In this talk, I will focus on a Fokker-Planck equation modeling interacting neurons in a network where each neuron is governed by an Integrate and Fire dynamic type. When the network is excitatory, neurons that discharge, instantaneously increased the membrane potential of the neurons of the network with a speed which is proportional to the amplitude of the global activity of the network. The self-excitable nature of these neurons in the case of excitatory networks leads to phenomena of blow-up, once the proportion of neurons that are close to their action potential is too high. In this talk, we are interested in understanding the regimes where solutions globally exist. By new methods of entropy and upper-solution, we give criteria where the phenomena of blow-up can not appear and specify, in some cases, the asymptotic behavior of the solution.

integrate-and-fire - neural networks - Fokker-Planck equation - blow-up[-]

In this talk I will present a stochastic model for the excitability of a neuron in a network. The neuron described by an Hodgkin-Huxley type model receives from the network a random input which is a perturbation of a periodic deterministic signal. For such a model we study ergodicity properties. Then, we prove limit theorems in order to be able to estimate characteristics of the sequence of spiking times. This talk is based on a joint work with R. Hoepfner (Univ. Mainz) and E. Loecherbach (Univ. Cergy-Pontoise).

Hodgkin-Huxley model - ergodicity - limit theorems - estimation[-]

The determination of the shape of an obstacle from its effects on known acoustic waves is an important problem in many technologies such as sonar, geophysical exploration and medical imaging. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint, because of its ill-posed and nonlinear nature. Its investigation requires the understanding of the theory for the associated direct scattering problem, and the mastery of the corresponding numerical solution methods. The main goal of this work is the development of an efficient procedure for retrieving the shape of an elastic obstacle from the knowledge of some scattered far-field patterns, and assuming certain characteristics of the surface of the obstacle. We propose a solution methodology based on a regularized Newton-type method. The solution of the considered IOP by the proposed iterative method incurs, at each iteration, the solution of a linear system whose entries are the Fréchet derivatives of the elasto-acoustic field with respect to the shape parameters. We prove that these derivatives are solutions of the same direct elasto-acoustic scattering problem that differs only in the transmission conditions on the surface of the scatterer. Furthermore, the computational efficiency of the IOP solver depends mainly on the computational efficiency of the solution of the forward problems that arise at each Newton iteration. We propose to solve the direct scattering-type problems using a finite-element method based on discontinuous Galerkin approximations equipped with curved element boundaries. Numerical results will be presented to illustrate the salient features of this computational methodology and highlight its performance characteristics.

acoustics - shape derivative - inverse obstacle problem - Fréchet derivatives - inverse elasto-acoustic scattering problems[-]