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

Bernhard A. Moser, Anna Werzi, Michael Lunglmayr

TL;DR

The paper addresses the mathematical link between Integrate-and-Fire (IF) and threshold-based sampling such as Send-on-Delta (SOD), extending IF to signals with Dirac impulses via a reset-to-mod variant. It recasts IF as a quantization operator under the Alexiewicz norm, proving a global quasi-isometry bound $\|g-f\|_A-2\vartheta < \|\mathrm{IF}_{\vartheta}^M(g)-\mathrm{IF}_{\vartheta}^M(f)\|_A < \|g-f\|_A+2\vartheta$ and establishing maximal sparsity with $\|\mathrm{IF}_{\vartheta}^M(f)\|_1 = \min\{\|s\|_1: s \in \mathring{B}^A_{\vartheta}(f) \cap \mathbb{S}_{\vartheta}\}$. It further develops reconstruction schemes for both SOD and IF/mod and connects these to sparse regularization frameworks, enabling stable signal recovery and a regularization interpretation via the Alexiewicz norm. Through acceleration-data experiments, the authors compare reset-to-mod with reset-by-subtraction, showing an optimal threshold $\vartheta^*$ and highlighting the superior stability of IF/mod at small thresholds. Overall, the work provides a rigorous geometric and sparsity-based foundation for event-based sensing and points to new high-frequency, SOD-like converters grounded in these metrics.

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.

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

TL;DR

The paper addresses the mathematical link between Integrate-and-Fire (IF) and threshold-based sampling such as Send-on-Delta (SOD), extending IF to signals with Dirac impulses via a reset-to-mod variant. It recasts IF as a quantization operator under the Alexiewicz norm, proving a global quasi-isometry bound and establishing maximal sparsity with . It further develops reconstruction schemes for both SOD and IF/mod and connects these to sparse regularization frameworks, enabling stable signal recovery and a regularization interpretation via the Alexiewicz norm. Through acceleration-data experiments, the authors compare reset-to-mod with reset-by-subtraction, showing an optimal threshold and highlighting the superior stability of IF/mod at small thresholds. Overall, the work provides a rigorous geometric and sparsity-based foundation for event-based sensing and points to new high-frequency, SOD-like converters grounded in these metrics.

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.
Paper Structure (10 sections, 4 theorems, 26 equations, 7 figures)

This paper contains 10 sections, 4 theorems, 26 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Mathematical Analysis of Integrate-and-Fire
  4. IF as Quantization Operator
  5. Quasi Isometry
  6. Maximal Sparsity Property
  7. Signal Reconstruction
  8. Sparse Regularization
  9. Examples and Evaluations
  10. Conclusion

Key Result

Theorem 1

For any integrable real function $f$ with superposition of finitely many Dirac pulses, i.e., $f$ is of the type $f(t) = \widetilde{f}(t) + \sum_k a_k\delta(t- t_k)$, where $\widetilde{f}:(t_a, t_b] \rightarrow \mathbb{R}$ is bounded and integrable and $t_j \in (t_a, t_b]$, there holds where the standard quantization operation $q$, see (eq:q), is applied point-wise and the accumulating sum operato

Figures (7)

  • Figure 1: Schematic description of integration within an IF neuron with spike feedback connection leading to signals with superimposed Dirac pulses.
  • Figure 2: Standard quantization by truncation $q(x):=q_{\vartheta}(x)$ for $\vartheta = 1$, i.e., $q_{\vartheta}(x)= \vartheta q(x/\vartheta)$.
  • Figure 3: Illustration of (\ref{['eq:TVreg']}) with typical curves $\alpha$, $\beta$ of opposite monotonicity behavior. $u$ denotes the quantization step in (\ref{['eq:TVreg']}). The simulation is based on the velocity data of Fig. \ref{['fig:IFVel02']} with $\lambda=0.002$. As the data is discrete, $\alpha(u)=u$ is only approximately valid.
  • Figure 4: $100$Hz acceleration data and its sparse representation by spikes obtained by integrate-and-fire (\ref{['eq:IF']}) based on reset-to-mod and its induced reconstruction \ref{['eq:IFrec']}). Here, spike amplitudes are multiples of the threshold. Due to (\ref{['eq:quant']}) and (\ref{['eq:IFrec']}) we have $\|f - \hbox{IF}^M_{\vartheta}(f)\|_A \leq \vartheta$ and $\|f - \widetilde{f}^M\|_A \leq 2\vartheta$.
  • Figure 5: Like Fig. \ref{['fig:IFmodAcc2']}, but now with IF based on reset-by-subtraction and its reconstruction \ref{['eq:IFSubrec']}). Here, spike amplitudes are restricted to $\pm \vartheta$. $\|f - \widetilde{f}_{\hbox{\tiny{IF/sub}}}\|_A \leq \vartheta$ is not valid any more in general.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 1: Quantization Operator
  • Theorem 2: Maximal Sparsity Property of IF/mod
  • Theorem 3: IF/mod-Reconstruction
  • Theorem 4: SOD as sparse-regularization solver