  The Nonlinear Noisy Leaky Integrate and Fire neuronal models are mathematical
models that describe the activity of neural networks. These models have been
studied at a microscopic level, using Stochastic Differential Equations, and at
a mesoscopic/macroscopic level, through the mean field limits using
Fokker-Planck type equations. The aim of this paper is to improve their
understanding, using a numerical study of their particle systems. We analyse in
depth the behaviour of the classical and physical solutions of the Stochastic
Differential Equations and, we compare it with what is already known about the
Fokker-Planck equation. This allows us to better understand what happens in the
neural network when an explosion occurs in finite time. After firing all
neurons at the same time, if the system is weakly connected, the neural network
converges towards its unique steady state. Otherwise, its behaviour is more
complex, because it can tend towards a stationary state or a "plateau"
distribution.
