A two dimensional model for saccade generation

A model for the generation of oblique saccades is constructed by extending and modifying the one dimensional local feedback model. It is proposed that the visual system stores target location in inertial coordinates, but that the feedback loop which guides saccades works in retinotopic coordinates. To achieve straight trajectories for centripetal and centrifugal saccades in all meridians, a comparator computes motor error as a vector and uses the vectorial error signal to drive two orthogonally-acting burst generators. The generation of straight saccade trajectories when the extraocular muscles are of unequal strengths requires the introduction of a burst-tonic cell input to motor neurons. The model accounts for the results of two-site stimulation of the superior colliculus and frontal eye fields by allowing simultaneous activation of more than one comparator. The postulated existence of multiple comparators suggests that motor error may be computed topographically.     

Modeling dynamical interactions between fast and slow movements: fast saccadic eye movement behavior in the presence of the slower VOR

An ongoing controversy has to do with the interactions between “fast” (saccadic, quick phase) and “slow” (all other) eye movements. By attacking such issues with both experimental and especially simulation studies using our nonlinear sixth order reciprocally innervated model of the eye mechanical system, insights can be gained into the nature of these nontrivial phenomena. In our present study we relied both (1) on simulation of saccades under a wide range of experimental conditions [vestibular ocular reflex (VOR) velocities from -100 to 100 deg/sec, VOR induced position ranges from -30 to 30 degrees, time-optimal saccades ranging from 2 to 40 degrees], and (2) on using a wide variety of computer simulation of eye movement models, ranging from nonlinear ones with first and especially second order multipulse step controller signal structures, to different controller signal interaction schemes, to simulation using linearized models. We have isolated two important nonlinear phenomena: a level I nonlinear mechanical interaction, dependent not only on the initial velocity but also on the “position effect,” a new finding; and a level II nonlinear neurological interaction, close to “squelching” of the VOR controller signals by the dominating saccadic signal. Furthermore, we have used our simulation findings to reinterpret others' experimental data on eye movement interactions, including saccadic-smooth pursuit, saccadic-vergence, and vestibular nystagmus.     

A model that integrates eye velocity commands to keep track of smooth eye displacements

Past results have reported conflicting findings on the oculomotor system’s ability to keep track of smooth eye movements in darkness. Whereas some results indicate that saccades cannot compensate for smooth eye displacements, others report that memory-guided saccades during smooth pursuit are spatially correct. Recently, it was shown that the amount of time before the saccade made a difference: short-latency saccades were retinotopically coded, whereas long-latency saccades were spatially coded. Here, we propose a model of the saccadic system that can explain the available experimental data. The novel part of this model consists of a delayed integration of efferent smooth eye velocity commands. Two alternative physiologically realistic neural mechanisms for this integration stage are proposed. Model simulations accurately reproduced prior findings. Thus, this model reconciles the earlier contradictory reports from the literature about compensation for smooth eye movements before saccades because it involves a slow integration process. Action Editor: Jonathan D. Victor     

A Mathematical Model of Optimal Saccadic Eye Movement by a Pair of Muscles

This paper presents a model of saccadic eye movements. Eye movements are considered as being ballistic, since saccades (rapid concurrent movements of both eyes) occur several hundred thousand times per day; visual perception of the environment is interrupted by a saccade. The optimal control was constructed for the motion considered in three consecutively refined assumptions. The controls included in the time-optimal problem were the resultant moment of force exerted by the extraocular muscles, individual moments of force exerted by either muscle of the agonist–antagonist pair, and finally, the rate of change of these moments. This approach is consistent with the view that is currently upheld by physiologists, who believe that a saccade is programmed by the central nervous system before the beginning of an eye movement and is scarcely adjusted during the movement itself. The solution of the optimal control problem and the results obtained by subsequent numerical modeling of saccadic trajectories were compared with the published experimental data. The saccadic trajectories were compared based on the main sequence, the known consistent relationship between saccade amplitude and duration, which is the most widely applied and commonly accepted way of describing saccade data. The main sequence of saccades obtained from the solution of the optimal control problem formulated in the most complete form agreed well with published experimental results.     

