Submitted

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.

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.

Place cells are believed to organize memory across space and time, inspiring the idea of the cognitive map. Yet unlike the structured activity in the associated grid and head-direction cells, they remain an enigma: their responses have been difficult to predict and are complex enough to be statistically well-described by a random process. Here we report one step toward the ultimate goal of understanding place cells well enough to predict their fields. Within a theoretical framework in which place fields are derived as a conjunction of external cues with internal grid cell inputs, we predict that even apparently random place cell responses should reflect the structure of their grid inputs and that this structure can be unmasked if probed in sufficiently large neural populations and large environments. To test the theory, we design experiments in long, locally featureless spaces to demonstrate that structured scaffolds undergird place cell responses. Our findings, together with other theoretical and experimental results, suggest that place cells build memories of external inputs by attaching them to a largely prespecified grid scaffold.