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.

翻译：积分点火（IF）是生物神经元脉冲触发机制的一种理想化模型，它被用于神经形态计算中实现受生物启发的基于事件的信息处理原理。我们证明，IF与阈值采样中使用的Send-on-Delta（SOD）概念密切相关。结果表明，可以通过调整IF模型，使SOD被理解为IF的微分形式。由此，我们基于Alexiewicz范数深入理解了其内在的度量结构，这有助于澄清相关的信号空间（包括有界可积信号与有限个狄拉克脉冲的叠加），识别最大稀疏特性，推导信号重构的误差界，并通过稀疏正则化进行特征描述。