How can head-direction cells represent all possible orientations when they have preferred directions?

[Bitking]

Yes, this seems to be the rule in all grid modules.
I have a collection of papers on this here:

Number encoder based off of entoehinal grid cells

Computer people like powers of two. Nature generally does not. The ratio between adjacent fields as you go from dorsal to ventral follows a somewhat different distribution - more of a logarithmic spacing. A commonly measured value is 1.14 (square root of 2) See if this helps: Buried in this is an amazing gem - " This spatial gradient mechanism is homologous to a gradient mechanism for temporal learning in the lateral entorhinal cortex and its hippocampal projections. Spatial and temporal …