Lecture 3

The Early Geometry of Visuomotor Transformations

Once cannot understand the spatial aspects of visuomotor transformation without first understanding the spatial geometry of the system we are dealing with. This is important for several reasons: 1) one needs to know the geometry that dictates how visual targets project onto the retina, in order to understand how these targets are initially coded in an eye-fixed frame. 2) One needs to know the geometry of the controlled systems to understand the desired movements and positions. 3) One needs to know the interlinking geometry of the system in order to know what additional transformations are required to map #1 onto #2. These factors might seem obvious –even biomechanically trivial- but in fact this geometry is complex and all to often crucial aspects of this geometry are ignored when neuroscientists consider visuomotor transformations.

We will approach this subjects through a series of definitions:

1). Gaze. Also known as gaze direction or the visual axis; this is the line passing from the fovea through the optical centre of the eye to the object of current regard. This is important from a sensory respective because only the fovea provides highest acuity vision (in lighted conditions). It is important from a motor perspective since the main purpose of gaze shifts is to redirect the fovea toward a new object of regard.

2) Retinal / eye-fixed / eye centered frame. We use these terms interchangeably to define a spherical coordinate system centered at the centre of rotation / optical centre of the eye, where the reference ‘0’ vector is aligned with current gaze direction, visual direction can be defined as a unit vector pointing forward along the line between any other point on the retina through the optical centre toward the corresponding point in space. The centre of rotation of the eye is not identical to the optic centre, but for targets beyond a few centimeters distance from the eye the difference is negligible.

The location of visual space that corresponds to the fovea can be described by just a two-dimensional gaze vector, with horizontal and vertical components. However, the point on the retinal stimulated by every other location is also influenced by the torsion of the eye about the visual axis. Moreover, the orientation of a complex stimulus on the retina is also determined by torsion (for example, 180 degrees torsion would turn everything upside down). Thus, one needs to know 3-D eye position in space order to establish the mapping between space and the retina.

3) Eye orientation vectors. 3-D eye orientation is normally defined as a rotation relative to some central reference orientation in a head-fixed frame. This can be expressed as a vector parallel to the axis of this rotation, directed according to the right hand rule, and scaled by the angle of rotation (or some function of this angle, such as sin[angle/2] in quaternions). Note that this vector does not tell how the eye rotated to get there – it just describes where the eye is right now relative to the reference position. To describe the motion of the eye, another type of vector is required, such as angular velocity.

4) Donders’ law. This law states that for any one gaze direction, the eye only assumes one 3-D orientation, regardless of the path that brought it there.

5) Listing’s law. Listing’s law describes a specific form of Donders’ law. Listing’s law states that the eye orientation vectors described in ‘3’ form a plane. For one particular reference orientation ‘primary position’, gaze direction is orthogonal to the associated plane, called Listing’s plane. Primary position can only be measured experimentally and is generally near, but not exactly at the mechanical centre of the head. If one defines the primary gaze direction as the torsional axis (where torsion corresponds to clockwise and counterclockwise rotation), and takes the horizontal and vertical axes within Listing’s plane, then one has a convenient head-fixed coordinate system for describing eye orientation. In these ‘Listing’s coordinates’, Listing’s law simply states that torsion is kept at zero. Experiments show that this is true within ± one degree in monkeys, and nearly as good in humans (although this is hard to measure with the same precision).

One misconception that is easy to make about Listing’s law is that it means the eye only rotates about axes in a plane. This is not true- the laws of rotational kinematics (reviewed above) dictate that in order to keep eye orientation in Listing’s plane, axes of eye rotation during movement must tilt systematically out of Listing’s plane according to the half angle rule. We will return to this subject we discuss implementation of LL.

It has been shown that Listing’s law is obeyed when the head is upright and either motionless or moving during saccades and smooth pursuit. It is also obeyed when the head translates without rotating and gaze remains fixed on some target, and during fixations between eye + head gaze shifts.

