Introduction
For centuries it has been known that the eyes are capable of rotating in three dimensions (3D): horizontally (left-right), vertically (up-down), and torsionally (clockwise-counterclockwise, which for now we can define as rotation of the eye about the visual axis when looking straight ahead). For obvious reasons, visual and oculomotor scientists have tended to focus on the horizontal and vertical rotations because these have the most effect on visual gaze direction and ocular torsion is hard to see, let alone measure. Indeed, the 19th century scientists cited in the next section of this chapter discovered oculomotor rules that seem to purposefully minimize torsion (and thus tilting of retina relative to the world), presumably to simplify oculomotor control and minimize distortions in perception. This may have contributed to the notion that torsion is the “poor cousin” that could be ignored or added on as something of an afterthought. This perspective fit well with 20th century technologies for recording eye position and the highly influential control system theories of oculomotor control, espoused by David A. Robinson and others in the 20th century, because it fits well with linear systems math and analysis.
In parallel to those “two-dimensional (2D)” developments, there was a lesser but increasing awareness of 3D aspects of eye motion, such as the ocular torsion observed during head tilt or brainstem damage. This advanced in the 1980s with the development of technologies for recording 3D eye orientation and axes of eye rotation. , This led to an increased awareness that the eye moves either with or without torsion, depending on the behavioral circumstances. Perhaps more importantly, theoretician Douglas Tweed pointed out that the mathematics of rotations are noncommutative, and thus (even during 2D motion) require modifications to standard models of oculomotor control. This led to surprisingly fierce debates in the 1990s that largely focused on the role of the “oculomotor plant”—the tissues and muscles surrounding the eye—and whether this can rescue the brain from the problems of 3D rotational math. Although seemingly esoteric, this has important implications for neural control and the clinic because disorders affecting the vertical-pulling eye muscles have equal influence on torsion of the eye.
Although these debates have settled down and we have learned much in the 21st century, misunderstandings and knowledge gaps remain. The purpose of this chapter is to explain, in the simplest way possible (i.e., without math), how the eyes rotate in three dimensions, how the brain controls this, and the normal/pathologic consequences on vision. One take-home message is that, just as in 2D eye control, both muscles and neurons make important and necessary contributions to the special aspects of 3D eye control.
Three-dimensional ocular kinematics
To measure eye motion and use these data to understand oculomotor control, and diagnose oculomotor deficits and their perceptual consequences, it is important to know something about rotational kinematics . Kinematics is the study of motion, typically measured in terms of position and velocity. In the case of 3D eye kinematics, we are mostly concerned with eye orientation , and angular motion about some axis of rotation. The eyes also translate very slightly within the head (and quite a bit in space when the head moves), but we will confine this chapter to rotations. To discuss 3D kinematics it is also necessary to define coordinates and reference frames. Coordinates can here be thought of as a set of mutually orthogonal axes (vertical axis for horizontal rotation, horizontal axis for vertical rotation, and a torsional axis) that can be used to express the components of position or motion. Sometimes in 3D eye recordings, orientations are expressed as rotation vectors relative to some central reference position (the origin in the coordinate system), where the rotation vector is parallel to the axis of rotation, scaled by the amount of rotation, and directed according to the right-hand rule ( Fig. 9.1A ; where the fingers curl in the direction of rotation and the thumb then points in the vector direction). These are then fixed within some rigid body called the reference frame. For example, for the retina, the natural reference frame is the eye. The eye muscles contract between fixed points on the eyes and skull, but being much lower in inertia, the eyes rotate and the head stays fixed, making the head the natural reference frame for eye motion. Using these concepts, we will first consider eye-in-head motion (with the head stable), and then briefly touch on what happens when the head moves freely.
Eye-in-head motion
Mechanically, the eyes are capable of rotating approximately ±50-degrees horizontally, ±45-degrees vertically, and ±30-degrees torsionally with the head. However, Donders, a 19th century Dutch physiologist, observed that only one torsional eye component is used for any given 2D gaze position when participants looked around with the head motionless. Listing, a German mathematician, then somehow intuited the specific rule for this, now called Listing’s law (LL). A simple definition is that the eyes only assume orientations that could be reached from a central primary position through a rotation about a fixed axis that lies in Listing’s plane , orthogonal to the primary gaze direction. Thus, if we use rotation vectors (defined using the right-hand rule) to express eye orientations as rotations relative to primary position (the reference position), these vectors will line up in Listing’s plane ( Fig. 9.1B ). This also provides a convenient coordinate system for describing eye orientations, where the horizontal and vertical axes lay in Listing’s plane and the orthogonal torsional axis aligns with the primary gaze direction. Note that for now, these axes are fixed in the head, with the assumption that the head itself is motionless in some central position. An alternative is to define torsion as aligning with the visual axis, but this coordinate system results in so-called false torsion : clockwise rotation about the visual axis for up-left/down-right eye positions, counterclockwise rotation for up-right/down-left eye positions.
