The most popular theoretical framework in use currently, in the neuromorphic (brain-inspired) computing community is the Neural Engineering Framework (NEF). Neural Engineering by Chris Eliasmith and Charles Anderson explains the framework comprehensively.
As a follow up to that, How to Build a Brain by Chris Eliasmith describes the more recent and more high-level description of how to get spiking neural networks to actually perform multiple functions: the Semantic Pointer Architecture (SPA).
If you're looking for hardware descriptions too, the research publications on the chips Neurogrid, Brainscales, Braindrop, SpiNNaker, Loihi, TrueNorth etc. provide some good high-level descriptions of how to actually build the aforementioned chips.