New developments in the geometry and topology of 3D head-direction coding
The head-direction (HD) system is a neuronal compass that represents head orientation in 3D. Its encoding of 3D head orientation can be decomposed into a multiplexed representation of 2D head tilt relative to gravity and 1D head orientation in the horizontal plane (azimuth). Defining azimuth during 3D motion is not trivial, and recent studies have indicated that the HD system implements a so-called dual-axis/tilted-azimuth (DA/TA) rule, a representation that is well supported experimentally but remains mathematically and mechanistically difficult to interpret. Here, I introduce a new formulation of DA/TA based directly on the elements of a rotation matrix describing 3D head orientation. In this formulation, TA is represented by a 2D vector, whose direction encode TA and whose norm depends on tilt, consistent with a tilt-dependent gain of HD tuning reported by experimental studies. I provide a geometrical interpretation of this formulation and use it to characterize the topology of the resulting representation. More generally, I introduce a framework in which head orientation is represented by a rotation matrix and neuronal responses arise from linear combinations of its elements followed by nonlinear transformations. Within this framework, the linear stage specifies which aspects of 3D orientation are encoded, and therefore allow relating neuronal representation to linear algebra concepts. Classical HD tuning, gravity tuning, and DA/TA can all be described within this common framework. Together, these results provide a new mathematical, geometrical, and topological perspective on 3D HD coding.