Digital computers have advanced to rival human-level intelligence across creative reasoning and complex problem-solving tasks. Yet, theoretical comparisons with unconventional computational substrates suggest we have only scratched the surface of computational potential. Biological nervous systems have long been studied for their efficiency and robustness, inspiring the invention of neuromorphic hardware. Neuromorphic systems have yet to outperform digital computers, likely due to our limited understanding of the governing computational principles.

This work investigates these computational principles using two modes of inquiry. First, axioms for mixed-signal neural networks are studied by induction as necessary conditions for provably correct neuromorphic computations. Part of the axioms are applied in the Neuromorphic Intermediate Representation, a set of neuromorphic primitives, which is demonstrated to work across more than 12 neuromorphic software and hardware platforms. A second inquiry is made into geometric approaches to event-based vision by deduction. By establishing a direct relationship between neuromorphic primitives and signal transformations, it is demonstrated how neural networks can be imbued with covariance properties that enable them to outperform conventional networks of similar complexity in object tracking tasks. The complementary inductive-deductive approaches provide a more complete lens from which to understand and implement neuromorphic computation.

Additionally, a chapter is dedicated to several openly accessible software projects for evaluating neuromorphic systems on commodity hardware. Apart from being the backbone for the research in this thesis, the accessibility and reproducibility may propagate the research and catalyze community efforts.

Finally, the thesis concludes with a discussion on the broader implications of the above findings and the future trajectory of neuromorphic computation.