Xu'an Dou

Xu'an Dou, Frank Chen, Kevin K Lin et al.

Leaky integrate-and-fire (LIF) networks are canonical models in computational neuroscience and a standard substrate for neuromorphic AI. We study Euler-Maruyama simulation of current-based LIF networks with exponential synapses and instantaneous resets. Since diffusion enters only through the synaptic current, each spike is a threshold hit for a deterministically advected voltage with random crossing speed $A$. Hence, numerical error is concentrated at event times. For layered feedforward networks we prove finite-horizon ($T$) mean-square strong bounds and weak bounds. With time grid $h$, the strong analysis uses a pruning-and-balance strategy: path space is split into a good set, where exact and numerical spike histories match and each matched spike satisfies crossing speed $A\geq α$, and a bad set containing near-tangential crossings and spike-count mismatch. On the good set, spike-time error is the local Euler-Maruyama error $h^{1/2}/A$. A threshold-flux estimate gives $E[A^{-2}\mathbf{1}_{\{A\geα\}}]\lesssim α_0^{-2}+Tρ_{\max}\log(α_0/α)$ for any $α_0>α$, while near-tangential probability is $O(Tα^2)$. Balancing these terms yields mean-square error $h$ times polylogarithmic factors, with explicit dependence on time, depth, and weights; away from spike mismatch, this matches the classical Euler-Maruyama $1/2$ rate up to logarithmic losses. For weak error, a backward-Kolmogorov argument adapted to resets splits the one-step defect into an interior Taylor term and a boundary-strip term for spikes, yielding order $O(Th)$. We also derive a Lyapunov exponent formula coupling stationary threshold flux to the reset saltation factor. Based on the results for feedforward networks, we also outline extensions to recurrent networks. This includes loop-truncated strong/weak bounds controlled by synaptic cycles and rate \& synchronicity estimates.

Xu'an Dou, Louis Tao, Zhe Xue et al.

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.