An alternative three-dimensional interpretation of Hering's equal-innervation law for version and vergence eye movements

In the context of Hering's equal-innervation law, this paper discusses the problem of how the three-dimensional positions of the two eyes, each expressed by a rotation vector, can be separated into contributions of the version and vergence system. As proposed by Van Rijn and Van den Berg [(1993) Vision Research, 33, 691-708], this can be done by taking the sum and difference of the position rotation vectors of each eye. In our alternative procedure the vergence signal is defined as the rotation which transforms the left eye position into the right eye position and the version signal is the common factor in both eye positions that remains after removing the vergence signal. The version and vergence contributions, defined in this way, can be interpreted straightforwardly as rotations. When Van Rijn and Van den Berg applied their definitions to their own data, they obtained the interesting result that the reconstructed version and vergence contributions were effectively limited to two dimensions (2D). The version signal was confined to Listing's plane (no torsion) whereas the vergence signal remained within a horizontal-torsional plane (no vertical vergence). They showed theoretically that a model based on 2D version/2D vergence control will indeed produce the torsional eye positions in near fixations found in their experiments. This model cannot account for a second set of data in the literature [Mok, Ro, Cadera, Crawford & Vilis (1992) Vision Research, 32, 2055-2064]. With our definitions, we found that the simple 2D version/2D vergence control strategy cannot account for the Van Rijn and Van den Berg (1993) data but is nicely compatible with the considerably smaller amount of cyclotorsion in the data collected by Mok et al. (1992). We also show that, in such a system, having 2D vergence control is compatible with minimization of torsional disparity and provides the cyclovergence signals suitable for stabilizing the eyes in the non-Listing positions caused by a vertical saccade in near vision.