The viscoelastic properties of passive eye muscle in primates. II: testing the quasi-linear theory

Oculomotor plant and neural dynamics suggest gaze control requires integration on distributed timescales

A fundamental principle of biological motor control is that the neural commands driving movement must conform to the response properties of the motor plants they control. In the oculomotor system, characterizations of oculomotor plant dynamics traditionally supported models in which the plant responds to neural drive to extraocular muscles on exclusively short, subsecond timescales. These models predict that the stabilization of gaze during fixations between saccades requires neural drive that approximates eye position on longer timescales and is generated through the temporal integration of brief eye velocity-encoding signals that cause saccades. However, recent measurements of oculomotor plant behaviour have revealed responses on longer timescales. Furthermore, measurements of firing patterns in the oculomotor integrator have revealed a more complex encoding of eye movement dynamics. Yet, the link between these observations has remained unclear. Here we use measurements from the larval zebrafish to link dynamics in the oculomotor plant to dynamics in the neural integrator. The oculomotor plant in both anaesthetized and awake larval zebrafish was characterized by a broad distribution of response timescales, including those much longer than 1 s. Analysis of the firing patterns of oculomotor integrator neurons, which exhibited a broadly distributed range of decay time constants, demonstrates the sufficiency of this activity for stabilizing gaze given an oculomotor plant with distributed response timescales. This work suggests that leaky integration on multiple, distributed timescales by the oculomotor integrator reflects an inverse model for generating oculomotor commands, and that multi-timescale dynamics may be a general feature of motor circuitry. KEY POINTS: Recent observations of oculomotor plant response properties and neural activity across the oculomotor system have called into question classical formulations of both the oculomotor plant and the oculomotor integrator. Here we use measurements from new and published experiments in the larval zebrafish together with modelling to reconcile recent oculomotor plant observations with oculomotor integrator function. We developed computational techniques to characterize oculomotor plant responses over several seconds in awake animals, demonstrating that long timescale responses seen in anaesthetized animals extend to the awake state. Analysis of firing patterns of oculomotor integrator neurons demonstrates the sufficiency of this activity for stabilizing gaze given an oculomotor plant with multiple, distributed response timescales. Our results support a formulation of gaze stabilization by the oculomotor system in which commands for stabilizing gaze are generated through integration on multiple, distributed timescales.

Dynamics of plant mechanics

Muscle and plant dynamics are most important during the high acceleration of saccades. Models have been developed to characterize muscle and plant dynamics. Building these models require an understanding of the length-tension (elastic) and force-velocity (viscous) relationships. Much work has been done to characterize these nonlinear functions, as they are influenced by innervation. However, the active force generator (active-state tension) in the muscle is still poorly understood. Thus, these models serve more to reveal where new studies of muscle behavior are needed than to explain what happens during a saccade.

Mathematical models and human disease

Mathematical models of brain function are built from data covering anatomy, physiology, biophysics and behavior. In almost all cases, many possible models could fit the available data. Theoreticians make assumptions that allow them to constrain the number of possible model structures. However, a model that was more useful clinically would result if the constraints came from lesion studies in animals or clinical disorders. Here, we show a few examples of how clinical disorders have led to improvements in models. We also show a few examples of how models could lead to neural prostheses for patients. The best outcomes result when clinicians, basic scientists and theoreticians work together to understand brain function.

