European Journal of Neuroscience, Vol. 19, pp. 1±10, 2004

ß Federation of European Neuroscience Societies

REVIEW ARTICLE Control of eye orientation: where does the brain's role end and the muscle's begin? Dora E. Angelaki1 and Bernhard J. M. Hess2 1 2

Department of Neurobiology, Washington University School of Medicine, 660 South Euclid Avenue, St Louis, MO 63110, USA Department of Neurology, University Hospital Zurich, CH-8091 Zurich, Switzerland

Keywords: computation, eye movement, kinematics, modelling, motor control, pursuit, saccades, torsion, vestibuloocular re¯ex

Abstract Our understanding of how the brain controls eye movements has bene®ted enormously from the comparison of neuronal activity with eye movements and the quanti®cation 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 in¯uence 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 suf®cient to generate correct 3-D eye orientation and movement. In contrast to this apparent simpli®cation 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 re¯exes, 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 ®rst 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.

Introduction Understanding the neural organization of motor systems has bene®ted enormously from a combined approach of modelling, behavioural and single-unit electrophysiological techniques. The close interplay between system analysis approaches and neurophysiology has led in the 1970s and 80s to a revolution in our understanding of the neural control of eye movements (Robinson, 1968, 1970, 1981). The easy accessibility of oculomotor-related brainstem and cerebellar areas using standard electrophysiological recording techniques in alert primates has provided a model motor system whose organization has become well understood in both health and disease. In most of these early pioneering neurophysiological and modelling studies, emphasis was primarily placed on the dynamic aspects of eye movements, largely ignoring their spatial organization. Accordingly, most quantitative motoneuron and premotor single-neuron studies have concentrated on either horizontal or vertical eye movements (Fuchs & Luschei, 1970, 1971; Robinson, 1970; Henn & Cohen, 1973; Pola & Robinson, 1978; Van Gisbergen et al., 1981; Henn et al., 1982; Tomlinson & Robinson, 1984; Fuchs et al., 1985, 1988; Hepp & Henn, 1985; Delgado-Garcia et al., 1986; De la Cruz et al., 1989; Scudder & Fuchs, 1992; Cullen & Guitton, 1997; Sylvestre & Cullen, 1999; Cullen et al., 2000). The restriction to the horizontal±vertical dimension, however, does not do justice to the full movement range of the Correspondence: Dr Dora Angelaki, as above. E-mail: [email protected] Received 27 June 2003, revised 3 October 2003, accepted 3 October 2003 doi:10.1046/j.1460-9568.2003.03068.x

eyes, each of which is controlled by three pairs of muscles and capable of rotating in three dimensions (3-D). Only a handful of neurophysiological studies have taken into account 3-D eye orientation and movement when evaluating neuronal discharges (Angelaki & Dickman, 2003; Van Opstal et al., 1991; Hepp et al., 1993; Van Opstal et al., 1996; Suzuki et al., 1999; Scherberger et al., 2001). Whereas a onedimensional dynamic system analysis approach was suf®cient to bridge the gap between neural recording and the observed 2-D oculomotor behaviour, it falls short of addressing the more complicated kinematics of 3-D rotations (Goldstein, 1980). Although the computationally intriguing but nonintuitive 3-D kinematic issues have motivated numerous experimental and theoretical studies, they still remain in the forefront of current controversies in recent oculomotor research. In this review we attempt to summarize the current status of the accumulated knowledge and address the controversies related to the control of 3-D eye orientation and movement.

Strategies specifying the third degree of freedom of ocular rotations: full field vs. foveal vision For ocular movements with the goal to stabilize images on the whole retina, the brain has to unambiguously specify all three degrees of freedom of the eye, which determine 3-D ocular orientation. One such example is the rotational vestibulo-ocular re¯ex (RVOR), which causes a compensatory eye rotation that is matched in angular velocity and direction to head rotation. Accordingly, not just gaze direction, which is de®ned by the horizontal and vertical components of the eye

2 D. E. Angelaki and B. J. M. Hess movement, but also the torsion of the eyes, are proportional to the respective three-rotational components of the head movement (Misslisch et al., 1994; Misslisch & Hess, 2000). Such a strategy results in stabilization of the full visual ®eld. On the other hand, if the functional goal of an eye movement is to stabilize binocular gaze and keep images in a particular depth plane stable on the foveae, such as during pursuit of a small moving target, it suf®ces to control the two degrees of freedom of gaze direction (of each eye), leaving ocular torsion unspeci®ed. Among the in®nite possible 3-D orientations which the eyeball could adopt, it is now well-established that the torsional orientation of the eye is uniquely speci®ed by gaze direction and vergence, a strategy which is appropriate for foveal and stereo vision (Schreiber et al., 2001). Accordingly, both smooth pursuit and saccadic eye movements obey robust kinematic constraints known as Listing's law (Tweed & Vilis, 1987, 1990; Haslwanter et al., 1991; Tweed et al., 1992): when expressed in a head-®xed coordinate system, all eye orientations during pursuit and saccades are con®ned to a single plane, referred to as `Listing's plane' (Ferman et al., 1987; Tweed & Vilis, 1987, 1990). The orientation of this plane depends on the reference position used to express eye orientations. Among all possible eye orientations, there is a unique reference position, known as `primary position', for which the plane orientation coincides with the horizontal±vertical plane (i.e. torsion is zero; Fig. 1). An example of the spatial distribution of eye positions for pursuit and saccades is illustrated in Fig. 2, where 3-D eye orientation during spontaneous eye movements in the light (Fig. 2A) and visually guided pursuit at different vertical or horizontal eccentricities (Fig. 2B) are plotted as the corresponding projections onto the pitch (top) or frontal (bottom) plane. During both saccades and pursuit, the torsional excursions are small (having a SD1), unlike the large oculomotor range in horizontal and vertical directions. The orientation of eye position planes during both pursuit and saccades exhibits a small but systematic dependence on static and dynamic head orientation relative to gravity (Haslwanter et al., 1992; Hess & Angelaki, 1997a,b, 2003). The Listing's law strategy provides a solution to the unspeci®ed thirddegree of freedom problem during fovea-related eye movements whose goal is to control the direction of gaze (de®ned only by the

