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 con-structed. 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. the time-changed of with unit and at . For , such dynamics admits a time-dependent density x 7→ q ( σ,x ) that is smooth on (0 , ∞ ) , except for a slope discontinuity at the reset site Λ . Moreover, the density x 7→ q ( σ,x ) is locally differentiable in 0 + and the time-changed rate of inactivation g ( σ ) is determined as the instantaneous flux: g ( σ ) = ∂ x q ( σ, 0) / 2 . For large enough λ , the flux g crosses the level 1 /λ as a locally strictly convex increasing function at time S 1 . In principle, this would correspond to the inverse time change Φ : σ 7→ ( σ − λG ( σ )) /ν admitting a strict local maximum at S 1 . However, such a behavior is not allowed as it would indicate that T 1 = Φ( S 1 ) is a time-reversal point for the original dPMF dynamics. This reveals T 1 as a blowup time for the original dynamics and S 1 as a blowup trigger time for the time-changed dynamics. During a blowup episode, the original time freezes, while the time changed dynamics