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

Bernhard A. Moser, Michael Lunglmayr

TL;DR

This work builds a mathematical foundation for spiking neural networks by placing spike trains in the Alexiewicz topology, enabling rigorous analysis of error propagation through LIF neurons. It establishes that the Leaky Integrate-and-Fire operator acts as a spike-train quantizer in the $\|\cdot\|_{A,\alpha}$ norm, deriving tight quantization bounds and a quasi-isometry relation between inputs and outputs. It further develops additive-error bounds, identifies a resonance-type phenomenon, and provides global Lipschitz-type bounds for feedforward SNNs, all while advocating reset-to-mod as a robust reinitialization mode. The results offer principled, topology-driven guarantees for error propagation in neuromorphic computing and point to a unified framework for event-based sampling and learning in SNNs.

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.

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

TL;DR

This work builds a mathematical foundation for spiking neural networks by placing spike trains in the Alexiewicz topology, enabling rigorous analysis of error propagation through LIF neurons. It establishes that the Leaky Integrate-and-Fire operator acts as a spike-train quantizer in the norm, deriving tight quantization bounds and a quasi-isometry relation between inputs and outputs. It further develops additive-error bounds, identifies a resonance-type phenomenon, and provides global Lipschitz-type bounds for feedforward SNNs, all while advocating reset-to-mod as a robust reinitialization mode. The results offer principled, topology-driven guarantees for error propagation in neuromorphic computing and point to a unified framework for event-based sampling and learning in SNNs.

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.
Paper Structure (22 sections, 10 theorems, 57 equations, 17 figures, 2 algorithms)

This paper contains 22 sections, 10 theorems, 57 equations, 17 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Which Topology for the Space of Spike Trains is Appropriate?
  4. Alexiewicz Topology
  5. Spike Train Quantization
  6. Proof.
  7. Proof.
  8. Additive Spike Errors and a Resonance Phenomenon
  9. Proof.
  10. Evaluation
  11. Spike Train Quantization due to (\ref{['eq:quantization']})
  12. Error Bounds regarding Postulates 1 and 2 and Quasi Isometry
  13. Lipschitz-Style Upper Bound for LIF and SNNs
  14. Outlook and Conclusion
  15. Proof of Theorem \ref{['th:closureIF']}
  16. ...and 7 more sections

Key Result

Lemma 1

For a leaky integrate-and-fire neuron $\hbox{LIF}_{\vartheta, \alpha}: \mathbb{S} \rightarrow \mathbb{S}$ with $0 \leq \alpha < \infty$ we have:

Figures (17)

  • Figure 1: Paradigm shift in information encoding of uniform (left) versus threshold-based (right) sampling.
  • Figure 2: LIF in continuous time with reset-by-subtraction, resp. reset-to-mod; at $t_6$ the amplitude $a_6 \in (2\vartheta,3\vartheta)$ of the input spike, which causes a two times cascaded reset-by-subtraction resulting in an output spike amplitude $b_6 = 2 \vartheta = [a_6/\vartheta]\,\vartheta$.
  • Figure 3: SNN as weighted directed acyclic graph.
  • Figure 4: Postulates for an adequate metric $d(.,.)$ for spike strains.
  • Figure 5: $\|.\|_{A,\alpha}$-unit balls for spike trains $\eta = a_1 \delta_{0} + a_2 \delta_{1}$ for $\|.\|_{A,\alpha}$ with $\alpha = 0$ (solid black), $\alpha = \infty$ (dashed), $\alpha = 1$ and $\alpha = 2$ (gray)
  • ...and 12 more figures

Theorems & Definitions (11)

  • Lemma 1
  • Theorem 1
  • Corollary 1: LIF Quasi Isometry
  • Corollary 2: Error Bound on Lag
  • Corollary 3: Error Bound on Threshold Perturbation
  • Theorem 2: Spike Train Decomposition
  • Lemma 2: Additive Error Bound for Integrate-and-Fire
  • Theorem 3: Liptschitz-Style Upper Bound for the LIF model
  • Example 1
  • Theorem 4: Global Lipschitz-Style Bound for SNNs
  • ...and 1 more