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arXiv · 2206.06972

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

Xu'an Dou·Zhennan Zhou
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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.

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Xu'an Dou, Zhennan Zhou. 2022-06-14. Dilating blow-up time: A generalized solution of the NNLIF neuron model and its global well-posedness. https://arxiv.org/abs/2206.06972

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