Resolving the Blow-Up: A Time-Dilated Numerical Framework for Multiple Firing Events in Mean-Field Neuronal Networks

Abstract

In large-scale excitatory neuronal networks, rapid synchronization manifests as {multiple firing events (MFEs)}, mathematically characterized by a finite-time blow-up of the neuronal firing rate in the mean-field Fokker-Planck equation. Standard numerical methods struggle to resolve this singularity due to the divergent boundary flux and the instantaneous nature of the population voltage reset. In this work, we propose a robust {multiscale numerical framework based on time dilation}. By transforming the governing equation into a dilated timescale proportional to the firing activity, we desingularize the blow-up, effectively stretching the instantaneous synchronization event into a resolved mesoscopic process. This approach is shown to be physically consistent with the {microscopic cascade mechanism} underlying MFEs and the system's inherent fragility. To implement this numerically, we develop a hybrid scheme that utilizes a {mesh-independent flux criterion} to switch between timescales and a semi-analytical ``moving Gaussian'' method to accurately evolve the post-blowup Dirac mass. Numerical benchmarks demonstrate that our solver not only captures steady states with high accuracy but also efficiently reproduces periodic MFEs, matching Monte Carlo simulations without the severe time-step restrictions associated with particle cascades.

Figures (17)

• Figure 1: Schematic of the time-dilation framework and the blow-up resolution mechanism. Left: The global time mapping, where the singularity at $t^*$ (infinite slope) is unfolded into a finite jump $\Delta m$ in the dilated time $\tau$. Right: The mesoscopic dynamics in the voltage domain $v$. Within the dilated interval, the probability density $\tilde{p}$ (blue) is transported across the threshold $V_F$. Crucially, at the onset of blow-up, the density at the boundary satisfies $\tilde{p}(V_F) > 0$. The mass flux exiting $V_F$ is instantaneously re-injected at the reset potential $V_R$ (red dashed path), accumulating as a Dirac delta function.
• Figure 2: The case $b = 0$. The left column displays the steady-state density profiles; the top-right panels show the corresponding $N(t)$, and the bottom-right panels present the auxiliary function $M(\tau)$ on the $\tau$-time scale.
• Figure 3: The case $b = 1/2$. The left column displays the steady-state density profiles; the top-right panels show the corresponding $N(t)$, and the bottom-right panels present the auxiliary function $M(\tau)$ on the $\tau$-time scale.
• Figure 4: Evolution of the probability density $p(v, t)$ in the critical regime ($b = 1$). This figure illustrates the process where the system concentrates from a smooth initial distribution toward the threshold $V_F = 1$ at the critical connectivity strength. Driven by excitatory feedback, the system triggers a single synchronization. The sharp changes in density during the explosion are precisely captured via the time-dilation technique. Following the synchronization, the system returns to a stable regime, with the probability density gradually approaching a new steady-state distribution. Top-left: The initial distribution and the steady state distribution. Down-left: The pre-blowup and post-blowup distribution and the corresponding Dirac mass. Top-right: The profile of $N(t)$. Down-right: The profile of auxiliary function $M(\tau)$.
• Figure 5: Dynamical behavior of the probability density $p(v,t)$ at moderate connectivity strength ($b=2$). As the connectivity strength increases to $b=2$, the system exhibits more pronounced nonlinear characteristics. The figure demonstrates the effectiveness of the numerical algorithm in handling the formation and evolution of the Dirac mass at the moment of blow-up, reflecting the rapid recovery process of the system after a massive discharge event. Top-left: The second and the third pre-blowup profiles. Down-left: The second pre-blowup and post-blowup distribution and the corresponding Dirac mass. Top-right: The profile of $N(t)$. Down-right: The profile of auxiliary function $M(\tau)$.

Theorems & Definitions (21)

• Remark 1
• Remark 2: Direct properties of $\Delta m$
• Remark 3: Mass Conservation
• Remark 4
• Proposition 1: See also delarue2015particle or zhang2014distribution
• proof
• Proposition 2
• proof
• Remark 5: Bound on Blow-up Size
• Remark 6: Physical Interpretation of $M(\tau)$