Commutative Properties of Head Direction Cells during Locomotion in 3D: Are All Routes Equal?

Navigation often requires movement in three-dimensional (3D) space. Recent studies have postulated two different models for how head direction (HD) cells encode 3D space: the rotational plane hypothesis and the dual-axis model. To distinguish these models, we recorded HD cells in female rats while they traveled different routes along both horizontal and vertical surfaces from an elevated platform to the top of a cuboidal apparatus. We compared HD cell preferred firing directions (PFDs) in different planes and addressed the issue of whether HD cell firing is commutative-does the order of the animal's route affect the final outcome of the cell's PFD? Rats locomoted a direct or indirect route from the floor to the cube top via one, two, or three vertical walls. Whereas the rotational plane hypothesis accounted for PFD shifts when the animal traversed horizontal corners, the cell's PFD was better explained by the dual-axis model when the animal traversed vertical corners. Responses also followed the dual-axis model (1) under dark conditions, (2) for passive movement of the rat, (3) following apparatus rotation, (4) for movement around inside vertical corners, and (5) across a 45° outside vertical corner. The order in which the animal traversed the different planes did not affect the outcome of the cell's PFD, indicating that responses were commutative. HD cell peak firing rates were generally equivalent along each surface. These findings indicate that the animal's orientation with respect to gravity plays an important role in determining a cell's PFD, and that vestibular and proprioceptive cues drive these computations.SIGNIFICANCE STATEMENT Navigating in a three-dimensional (3D) world is a complex task that requires one to maintain a proper sense of orientation relative to both local and global cues. Rodent head direction (HD) cells have been suggested to subserve this sense of orientation, but most HD cell studies have focused on navigation in 2D environments. We investigated the responses of HD cells as rats moved between multiple vertically and horizontally oriented planar surfaces, demonstrating that HD cells align their directional representations to both local (current plane of locomotion) and global (gravity) cues across several experimental conditions, including darkness and passive movement. These findings offer critical insights into the processing of 3D space in the mammalian brain.