Considerations on Listing's Law and the primary position by means of a matrix description of eye position control

Klier, Eliana M. and J. Douglas Crawford. Human oculomotor system accounts for 3-D eye orientation in the visual-motor transformation for saccades. J. Neurophysiol. 80: 2274–2294, 1998. A recent theoretical investigation has demonstrated that three-dimensional (3-D) eye position dependencies in the geometry of retinal stimulation must be accounted for neurally (i.e., in a visuomotor reference frame transformation) if saccades are to be both accurate and obey Listing's law from all initial eye positions. Our goal was to determine whether the human saccade generator correctly implements this eye-to-head reference frame transformation (RFT), or if it approximates this function with a visuomotor look-up table (LT). Six head-fixed subjects participated in three experiments in complete darkness. We recorded 60° horizontal saccades between five parallel pairs of lights, over a vertical range of ±40° ( experiment 1), and 30° radial saccades from a central target, with the head upright or tilted 45° clockwise/counterclockwise to induce torsional ocular counterroll, under both binocular and monocular viewing conditions ( experiments 2 and 3). 3-D eye orientation and oculocentric target direction (i.e., retinal error) were computed from search coil signals in the right eye. Experiment 1: as predicted, retinal error was a nontrivial function of both target displacement in space and 3-D eye orientation (e.g., horizontally displaced targets could induce horizontal or oblique retinal errors, depending on eye position). These data were input to a 3-D visuomotor LT model, which implemented Listing's law, but predicted position-dependent errors in final gaze direction of up to 19.8°. Actual saccades obeyed Listing's law but did not show the predicted pattern of inaccuracies in final gaze direction, i.e., the slope of actual error, as a function of predicted error, was only −0.01 ± 0.14 (compared with 0 for RFT model and 1.0 for LT model), suggesting near-perfect compensation for eye position. Experiments 2 and 3: actual directional errors from initial torsional eye positions were only a fraction of those predicted by the LT model (e.g., 32% for clockwise and 33% for counterclockwise counterroll during binocular viewing). Furthermore, any residual errors were immediately reduced when visual feedback was provided during saccades. Thus, other than sporadic miscalibrations for torsion, saccades were accurate from all 3-D eye positions. We conclude that 1) the hypothesis of a visuomotor look-up table for saccades fails to account even for saccades made directly toward visual targets, but rather, 2) the oculomotor system takes 3-D eye orientation into account in a visuomotor reference frame transformation. This transformation is probably implemented physiologically between retinotopically organized saccade centers (in cortex and superior colliculus) and the brain stem burst generator.

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Three-dimensional visuo-motor control of saccades

Journal of Neurophysiology ◽

10.1152/jn.00513.2012 ◽

2013 ◽

Vol 109 (1) ◽

pp. 183-192 ◽

Cited By ~ 5

Author(s):

Bernhard J. M. Hess

Keyword(s):

Motor Control ◽

Reverse Engineering ◽

Visual Information ◽

Three Dimensional ◽

Engineering Problem ◽

Line Of Sight ◽

Eye Position ◽

Listing's Law ◽

Saccade Onset ◽

Listing’S Law

Although the motion of the line of sight is a straightforward consequence of a particular rotation of the eye, it is much trickier to predict the rotation underlying a particular motion of the line of sight in accordance with Listing's law. Helmholtz's notion of the direction-circle together with the notion of primary and secondary reference directions in visual space provide an elegant solution to this reverse engineering problem, which the brain is faced with whenever generating a saccade. To test whether these notions indeed apply for saccades, we analyzed three-dimensional eye movements recorded in four rhesus monkeys. We found that on average saccade trajectories closely matched with the associated direction-circles. Torsional, vertical, and horizontal eye position of saccades scattered around the position predicted by the associated direction-circles with standard deviations of 0.5°, 0.3°, and 0.4°, respectively. Comparison of saccade trajectories with the likewise predicted fixed-axis rotations yielded mean coefficients of determinations (±SD) of 0.72 (±0.26) for torsion, 0.97 (±0.10) for vertical, and 0.96 (±0.11) for horizontal eye position. Reverse engineering of three-dimensional saccadic rotations based on visual information suggests that motor control of saccades, compatible with Listing's law, not only uses information on the fixation directions at saccade onset and offset but also relies on the computation of secondary reference positions that vary from saccade to saccade.

