INTRODUCTION

Top Abstract Introduction Methods Results Discussion Appendix References

Horizontal eye position¹ is controlled principally by the actions of the two horizontal recti muscles, namely the medial and lateral rectus. Stable eye position is achieved when the forces exerted by these muscles balance the passive elastic force of the orbital tissues (which acts to restore the eye to near the primary position) so that there is no net torque on the eyeball. However, the requirement on the horizontal recti to counteract the passive torque only constrains the difference between the forces that are exerted by the two muscles not their individual magnitudes. This can be seen clearly in the case of the primary position itself: here the passive torque is essentially zero, so provided the force exerted by the medial rectus is equal to that of the lateral rectus, the eye will look straight ahead. For human eye muscles, the actual value of the force in the primary position is ~10 g (e.g., Carpenter 1988). How has the oculomotor control system arrived at this value?

Answering this question is necessary for understanding the principles underlying the control of eye position, and as such is relevant both to normal function and to those clinical conditions in which the control of eye position is not normal. Also the selection of force magnitudes in the extraocular muscles (EOMs) is an example of a fundamental and widespread control problem that arises whenever there are more muscles acting on a joint than there are degrees of freedom through which the joint can move. It is possible that the strategy used by the oculomotor system to solve this redundancy problem may have general application.

A quantitative solution to the redundancy problem for EOMs has been proposed by Daunicht (1988, 1991). Daunicht considered the task faced by the control system in maintaining the eye a small distance from its current position, for example the primary position. Increasing the command signals to both horizontal recti, thereby producing cocontraction, would not be efficient for this purpose: in the worst case, the change in motor commands would produce no change in position at all. In contrast, the most efficient change in motor commands is the one that gives the maximum change in eye position. Daunicht proposed that the oculomotor system uses the most efficient changes in motor command, which in effect means that any particular change in eye position is maintained by the smallest possible change in motor commands. "Smallest possible" here refers to the smallest possible sum of the squared changes in motor commands (see METHODS): if the changes in motor command are considered to form a vector, this corresponds to the minimum norm or magnitude of the vector. "Minimum norm" control is therefore the term used in the present study for the proposed control principle, in preference to the possibly confusing term "minimum effort" (Daunicht 1988).

Daunicht's scheme is, to our knowledge, the most developed quantitative solution so far suggested for the redundancy problem in extraocular motor commands. Moreover, its underlying principle (sometimes referred to as pseudoinverse control) has been proposed as a general method for both biological and artificial motor systems (e.g., Klein and Huang 1983; Pellionisz 1984). The present study therefore sought to determine whether Daunicht's scheme is in fact consistent with published data on the behavior of EOMs and ocular motoneurons (OMNs). The study's scope is restricted to consideration of horizontal eye position despite the fact that Daunicht's proposal deals with three-dimensional eye position, because even in the simplified case relating the minimum-norm rule to experimental data are far from straightforward. For the same reason, this study deals only with the motor commands that relate to conjugate eye position.

For horizontal eye position, the minimum-norm principle predicts that a small change in position will be associated with an increase in the motor command to the agonist muscle and a decrease in the command to the antagonist muscle and that the magnitude of the changes in command will be directly proportional to the strength of the muscle in each case. The first of these is a qualitative prediction, long known as reciprocal innervation and possibly attributable to Descartes (Sherrington 1947). It is the second, quantitative, prediction that is the distinctive contribution of the minimum-norm principle and one that requires operational definitions of the terms motor command and muscle strength.

The original definition of change in motor command was "motor activity change" (Daunicht 1988). An obvious referent for this phrase would be the change in summed activity of the efferent nerves to a given muscle, corresponding to the change in summed activity of the parent pool of OMNs. The firing rates of OMNs in relation to eye position have been obtained by electrophysiological recording in awake monkeys (Fuchs et al. 1988; Gamlin and Mays 1992; Keller 1981; King et al. 1981; Van Gisbergen and Van Opstal 1989). A striking feature of OMN pool activity, not explicitly mentioned by Daunicht, is the way it increases nonlinearly when the eye moves into the ON direction of the relevant muscle as individual OMNs and their associated motor units are recruited. Because of recruitment, the change in motor command (as interpreted here) that corresponds to a fixed small change in eye position varies very markedly over the oculomotor range.

