  We recently introduced idealized mean-field models for networks of
integrate-and-fire neurons with impulse-like interactions -- the so-called
delayed Poissonian mean-field models. Such models 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. Due to
the reset mechanism of integrate-and-fire neurons, these blowups can happen
repeatedly, at least in principle. A benefit of considering Poissonian
mean-field models is that one can resolve blowups analytically by mapping the
original singular dynamics onto uniformly regular dynamics via a time change.
Resolving a blowup then amounts to solving the fixed-point problem that
implicitly defines the time change, which can be done consistently for a single
blowup and for nonzero delays. Here we extend this time-change analysis in two
ways: First, we exhibit the existence and uniqueness of explosive solutions
with a countable infinity of blowups in the large interaction regime. Second,
we show that these delayed solutions specify "physical" explosive solutions in
the limit of vanishing delays, which in turn can be explicitly constructed. The
first result relies on the fact that blowups are self-sustaining but
nonoverlapping in the time-changed picture. The second result follows from the
continuity of blowups in the time-changed picture and incidentally implies the
existence of periodic solutions. These results are useful to study the
emergence of synchrony in neural network models.
