Time-optimal control of saccadic eye movements

Modeling kinematic variability reveals displacement and velocity based dual control of saccadic eye movements

Noise is a ubiquitous component of motor systems that leads to behavioral variability of all types of movements. Nonetheless, systems-based models investigating human movements are generally deterministic and explain only the central tendencies like mean trajectories. In this paper, a novel approach to modeling kinematic variability of movements is presented and tested on the oculomotor system. This approach reconciles the two prominent philosophies of saccade control: displacement-based control versus velocity-based control. This was achieved by quantifying the variability in saccadic eye movements and developing a stochastic model of its control. The proposed stochastic dual model generated significantly better fits of inter-trial variances of the saccade trajectories compared to existing models. These results suggest that the saccadic system can flexibly use the information of both desired displacement and velocity for its control. This study presents a potential framework for investigating computational principles of motor control in the presence of noise utilizing stochastic modeling of kinematic variability.

Enhancing Prediction of Forelimb Movement Trajectory through a Calibrating-Feedback Paradigm Incorporating RAT Primary Motor and Agranular Cortical Ensemble Activity in the Goal-Directed Reaching Task

Complete reaching movements involve target sensing, motor planning, and arm movement execution, and this process requires the integration and communication of various brain regions. Previously, reaching movements have been decoded successfully from the motor cortex (M1) and applied to prosthetic control. However, most studies attempted to decode neural activities from a single brain region, resulting in reduced decoding accuracy during visually guided reaching motions. To enhance the decoding accuracy of visually guided forelimb reaching movements, we propose a parallel computing neural network using both M1 and medial agranular cortex (AGm) neural activities of rats to predict forelimb-reaching movements. The proposed network decodes M1 neural activities into the primary components of the forelimb movement and decodes AGm neural activities into internal feedforward information to calibrate the forelimb movement in a goal-reaching movement. We demonstrate that using AGm neural activity to calibrate M1 predicted forelimb movement can improve decoding performance significantly compared to neural decoders without calibration. We also show that the M1 and AGm neural activities contribute to controlling forelimb movement during goal-reaching movements, and we report an increase in the power of the local field potential (LFP) in beta and gamma bands over AGm in response to a change in the target distance, which may involve sensorimotor transformation and communication between the visual cortex and AGm when preparing for an upcoming reaching movement. The proposed parallel computing neural network with the internal feedback model improves prediction accuracy for goal-reaching movements.

Slow saccades in cerebellar disease

Eye movements are frequently considered diagnostic markers indicating involvement of the cerebellum. Impaired amplitude of saccades (saccade dysmetria), impaired gaze holding function (horizontal or downbeat nystagmus), and interrupted (choppy) pursuit are typically considered hallmarks of cerebellar disorders. While saccade dysmetria is a frequently considered abnormality, the velocity of saccades are rarely considered part of the constellation of cerebellar involvement. Reduced saccade velocity, frequently called “slow saccades” are typically seen in a classic disorder of the midbrain called progressive supranuclear palsy. It is also traditionally diagnostic of spinocerebellar ataxia type 2. In addition to its common causes, the slowness of vertical saccades is not rare in cerebellar disorders. Frequently this phenomenology is seen in multisystem involvement that substantially involves the cerebellum. In this review we will first discuss the physiological basis and the biological need for high saccade velocities. In subsequent sections we will discuss disorders of cerebellum that are known to cause slowing of saccades. We will then discuss possible pathology and novel therapeutic strategies.

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

A linear homeomorphic eye movement model that produces 3D saccadic eye movements consistent with anatomical and physiological evidence is introduced in this second part of a two-paper sequence. Central to the model is the implementation of a time-optimal neural control strategy involving six linear muscle models that faithfully represent the dynamic characteristics of 3D saccades. The 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]. The neural input for each muscle is separately maintained while the effective pulling direction is modulated by its respective pulley. The results demonstrate that a time-optimal, 2D commutative neural controller, together with the pulley system, actively functions to implement Listing's law during both static and dynamic simulations and provide an excellent match with the experimental data. The parameters and neural input to the muscles are estimated using a time domain system identification technique from saccade data, with an excellent match between the model estimates and the data. A total of 20 horizontal, 5 vertical and 62 oblique saccades are analyzed.

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.

