Binocular eye orientation during fixations: Listing's law extended to include eye vergence

Any eye position can be reached from a position called the primary position by rotation about a single axis. Listing's law states that for targets at optical infinity all rotation axes form a plane; the so-called Listing plane. Listing's law is not valid for fixation of nearby targets. To document these deviations of Listing's law we studied binocular eye positions during fixations of point targets in the dark. We tested both symmetric (targets in a sagittal plane) and asymmetric vergence conditions. For upward fixation both eyes showed intorsion relative to the position that would have been taken if each eye followed Listing's law. For downward fixation we found extorsion. The in- or extorsion increased approximately linearly with the vergence angle. The direction of the Listing axis and the turn angle about this axis can be described by rotation vectors. Our observations indicate that for fixation of nearby targets the rotation vectors of the two eyes become different and are no longer located in a single plane. However, we find that it is possible to decomose the rotation vector of each eye into the sum of a symmetric and an anti-symmetric part, each with its own properties. (1) The symmetric part is associated with eye version. This component of the rotation vector is identical for both eyes and lies in Listing's plane. In contrast to the classical form of Listing's law, this part of the rotation vector lies in Listing's plane irrespective of the fixation distance. (2) The anti-symmetric part of the rotation vector is related to eye vergence. This component is of equal magnitude but oppositely directed in each eye. The anti-symmetric part lies in the mid-sagittal plane, also irrespective of fixation distance. For fixation of targets at optical infinity the anti-symmetric part equals zero and the eye positions obey the classical form of Listing's law. Thus, the symmetric and anti-symmetric parts of the rotation vectors are restricted to two perpendicular planes. Combining these restrictions in a model, with the additional restriction that the vertical vergence equals zero during fixation of point targets, we arrive at the prediction that the cyclovergence is proportional to the product of elevation and horizontal vergence angles. This was well born out by the data. The model allows to describe the binocular eye position for fixation of any target position in terms of the bipolar coordinates of the target only (i.e. using only three degrees of freedom instead of the six needed for two eyes).