Fig. 1. Primary position and Listing's plane. There is a unique orientation of the eye, called `primary position' or `primary gaze direction' (direction parallel to x-axis), such that pure vertical and pure horizontal eye movements that move the eye or gaze line from primary to secondary positions, do not change ocular torsion (eye rotations along the respective meridians through A or through C). Similarly, any movement that rotates the eye/gaze line from primary to tertiary positions in oblique meridian planes does not change torsion (e.g. movements along the meridians through B or D). The axes of single rotations that move the eye from primary to secondary or tertiary positions lie all in one plane, called Listing's plane (plane containing y- and z-axes). Tertiary positions cannot be reached from secondary positions by any combination of horizontal and vertical ocular rotations (a torsional component is also needed; see half-angle rule).

Fig. 2. Spatial plots of three-dimensional eye positions during (A) spontaneous ®xations and/or saccades and (B) smooth pursuit eye movements. (Top) Sagittal plane, where horizontal (Ehor) are plotted vs. torsional (Etor) eye positions. (Bottom) Frontal plane, where horizontal (Ehor) are plotted vs. vertical (Ever) eye positions (see schematic monkey heads). In B, eye movements were recorded during horizontal and vertical pursuit (0.1 Hz, 158) at different eccentricities. All eye positions have been expressed relative to the primary position (dashed lines) which is in general different from the straight-ahead position.

horizontal and vertical components of eye position) with little regard for the peripheral retina. In contrast, because of the goal to stabilize images on the whole retina, the RVOR does not follow Listing's law (Misslisch et al., 1994; Misslisch & Hess, 2000). Perhaps the most fundamental but relatively nonintuitive aspect of 3-D kinematics is that in general a gaze-dependent torsional eye velocity component is necessary to keep ocular orientation in Listing's plane during eye movements. Whenever gaze changes direction during saccades or smooth pursuit, the axis about which the eye actually rotates tilts in an eye-position-dependent manner out of Listing's plane with the effect of keeping ocular orientation aligned with Listing's plane. This important kinematic effect has been called the velocity criterion of Listing's law (von Helmholtz, 1867) and is usually referred to as the `half-angle rule' (Fig. 3; Tweed & Vilis, 1987, 1990; Tweed et al., 1992; Misslisch et al., 1994). It states that, during eye movements from nonprimary eye positions, the angular velocity axis of the eye tilts out of Listing's plane by approximately half the deviation of gaze from primary position (Fig. 3). Related to these eye-positiondependent effects is the fact that the time derivative of eye orientation (EÇ ˆ dE/dt) differs from the angular velocity (V) in 3-D, unless the eye is in the primary position. (The concept of angular velocity describes the velocity of rotation of the eye about a ®xed axis.) Mathematically, the relationship between eye position and angular velocity in 3-D is captured in the following equation (Goldstein, 1980;

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Control of 3-D eye orientation

Fig. 3. According to Listing's law, the axis of rotation of the eye (V) is neither head-®xed nor eye-®xed, but rotates in the same direction as gaze through half the gaze angle (u/2; half-angle rule). Thus, at eccentric eye positions, during a horizontal saccade or pursuit eye movement, the axis of rotation of the eye is not purely horizontal (head-vertical dashed line) but also has a torsional component (head-horizontal dashed line).

Tweed & Vilis, 1987):

1 E_ ˆ V o E 2

…1†

where the eye orientation, E, the rate of change (derivative) of eye orientation, EÇ, and angular velocity, V, are expressed as quaternions (Tweed & Vilis, 1987) and `o' denotes a quaternion product. Quaternions can be looked upon as four-dimensional vectors. Thus, eye orientation E can conveniently be represented as a 4-D vector E ˆ (E0 E1 E2 E3)0 (the prime indicates that E is a column vector). Based on the Euler axis-angle representation (Goldstein, 1980), any ocular orientation can be uniquely described with a 4-D vector by setting E0 ˆ cos (r/2) and ~ E ˆ (E1 E2 E3)0 ˆ~ n sin(r/2) where v 1 0 u 3 uX ~ Ei2 A n ˆ @~ E t 1

(i.e. ~ n is a vector of unitary length). With this tool at hand, we can associate any orientation of the eye relative to a reference orientation (usually primary position) by specifying just four parameters, an angle (r) and axis ~ n (three parameters) about which the eye has been rotated relative to the reference position. If we associate the 3-D vector of angular velocity with the quaternion V ˆ (0 V1 V2 V3) we can write the quaternion product (denoted by `o') in eqn (1) in the more familiar form of a matrixvector product: 0

E0 B _E ˆ 1BE1 2@E2 E3

E1 E0 E3 E2

E2 E3 E0 E1

10 1 0 0 E3 B C B E2 C CBO1 C ˆ 1B E1 A@O2 A 2@ E0 O3

1 E1 E2 E3 0 1 O1 E0 E3 E2 C C@O2 A …2† A E3 E0 E1 O3 E2 E1 E0

Reversing the position of V and E in eqn (1) and calculating the matrix±vector product according to eqn (2) shows that this product is not commutative (due to the minus signs in rows 2±4). This relation-

