A simple control law generates Listing's positions in a detailed model of the extraocular muscle system

Static Characteristics of a New Three-Dimensional Linear Homeomorphic Saccade Model

A linear homeomorphic saccade model that produces 3D saccadic eye movements consistent with physiological and anatomical evidence is introduced. Central to the model is the implementation of a time-optimal controller with six linear muscles and pulleys that represent the saccade oculomotor plant. Each muscle is modeled as a parallel combination of viscosity [Formula: see text] and series elasticity [Formula: see text] connected to the parallel combination of active-state tension generator [Formula: see text], viscosity element [Formula: see text], and length tension elastic element [Formula: see text]. Additionally, passive tissues involving the eyeball include a viscosity element [Formula: see text], elastic element [Formula: see text], and moment of inertia [Formula: see text]. The neural input for each muscle is separately maintained, whereas the effective pulling direction is modulated by its respective mid-orbital constraint from the pulleys. Initial parameter values for the oculomotor plant are based on anatomical and physiological evidence. The oculomotor plant uses a time-optimal, 2D commutative neural controller, together with the pulley system that actively functions to implement Listing's law during both static and dynamic conditions. In a companion paper, the dynamic characteristics of the saccade model is analyzed using a time domain system identification technique to estimate the final parameter values and neural inputs from saccade data. An excellent match between the model estimates and the data is observed, whereby a total of 20 horizontal, 5 vertical, and 64 oblique saccades are analyzed.

Optic Nerve Sheath as a Novel Mechanical Load on the Globe in Ocular Duction

Purpose

The optic nerve (ON) sheath's role in limiting duction has been previously unappreciated. This study employed magnetic resonance imaging (MRI) to demonstrate this constraint on adduction.

Methods

High-resolution, surface coil axial MRI was obtained in 11 normal adults, 14 subjects with esotropia (ET) having normal axial length (AL) < 25.8 mm, 13 myopic subjects with ET and mean AL 29.3 ± 3.3 (SD) mm, and 7 subjects with exotropia (XT). Gaze angles and ON lengths were measured for scans employing eccentric lateral fixation in which an ON became completely straightened.

Results

In all groups, ON straightening occurred only in the adducting, not abducting, eye. Adduction at ON straightening was 26.0 ± 8.8° in normal subjects, not significantly different from XT at 22.2 ± 11.8°. However, there was significant increase in comparable adduction in ET to 36.3 ± 9.3°, and in myopic ET to 33.6 ± 10.7° (P < 0.04). Optic nerve length at straightening was 27.6 ± 2.7 mm in normals, not significantly different from 28.2 ± 2.8 mm in ET and 27.8 ± 2.7 mm in XT. In myopic ET, ON length at straightening was significantly reduced to 24.0 ± 2.9 mm (P < 0.002) and was associated with globe retraction in adduction, suggesting ON tethering.

Conclusions

Large adduction may exhaust length redundancy in the normally sinuous ON and sheath, so that additional adduction must stretch the sheath and retract or deform the globe. These mechanical effects are most significant in ET with axial myopia, but may also exert traction on the posterior sclera absent strabismus or myopia. Tethering by the ON sheath in adduction is an important, novel mechanical load on the globe.

An updated time-optimal 3rd-order linear saccadic eye plant model

A linear third-order model of the ocular motor plant for horizontal saccadic eye movements is presented consisting of a linear ocular motor plant and a time-optimal saccadic controller based on physiological considerations. The ocular motor plant consists of the eyeball and two extraocular muscles. All parameters and initial conditions are estimated or measured from physiological data. The neural inputs are described by pulse-slide-step waveforms with a post inhibitory rebound burst and based on a time-optimal controller. Model parameters are estimated using the system identification technique. The static and dynamic behaviors of the model are in excellent agreement with the experimental data.