Contrasting speed-accuracy tradeoffs for eye and hand movements reveal the optimal nature of saccade kinematics

In contrast to hand movements, the existence of a neural representation of saccade kinematics is unclear. Saccade kinematics is typically thought to be specified by motor error/desired displacement and generated by brain stem circuits that are not penetrable to voluntary control. We studied the influence of instructed hand movement velocity on the kinematics of saccades executed without explicit instructions. When the hand movement was slow the saccade velocity decreased, independent of saccade amplitude. We leveraged this modulation of saccade velocity to study the optimality of saccades (in terms of velocity and endpoint accuracy) in relation to the well-known speed-accuracy tradeoff that governs voluntary movements (Fitts' law). In contrast to hand movements that obeyed Fitts' law, normometric saccades exhibited the greatest endpoint accuracy and lower reaction times, relative to saccades accompanying slow and fast hand movements. In the slow condition, where saccade endpoint accuracy suffered, we observed that targets were more likely to be foveated by two saccades resulting in step-saccades. Interestingly, the endpoint accuracy was higher in two-saccade trials, compared with one-saccade trials in both the slow and fast conditions. This indicates that step-saccades are a part of the kinematic plan for optimal control of endpoint accuracy. Taken together, these findings suggest normometric saccades are already optimized to maximize endpoint accuracy and the modulation of saccade velocity by hand velocity is likely to reflect the sharing of kinematic plans between the two effectors.NEW & NOTEWORTHY The optimality of saccade kinematics has been suggested by modeling studies but experimental evidence is lacking. However, we observed that, when subjects voluntarily modulated their hand velocity, the velocity of saccades accompanying these hand movements was also modulated, suggesting a shared kinematic plan for eye and hand movements. We leveraged this modulation to show that saccades had less endpoint accuracy when their velocity decreased, illustrating that normometric saccades have optimal speed and accuracy.

Dynamics-based motion de-blurring for a PZT-driven, compliant camera orientation mechanism

This paper proposes a method for removing motion blur from images captured by a fast-moving robot eye. Existing image techniques focused on recovering blurry images due to camera shake with long exposure time. In addition, previous studies relied solely on properties of the images or used external sensors to estimate a blur kernel, or point spread function (PSF). This paper focuses on estimating a latent image from the blur images taken by the robotic camera orientation system. A PZT-driven, compliant camera orientation system was employed to demonstrate the effectiveness of this approach. Discrete switching commands were given to the robotic system to create a rapid point-to-point motion while suppressing the vibration with a faster response. The blurry images were obtained when the robotic system created a rapid point-to-point motion, like human saccadic motion. This paper proposes a method for estimating the PSF in knowledge of system dynamics and input commands, resulting in a faster estimation. The proposed method was investigated under various motion conditions using the single-degree-of-freedom camera orientation system to verify the effectiveness and was compared with other approaches quantitatively and qualitatively. The experiment results show that overall the performance metric of the proposed method was 27.77% better than conventional methods. The computation time of the proposed method was 50 times faster than that of conventional methods.

A physiological neural controller of a muscle fiber oculomotor plant in horizontal monkey saccades

A neural network model of biophysical neurons in the midbrain is presented to drive a muscle fiber oculomotor plant during horizontal monkey saccades. Neural circuitry, including omnipause neuron, premotor excitatory and inhibitory burst neurons, long lead burst neuron, tonic neuron, interneuron, abducens nucleus, and oculomotor nucleus, is developed to examine saccade dynamics. The time-optimal control strategy by realization of agonist and antagonist controller models is investigated. In consequence, each agonist muscle fiber is stimulated by an agonist neuron, while an antagonist muscle fiber is unstimulated by a pause and step from the antagonist neuron. It is concluded that the neural network is constrained by a minimum duration of the agonist pulse and that the most dominant factor in determining the saccade magnitude is the number of active neurons for the small saccades. For the large saccades, however, the duration of agonist burst firing significantly affects the control of saccades. The proposed saccadic circuitry establishes a complete model of saccade generation since it not only includes the neural circuits at both the premotor and motor stages of the saccade generator, but also uses a time-optimal controller to yield the desired saccade magnitude.

