Can rodents conceive hyperbolic spaces?

TL;DR

This paper explores whether rodents can develop neural representations of hyperbolic spaces, predicting that self-organizing grid cells would form distinctive firing patterns in non-Euclidean environments, challenging the assumption of Euclidean spatial encoding.

Contribution

It demonstrates that self-organizing models predict rodents can form hyperbolic grid patterns, suggesting neural spatial metrics are adaptable beyond Euclidean geometry.

Findings

Hyperbolic grids show multi-peaked firing maps with seven neighbors.

Predicted grid patterns depend on the curvature of the environment.

Testable predictions for neural responses in hyperbolic spaces.

Abstract

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.