Séminaire Jonathan Touboul

Jeudi 23 novembre 2017 (15h00, Salle G001)

Jonathan Touboul, Collège de France.

Titre : Around the dynamics of nonlinear integrate-and-fire neurons.

Résumé : Neurons communicate information through the emission of stereotyped impulses, called spikes, which are emitted in response to increases of the electrical voltage of the cell. Integrate-and-fire models, a central class of models of neurons, decompose nerve cells activity into an integration phase and the spike emission with an instantaneous reset. They thus form a class of hybrid dynamical systems that combine a nonlinear differential equation and a discrete dynamical systems associated to spikes. I will present the analysis of the dynamics of these models of neurons. I will start by focusing on the geometry of the continuous (subthreshold) dynamics, a key to understand the way neurons respond to an input. I will next show that sequences of spikes can be described as iterates of a discrete map, called the firing map, which is a continuous unimodal map when the subthreshold dynamics has no fixed points, and which may be discontinuous with infinite left- and right-derivatives at the discontinuity points otherwise. I will show the relationship between fixed points, periodic and chaotic orbits of the firing map and regular spiking, bursting and chaotic spiking of the neuron model. Furthermore, I will exhibit the purely geometric hybrid mechanism supporting the emergence of MMOs in these systems (in the absence of explicit slow-fast structure) or a period-adding bifurcation structure and chaos. This talk relies on joint works with R. Brette, J. Rubin, J. Signerska and A. Vidal.