Numerical analysis for leaky-integrate-fire networks under Euler-Maruyama

TL;DR

For layered feedforward networks, this work studies Euler-Maruyama simulation of current-based LIF networks with exponential synapses and instantaneous resets and derives a Lyapunov exponent formula coupling stationary threshold flux to the reset saltation factor.

Abstract

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.

Figures (1)

• Figure 1: A feedforward network arthitecture and strong error proof strategy. (A) Feedforward network architecture with noisy currents in every layer, feedforward weights $W^{\ell-1,\ell}$, and temporally exponential synaptic kernel. (B) Illustration of fast and slow threshold crossings. Blue: true solution (reference); red: numerical solution (perturbed); yellow: pathwise error ($\Delta v$); purple: spike time error ($e_m$). (C) Pruning decomposition of the mean-square strong error into a good set $G_L$ and a bad set $B_L$.

Theorems & Definitions (103)

• Remark 2.1: Threshold current and crossing speed
• Theorem : Strong error (pruning bound; "almost" $1/2$ rate)
• Theorem : Weak error (order one)
• Theorem : Lyapunov exponents (saltation/flux and feedforward propagation)
• Theorem : Recurrent extensions (deterministic and noisy)
• Theorem 3.1: Warm-up: pruning strong bound for voltage-noise LIF
• proof
• Remark 3.2: How is this warm-up example related to current-noise LIF
• Remark 3.3: On the $\alpha$--$h$ roles in the good/bad split
• Lemma 3.4: Transversal sensitivity: STE is controlled by voltage error and the crossing speed