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

arXiv: https://arxiv.org/abs/2305.05772 · HTML: https://mlsanity.com/paper/2305.05772/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 May 2023.

## Built on by (most cited first, 2 of 2)

- [Quantization in Spiking Neural Networks](https://mlsanity.com/paper/2305.08012/) (2023)
- [On Leaky-Integrate-and Fire as Spike-Train-Quantization Operator on Dirac-Superimposed Continuous-Time Signals](https://mlsanity.com/paper/2402.07954/) (2024)

## Abstract (arXiv)

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.