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

TL;DR

This paper analyzes the numerical accuracy of Euler-Maruyama simulations for leaky integrate-and-fire neural networks, providing error bounds and conditions for both strong and weak convergence.

Contribution

It offers the first rigorous error bounds for Euler-Maruyama simulation of LIF networks, including conditions for strong and weak convergence, and discusses extensions to recurrent networks.

Findings

Strong error order is approximately h up to polylogarithmic factors.

Weak order of convergence is 1 for smooth observables.

Error bounds depend on spike history, rate, and spike-time tail controls.

Abstract

Leaky integrate-and-fire (LIF) networks are standard reduced models for spike-based neural dynamics and a natural substrate for neuromorphic computation. We study time-driven Euler--Maruyama simulation of current-based LIF networks with exponentially decaying synapses and instantaneous resets. Because diffusion acts through the synaptic current rather than directly through the voltage, numerical error is concentrated at threshold events. It is therefore driven by spike-time perturbations and by grid-induced spike-count mismatch. For layered feedforward networks, under suitable density, rate, regularity, and one-step boundary-layer assumptions, we prove finite-horizon strong and weak error bounds. For the strong error, we first condition on spike histories that match up to the observation horizon. On this matched event, we combine a conditional single-spike hitting-time comparison with direct averaging of the induced synaptic-impact kernel against the boundary flux of crossing speeds. This yields matched-trajectory mean-square strong error of order $h$ up to polylogarithmic factors. We then control spike-count mismatch separately by local rate, dense-spike, and spike-time-tail bounds. For weak error, we use an averaged backward-semigroup argument. Assuming a backward transmission problem and one-step rate, strip, and factorial-moment controls for the numerical marginals, we obtain weak order $1$ for smooth spike-map-compatible observables, with constants explicit in rate and weight bounds. We also outline deterministic and noisy recurrent extensions, motivated by a Lyapunov-exponent formula that couples the stationary threshold flux to the reset saltation factor. These results distinguish single-trial spike-train fidelity from observable-level accuracy and clarify which notion of numerical accuracy is most relevant for mechanistic spiking models and spike-based computation.