The Journal of Neurophysiology Vol. 79 No. 6 June 1998, pp. 3197-3215
Copyright ©1998 by the American Physiological Society

Commutative Saccadic Generator Is Sufficient to Control a 3-D Ocular Plant With Pulleys

Christian Quaia and Lance M. Optican

Laboratory of Sensorimotor Research, National Eye Institute, Bethesda, Maryland 20892

	ABSTRACT

Abstract Introduction Methods Results Discussion References

Quaia, Christian and Lance M. Optican. Commutative saccadic generator is sufficient to control a 3-D ocular plant with pulleys. J. Neurophysiol. 79: 3197-3215, 1998. One-dimensional models of oculomotor control rely on the fact that, when rotations around only one axis are considered, angular velocity is the derivative of orientation. However, when rotations around arbitrary axes [3-dimensional (3-D) rotations] are considered, this property does not hold, because 3-D rotations are noncommutative. The noncommutativity of rotations has prompted a long debate over whether or not the oculomotor system has to account for this property of rotations by employing noncommutative operators. Recently, Raphan presented a model of the ocular plant that incorporates the orbital pulleys discovered, and qualitatively modeled, by Miller and colleagues. Using one simulation, Raphan showed that the pulley model could produce realistic saccades even when the neural controller is commutative. However, no proof was offered that the good behavior of the Raphan-Miller pulley model holds for saccades different from those simulated. We demonstrate mathematically that the Raphan-Miller pulley model always produces movements that have an accurate dynamic behavior. This is possible because, if the pulleys are properly placed, the oculomotor plant (extraocular muscles, orbital pulleys, and eyeball) in a sense appears commutative to the neural controller. We demonstrate this finding by studying the effect that the pulleys have on the different components of the innervation signal provided by the brain to the extraocular muscles. Because the pulleys make the axes of action of the extraocular muscles dependent on eye orientation, the effect of the innervation signals varies correspondingly as a function of eye orientation. In particular, the Pulse of innervation, which in classical models of the saccadic system encoded eye velocity, here encodes a different signal, which is very close to the derivative of eye orientation. In contrast, the Step of innervation always encodes orientation, whether or not the plant contains pulleys. Thus the Step can be produced by simply integrating the Pulse. Particular care will be given to describing how the pulleys can have this differential effect on the Pulse and the Step. We will show that, if orbital pulleys are properly located, the neural control of saccades can be greatly simplified. Furthermore, the neural implementation of Listing's Law is simplified: eye orientation will lie in Listing's Plane as long as the Pulse is generated in that plane. These results also have implications for the surgical treatment of strabismus.

	INTRODUCTION

Abstract Introduction Methods Results Discussion References

Early models of oculomotor control (e.g., Robinson 1975; Zee et al. 1976) focused on rotations around a single axis (e.g., a vertical axis for horizontal movements); in one dimension, eye velocity is the derivative of eye position, and models relied on this fact. To extend models of oculomotor control from one to three dimensions, several problems must be addressed. First, when rotations around arbitrary axes (all passing through 1 point fixed in space) are considered, the concept of position must be replaced by the concept of orientation, which is less intuitive and more difficult to define mathematically. Second, the concept of velocity must be replaced by the concept of angular velocity. Third, and most important, it must be kept in mind that, for rotations around arbitrary axes, the derivative of orientation is not angular velocity (Goldstein 1980).

This last property, which applies to any rigid body rotating around a point fixed in space, is due to the noncommutativity of rotations, which can be described geometrically as follows: starting from the same initial orientation, a rotation of ° around an axis followed by a rotation of ° around (with , , , and arbitrary, as long as is not parallel to ) does not produce the same final orientation obtained when the order of the rotations is reversed. For example, in the two panels of Fig. 1, a camera, starting from the same initial orientation (left column), is rotated around the same pair of axes (arrows in the figure) but in different order; clearly, the final orientations (right column) are different for the two sequences of rotations.

View larger version (66K): [in this window] [in a new window]   FIG. 1. Geometric definition of noncommutativity is illustrated. Arrows indicate that the image on the right of each arrow is obtained by rotating the image on the left around an axis collinear with the arrow. The direction of rotation corresponds to the direction in which a right-hand screw advances. A and B: the camera, starting from the same initial orientation, undergoes 2 sequences of rotations. A: the camera first rotates 90° around a vertical axis, and then 90° around a horizontal axis. B: the order of rotations is reversed. The final orientation of the camera is clearly different in the 2 cases.

Because rotations around a single axis (i.e., when and are parallel) are commutative, in his model Robinson could use a simple integrator to transform eye velocity into eye position. However, because of the noncommutativity of arbitrary rotations, it is logical to conclude that a model that relies on the angular velocity being the derivative of orientation cannot be used to control a rotational plant in three dimensions. Consequently, to extend Robinson's model from control of the eye around one axis to control around all three of its axes, Tweed and Vilis (1987) developed a model that uses noncommutative, rotational operators to generate the innervation signals. Subsequently, Schnabolk and Raphan (1994a,b) proposed that, in fact, a noncommutative neural system was not needed to control eye movements. Schnabolk and Raphan argued that the noncommutativity of rotations was not relevant because the innervation signals determine muscle torques (which are vectors, and thus commute), not eye orientation.

