On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding
Dehong Xu Ruiqi Gao Wenhao Zhang Xue-Xin Wei Ying Nian Wu
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.
grid cells, conformal isometry, distance-preserving, position embedding, representation learning
We investigate the conformal isometry hypothesis that leads to the emergence of hexagon periodic patterns in grid cells, showing that learning a maximally distance-preserving position embedding naturally leads to these patterns.