Grid cell activity and path integration on 2-D manifolds in 3-D space

Spatial navigation relies on various types of neurons to form an internal representation in the brain of the external world. Among them, grid cells are generally believed to serve as an invariant metric system by their spatially periodic firing fields. But how this metrical coordinate system is organized in three-dimensional (3-D) real world remains a mystery, since most researches merely concerned the encoding scheme on the horizontal plane. We computationally explored the activity pattern of grid cell in the medial entorhinal cortex of crawling animal in 3-D space. By including the presumably referring signals of gravity and animal’s body plane, grid cell firing fields on curved surfaces were produced based on the novel gravity-modulated oscillatory model. The results can account for the known experimental recordings and predict a mosaic-type grid code consisting of dynamically rotated planar arrangements. We further analyzed the path integration mechanism and derived the condition to ensure the invariant grid fields on any curved surface in 3-D space. It turns out that if the grid code is indeed not fully volumetric, it may become trajectory-dependent in 3-D space. And thus for crawling animals, 3-D grid fields could be degenerated and impaired, causing the path integration and distance measurement inaccurate. Besides, the volumetric firing fields were also discussed, although it is more suitable for flying or aquatic animals. This work can help us understand the intrinsically different spatial codes and navigating abilities among species with various locomotion modes and provide new insights of how the actual physical world is represented in the brain.