Opto-mechanical characterization of sclera by polarization sensitive optical coherence tomography

Polarization sensitive optical coherence tomography (PSOCT) is an interferometric technique sensitive to birefringence. Since mechanical loading alters the orientation of birefringent collagen fibrils, we asked if PSOCT can be used to measure local mechanical properties of sclera. Infrared (1300 nm) PSOCT was performed during uniaxial tensile loading of fresh scleral specimens of rabbits, cows, and humans from limbal, equatorial, and peripapillary regions. Specimens from 8 human eyes were obtained. Specimens were stretched to failure at 0.01 mm/s constant rate under physiological conditions of temperature and humidity while birefringence was computed every 117 ms from cross-sectional PSOCT. Birefringence modulus (BM) was defined as the rate of birefringence change with strain, and tensile modulus (TM) as the rate of stress change between 0 and 9% strain. In cow and rabbit, BM and TM were positively correlated with slopes of 0.17 and 0.10 GPa, and with correlation coefficients 0.63 and 0.64 (P < 0.05), respectively, following stress-optic coefficients 4.69, and 4.20 GPa^-1. In human sclera, BM and TM were also positively correlated with slopes of 0.24 GPa for the limbal, 0.26 GPa for the equatorial, and 0.31 GPa for the peripapillary regions. Pearson correlation coefficients were significant at 0.51, 0.58, and 0.69 for each region, respectively (<0.001). Mean BM decreased proportionately to TM from the limbal to equatorial to peripapillary regions, as stress-optic coefficients were estimated as 2.19, 2.42, and 4.59 GPa^-1, respectively. Since birefringence and tensile elastic moduli correlate differently in cow, rabbit, and various regions of human sclera, it might be possible to mechanically characterize the sclera in vivo using PSOCT.

Atomic force microscopy determination of Young׳s modulus of bovine extra-ocular tendon fiber bundles

Extra-ocular tendons (EOTs) transmit the oculorotary force of the muscles to the eyeball to generate dynamic eye movements and align the eyes, yet the mechanical properties of the EOTs remain undefined. The EOTs are known to be composed of parallel bundles of small fibers whose mechanical properties must be determined in order to characterize the overall behavior of EOTs. The current study aimed to investigate the transverse Young׳s modulus of EOT fiber bundles using atomic force microscopy (AFM). Fresh bovine EOT fiber bundle specimens were maintained under temperature and humidity control, and indented 100nm by the inverted pyramid tip of an AFM (Veeco Digital Instruments, NY). Ten indentations were conducted for each of 3 different locations of 10 different specimens from each of 6 EOTs, comprising a total of 1800 indentations. Young׳s modulus for each EOT was determined using a Hertzian contact model. Young׳s moduli for fiber bundles from all six EOTs were determined. Mean Young׳s moduli for fiber bundles were similar for the six anatomical EOTs: lateral rectus 60.12±2.69 (±SD)MPa, inferior rectus 59.69±5.34MPa, medial rectus 56.92±1.91MPa, superior rectus 59.66±2.64MPa, inferior oblique 57.7±1.36MPa, and superior oblique 59.15±2.03. Variation in Young׳s moduli among the six EOTs was not significant (P>0.25). The Young׳s modulus of bovine EOT fibers is highly uniform among the six extraocular muscles, suggesting that each EOT is assembled from fiber bundles representing the same biomechanical elements. This uniformity will simplify overall modeling.

Determination of poisson ratio of bovine extraocular muscle by computed X-ray tomography

The Poisson ratio (PR) is a fundamental mechanical parameter that approximates the ratio of relative change in cross sectional area to tensile elongation. However, the PR of extraocular muscle (EOM) is almost never measured because of experimental constraints. The problem was overcome by determining changes in EOM dimensions using computed X-ray tomography (CT) at microscopic resolution during tensile elongation to determine transverse strain indicated by the change in cross-section. Fresh bovine EOM specimens were prepared. Specimens were clamped in a tensile fixture within a CT scanner (SkyScan, Belgium) with temperature and humidity control and stretched up to 35% of initial length. Sets of 500–800 contiguous CT images were obtained at 10-micron resolution before and after tensile loading. Digital 3D models were then built and discretized into 6–8-micron-thick elements. Changes in longitudinal thickness of each microscopic element were determined to calculate strain. Green's theorem was used to calculate areal strain in transverse directions orthogonal to the stretching direction. The mean PR from discretized 3D models for every microscopic element in 14 EOM specimens averaged 0.457 ± 0.004 (SD). The measured PR of bovine EOM is thus near the limit of incompressibility.

