2-D to 3-D transformations

The neural signals we have discussed so far are not sufficient to constrain the desired configurations of the effectors being controlled. For example, desired gaze direction (in any frame) does not specify the 3-D orientation (i.e. torsion) of the eye or head . And even if the desired 3-D translational location of the hand is specified, this does not determine the final configuration of the arm, which has 7 degrees of rotational freedom in its joint structure. This is known as ‘the degrees of freedom problem’.

As we have already seen, the remaining degrees of freedom are determined by Listing’s law (for the eye), the Fick strategy (for the head), and similar strategies for the arm. But what are the neural mechanisms that derive these multi-dimensional laws from lesser-dimensional inputs?

The ‘2-D to 3-D’ transformation is best understood for the gaze control system, especially for the eye movement component. First, it appears that high-level gaze control centers (superior colliculus, frontal eye fields, and supplementary eye fields) only encode the desired 2-D direction of gaze, in the frames discussed above. We know this primarily because electrical stimulation of these structures with the head immobilized produces saccades within Listing’s plane, whereas stimulation with the head un-immobilized produces coordinated eye-head gaze shifts that include Fick-like head patterns and torsional eye coordination that eventually lands the eye back in Listing’s plane ().

The story is different for the premotor structures that control eye saccades, including the reticular formation burst generator, which provides the velocity signal for ocular motoneurons and the neural integrator that converts this into an orientation signal. These velocity and position structures are divided into a horizontal controller (the paramedian pontine reticular formation [PPRF] and nucleus prepositus hypologossi [nPH] respectively, and a torsional-vertical controller in the midbrain (the rostral interstitial nucleus of the medial longitudinal fasciculus [riMLF] and interstitial nucleus of Cajal [INC] respectively]. Moreover, these structures are subdivided into sub-populations with directional control similar to that of the semi-circular canals and extra-ocular muscles.

So how is torsional control managed? Importantly for this topic, both the burst generator () and the neural integrator () coordinates appear to align with Listing’s plane, i.e., the vertical axis that controls horizontal eye rotations falls within LP, whereas the vertical-torsional axes are symmetric across Listing’s plane (Fig.). Moreover, these population coordinates are arranged such that symmetric activation of the clockwise and counterclockwise rotaters on the right and left brainstem, respectively, will give zero torsion in Listing’s coordinates. Preliminary evidence suggests that there may be a similar arrangement for head movements, only using Fick coordinates ().

This means that if the default connections from higher-level structures like the SC project symmetrically to the premotor centers, they will produce movements that obey Listing’s law and the Fick strategy. Another mechanism is then needed to provide a-symetric activation for torsional components of saccades during corrective movements or coordination with the VOR during head-unimmoblized gaze shifts. This may involve the cerebellum, specifically signals passing through the paramedian pontine reticular formation, which possesses units that fire during torsional movements orthogonal to Listing’s plane.

Finally, it has been shown that for saccades in Listing’s plane, ocular motoneurons only encode the movement vector within LIP (). It thus appears that the position-dependent axis tilts required to keep the eye in Listing’s plane are implemented mechanically; possibly by the effects of orbital tissues on pulling directions of the extra-ocular muscles (), or by side-slip of those muscles. Analagous mechanical factors may facilitate the Fick constraint for head movement (), although clearly the brain can choose to overcome these factors when it wishes to violate these laws. Ultimately, these laws are the product of both neural and mechanical factors.

The analogous transformations for reach movements are clearly far more complex, and much less understood at the present time, but one hopes that (as with other areas described above like spatial updating and 3-D reference frame transformations) some of the principles observed in the gaze-control system might hold here as well.