Section: New Results

Mean field theory and stochastic processes

Emergence of collective phenomena in a population of neurons

Participants : Benjamin Aymard, Fabien Campillo, Romain Veltz.

In this work, we propose a new model of biological neural network, combining a two-dimensional integrate-and-fire neuron model with a deterministic model of electrical synapse, and a stochastic model of chemical synapse. We describe the dynamics of a population of neurons in interaction as a piecewise deterministic Markov process. We prove the weak convergence of the associated empirical process, as the population size tends to infinity, towards a McKean-Vlasov type process and we describe the associated PDE. We are also interested in the simulation of these dynamics, in particular by comparing “detailed” simulations of a finite population of neurons with a simulation of the system with infinite population. Benjamin Aymard has the adapted toolkit to attack these questions numerically. The mean field equations studied by Benjamin are of transport type for which numerical methods are technical. However, they are the domain of expertise of Benjamin. His postdoc is funded by the Flagship Human Brain Project.

Off-line numerical Bayes identification of dynamical systems for life sciences

Participants : Fabien Campillo, Vivien Rossi [CIRAD] .

In this project, we develop Monte Carlo algorithms for the identification of parameters and hidden components for dynamic systems used in the life sciences. The peculiarity of these systems and they do not require online processing and they call for data of various natures and sometimes low quality. We use particle filtering techniques so that we try to improve the prediction phases using MCMC techniques.

On the variations of the principal eigenvalue with respect to a parameter in growth-fragmentation models

Participants : Fabien Campillo, Nicolas Champagnat [Inria, project-team TOSCA, Nancy] , Coralie Fritsch [Inria, project-team TOSCA, Nancy] .

We study the variations of the principal eigenvalue associated to a growth-fragmentation-death equation with respect to a parameter acting on growth and fragmentation. To this aim, we use the probabilistic individual-based interpretation of the model. We study the variations of the survival probability of the stochastic model, using a generation by generation approach. Then, making use of the link between the survival probability and the principal eigenvalue established in a previous work, we deduce the variations of the eigenvalue with respect to the parameter of the model.

This work has been published in Communications in Mathematical Sciences and is available as [18].

Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics

Participants : Audric Drogoul [Thales, France] , Romain Veltz.

In this work, we provide three different numerical evidences for the occurrence of a Hopf bifurcation in a recently derived mean field limit of a stochastic network of excitatory spiking neurons [40], [46]. The mean field limit is a challenging nonlocal nonlinear transport equation with boundary conditions. The first evidence relies on the computation of the spectrum of the linearized equation. The second stems from the simulation of the full mean field. Finally, the last evidence comes from the simulation of the network for a large number of neurons. We provide a “recipe” to find such bifurcation which nicely complements the works in [40], [46]. This suggests in return to revisit theoretically these mean field equations from a dynamical point of view. Finally, this work shows how the noise level impacts the transition from asynchronous activity to partial synchronization in excitatory globally pulse-coupled networks.

This work has been published in Chaos and is available as [22].

Mathematical statistical physics applied to neural populations

Participants : Émilie Soret, Olivier Faugeras, Étienne Tanré [Inria, project-team TOSCA, Sophia-Antipolis] .

This project focuses on Mean-Field descriptions or thermodynamics limits of large populations of neurons. They study a system of Stochastic Differential Equations (SDEs) which describes the evolution of membrane potential of each neuron over the time when the synaptic weights are random variables (not assumed to be independent). This setup is well suited to Émilie, who has worked during her PhD and first postdoc on mathematical statistical physics and stochastic processes. Her postdoc is funded by the Flagship Human Brain Project.

A numerical approach to determine mutant invasion fitness and evolutionary singular strategies

Participants : Coralie Fritsch [Inria, project-team TOSCA, Nancy] , Fabien Campillo, Otso Ovaskainen [University of Helsinki, Finland] .

We propose a numerical approach to study the invasion fitness of a mutant and to determine evolutionary singular strategies in evolutionary structured models in which the competitive exclusion principle holds. Our approach is based on a dual representation, which consists of the modelling of the small size mutant population by a stochastic model and the computation of its corresponding deterministic model. The use of the deterministic model greatly facilitates the numerical determination of the feasibility of invasion as well as the convergence-stability of the evolutionary singular strategy. Our approach combines standard adaptive dynamics with the link between the mutant survival criterion in the stochastic model and the sign of the eigenvalue in the corresponding deterministic model. We present our method in the context of a mass-structured individual-based chemostat model. We exploit a previously derived mathematical relationship between stochastic and deterministic representations of the mutant population in the chemostat model to derive a general numerical method for analyzing the invasion fitness in the stochastic models. Our method can be applied to the broad class of evolutionary models for which a link between the stochastic and deterministic invasion fitnesses can be established.

This work has been published in Theoretical Population Biology and is available as [23].