The original definition of muscle strength was as a "coefficient ...  of ... neuromuscular transmission" (Daunicht 1991), which could be derived from the slope of the relation between muscle tension and neural activity (Daunicht 1988). Given the preceding interpretation of motor command, the strength of an EOM at a given eye position corresponds to the isometric change in muscle force produced by a unit change in summed activity of the OMN pool at that position. A crucial feature of muscle strength defined in this way is that its magnitude is determined by the manner in which motor units are recruited (as described in detail in METHODS). For example, if motor units are recruited in order of increasing strength (the size principle,²) (e.g., Henneman and Mendell 1981; Henneman et al. 1965), then the "muscle strength" of an EOM also will increase as the eye moves to positions further in the EOM's direction of action.

One consequence of the dependence of muscle strength on recruitment is that the minimum-norm principle is in fact making precise predictions about motor-unit recruitment in EOMs. A second consequence is that direct measurements of EOM muscle strength as a function of eye position are not available (see METHODS). However, the values of isometric-force gradient needed for the estimation of muscle strength can be derived from measurements of extraocular muscle tension as a function of muscle length and fixation command in people (e.g., Miller and Demer 1994; Miller and Robinson 1984; Robinson 1975; Simonsz and Spekreise 1996), on the assumption that monkey and human do not differ significantly with regard to OMN activity or extraocular muscle properties. The present study obtains estimates for the range of horizontal eye positions over which the minimum-norm rule holds by using the muscle-force measurements in combination with Daunicht's proposal to generate predicted changes in motor commands. These then are compared with the actual changes observed electrophysiologically. Preliminary findings have been published previously in abstract form (Dean et al. 1996).

It should be emphasized that, although Daunicht's minimum-norm rule relates changes in motor command to changes in position, it is not concerned with the movements by which those changes are achieved. Minimum-norm control deals only with commands to the eye muscles when the eye is in static equilibrium. Movement commands, especially those for fast saccadic movements, may be far from minimum norm. One advantage of restricting the problem in this way is that complex issues of plant dynamics can be ignored.

	METHODS

Top Abstract Introduction Methods Results Discussion Appendix References

This section is divided into five parts. The first describes the basic relationships among muscle force, length and fixation command in EOM. The second indicates how Daunicht's minimum-norm rule can be derived for the one-dimensional case. The third and fourth sections consider how the two key terms, motor command and muscle strength, in the derivation should be interpreted in the light of the data on the recruitment of motor units in the EOMs. Finally, methods for estimating isometric-force gradients are outlined.

Length-tension relationships for EOM

The static force exerted by an EOM is a function of both its length and the fixation command signal delivered by its parent nerve. One method for measuring this function in the human lateral rectus muscle (Robinson et al. 1969) is shown schematically in Fig. 1A.

View larger version (16K): [in this window] [in a new window]   Fig. 1. A: diagram of experimental design for measuring tension in the left lateral rectus muscle in awake human subjects (Robinson et al. 1969). Lateral and medial rectus muscles are detached from their insertions into the globe of the left eye, and the lateral rectus is fixed to a strain gauge. This allows its length to be varied and the resultant tension to be measured. Subjects are instructed to fixate a target at displacement  with their right eye. B: pattern of results typically obtained from the design shown in A. y axis shows total tension in the left lateral rectus muscle. x axis shows muscle length, converted into its equivalent of angular displacement of the eye, .  = 0 is the muscle length corresponding to the primary position, and  > 0 means muscle lengths shorter than the primary-position length, i.e., the eyeball rotated left in the pulling direction of the muscle. Five curves shown are produced by different levels of innervation to the left lateral rectus as specified by the fixation commands corresponding to target displacements .

