Many integro-differential equations are used to describe neuronal networks or neural assemblies. Among them, the Wilson-Cowan equations are the most well-known and describe spiking rates in different locations. Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. It has also been proposed to 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 time s elapsed since its last discharge. We will compare these models and perform some mathematical analysis. A striking observation is that solutions to the I\&F can blow-up in finite time, a form of strong desynchronization. We can also show that for small or large connectivity the 'elapsed time model' leads to desynchronization. For intermediate regimes, sustained periodic activity occurs and its shape is compatible with observations. A common tool is the use of the relative entropy method. This talk is based on works with K. Pakdaman and D. Salort, M. Caceres and J. A. Carrillo.

报告人简介:  Professor Benoit Perthame, affiliated with the University of P. and M. Curie (Paris 6) and Institut Universitaire de France, is one of the leading applied mathematicians in the world. His name is associated with a number of important mathematical results like the Averaging Lemma. In recent years, his main research interests focus on nonlinear partial differential equations and applications to biology. He has been an invited speaker at many important international occasions, like the International Congress of Mathematicians (Zurich, 1994) and ICIAM 2011 (Vancouver), and in the editorial boards of many famous international journals. He has received many international awards and has published 2 books and more than 200 papers.