Spiking neural networks in the Alexiewicz topology: A new perspective on analysis and error bounds

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.