LL is modified during some behaviors. When the eyes converge toward a near target, their two LPs tilt outward ‘like saloon doors’. When the head tilts vertically, LP tilts slightly in the opposite direction and when it tilts torsionally LP shifts torsionally slightly in the opposition direction.

In other cases, LL breaks down altogether. During head rotations, the VOR or OKN counter-rotate about approximately the same axis, causing systematic accumulation of eye torsion. During head-free gaze shifts, saccades also take on torsional components in order to predictively compensate for VOR torsion.

These rules are relevant for spatial representations and transformations because they determine (just as much as the physical location of visual stimuli) the pattern of retinal stimulation and thus the direction of these stimuli in an eye-fixed frame. As we shall discuss below these patterns are important for binocular vision. Moreover, they are important when we come to the topic of reference frame transformations for action.

6) Head orientation and the Fick Constraint. Measurements of head orientation during and between gaze shifts have revealed that the head obeys a form of Donder’s law, although with less precision than the eye. The resulting orientation vectors form a quasi-two-dimensional range, but when plotted in an orthogonal space-fixed coordinate system the range is not flat like Listing’s plane, but rather twisted at the corners. This is the range of orientations that one sees when torsion is held at zero in ‘Fick coordinates’, where the vertical axis (for horizontal rotation) is fixed in the body and the horizontal axis (for vertical rotation) is fixed in the head and thus rotates around the vertical axis, and torsion is constrained about an axis roughly aligned with the pointing direction of the nose.

7) Eye orientation in space. During gaze fixations, eye orientation in space is the geometric product of Listing’s of the eye-in-head and the Fick constraint of the head-on-body. But since the head tends to contribute more toward horizontal components (about a body-fixed axis), and the eye tends to contribute more to the vertical component (about a head-fixed axis) the Donders’ law for the eye in space resembles the Fick constraint, where the torsional variance (being the sum of both eye and head variance) is the highest.

One thing to note for visual consequences: when the head is fixed and torsion kept at zero in Listing’s coordinates, the eye shows systematic ‘false torsion’ (rotation of the eye about the visual axis) when looking in oblique directions. However, when the head is allowed to contribute naturally to gaze, this is reduced (since torsion is minimized about an axis more closely aligned with gaze).

8) Translational geometry of the eye and head. Although eye orientation is determined by the orientations of the eyes and head, both of these bodies translate through space. Since the cervical column joins the skull near the bottom-back of the head (rather than its geometric centre, like the eye), even rotations about this point cause the forward aspects of the skull, in particular both eyes, to translate through space. This is approximately the case for horizontal head rotations and small vertical head rotations. Moreover, for larger vertical head rotations, joints progressively further down the neck must be recruited, causing the overall centre of rotation to drop, the neck to bend, and the entire head to translate through space (down-forward, and up-backward). These translations are not important for gaze fixations on infinitely distant targets, but they are important for near targets, because of the geometry of motion parallax. This has complex implications for visuomotor control, for example in the translation VOR, and as well shall see, for updating representations of visual direction during self motion.

9) Shoulder-arm geometry. Since the focus of this course is on the early geometry of visuomotor transformations, we will primarily (although not always) consider reaches and pointing movements toward goals with the arm fully extended. Here the arm can be simplified to a rigid cylinder that rotates in 3-D about the shoulder joint. It has been shown that in this situation (except when upper arm rotation must be recruited to orient grasp) the arm obeys a set of constraints similar to Lsiting’s law. However, when we consider visuomotor transformations for reach, we must account for two important factors: the reference frame for the arm is the torso, which may vary considerably in relative orientation to the eye during gaze shifts. Second, the centre of shoulder rotation is significantly shifted from that of the head, which in turn is shifted with respect to that of the eye (see #7). Each of these factors must be accounted for, in order to generate an accurate reach plan (we shall take this topic up again in a later lecture).