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

Spiking Neural Networks in the Alexiewicz Topology: A New Perspective on Analysis and Error Bounds