On Leaky-Integrate-and Fire as Spike-Train-Quantization Operator on Dirac-Superimposed Continuous-Time Signals

TL;DR

The paper treats Leaky-Integrate-and-Fire (LIF) as a nonlinear operator that maps locally HK-integrable signals with possible Dirac pulses to spike trains, and it quantifies information preservation using the weighted Alexiewicz norm $\|\cdot\|_{A,\alpha}$. It proves that for reset-to-mod (the zero-refractory-time limit) the spike train produced by LIF satisfies the quantization bound $\|\text{LIF}_{\alpha,\vartheta}(f) - f\|_{A,\alpha} < \vartheta$, under general signal conditions C1–C4, with extensions to broader signal classes. The discrete lemma is extended to continuous-time signals, showing LIF acts as a $\vartheta$-quantizer in the $\|\cdot\|_{A,\alpha}$ geometry; evaluations compare reset modes and demonstrate the robustness of reset-to-mod, especially for Dirac-impulse weights that approach or exceed the threshold. Overall, the work provides a rigorous mathematical foundation for analogue-to-spike conversion and informs reinitialization choices in SNNs, with potential implications for signal reconstruction and learning in neuromorphic systems.

Abstract

Leaky-integrate-and-fire (LIF) is studied as a non-linear operator that maps an integrable signal $f$ to a sequence $η_f$ of discrete events, the spikes. In the case without any Dirac pulses in the input, it makes no difference whether to set the neuron's potential to zero or to subtract the threshold $\vartheta$ immediately after a spike triggering event. However, in the case of superimpose Dirac pulses the situation is different which raises the question of a mathematical justification of each of the proposed reset variants. In the limit case of zero refractory time the standard reset scheme based on threshold subtraction results in a modulo-based reset scheme which allows to characterize LIF as a quantization operator based on a weighted Alexiewicz norm $\|.\|_{A, α}$ with leaky parameter $α$. We prove the quantization formula $\|η_f - f\|_{A, α} < \vartheta$ under the general condition of local integrability, almost everywhere boundedness and locally finitely many superimposed weighted Dirac pulses which provides a much larger signal space and more flexible sparse signal representation than manageable by classical signal processing.

Figures (5)

• Figure 1: Schematic description of integration within a LIF neuron with spike feedback connection leading to signals with superimposed Dirac pulses.
• Figure 2: Examples of Dirac-superimposed signals $f$ (waves (dotted line) with injected Dirac impulses) that are processed by LIF with refractory time $t_r = \hbox{discretization time step}\,\, \Delta t = 0.01$ showing $\hbox{err}(T):=\int_0^T e^{-\alpha (T - t)}(\hbox{LIF}_{\alpha, \vartheta}(f)(t)-f(t)) dt$ as magenta curve. Due to our theory $\max_t \hbox{err}(t)$ keeps below the threshold for reset-to-mod; reset-by-subtraction shows a similar $\hbox{err}$-profile except in the neighborhood of the Dirac impulses while the $\hbox{err}$-profile of reset-to-zero derails after the first Dirac impulse.
• Figure 3: Evaluation of (\ref{['eq:qFormula']}) for reset-to-mod, reset-by-subtraction and reset-to-zero (1st/2nd/3rd column), based on spike trains with spike amplitudes in $[-\vartheta, \vartheta]$ and $100$ runs. The 1st row refers to $\alpha = 1$ and the 2nd row to $\alpha = 0.1$.
• Figure 4: The same as in Fig. \ref{['fig:QuantizationError1']} but with spike amplitudes in $[-3/2 \vartheta, 3/2 \vartheta]$.
• Figure 5: Illustration of Eqn. (\ref{['eq:quantDeltaRecursion']}). The red arrows indicate reset by reset-to-mod.

Theorems & Definitions (3)

• Proposition 1: Well-Definedness of LIF on Signal Space
• Lemma 1
• Theorem 1: LIF with reset-to-mod as $\|.\|_{A,\alpha}$-Quantization for C1-4 Signals