The lateral and medial recti from one eye (shown in Fig. 1A as the left eye) are detached in preparation for strabismus surgery. The lateral rectus is attached to a strain gauge, enabling its length (L) to be fixed and its tension (T) to be measured while the subject fixates a target at location  with the right eye. Assuming Hering's law of equal innervation by which the same fixation command also is passed to the left eye, it is possible to plot T as a function of L at different values of fixation command (Fig. 1B). In this figure, the length of the left lateral rectus L is transformed into an equivalent eye position  deg. The sign convention adopted in Fig. 1B is that left is positive, i.e., stretching the left lateral rectus corresponds to a decrease in , whereas looking further to the left corresponds to an increase in .

The behavior shown in Fig. 1B can be described by a family of hyperbolic curves as suggested by Robinson (1975). The version of Robinson's equation used here is

(1)

The relation between tension T and muscle length  depends on the three parameters k, a, and e. The hyperbolic shape of the length-tension curves reflects the observations that when an EOM is stretched far enough (Fig. 1B, left) at a given level of fixation command, it behaves like a spring with constant elasticity, i.e., tension proportional to length (parameter k). When the muscle is unstretched, its tension is zero. The transition between the two states is gradual rather than abrupt, as indicated by the curvature parameter a (deg). Changing the fixation command to the muscle preserves the overall pattern of response but alters the point at which the transition to "stretched" behavior occurs. The "innervation" parameter e (deg) corresponds to the amount the basic curve is shifted along the x axis as the fixation command  changes. Determination of the relationship between e and  is described later (Estimation of isometric force gradients).

Minimum-norm rule for horizontal eye position

Daunicht's derivation of the minimum-norm rule for all six EOMs (Daunicht 1988, 1991) is simplified here to the case of the two horizontal recti controlling horizontal eye position. (Steps to assess the effects of this simplification and of Daunicht's own simplifications concerning the mechanical details of the EOM system are described in the final part METHODS). Figure 2A shows the eye at equilibrium, looking  deg to the left. Because the eye is not moving, the torques acting on it must balance out. If the forces concerned all act tangentially at the surface of the globe, then this balance can be represented by Eq. 2

(2)

where f₁ is the force exerted by the agonist muscle, f₂ is the force exerted by the antagonist muscle, and F_OT is the force exerted by the stretched orbital tissues (forces pulling to the left are positive). F_OT is a possibly nonlinear function of eye position: the fs are the nonlinear functions of muscle length and fixation command sent to the muscle described by Eq. 1.

View larger version (18K): [in this window] [in a new window]   Fig. 2. A: diagram of eyeball held in equilibrium at horizontal rotation  from the primary position. Two forces act on the globe to rotate it back toward the primary position: the orbital restoring force FOT and the force f2 exerted by the antagonist muscle. Force f1 by the agonist muscle acts to rotate the muscle away from the primary position. Magnitude of the orbital restoring force varies with , and the magnitude of a muscle force depends on the muscle's length p, strength z and motor command m. B: diagram of eyeball held in equilibrium at a new horizontal rotation  + , where represents a very small additional rotation from the primary position. The change in eye position is produced by changes in the motor commands m1 and m2 to the agonist and antagonist muscles. Changes in muscle forces f1 and f2 result from m1 and m2 combined with the changes in muscle length p1 and p2 that are determined by . Orbital restoring force FOT increases by an amount L, where L is the coefficient of elasticity of the orbital tissues at eye position .

Daunicht (1991) made use of the fact that for a very small displacement of the eye from equilibrium (Fig. 2B), the system behaves linearly (see APPENDIX: Derivation of minimum-norm rule for horizontal eye position). It is therefore possible to calculate the change in eye position produced by two (small) arbitrary changes in motor command to the two muscles, m₁ and m₂ (Eq. 3: derived as Eq. A6 in the APPENDIX)

(3)

In this equation, L is the coefficient of elasticity of the orbital tissues measured at the eye position , and ₁ and ₂ are the coefficients of elasticity of the two muscles measured at eye position  and baseline levels of motor command m₁ and m₂, respectively. The terms z₁ and z₂ represent the muscle strengths, which correspond to the isometric force gradients also measured at eye position  and baseline levels of motor command m₁ and m₂ respectively (see APPENDIX).

