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

Bernhard A. Moser; Michael Lunglmayr

Home/Archive/arXiv

arXiv · 2305.05772

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

Bernhard A. Moser·Michael Lunglmayr

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.NE cs.DM eess.SP math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Bernhard A. Moser, Michael Lunglmayr. 2023-05-09. Spiking Neural Networks in the Alexiewicz Topology: A New Perspective on Analysis and Error Bounds. https://arxiv.org/abs/2305.05772

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Breaking the Central Bias: Spatially Partitioned Experts for Coordinate-Based Neuroevolution

Evolvable-Substrate HyperNEAT (ES-HyperNEAT), a bio-inspired indirect encoding that determines neuron placement and connection weights from spatial coordinates, exhibits a failure mode on MNIST as a diagnostic benchmark. Because input pixels map to a coordinate space centered at the origin, evolved networks converge on a small central cluster of input pixels, a spatial-concentration bias; prior work observed only 21% mean accuracy in this regime. Is this bias an optimization artifact or an architectural ceiling? Inspired by Mixture-of-Experts (MoE) principles, we partition the input into non-overlapping spatial segments, each assigned to a separately evolved specialist network. With 13 such experts, this design reaches 43% mean accuracy, a 106% relative improvement over the baseline. The architectural gain does not depend on data-driven aggregation: equal-weighted averaging, which uses no validation data, already yields a 70% improvement; the gain comes from partitioning, not the weighting. Receptive-field analysis shows the mechanism: partitioning forces evolution to discover features across the entire image, expanding active pixel coverage from 4% to 79%. Absolute accuracy stays below gradient-trained baselines, but the relative gain points to central bias, not the evolutionary search. Two tools are designed to generalize beyond MNIST: a receptive-field diagnostic for silent input-coverage collapse, and a spatial-partitioning remedy that restores coverage.

cs.NE↗

arXiv2026

A Bio-Plausible Visual Neural Network for Locust-Inspired Collision Perception

Locust visual systems have long served as an important biological paradigm for studying looming perception and collision avoidance. Numerous computational models have successfully reproduced the selective responses of Lobula Giant Movement Detector (LGMD) neurons to approaching objects, thereby emulating the fundamental functionality of the biological system. However, existing models remain limited in biological plausibility and robustness when operating in complex and dynamic visual environments. To address these limitations, we propose a biologically plausible neural network for locust-inspired looming detection. The proposed framework incorporates a spatially isotropic sampling strategy that mimics the ommatidial organization of the locust compound eye, a population-voting mechanism inspired by population coding in biological neural systems, and leaky integrate-and-fire neuronal dynamics to replace conventional sigmoid-based membrane activation. Systematic experiments on synthetic stimuli, laboratory sequences, and real-world driving scenarios demonstrate that the proposed model improves robustness under challenging visual conditions while preserving computational efficiency and enhancing biological fidelity. These results highlight the potential of biologically grounded neural computation for robust and efficient collision perception.

cs.NE↗

arXiv2026

Structural Fusion of Bayesian Networks with Limited Treewidth Using Genetic Algorithms

This paper introduces an evolutionary computation approach for consensus in structural Bayesian Network (BN) fusion under the constraint of limited treewidth. The consensus BN aims to reconcile multiple input BNs into a single one that retains key structural features present in the original networks. Treewidth, a graph-based parameter associated with computationally tractable inference, is utilized to restrict the complexity of the resulting network. A genetic algorithm is proposed to look for a BN that codifies as much information about the unrestricted fusion as possible while ensuring the treewidth restriction. Experimental evaluation demonstrates the genetic algorithm's ability to obtain consensus BNs with limited treewidth, providing a valuable tool for aggregating information from diverse sources while returning a computationally actionable model.

cs.NE↗