3

ship between eye orientation, E, and angular velocity, V, relates to the mathematical fact that 3-D rotations do not commute (Goldstein, 1980). One consequence of this noncommutativity is illustrated in Fig. 4, which shows the fact that ®nal gaze direction differs if one commutes the order of a pure horizontal eye movement followed by a pure vertical eye movement of equal magnitude. The noncommutativity of eqn (1) entails the following paradox: eye orientations (E) in Listing's plane have zero torsion (e.g. Fig. 2), but eye movements from any orientation other than the primary position need to adjust torsion and thus require an eye velocity (V) with a nonzero torsional component (e.g. Fig. 3) to stay in Listing's plane. This nonintuitive property represents a fundamental feature of 3-D ocular kinematics. An example of such an eye-position-dependent angular velocity is illustrated in Fig. 5 for smooth pursuit eye movements. During horizontal smooth pursuit with gaze straight ahead, for example, eye movements are purely horizontal with negligible modulation in torsional eye velocity (Fig. 5, middle). Because of the kinematic constraints of Listing's law, however, a combination of both horizontal and torsional eye velocity is observed during pursuit of a target which is moving horizontally in eccentric positions. During horizontal pursuit with gaze up, for example, a negative (counterclockwise relative to the animal) torsion accompanies the positive (leftward) component of pursuit (Fig. 5, top). The opposite is true for down gaze (Fig. 5, bottom). As a result, the instantaneous axis of rotation of the eye tilts away from a purely head-horizontal axis in the same direction as gaze, through approximately half the gaze angle, as illustrated when eye velocity is plotted in head coordinates (Fig. 5, right). All fovea-related eye movements, including both smooth pursuit and saccades (Tweed & Vilis, 1987, 1990; Haslwanter et al., 1991; Tweed et al., 1992), have been shown to follow Listing's law and exhibit this eye-positiondependent tilt. The generation of such an eye-position-dependent velocity has been the subject of considerable debate and continues to be at the forefront of contemporary oculomotor research in recent years. The controversy has primarily centred on issues of neural vs. mechanical contributions to the control of torsion during saccadic eye movements (Demer et al., 2000; Haustein, 1989; Schnabolk & Raphan, 1994; Tweed et al., 1994; Quaia & Optican, 1998; Raphan, 1998). This review addresses the two con¯icting views on this question and attempts to summarize the pros and cons of each based on recent experimental and modelling efforts.

The controversy about ocular torsion Several recent studies have focused on how torsional eye velocity components, necessary to keep eye position in Listing's plane during eye movements, are generated. Tweed & Vilis (1987, 1990) ®rst addressed the problem in the late 1980s. These authors pointed out that traditional oculomotor concepts that were well established for the horizontal system, like that of the velocity-to-position neural integrator (Cannon & Robinson, 1987; Skavenski & Robinson, 1973; Robinson, 1981), needed to be re-evaluated when considering 3-D eye movements. They proposed the following two generalizations: ®rst, replacing the traditional one- or two-parametric descriptions (i.e. horizontal±vertical) of eye position with the concept of ocular orientation based on Euler's theorem about the motion of rigid bodies (Goldstein, 1980); second, for describing eye velocity they proposed using the natural generalization of angular velocity from 2-D to 3-D. Based on these seemingly innocuous generalizations, rather incisive revisions in widely used oculomotor models became necessary simply because, as Tweed & Vilis (1987) pointed out, angular velocity is not the derivative of eye orientation in 3-D [(V 6ˆ EÇ; eqn (1)]. Accordingly, the extension of the neural integration concept in 3-D requires the

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4 D. E. Angelaki and B. J. M. Hess

Fig. 4. Non-commutativity of rotations. Schematic 3-D view of the horizontal and vertical oculomotor range (left): horizontal gaze movements rotate the eye or gaze line in the horizontal meridian plane (x±y plane) or in, respectively, parallel planes by simultaneously changing vertical eccentricity (for example, along the arcs F0 ±G0 or E0 ±H0 ); in contrast, movements at constant vertical eccentricity are mixed horizontal±torsional movements which rotate the eye in meridian planes (for example, along the arc F±G or E±H). Vertical gaze movements rotate the eye or gaze line in the vertical meridian plane (x±z pane) or in, respectively, parallel planes by changing simultaneously horizontal eccentricity (for example, along the arcs E±F or H±G); in contrast, movements at constant horizontal eccentricity are mixed vertical± torsional movements which rotate the eye in meridian planes (for example, along the arc E0 ±F0 or H0 ±G0 ). As a consequence, combinations of horizontal and vertical gaze shifts lead to different ®nal gaze orientations, depending on the order of rotation (compare gaze vectors OG and OG0 ; O, centre of sphere; G and G0 , points on surface of unit sphere). (Right) Same as drawing on the left, but frontal view, illustrating the end position of the gaze vector (black and grey dots) after commuting the horizontal and vertical movement components of equal amplitude. Horizontal rotations (about the z-axis), followed by vertical rotations (about the y-axis) result in ®nal gaze directions (black doted traces and dots) which deviate from those obtained by reversing the order of rotations (grey traces and dots). This difference increases as gaze eccentricity increases. For example, reversing the order of vertical and horizontal rotations through 10, 20 and 308 results in ®nal gaze directions which differ from each other by 0.2, 1.7 and 5.48, respectively. Note „that the ®nal gaze directions differ from each other even though the integrals of the respective „ _ _ angular velocities are identical, i.e. a ‡ b ˆ …a_ ‡ b†dt ˆ b ‡ a ˆ …b_ ‡ a†dt (where a_ is the horizontal and b_ the vertical angular velocity).