The lognormal handwriter: learning, performing, and declining

The generation of handwriting is a complex neuromotor skill requiring the interaction of many cognitive processes. It aims at producing a message to be imprinted as an ink trace left on a writing medium. The generated trajectory of the pen tip is made up of strokes superimposed over time. The Kinematic Theory of rapid human movements and its family of lognormal models provide analytical representations of these strokes, often considered as the basic unit of handwriting. This paradigm has not only been experimentally confirmed in numerous predictive and physiologically significant tests but it has also been shown to be the ideal mathematical description for the impulse response of a neuromuscular system. This latter demonstration suggests that the lognormality of the velocity patterns can be interpreted as reflecting the behavior of subjects who are in perfect control of their movements. To illustrate this interpretation, we present a short overview of the main concepts behind the Kinematic Theory and briefly describe how its models can be exploited, using various software tools, to investigate these ideal lognormal behaviors. We emphasize that the parameters extracted during various tasks can be used to analyze some underlying processes associated with their realization. To investigate the operational convergence hypothesis, we report on two original studies. First, we focus on the early steps of the motor learning process as seen as a converging behavior toward the production of more precise lognormal patterns as young children practicing handwriting start to become more fluent writers. Second, we illustrate how aging affects handwriting by pointing out the increasing departure from the ideal lognormal behavior as the control of the fine motricity begins to decline. Overall, the paper highlights this developmental process of merging toward a lognormal behavior with learning, mastering this behavior to succeed in performing a given task, and then gradually deviating from it with aging.

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.

Human beings and many other species redirect their gaze towards targets of interest through rapid gaze shifts known as saccades. These are made approximately three to four times every second, and larger saccades result from fast and concurrent movement of the animal's eyes and head. Experimental studies have revealed that during saccades, the motor system follows certain principles such as respecting a specific relationship between the relative contribution of eye and head motor systems to total gaze shift. Various researchers have hypothesized that these principles are implications of some optimality criteria in the brain, but it remains unclear how the brain can learn such an optimal behavior. We propose a new model that uses a plausible learning mechanism to satisfy an optimality criterion. We show that after learning, the model is able to reproduce motor behavior with biologically plausible properties. In addition, it predicts the nature of the learning signals. Further experimental research is necessary to test the validity of our model.

Modeling trade-off between time-optimal and minimum energy in saccade main sequence

Saccadic eye movement is highly stereotyped and commonly believed to be governed by an open-loop control mechanism. We propose a principle combining time-optimal and minimum control energy criteria to account for the saccade main sequence as observed from empirical data. The model prediction revealed that the weighting factor of the energy conservation becomes more dominant than the time-optimal when the saccade amplitude is large. We demonstrate that the proposed model is a general form synthesizing the time-optimum, minimum torque change, and minimum control effort models. In addition, we show the connection between our model and the stochastic minimum variance models from the aspect of optimization.

The limit profile of a rapid movement velocity

In motor control, various theories and computational models have been developed to explain and model the stereotypical velocity profiles of skilled rapid movements. According to the fact that these theories aim at describing the same physical pattern (a velocity profile) with different mathematical expressions, some relationships between these various representation schemes should exist. This paper presents a comparative study of two motor control theories that have put forward analytical expressions to describe the stereotypical velocity profiles of rapid movements: the Kinematic Theory and the Minimization Theory. Among the various forms of the latter, the Minimum-Square-Derivatives (MSD) principle and the Minimum-Time model are analyzed. It is shown that their concepts are linked and describe, with different arguments, a paradigm similar to the one used in the Kinematic Theory to model a velocity profile with a Delta-Lognormal equation. This unifying paradigm represents the functioning of a neuromuscular system by the convolution product of an infinite number of subsystem impulse responses. A second finding emerging from the present study is that the analytical models of velocity profiles, as described by the minimum principles under study, correspond, with more or less accuracy, to an approximation of the Delta-Lognormal equation. Overall, the Kinematic Theory can be seen as relying on a general optimization principle and the use of the Minimization Theory in motor control gets new insights.

A new algorithm and system for the characterization of handwriting strokes with delta-lognormal parameters

In this paper, we present a new analytical method for estimating the parameters of Delta-Lognormal functions and characterizing handwriting strokes. According to the Kinematic Theory of rapid human movements, these parameters contain information on both the motor commands and the timing properties of a neuromuscular system. The new algorithm, called XZERO, exploits relationships between the zero crossings of the first and second time derivatives of a lognormal function and its four basic parameters. The methodology is described and then evaluated under various testing conditions. The new tool allows a greater variety of stroke patterns to be processed automatically. Furthermore, for the first time, the extraction accuracy is quantified empirically, taking advantage of the exponential relationships that link the dispersion of the extraction errors with its signal-to-noise ratio. A new extraction system which combines this algorithm with two other previously published methods is also described and evaluated. This system provides researchers involved in various domains of pattern analysis and artificial intelligence with new tools for the basic study of single strokes as primitives for understanding rapid human movements.