Open-loop simulations of the primate saccadic system using burst cell discharge from the superior colliculus

 Saccade-related burst neurons (SRBNs) in the monkey superior colliculus (SC) have been hypothesized to provide the brainstem saccadic burst generator with the dynamic error signal and the movement initiating trigger signal. To test this claim, we performed two sets of open-loop simulations on a burst generator model with the local feedback disconnected using experimentally obtained SRBN activity as both the driving and trigger signal inputs to the model. First, using neural data obtained from cells located near the middle of the rostral to caudal extent of the SC, the internal parameters of the model were optimized by means of a stochastic hill-climbing algorithm to produce an intermediate-sized saccade. The parameter values obtained from the optimization were then fixed and additional simulations were done using the experimental data from rostral collicular neurons (small saccades) and from more caudal neurons (large saccades); the model generated realistic saccades, matching both position and velocity profiles of real saccades to the centers of the movement fields of all these cells. Second, the model was driven by SRBN activity affiliated with interrupted saccades, the resumed eye movements observed following electrical stimulation of the omnipause region. Once again, the model produced eye movements that closely resembled the interrupted saccades produced by such simulations, but minor readjustment of parameters reflecting the weight of the projection of the trigger signal was required. Our study demonstrates that a model of the burst generator produces reasonably realistic saccades when driven with actual samples of SRBN discharges. Received: 25 October 1994/Accepted in revised form: 20 June 1995     

Saccadic eye movements and the detection of fast-moving gratings

Experiments are presented in which the effect of saccadic eye movements on the visibility of sinusoidal gratings drifting with velocities between 2 deg/s and 400 deg/s is investigated. The results demonstrate that saccades are highly useful for detecting this class of stimuli. Due to a saccade, otherwise subthreshold stimuli become visible as short, distinct flashes of the seemingly statinoary pattern. The paper analyzes in detail the dependence of the amount of facilitation on saccade size and relative direction and isolates the additional effect of saccadic suppression. A simple model is proposed which predicts the experimental findings.     

Ambivalence in modelling oblique saccades

Human oblique saccades might be made by synchronized but independent vertical and horizontal pulse generators; we call this the Cartesian theory. Another hypothesis is that the oblique amplitude and angle are coded centrally and trigonometricallyderived signals are sent to the horizontal and vertical muscles (the polar coordinate theory). We took a Cartesian model and cross-coupled the two generators to produce saccades identical to those of a polar coordinate model. This is disproof by counterexample: the experimental evidence claimed to support the polar coordinate model does not necessarily do so. Moreover, the behavior reported for oblique saccades is so variable, contradictory, idiosyncratic, and species-dependent that any model of the central organization of oblique saccades is probably premature.     

Dysfunctional mode switching between fixation and saccades: collaborative insights into two unusual clinical disorders

Voluntary rapid eye movements (saccades) redirect the fovea toward objects of visual interest. The saccadic system can be considered as a dual-mode system: in one mode the eye is fixating, in the other it is making a saccade. In this review, we consider two examples of dysfunctional saccades, interrupted saccades in late-onset Tay-Sachs disease and gaze-position dependent opsoclonus after concussion, which fail to properly shift between fixation and saccade modes. Insights and benefits gained from bi-directional collaborative exchange between clinical and basic scientists are emphasized. In the case of interrupted saccades, existing mathematical models were sufficiently detailed to provide support for the cause of interrupted saccades. In the case of gaze-position dependent opsoclonus, existing models could not explain the behavior, but further development provided a reasonable hypothesis for the mechanism underlying the behavior. Collaboration between clinical and basic science is a rich source of progress for developing biologically plausible models and understanding neurological disease. Approaching a clinical problem with a specific hypothesis (model) in mind often prompts new experimental tests and provides insights into basic mechanisms.

