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The nonlinear noisy leaky integrate-and-fire (NNLIF) model is a popularmean-field description of a large number of interacting neurons, which hasattracted mathematicians to study from various aspects. A core property of thismodel is the finite time blow-up of the firing rate, which scientificallycorresponds to the synchronization of a neuron network, and mathematicallyprevents the existence of a global classical solution. In this work, we proposea new generalized solution based on reformulating the PDE model with a specificchange of variable in time. A firing rate dependent timescale is introduced, inwhich the transformed equation can be shown to be globally well-posed for anyconnectivity parameter even in the event of the blow-up. The generalizedsolution is then defined via the backward change of timescale, and it may havea jump when the firing rate blows up. We establish properties of thegeneralized solution including the characterization of blow-ups and the globalwell-posedness in the original timescale. The generalized solution provides anew perspective to understand the dynamics when the firing rate blows up aswell as the continuation of the solution after a blow-up.

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