Random Spiking Neural Networks are Stable and Spectrally Simple

TL;DR

This paper analyzes the stability and spectral properties of spiking neural networks using Boolean function analysis, revealing that wide LIF-SNNs are stable and spectrally simple, which explains their robustness.

Contribution

It introduces the concept of spectral simplicity for SNNs and demonstrates that random LIF-SNNs are biased toward simple functions, linking spectral properties to stability.

Findings

Wide LIF-SNN classifiers are stable on average.

Fourier spectrum concentrates on low-frequency components.

Random LIF-SNNs are biased toward simple functions.

Abstract

Spiking neural networks (SNNs) are a promising paradigm for energy-efficient computation, yet their theoretical foundations-especially regarding stability and robustness-remain limited compared to artificial neural networks. In this work, we study discrete-time leaky integrate-and-fire (LIF) SNNs through the lens of Boolean function analysis. We focus on noise sensitivity and stability in classification tasks, quantifying how input perturbations affect outputs. Our main result shows that wide LIF-SNN classifiers are stable on average, a property explained by the concentration of their Fourier spectrum on low-frequency components. Motivated by this, we introduce the notion of spectral simplicity, which formalizes simplicity in terms of Fourier spectrum concentration and connects our analysis to the simplicity bias observed in deep networks. Within this framework, we show that random LIF-SNNs are biased toward simple functions. Experiments on trained networks confirm that these stability properties persist in practice. Together, these results provide new insights into the stability and robustness properties of SNNs.