Integrate-and-Fire from a Mathematical and Signal Processing Perspective

TL;DR

This paper explores the mathematical and signal processing foundations of the Integrate-and-Fire model, revealing its relation to Send-on-Delta sampling and providing insights into signal space, sparsity, and reconstruction error bounds.

Contribution

It establishes a connection between IF and SOD, offering a new perspective on the metric structure and signal space in neuromorphic computing.

Findings

IF relates to Send-on-Delta as a differential form.

Insights into the signal space include bounded signals with Dirac impulses.

Provides bounds for signal reconstruction and sparsity properties.

Abstract

Integrate-and-Fire (IF) is an idealized model of the spike-triggering mechanism of a biological neuron. It is used to realize the bio-inspired event-based principle of information processing in neuromorphic computing. We show that IF is closely related to the concept of Send-on-Delta (SOD) as used in threshold-based sampling. It turns out that the IF model can be adjusted in a way that SOD can be understood as differential version of IF. As a result, we gain insight into the underlying metric structure based on the Alexiewicz norm with consequences for clarifying the underlying signal space including bounded integrable signals with superpositions of finitely many Dirac impulses, the identification of a maximum sparsity property, error bounds for signal reconstruction and a characterization in terms of sparse regularization.