There is a fundamental difference in the behavior of the two models. The model developed by Tweed and Vilis (1987) is the correct extension to three dimensions of the model proposed by Robinson in one dimension (Robinson 1975), because eye orientation and angular velocity are neurally represented. Furthermore, this model assumes that the goal of the saccadic system is not only to move the eye from its current to a new orientation, but also to accomplish this as quickly as possible. In particular, any slow movement that follows a saccade and that is due to a mismatch between the tonic innervation signal present after the end of the saccade and the orientation of the eye (Bahill et al. 1975), is avoided. These slow movements, which are called postsaccadic drifts (or glissades), can degrade visual perception (Westheimer and McKee 1975) and can be adaptively minimized by the brain (Optican and Miles 1985); by definition (Bahill et al. 1975), they encompass also the so-called torsional transients or blips.

Thus the Tweed-Vilis model (as well as the Robinson model) focused not only on the steady-state conditions (i.e., after stabilization of eye orientation) but also on the dynamics of the movement used to foveate the target. In contrast, Schnabolk and Raphan concentrated exclusively on the issue of eventually acquiring the target, without making any attempt to make a model that avoids (or minimizes) postsaccadic drifts. In fact, they explicitly stated: "there is of necessity a mismatch between the plant dynamics and the pulse-step driving it" (Schnabolk and Raphan 1994a, p. 634).

However, there is compelling physiological evidence that suggests that a model that does not account appropriately for the dynamics of the movement (i.e., that produces movements with large postsaccadic drift) is not a good approximation of the system implemented by the brain to control saccades. In fact, movements produced by normal subjects (either primates or humans) have very little postsaccadic drift (see Tweed et al. 1994a), and when the drift is induced artificially the innervation signals are adaptively modified to reduce the retinal slip (Optican and Miles 1985). Accordingly, the fact that a commutative controller is sufficient to guarantee a good steady-state behavior of the system does not imply that the brain can use such a strategy to drive the eye plant, as erroneously concluded by Schnabolk and Raphan.

Nevertheless, the need to account for the dynamics of the plant does not necessarily imply that the neural controller must be noncommutative. In fact, a noncommutative controller is necessary only if both orientation and angular velocity are encoded neurally, as was the case in the Tweed and Vilis (1987) and Robinson models. However, Tweed and colleagues (Tweed et al. 1994a; Tweed and Vilis 1990) pointed out that, if 1) eye orientation and 2) its derivative, but not angular velocity, were encoded neurally, the first signal could be computed by simply integrating the second. Under such an assumption, an essentially commutative saccadic generator could be used, and it would then be up to the plant to convert the innervation signal into the appropriate rotation of the eye; however, Tweed et al. (1994a) did not suggest any mechanism that would allow the plant to perform such conversion. Nonetheless, several recent studies have relied on the hypothesis that the derivative of eye orientation, as opposed to eye velocity, is neurally encoded (Crawford 1994, 1997; Crawford and Guitton 1997).

Recently, Raphan (1997) introduced a model of the ocular plant that incorporates the orbital pulleys discovered, and qualitatively modeled, by Miller and colleagues (Demer et al. 1995; Miller 1989; Miller et al. 1993; Miller and Robins 1987). As pointed out by Miller and colleagues (1993, 1997), the pulleys make the axes of action of the extraocular muscles depend on the orientation of the eye. Raphan proposed a simple mathematical approximation of the action of the pulleys. He used simulations to show that a saccade produced by a commutative controller driving a plant without pulleys had a large postsaccadic drift, whereas the same saccade made using a plant model with pulleys had a much smaller drift. However, no proof was offered that the improved dynamic behavior of the Raphan-Miller pulley model holds in general, i.e., for saccades different from those simulated. Furthermore, Crawford and Guitton (1997) pointed out that the pulleys have to react differently to phasic and tonic inputs, but neither they nor Raphan have offered a solution to this problem, which is critical for the pulley hypothesis.

The first goal of this paper is to verify that the Raphan-Miller pulley model always produces movements that have an accurate dynamic behavior. Using a novel quantification of the dynamic errors associated with the use of a commutative controller to drive a noncommutative plant, we compare, in a rigorous, mathematical way, the behavior of the Schnabolk-Raphan model with that of the Raphan-Miller pulley model. This is a departure from earlier approaches, when inferences drawn from the results of simulations of single movements (e.g., Schnabolk and Raphan 1994a; Straumann et al. 1995) were then proven incorrect by probing the behavior of the same models for other movements (Tweed 1997b; Tweed et al. 1994b). Using this quantitative approach, we demonstrate mathematically that the good dynamic behavior of the Raphan-Miller pulley model holds for all movements and that the saccadic generator (i.e., the network that converts the desired rotation of the eyes into the signals needed to produce that rotation) can be commutative.