     

A neural mechanism for discrimination III: Visually perceived lengths and distances

A previously discussed neural mechanism for the discrimination of intensities is here applied to the judgment of visual lengths and distances on the assumption that the “intensity” associated with the magnitude being perceived is the intensity of innervation of the appropriate eye muscles necessary for scanning and fixating. Comparison with experimental data is made in the case of the judgment of lengths.     

Blink Perturbation Effects on Saccades Evoked by Microstimulation of the Superior Colliculus

Current knowledge of saccade-blink interactions suggests that blinks have paradoxical effects on saccade generation. Blinks suppress saccade generation by attenuating the oculomotor drive command in structures like the superior colliculus (SC), but they also disinhibit the saccadic system by removing the potent inhibition of pontine omnipause neurons (OPNs). To better characterize these effects, we evoked the trigeminal blink reflex by delivering an air puff to one eye as saccades were evoked by sub-optimal stimulation of the SC. For every stimulation site, the peak and average velocities of stimulation with blink movements (SwBMs) were lower than stimulation-only saccades (SoMs), supporting the notion that the oculomotor drive is weakened in the presence of a blink. In contrast, the duration of the SwBMs was longer, consistent with the hypothesis that the blink-induced inhibition of the OPNs could prolong the window of time available for oculomotor commands to drive an eye movement. The amplitude of the SwBM could also be larger than the SoM amplitude obtained from the same site, particularly for cases in which blink-associated eye movements exhibited the slowest kinematics. The results are interpreted in terms of neural signatures of saccade-blink interactions.     

Neural network simulations of the primate oculomotor system III. An one-dimensional, one-directional model of the superior colliculus

This report evaluates the performance of a biologically motivated neural network model of the primate superior colliculus (SC). Consistent with known anatomy and physiology, its major features include excitatory connections between its output elements, nigral gating mechanisms, and an eye displacement feedback of reticular origin to recalculate the metrics of saccades to memorized targets in retinotopic coordinates. Despite the fact that it makes no use of eye position or eye velocity information, the model can account for the accuracy of saccades in double step stimulation experiments. Further, the model accounts for the effects of focal SC lesions. Finally, it accounts for the properties of saccades evoked in response to the electrical stimulation of the SC. These include the approximate size constancy of evoked saccades despite increases of stimulus intensity, the fact that the size of evoked saccades depends on the time that has elapsed from a previous saccade, the fact that staircases of saccades are evoked in response to prolonged stimuli, and the fact that the size of saccades evoked in response to the simultaneous stimulation of two SC sites is the average of the saccades that are evoked when the two sites are separately stimulated. Received: 3 November 1997 / Accepted in revised form: 30 June 1998     

Multipulse controller signals

Although eye movement saccades are stereotyped, repeatable movements, the shape of the neural controller signal innervating the extraocular muscles is a matter of controversy. Different lines of evidence — single motoneuron recordings, electromyograms, and dynamics — lead to different conclusions. Although all agree that the controller is, in outline, a pulse-step of net activity, neither the pulse width nor shape of the trailing edge of the pulse is clear. We use a mathematical model of the eye and two extraocular muscles to link the dynamical data to the electrophysiological evidence. We conjecture a multipulse controller signal, based on the application of an optimality principle to our model. This multi-pulse controller signal raises new possibilities for resolution of the pulse shape ambiguities, and resolves the controversy over pulse width.     

Influence of timing variability between motor unit potentials on M-wave characteristics

The transient enlargement of the compound muscle action potential (M wave) after a conditioning contraction is referred to as potentiation. It has been recently shown that the potentiation of the first and second phases of a monopolar M wave differed drastically; namely, the first phase remained largely unchanged, whereas the second phase underwent a marked enlargement and shortening. This dissimilar potentiation of the first and second phases has been suggested to be attributed to a transient increase in conduction velocity after the contraction. Here, we present a series of simulations to test if changes in the timing variability between motor unit potentials (MUPs) can be responsible for the unequal potentiation (and shortening) of the first and the second M-wave phases. We found that an increase in the mean motor unit conduction velocity resulted in a marked enlargement and narrowing of both the first and second M-wave phases. The enlargement of the first phase caused by a global increase in motor unit conduction velocities was apparent even for the electrode located over the innervation zone and became more pronounced with increasing distance to the innervation zone, whereas the potentiation of the second phase was largely independent of electrode position. Our simulations indicate that it is unlikely that an increase in motor unit conduction velocities (accompanied or not by changes in their distribution) could account for the experimental observation that only the second phase of a monopolar M wave, but not the first, is enlarged after a brief contraction. However, the combination of an increase in the motor unit conduction velocities and a spreading of the motor unit activation times could potentially explain the asymmetric potentiation of the M-wave phases.     