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Measurements of stiffness of extraocular muscles of the rabbit

Journal of Neurophysiology ◽

10.1152/jn.1976.39.5.1009 ◽

1976 ◽

Vol 39 (5) ◽

pp. 1009-1019 ◽

Cited By ~ 18

Author(s):

N. H. Barmack

Keyword(s):

Electrical Stimulation ◽

Muscle Activation ◽

Position Control ◽

Human Subjects ◽

Extraocular Muscles ◽

Eye Position ◽

Vestibuloocular Reflex ◽

Primary Position ◽

Order Of Magnitude ◽

Contralateral Eye

1. The stiffness of the eye and the extraocular muscles of the conscious rabbit was measured and compared with similar measurements made on human subjects. 2. The extraocular muscles of the rabbit developed less than 0.5 g at optimal isometric length when the contralateral eye was in "primary position". Deviations of the eye of the rabbit from primary position were accomplished by an increase in force of the agonist muscle with only a nominal decrease in force in the antagonist. 3. With all muscles attached to the globe, the stiffness of the eye for displacements of up to 35 degrees from primary position was 0.11 +/- 0.03 g/deg. 4. The stiffness of the human eye to lateral rotations was 1.09 +/- 0.24 g/deg. Hence, identical disturbances in force would cause deviations in the position of the eye of the rabbit which are nearly an order of magnitude greater than those of man. 5. The stiffness of the extraocular muscles of the rabbit was sensitive to levels of muscle activation produced either by indirect electrical stimulation or by the vestibuloocular reflex. 6. The implications of these findings for the development of eye position control in animals with good binocular vision are discussed.

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Chameleon eye position obeys Listing's law

Vision Research ◽

10.1016/s0042-6989(01)00111-0 ◽

2001 ◽

Vol 41 (17) ◽

pp. 2245-2251 ◽

Cited By ~ 8

Author(s):

Peter S Sándor ◽

Maarten A Frens ◽

Volker Henn

Keyword(s):

Eye Position ◽

Listing's Law ◽

Listing’S Law

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V1 neurons encode the perceptual compensation of “false torsion” arising from Listing’s law

10.1101/428441 ◽

2018 ◽

Author(s):

Mohammad Farhan Khazali ◽

Peter Thier

Keyword(s):

Eye Movements ◽

Visual System ◽

Eye Position ◽

V1 Neurons ◽

Listing's Law ◽

Listing’S Law ◽

Torsional Component

AbstractWe try to deploy the retinal fovea to optimally scrutinize an object of interest by directing our eyes to it. Horizontal and vertical components of these fixation eye movements are determined by the object’s location. However, fixation eye movements also involve a torsional component, which according to Listing’s law is fully determined by the 2D eye position acquired. According to Von Helmholtz knowledge of the torsion provided by this law alleviates the perceptual interpretation of the image tilt that changes with fixation, a view supported by psychophysical experiments he pioneered. We address the question where and how Listing’s law is implemented in the visual system and we show that neurons in monkey area V1 use knowledge of torsion to compensate the image tilt associated with specific eye positions as set by Listing’s law.

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Revealing the Kinematics of the Oculomotor Plant With Tertiary Eye Positions and Ocular Counterroll

Journal of Neurophysiology ◽

10.1152/jn.00737.2010 ◽

2011 ◽

Vol 105 (2) ◽

pp. 640-649 ◽

Cited By ~ 9

Author(s):

Eliana M. Klier ◽

Hui Meng ◽

Dora E. Angelaki

Keyword(s):

Eye Movements ◽

Degrees Of Freedom ◽

Three Dimensional ◽

Abducens Nerve ◽

Oculomotor System ◽

Primary Position ◽

Listing's Law ◽

Listing’S Law ◽

Ocular Counterroll ◽

Half Angle

Retinal information is two-dimensional, whereas eye movements are three-dimensional. The oculomotor system solves this degrees-of-freedom problem by constraining eye positions to zero torsion (Listing's law) and determining how eye velocities change with eye position (half-angle rule). Here we test whether the oculomotor plant, in the absence of well-defined neural commands, can implement these constrains mechanically, not just in a primary position but for all eye and head orientations. We stimulated the abducens nerve at tertiary eye positions and when ocular counterroll was induced at tilted head orientations. Stimulation-induced eye velocities follow the half-angle rule, even for tertiary eye positions, and microstimulation at tilted head orientations elicits eye positions that adhere to torsionally shifted planes, similar to naturally occurring eye movements. These results support the notion that oculomotor plant can continuously apply these three-dimensional rules correctly and appropriately for all eye and head orientations that obey Listing's law, demonstrating a major role of peripheral biomechanics in motor control.

