Dilating blow-up time: A generalized solution of the NNLIF neuron model and its global well-posedness

TL;DR

This paper introduces a generalized solution framework for the NNLIF neuron model that remains well-defined even during firing rate blow-ups, offering new insights into neuron network synchronization and solution continuation.

Contribution

A novel change of variable in time is proposed to define a generalized solution that is globally well-posed despite blow-ups in the firing rate.

Findings

The transformed PDE is globally well-posed for any connectivity parameter.

The generalized solution can have jumps at blow-up times.

The approach provides a new perspective on solution dynamics during blow-ups.

Abstract

The nonlinear noisy leaky integrate-and-fire (NNLIF) model is a popular mean-field description of a large number of interacting neurons, which has attracted mathematicians to study from various aspects. A core property of this model is the finite time blow-up of the firing rate, which scientifically corresponds to the synchronization of a neuron network, and mathematically prevents the existence of a global classical solution. In this work, we propose a new generalized solution based on reformulating the PDE model with a specific change of variable in time. A firing rate dependent timescale is introduced, in which the transformed equation can be shown to be globally well-posed for any connectivity parameter even in the event of the blow-up. The generalized solution is then defined via the backward change of timescale, and it may have a jump when the firing rate blows up. We establish properties of the generalized solution including the characterization of blow-ups and the global well-posedness in the original timescale. The generalized solution provides a new perspective to understand the dynamics when the firing rate blows up as well as the continuation of the solution after a blow-up.