Helmholtz confirmed that visual afterimages followed patterns predicted by LL, (i.e., tilting with false torsion), and 20th century behavioral recordings allowed us to confirm and directly visualize Listing’s plane (as 3D rotation vectors) during head-restrained saccades ( Fig. 9.1C ). Human eye torsion varies within approximately 2 degrees of Listing’s plane, whereas monkeys are even more precise (~1 degrees) and have been the most-used experimental model for studying the physiology of 3D eye rotation. It is noteworthy that the primary position, although near the center of the eye’s mechanical range, varies considerably in humans and tends to be above center in monkeys (such that Listing’s plane tilts back).
An important observation is that the actual axes of rotation (measured in angular velocity vectors, also following the right-hand rule convention) only align with Listing’s plane for movements toward or away from primary position. For saccades passing orthogonal to primary position, the half angle rule applies, for example, the axis of a horizontal saccade must tilt by half the angle of gaze above or below primary position ( Fig. 9.1D ).
The latter, somewhat counterintuitive observation has caused a great deal of confusion, i.e., why would saccade axes tilt out of Listing’s plane to keep eye position in Listing’s plane? This tilt compensates for the noncommutative nature of eye rotations. For example, a horizontal rotation followed by a vertical rotation results in different final eye position than rotations in the opposite order. All of this can be proven mathematically (or intuitively by careful manipulation of objects) but for our purposes it must suffice to say that these observations have been confirmed experimentally many times.
LL thus provides an important organizing principle for understanding 3D eye movements and the perceptual consequences considered later in this chapter. As already noted, LL holds during head-restrained saccades and also smooth pursuit. , LL also holds during the translational vestibulo-ocular reflex (VOR), where the binocular gaze point must remain fixed on some external point when the head shifts position. The common element is that these movements use a 3D system to aim 2D gaze, a classic “degrees of freedom problem” in motor control. LL is a control strategy that solves this problem in an orderly fashion.
In other oculomotor systems where there is no degrees of freedom problem, there is no LL. The ideal rotational VOR and optokinetic reflex should stabilize the retinal image by rotating the eye about the same 3D axis as the head but in the opposite direction, and they essentially do, with a somewhat lower torsional “gain” (amount of eye rotation/head rotation) and minor tilts/wiggles of the axis. , , In other eye movements LL is modified. This obviously violates LL when the head rotates torsionally, but even pure horizontal/vertical rotation results in eye position-dependent violations, owing to the lack of a half angle rule. Ocular counterroll shifts Listing’s plane torsionally by about 10% of the amount of static head tilt, whereas convergence causes the Listing’s planes of the two eyes to rotate outward (counterintuitively) like “saloon doors.” ,
Three-dimensional kinematics during head-unrestrained gaze shifts
Finally, we must consider what happens to these rules in the real-world situation where both eye and head rotation contribute to final gaze position ( Fig. 9.2 ). Here, Donders’ law remains in place, with two exceptions. ,
First, the position plane becomes a twisted surface, where one opposing pair of corners (e.g., up-left, down-right) shift in one torsional direction and the other pair (up-right, down-left) in the other torsional direction ( Fig. 9.2A ). The specific pattern resembles the torsion produced by a Fick Gimbal, where the vertical axis of rotation is fixed in space but the horizontal axis rotates about this (like a telescope mount). This is in part because the head behaves like a Fick Gimbal, and in part because the head is used relatively more for horizontal range and the eyes for vertical range, again acting like a Fick Gimbal. One effect of this is to minimize torsional rotation of the eye about the visual axis, thus reducing “false torsion.”
Second, the torsional eye orientation in space becomes much more variable (from about ±1 degree to ±5 degrees). This is partially because head torsion is more variable, and partially because eye-in-head control gets more complicated: saccades are typically followed by head rotation, and this is accompanied by VOR movements ( Fig. 9.2 B ), which include torsional components. Rather than wait till the end of gaze shift to correct these, the system generates anticipatory torsional components (out of Listing’s plane) in the opposite direction ( Fig. 9.2C ). This coordination strategy causes eye-in-head position to leave and then return to Listing’s plane during each head-unrestrained gaze shift but has the important advantage (for vision) of gaze shifts ending with the system stable and torsion minimized.
