Do head direction cells represent 2D or 3D direction?

I’ve been doing some googling on head direction cells, and everything I’ve found so far depict them as a ring (essentially a continuous 1D GCM in the shape of a circle). However, that alone doesn’t seem like it would provide a true orientation in three dimensions. I figure I can reasonably assume gravity provides the anchor for the down direction. If so, then a ring structure for head direction would depict a rat’s heading along a flat surface in 2D. However, it is missing one axis of rotation.

Has anyone proposed head direction cells which depict the surface of a sphere in grid cells (where rat is in the center of the sphere and can look up, for example)? Or is there another mechanism I should be reading up on related to acquiring the third axis of rotation?

2D.
Grid cells on steeply sloping terrain - evidence for planar rather than volumetric encoding

Neural encoding of large-scale three-dimensional space—properties and constraints

Neuroscience: How Is Three-Dimensional Space Encoded in the Brain?

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Cool, that aligns with what I have been reading. It does seem to indicate that a piece of the puzzle is missing for depicting a true 3D heading though.

One possibility I thought of is that if I stop trying to think in terms of traditional coordinate system axes (in particular normalized right angles), then that would open up one possibile solution. “Head direction” (which in this case should have another name) would always be perpendicular to “down direction”, and then one could add a “look direction” which is not perpendicular to the other two directions.

This is certainly not the traditional way of thinking about an orientation (and pure speculation on my part), but in theory it would solve the problem of depicting orientation in 3D. It however leads to some interesting questions such as how the head direction cells would behave when, for example, a rat is standing on a slope and lifts its head up enough that the “head direction” cells should indicate it is facing the opposite direction…

Wow, definitely some interesting observations. These in particular are pretty illuminating:

as animals move from a horizontal surface to a vertically oriented wall, head direction cells maintain their directional firing such that the preferred firing direction of each cell on the floor is directly translated onto the wall, as if the wall were simply an extension of the floor. For example, a cell that is firing as the animal moves toward a vertical wall will continue firing as the animal moves onto the wall even though it now faces upward.

One of the most surprising findings from the Finkelstein et al study was what happened when the bats turned upside-down. Unlike the general degradation of directional firing in the inverted rats of Calton and Taube (2005) those head direction cells in the bats that maintained directional firing (about 40%) showed an unexpected reversal of firing direction, such that a cell that fired when the animal’s head pointed (say) North in the upright posture would fire when the head faced South in the inverted posture.

This definitely gives a pretty good explanation for the problem that head direction is trying to solve. Basically it is attempting to plot a continuous direction over a surface regardless of the surface topology. Sort of like the effect of a laser level when it encounters corners.

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Could it be more of an indicator than an anchor?

I’m not sure if head direction cells work the same in virtual environments (e.g. a human playing a video game), but if they did it would be interesting to see how they would behave without the vestibular or proprioceptive cues (@Bitking?)

I can think of a bunch of games where you have to adapt to some unusual physics constraints in order to navigate a 3D environment, like running on ceilings etc.

Assuming that the substrate for head direction is based on the posture control system the chart on page 222 in this reference should shed some light on the relationship between the vestibular system and head position:
http://cw.tandf.co.uk/neurophysiology/sample-material/Neurophysiology-5E_Chapter-11.pdf

I have copious references to the basal ganglia and related system as part of the posture control system. These are all older brain structures - I expect rich connections to the older brain spatial systems without even looking.

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