Neurons exchange informations via discharges, propagated by membrane potential, which trigger firing of the many connected neurons. How to describe large networks of such neurons? What are the properties of these mean-field equations? How can such a network generate a spontaneous activity? Such questions can be tackled using nonlinear integro-differential equations. These are now classically used in the neuroscience community to describe neuronal networks or neural assemblies. Among them, the best known is certainly Wilson-Cowan's equation which describe spiking rates arising in different brain locations.

Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. Several mathematical results will be presented concerning existence, blow-up, convergence to steady state, for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed. One can also describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time t the probability to find a neuron with times elapsed since its last discharge. Here, we can show that small or large connectivity leads to desynchronization. For intermediate regimes, sustained periodic activity occurs. A common mathematical tool is the use of the relative entropy method.

This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo, D. Smets and G. Wainrib.
1. The single neuron, slow-fast dynamics;
2. The Integrate-and-Fire model, role of noise;
3. Analysis of I&F model;
4. Time elapsed models.

Benoit Perthame教授是巴黎第六大学（Université Pierre et Marie Curie）最高级别教授（exceptional class, the highest in France）、法国科学院院士 （Member of Académie des Sciences）、欧洲科学院院士（Member of Academia Europaea）。他曾是国际数学家大会的大会报告人（2014年，一小时）和邀请报告人（1994年），也是国际工业与应用数学大会的大会报告人（2013年）。他在微分方程、生物数学、计算数学等多方面做出了杰出的贡献，是当今世界最有影响力的数学家之一。