Optimal control of gaze shifts

To explore the visible world, human beings and other primates often rely on gaze shifts. These are coordinated movements of the eyes and head characterized by stereotypical metrics and kinematics. It is possible to determine the rules that the effectors must obey to execute them rapidly and accurately and the neural commands needed to implement these rules with the help of optimal control theory. In this study, we demonstrate that head-fixed saccades and head-free gaze shifts obey a simple physical principle, "the minimum effort rule." By direct comparison with existing models of the neural control of gaze shifts, we conclude that the neural circuitry that implements the minimum effort rule is one that uses inhibitory cross talk between independent eye and head controllers.

Saccadic eye movements minimize the consequences of motor noise

The durations and trajectories of our saccadic eye movements are remarkably stereotyped. We have no voluntary control over these properties but they are determined by the movement amplitude and, to a smaller extent, also by the movement direction and initial eye orientation. Here we show that the stereotyped durations and trajectories are optimal for minimizing the variability in saccade endpoints that is caused by motor noise. The optimal duration can be understood from the nature of the motor noise, which is a combination of signal-dependent noise favoring long durations, and constant noise, which prefers short durations. The different durations of horizontal vs. vertical and of centripetal vs. centrifugal saccades, and the somewhat surprising properties of saccades in oblique directions are also accurately predicted by the principle of minimizing movement variability. The simple and sensible principle of minimizing the consequences of motor noise thus explains the full stereotypy of saccadic eye movements. This suggests that saccades are so stereotyped because that is the best strategy to minimize movement errors for an open-loop motor system.

The main sequence of saccades optimizes speed-accuracy trade-off

In primates, it is well known that there is a consistent relationship between the duration, peak velocity and amplitude of saccadic eye movements, known as the 'main sequence'. The reason why such a stereotyped relationship evolved is unknown. We propose that a fundamental constraint on the deployment of foveal vision lies in the motor system that is perturbed by signal-dependent noise (proportional noise) on the motor command. This noise imposes a compromise between the speed and accuracy of an eye movement. We propose that saccade trajectories have evolved to optimize a trade-off between the accuracy and duration of the movement. Taking a semi-analytical approach we use Pontryagin's minimum principle to show that there is an optimal trajectory for a given amplitude and duration; and that there is an optimal duration for a given amplitude. It follows that the peak velocity is also fixed for a given amplitude. These predictions are in good agreement with observed saccade trajectories and the main sequence. Moreover, this model predicts a small saccadic dead-zone in which it is better to stay eccentric of target than make a saccade onto target. We conclude that the main sequence has evolved as a strategy to optimize the trade-off between accuracy and speed.

An optimization principle for determining movement duration

Movement duration is an integral component of motor control, but nearly all extant optimization models of motor planning prefix duration instead of explaining it. Here we propose a new optimization principle that predicts movement duration. The model assumes that the brain attempts to minimize movement duration under the constraint of meeting an accuracy criterion. The criterion is task and context dependent but is fixed for a given task and context. The model determines a unique duration as a trade-off between speed (time optimality) and accuracy (acceptable endpoint scatter). We analyzed the model for a linear motor plant, and obtained a closed-form equation for determining movement duration. By solving the equation numerically with specific plant parameters for the eye and arm, we found that the model can reproduce saccade duration as a function of amplitude (the main sequence), and arm-movement duration as a function of the ratio of target distance to size (Fitts's law). In addition, it explains the dependency of peak saccadic speed on amplitude and the dependency of saccadic duration on initial eye position. Furthermore, for arm movements, the model predicts a scaling relationship between peak velocity and distance and a reduction in movement duration with a moderate increase in viscosity. Finally, for a linear plant, our model predicts a neural control signal identical to that of the minimum-variance model set to the same movement duration. This control signal is a smooth function of time (except at the endpoint), in contrast to the discontinuous bang-bang control found in the time-optimal control literature. We suggest that one aspect of movement planning, as revealed by movement duration, may be to assign an endpoint accuracy criterion for a given task and context.

3D dynamic computer model of the head-neck complex

A 3D dynamic computer model for the movement of the head is presented that incorporates anatomically correct information about the diverse elements forming the system. The skeleton is considered as a set of interconnected rigid 3D bodies following the Newton-Euler laws of movement. The muscles are modeled using Enderle's linear model. Finally, the soft tissues, namely the ligaments, intervertebral disks, and zigapophysial joints, are modeled using the finite elements approach. The model is intended to study the neural network that controls movement and maintains the balance of the head-neck complex during eye movements.