The second goal of the paper is to explain, in an intuitive way, why this is possible and what the consequences of these results are for the neural signals that are generated by the oculomotor controller. The bottom line is that, if the pulleys are properly placed, the oculomotor plant (extraocular muscles, orbital pulleys, and eyeball) is seen by the saccadic controller as an essentially commutative system. We will elucidate the mechanisms underlying this surprising outcome, which gives new significance to the role of orbital pulleys; particular care will be given to explaining the effects of the pulleys on the different components (Pulse and Step) of the innervation signal.

Furthermore, we will discuss the consequences that the presence of the pulleys have for the overall organization of the saccadic system and for the neural implementation of Listing's Law, which is simplified. We will also consider the import that these results have for strabismus, a disorder of ocular alignment, with regard to both its development and surgical treatment. Finally, we will propose several experimental paradigms that could be used to test the pulley hypothesis.

A brief description of these results appeared elsewhere (Quaia and Optican 1997a).

	METHODS

Abstract Introduction Methods Results Discussion References

Representation of orientation

The eyeball can be modeled as a sphere capable of rotating around any axis through its center, which is fixed in space. When an object can rotate around arbitrary axes, there are many different ways to define its orientation. For example, it is possible to describe the orientation by a sequence of three rotations around particular axes, as in the Fick and Helmholtz coordinate systems, with rotation matrices, or with quaternions (for review see Haslwanter 1995).

Of the many mathematically equivalent descriptions of rotations, we prefer the axis-angle form, which follows from Euler's theorem. This theorem states that any orientation of a rigid body with one point fixed can be achieved, starting from a reference orientation, by a single rotation about an axis (through the fixed point) along a unit-length vector through a positive angle  (Goldstein 1980). For example, in Fig. 2 we replot the left and right panels of Fig. 1; the vectors in the middle column are collinear with the Euler axes around which the cameras in the left column have to be rotated to assume the orientation represented in the right column through a rotation around a single axis. Thus the vectors in the middle column represent the Euler axes that describe the orientation of the cameras on the right; the top and the bottom vectors are different, as are the orientations of the rightmost cameras. The advantage of using this representation over others in studying the oculomotor system is that it represents the shortest path from the current orientation to the primary orientation (Nakayama and Balliet 1977; Schnabolk and Raphan 1994a). Consequently, we will use the Euler axis-angle form to represent orientation throughout the paper.

View larger version (76K): [in this window] [in a new window]   FIG. 2. Initial and final orientations of the cameras plotted in Fig. 1 are repeated here. Arrows indicate the Euler axes that describe the orientation of the objects on the right. The Euler axis is the axis around which the camera, starting from the orientation depicted on the left, has to be rotated in order to assume the orientation on the right by means of a single rotation. The final orientations in A and B are different, and the Euler axes reflect this difference.

Simplifying assumptions and limitations

In this paper we treat each agonist-antagonist pair of muscles as a single ideal muscle, able to apply either a positive or a negative torque. With this simplification, which is, explicitly or implicitly, used also in the models developed by Tweed and Vilis and by Schnabolk and Raphan, the tension that the muscles develop is linearly proportional to their innervation (Haustein 1989), i.e., the tension-innervation ratio is constant. We introduce other simplifications, also used by Tweed and Vilis (1987), Schnabolk and Raphan (1994a), and Raphan (1997): first, we suppose that the three pairs of muscles act in orthogonal planes; second, we assume that straight ahead is the mechanical resting point for the eye; and third, we assume that the neural coordinate systems are orthogonal and aligned with, or at least symmetrical about, Listing's Plane. Note that this last assumption is experimentally supported (Crawford 1994; Crawford and Vilis 1992; Hepp et al. 1993), whereas the others, even though commonly used, have no experimental support. Finally, we use a first-order approximation for the plant [the Schnabolk and Raphan model also includes the inertia of the globe, which has a negligible effect (Tweed et al. 1994a)], ignoring any nonlinearity present in the system.

Even though these approximations somehow limit the accuracy of the model presented here, we considered it necessary to use the same simplifications introduced by the other workers who addressed the subject. Moreover, the results reported here are general and could be extended to other rotational joints (natural or artificial), so that a general treatment of the subject, not too strictly related to the exact geometry and properties of the ocular plant, is actually desirable.

The simulations reported in this paper were performed using MATLAB (The Mathworks, Natick, MA) and "C" programs running on a Challenge-L computer (Silicon Graphics, Mountain View, CA).

	RESULTS

Abstract Introduction Methods Results Discussion References

We developed a novel quantitative method to estimate dynamic errors (i.e., the magnitude of the postsaccadic drifts) that occur when a commutative controller is used to drive a noncommutative plant, and thus to decide whether a commutative controller can be used to generate the appropriate innervation signals for the oculomotor plant. Because of the novelty of the method and of the complexity of the subject, we deem it necessary to start by applying our reasoning to a simpler, better known, case, i.e., the saccadic system in one dimension.

