Computational modeling of head direction cells in three-dimensional space: directional encoding and visual cue manipulation.

Head direction (HD) cells can fire selectively as a function of the animal's head azimuth direction and form an internal compass for navigation. They were found in the mammalian limbic system including the dorsal presubiculum and entorhinal cortex. The underlying network updates its directional estimate in a self-organized fashion and can be recalibrated by external sensory cues. Although ring-attractor models were proposed to account for azimuth coding on horizontal planes, they cannot explain conjunctive azimuth-and-pitch tuning observed in HD cells of the presubiculum of flying bats navigating in three-dimensional (3-D) space. Based on the 3-D electrophysiological recordings, we developed a toroidal continuous attractor network that jointly encodes horizontal azimuth and vertical pitch angle of head direction. The model can reproduce the experimentally recorded tuning curves of individual HD cells, and accurately encode the azimuth-pitch dynamic head direction of bat by HD cell population. The model also simulate the influence of horizontal visual cue manipulation on the HD system in 3-D space and predicts how horizontal landmark rotation induces a sustained azimuthal offset that persists after the cue is removed, which is comparable to two-dimensional experimental findings. This research clarifies how conjunctive 3-D direction codes are generated and modified by vestibular input, visual information and recurrent connectivity. It also uncovers the computational principles and properties of the brain's navigation functions in realistic 3-D environments and offers new theoretical reference for future studies.