Neuromuscular plasticity and rehabilitation of the ocular near response

The near response is composed of cross-coupled interactions between convergence and other distance-related oculomotor responses including accommodation, vertical vergence, and cyclovergence. The cross-coupling interactions are analogous to the body postural reflexes that maintain balance. Near-response couplings guide involuntary motor responses during voluntary shifts of distance and direction of gaze without feedback from defocus or retinal-image disparity. They optimize the disparity stimulus for stereoscopic depth perception and can be modified by optically induced sensory demands placed on binocular vision. In natural viewing conditions, the near response is determined by passive orbital mechanics and active-adaptable tonic components. For example, the normal coupling of vertical vergence with convergence in tertiary gaze is partly a byproduct of passive orbital mechanics. Both, adapted changes of vertical vergence in response to anisophoria, produced by unequal ocular magnification (aniseikonia), and adapted changes in the orientation of Listing's plane in response to torsional disparities can be achieved by a combination of passive orbital mechanics and neural adjustments for the control of the vertical vergence and cyclovergence. Adaptive adjustments are coupled with gaze direction, convergence angle, and head tilt. Several adaptation studies suggest that it is possible to achieve non-linear changes in the coupling of both vertical vergence and cyclovergence with gaze direction. This coupling can be achieved with changes in neural control signals of ocular elevator muscles that are cross-coupled with both convergence and direction of tertiary gaze. These linear and non-linear coupling interactions can be adapted to compensate for (1) anisophoria induced by spectacle corrections for anisometropia, (2) accommodative esotropia, (3) convergence excess and insufficiency, and (4) non-concomitant deviations with ocular torticollis associated with trochlear palsy. The adaptable near-response couplings form the basis of an area of orthoptics that optimizes visual performance by facilitating our natural ability to calibrate neural pathways underlying binocular postural reflexes.

Dynamics of primate oculomotor plant revealed by effects of abducens microstimulation

Despite their importance for deciphering oculomotor commands, the mechanics of the extraocular muscles and orbital tissues (oculomotor plant) are poorly understood. In particular, the significance of plant nonlinearities is uncertain. Here primate plant dynamics were investigated by measuring the eye movements produced by stimulating the abducens nucleus with brief pulse trains of varying frequency. Statistical analysis of these movements indicated that the effects of stimulation lasted about 40 ms after the final pulse, after which the eye returned passively toward its position before stimulation. Behavior during the passive phase could be approximated by a linear plant model, corresponding to Voigt elements in series, with properties independent of initial eye position. In contrast, behavior during the stimulation phase revealed a sigmoidal relation between stimulation frequency and estimated steady-state tetanic tension, together with a frequency-dependent rate of tension increase, that appeared very similar to the nonlinearities previously found for isometric-force production in primate lateral rectus muscle. These results suggest that the dynamics of the oculomotor plant have an approximately linear component related to steady-state viscoelasticity and a nonlinear component related to changes in muscle activation. The latter may in part account for the nonlinear relations observed between eye-movement parameters and single-unit firing patterns in the abducens nucleus. These findings point to the importance of recruitment as a simplifying factor for motor control with nonlinear plants.

Acute superior oblique palsy in the monkey: effects of viewing conditions on ocular alignment and modelling of the ocular motor plant

We investigated the immediate and long-term changes in static eye alignment with acute superior oblique palsy (SOP) in the monkey. When the paretic eye was patched immediately after the lesion for 6-9 days, vertical alignment slowly improved. When the patch was removed and binocular viewing was allowed, alignment slowly worsened. In contrast when a monkey was not patched immediately after the lesion vertical alignment did not improve. We also show that a model of the eye plant can reproduce the observed acute deficit induced by SOP, but only by abandoning Robinson's symmetric simplification of the reciprocal innervation relationship within pairs of agonist-antagonist muscles. The model also demonstrated that physiologic variability in orbital geometry can have a large impact on SOP deficits.