Saccadic system in one dimension (rotations around a single axis)

In a first-order approximation, the oculomotor plant in one dimension (1 pair of muscles pulling around a single axis) can be described as a Voigt element (i.e., the parallel connection of a viscous element and an elastic element), which represents the passive characteristics of the plant, and a tension applied in parallel to it, which represents the action of the contractile elements (Robinson 1964). In theory, the inertia of the globe should also be taken into account, but its value is so small (Robinson 1964) that for our purposes it can safely be ignored (Tweed et al. 1994a).

Newton's law states that the torque T(t) exerted by the muscles is the sum of the viscous and the elastic torque. If we indicate with (t) eye orientation at time t and with (t) eye angular velocity at time t

(1)

where B is the viscosity and K the stiffness of the plant (muscles and orbital tissues).

In one dimension, rotations and translations are essentially equivalent, and angular velocity is the derivative of orientation. Thus the transfer function of the system in the Laplace domain is

(2)

The system has a single pole, and the step response of the system is an exponential with time constant T_c, equal to the ratio of the viscosity to the stiffness. As shown by Robinson (1975), the value of this time constant in the monkey is ~200 ms.

The torque T(t) produced by the muscles is related to the innervation I(t) that the muscles receive from the motoneurons by the factor S, called the tension-innervation ratio (see METHODS)

(3)

Both Eqs. 1 and 3 define the same torque; consequently, at each instant of time

(4)

where B_S = B/S and K_S = K/S.

To rotate the eye from its initial orientation ₀ to a new orientation ₁, it would be sufficient to change the innervation signal in a stepwise manner, changing its value from K_S·₀ to K_S·₁. However, this would produce a slow movement, with the eye orientation changing from ₀ to ₁ with an exponential time course (step response of a single-pole system; Fig. 3A). For any movement, at least 600 ms (3·T_c) would be needed to foveate the target. This is clearly undesirable, because retinal slip during this time could strongly degrade vision (Westheimer and McKee 1975).

View larger version (14K): [in this window] [in a new window]   FIG. 3. A: when a step torque is applied to the system described by Eq. 1, a slow exponential movement is produced, and >600 ms (3 times the time constant of the system) are needed to attain the final orientation. B: in contrast, if a Pulse-Step (e.g., the Pulse is a half cycle of a sine wave and the Step is obtained by integrating it) torque signal is applied, the attainment of the final orientation is much quicker. C: however, the Pulse and the Step must be appropriately matched, otherwise a Pulse-Step mismatch occurs and the fast part of the movement is followed by a slow drift (called postsaccadic drift) toward the final orientation.

Recordings in the motoneurons of the extraocular muscles show that, fortunately, the innervation signal does not change in a stepwise manner during saccades (Fuchs et al. 1988; Luschei and Fuchs 1972; Robinson 1970; Robinson and Keller 1972). In fact, the motoneurons' activity is composed of a tonic (Step) and a phasic (Pulse) component. During periods of fixation, only the Step is present, whereas during saccades both the Step and the Pulse are present, and the movement is considerably faster (Fig. 3B). Moreover, recordings in the brain stem and anatomic studies (for a review see Hepp et al. 1989) revealed that these two components are generated in different neural structures and that they are summed together at the level of the motoneurons. Thus, as far as the saccadic system is concerned, the innervation signal can be expressed as

(5)

When the eye is not moving (steady state,  = 0), only the Step component is present in the motoneurons' activity. Because both Eqs. 4 and 5 define the same signal (innervation), it follows that, in steady state

(6)

Thus, at least in steady state, the Step encodes eye orientation (). However, the above considerations do not guarantee that the Step encodes the orientation during the movement, when  is not zero. In fact, Eqs. 3 and 5, which describe innervation in two different ways, guarantee only that the sum of Step and Pulse is proportional to the mechanical torque, but do not impose any constraint on the relative contribution of Pulse and Step to the innervation signal. For example, the Step could assume an arbitrary value during the movement, and its final value could be calculated directly from the desired final orientation ₁ (i.e., target position) using Eq. 6. Nevertheless, if the orientation of the eye at the end of the Pulse is not equal to the orientation associated with the value of the Step (which, by Eq. 6, always determines the steady-state condition), the eye would drift to ₁ with a time constant T_c (Fig. 3C). Such slow drift (called postsaccadic drift or glissade) is caused by a Pulse-Step mismatch (Bahill et al. 1975; Optican and Robinson 1980), and when it occurs the time during which the eye is not stable is increased. To avoid glissades the Pulse and the Step must be precisely matched to each other.

The two signals (Pulse and Step) could be generated independently, as long as they were appropriately matched, but it would be parsimonious to generate one from the other. There is compelling physiological evidence that the Step is generated from the Pulse. For example, when saccades are interrupted in midflight by stimulation of the omnipause neurons, which inhibit the pulse generators, the eye does not drift toward the goal or the initial orientation, but stays still (Keller 1974). Furthermore, when the Pulse is produced by electrically stimulating the brain areas that carry the Pulse signal, no target is specified. In this case, the eye displacement is a function of the duration and intensity of the stimulation; however, when the stimulation is over, the eye does not drift but maintains its orientation (Cohen and Komatsuzaki 1972; Crawford et al. 1991; Crawford and Vilis 1992; Keller 1974). In other words, the Step is always appropriate to keep the eyes where they are, even when a Pulse is artificially generated or modified; we infer from this that the Step is calculated dynamically from the Pulse. This implies that the Step always encodes eye orientation, even during the movement, when  is not zero (i.e., Eq. 6 holds all the time). This is in sharp contrast to the statement by Schnabolk and Raphan (1994a, p. 624), that the Step encodes orientation only in steady state.