incorporation of nonlinear (multiplicative) mathematical operations in order to implement eqn (1) under the assumption that the directions of action of extraocular muscles remain head-®xed and independent of eye position (Tweed & Vilis, 1987, 1990). The 3-D integrator scheme proposed by Tweed and Vilis has been illustrated in Fig. 6. This revolutionary proposal assigned the computational load imposed by the 3-D kinematics to premotor neural networks (Fig. 7a). Furthermore, it assumed that premotor and motor neurons encode the 3-D angular velocity of the eye, an assumption which, as will be summarized below, has a ®rm basis for the vestibulo-ocular re¯ex and smooth pursuit eye movements, although not necessarily for the saccadic system. Indeed, both of these hypotheses have recently been challenged, at least for saccadic eye movements. First, it has been proposed that mobile soft-tissue sheaths or `pulleys' in the orbit in¯uence the pulling direction of the extraocular muscles (Demer et al., 2000; Kono et al., 2002). Second, Henn and colleagues failed to ®nd clear evidence for a neural representation of 3-D angular velocity (V) in the premotor pathway for saccadic eye movements (van Opstal et al., 1991, 1996; Hepp et al., 1993, 1999; Scherberger et al., 2001). The former result has fuelled a mechanical hypothesis for the implementation of 3-D kinematics (Fig. 7b) which, at least in its original formulation, was diametrically opposite to the 3-D neural controller hypothesis of Tweed & Vilis (1987).

Passive and active pulley hypotheses There is growing evidence that most current oculomotor models oversimplify the mechanics of the peripheral ocular plant. Speci®cally, several recent studies have reported the existence of mobile soft-tissue sheaths or pulleys in the orbit, composed of collagen, elastin and smooth muscle ®bres which appear to in¯uence the pulling direction of the extraocular muscles (Demer et al., 1995, 2000; Miller, 1989; Miller

et al., 1993; Sappey, 2001; Simonsz, 2001). Based on theoretical arguments, it was proposed that pulleys might simplify the brain's work in implementing Listing's law (Quaia & Optican, 1998; Raphan, 1997, 1998). In fact, a well-founded theoretical study by Quaia & Optican (1998) has shown that appropriately placed pulleys (Fig. 8) and an idealized model of the eye plant (see later section) can both generate physiologically realistic saccades and implement the halfangle rule without a need to calculate a `neural angular velocity signal' as proposed by Tweed & Vilis (1987, 1990). This astounding ®nding comes about because, as elegantly explained by Quaia & Optican (1998), `correctly' located pulleys within an idealized plant make the mathematical integral of the saccadic pulse, modelled as time derivative of orientation, approximately equal to the saccadic step which maintains ocular orientation. Accordingly, the authors proposed a simple approximation of the saccadic step from the pulse signal, a hypothesis which at least super®cially seems to solve the problem of the 3-D velocity-to-position transformation without directly dealing with the noncommutativity issue. The original formulation of the functional signi®cance of the muscle pulleys assumed that they remain ®xed relative to the eye (`passive pulley hypothesis'). Under this assumption, the brain's work in generating kinematically appropriate saccades would be greatly simpli®ed (Robinson, 1975; Miller & Robins, 1987; Raphan, 1997, 1998). However, it soon became clear that for an appropriate approximation of the saccadic step from the pulse signal, pulleys had to be neurally controlled by actively changing their position relative to the eyeball for eccentric gaze directions (Quaia & Optican, 1998). Indeed, magnetic resonance imaging of rectus muscle paths has shown that the arrangement of the pulleys is consistent with an oculomotor plant which could implement the eye-position dependence of Listing's law (`active pulley hypothesis', Demer et al., 2000; Kono et al., 2002). Interestingly, the mere fact that active neural control of the pulleys was

ß 2003 Federation of European Neuroscience Societies, European Journal of Neuroscience, 19, 1±10

Control of 3-D eye orientation

5

Fig. 7. Three alternative schemes for the control of eye orientation in 3-D. (a) The original hypothesis of Tweed & Vilis (1987), as proposed prior to the experimental demonstration of muscle `pulleys', while it was generally accepted that the pulling direction of the muscles remained head-®xed. (b) The active pulley hypothesis (APH; Demer et al. 1995, 2000; Raphan, 1997, 1998; Quaia & Optican, 1998). (c) The hypothesis of a noncommutative 3-D premotor neural controller in the presence of an eye plant with pulleys.

Fig. 5. Dependence of horizontal smooth pursuit eye movements on vertical gaze. From top to bottom, 3-D eye position (Etor, Ever and Ehor) and angular velocity (Vtor, Vver, Vhor) for up, centre and down gaze during 0.5 Hz ( 38) pursuit. Fast phases have been eliminated (modi®ed from Angelaki et al., 2003).

necessary dashes the conjecture of simplicity, despite a popular belief to the contrary. Because the pulleys themselves must be neurally controlled, the brain must now generate neural commands to control both the muscles and the pulleys. To date there have been no suggestions about how this control might be achieved, but it is hard to see how it could be `simple'. It is important to point out that the modelling conclusions of Quaia & Optican (1998) were based on the assumption of an idealized plant (`linear plant model' proposed by Tweed et al., 1994; Tweed, 1997), where several unrealistic simpli®cations were made, including: (i) the six muscles had to be reduced to three by treating agonist±antagonist pairs as single muscles able to apply a positive or negative torque; (ii) the passive elasticity and viscosity properties of the muscles were lumped and assumed to be isotropic; (iii) the tension innervation ratio

Fig. 6. The Tweed & Vilis (1987) hypothesis for the neural integration of a 3-D controller. The 3-D angular velocity vector, V, is multiplied with the ocular orientation signals, E, represented as a 4-D-vector [see eqns (1) and (2)].

was assumed to be independent of muscle length; (iv) orthogonal planes of action were assumed for each muscle pair. There is growing evidence that these gross simpli®cations of the linear plant models can be misleading. For example, the torques exerted by a muscle, in addition to having a component proportional to innervation, are also characterized by a complicated nonlinear length±tension relationship (Simonsz, 1994). Because of their complexity, simulations of biomechanically realistic models have so far been restricted to static ®xations (Porrill et al., 2000; Quaia & Optican, 2003). A discrepancy in the predictions of idealized and biomechanically realistic models has already been noted for the static control of 3-D eye position (Porrill et al., 2000; Haslwanter, 2002). Therefore, an extension of biomechanical models to include eye movement dynamics (thus incorporating the complex properties of muscle force generation; Goldberg et al., 1998; Miller & Robins, 1992; Goldberg & Shall, 1999; BuÈttner-

Fig. 8. Schematic explanation of how horizontal rectus pulleys can implement the half-angle rule. If the path of the muscles through the orbit is constrained by pulleys, the muscular path from the origin to the pulleys is essentially constant in the orbit, regardless of eye orientation. However, the axis of action of the muscle changes with eye orientation (modi®ed from Quaia & Optican, 1998).

