Section: New Results

Mean field methods

Noise-induced behaviors in neural mean field dynamics

Participants : Jonathan Touboul, Geoffroy Hermann, Olivier Faugeras.

The collective behavior of cortical neurons is strongly affected by the presence of noise at the level of individual cells. In order to study these phenomena in large-scale assemblies of neurons, we consider networks of firing-rate neurons with linear intrinsic dynamics and nonlinear coupling, belonging to a few types of cell populations and receiving noisy currents. Asymptotic equations as the number of neurons tends to infinity (mean field equations) are rigorously derived based on a probabilistic approach. These equations are implicit on the probability distribution of the solutions which generally makes their direct analysis diffcult. However, in our case, the solutions are Gaussian, and their moments satisfy a closed system of nonlinear ordinary differential equations (ODEs), which are much easier to study than the original stochastic network equations, and the statistics of the empirical process uniformly converge towards the solutions of these ODEs. Based on this description, we analytically and numerically study the influence of noise on the collective behaviors, and compare these asymptotic regimes to simulations of the network. We observe that the mean field equations provide an accurate description of the solutions of the network equations for network sizes as small as a few hundreds of neurons. In particular, we observe that the level of noise in the system qualitatively modifies its collective behavior, producing for instance synchronized oscillations of the whole network, desynchronization of oscillating regimes, and stabilization or destabilization of stationary solutions. These results shed a new light on the role of noise in shaping collective dynamics of neurons, and gives us clues for understanding similar phenomena observed in biological networks.

This work has been accepted for publication in the SIAM Journal on Applied dynamical Systems [72] .

Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons

Participants : Javier Baladron, Diego Fasoli, Olivier Faugeras, Jonathan Touboul.

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the Fitzhugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes places, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations, or non-local partial differential equations resembling the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of these equations, i.e. the existence and uniqueness of a solution. We also show the results of some preliminary numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a infinite size network, even for modest sizes. These experiment also indicate that the McKean-Vlasov-Fokker-Planck equations may be a good way to understand the mean-field dynamics through, e.g., a bifurcation analysis.

This work has been submitted for publication in the Journal of Mathematical Neuroscience [55] .

Three applications of GPU computing in neuroscience

Participants : Javier Baladron, Olivier Faugeras.

GPUs are low cost highly parallel devices that are now not only used for graphics but also for numerical simulation. We present three applications of a computer system with multiple GPUs to the domain of theoretical neuroscience. The first application is to a continuous model of the primary visual area, the second to the simulation of a stochastic neural network, and the third to the computation of the probability distribution on the possible states of a network. In all three cases we show that the speed-up obtained by the use of GPUs has considerably helped answering a scientific or technological question.

This work has been accepted for publication in Computing in Science and Engineering [63] .