Components of the neural signal underlying congenital nystagmus

Congenital nystagmus is an involuntary bilateral horizontal oscillation of the eyes that develops soon after birth. In this study, the time constants of each of the components of the neural signal underlying congenital nystagmus were obtained by time series analysis and interpreted by comparison with those of the normal oculomotor system. In the neighbourhood of the fixation position, the system generating the neural signal is approximately linear with 3 degrees of freedom. The shortest time constant was in the range of 7-9 ms and corresponds to a normal saccadic burst signal. The other stable time constant was in the range of 22-70 ms and corresponds to the slide signal. The final time constant characterises the unidentified neural mechanism underlying the unstable drift component of the oscillation cycle and ranges between 31 and 32 ms across waveforms. The characterisation of this unstable time constant poses a challenge for the modelling of both the normal and abnormal oculomotor control system. We tentatively identify the unstable component with the eye position signal supplied to the superior colliculus in the normal eye movement system and explore some of the implications of this hypothesis.

A nonlinear model of passive muscle viscosity

The material properties of passive skeletal muscle are critical to proper function and are frequently a target for therapeutic and interventional strategies. Investigations into the passive viscoelasticity of muscle have primarily focused on characterizing the elastic behavior, largely neglecting the viscous component. However, viscosity is a sizeable contributor to muscle stress and extensibility during passive stretch and thus there is a need for characterization of the viscous as well as the elastic components of muscle viscoelasticity. Single mouse muscle fibers were subjected to incremental stress relaxation tests to characterize the dependence of passive muscle stress on time, strain and strain rate. A model was then developed to describe fiber viscoelasticity incorporating the observed nonlinearities. The results of this model were compared with two commonly used linear viscoelastic models in their ability to represent fiber stress relaxation and strain rate sensitivity. The viscous component of mouse muscle fiber stress was not linear as is typically assumed, but rather a more complex function of time, strain and strain rate. The model developed here, which incorporates these nonlinearities, was better able to represent the stress relaxation behavior of fibers under the conditions tested than commonly used models with linear viscosity. It presents a new tool to investigate the changes in muscle viscous stresses with age, injury and disuse.

The nonlinearity of passive extraocular muscles

Passive extraocular muscles (EOMs), like most biological tissues, are hyperelastic, that is, their stiffness increases as they are stretched. It has always been assumed, and in a few occasions argued, that this is their only nonlinearity and that it can be ignored in central gaze. However, using novel measurement techniques in anesthetized paralyzed monkeys, we have recently demonstrated that EOMs are characterized by another prominent nonlinearity: the forces induced by sequences of stretches do not sum. Thus, superposition, a central tenet of linear and quasi-linear models, does not hold in passive EOMs. Here, we outline the implications of this finding, especially in light of the common assumption that it is easier for the brain to control a linear than a nonlinear plant. We argue against this common belief: the specific nonlinearity of passive EOMs may actually make it easier for the brain to control the plant than if muscles were linear.

Creep behavior of passive bovine extraocular muscle

This paper characterized bovine extraocular muscles (EOMs) using creep, which represents long-term stretching induced by a constant force. After preliminary optimization of testing conditions, 20 fresh EOM samples were subjected to four different loading rates of 1.67, 3.33, 8.33, and 16.67%/s, after which creep was observed for 1,500 s. A published quasilinear viscoelastic (QLV) relaxation function was transformed to a creep function that was compared with data. Repeatable creep was observed for each loading rate and was similar among all six anatomical EOMs. The mean creep coefficient after 1,500 seconds for a wide range of initial loading rates was at 1.37 ± 0.03 (standard deviation, SD). The creep function derived from the relaxation-based QLV model agreed with observed creep to within 2.7% following 16.67%/s ramp loading. Measured creep agrees closely with a derived QLV model of EOM relaxation, validating a previous QLV model for characterization of EOM biomechanics.