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6 D. E. Angelaki and B. J. M. Hess Ennever & Horn, 2002a,b) is fundamental before modelling studies can be trusted to provide guidance in the quest for the functional role of the pulleys. Despite these uncertainties in interpretation, Demer's functional imaging results (Demer et al., 2000), combined with the modelling studies of Quaia & Optican (1998; also Raphan, 1998), have been interpreted to provide an alternative view to the proposal of Tweed & Vilis (1987, 1990), whereby a commutative 2-D (horizontal±vertical) neural controller could be suf®cient to generate accurate saccades in Listing's plane (Fig. 7b). The commutative controller hypothesis is attractive because, at a super®cial glance, it would simplify the premotor processing of eye movements such that the traditional `Robinsonian' concepts of dynamic premotor processing, well established for the horizontal system, could continue to be used for controlling gaze, assigning the problems of controlling 3-D eye orientation to an `intelligent' oculomotor plant. This idea has become surprisingly popular among many eye movement researchers. However, there are two problems with a quick excitement about this widely acclaimed simple solution. First, other than the fMRI results of Demer and colleagues (Demer et al., 2000; Kono et al., 2002) there is little direct evidence to date let alone a straightforward experimental proof of the hypothesis regarding the functional role of rectus pulleys. The second problem with overinterpreting the implications of these results lies on the fact that although muscle pulleys appear to simplify the task for head-®xed saccades, this simpli®cation cannot be easily generalized to either head movements (e.g. the VOR) or head-free gaze shifts. In fact, the hypothesized action of the pulleys neither eliminates the necessity for a 3-D controller nor proves the existence of a 2-D neural controller. This will become clearer in the following paragraphs.

Vestibulo-ocular reflex (VOR) Whereas the proposed pulley arrangement could implement the halfangle rule of Listing's law for pursuit and saccades, their possible function during the RVOR is much less clear. Speci®cally, the RVOR does not follow Listing's law and the half-angle rule (Angelaki et al., 2003; Crawford & Vilis, 1991; Misslisch et al., 1994; Misslisch & Hess, 2000). If the extraocular muscle pulleys implement the halfangle rule by imposing an eye-position-dependent pulling direction for the rectus muscles, how is this property `undone' during the RVOR? It was originally proposed that the pulleys might advance and retract along their muscle paths, adopting one arrangement for Listing's law and another for the RVOR (Demer et al., 2000; Thurtell et al., 1999, 2000). This theory, however, was recently shown to be incorrect, as the required retraction to explain the RVOR would have to be so large as to be nonphysiological. Furthermore, the proposed retraction, no matter how large, could not explain the full pattern of rotation axes seen in the RVOR (Misslisch & Tweed, 2001). The fact that the active pulley hypothesis, at least in its originally proposed form, is incorrect has gained further support by a recent study where the RVOR was elicited together with the translational VOR (TVOR; Angelaki et al., 2003). Unlike the RVOR, the TVOR follows a 3-D kinematic behaviour which is more similar to visually guided, fovea-related eye movements such as pursuit (Angelaki et al., 2000, 2003). Contrary to the RVOR where peripheral image stability is functionally important, the TVOR, like pursuit and saccades, seeks to stabilize images on the fovea by disregarding global image features seen by the whole retina (Angelaki et al., 2003). How the rotational and translational VORs combine was recently investigated during eccentric yaw rotations. The combined VOR did not follow a ®xed eye-position dependence which would be consistent with the postulated mechanical constraints imposed by extraocular muscle pulleys. Instead, a variable

eye-position-dependent torsion was observed, which was as much as 10 larger than what would be expected based on extraocular muscle pulleys. In addition, eye velocity often tilted opposite to the direction of gaze, an observation which is again inconsistent with a simple mechanical origin of 3-D ocular kinematics. Quantitative analyses revealed that the combined VOR behaved as if the eye-positiondependent torsion was computed separately for the RVOR and the TVOR components, implying a neural origin upstream of the site of RVOR±TVOR convergence (thus excluding a purely mechanical solution). Thus, although attractive and convenient for head-®xed saccades, the active pulley retraction hypothesis could not account for behavioural observations regarding the 3-D properties of the VOR. These ®ndings prompted the next twist in a series of continuing modi®cations of the original pulley hypothesis.

