Partial coherence and frustration in self-organizing spherical grids

Nearby grid cells have been observed to express a remarkable degree of long-rangeorder, which is often idealized as extending potentially to infinity. Yet their strict peri-odic firing and ensemble coherence are theoretically possible only in flat environ-ments, much unlike the burrows which rodents usually live in. Are the symmetrical,coherent grid maps inferred in the lab relevant to chart their way in their natural hab-itat? We consider spheres as simple models of curved environments and waiting forthe appropriate experiments to be performed, we use our adaptation model to pre-dict what grid maps would emerge in a network with the same type of recurrent con-nections, which on the plane produce coherence among the units. We find that onthe sphere such connections distort the maps that single grid units would express ontheir own, and aggregate them into clusters. When remapping to a different sphericalenvironment, units in each cluster maintain only partial coherence, similar to what isobserved in disordered materials, such as spin glasses.