Exploring smoothness and discontinuities in human motor behaviour with Fourier analysis

The popular notion that human movements are smooth appears to be in contradiction to the fact that point-to-point movements must necessarily have discontinuities of some finite order at their onset and possibly offset. We explore discontinuities in the Fourier domain and show that the order and total strength of discontinuities in a trajectory can be measured from the slope and intercept of the envelope of the energy spectrum at high frequencies. In linear system models, the order of discontinuity is constrained by the motor command discontinuity and the order of the motor plant. We deduce that trajectories such as the minimum jerk are not smooth, and may even be the least smooth trajectories possible for biologically plausible motor plants. We further examine the role of discontinuities in optimal control and show that minimum square derivative profiles (such as minimum jerk) are time-optimal trajectories. This leads to the notion that point-to-point movements are a trade-off between duration and discontinuity strength, possibly reflecting neural command intensity or signal-dependent noise.

Neural control of saccades

Quantitative models of the oculomotor plant and control of the saccadic eye movement system are presented in this chapter. Oculomotor plant models described here are linear, including a second-order model by Westheimer (1954), Bahill et al. (1980) and Enderle et al. (2000). The model of the saccade generator is initiated by the superior colliculus and terminated by the cerebellar fastigial nucleus that operates under a time optimal control strategy. A common mechanism for all types of saccades is described, including those with dynamic overshoot and glissadic behavior. Conflicting evidence exists regarding the operation of the excitatory burst neuron during saccades. The excitatory burst neuron operates within two states: complete inhibition, and without inhibition that is characterized by high firing at rates of up to 1000 Hz. While there is direct evidence of projections from the superior colliculus to the paramedian pontine reticular formation, there is conflictory evidence regarding the connections from the superior colliculus to the excitatory burst neuron, with the most recent experimental results supporting no direct connections. A model of the excitatory burst neuron is described using a Hodgkin-Huxley model of the neuron that fires at 1000 Hz automatically and without stimulation when released from inhibition. SIMULINK simulations using this neuron model have all of the characteristics of the excitatory burst neuron firing rate during a saccade. This model eliminates the need to introduce BIAS inputs that causes bursting in some models of the saccade generator. Such a model is also appropriate for modeling the Omnipause neurons.

Role of uncertainty in sensorimotor control

Neural signals are corrupted by noise and this places limits on information processing. We review the processes involved in goal-directed movements and how neural noise and uncertainty determine aspects of our behaviour. First, noise in sensory signals limits perception. We show that, when localizing our hand, the central nervous system (CNS) integrates visual and proprioceptive information, each with different noise properties, in a way that minimizes the uncertainty in the overall estimate. Second, noise in motor commands leads to inaccurate movements. We review an optimal-control framework, known as 'task optimization in the presence of signal-dependent noise', which assumes that movements are planned so as to minimize the deleterious consequences of noise and thereby minimize inaccuracy. Third, during movement, sensory and motor signals have to be integrated to allow estimation of the body's state. Models are presented that show how these signals are optimally combined. Finally, we review how the CNS deals with noise at the neural and network levels. In all of these processes, the CNS carries out the tasks in such a way that the detrimental effects of noise are minimized. This shows that it is important to consider effects at the neural level in order to understand performance at the behavioural level.

The influence of repetitive eye movements on vergence performance

We measured the peak velocity of convergence eye movement responses in four normal subjects before and after a large number of either repetitive vergence or repetitive saccadic eye movements. A 20% decrease in the mean value of peak velocity was observed in vergence responses after 100 repetitive step vergence eye movements. However, 100 cycles of slow sinusoidal vergence tracking did not induce any notable change in vergence dynamics. Five hundred repetitive saccadic eye movements also caused an approximately 20% decrease in peak velocity. The reduction in peak velocity was related to the number of repetitions for both vergence and saccadic fatiguing stimuli. The frequency of occurrence of double-vergences was also used as an index to monitor the influence of repetitive eye movements on convergence performance. Results showed that repetitive step convergence movements could double, or even triple, the frequency of the occurrence of double-vergence responses, while slow sinusoidal vergence tracking or repetitive saccades had no influence on the frequency of response doubles.

