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Model agnostic signal encoding by leaky integrate and fire, performance and uncertainty

Diana Carbajal, José Luis Romero

TL;DR

This work analyzes the performance of a leaky integrate-and-fire time-encoding machine (IF-TEM) as a model-agnostic encoder of continuous-time signals. It introduces a bandwidth-based Ansatz to quantify approximate encoding guarantees that hold across broad signal classes, while explicitly accounting for uncertainties in leakage, past/future content, and spike timing, using the Wasserstein-1 distance to measure spike-discrepancies. The authors prove an explicit error bound (Theorem 1) relating the reconstruction error to the firing threshold $\theta$, target bandwidth $\Omega$, spike count $n$, and disturbance parameters, and they establish a stability result (Corollary 1) for signals compared via their bandwidth Ansatz. The paper also provides concrete examples (shift-invariant and spline-like spaces) and numerical experiments that validate the theory and illustrate robust performance under practical imperfections, offering a principled initialization for iterative, model-specific reconstructions. Overall, the results deliver model-agnostic guarantees for effective time-encoding and reconstruction in neuromorphic and time-encoded sensing contexts, with clear implications for low-power, spike-based signal processing.

Abstract

Integrate-and-fire is a resource efficient time-encoding mechanism that summarizes into a signed spike train those time intervals where a signal's charge exceeds a certain threshold. We analyze the IF encoder in terms of a very general notion of approximate bandwidth, which is shared by most commonly-used signal models. This complements results on exact encoding that may be overly adapted to a particular signal model. We take into account, possibly for the first time, the effect of uncertainty in the exact location of the spikes (as may arise by decimation), uncertainty of integration leakage (as may arise in realistic manufacturing), and boundary effects inherent to finite periods of exposure to the measurement device. The analysis is done by means of a concrete bandwidth-based Ansatz that can also be useful to initialize more sophisticated model specific reconstruction algorithms, and uses the earth mover's (Wasserstein) distance to measure spike discrepancy.

Model agnostic signal encoding by leaky integrate and fire, performance and uncertainty

TL;DR

This work analyzes the performance of a leaky integrate-and-fire time-encoding machine (IF-TEM) as a model-agnostic encoder of continuous-time signals. It introduces a bandwidth-based Ansatz to quantify approximate encoding guarantees that hold across broad signal classes, while explicitly accounting for uncertainties in leakage, past/future content, and spike timing, using the Wasserstein-1 distance to measure spike-discrepancies. The authors prove an explicit error bound (Theorem 1) relating the reconstruction error to the firing threshold , target bandwidth , spike count , and disturbance parameters, and they establish a stability result (Corollary 1) for signals compared via their bandwidth Ansatz. The paper also provides concrete examples (shift-invariant and spline-like spaces) and numerical experiments that validate the theory and illustrate robust performance under practical imperfections, offering a principled initialization for iterative, model-specific reconstructions. Overall, the results deliver model-agnostic guarantees for effective time-encoding and reconstruction in neuromorphic and time-encoded sensing contexts, with clear implications for low-power, spike-based signal processing.

Abstract

Integrate-and-fire is a resource efficient time-encoding mechanism that summarizes into a signed spike train those time intervals where a signal's charge exceeds a certain threshold. We analyze the IF encoder in terms of a very general notion of approximate bandwidth, which is shared by most commonly-used signal models. This complements results on exact encoding that may be overly adapted to a particular signal model. We take into account, possibly for the first time, the effect of uncertainty in the exact location of the spikes (as may arise by decimation), uncertainty of integration leakage (as may arise in realistic manufacturing), and boundary effects inherent to finite periods of exposure to the measurement device. The analysis is done by means of a concrete bandwidth-based Ansatz that can also be useful to initialize more sophisticated model specific reconstruction algorithms, and uses the earth mover's (Wasserstein) distance to measure spike discrepancy.

Paper Structure

This paper contains 28 sections, 9 theorems, 90 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Time-encoding via integrate-and-fire
  3. Contribution
  4. Results
  5. The leaky integrate-and-fire sampler
  6. Goal
  7. Sources of uncertainty
  8. Specifications for the leakage intensity
  9. The signal's past and future
  10. The location of the spikes
  11. Signal's bandwidth
  12. The bandwidth-based Ansatz
  13. Signal models and quantification of their uncertainty
  14. Proof of main result
  15. Sensitivity to uncertainties
  16. ...and 13 more sections

Key Result

Theorem 2.1

Let a signal $f$ with $\hat{f} \in L^1(\mathbb{R})$ be measured on the observation window $[0,T]$ by an integrate-and-fire sampler with firing threshold $\theta$. Let $f$ have bandwidth $\Omega \geq 1$ with tolerance $\Delta_\Omega=\theta$ and define the inference window $[T_1,T_2]$ by Then the bandwidth-based Ansatz eq_ba satisfies for an absolute constant $C>0$.

Figures (6)

  • Figure 1: Block diagram of a leaky IF sampler with firing threshold $\theta>0$ and leakage intensity $\alpha>0$.
  • Figure 2: A signal and its IF output; see also Section \ref{['sec_num']}.
  • Figure 3: The frequency cut-off function $\hat{\psi}$
  • Figure 4: Reconstructions of the test signal with different values of $\theta$.
  • Figure 5: $L^\infty$ reconstruction error for different values of $\theta$.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1: Sensitivity to the leakage intensity
  • proof
  • Lemma 3.2: Sensitivity to firing times
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 8 more