Abstract

in the 1960s, which still is the basis for models of the ocular motor system. The focus is on the motor system, therefore, models of visual cortical mechanisms such as computation of motion from retinal sensory inputs will only briefly be touched upon. However, one should not forget that the question of how retinal input represented on retinotopic maps is neurally transformed by the brain to finally result in a motor command for an eye movement is an important aspect which should not be neglected. In the following, the various models will be presented in the reverse order, that is, the chapter begins with models focused on the biomechanics of the eye. Subsequently, models of the neural velocity-to- position integrator, which is common to all types of eye movements, are considered. Finally, models of the various types of eye movements and their neural control are presented.

Eye Plant

The term ‘eye plant’ covers the kinematic and dynamic behavior of the eye. Thus, models of the eye plant (for review, see also [1]) focus on the relationship between a motor command generated in the ocular motor nuclei of the brainstem and the resulting eye movement. Evidently, this transformation from motor command to eye movement is determined by the biomechanics of the eye globe, the extraocular eye muscles, the muscle pulleys (connective tissue pulleys that serve as the functional mechanical origin of the muscles), and the orbital tissues [see Demer, this vol, pp 132–157]. Most models focusing on the eye plant explicitly deal with the 3-D geometry and kinematics of the eye, and with specific properties of the plant such as the force-length relationship of the muscles or the placement of the pulleys. In contrast, models dealing with the neural control implemented in brainstem structures and above very often treat the eye plant as a lumped element. Two types of eye plant models can be distinguished: static models, concerned with the anatomy of the eye plant, and dynamic models, also considering the temporal properties involved (e.g. time constants of the eye plant).

Static models, derived from Robinson’s work [2, 3] have resulted in software packages, i.e. Orbit [4], SEE [5], designed to help the ophthalmologist, for example, in strabismus surgery. Other authors have designed static models to evaluate the role of the eye plant in Listing’s law [6–9]. For a review on Listing’s law, see Wong [10]. This question is closely related to the problem of noncommutativity of 3-D rotations. From these theoretical studies, especially after the existence of muscle pulleys was established [see Demer, this vol, pp 132–157], it was concluded that, given specific pulley configurations, Listing’s law (i.e. if eye orientation is expressed as rotation vectors or quaternions,

torsion depends linearly on gaze direction) may be implemented by the eye plant. In other words, a 2-D innervation of the six extraocular eye muscles would be sufficient to achieve the torsional eye orientations required by Listing’s law (see also below) in tertiary positions (off the horizontal and vertical meridians). This view has recently been supported by recordings from the motoneurons during smooth pursuit [11]. This does not mean that the eye plant constricts eye movements to obey Listing’s law, but it simplifies its implementation to a great extent.

Dynamics have been implemented mostly in simplified, lumped eye plant models [12–16], since detailed experimental studies of the 3-D dynamics have been missing. Recently, the dynamics of the eye plant have been re-evaluated [17], suggesting that in contrast to previous assumptions of a dominant time constant of 200 ms, the dynamics have to be described by a wide range of time constants ranging from about 10 ms to 10 s. A possibly more severe shortcoming of the lumped eye plant models is that they do not account for the fact that muscle force is a function of innervation and length. According to a more realistic model of 3-D dynamics [1], this leads to passive eye position-dependent torque that has to be compensated for by additional innervation. Thus, while models using simplified eye plant approximations are useful and valid in many cases, a more adequate implementation of the eye plant will be necessary to fully understand the neural mechanisms controlling eye movements.

The Neural Velocity-to-Position Integrator

Together with the ocular motor nuclei in the brainstem, the neural velocity- to-position integrator [for review, see 18] forms the final neural structure common to all types of eye movements. The neural commands for eye movements, which are also sent to the ocular motor nuclei, consist of phasic signals coding eye velocity (e.g. the saccadic burst command). However, if this were the only signal sent to the muscles, the eye would not remain in an eccentric position, but drift back to the equilibrium position determined by the eye plant. Therefore, an additional signal is necessary to generate the tonic muscle force to hold the eye. This signal comes from the neural velocity-to-position integrators located in the brainstem (nucleus prepositus hypoglossi and medial vestibular nucleus) for horizontal eye movements and the midbrain (interstitial nucleus of Cajal) for vertical eye movements. Additionally, the cerebellar flocculus plays an important role in neural integration in mammals, as shown by lesion experiments in different species such as rats, cats, and nonhuman primates [19]. As for the eye plant, many models consider the neural integrator as lumped element, which is described by a so-called leaky integrator with a time constant of more than 2 s

for primates, which determines the residual centripetal drift. This lumped description is useful and valid for models interested in other aspects of the ocular motor system. However, it does not allude as to how the integrator is implemented neurally, or which additional properties it may need.

Specifically when considering 3-D eye movements, it has been shown that simply using three leaky integrators (as an extension to 1-D models) may not suffice depending on the coding of velocity information to be integrated, because 3-D rotations do not commute. This poses a problem especially for the vestibulo-ocular reflex (VOR): the semicircular canal afferent signal codes angular velocity, but the integral of angular velocity does not yield orientation [15]. This problem can, however, be circumvented if the signal to be integrated is first converted to the derivative of eye orientation (which is not angular velocity). Thus, in such case, a commutative integrator composed of three parallel 1-D integrators can be used [13, 15, 16], and will produce a correct tonic signal to hold the eye eccentrically, given that the eye plant has the property of converting this neural command to actual eye orientation. Such a configuration will also maintain the eye orientation in Listing’s plane if the command is 2-D. Notably, as mentioned above, eye movements violating Listing’s law (e.g. during the VOR, or during active eye-head gaze shifts) are still possible, but necessarily require a full 3-D neural command. Additionally, Listing’s law is modified by vergence and head tilt. Such a modification requires changes in the central nervous commands, either by altering the pulley configuration or the commands sent to the extraocular muscles. Therefore, an extension to the neural integrator scheme has been proposed which incorporates additional input from the otoliths to achieve accurate fixations during head tilt [20, 21].

The neural implementation of the integration is the topic of a considerable number of studies. It has been suggested that a network of reciprocal inhibition forms a positive feedback loop which effectively prolongs the short time constants of single neurons to the desired long time constant of the integrating network [18, 22, 23]. Other related models proposed that the positive feedback loop forming the integrator is excitatory and contains an internal model of the eye plant dynamics [24, 25]. One of the problems of the original reciprocal feedback hypothesis was that fine tuning of the synaptic strength is implausible given that membrane time constants of about 5 ms have to be extended to the 20 s of the network [26]. A possible solution [27] is that the intrinsic time constant of processing is determined by synaptic time constants with values around 100 ms (corresponding to NMDA receptors). Alternative models suggest that single cell properties determine integration [28, 29].

While the models above mostly assume that the known integrator brainstem regions exclusively perform the integration, it has been shown by several studies that, in mammals, lesions of the cerebellar floccular lobe or the parts of