3D geometrical transformations for visually guided reaching 

G. Blohm¹ and J.D. Crawford^1,2

¹Centre for Vision Research, York University, Toronto, Ontario, Canada

²Canadian Institutes for Health Research Group on Action and Perception

 Reaching to visual targets requires a transformation of retinal signals into motor commands for the arm. It is thought that the Posterior Parietal Cortex encodes hand and target positions in gaze centered coordinates, whereas the Primary Motor Cortex probably issues body-centered motor commands to the limb plant. The transformation between these structures could be direct, i.e. a movement vector in retinal coordinates would correspond directly to the same movement vector in motor coordinates. However, the complex and nonlinear linkage geometry of the eye with respect to the head and body in real 3-D space suggests that a more sophisticated transformation is required.

Here we developed an analytical model handling both translational and rotational components of movement. In our model, all eye movements comply with Listing’s law and static VOR (ocular counter-roll and the gravitational modulation of the Listing plane). The model also accounted for the complete linkage structure between the eye and the shoulder. This model allowed us for the first time to analyze the complete early geometry of visually guided reaching in head-free conditions.

In our model, the visuomotor transformation computes the sensory difference vector between current hand position and desired target position, in eye-fixed coordinates, and then transforms this into a motor vector command in shoulder coordinates. We compared this ‘optimal transformation model’ (OTM) accounting for different eye-head-body configurations, with the ‘direct transformation hypothesis’ (DTH) that mapped each sensory vector onto a unique motor vector - computed with the eye and head in their primary position.

As expected, both models performed equally well when the eye and head were in their primary positions. However, once the eye-in-head and/or head-in-space configuration deviated from the primary position, OTM continued to perform perfectly while DTH lead to progressively larger errors. Two situations illustrate this best: (1) Oblique gaze in primary head position. If the gaze headed 35° up-left (= around 27° up and 27° left) and the arm had to reach from a 25cm rightward spatial position to a 25cm leftward spatial position (both located 12.5cm down and at 50cm visual depth), the reaching movement predicted by DTH would miss the reach target by almost 22cm. This was essentially due to ‘false torsion’ (eye movements comply to Listing’s law) and to the retinal curvature of horizontal lines when the eyes were vertically offset. (2) Head roll with ‘straight ahead’ eye-in-head position. Consider the head rolling leftward by 20° while the eye-in-head stayed at its primary position and the hand-target configuration was the same as for the previous example. In this case DTH fails to account for the rotation of the visual vector with respect to the movement vector, and produced reach errors of approximately 17cm.

These examples show that in real 3-D space, DTH fails to provide accurate movement commands. In other words, even when both hand and target position are calculated in retinal coordinates, the feed-forward transformations for visually guided arm movements must incorporate comparisons with initial eye and head configuration.

Citation: Blohm G, Crawford JD, "3D geometrical transformations for visually guided reaching", 15th Annual Meeting of the Neural Control of Movement Society, Key Biscayne, Florida, USA, 2005