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的误差传播分析并深化对脉冲神经网络（SNN）的理解，本文探讨了将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的问题。核心问题在于脉冲序列空间的适当结构及其对SNN误差度量设计的启示，包括时间延迟、阈值偏差以及漏积分激发（LIF）神经元模型重置模式的设计。首先，通过分析LIF模型所有亚阈值信号的闭包，我们确定了底层拓扑结构。对于零漏电情形，该方法导出Alexiewicz拓扑，随后将其推广至具有任意正漏电的LIF神经元。由此，LIF可理解为相应范数下的脉冲序列量化。通过这一途径，我们获得了多种误差界限与不等式，例如输入与输出脉冲序列间的拟等距关系。另一结果是关于误差传播的Lipschitz型全局上界及相关的共振型现象。