The spectral main sequence of human saccades

Despite the many models of saccadic eye movements, little attention has been paid to the shape of saccade trajectories. Some investigators have argued that saccades are driven by a rectangular "bang-bang" neural control signal, whereas others have emphasized the similarity to fast arm movement trajectories, such as the "minimum jerk" profile. However, models have not been tested rigorously against empirical trajectories. We examined the Fourier transforms of saccades and compared them with theoretical models. Horizontal saccades were recorded from 10 healthy subjects. The Fourier transform of each saccade was accurately computed using a padded fast Fourier transform (FFT), and the frequencies of the first three minima (M1, M2, M3) in each energy spectrum were measured to a precision of 0.12 Hz. Each subject showed near-linear trends in the relationships among M1, M2, and M3 and the reciprocal of duration (1/T), which we call the "spectral main sequence." Extrapolation of plots did not pass through the origin, indicating a subtle departure from self-similarity. Bivariate confidence regions were established to allow for slope-intercept variability. The nonharmonic relationships seen cannot arise from a rectangular saccadic pulse driving a linear ocular plant. The relationships are also incompatible with minimum acceleration, minimum jerk, or higher-order minimum square derivative trajectories. The best fits were made by trajectories that minimize postmovement variance with signal-dependent noise (). It is concluded that the spectral main sequence is exquisitely sensitive to the saccade trajectory and should be used to test objectively all present and future models of saccades.

On the optimal control of behaviour: a stochastic perspective

Evolution is a closed stochastic optimisation process driven by the interaction between behaviour and environment towards local maxima in fitness. It is inferred that nervous systems are selected to provide optimal control of behaviour (the 'assumption of optimality'), such that for some behaviours, the expectation of future hazards to survival are minimised. This is illustrated by goal-directed saccades in which minimising total flight-time of primary and secondary movements provides a better fit to observations than simply minimising the error of the primary movement. This optimisation is extended to intra-movement trajectories, where low-bandwidth (smooth) velocity profiles provide a more satisfactory description of observations than simple bang-bang control. Since minimum-time behaviours cannot be controlled by error feedback, it is concluded that the cerebellum must be executing a real-time unreferenced optimisation process. This requires explorative as well as exploitative behaviour. Stochastic gradient descent is discussed as a possible means by which the cerebellum may optimise behaviour.

Signal-dependent noise determines motor planning

When we make saccadic eye movements or goal-directed arm movements, there is an infinite number of possible trajectories that the eye or arm could take to reach the target. However, humans show highly stereotyped trajectories in which velocity profiles of both the eye and hand are smooth and symmetric for brief movements. Here we present a unifying theory of eye and arm movements based on the single physiological assumption that the neural control signals are corrupted by noise whose variance increases with the size of the control signal. We propose that in the presence of such signal-dependent noise, the shape of a trajectory is selected to minimize the variance of the final eye or arm position. This minimum-variance theory accurately predicts the trajectories of both saccades and arm movements and the speed-accuracy trade-off described by Fitt's law. These profiles are robust to changes in the dynamics of the eye or arm, as found empirically. Moreover, the relation between path curvature and hand velocity during drawing movements reproduces the empirical 'two-thirds power law. This theory provides a simple and powerful unifying perspective for both eye and arm movement control.

Frequency response analysis of human saccadic eye movements: estimation of stochastic muscle forces

A frequency response method is used to estimate parameters of a fourth-order model of the oculomotor system and the active state tensions during a saccadic eye movement. The lateral and medial rectus muscle of each eye is modeled as a parallel combination of an active state tension generator with a viscosity and elastic element, connected to a series elastic element. The eyeball is modeled as a sphere connected to a viscosity and elastic element. Each of these elements is assumed to be ideal and linear. The active state tension for each muscle is modeled by a low-pass filtered pulse-step waveform. Initial estimates of the oculomotor mechanical components are based on physiological evidence. Initial estimates of the active state tension are based on an extrapolation of the eye movement trajectory. Horizontal saccadic eye movements were recorded from infrared signals reflected from the anterior surface of the cornea and then digitized. Parameter estimates were calculated for the model by using a conjugate gradient search program which minimizes the integral of the absolute value of the squared error between the model and the data. The predictions of the model are shown to be in good agreement with the data. Final estimates of motoneuronal activity demonstrate that the agonist muscle is maximally stimulated during the early portion of a saccadic eye movement regardless of the amplitude of the saccade; only the duration of the maximal stimulation affects the size of the saccade. The antagonist muscle is completely inhibited during the period of maximum agonist muscle stimulation. Furthermore, it is demonstrated that saccade motoneuronal activity is a stochastic phenomenon.