  The grid cells discovered in the rodent medial entorhinal cortex have been
proposed to provide a metric for Euclidean space, possibly even hardwired in
the embryo. Yet one class of models describing the formation of grid unit
selectivity is entirely based on developmental self-organization, and as such
it predicts that the metric it expresses should reflect the environment to
which the animal has adapted. We show that, according to self-organizing
models, if raised in a non-Euclidean hyperbolic cage rats should be able to
form hyperbolic grids. For a given range of grid spacing relative to the radius
of negative curvature of the hyperbolic surface, such grids are predicted to
appear as multi-peaked firing maps, in which each peak has seven neighbours
instead of the Euclidean six, a prediction that can be tested in experiments.
We thus demonstrate that a useful universal neuronal metric, in the sense of a
multi-scale ruler and compass that remain unaltered when changing environments,
can be extended to other than the standard Euclidean plane.
