Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up… (More)
We study the asymptotic behavior of solutions of the initial– boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution approaches exponentially its mean as time tends to… (More)
We prove the nonlinear stability in L p , with 1 p 2, of particular steady solutions of the Vlasov-Poisson system for charged particles in the whole space IR 6. Our main tool is a functional associated to the relative entropy or Casimir-energy functional.
We model a nanoMOSFET by a mesoscopic, time-dependent, coupled quantum-classical system based on a sub-band decomposition and a simple scattering operator. We first compute the sub-band decomposition and electrostatic force field described by a Schrödinger-Poisson coupled system solved by a Newton-Raphson iteration using the eigenvalue/eigenfunction… (More)
To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13, 12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neu-rons.… (More)
The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differential equation models. The main aim of the present article is to build a bridge between several point… (More)
The Network Noisy Leaky Integrate and Fire equation is among the simplest model allowing for a self-consistent description of neural networks and gives a rule to determine the probability to find a neuron at the potential v. However, its mathematical structure is still poorly understood and, concerning its solutions, very few results are available. In the… (More)
We prove the nonlinear stability in L p , with 1 ≤ p ≤ 2, of particular steady solutions of the Vlasov–Poisson system for charged particles in the whole space R 6. Our main tool is a functional associated to the relative entropy or Casimir-energy functional. 1. Introduction. We consider a gas of charged particles described by a distribution function f (t,… (More)