Cian O'Donnell, a British neuroscientist, originally asked this question on Twitter: https://twitter.com/cian_neuro/status/1075432086692089857. I am not a biophysicist by training but I wonder whether there might be existing publications that have shed light on this question.
My primitive back-of-the-envelope analysis involves the following:
- If we suppose that most neurons are limited to a volume V, give or take epsilon volume, and the function of axons is to carry a signal over a large distance whereas the function of dendrites is to integrate information from different sources then a semi-spherical radiation of dendrites should be the norm.
- This becomes an extremal optimisation problem where there are specific constraints on the minimal axon thickness, the minimal dendrite thickness and an upper-bound on the average number of dendrites that fan out of a given axon.
- This becomes an extremal optimisation problem where there are specific constraints on the minimal axon thickness, the minimal dendrite thickness and an upper-bound on the average number of dendrites that fan out of a given axon.
This is a very sloppy analysis which ignores biophysical constraints but I think that we have an extremal optimisation problem here.