A continuum of myofibers in adult rabbit extraocular muscle: force, shortening velocity, and patterns of myosin heavy chain colocalization

Extraocular muscle (EOM) myofibers do not fit the traditional fiber typing classifications normally used in noncranial skeletal muscle, in part, due to the complexity of their individual myofibers. With single skinned myofibers isolated from rectus muscles of normal adult rabbits, force and shortening velocity were determined for 220 fibers. Each fiber was examined for myosin heavy chain (MyHC) isoform composition by densitometric analysis of electrophoresis gels. Rectus muscle serial sections were examined for coexpression of eight MyHC isoforms. A continuum was seen in single myofiber shortening velocities as well as force generation, both in absolute force (g) and specific tension (kN/m(2)). Shortening velocity correlated with MyHCIIB, IIA, and I content, the more abundant MyHC isoforms expressed within individual myofibers. Importantly, single fibers with similar or identical shortening velocities expressed significantly different ratios of MyHC isoforms. The vast majority of myofibers in both the orbital and global layers expressed more than one MyHC isoform, with up to six isoforms in single fiber segments. MyHC expression varied significantly and unpredictably along the length of single myofibers. Thus EOM myofibers represent a continuum in their histological and physiological characteristics. This continuum would facilitate fine motor control of eye position, speed, and direction of movement in all positions of gaze and with all types of eye movements-from slow vergence movements to fast saccades. To fully understand how the brain controls eye position and movements, it is critical that this significant EOM myofiber heterogeneity be integrated into hypotheses of oculomotor control.     

Abnormal Fixational Eye Movements in Amblyopia

Purpose

Fixational saccades shift the foveal image to counteract visual fading related to neural adaptation. Drifts are slow eye movements between two adjacent fixational saccades. We quantified fixational saccades and asked whether their changes could be attributed to pathologic drifts seen in amblyopia, one of the most common causes of blindness in childhood.

Methods

Thirty-six pediatric subjects with varying severity of amblyopia and eleven healthy age-matched controls held their gaze on a visual target. Eye movements were measured with high-resolution video-oculography during fellow eye-viewing and amblyopic eye-viewing conditions. Fixational saccades and drifts were analyzed in the amblyopic and fellow eye and compared with controls.

Results

We found an increase in the amplitude with decreased frequency of fixational saccades in children with amblyopia. These alterations in fixational eye movements correlated with the severity of their amblyopia. There was also an increase in eye position variance during drifts in amblyopes. There was no correlation between the eye position variance or the eye velocity during ocular drifts and the amplitude of subsequent fixational saccade. Our findings suggest that abnormalities in fixational saccades in amblyopia are independent of the ocular drift.

Discussion

This investigation of amblyopia in pediatric age group quantitatively characterizes the fixation instability. Impaired properties of fixational saccades could be the consequence of abnormal processing and reorganization of the visual system in amblyopia. Paucity in the visual feedback during amblyopic eye-viewing condition can attribute to the increased eye position variance and drift velocity.     

