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

2025-01-20

AI Summary

This work addresses the weak theoretical foundation for spiking neuron modeling in neuromorphic computing. We propose a unified mathematical framework integrating the integrate-and-fire (IF) neuron model with the send-on-delta (SOD) sampling mechanism. By introducing the Alexiewicz norm, we rigorously establish a differential-integral duality between their threshold-triggered dynamics. We further construct the first complete characterization of the Dirac impulse superposition signal space, introduce a maximum sparsity metric, and develop a sparse regularization modeling approach. We derive tight bounds on signal reconstruction error, proving that IF/SOD is theoretically optimal for bounded-variation signals and finite Dirac impulse superpositions. These results formally establish IF/SOD as an optimal sparse sampling mechanism and provide a rigorous signal-processing foundation for event-driven sensing and brain-inspired computing.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Signal Processing
• Neuromorphic Computing
• Integrate-and-Fire Models

Innovation

• Integrate-and-Fire Model
• Send-on-Delta Concept
• Signal Processing in Neuromorphic Computing