Equation 3 is a precise representation of the redundancy problem for horizontal eye position, showing not only that a given small change in position can be brought about by (infinitely) many combinations of change in motor command but also giving the actual change in eye position that any given combination will produce. Two particular sets of combination are relevant to the present study. The first is the set that produces no change in eye position. Setting to 0 in Eq. 3 gives Eq. 4

(4)

The equation is represented diagrammatically in Fig. 3A, which plots one motor command against the other and draws a line through those points that produce the particular eye position  if the values of the zs do not change (cf. Feldman 1981). The gradient of this "iso-position" line is z₁/z₂. This implies that for keeping the eye in one place, the commands to the two eye muscles must be both increased (or decreased) and that the change in command to the weaker muscle must be larger than that to the stronger muscle to maintain the balance.

View larger version (12K): [in this window] [in a new window]   Fig. 3. A: schematic plot of the motor command to the antagonist muscle (muscle 2) against the motor command to the agonist muscle (muscle 1) showing the values of the motor command pairs that produce a given eye rotation . Line on which these values lie is termed an iso-position line. Plot shows a small segment of an iso-position line, small enough that the strengths of the muscles z1 and z2 can be considered constant. Two points on the segment are illustrated, corresponding to the pair of motor commands (m1, m2) and a nearby pair (m1+m1, m2+m2). Slope of the line is given by m2/m1, which can be shown to equal the ratio of the muscle strengths z1/z2 (see text). B: same schematic plot as A with the addition of a new iso-position line  +  representing a very small change in position from the original iso-position line . The point (m1+m1, m2+m2) lies on the new iso-position line. If the change in motor commands m1 and m2 that achieves also minimizes norm, it can be shown that m2/m1 is equal to z2/z1 (see text). Because this is the gradient of the line joining the points (m1, m2) and (m1+m1, m2+m2), that line is at right angles to the isop-osition line. Correspondingly, the point (m1+m1, m2+m2) is the point on the iso-position line  +  that is closest to (m1, m2).

The second important set of motor-command changes is illustrated in Fig. 3B, which shows the iso-position line from Fig. 3A (labeled ) together with a very close iso-position line labeled  + . Here the change in motor commands has been chosen to give a point on the new iso-position line such that the line segment that joins points (m₁, m₂) and (m₁ + m₁, m₂ + m₂) is at right angles to the original iso-position line. Two consequences follow from this choice. The first is that because of the right angle, the gradient of the line segment can be deduced from the gradient of the original iso-position line shown in Eq. 3. It is

(5)

The second consequence is the line segment represents the shortest distance between the point (m₁, m₂) and the new iso-position line. The set of motor command changes (m₁, m₂) are therefore the smallest possible that will produce (see APPENDIX for formal derivation) or in other words they are the minimum-norm changes in command. In contrast to the changes in motor commands that maintain position, the most effective changes in commands for producing change in position are of opposite sign to the two muscles (reciprocal innervation), with the stronger muscle getting the bigger change. A more general method used by Daunicht to derive minimum-norm commands for eye position in three dimensions is outlined in the APPENDIX (Pseudoinverse control).

Interpretation and measurement of m

To test whether the prediction of Eq. 5 is borne out experimentally, the terms of the equation have to be defined operationally. The original definition of m was as a "motor activity change" (Daunicht 1988). As indicated in INTRODUCTION, an obvious interpretation of this phrase would be the change in activity of neurons within the relevant pool of OMNs, as illustrated for the schematic distributed model of Fig. 4 (cf. Dean 1996). In response to a change in fixation command , the firing rate of the ith OMN in a pool of n OMNs changes by (FR_i). The sum of these changes in firing rate would then correspond to

(6)

