Physical blowups via buffered time change in a mean-field neural network

Abstract

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. Such a treatment is possible via time change techniques for a Poissonian variation of the classically considered integrate-and-fire dynamics. However, just as for shock solutions for nonlinear conservation laws, there are several admissible blowup solutions to the corresponding McKean-Vlasov equations. Because of this ambiguity, it is unclear which notion of blowup solutions shall be adopted. Here, we unambiguously define physical blowup dynamics as solutions to a fixed-point problem bearing on the time change associated to the McKean-Vlasov equation. To justify the physicality of this formulation, we introduce a buffering mechanism that regularizes blowup dynamics, while satisfying the same conservation principles as the finite-dimensional particle-system dynamics. We then show that physical blowup dynamics are recovered from these regularized solutions in the limit of vanishing buffering. Our approach also shows that these physical solutions are unique, globally defined, and avoid the so-called eternal blowup phenomenon, as long as the neurons exhibit nonzero, distributed refractory periods, a reasonable modeling assumption.