Time to Spike? Understanding the Representational Power of Spiking Neural Networks in Discrete Time

2025-05-23

AI Summary

This paper addresses the weak theoretical foundation of discrete-time leaky integrate-and-fire spiking neural networks (LIF-SNNs) by establishing the first rigorous characterization of their function approximation capability and input-space partitioning complexity. Methodologically, we introduce a theoretical framework based on piecewise-constant function modeling and polyhedral partition analysis, deriving tight lower bounds on the minimum network size required to approximate continuous functions, and quantifying the synergistic impact of temporal depth (number of time steps) and architectural depth (number of layers) on representational power. Key contributions are: (1) the first provable function approximation bounds for discrete-time LIF-SNNs; (2) the identification of temporal depth—not activation nonlinearity—as the primary driver of expressive capacity, distinguishing SNNs fundamentally from artificial neural networks (ANNs); and (3) empirical validation of the theoretically predicted exponential growth in partition complexity with temporal depth. These results establish a new paradigm for principled architecture design and theoretical analysis of SNNs.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Deep Neural Architectures and Foundation Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Recent years have seen significant progress in developing spiking neural networks (SNNs) as a potential solution to the energy challenges posed by conventional artificial neural networks (ANNs). However, our theoretical understanding of SNNs remains relatively limited compared to the ever-growing body of literature on ANNs. In this paper, we study a discrete-time model of SNNs based on leaky integrate-and-fire (LIF) neurons, referred to as discrete-time LIF-SNNs, a widely used framework that still lacks solid theoretical foundations. We demonstrate that discrete-time LIF-SNNs with static inputs and outputs realize piecewise constant functions defined on polyhedral regions, and more importantly, we quantify the network size required to approximate continuous functions. Moreover, we investigate the impact of latency (number of time steps) and depth (number of layers) on the complexity of the input space partitioning induced by discrete-time LIF-SNNs. Our analysis highlights the importance of latency and contrasts these networks with ANNs employing piecewise linear activation functions. Finally, we present numerical experiments to support our theoretical findings.

Problem

• Theoretical understanding of discrete-time LIF-SNNs is limited
• Quantify network size needed for continuous function approximation
• Impact of latency and depth on input space partitioning

Innovation

• Discrete-time LIF-SNNs model for theoretical analysis
• Quantify network size for continuous function approximation
• Analyze latency and depth impact on partitioning