Spiking Neural Networks in the Alexiewicz Topology: A New Perspective on Analysis and Error Bounds

TL;DR

This paper introduces a new mathematical framework using the Alexiewicz topology to analyze error propagation in spiking neural networks, providing bounds and insights into their stability and behavior.

Contribution

It develops a novel topological approach to model SNNs as endomorphisms, enabling precise error measurement and analysis of error bounds in neuromorphic computing.

Findings

Established the Alexiewicz topology for LIF neuron models with leakage

Derived error bounds and inequalities for spike train transformations

Identified a Lipschitz-style upper bound for error propagation

Abstract

In order to ease the analysis of error propagation in neuromorphic computing and to get a better understanding of spiking neural networks (SNN), we address the problem of mathematical analysis of SNNs as endomorphisms that map spike trains to spike trains. A central question is the adequate structure for a space of spike trains and its implication for the design of error measurements of SNNs including time delay, threshold deviations, and the design of the reinitialization mode of the leaky-integrate-and-fire (LIF) neuron model. First we identify the underlying topology by analyzing the closure of all sub-threshold signals of a LIF model. For zero leakage this approach yields the Alexiewicz topology, which we adopt to LIF neurons with arbitrary positive leakage. As a result LIF can be understood as spike train quantization in the corresponding norm. This way we obtain various error bounds and inequalities such as a quasi isometry relation between incoming and outgoing spike trains. Another result is a Lipschitz-style global upper bound for the error propagation and a related resonance-type phenomenon.