**Speakers**

María J. Cáceres

Michael Fischer

Maxime Herda

Hugo Martin

**Abstract**

**María J. Cáceres**, The blow-up phenomenon and the “plateau” states for Nonlinear Noisy Leaky Integrate and Fire Neuronal Models

There are a lot of types of mathematical models to describe the behaviour of neural networks. One of the simplest self-contained mean-field models considered to this purpose is the family of Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models (based on nonlinear systems of PDEs of Fokker-Planck type). These models have also been studied at a microscopic level using Stochastic Differential Equations. In all of them, neurons are described at the level of their membrane potential.

In this talk we analyse this family of self-contained mean field systems. We will focus on the blow-up phenomenon. Through a numerical study of their particle systems, we will answer the question of what happens to the system when all neurons fire at the same time. We find that the neural network converges towards its unique steady state, if the system is weakly connected. Otherwise, its behaviour is more complex, tending towards 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.

This talk is based on a paper in collaboration with Ramos-Lora and previous works in collaboration with Carrillo, Perthame, Roux, Schneider and Salort.

**Michael Fischer**, Performance fluctuation in the ELO Rating, a kinetic approach

For chess, Arpad Elo developed an objective rating system in 1960, which was adopted by the world chess federation (FIDE) at the congress in Siegen in 1970. In the meantime, adaptations of the ELO rating are also used in many other sports, for example football.

Elo himself verified the reliability of his model with microscopic simulations. In 2015, Junca and Jabin derived a kinetic model from the microscopic interactions and proved the convergence of the rating *R* against the strength* ρ* under certain assumptions in the long-time behaviour.

While follow-up work dealt with learning effects, thus an increase in *ρ*, we investigate the influence of strong performance fluctuations motivated by microscopic considerations. We derive a more general model and can find conditions for the occurring parameters under which the convergence, now *R* against the expected value *θ*, can be proven.

**Maxime Herda**, A Fokker-Planck approach to the study of robustness in gene expression

In this talk, we will present a simple model describing a gene regulatory network in biology. It consists of Fokker-Planck equations which, at the microscopic level, arise from a stochastic chemical kinetic system. The densities solving the Fokker-Planck equations describe the joint distribution of the messenger RNA (mRNA) and micro-RNA (µRNA) content in a cell. Through theoretical and numerical analysis of the model, we will investigate the increase of robustness in gene expression due to the presence of μRNA.

This is a work in collaboration with Pierre Degond and Sepideh Mirrahimi.

**Hugo Martin**, Periodic asymptotic dynamics of the measure solutions to a growth-fragmentation equation in a critical case

In the last years, measure solutions to PDE, in particular those modeling populations, have drawn much attention. The talk will be devoted to the presentation of a recent, unusual result in this field, that we obtained with Pierre Gabriel. First, I will introduce the model and recall results obtained in a L^p framework. Then, I will define the notion of solution in the measure framework we are using. Moving to the proof itself, I will present the general ideas of the proof of the wellposedness of the problem, that relies on a duality relation used to build a solution expressed as a semigroup acting on an initial measure. Then, I will go a little more into details of the demonstration of the asymptotic behaviour. In particular, I will exhibit how we used Harris’ ergodic theorem to obtain a uniform exponential convergence in (weighted) total variation norm toward an oscillating measure.