Abstract

This talk concerns mean-field PDE descriptions of a network of neurons. We present two works, which both explore the perspective of introducing a new timescale, called the dilated timescale. The dilated timescale offers a framework for defining generalized solutions which allow blow-ups, and might also help understand solution behavior elusive in the original timescale, even in the case without blow-ups. 

In the first part of the talk, we focus on the NNLIF (nonlinear noisy leaky integrate-and-fire) model, which gives a mean-field Fokker-Planck equation for spiking neurons. A crucial mathematical issue for the model is the blow-up of the classical solution, which scientifically connects to the synchronization of a neuron network. We propose a new generalized solution which allows blow-ups. The idea is to introduce a new timescale in which the blow-up event is ``dilated and unfolded’’. We establish properties of the generalized solution including the characterization of blow-ups and the global well-posedness. The generalized solution provides a new perspective to understand the dynamics in the presence of blow-ups as well as the continuation of the solution after a blow-up. Joint work with Zhennan Zhou (https://arxiv.org/abs/2206.06972).

The second part of the talk considers the mean-field equation for pulsed-coupled oscillators. We show that the solution exhibits a contraction property in the new dilated timescale, but not in the original timescale. This suggests that it may be useful to look into the dilated timescale, even in the case without blow-ups. This is an on-going work joint with José Carrillo, Pierre Roux and Zhennan Zhou.