Loved the ending of the discussion, on why the head-direction circuit is organised in 8 columns:
"not all numbers of neurons enable a working circuit. The circuits for N=2 and N=4 are degenerate – either producing a single dimensional encoding, or two disconnected circuits that do not enforce the required circular topology. N=8 is the smallest power of two that could result in a non-degenerate circuit. This hints at the possibility that the eight-column architecture is not a chance evolutionary artefact, but rather that it is the genetically simplest circuit capable of performing heading integration."
"we illustrate that in our theory, the consistent presence of the eight-column organisation of head direction circuits across multiple insect species is not a chance artefact but instead can be explained by basic evolutionary principles."
Interesting take on possible constraints that defined the circuit architecture of the #Drosophila central complex in its role as a brain centre for spatial navigation.