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

arXiv: https://arxiv.org/abs/2402.07954 · HTML: https://mlsanity.com/paper/2402.07954/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 9 references that are in the map.
- First posted on arXiv 10 Feb 2024.

## Builds on (most cited first, 3 of 3)

- [Training Spiking Neural Networks Using Lessons From Deep Learning](https://mlsanity.com/paper/2109.12894/) (2021)
- [On Quasi-Isometry of Threshold-Based Sampling](https://mlsanity.com/paper/1811.01436/) (2018)
- [Spiking Neural Networks in the Alexiewicz Topology: A New Perspective on Analysis and Error Bounds](https://mlsanity.com/paper/2305.05772/) (2023)

## Abstract (arXiv)

Leaky-integrate-and-fire (LIF) is studied as a non-linear operator that maps an integrable signal f to a sequence ηf 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 |ηf - 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.