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Visual-Motor Transformations Required for Accurate and Kinematically Correct Saccades

Journal of Neurophysiology ◽

10.1152/jn.1997.78.3.1447 ◽

1997 ◽

Vol 78 (3) ◽

pp. 1447-1467 ◽

Cited By ~ 65

Author(s):

J. Douglas Crawford ◽

Daniel Guitton

Keyword(s):

Reference Frame ◽

Eye Position ◽

Target Displacement ◽

Listing's Law ◽

Listing's Plane ◽

Listing’S Law ◽

Reference Frame Transformation ◽

Visual Motor ◽

Frame Transformation ◽

Listing’S Plane

Crawford, J. Douglas and Daniel Guitton. Visual-motor transformations required for accurate and kinematically correct saccades. J. Neurophysiol. 78: 1447–1467, 1997. The goal of this study was to identify and model the three-dimensional (3-D) geometric transformations required for accurate saccades to distant visual targets from arbitrary initial eye positions. In abstract 2-D models, target displacement in space, retinal error (RE), and saccade vectors are trivially interchangeable. However, in real 3-D space, RE is a nontrivial function of objective target displacement and 3-D eye position. To determine the physiological implications of this, a visuomotor “lookup table” was modeled by mapping the horizontal/vertical components of RE onto the corresponding vector components of eye displacement in Listing's plane. This provided the motor error (ME) command for a 3-D displacement-feedback loop. The output of this loop controlled an oculomotor plant that mechanically implemented the position-dependent saccade axis tilts required for Listing's law. This model correctly maintained Listing's law but was unable to correct torsional position deviations from Listing's plane. Moreover, the model also generated systematic errors in saccade direction (as a function of eye position components orthogonal to RE), predicting errors in final gaze direction of up to 25° in the oculomotor range. Plant modifications could not solve these problems, because the intrisic oculomotor input-output geometry forced a fixed visuomotor mapping to choose between either accuracy or Listing's law. This was reflected internally by a sensorimotor divergence between input-defined visual displacement signals (inherently 2-D and defined in reference to the eye) and output-defined motor displacement signals (inherently 3-D and defined in reference to the head). These problems were solved by rotating RE by estimated 3-D eye position (i.e., a reference frame transformation), inputting the result into a 2-D–to–3-D “Listing's law operator,” and then finally subtracting initial 3-D eye position to yield the correct ME. This model was accurate and upheld Listing's law from all initial positions. Moreover, it suggested specific experiments to invasively distinguish visual and motor displacement codes, predicting a systematic position dependence in the directional tuning of RE versus a fixed-vector tuning in ME. We conclude that visual and motor displacement spaces are geometrically distinct such that a fixed visual-motor mapping will produce systematic and measurable behavioral errors. To avoid these errors, the brain would need to implement both a 3-D position-dependent reference frame transformation and nontrivial 2-D–to–3-D transformation. Furthermore, our simulations provide new experimental paradigms to invasively identify the physiological progression of these spatial transformations by reexamining the position-dependent geometry of displacement code directions in the superior colliculus, cerebellum, and various cortical visuomotor areas.

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Three-Dimensional Vestibuloocular Reflex of the Monkey: Optimal Retinal Image Stabilization Versus Listing's Law

Journal of Neurophysiology ◽

10.1152/jn.2000.83.6.3264 ◽

2000 ◽

Vol 83 (6) ◽

pp. 3264-3276 ◽

Cited By ~ 37

Author(s):

Hubert Misslisch ◽

Bernhard J. M. Hess

Keyword(s):

