  Idealized networks of integrate-and-fire neurons with impulse-like
interactions obey McKean-Vlasov diffusion equations in the mean-field limit.
These equations are prone to blowups: for a strong enough interaction coupling,
the mean-field rate of interaction diverges in finite time with a finite
fraction of neurons spiking simultaneously, thereby marking a macroscopic
synchronous event. Characterizing these blowup singularities analytically is
the key to understanding the emergence and persistence of spiking synchrony in
mean-field neural models. However, such a resolution is hindered by the
first-passage nature of the mean-field interaction in classically considered
dynamics. Here, we introduce a delayed Poissonian variation of the classical
integrate-and-fire dynamics for which blowups are analytically well defined in
the mean-field limit. Albeit fundamentally nonlinear, we show that this delayed
Poissonian dynamics can be transformed into a noninteracting linear dynamics
via a deterministic time change. We specify this time change as the solution of
a nonlinear, delayed integral equation via renewal analysis of first-passage
problems. This formulation also reveals that the fraction of simultaneously
spiking neurons can be determined via a self-consistent,
probability-conservation principle about the time-changed linear dynamics. We
utilize the proposed framework in a companion paper to show analytically the
existence of singular mean-field dynamics with sustained synchrony for large
enough interaction coupling.