A problem with this definition is that the changes in firing rates are produced by a change in fixation command , the nature of which is not directly known. However, provided the oculomotor system is functioning normally, for conjugate eye positions there is a fixed relationship among , the central fixation command to any one of the six OMN pools, and three of its effects: eye position , OMN firing rates, and force in the corresponding extraocular muscle. This means that there is also a fixed relationship between any two effects of , in this instance OMN firing rates and eye position . The relation of OMN firing rate to eye position is

(7)

which indicates that firing rate varies linearly with eye position above the OMN's threshold position  e.g. (Keller 1981). K represents the slope of the line. This relationship can be used to substitute for (FR_i) in the definition of m (Eq. 6)

(8)

A fundamental feature of OMN pools is that the threshold  varies between OMNs, so that as the eye moves into the ON direction of the muscle controlled by the pool, the number of active units increases. In other words, a basic feature of the fixation command  is that it can recruit OMNs as it gets stronger. The change in motor command m corresponding to a fixed change in position necessarily increases as  increases

(9)

This quantity can be conveniently measured using the gradient of the summed firing rates of the OMN pool when plotted against eye position

(10)

A number of data sets is available for the firing rates of OMN pools. The initial set used in the present study was that shown in Fig. 1 of the review by Van Gisbergen and Van Opstal (1989). This figure plots the slope K against threshold  for 87 OMNs, the firing rate characteristics of which were measured in four previous studies. One of these (Robinson 1970) was of the primate oculomotor nucleus, the remaining three (Fuchs and Luschei 1970; Goldstein 1983; Skavenski and Robinson 1973) were of the primate abducens nucleus. All the studies found that the OMN firing slopes K_i increased as _i increased, that is, as recruitment proceeded. The graph points were converted with Flexitrace software to numerical data, which then were used to calculate the gradient of total OMN firing rate with respect to position, corresponding to the measurement of m/ in Eq. 10 (e.g., Van Opstal and Van Gisbergen 1989).

View larger version (21K): [in this window] [in a new window]   Fig. 4. Simple distributed model of an extraocular muscle (EOM) and its associated ocular motoneurons (OMNs). In response to a change in fixation command , each OMN changes its firing rate by an amount d(FRi) (for the ith OMN) that depends on its firing-rate threshold i and slope Ki. Change in firing rate produces a change in the force fi exerted by the motor unit belonging to each OMN, which depends on the muscle unit's strength xi (as well as on its length, which is determined by the position of the eye ). Total change of force exerted by the EOM is assumed to be the sum the unit force change fi combined with the change of force in the antagonist EOM that also is produced by , it causes the eye to change position by .

Four other data sets, all on primate OMNs, also were used for estimation of m/ so that the sensitivity of the minimum-norm prediction to measurement variation could be assessed.

1) The thresholds for 160 OMNs are shown in a histogram in Fig. 2C of a review by Keller (1981). These are taken from four experiments, two of which (Fuchs and Luschei 1970; Robinson 1970) also contributed data to the set of Van Gisbergen and Van Opstal (1989) described earlier. The two additional studies were by Schiller (1970) on OMNs in the oculomotor and abducens nuclei and by Keller and Robinson (1971) on the abducens nucleus. The review does not give explicit values for the slope K of these units, so the relationship between K and threshold  estimated by Van Gisbergen and Van Opstal (1989) was used to estimate K. An additional estimate was made of the thresholds of the units lumped in the histogram bin labeled ">40 (deg) off," on the assumption derived from the remainder of the data that the number of units per 5° bin decreased linearly as the threshold increased.

2) Slope is plotted against threshold for 78 OMNs in Fig. 2, A and B, of the study by King et al. (1981). The units were recorded from the oculomotor and trochlear nuclei.

3) Slope is plotted against threshold for 81 OMNs in Fig. 5 of the study by Fuchs et al. (1988). These were recorded in the abducens nucleus, and were identified as motoneurons by spike-triggered averaging of the lateral rectus electromyographic (EMG).