Retinal Image ◽

Three Dimensional ◽

Oculomotor Control ◽

Slow Phase ◽

Eye Position ◽

Vestibuloocular Reflex ◽

Image Stabilization ◽

Listing's Law ◽

Listing’S Law ◽

Visual Background

If the rotational vestibuloocular reflex (VOR) were to achieve optimal retinal image stabilization during head rotations in three-dimensional space, it must turn the eye around the same axis as the head, with equal velocity but in the opposite direction. This optimal VOR strategy implies that the position of the eye in the orbit must not affect the VOR. However, if the VOR were to follow Listing's law, then the slow-phase eye rotation axis should tilt as a function of current eye position. We trained animals to fixate visual targets placed straight ahead or 20° up, down, left or right while being oscillated in yaw, pitch, and roll at 0.5–4 Hz, either with or without a full-field visual background. Our main result was that the visually assisted VOR of normal monkeys invariantly rotated the eye around the same axis as the head during yaw, pitch, and roll (optimal VOR). In the absence of a visual background, eccentric eye positions evoked small axis tilts of slow phases in normal animals. Under the same visual condition, a prominent effect of eye position was found during roll but not during pitch or yaw in animals with low torsional and vertical gains following plugging of the vertical semicircular canals. This result was in accordance with a model incorporating a specific compromise between an optimal VOR and a VOR that perfectly obeys Listing's law. We conclude that the visually assisted VOR of the normal monkey optimally stabilizes foveal as well as peripheral retinal images. The finding of optimal VOR performance challenges a dominant role of plant mechanics and supports the notion of noncommutative operations in the oculomotor control system.

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Axes of eye rotation and Listing's law during rotations of the head

Journal of Neurophysiology ◽

10.1152/jn.1991.65.3.407 ◽

1991 ◽

Vol 65 (3) ◽

pp. 407-423 ◽

Cited By ~ 138

Author(s):

J. D. Crawford ◽

T. Vilis

Keyword(s):

Head Rotation ◽

Three Dimensions ◽

Eye Position ◽

Gaze Stabilization ◽

Listing's Law ◽

Listing's Plane ◽

Listing’S Law ◽

Eye Rotation ◽

Indirect Path ◽

Listing’S Plane

1. The vestibuloocular reflex (VOR) was examined in four alert monkeys during rotations of the head about torsional, vertical, horizontal, and intermediate axes. Eye positions and axes were recorded in three dimensions (3-D). Visual targets were used to optimize gaze stabilization. 2. Axes of eye rotation during slow phases showed small but systematic deviations from collinearity with the axes of head rotation. These noncollinearities apparently resulted from vector summation of torsional, vertical, and horizontal VOR components with different gains. 3. VOR gain was lowest about a head-fixed torsional axis that was correlated with the primary gaze direction, as determined by Listing's law for saccades. As a result, rotation of the head about a partially torsional axis produced noncollinear slow phases, with axes that tilted toward Listing's plane. 4. During slow phases, eye position changed not only in the direction of rotation, but also systematically in other directions. Even axes of eye rotation within Listing's plane caused eye position to move out of the plane to a torsional position that was then held. Thus Listing's law for saccades cannot be a product of plant mechanics. 5. VOR slow phases were simulated with the use of a model that incorporated 3-D rotational kinematics into the indirect path and the oculomotor plant. This demonstrated that the observed pattern of position changes is the expected consequence of rotating the eye about a fixed axis and that to hold these positions the indirect path must employ a 3-D velocity-to-position transformation. 6. Quick phases not only corrected the violations of Listing's law produced by slow phases but anticipated them by directing the eye toward a plane rotated in the direction of head rotation. This was modeled by inputting the vestibular signal to a Listing's law operator that is shared by the quick phase and saccadic systems.

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V1 neurons encode the perceptual compensation of false torsion arising from Listing’s law

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.2007644117 ◽

2020 ◽

Vol 117 (31) ◽

pp. 18799-18809

Author(s):

Mohammad Farhan Khazali ◽

Hamidreza Ramezanpour ◽

Peter Thier

Keyword(s):

Visual System ◽

Eye Position ◽

Two Dimensional ◽

V1 Neurons ◽

Listing's Law ◽

Listing’S Law ◽

Torsional Component ◽

Eye Torsion

We try to deploy the retinal fovea to optimally scrutinize an object of interest by directing our eyes to it. The horizontal and vertical components of eye positions acquired by goal-directed saccades are determined by the object’s location. However, the eccentric eye positions also involve a torsional component, which according to Donder’s law is fully determined by the two-dimensional (2D) eye position acquired. According to von Helmholtz, knowledge of the amount of torsion provided by Listing’s law, an extension of Donder’s law, alleviates the perceptual interpretation of the image tilt that changes with 2D eye position, a view supported by psychophysical experiments he pioneered. We address the question of where and how Listing’s law is implemented in the visual system and we show that neurons in monkey area V1 use knowledge of eye torsion to compensate the image tilt associated with specific eye positions as set by Listing’s law.

Download Full-text