Evidence from retractor bulbi EMG for linearized motor control of conditioned nictitating membrane responses

Classical conditioning of nictitating membrane (NM) responses in rabbits is a robust model learning system, and experimental evidence indicates that conditioned responses (CRs) are controlled by the cerebellum. It is unknown whether cerebellar control signals deal directly with the complex nonlinearities of the plant (blink-related muscles and peripheral tissues) or whether the plant is linearized to ensure a simple relation between cerebellar neuronal firing and CR profile. To study this question, the retractor bulbi muscle EMG was recorded with implanted electrodes during NM conditioning. Pooled activity in accessory abducens motoneurons was estimated from spike trains extracted from the EMG traces, and its temporal profile was found to have an approximately Gaussian shape with peak amplitude linearly related to CR amplitude. The relation between motoneuron activity and CR profiles was accurately fitted by a first-order linear filter, with each spike input producing an exponentially decaying impulse response with time constant of order 0.1 s. Application of this first-order plant model to CR data from other laboratories suggested that, in these cases also, motoneuron activity had a Gaussian profile, with time-of-peak close to unconditioned stimulus (US) onset and SD proportional to the interval between conditioned stimulus and US onsets. These results suggest that for conditioned NM responses the cerebellum is presented with a simplified "virtual" plant that is a linearized version of the underlying nonlinear biological system. Analysis of a detailed plant model suggests that one method for linearising the plant would be appropriate recruitment of motor units.

Evidence supporting extraocular muscle pulleys: refuting the platygean view of extraocular muscle mechanics

BACKGROUND

Late in the 20th century, it was recognized that connective tissue structures in the orbit influence the paths of the extraocular muscles and constitute their functional origins. Targeted investigations of these connective tissue "pulleys" led to the formulation of the active pulley hypothesis, which proposes that pulling directions of the rectus extraocular muscles are actively controlled via connective tissues.

PURPOSE

This review rebuts a series of criticisms of the active pulley hypothesis published by Jampel, and Jampel and Shi, in which these authors have disputed the existence and function of the pulleys.

METHODS

This article reviews published evidence for the existence of orbital pulleys, the active pulley hypothesis, and physiological tests of the active pulley hypothesis. Magnetic resonance imaging in a living subject and histological examination of a human cadaver directly illustrate the relationship of pulleys to extraocular muscles.

RESULTS

Strong scientific evidence is cited that supports the existence of orbital pulleys and their role in ocular motility. The criticisms of the hypothesis have ignored mathematical truisms and strong scientific evidence.

CONCLUSIONS

Actively control led orbital pulleys play a fundamental role in ocular motility. Pulleys profoundly influence the neural commands required to control eye movements and binocular alignment. Familiarity with the anatomy and physiology of the pulleys is requisite for a rational approach to diagnosing and treating strabismus using emerging methods. Conversely, approaches that deny or ignore the pulleys risk the sorts of errors that arise in geography and navigation from incorrect assumptions such as those of a flat ("platygean") earth.

Current concepts of mechanical and neural factors in ocular motility

PURPOSE OF REVIEW

The oculomotor periphery was classically regarded as a simple mechanism executing complex behaviors specified explicitly by neural commands. A competing view has emerged that many important aspects of ocular motility are properties of the extraocular muscles and their associated connective tissue pulleys. This review considers current concepts regarding aspects of ocular motility that are mechanically determined versus those that are specified explicitly as innervation.

RECENT FINDINGS

While it was established several years ago that the rectus extraocular muscles have connective tissue pulleys, recent functional imaging and histology has suggested that the rectus pulley array constitutes an inner mechanism, analogous to a gimbal, that is rotated torsionally around the orbital axis by an outer mechanism driven by the oblique extraocular muscles. This arrangement may account mechanically for several commutative aspects of ocular motor control, including Listing's Law, yet permits implementation of non-commutative motility. Recent human behavioral studies, as well as neurophysiology in monkeys, are consistent with implementation of Listing's Law in the oculomotor periphery, rather than centrally.

SUMMARY

Varied evidence now strongly supports the conclusion that Listing's Law and other important ocular kinematics are mechanically determined. This finding implies more limited possibilities for neural adaptation to some ocular motor pathologies, but indicates possibilities for surgical treatments.

Do motoneurons encode the noncommutativity of ocular rotations?