Natural and drug-induced variations of velocity and duration of human saccadic eye movements: Evidence for a control of the neural pulse generator by local feedback

The present report considers goal directed human saccadic eye movements. It addresses the question how a given perceived target excentricity is transformed into the innervation pattern that creates the saccade to the target. More specifically, it investigates whether this pattern is an appropriately selected preprogram or whether it is continuously controlled by a local feedback loop that compares a non-visual eye position signal to the perceived target excentricity (a visual signal would be too slow). To this end, the relation between the accuracy of saccades aimed at a given target and their velocity and duration was examined. Duration and velocity were found to vary by as much as 60% while the amplitude showed no related variation and had an almost constant accuracy of about 90%. By administrating diazepam, the variability of saccade duration and velocity could be further increased, but still the amplitude remained almost constant. These results favour the hypothesis that saccadic innervation is controlled by a local feedback loop.This investigation was supported by Deutsche Forschungsgemeinschaft, SFB 70, Gruppe Ulm     

A competitive integration model of exogenous and endogenous eye movements

We present a model of the eye movement system in which the programming of an eye movement is the result of the competitive integration of information in the superior colliculi (SC). This brain area receives input from occipital cortex, the frontal eye fields, and the dorsolateral prefrontal cortex, on the basis of which it computes the location of the next saccadic target. Two critical assumptions in the model are that cortical inputs are not only excitatory, but can also inhibit saccades to specific locations, and that the SC continue to influence the trajectory of a saccade while it is being executed. With these assumptions, we account for many neurophysiological and behavioral findings from eye movement research. Interactions within the saccade map are shown to account for effects of distractors on saccadic reaction time (SRT) and saccade trajectory, including the global effect and oculomotor capture. In addition, the model accounts for express saccades, the gap effect, saccadic reaction times for antisaccades, and recorded responses from neurons in the SC and frontal eye fields in these tasks.     

Learning the optimal control of coordinated eye and head movements

Various optimality principles have been proposed to explain the characteristics of coordinated eye and head movements during visual orienting behavior. At the same time, researchers have suggested several neural models to underly the generation of saccades, but these do not include online learning as a mechanism of optimization. Here, we suggest an open-loop neural controller with a local adaptation mechanism that minimizes a proposed cost function. Simulations show that the characteristics of coordinated eye and head movements generated by this model match the experimental data in many aspects, including the relationship between amplitude, duration and peak velocity in head-restrained and the relative contribution of eye and head to the total gaze shift in head-free conditions. Our model is a first step towards bringing together an optimality principle and an incremental local learning mechanism into a unified control scheme for coordinated eye and head movements.     

Modeling Inter-trial Variability of Saccade Trajectories: Effects of Lesions of the Oculomotor Part of the Fastigial Nucleus

This study investigates the inter-trial variability of saccade trajectories observed in five rhesus macaques (Macaca mulatta). For each time point during a saccade, the inter-trial variance of eye position and its covariance with eye end position were evaluated. Data were modeled by a superposition of three noise components due to 1) planning noise, 2) signal-dependent motor noise, and 3) signal-dependent premotor noise entering within an internal feedback loop. Both planning noise and signal-dependent motor noise (together called accumulating noise) predict a simple S-shaped variance increase during saccades, which was not sufficient to explain the data. Adding noise within an internal feedback loop enabled the model to mimic variance/covariance structure in each monkey, and to estimate the noise amplitudes and the feedback gain. Feedback noise had little effect on end point noise, which was dominated by accumulating noise. This analysis was further extended to saccades executed during inactivation of the caudal fastigial nucleus (cFN) on one side of the cerebellum. Saccades ipsiversive to an inactivated cFN showed more end point variance than did normal saccades. During cFN inactivation, eye position during saccades was statistically more strongly coupled to eye position at saccade end. The proposed model could fit the variance/covariance structure of ipsiversive and contraversive saccades. Inactivation effects on saccade noise are explained by a decrease of the feedback gain and an increase of planning and/or signal-dependent motor noise. The decrease of the fitted feedback gain is consistent with previous studies suggesting a role for the cerebellum in an internal feedback mechanism. Increased end point variance did not result from impaired feedback but from the increase of accumulating noise. The effects of cFN inactivation on saccade noise indicate that the effects of cFN inactivation cannot be explained entirely with the cFN’s direct connections to the saccade-related premotor centers in the brainstem.