Now, because the Step encodes orientation, from Eqs. 4-6 it follows that

(7)

Now that we have determined the value of the Pulse and the Step, we have to discover how the Step can be generated from the Pulse, i.e., we must find a function f such that

(8)

In the case of rotations around one axis, (t) is the derivative of (t), and the problem of determining f is trivial

(9)

So, the Step can be obtained by simply integrating (in the mathematical sense) the Pulse, with an appropriate gain. This automatically guarantees a Pulse-Step match, and thus the absence of postsaccadic drift (Robinson 1975).

Effect of a Pulse-Step mismatch and estimation of the overall mismatch

Before going on to show how the Step can be computed for rotations around arbitrary axes, it is important to see what happens when Eq. 9 is not obeyed, i.e., when an incorrect function is used to derive the Step from the Pulse. Clearly a postsaccadic drift is expected, but predicting its magnitude is not straightforward. For example, suppose that at a given point the eyes are still (i.e., the Step is appropriate to maintain eye orientation) and their orientation is ₀. To make a saccade, a Pulse is generated; at the very beginning of the saccade, the Step encodes orientation by hypothesis, and, thus (Eq. 5) the Pulse must encode eye velocity. However, suppose now that, for some reason, the gain of the integrator (Eq. 9) is not appropriate, i.e.

(10)

with k  1. Clearly the Step no longer encodes orientation; however, regardless of the value of Pulse and Step, the total torque is always proportional to the innervation (Eqs. 1, 3, and 5), i.e.

(11)

Because the Step accounts for only a fraction of the orientation (Eq. 10), the Pulse must account both for the velocity and for the remaining part of the orientation. Thus we can rewrite Eq. 11 as follows

(12)

with (t)  0. Now, from Eq. 12 it is clear that the Pulse does not simply encode the velocity (t), but some different signal that is not proportional to (t). However, if the viscous torque [B·(t)] is much larger than the elastic torque [K·(t)], B·(t)(t)·K·(t), and thus the Pulse can still be considered as proportional to (t). In this case, Eq. 10 essentially holds (i.e., the Pulse always encodes velocity), and the instantaneous Pulse-Step mismatch (i.e., k; it will become clear later on why the term instantaneous mismatch is used) is a good estimate of the overall Pulse-Step mismatch. For example, if k is equal to 0.9, during the movement the Step will change by only 90% of the change in orientation (Eq. 10), and thus the overall mismatch (i.e., the magnitude of the postsaccadic drift) will be 10% of the amplitude of the movement.

In contrast, if the viscous component of the torque is smaller than (or comparable with) the elastic torque (i.e., the velocity of the movement is low), Eq. 12 implies that the Pulse does not encode angular velocity (i.e., Eq. 10 does not hold anymore), and it is thus impossible to evaluate the overall Pulse-Step mismatch from the instantaneous mismatch. Nonetheless, note that when the innervation signal changes very slowly (i.e., Pulse  0 and  0), Eqs. 1 and 4 can be reduced to Eq. 6; in other words, the orientation matches the Step, and the Pulse-Step mismatch, which by definition (Bahill et al. 1975) is the difference between the orientation of the eye and the Step of innervation, is null, regardless of the value assumed by the instantaneous Pulse-Step mismatch (i.e., k).

To summarize, when   , the overall mismatch coincides with the instantaneous mismatch, and, when   0, the overall mismatch is always null. For all intermediate values of , the overall mismatch is then smaller than the instantaneous mismatch, because part of the drift occurs during the movement. Consequently, the instantaneous mismatch provides an upper bound for the overall mismatch: if the instantaneous mismatch is small, the overall mismatch will be small, always. However, if the instantaneous mismatch is large, that does not necessarily mean that the overall mismatch will be large: if the speed of the movement is small, the overall mismatch can still be small. Nevertheless, because of the high speeds that characterize saccadic eye movements, the viscous torque is larger than the elastic torque; accordingly, when saccades are considered, the instantaneous mismatch is only slightly larger than the actual overall mismatch. In contrast, when slow movements (e.g., smooth pursuit movements) are considered, the instantaneous mismatch overestimates the overall mismatch.

Rotations around arbitrary axes

As we already pointed out (see METHODS), the Euler representation of orientation (,) represents the shortest path taking the eye between primary orientation and a given orientation. Thus it is reasonable to say that the restoring torque due to the stiffness K of the plant will tend to realign the visual axis with the primary position acting along the unit-length axis , with an intensity proportional to  (Schnabolk and Raphan 1994a). Accordingly, if the axis-angle form is used, expressing the torque in three dimensions is straightforward

(13)

where (t) is the torque exerted by the muscles and (t) the angular velocity (both are 3-D vectors).

