Ykiun Res. Vol. 32. No. 7, pp. 1225-1238, 1992 Printed in Great Britain. All rights reserved
Copyright
Three-Dimensional Properties Pursuit Eye Movements D. TWEED,*
M. FETTER,*
Received 3 December
S. ANDREADAKI,*
E. KOENIG,*
0042.6989192 $5.00 + 0.00 c’ 1992 Pergamon Press Ltd
of Human J. DICHGANS*
1990; in revised form 20 December 1991
For any given location and velocity of a point target, there are infinitely many different eye velocities that the pursuit system could use to track the target perfectly. Three-dimensional recordings of eye position and velocity in 8 normal human subjects showed that the system chooses the unique tracking velocity that keeps eye position vectors (a particular mathematical representation of three-dimensional eye orientation) confined to a single plane, i.e. pursuit obeys Listing’s law. One advantage of this strategy over other possible ones, such as choosing the smallest eye velocity compatible with perfect tracking, is that it permits continuous pursuit without accumulation of ocular torsion. For nonpoint targets, there is at most one eye velocity compatible with perfect retinal image stabilisation, and the optimal velocity may not fit Listing’s law; we observed small but consistent deviations from the law during pursuit of rotating line targets. Eye movements
Pursuit
Listing’s
law
Eye torsion
INTRODUCTION
Pursuit eye movements are used in tracking small moving targets. This study examines the properties of threedimensional eye position and velocity during pursuit in normal human subjects. The starting point for this work was the geometrical observation that the required output of the pursuit system is underdetermined by its input. That is, given any instantaneous location and velocity of a point target, there are infinitely many different speeds and directions of eye rotation that would keep the target image stationary on the retina. The aim of this study was to find the rule used by the pursuit system in choosing from among these possible eye motions. How can infinitely many different speeds and directions of eye rotation all be compatible with perfect tracking of a single target? To deal with this question, it will be convenient for us to bundle together the concepts of speed and direction into a single package: the angular velocity vector, which is the usual mathematical way of describing rotatory motion. The definition of this vector is based on the fact that any rotating body has, at any instant, a unique axis of rotation. The angular velocity vector is defined to lie along this axis, and the vector’s length equals the speed with which the body is spinning about the axis in, say, deg/sec. In order that the velocity vector uniquely specify the direction of rotation, it is conventional to introduce the right hand rule, which states that the angular velocity vector is so-directed along the spin axis that, if you point your right thumb in the direction of the vector, your fingers curl round in
*Department Strasse 3.
of Neurology, 7400
Tiibingen,
University of Tiibingen, Hoppe-Seyler Fed. Rep. Germany.
Angular
velocity
the direction the body is turning. For example, if the body is spinning to the left about a vertical axis, then the velocity vector points upward, because if you point your thumb upward, your fingers curl round to the left (the direction of spin). Two special cases that will be important throughout this paper occur when the angular velocity vector of the eye is parallel with or orthogonal (i.e. at a right angle) to the line of sight. If the velocity vector is parallel with the line of sight, this means that the eye is spinning about its own gaze line, and the direction of gaze is therefore unchanging. It is as if you were to twirl a flashlight about its own long axis: the light spot projected on the wall would not move. Given any angular velocity vector of the eye, it is always possible to express it as the sum of two components, one parallel with the line of sight and one orthogonal. If the velocity vector as a whole is causing any change in the gaze direction, it is the orthogonal component alone that is responsible; the parallel component, large or small, is irrelevant. In the special case where the velocity vector is exactly orthogonal to the gaze line, then its component along the gaze direction has length zero; we say the vector has no component parallel with the line of sight. In this case, the entire velocity vector contributes to sweeping the gaze direction along; there is no “wasted” component of velocity that merely spins the eye about the gaze line. The answer to our earlier question-how can infinitely many different eye velocities permit perfect tracking of a single target?-is now apparent: if the eye achieves perfect instantaneous tracking with an angular velocity vector w, it will also achieve perfect instantaneous tracking with angular velocity w +p. where p is any
