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On the Sampling Sparsity of Neuromorphic Analog-to-Spike Conversion based on Leaky Integrate-and-Fire

Bernhard A. Moser, Michael Lunglmayr

TL;DR

It is shown in a rigorous mathematical way that information encoding by means of Threshold-Based Representation based on either Leaky Integrate-and-Fire (LIF) or Send-on-Delta (SOD) is linked to an analog-to-spike conversion that guarantees maximum sparsity while satisfying an approximation condition based on the Alexiewicz norm.

Abstract

In contrast to the traditional principle of periodic sensing neuromorphic engineering pursues a paradigm shift towards bio-inspired event-based sensing, where events are primarily triggered by a change in the perceived stimulus. We show in a rigorous mathematical way that information encoding by means of Threshold-Based Representation based on either Leaky Integrate-and-Fire (LIF) or Send-on-Delta (SOD) is linked to an analog-to-spike conversion that guarantees maximum sparsity while satisfying an approximation condition based on the Alexiewicz norm.

On the Sampling Sparsity of Neuromorphic Analog-to-Spike Conversion based on Leaky Integrate-and-Fire

TL;DR

It is shown in a rigorous mathematical way that information encoding by means of Threshold-Based Representation based on either Leaky Integrate-and-Fire (LIF) or Send-on-Delta (SOD) is linked to an analog-to-spike conversion that guarantees maximum sparsity while satisfying an approximation condition based on the Alexiewicz norm.

Abstract

In contrast to the traditional principle of periodic sensing neuromorphic engineering pursues a paradigm shift towards bio-inspired event-based sensing, where events are primarily triggered by a change in the perceived stimulus. We show in a rigorous mathematical way that information encoding by means of Threshold-Based Representation based on either Leaky Integrate-and-Fire (LIF) or Send-on-Delta (SOD) is linked to an analog-to-spike conversion that guarantees maximum sparsity while satisfying an approximation condition based on the Alexiewicz norm.

Paper Structure

This paper contains 15 sections, 7 theorems, 38 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Spike-Train Quantization in the Alexiewicz Norm
  4. Geometry of the Alexiewicz Ball
  5. Proof.
  6. Sparsity Bounds for Leaky Integrate-and-Fire
  7. Proof.
  8. Extremal Sparsity Property of Integrate-and-Fire
  9. Proof.
  10. Extremal Sparsity Property of Integrate-and-Fire (IF) and Send-on-Delta (SOD)
  11. Sparsity in the General Case
  12. Conclusion
  13. Proof.
  14. Proof.
  15. Proof.

Key Result

Lemma 1

where for which the inverse is found (by using the product rule) to be $f(t) = A_{\alpha}^{-1}(\hat{f}):= \alpha\, \hat{f}(t) + \frac{d}{dt}\hat{f}(t)$, where $\frac{d}{dt}\hat{f}(t)$ denotes the distributional derivative of $\hat{f}$.

Figures (13)

  • Figure 1: Illustration of standard quantization function $q(z):=q_{\vartheta}(z)$ for $\vartheta = 1$
  • Figure 2: In contrast to the standard $l_p$ norms (left), for the Alexiewicz nom $\|.\|_{A, \alpha}$, $\beta = e^{-\alpha}$, time ordering matters resulting in a less symmetric geometry of its unit ball (right).
  • Figure 3: Illustration of spike trains $s = s_1\, \delta(t-t_1) + s_2\, \delta(t-t_2)$ consisting of just $2$ spikes $(s_1, s_2)$, represented as 2D points. $l_1$-ball in the center (black dashed line) and 2D unit Alexiewicz balls (blue) centered at input signals $f$ (black points), with closest $l_1$-points (gray points) and LIF spike train (red points). In this setting LIF satisfies the extremal sparsity property in the sense that the red points are those grid points within the Alexiewics balls that have minimal $l_1$-norm.
  • Figure 4: Distribution of $\lambda = (\|s\|_1 - c)/(\|f\|_1 - c)$ for $f$ satisfying the condition indicated in the title, where $s = \hbox{LIF}_{\alpha, \vartheta}(f)$ and $c = \|B_{\alpha, \vartheta}(f)\|_1$.
  • Figure 5: Like Fig. \ref{['fig:ratio1']} for $f$ satisfying the condition indicated in the title.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 1
  • Theorem 2: Quantization Representation of Integrate-and-Fire
  • Theorem 3: Extremal Sparsity Property of Integrate-and-Fire
  • Corollary 1: Extremal Sparsity Property of SOD
  • Lemma 2
  • Lemma 3