An Investigation of Conformal Isometry Hypothesis for Grid Cells

This paper investigates the conformal isometry hypothesis as a potential explanation for hexagonal periodic patterns in grid cell response maps. The hypothesis posits that grid cell activity forms 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 suggests that this neural manifold is a conformally isometric embedding of physical space, where local displacements in neural space are proportional to those in physical space. In this paper, we conduct numerical experiments to show that this hypothesis leads to the hexagon periodic patterns of grid cells, agnostic to the choice of transformation models. Furthermore, we present a theoretical understanding that hexagon patterns emerge by minimizing our loss function because hexagon flat torus exhibits minimal deviation from local conformal isometry. In addition, we propose a conformal modulation of the agent's input velocity, enabling the recurrent neural network of grid cells to satisfy the conformal isometry hypothesis automatically.