The torque is applied to the eyeball by appropriately innervating three pairs of muscles. Using the simplification that each pair of muscles is collapsed into an equivalent ideal muscle (see METHODS), from now on we will assume that three innervational signals are generated, forming a vector of innervation  = [I₁ I₂ I₃]^T. The torque exerted by the muscles can be evaluated by multiplying the vector of action of each pair of muscles, , (the unit-length vector along which the globe rotates under the action of a pair of muscles), by the corresponding innervation and by the innervation/tension ratio

(14)

The total torque applied to the globe is then the sum of the three vectors obtained applying Eq. 14 to each pair of muscles

(15)

This is mathematically equivalent to multiplying the matrix that has as columns the vectors of action of the three muscles, by the vector of innervation

(16)

where both the Step and Pulse are vectors of three components. As stated in METHODS, we will suppose that the three pairs of muscles act in orthogonal planes and that the matrix is the identity matrix.

Now, using the same line of reasoning used for the one-dimensional case, it can be shown that, to avoid a Pulse-Step mismatch, from Eqs. 13 and 16, which both define the torque vector, it follows that

(17)

(18)

At this point we are again facing the problem of how to generate the Step from the Pulse

(19)

Intuitively, the simplest thing to do to extend the model described by Eq. 9 [i.e., the Step is the integral of the Pulse, as proposed by Robinson (1975)] from one dimension to three dimensions, is to use three integrators, one for each pair of muscles. However, as pointed out in the INTRODUCTION, rotations along arbitrary axes do not commute, and thus the orientation can not be obtained simply by integrating the angular velocity, as assumed in the one-dimensional case to compute the Step from the Pulse (i.e., to derive Eq. 9 from Eq. 8).

To account for this fact, Tweed and Vilis (1987) developed a model (the quaternion model) that uses noncommutative, rotational operators to generate the Pulse and to compute the Step from the Pulse. The quaternion model is a mathematically correct extension of the Robinson model, and, if the appropriate weights are selected, it produces perfect three-dimensional movements, without any postsaccadic drift (the Pulse and the Step are perfectly matched). Although the mathematics of the model are fairly complicated, it is conceptually straightforward: a Pulse is generated and the Step is evaluated as a function of the Pulse and of the current eye orientation (i.e., of the Step itself), by means of a multiplicative feedback loop (Tweed and Vilis 1987).

In contrast, Schnabolk and Raphan (1994a) proposed that, because the innervation signals applied to the muscles produce torques, which commute, a commutative controller, in which the Step is obtained by integration of the Pulse, can be used. Now, it is true that torques commute, and thus the steady-state orientation is determined by the summation of the torques in any order. In fact, regardless of the dynamics of the movement, in steady-state (Pulse and angular velocity both null), the orientation will be determined by Eq. 17, which simply relates the orientation of the eye with the value of the Step, irrespective of past movements and innervation signals. However, the commutativity of torques guarantees only what happens in steady state, but does not enforce Pulse-Step matching. Nonetheless, using simulations, Schnabolk and Raphan showed that their model also seems to behave fairly well dynamically, at least when small movements are considered. To solve the paradox that a noncommutative plant can apparently be controlled by a commutative controller, a very simple question has to be addressed: just how large is the postsaccadic drift when a commutative controller drives the oculomotor plant?

How noncommutative is the plant, as seen by the controller?

The only information needed to decide whether a commutative controller is sufficient for a 3-D ocular plant, is the magnitude of the Pulse-Step mismatch that occurs when a commutative neural controller is used. In fact, in normal movements small drifts, due to Pulse-Step mismatch, are sometimes present (Bahill et al. 1975); however, they are, and they must be, small.

If the plant were commutative, the integral of vectorial velocity would be orientation and thus, in analogy to Eq. 9, we could write

(20)

So, ignoring the scaling factor K/B, the system would be commutative if and only if

(21)

If Eq. 21 holds, the Step (which by Eq. 17 should encode orientation) can be computed by simply integrating the Pulse (which by Eq. 18 encodes angular velocity), with an opportune gain, and a commutative controller would be sufficient to produce movements with correct dynamics. Thus one easy way to estimate the magnitude of the postsaccadic drift is to check how inaccurate Eq. 21 is. So, if we compute

(22)

we can define the following parameter

(23)

where the operator · indicates the Euclidean norm of a vector. We call  the instantaneous Pulse-Step mismatch (or, more concisely, the instantaneous mismatch), because it represents the difference between the derivative of the orientation and the Pulse (at least when the speed of the movement is not too low, as explained earlier). If  is small, the integral of the Pulse is very close to the orientation, and thus a simple integrator (i.e., a commutative controller) can be used to compute the Step from the Pulse. It can be shown (see APPENDIX A) that  can be expressed as a function of the eccentricity  and of the angle  comprised between the orientation vector and the instantaneous velocity vector

(24)

where  is expressed in radians.