Characterization of ocular tissues using microindentation and hertzian viscoelastic models

PURPOSE

The authors applied a novel microindentation technique to characterize biomechanical properties of small ocular and orbital tissue specimens using the hertzian viscoelastic formulation, which defines material viscoelasticity in terms of the contact pressure required to maintain deformation by a harder body.

METHODS

They used a hard spherical indenter having 100 nm displacement and 100 μg force precision to impose small deformations on fresh bovine sclera, iris, crystalline lens, kidney fat, orbital pulley tissue, and orbital fatty tissue; normal human orbital fat, eyelid fat, and dermal fat; and orbital fat associated with thyroid eye disease. For each tissue, stress relaxation testing was performed using a range of ramp displacements. Results for single displacements were used to build quantitative hertzian models that were, in turn, compared with behavior for other displacements. Findings in orbital tissues were correlated with quantitative histology.

RESULTS

Viscoelastic properties of small specimens of orbital and ocular tissues were reliably characterized over a wide range of rates and displacements by microindentation using the hertzian formulation. Bovine and human orbital fatty tissues exhibited highly similar elastic and viscous behaviors, but all other orbital tissues exhibited a wide range of biomechanical properties. Stiffness of fatty tissues tissue depended strongly on the connective tissue content.

CONCLUSIONS

Relaxation testing by microindentation is a powerful method for characterization of time-dependent behaviors of a wide range of ocular and orbital tissues using small specimens, and provides data suitable to define finite element models of a wide range of tissue interactions.

Disorders of vergence eye movements

PURPOSE OF REVIEW

The aim is to re-interpret disorders of vergence in the light of recent studies that view disjunctive eye movements as but one component of three-dimensional gaze control.

RECENT FINDINGS

Most natural eye movements combine vergence with saccades, pursuit and vestibular eye movements. Electrophysiological studies in epileptic patients, as well as evidence from monkeys, indicate that frontal and parietal cortex govern vergence as a component of three-dimensional gaze. Clinicians apply Hering's law of equal innervation to interpret disjunctive movements as the superposition of conjugate and vergence commands. However, electrophysiological studies indicate that disjunctive saccades are achieved by programming each eye's movement independently. Patients with internuclear ophthalmoplegia (INO) may have preserved vergence, which can be recruited to compensate for loss of conjugacy. Vergence may also enable gaze shifts in saccadic palsy. Some forms of nystagmus suppress or change with convergence; co-contraction of the horizontal rectus muscles does not appear to be the explanation. Rather, effects of near viewing on central vestibular mechanisms or differential activation of specific types of extra-ocular muscle fiber may be responsible.

SUMMARY

Interpretation of disorders of vergence is aided by applying a scheme in which their contributions to three-dimensional gaze control is considered.

The viscoelastic properties of passive eye muscle in primates. III: force elicited by natural elongations

We have recently shown that in monkey passive extraocular muscles the force induced by a stretch does not depend on the entire length history, but to a great extent is only a function of the last elongation applied. This led us to conclude that Fung's quasi-linear viscoelastic (QLV) model, and more general nonlinear models based on a single convolution integral, cannot faithfully mimic passive eye muscles. Here we present additional data about the mechanical properties of passive eye muscles in deeply anesthetized monkeys. We show that, in addition to the aforementioned failures, previous models also grossly overestimate the force exerted by passive eye muscles during smooth elongations similar to those experienced during normal eye movements. Importantly, we also show that the force exerted by a muscle following an elongation is largely independent of the elongation itself, and it is mostly determined by the final muscle length. These additional findings conclusively rule out the use of classical viscoelastic models to mimic the mechanical properties of passive eye muscles. We describe here a new model that extends previous ones using principles derived from research on thixotropic materials. This model is able to account reasonably well for our data, and could thus be incorporated into models of the eye plant.