. Otherwise, its behaviour is more complex, tendingtowards a stationary state or a "plateau" distribution (membrane potentials are uniformly distributed between reset and threshold values). To our knowledge, these distributions have not been described before for these nonlinear models. In addition, weanalyse in depth the behaviour of the classical and physical solutions of the StochasticDifferential Equations and, we compare it with what is already known about the classical solutions of Fokker-Planck equation. In this way, our numerical analysis, based onthe microscopic scale, allows us to explain not only what happens after the explosionphenomenon, but also, how the physical solutions of the Fokker-Planck equation are.This notion of solution, for the Fokker-Planck equation, has not been studied to date.
Keywords: Neuronal networks; Leaky Integrate and Fire models; Fokker-Planck equation; stochastic differential equation; blow-up phenomena; plateau distribution.