In order to ease the analysis of error propagation in neuromorphic computing and to get a better understanding of spiking neural networks (SNN), we address the problem of mathematical analysis of SNNs as endomorphisms that map spike trains to spike trains. A central question is the adequate structure for a space of spike trains and its implication for the design of error measurements of SNNs including time delay, threshold deviations, and the design of the reinitialization mode of the leaky-integrate-and-fire (LIF) neuron model. First we identify the underlying topology by analyzing the closure of all sub-threshold signals of a LIF model. For zero leakage this approach yields the Alexiewicz topology, which we adopt to LIF neurons with arbitrary positive leakage. As a result LIF can be understood as spike train quantization in the corresponding norm. This way we obtain various error bounds and inequalities such as a quasi isometry relation between incoming and outgoing spike trains. Another result is a Lipschitz-style global upper bound for the error propagation and a related resonance-type phenomenon.

翻译：为简化神经形态计算中误差传播的分析，并深化对脉冲神经网络(SNN)的理解，本文将SNN作为将脉冲序列映射到脉冲序列的自同态进行数学分析。核心问题包括：脉冲序列空间的合理结构，及其对脉冲神经网络误差度量设计（涉及时间延迟、阈值偏移及漏积分点火(LIF)神经元模型重置模式设计）的启示。首先通过分析LIF模型所有阈下信号的闭包，确定其底层拓扑结构。在零泄漏情况下，该方法导出阿列克谢维奇拓扑，我们将其推广至任意正泄漏的LIF神经元。由此，LIF可被理解为相应范数下的脉冲序列量化。基于此，我们推导出各类误差界与不等式，例如输入与输出脉冲序列间的准等距关系。此外，还获得一个李普希茨型全局误差传播上界及相关的共振型现象。