Section: New Results

Modeling in computational neurosciences

Participants : Maria Caceres [Univ. Granada] , Jose Carrillo [ICREA Barcelona] , Benoît Perthame, Jonathan Touboul.

Networks of interacting neurons can be well described by nonlinear PDEs like the Noisy Integrate and Fire model. These are 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. In [10] , we analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.

At a larger scale, neurons form large-scale spatially extended populations receiving similar input and interconnected in a specific way. Each neuron receives noisy inputs, and as such their membrane potential is adequately described as the solution of stochastic network equations. In [58] we study the asymptotic regimes of such spatially extended networks with delays and obtain a complex mean-field equation the dynamics of which is analyze in [57] . We observe that noise induces transitions from stationary spatially homogeneous solutions to oscillatory solutions, and the transition is characterized by chaotic Turing patterns of activity.