Revised active pulley hypothesis for ocular mechanics Indeed, the way in which the `active pulley hypothesis' deals with the RVOR has recently been revised. Demer (2002) has provided an alternative hypothesis as to what happens during the RVOR which might in fact be the beginning of a compromise to the often heated controversy of the past decade. Demer now proposes that the violation of Listing's law during the RVOR might be mediated by the oblique extraocular muscles whose velocity axes are actively maintained orthogonal to the Listing's velocity axes of the rectus muscles. This new proposal was based on experimental ®ndings during ocular convergence and static tilt. Speci®cally, a temporal rotation of Listing's plane is observed with convergence (Mok et al., 1992; Van Rijn & Van den Berg, 1993; Minken & van Gisbergen, 1994). It was found that, during ocular convergence, rectus pulleys rotated in the orbit but remained ®xed relative to the pulling direction of the obliques (Demer et al., 2003). A similar result was observed for the shift in Listing's plane during static ocular counterrolling (Demer & Clark, 2003). According to this revised pulley hypothesis, `appropriate' neural signals during convergence and counterrolling (and presumably also RVOR) activate the oblique muscles, causing not only the eye to rotate torsionally but also to simultaneously rotate the rectus pulleys as well. As a result, the rectus muscle pulling directions would remain ®xed relative to the obliques but change relative to the head. Thus, in contrast to its promised simplicity when it was originally conceived (Demer et al., 1995, 2000; Raphan, 1997, 1998), the hypothesized role of the pulleys in 3-D eye movement control becomes increasingly complex. This revised hypothesis could account nicely for the alterations in Listing's plane orientation under static conditions, e.g. the rotation or translation of Listing's plane during static head tilts and ocular convergence (Haslwanter et al., 1992; Mok et al., 1992; van Rijn & van den Berg, 1993; Minken & van Gisbergen, 1994). It is also in line with the shift in the preferred direction of burst neurons during saccades in static roll head positions (Scherberger et al., 2001). However, what happens during the RVOR remains purely speculative at present. As functional imaging cannot provide information about the movement of the eyes during the RVOR, this hypothesis can only be addressed with complementary electrophysiological experiments of motoneuron discharge rates. Indeed, a straightforward experiment which would directly address the active pulley hypothesis (APH) would be to quantitatively investigate if motoneuron ®ring rates correlate best with angular velocity (V) or the derivative of eye orientation (EÇ). As implied by the half-angle rule, the larger the eye position eccentricity relative to primary position the larger the difference between EÇ and V. Unfortunately, the single 3-D study of

ß 2003 Federation of European Neuroscience Societies, European Journal of Neuroscience, 19, 1±10

Control of 3-D eye orientation motoneuron ®ring rates only dealt with static ®xations (Suzuki et al., 1999). The APH would predict that the dynamic component in motoneuron ®ring rates would best correlate with EÇ rather than V during all eye movements (VOR, saccades and pursuit; Fig. 7b and c). For example, the burst activity of motoneurons during saccades should not change in relation to the eye-position-dependent change in angular eye velocity associated with different eye eccentricities. Alternatively, if it were found that motoneuron ®ring rates best correlate with V rather than EÇ (Fig. 7a), the validity of the APH would become questionable. Such studies of motoneuron ®ring rates during changes in eye position brought about during saccades, pursuit and the VOR have yet to be performed.

Commutativity vs. noncommutativity in oculomotor control: where does the debate lie? The discussions about the implications of extraocular muscle pulleys often gets mixed up with the controversy about commutativity or noncommutativity in the premotor control of eye movements. A `commutative (2-D) neural controller' implies that the premotor neural commands for a horizontal eye movement are independent of vertical eye position and vice versa. For example, when considering horizontal saccades, the number of spikes in a burst should be the same when the saccade is executed from primary or eccentric vertical eye positions (neural ®ring rate modulation should not depend on eye position). Mathematically, the debate between a noncommutative vs. a commutative neural controller is equivalent to the question of whether premotor neurons encode the angular velocity (V) or the rate of change of eye position (EÇ) (Fig. 7). For eye movements in Listing's plane, V must be an eye-position-dependent 3-D vector (i.e. with a nonzero torsional component), whereas EÇ is 2-D. However, the opposite is true for the RVOR. The expected differences between neural coding of EÇ and V for different eye movements are summarized in Table 1. For eye movements in Listing's plane (pursuit, saccades and the TVOR), EÇ coding implies that neural ®ring rates exhibit no eye-position dependence (`no EPD' in Table 1) and can be best described by a 2-D (horizontal± vertical) model. In contrast, a neural sensitivity to V should be best described with a 3-D model and a clear eye position-dependence of ®ring rates (`EPD' in Table 1). For the RVOR which does not follow Listing's law, the reverse would be true. Assuming rotation axes in Listing's plane (yaw and pitch), neurons encoding V should exhibit neural ®ring rate modulations which are independent of eye position and best described by a 2-D model. In contrast, EÇ-coding would expect a clear eye-position dependence in the ®ring rate modulation of the cell, as well as a better ®t with a 3-D model (Smith & Crawford, 1998). The recently proposed APH models (Quaia & Optican, 1998; Raphan, 1998) have claimed that, simply because motoneurons are hypothesized to encode EÇ, a commutative controller (using, according to Raphan, 1998, only a horizontal±vertical but no torsional neural signal) is automatically implied for premotor processing (Fig. 7b). Table 1. The expected differences between EÇ and V for different eye movements

Pursuit Saccades TVOR RVOR (yaw, pitch) EPD, eye-position dependence.