1225
1226
L>. TWEED L’/
Copyright
Three-Dimensional Properties Pursuit Eye Movements D. TWEED,*
M. FETTER,*
Received 3 December
S. ANDREADAKI,*
E. KOENIG,*
0042.6989192 $5.00 + 0.00 c’ 1992 Pergamon Press Ltd
of Human J. DICHGANS*
1990; in revised form 20 December 1991
For any given location and velocity of a point target, there are infinitely many different eye velocities that the pursuit system could use to track the target perfectly. Three-dimensional recordings of eye position and velocity in 8 normal human subjects showed that the system chooses the unique tracking velocity that keeps eye position vectors (a particular mathematical representation of three-dimensional eye orientation) confined to a single plane, i.e. pursuit obeys Listing’s law. One advantage of this strategy over other possible ones, such as choosing the smallest eye velocity compatible with perfect tracking, is that it permits continuous pursuit without accumulation of ocular torsion. For nonpoint targets, there is at most one eye velocity compatible with perfect retinal image stabilisation, and the optimal velocity may not fit Listing’s law; we observed small but consistent deviations from the law during pursuit of rotating line targets. Eye movements
Pursuit
Listing’s
law
Eye torsion
INTRODUCTION
Pursuit eye movements are used in tracking small moving targets. This study examines the properties of threedimensional eye position and velocity during pursuit in normal human subjects. The starting point for this work was the geometrical observation that the required output of the pursuit system is underdetermined by its input. That is, given any instantaneous location and velocity of a point target, there are infinitely many different speeds and directions of eye rotation that would keep the target image stationary on the retina. The aim of this study was to find the rule used by the pursuit system in choosing from among these possible eye motions. How can infinitely many different speeds and directions of eye rotation all be compatible with perfect tracking of a single target? To deal with this question, it will be convenient for us to bundle together the concepts of speed and direction into a single package: the angular velocity vector, which is the usual mathematical way of describing rotatory motion. The definition of this vector is based on the fact that any rotating body has, at any instant, a unique axis of rotation. The angular velocity vector is defined to lie along this axis, and the vector’s length equals the speed with which the body is spinning about the axis in, say, deg/sec. In order that the velocity vector uniquely specify the direction of rotation, it is conventional to introduce the right hand rule, which states that the angular velocity vector is so-directed along the spin axis that, if you point your right thumb in the direction of the vector, your fingers curl round in
*Department Strasse 3.
of Neurology, 7400
Tiibingen,
University of Tiibingen, Hoppe-Seyler Fed. Rep. Germany.
Angular
velocity
the direction the body is turning. For example, if the body is spinning to the left about a vertical axis, then the velocity vector points upward, because if you point your thumb upward, your fingers curl round to the left (the direction of spin). Two special cases that will be important throughout this paper occur when the angular velocity vector of the eye is parallel with or orthogonal (i.e. at a right angle) to the line of sight. If the velocity vector is parallel with the line of sight, this means that the eye is spinning about its own gaze line, and the direction of gaze is therefore unchanging. It is as if you were to twirl a flashlight about its own long axis: the light spot projected on the wall would not move. Given any angular velocity vector of the eye, it is always possible to express it as the sum of two components, one parallel with the line of sight and one orthogonal. If the velocity vector as a whole is causing any change in the gaze direction, it is the orthogonal component alone that is responsible; the parallel component, large or small, is irrelevant. In the special case where the velocity vector is exactly orthogonal to the gaze line, then its component along the gaze direction has length zero; we say the vector has no component parallel with the line of sight. In this case, the entire velocity vector contributes to sweeping the gaze direction along; there is no “wasted” component of velocity that merely spins the eye about the gaze line. The answer to our earlier question-how can infinitely many different eye velocities permit perfect tracking of a single target?-is now apparent: if the eye achieves perfect instantaneous tracking with an angular velocity vector w, it will also achieve perfect instantaneous tracking with angular velocity w +p. where p is any
1225
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L>. TWEED L’/