As previously explained (see section entitled Effect of a Pulse-Step mismatch and estimation of the overall mismatch),  is an upper bound for the overall mismatch. So, if, for example, during a movement,  is always equal to 0.2, it means that the overall Pulse-Step mismatch will be smaller than 20% of the amplitude of the movement, but because of the high speed of saccades (see above), only slightly smaller.

Equation 24 shows that the instantaneous mismatch (which in this case coincides with the relative difference between the angular velocity and the derivative of orientation) increases almost linearly with the eccentricity , and it is a function of the angle between orientation and angular velocity. Because  is not a constant, and it varies with eccentricity, a simple change in the gain of the integrator is not sufficient to fix the problem. It is worth noting that, because  is a function of eccentricity, the same movement (fixed axes rotation) executed at different eccentricities will have different Pulse-Step mismatches. When rotations around one axis are considered,  is always equal to zero, the system is commutative, and  is always zero.

In Fig. 4 we plot  as a function of the eccentricity  when  is equal to 90° (i.e., the worst case scenario for a given eccentricity), e.g., at the beginning of an upward saccade made from a rightward orientation. For small eccentricities (<15°),  < 10%, so the system is almost commutative. And this is exactly why the movements simulated by Schnabolk and Raphan (1994a), which had small amplitudes, were characterized by a fairly good dynamic behavior, with a limited, but not zero, Pulse-Step mismatch. However, for eccentricities >15°,  can become large, and large Pulse-Step mismatches are expected. And that is exactly why when Tweed et al. (1994a) simulated the model proposed by Schnabolk and Raphan using large movements and eccentricities, they obtained movements with large postsaccadic drifts, which are not observed experimentally.

View larger version (13K): [in this window] [in a new window]   FIG. 4. Instantaneous mismatch  is plotted as a function of ocular eccentricity.  is defined as the norm of the difference between the angular velocity and the derivative of orientation normalized by the norm of the angular velocity (Eq. 23). Here the case when the angular velocity is orthogonal to the orientation (worst case scenario, see text) is plotted. The instantaneous mismatch increases almost linearly with the eccentricity.

These observations led Tweed and co-workers to conclude that such a model cannot be right, and that the noncommutativity of rotations (more precisely the derivative of orientation not being angular velocity) must be accounted for, either neurally or mechanically. The neural solution to the problem is to use the model developed by Tweed and Vilis (1987); as a mechanical solution, Tweed and colleagues (1994a) proposed the so-called linear plant model, in which eye orientation and its derivative, but not angular velocity, are encoded neurally (and thus one can be computed by simply integrating the other), and then the plant somehow carries out the necessary transformations. However, Tweed and colleagues did not suggest any scheme regarding the physical implementation of the "linear plant."

Straumann and colleagues (1995) made another attempt to solve the problem mechanically, using a commutative neural controller. They showed that if a second-order plant is paired with a neural controller that generates a Slide component in addition to the Step and the Pulse, the postsaccadic drift produced by Schnabolk and Raphan's model can be reduced. However, this finding was based on a very restricted subset of simulations, and it has been recently shown (Tweed 1997b) that it does not hold for arbitrary movements.

Recently, Raphan (1997) replied to the observation of Tweed et al. (1994a), noting that the simulations by Schnabolk and Raphan assumed that the planes of action of the muscles do not change with the orientation of the eye, whereas Miller et al. (1993) showed that they do, as a consequence of passing through pulleys coupled to the orbit. Miller showed that the muscles do not change their path from their origin point to some point behind the insertion point; from there they go straight to the insertion point. This intermediate point corresponds to the location of orbital tissues that act on the muscles as pulleys, constraining their movements. As Miller and co-workers (1993) pointed out " ... orbital mechanics is fundamentally different under a pulley model. Here, the axis of rotation is determined by the center of rotation, the effective location of the pulley, and the anatomic insertion. Unlike the conventional model, the pulley model predicts that gaze movements out of the muscle plane will cause the axis of rotation to tilt with the globe ... ."

In Fig. 5 this concept is expressed graphically with a scale model of the human orbit (generated using anatomic data from Miller and Robinson 1984). If no pulleys were present (Fig. 5A), the muscles would be able to move freely in the orbit, and an elevation of the eyes by 45° would cause a mild backward tilt of the axis of action of the horizontal recti. This change in the vector of action (i.e., the angle around which a pair of muscles rotates the globe) is due to the change of the position in space of the insertion point of the muscles on the globe, and it has always been ignored. If pulleys are introduced (Fig. 5B), the geometry changes dramatically, and the axes of action of the muscles can change considerably with the orientation of the eyes. This enlarged effect (which cannot be ignored) is due to the reduced distance between the insertion point and the fixed point, which corresponds not to the muscle origin but to the pulley position. Thus for the same innervation, the rotation produced by the muscles varies as a function of eye orientation.