Derivative of eye position, EÇ

Angular velocity, V

2-D, 2-D, 2-D, 3-D,

3-D, 3-D, 3-D, 2-D,

no EPD no EPD no EPD EPD

EPD EPD EPD no EPD

7

However, contrary to this widely held opinion, there is growing evidence that a noncommutative neural controller is still necessary, despite the existence of appropriately placed pulleys (Fig. 7c). In fact, several studies supporting the need for a 3-D neural controller have incorporated muscle pulleys in their models (Smith & Crawford, 1998; Tweed et al., 1999). This is so because premotor cells do not necessarily have to encode EÇ, even though motoneurons might do so according to the APH. There are multiple ®ndings arguing for the existence of a 3-D premotor neural controller. (i) Different eye movements exhibit different 3-D kinematics. Speci®cally, as summarized above, Listing's law is relaxed or abandoned during RVOR and optokinetic responses (Crawford & Vilis, 1991; Fetter et al., 1992; Misslisch et al., 1994). Furthermore, during eccentric rotations, experimental evidence suggests that the eye-position-dependent torsion is neurally computed prior to the rotational± translational VOR signal convergence (Angelaki, 2003). In addition, Listing's planes are modi®ed during convergence and static head tilt (Haslwanter et al., 1991; Mok et al., 1992; van Rijn & van den Berg, 1993; Minken & van Gisbergen, 1994). The revised APH, in fact, proposing that oblique muscle activation is necessary to generate static ocular counterrolling and a change in torsion during convergence (Demer, 2002; Demer & Clark, 2003), would require a 3-D premotor neural controller (otherwise, how would the appropriate torsional signal be generated?). However, because the revised APH remains purely qualitative at present, this fact has not as yet been explored. (ii) The simple and elegant experiments of Tweed et al. (1999; see also Tweed, 1997) showed that the vector integration approach (Raphan, 1997, 1998; Quaia & Optican, 1998) and, consequently, oculomotor models which use EÇ instead of angular velocity V to simplify the integration step, do not reproduce the noncommutative behaviour of the RVOR. Thus, although pulleys might minimize the pulse±step mismatch which occurs by integration of the time derivative of ocular orientation EÇ instead of angular velocity V, they do not make a commutative RVOR (Tweed et al., 1999). (iii) During head-free gaze shifts, slow phase movements are routinely preceded by a saccade, which normally drives the eyes out of Listing's plane in an anticipatory fashion, such that it ends up in Listing's plane only after the whole movement is complete (Crawford & Vilis, 1991; Crawford & Guitton, 1997; Tweed, 1997). The fact that the saccade drives eye position out of Listing's plane in an anticipation of the whole gaze shift cannot be explained by a 2-D neural controller and an eye plant with pulleys. (iv) Finally, other studies have also convincingly shown that the visuomotor transformations from retinal information into kinematically correct oculomotor signals require accurate neural control over all degrees of freedom of ocular orientation (Crawford & Guitton, 1997; Klier & Crawford, 1998). Thus, in summary, the often heated argument about commutativity vs. noncommutativity in the control of eye movements is not about pulleys (as often erroneously implied) but about whether the central nervous system uses a 2-D (horizontal±vertical) or a 3-D (horizontal± vertical±torsional) neural controller for eye movements. There is a tendency to assume that the active pulley hypothesis which states that the phasic component of motoneuron discharge encodes the derivative of eye position EÇ (and not angular velocity V), alleviates the need for a premotor coding of V (Fig. 7b) (Demer et al., 1995, 2000; Raphan, 1997, 1998). However, there is ample evidence which supports an alternative hypothesis, namely that, irrespective of the existence and function of muscle pulleys, premotor processing and the underlying sensorimotor transformations rely on a 3-D neural controller (Haslwanter, 1995; Crawford, 1998; Klier & Crawford, 1998; Smith & Crawford, 1998; Tweed et al., 1999; Klier et al., 2001). Such

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8 D. E. Angelaki and B. J. M. Hess evidence points to the existence of a 3-D neural controller in the premotor circuitry, even though angular velocity (V) signals might be transformed downstream into eye-orientation-derivative (EÇ) signals at the level of motoneurons, in order to control an eye plant with pulleys (Fig. 7c). Unfortunately, experiments to date have mainly focused on theoretical studies and behavioural input±output relationships, where arguments can provide only indirect evidence towards the existence of a 3-D premotor neural controller. Directly probing and quantitatively characterizing the properties of different premotor groups of neurons, as well as motoneurons (see above) is necessary. Such experiments are as critical in understanding the control of 3-D eye orientation as the neurophysiological studies in the 1970s and 80s have been for the pioneering discoveries regarding the neural control of horizontal and vertical eye movements.

Search in neural activities for a 3-D neural controller for saccades and pursuit The existence of a 2-D or 3-D premotor neural controller can only be directly investigated by examining whether premotor neurons encode the derivative of eye position, EÇ, or angular velocity, V [eqn (1)]. As explained in more detail in previous sections, although these two vectors, EÇ and V, are indistinguishable in the primary position, their amplitudes and directions deviate signi®cantly the larger the eye eccentricity from primary position. Thus, one must compare neural ®ring rates at different initial eye positions and quantitatively search for differences in cell responsiveness. The idea of searching for neurophysiological evidence in order to distinguish between a 2-D (commutative) or 3-D (noncommutative) neural controller has long been appreciated but, unfortunately, we still do not yet have many answers. The last several years of Volker Henn's research efforts, before his untimely death in the middle nineties, were devoted to addressing these questions. His efforts focused on the premotor coding of saccadic eye movements (Hepp et al., 1993, 1999; van Opstal et al., 1991, 1996; Scherberger et al., 2001). Henn and colleagues have clearly demonstrated that the collicular motor map is 2-D and conjectured that the reticular burst generators encode EÇ without excluding possible encoding of V by combining EÇ with an eye position signal (Hepp et al., 1993; Hepp et al., 1995). Using spontaneous saccades (as in the experiments of Henn and colleagues), however, might not be the best protocol to identify small differences between 3-D angular velocity and the 2-D derivative of eye position in burst activities (Hepp et al., 1999; Scherberger et al., 2001). In fact, small changes in neural ®ring rates related to 3-D signals might be more easily discernible during pursuit±VOR than during saccadic eye movements because the phasic component can be sustained for longer periods to give a better signal-to-noise ratio (Smith & Crawford, 1998). Although typically discussed for saccades, these issues had never until very recently been addressed for smooth pursuit eye movements. The same controversies outlined above for saccades also apply to pursuit eye movements. Speci®cally, similar to saccades, premotor pursuit neural activities could be restricted to a coding of the 2-D (horizontal±vertical) derivative of eye orientation (EÇ) rather than 3-D angular eye velocity (V). Alternatively, one could speculate that 3D coding of pursuit velocity might be advantageous, not only for the reasons mentioned above for saccades, but also if pursuit±VOR interactions were to share a common coordinate system (as semicircular canal afferents encode angular velocity, V). The premotor coding for pursuit would then be similar to the common coordinate system used for optokinetic±VOR convergence, as has been shown to exist in birds and rabbits (Graf et al., 1988; Tan et al., 1993; Wylie & Frost, 1993; Van der Steen et al., 1994). Indeed, a recent study in rhesus

