  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.
