Leaky-integrate-and-fire (LIF) is studied as a non-linear operator that maps an integrable signal $f$ to a sequence $\eta_f$ 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 $\|.\|_{A, \alpha}$ with leaky parameter $\alpha$. We prove the quantization formula $\|\eta_f - f\|_{A, \alpha} < \vartheta$ 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.

翻译：泄漏积分-激发（LIF）被研究为一种非线性算子，它将可积信号$f$映射为离散事件序列$\eta_f$（即尖峰）。在输入不含任何狄拉克脉冲的情况下，触发尖峰后将神经元电位置零与立即减去阈值$\vartheta$两种操作并无差异。然而，在狄拉克脉冲叠加的情况下，情况有所不同，这引发了对每种提出的重置方案进行数学论证的需求。在不应期为零的极限情况下，基于阈值减法的标准重置方案会导出一种模重置方案，从而可将LIF表征为基于带泄漏参数$\alpha$的加权Alexiewicz范数$\|.\|_{A, \alpha}$的量化算子。我们证明了在局部可积、几乎处处有界且局部仅有有限多个加权狄拉克脉冲叠加的一般条件，量化公式$\|\eta_f - f\|_{A, \alpha} < \vartheta$成立，这提供了比经典信号处理更广泛的信号空间和更灵活的稀疏信号表示。