The 3D geometry of visuomotor velocity transformations for pursuit eye movements

Gunnar Blohm, Pierre Daye, Philippe Lefevre


Saccade planning requires a geometric transformation between the retinal stimulus and the desired motor plan to acquire the target (Crawford & Guitton 1997). This reference frame transformation problem has, however, never been considered for velocity signals. Therefore we asked whether a separate 3D visuomotor transformation of velocity signals was theoretically required by modeling the underlying geometry. We then tested our model predictions in a series of smooth pursuit experiments.

We used quaternions to model the 3D eye-in-head geometry. Our model predicted that a visuomotor velocity transformation would require the use of extra-retinal 3D eye-in-head position to convert the retinal velocity input into spatially accurate behavior and should include three different components; (1) the same retinal velocity should result in different eye rotation axes depending on eye-in-head position, (2) false torsion due to off-axes eye positions must be compensated for and (3) ocular torsion (e.g. due to the VOR) must be accounted for.

We tested these 3 predictions on human subjects. Subjects were required either to pursue a moving target viewed under different vertical (prediction 1) or oblique (prediction 2) eye positions, or viewed under different head roll angles in order to obtain VOR-induced eye torsion (prediction 3). We measured 3D eye-in-head position (Chronos: 400Hz) and head-in-space position and orientation (Codamotion: 200Hz) and analyzed the open-loop gaze pursuit response, i.e. the first 100ms after pursuit onset (velocity threshold with backward extrapolation). We then compared the observed pursuit response to the predictions of the model: if no transformation was performed, pursuit direction should best correlate with the retinal target movement direction; a complete 3D velocity transformation would be reflected in spatially accurate pursuit.

We found that for all 3 predictions, the direction of pursuit initiation was spatially accurate and did not follow the retinal (no transformation) hypothesis. This suggests that the brain indeed performs a complete 3D visuomotor velocity transformation for smooth pursuit eye movements that is different from the previously described visuomotor transformation of position signals for saccades. Since pursuit direction was accurate even for torsional values outside of Listing’s plane in our head-roll condition (prediction 3), we rule out the possibility that the velocity transformation geometry we describe here could be accounted for by the mechanical properties of the plant, e.g. through pulleys.

Supported by: Marie Curie (EU), FNRS (Belgium), IAP (Belgium), ESA (EU), ARC (UCLouvain, Belgium), NSERC (Canada), Botterell Fund (Queen’s University, Canada)