View larger version (25K): [in this window] [in a new window]   FIG. 5. Axis of action of the horizontal recti for 2 different models of orbital mechanics. The schematics are a scaled version of an actual human orbit (Miller and Robinson 1984). A: if the muscles can move freely in the orbit, the muscular path does not change much whether the eye is in primary position (green solid line) or elevated by 45° (red solid line). Correspondingly, the axis of action (green dotted and red dashed lines) is approximately fixed in the orbit. B: 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 the orientation in the eye. However, the axis of action of the muscles changes dramatically with orientation; the magnitude of this change is clearly a function of the position of the pulleys.

Introducing a first-order mathematical approximation for the action of the pulleys, previously only modeled qualitatively by Miller and colleagues, Raphan (1997) showed that the same movement simulated by Tweed et al. (1994a) using the original Schnabolk and Raphan model, was essentially drift-free when the action of the pulleys was accounted for. However, the effect of pulleys on arbitrary movements remained to be determined, because simulations of individual movements had previously been found to be misleading (Schnabolk and Raphan 1994a; Straumann et al. 1995). Furthermore, an analysis of how the pulleys affect the physical significance of the innervation signals was not provided, and, as a consequence, some of the conclusions drawn were inaccurate. Finally, as Crawford and Guitton (1997) pointed out, for the pulley scheme to work, the pulleys have to react differently to phasic and tonic inputs, but neither Crawford and Guitton nor Raphan have shown how such a differential effect can emerge.

Rotations with pulley effect

In the case described above (rotations around arbitrary axes without pulleys), it was assumed that the axes of action of the three muscle pairs were orthogonal and the matrix was the identity matrix. We will now assume that the planes of action of the muscles are not fixed in the orbit but are a function of the instantaneous orientation of the eyes. For the sake of simplicity, suppose that the axes of action of the muscles rotate around the axis of rotation by an angle that is a fraction, K, of the angle of rotation  (Raphan 1997).

When such a partial muscular slip is introduced, the matrix , having as columns the vectors of action of the three pairs of muscles, becomes a function of the orientation of the eyes, and it corresponds to the rotation matrix associated with a rotation of  = K· degrees around the axis (Raphan 1997)

(25)

Clearly, Eq. 25 holds under the assumption that is equal to the identity matrix when the eye is in primary position. If this condition does not hold, is equal to the product of R[,] and ₀, the muscle matrix in primary position. We will assume that ₀ is equal to the identity matrix, as done throughout the paper (see METHODS). Thus Eq. 25 holds as it is.

So, the torque applied to the globe (Eq. 16) can be rewritten as

(26)

where both the Step and the Pulse components of the innervation are vectorial signals. With the use of this equation and the definition of the torque in terms of viscous and elastic components (Eq. 13), it is easy to show that, when the partial slip of the muscles is accounted for, the Pulse-Step mismatch would be null if and only if the Pulse and the Step are defined as follows

(27)

(28)

With the use of the following properties of rotation matrices ( and generic)

(29)

and taking into account Eq. 25, the Step and the Pulse are

(30)

(31)

From Eqs. 30 and 31 it is clear that the pulleys exert a different action on the Step and the Pulse. In particular, the Step encodes eye orientation even when the pulleys are considered. This very important finding will be discussed and illustrated in a more intuitive manner later on.

Going back to the quantification of the instantaneous mismatch, the problem is reduced to quantifying the difference between the derivative of the Step and the Pulse; if and only if this difference is small can a commutative controller be safely used. Ignoring the ratio between K and B as done before, we can define  as

(32)

In this case,  is not equivalent to the difference between angular velocity and the derivative of eye orientation, because the Pulse does not encode angular velocity in this model (see Eq. 31).

It can be demonstrated (see APPENDIX B) that  can be expressed as a function of the eccentricity , of the angle  and of  = K· (33) So, Eq. 24 can now be derived as a particular case ( = 0) of Eq. 33.

In Fig. 6 we plot the value assumed by  when  is equal to 90°, as was done in Fig. 4. We plot different curves, showing the value assumed by  for five different values of K (0, 0.25, 0.5, 0.75, and 1). It appears clear from Fig. 6 (and it can be easily confirmed by setting the derivative of  equal to 0) that the optimal value of K is 0.5. If the pulleys are located so as to produce a 50% muscular slip (K = 0.5), the value of  always stays below 0.025. This means (see above) that the overall mismatch will always be <2.5%, regardless of the movement. Furthermore, the reader should bear in mind that the curves reported in Fig. 6 represent the worst case scenario (when angular velocity is orthogonal to orientation) for a given eccentricity. When an actual movement is considered, the value of  varies during a saccade, and its average tends to be <0.02.

View larger version (16K): [in this window] [in a new window]   FIG. 6. Instantaneous mismatch is reported for the pulley model. As in Fig. 4, the worst case scenario is plotted (angular velocity orthogonal to orientation). Five values of K (0, 0.25, 0.5, 0.75, and 1) are considered, which correspond to different positions of the pulleys. The position of the pulleys has a dramatic effect on the instantaneous mismatch; for K = 0.5 the pulleys minimize postsaccadic drift, and the system, as seen by the brain, is essentially commutative.

We simulated the same eye movement (from a secondary to a tertiary position) coupling a c