4) The threshold and slope (for conjugate eye positions) is given for 74 medial rectus motoneurons in Table 1 of Gamlin and Mays (1992). As with other OMNs, these units showed a positive correlation between firing-rate slope Ki (for conjugate eye positions) and threshold i.

Interpretation and measurement of z

The original interpretation of z was as a "coefficient ... of ... neuromuscular transmission" (Daunicht 1991), which could be derived from the slope of the relation between tension and neural activity (Daunicht 1988). This interpretation does not deal explicitly with the problem of recruitment of motor units. Figure 4 illustrates how a change in fixation command produces a change in muscle force in a distributed model of the OMN pool and its associated motor units. The change in fixation command alters the firing rates of the j recruited OMNs in the pool [(FR₁) ... (FR_j)] (Eq. 9). The altered firing rates in turn change the forces delivered by the motor units (f₁ ... f_j), which in the simplified system of Fig. 4 are assumed to sum to the change in total muscle force f (for further details, see Dean 1996; for qualifications, see Goldberg and Shall 1997; Goldberg et al. 1997a). In normal operation, the change in motor command produces a change in eye position, so that f depends on both the change in fixation command and the change in position (Eq. 1: Eq. A3). The change due to fixation command alone is the change in isometric force, here termed F. The distributed model therefore gives rise to the following expression

(11)

where j motor units have been recruited, and F_i is the change in isometric force generated in the ith unit by the change (FR_i) of its parent OMN.

Measurements in cat indicate that for recruited units the isometric change in force is approximately linearly related to the change in stimulation frequency (e.g., Shall and Goldberg 1992)

(12)

where y_i can be regarded as a measure of the strength of the motor unit (cf. Dean 1996). Assuming simple summation of the changes in force generated by individual motor units, then the change in force for the whole muscle is given by Eq. 13.

(13)

Therefore

(14)

Equation 14 indicates that z is not an easy quantity to measure directly because it is not simply an intrinsic property of the muscle. It depends on the manner in which the fixation command recruits motor units. This would not be a problem if all motor units were of equal strength because z then would be a constant. However, it is known that EOM motor units are not of equal strength (for review, see Goldberg 1990). If (for example) y_i was to increase with recruitment, then so would z. Moreover, for any recruitment order, the influence on z of units recruited later is greater than those recruited earlier because z is a gradient measure and the OMN firing slopes Ki increase as recruitment proceeds (see preceding text). It is not therefore possible to measure z directly in primate EOM by using electrical stimulation of the parent nerve.

Fortunately, it is possible to bypass z when testing the prediction of the minimum-norm rule given in Eq. 5. A change in fixation command gives rise to three effects: a change in OMN firing rates m, a change in muscle force f, and a change in eye position . As mentioned above, in the normally functioning eye, the relation between these three effects is fixed. Thus the isometric component F of the change in individual muscle force can be related to change in firing rates, as in Eq. 11
It follows that
where the subscripts 1 and 2 refer to the two horizontal recti. But Eq. 5 shows that if the minimum-norm rule is obeyed, the ratio of the zs is the same as that of the ms. Thus
so that

(15)

Because m has been defined operationally in terms of firing-rate gradient for the oculomotor pool (Eq. 10), Eq. 15 formulates the prediction from the minimum-norm rule in terms of measurable quantities. The terms on the left-hand side derive from measurements of the length-tension curves in whole EOMs: the terms on the right-hand side derive from measurements of the firing rates of individual OMNs. The relationship between the two embodied in Eq. 15 is thus fundamental for testing the prediction of the minimum-norm rule.

Although the values of z are bypassed for the purpose of testing the minimum-norm prediction, they are important for interpreting the outcome of the test in terms of motor-unit recruitment (see INTRODUCTION). The values therefore are derived (as F/m) in a subsequent section of RESULTS.