As we look around, the orientation of our eyes depends on the order of the rotations that are carried out, a mathematical feature of rotatory motions known as noncommutativity. Theorists and experimentalists continue to debate how biological systems deal with this property when generating kinematically appropriate movements. Some believe that this is always done by neural commands to a simplified eye plant. Others have postulated that noncommutativity is implemented solely by the mechanical properties of the eyeball. Here we directly examined what the brain tells the muscles, by recording motoneuron activities as monkeys made eye movements. We found that vertical recti and superior/inferior oblique motoneurons, which drive sensory-generated torsional eye movements, do not modulate their firing rates according to the noncommutative-driven torsion during pursuit. We conclude that part of the solution for kinematically appropriate eye movements is found in the mechanical properties of the eyeball, although neural computations remain necessary and become increasingly important during head movements.

SEE++: a biomechanical model of the oculomotor plant

The consequences of changes in the oculomotor system on the three-dimensional eye movements are difficult to grasp. Although changes to the rectus muscles can still be approximately understood with simplified geometric models, this approach no longer works with the oblique muscles. It is shown how SEE++, a biomechanical model of the oculomotor plant that was built on the ideas of Miller and Robinson (1984) can improve the understanding of the effects of changes to the oblique eye muscles. By displaying only selected muscles, and by illustrating the relative contribution of these muscles through color-coding the bulb surface, the functional properties of the oblique muscles can be presented in a much clearer way. Investigating the effects of a hyperactive inferior oblique muscle shows that this type of model can help to clarify the functional cause of a pathology, which can otherwise be unclear, even for common pathologies.

Consistency of Listing's law and reciprocal innervation with pseudo-inverse control of eye position in 3-D

Pseudo-inverse kinematics, under which small movements are produced by the least possible sum square changes in motor command, has been proposed as a unifying principle for the elimination of redundancy in general biological motor control systems (Pellionsz 1984) and in particular in the oculomotor system (Daunicht 1988, 1991). We have noted elsewhere (Dean et al, 1999) that this principle is incomplete without first specifying a parameterisation of motor command space and we proposed that the relevant motor-command parameter is summed motor unit firing rate. Under this assumption we were able to show that pseudo-inverse control of the horizontal extraocular muscles is consistent with available motor pool firing rate data. In this paper we extend this result to three dimensions and six extraocular muscles, showing that pseudo-inverse control is consistent with published firing rate date for a realistic model of oculomotor kinematics. We suggest that pseudo-inverse control may represent a common currency for modular control of many degree of freedom systems while its implementation may be a consequences of the minimisation of a more ecologically relevant parameter such as post-saccadic retinal slip.

Control of eye orientation: where does the brain's role end and the muscle's begin?

Our understanding of how the brain controls eye movements has benefited enormously from the comparison of neuronal activity with eye movements and the quantification of these relationships with mathematical models. Although these early studies focused on horizontal and vertical eye movements, recent behavioural and modelling studies have illustrated the importance, but also the complexity, of extending previous conclusions to the problems of controlling eye and head orientation in three dimensions (3-D). An important facet in understanding 3-D eye orientation and movement has been the discovery of mobile, soft-tissue sheaths or 'pulleys' in the orbit which might influence the pulling direction of extraocular muscles. Appropriately placed pulleys could generate the eye-position-dependent tilt of the ocular rotation axes which are characteristic for eye movements which follow Listing's law. Based on such pulley models of the oculomotor plant it has recently been proposed that a simple two-dimensional (2-D) neural controller would be sufficient to generate correct 3-D eye orientation and movement. In contrast to this apparent simplification in oculomotor control, multiple behavioural observations suggest that the visuo-motor transformations, as well as the premotor circuitry for saccades, pursuit eye movements and the vestibulo-ocular reflexes, must include a neural controller which operates in 3-D, even when considering an eye plant with pulleys. This review summarizes the most recent work and ideas on this controversy. In addition, by proposing directly testable hypotheses, we point out that, in analogy to the previously successful steps towards elucidating the neural control of horizontal eye movements, we need a quantitative characterization first of motoneuron and next of premotor neuron properties in 3-D before we can succeed in gaining further insight into the neural control of 3-D motor behaviours.