monkeys has provided direct evidence that eye movement neurons in the rostral vestibular nuclei exhibit signi®cant changes in their ®ring rates which parallels the monotonic increase in torsional eye velocity as a function of eye position during smooth pursuit (Angelaki & Dickman, 2003). In fact, multiple linear regression analyses of cell responsiveness during pursuit at multiple eye orientations revealed that cell ®ring rates encode the eye-position-dependent torsional component of pursuit eye velocity. These observations would suggest that at least some neural ®ring rates are more closely related to the 3-D angular velocity, V, than to the eye-position-independent 2-D derivative of eye position, EÇ. Thus, in contrast to previous reports on the saccadic premotor pathways (van Opstal et al., 1991, 1996; Hepp et al., 1993, 1995, 1999; Scherberger et al., 2001), more direct evidence exists for a 3-D neural controller in the premotor processing of pursuit eye movements.

Neural coding during the VOR Compared to pursuit and saccades, consideration of these issues for the RVOR differs in two important aspects. First, the RVOR does not follow Listing's law. This is a trivial statement when the head's velocity axis has a torsional component (as the RVOR will generate an eye velocity equal and opposite to head velocity). However, even head rotation axes in Listing's plane (i.e. pitch±yaw) will cause torsional violations of Listing's law as a function of initial eye position (Crawford & Vilis, 1991). This is so because RVOR slow phase velocity axes do not tilt out of Listing's plane as a function of eye position (as required to keep eye positions in Listing's plane; Figs 3 and 5), but they tend to remain aligned with the head velocity axes in order to minimize image slip on the peripheral retina (Misslisch et al., 1994; Angelaki et al., 2003). The second difference is even more important. In contrast to visually driven pursuit and saccades whose sensory inputs are 2-D (retinal) signals and where the hypothesis of EÇ premotor coding is possible (although it requires experimental testing), the sensory signals driving the RVOR represent 3-D angular velocity. Thus, it is physically impossible for the canal activity vector to encode eye orientation derivatives (Tweed, 1997). Because the RVOR encodes V, at least at the sensory side, a V-to-EÇ transformation is implied in order to be consistent with an eye plant with pulleys which requires an `EÇ' input (Fig. 7b and c). As seen from eqns (1) and (2), this transformation requires a multiplication (see also Fig. 6). Whether these velocity signals are transformed into orientation velocity, which would facilitate integration to yield ocular orientation in a network shared by saccadic signals, is a completely open question (see Tweed, 1997). The need for this multiplication step for the V-to-EÇ transformation for the RVOR, despite the existence of an eye plant with pulleys, represents one of the arguments which has motivated several investigators to point out that the active pulley hypothesis does not solve the problem of noncommutativity (Tweed, 1997; Smith & Crawford, 1998; Tweed et al., 1999). Because no studies have systematically examined the 3-D properties of either premotor cells or motoneurons during rotation and compared these activities with those during pursuit and saccades, these issues have yet to be explored.

Conclusions In summary, much experimental effort has been devoted in recent years in order to understand the neural vs. mechanical aspects for the control of 3-D eye orientation and the associated kinematic behaviours. The controversies about 3-D eye movements during the past decade focus on the question of whether premotor neurons encode the derivative of eye position, EÇ, or the angular velocity, V [eqn (1)]. Arguments have been put forward for both alternatives. Although there is a tendency to

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Control of 3-D eye orientation assume that the active pulley hypothesis, which states that the phasic component of motoneuron discharge rate encodes the derivative of eye position, EÇ, (and not angular velocity, V), alleviates the need for a premotor coding of V (Demer et al., 1995, 2000; Raphan, 1997, 1998), multiple evidence exists to show otherwise. Under dynamic conditions (e.g. RVOR or combined eye±head movements), torsional eye movements are actively controlled, a process which cannot be explained by a horizontal±vertical premotor neural control system (Klier & Crawford, 1998; Tweed et al., 1998, 1999). Thus, irrespectively of muscle pulleys and their role in providing an eye-position-dependent extraocular muscle pulling direction, the premotor processing of the vestibuloocular re¯ex, head-free gaze shifts and the underlying sensorimotor transformations require a 3-D neural controller (Haslwanter, 1995; Crawford, 1998; Klier & Crawford, 1998; Smith & Crawford, 1998; Tweed et al., 1999; Klier et al., 2001). Most studies to date have primarily focused on behavioural and modelling arguments with very little effort being devoted to the understanding of the neural response properties of motoneurons. Thus, future work must focus heavily on the characterization of neural activities during 3-D eye movements and the determination of whether the ®ring rates of motoneurons (and different groups of premotor neurons) correlate better with the derivative of eye position, EÇ, or angular velocity, V. Similarly to the great advances of a combined modelling and neurophysiological approach for understanding the dynamic aspects for the neural control of horizontal and vertical eye movements, such a combined approach is even more critical for appreciating and understanding the biological aspects of the often complicated 3-D spatial organization of motor behaviours.

Acknowledgements Supported by NIH (EY12814) and Swiss National Science Foundation (SNF 31-57086.99).

Abbreviations 2-D, two dimensions/-dimensional; 3-D, three dimensions/-dimensional; APH, active pulley hypothesis; RVOR, rotational vestibulo-ocular re¯ex; TVOR, translational vestibulo-ocular re¯ex; VOR, vestibulo-ocular re¯ex.

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