Estimation of isometric force gradients

As indicated in Fig. 4, the fixation command produces a change in eye position in the normally functioning eye. Both itself, and the change in muscle length produced by , alter the force exerted by the muscle (f). If, however, the length of the muscle is fixed, then the change in muscle force (F) is caused only by the change in motor command. The quantity F can be estimated from the isometric-force gradient, measured with respect to fixation command (referred to hereafter simply as isometric-force gradient). The behavior of EOMs described by Eq. 1 allows this isometric-force gradient to be calculated (either analytically or numerically) for values of eye position and fixation command interpolated between those at which the original measurements were taken. As with the measurements of m (see preceding text), the present study used an initial set of values for the parameters in Eq. 1, then subsequently explored the effects of varying those parameters on the minimum-norm prediction.

The initial parameter values were derived from the model described in Robinson (1975), as follows.

The parameter k in Eq. 1 corresponds to the coefficient of elasticity of the stretched muscle with respect to change in eye position. Robinson (1975) gives these coefficients with respect to percentage change in muscle length. They are converted here to the coefficients for change in eye position, using the values for muscle length given in Robinson (1975) and assuming that a 1° change in eye position corresponds to a 0.2074 mm change in muscle length. The values are 0.76 g/° for the lateral rectus and 1.01 g/° for the medial rectus.

The parameter a determines how curved the transition is between the two straight line portions of the curve in Eq. 1. Its dimensions are those of force, and the value given in Robinson (1975) is 6.24 g for lateral rectus and 6.49 g for medial rectus.

The parameter e corresponds to the amount the basic curve in Eq. 1 is shifted along the x axis as the fixation command changes. It can be derived from the kind of data shown in Fig. 1B, and values so obtained are shown in Fig. 3 of Robinson (1975). These were from the horizontal recti of strabismic patients (Collins 1971; Collins et al. 1969; Robinson et al. 1969), described as an "average of all results to date provided by C. C. Collins and D. M. O'Meara" (Robinson 1975). However, these values (termed "primary innervation" by Robinson) require adjustment if the net muscle force of an agonist-antagonist pair is to equal the force exerted by the orbital tissues. This adjustment is termed "secondary innervation," and its derivation given in the APPENDIX (Secondary innervation and the parameter e). The best fit quartic curves to the resultant values of e₁ (for lateral rectus) and e₂ (medial rectus) are given in Eq. 16

(16)

and are illustrated in the APPENDIX (Fig. A1). Following the usage in Robinson (1975), the parameter e is expressed as percentage change in muscle length rather than as the equivalent change in eye position. The length-tension curves for the lateral rectus muscle using these values of e₁ with secondary innervation are shown in Fig. 5A (- - -). In addition to the experimentally obtained curves (cf. Fig. 1), Fig. 5A shows the "natural" length-tension curve (), which indicates the forces obtained when the muscle is at the length prescribed by the fixation command. A similar natural curve can be calculated for the medial rectus muscle (Fig. 5B): the difference between the two, adjusted for sign, closely matches the experimentally observed values for orbital force (APPENDIX: Eq. A16). This demonstrates that the method for deriving e₁ and e₂ was successful and that the resultant model is consistent. Finally, the isometric-force gradient with respect to fixation command, F/ can be calculated for each muscle from Eqs. 1 and 16, as described in the APPENDIX (Isometric force gradient with respect to fixation command).

View larger version (18K): [in this window] [in a new window]   Fig. 5. A: length-force curves (···) derived from the equation for extraocular muscle described in Robinson (1975). Values for the equation parameters k, a, and e are those for the lateral rectus muscle and are given in the text. Each ··· gives the length-force curve for a particular fixation command . Curves are plotted at 15° intervals for , ranging from 45° to + 45°. , joins those points at which the position of the eye is the same as the fixation command. This line therefore represents the force in the lateral rectus at different positions of the eye during normal behavior. B: muscle force plotted against eye position for medial and lateral rectus (···). Curve for the lateral rectus is replotted from A: the curve for the medial rectus is derived in a similar manner, using the values for the equation parameters k, a, and e given in the text for the medial rectus. , difference in force between the 2 muscles. , force exerted by the passive orbital tissues, derived from experimental measurements (details in text) but reversed in sign to facilitate comparison with the model. It can be seen that the muscle equations give rise to a net force that almost exactly balances the passive force of the orbital tissues.

