On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding

TL;DR

This paper explores the conformal isometry hypothesis as a mechanism for the hexagonal patterns in grid cells, proposing that neural embeddings preserve physical distances, which is crucial for navigation and path planning.

Contribution

It introduces a novel hypothesis that grid cell patterns result from a conformal isometric embedding of space, supported by numerical experiments and theoretical analysis.

Findings

Hexagonal grid patterns emerge from distance-preserving neural embeddings.

Neural embeddings can be learned independently of network architecture.

Hexagon flat torus maximizes distance preservation in the embedding.

Abstract

This paper investigates the conformal isometry hypothesis as a potential explanation for the hexagonal periodic patterns in grid cell response maps. We posit that grid cell activities form a high-dimensional vector in neural space, encoding the agent's position in 2D physical space. As the agent moves, this vector rotates within a 2D manifold in the neural space, driven by a recurrent neural network. The conformal hypothesis proposes that this neural manifold is a conformal isometric embedding of 2D physical space, where local physical distance is preserved by the embedding up to a scaling factor (or unit of metric). Such distance-preserving position embedding is indispensable for path planning in navigation, especially planning local straight path segments. We conduct numerical experiments to show that this hypothesis leads to the hexagonal grid firing patterns by learning maximally distance-preserving position embedding, agnostic to the choice of the recurrent neural network. Furthermore, we present a theoretical explanation of why hexagon periodic patterns emerge by minimizing our loss function by showing that hexagon flat torus is maximally distance preserving.