All of these kinematics observations (with and without head motion) have geometric consequences for vision, which we will return to later. But first, we will consider control mechanisms. What causes saccade axes to tile out of Listing’s plane? How does the brain decide when to use or not use LL? And how are these processes coordinated?
Three-dimensional control mechanisms
Motoneurons and muscles
The extraocular muscles are arranged in anatomic coordinate system where the horizontal recti control horizontal eye rotation, whereas the other muscles (vertical recti and obliques) and their corresponding motoneuron pools in the brainstem each control a combination of vertical and torsional rotation ( Fig. 9.3A , see Chapter 7 for details). If one defines torsional rotation about the visual axes, then the relative contributions of the latter muscles to vertical and torsional rotation varies with horizontal eye position, but if torsion is defined in LL coordinates as described, the arrangement is simpler: they appear to control roughly equal amounts of torsion. For example, in the right eye the directions would be superior rectus: up-counterclockwise, inferior oblique: up-clockwise, inferior rectus: down-clockwise, and superior oblique: down-counterclockwise (with torsional directions reversed in the left eye). Thus, assuming the moment that the horizontal recti control an axis of rotation parallel to Listing’s plane, the brain has to co-activate a vertical rectus + oblique muscle to get pure vertical (or pure torsional) eye rotation. This is important for interpreting the complex 3D oculomotor deficits resulting from eye muscle damage, but this cannot be entirely separated from potential compensatory mechanisms in the brain (we will return to the brain further in the chapter).
A second issue surrounding the contributions of motoneurons and muscles relates to the half angle rule described previously. The 1990s were witness to a theoretical debate, with Tweed espousing the view that the oculomotor system requires a 3D noncommutative controller, whereas Raphan led the opposing view that a 2D linear controller is sufficient for saccades. , , Over time, this debate coalesced to the question of whether the oculomotor plant itself implements the half angle rule, negating the need for the brain to compute its position-dependent axis tilts. , Klier and Angelaki ( Fig. 9.3 ) showed that motoneurons do not encode position-dependent saccade axis tilts, suggesting that the half angle rule is implemented mechanically somehow by the oculomotor plant itself. Mathematically, this means that the phasic component of motoneuron activity (the part that drives motion) does not encode angular velocity, but rather the derivative of eye orientation. This is a feature that has been implicitly or explicitly included in most 3D saccade models over the past 25 years. ,
The next obvious question is: What in the plant is causing these tilts? Surprisingly, we do not know enough about the physics and physiology of the plant to be certain. For example, it could be that the elastic properties of tissues surrounding the eye are responsible. However, the leading candidates for this function are the eye muscle “pulleys” proposed by J. Miller and J. Demer. Based on anatomic images, it appears that tissues surrounding the eye muscles could cause them to change pulling direction as a function of eye position in a way that is consistent with the half angle rule. At this time this is the accepted view, but to conclusively close this topic it will be important to verify the anatomical evidence for pulley actions with more functional physiology.
The preceding discussion might seem to resolve the LL issue, but this would be a mistake. There are several other issues to consider. First, as we have seen, the oculomotor plant itself is a 3D controller, and the brain must select the right balance of muscle activity to give zero torsion, even for head-restrained saccades. Further, we have seen that the brain seems to use or reduce this dimensionality at will, depending on current needs of the visual system (e.g., image stabilization for the VOR vs. gaze pointing for saccades) and can generate torsional saccade components (in or out of Listing’s plane) whenever required to modify, anticipate, or correct for LL ( Fig. 9.2C ). It is also known that LL breaks down during various brain pathologies (see next section). The upshot is that although plant mechanics seem to simplify one aspect of LL (the half angle rule), the question of how the brain chooses (or does not choose) to obey LL requires further consideration.
Brainstem oculomotor coordinates
The “fundamental theorem” of oculomotor control, proposed by D.A. Robinson, can be simplified as follows: eye movements are driven by velocity signals that (1) drive both the phasic component of motoneuron activity (mentioned previously) to contract eye muscles and move the eye and (2) are also input to a “neural integrator” that converts these velocity signals into a position signal that maintains motoneuron/muscle activity after a movement, thus holding eye position. In the case of the saccade system, the phasic signal derives from brainstem reticular formation burst neurons ( Fig. 9.4B ), whereas the rotational VOR signal originates in the semicircular canals themselves. If saccade burst neurons encode eye orientation derivatives, this can be input directly to both a neural integrator and the type of motoneurons described previously. However, for the VOR to work properly, semicircular canal outputs would have to be compared to 3D eye orientation to compute eye position derivatives, effectively canceling the half angle axis tilts added by the eye muscles.