Properties of 3D rotations and their relation to eye movement control

Rotations of the eye are generated by the torques that the eye muscles apply to the eye. The relationship between eye orientation and the direction of the torques generated by the extraocular muscles is therefore central to any understanding of the control of three-dimensional eye movements of any type. We review the geometrical properties that dictate the relationship between muscle pulling direction and 3D eye orientation. We then show how this relation can be used to test the validity of oculomotor control hypotheses. We test the common modeling assumption that the extraocular muscle pairs can be treated as single bidirectional muscles. Finally, we investigate the consequences of assuming fixed muscle pulley locations when modeling the control of eye movements.

Mechanics of eye movements: implications of the "orbital revolution"

Our understanding of the functional structure of extraocular muscles has undergone a profound change: while these muscles used to be represented by strings running straight from their origin in the posterior orbita to their insertion on the globe, we now know that their paths and pulling directions are dominated by fibromuscular pulley structures, keeping them close to the orbital wall for most of their path. An overview is presented of recent models that have been developed to understand the implications of muscle pulleys for the neural control of eye movements and the applications of such models to the interpretation of experimental data.

Adaptive control of vergence in humans

Vergence eye alignment minimizes horizontal, vertical, and cyclodisparities to optimize stereo-depth perception. Only the horizontal component of vergence is under voluntary control. Couplings with voluntary version and horizontal vergence guide vertical vergence and cyclovergence. Can these couplings be modified in response to sensory demands on binocular vision? We have modified vertical vergence and cyclovergence in response to optical changes in disparity. Vertical vergence was stimulated with aniseikonic lenses that exaggerated vertical disparity in tertiary gaze. Vertical vergence adapted in an hour to produce nonconcomitant changes in vertical phoria that varied with vertical eye position in tertiary gaze. Cyclovergence was stimulated with cyclodisparities that varied with gaze elevation and convergence angle. Cyclovergence adapted within 2 hours to produce nonconcomitant changes in cyclophoria that varied with gaze elevation and convergence. The adaptive couplings for vertical vergence and cyclovergence are modeled as a combination of passive orbital mechanics and active gain control of the vertical recti and obliques. Vergence adaptation is a calibration process that adjusts the innervation for horizontal, vertical, and torsion components of vergence to the physical constraints set by the extraocular muscles and orbital connective tissues. Passive orbital mechanics simplify the neural control for precise vertical vergence and cyclovergence that are needed to achieve binocular alignment under open-loop conditions in response to perceived spatial location.

Plasticity of convergence-dependent variations of cyclovergence with vertical gaze

Binocular alignment of foveal images is facilitated by cross-couplings of vergence eye movements with distance and direction of gaze. These couplings reduce horizontal, vertical and cyclodisparities at the fovea without using feedback from retinal image disparity. Horizontal vergence is coupled with accommodation. Vertical vergence that aligns tertiary targets in asymmetric convergence is thought to be coupled with convergence and horizontal gaze. Cyclovergence aligns the horizontal retinal meridians during gaze elevation in symmetrical convergence and is coupled with convergence and vertical gaze. The latter vergence-dependent changes of cyclovergence have been described in terms of the orientation of Listing's plane and have been referred to as the binocular extension of Listing's law. Can these couplings be modified? Plasticity has been demonstrated previously for two of the three dimensions of vergence (horizontal and vertical). The current study demonstrates that convergence-dependent changes of the orientation of Listing's plane can be adapted to either exaggerate or to reduce the cyclovergence that normally facilitates alignment of the horizontal meridians of the retinas with one another during gaze elevation in symmetrical convergence. The adaptability of cyclovergence demonstrates a neural mechanism that, in conjunction with the passive forces determined by biomechanical properties of the orbit, could play an active role in implementing Listing's extended law and provide a means for calibrating binocular eye alignment in three dimensions.