Subsequently, the values of the three parameters in this basic model were varied to test the robustness of the minimum-norm prediction.

PARAMETER k. Two main alterations have been proposed to the original model for this parameter. One is that the medial and horizontal recti are of similar stiffness (and length), as argued by Clement (1987) and Simonsz and Spekreise (1996). The second is that the original values for the stiffness were too high (Miller and Robinson 1984; Simonsz and Spekreise 1996). Isometric force gradients as a function of eye position therefore were calculated for two identical muscles and for half the original stiffness.

PARAMETER a. Relatively little attention has been paid to this parameter, possibly because the EOMs normally operate in regions of the length-tension curve that are approximately linear (Fig. 5A). Here, isometric-force gradients were calculated for values of a half or double the original values.

PARAMETER e. As described in the preceding text, one feature of the original model was its use of secondary innervation. The effects of this were examined by calculating isometric-force gradients for the original estimates of the parameter e, that is primary innervation (see APPENDIX, Eq. A14). Also, Clement (1985) has suggested slightly different values of h and w for the equation for secondary innervation (APPENDIX, Eq. A17). Isometric force gradients were therefore calculated for Clement's values, namely h = 5.5, w = 9.0.

1) The full three-dimensional EOM model of Miller and Robinson (1984) was implemented and adjusted so that its performance resembled that of ORBIT (Miller and Shamaeva 1995). The isometric-force gradients of the model to changes in horizontal fixation command then were measured. This was to check the assumption that the horizontal eye muscles operated essentially independently of the other muscles, at least over the range of operation of the model (±30°). It is also a check for the effects of other simplifying assumptions concerning the action of EOMs, for example, the tangential direction of the forces exerted.

2) All the variants considered so far are derived from the same original set of measurements, of length-tension curves in detached horizontal recti (see preceding text). A new set of measurements was obtained by Collins and coworkers (Collins et al. 1975) on four strabismus patients, using force transducers implanted in series between the tendon of a muscle and its severed insertion in the globe. Each transducer was calibrated by attaching a suture between the globe and an external precalibrated strain gauge and obtaining force measurements with both devices while the muscle was stretched out of its field of action and the other eye fixating a series of different target positions. Length-tension curves then were measured, with both the strain gauge and the implanted transducer, by restraining the eye in turn in each of seven positions while the unrestrained eye fixated a series of target positions. The mean data from four muscles (2 medial recti and 2 lateral recti) are plotted in Fig. 1 of the paper. The parameter estimates from these data required for the present study were obtained in two ways, by measurement from the curves plotted in Fig. 1, as described in Dean (1996), and by a MATLAB least-squares estimation procedure from the data points themselves. The two procedures were in substantial agreement. The values produced by the first procedure were k = 0.827 g/°; a = 4.57 g, and (17) This equation for e as a function of fixation command  differs from those derived from the earlier set of measurements in lacking cubic or higher terms, i.e., it is a simple quadratic function. It also differs in that the orbital force resulting when two symmetrical muscles using this equation are opposed is not zero, but a linear function of eye position with slope 0.4 g/°. Thus secondary innervation is not required to balance the orbital force. Isometric force gradients were calculated for these parameter values.

3) Simonsz and Spekreise (1996) describe a version of the original Robinson model simplified for easy clinical use. It employs symmetrical horizontal rectus muscles and lower values of stiffness, as described in the preceding text. In addition, the tension in passive muscle (i.e., with no innervation) increases exponentially with length when the muscle is stretched rather than linearly as in the original model. The basis for this exponential increase is a series of measurements (Simonsz 1994; Simonsz et al. 1986, 1988) on detached horizontal recti in strabismic patients under either general (n = 80) or local anesthetic (n = 32). However, the length-tension curve was linear for contracting muscle, whether the contraction was produced voluntarily (Simonsz 1994) or by injection of succinylcholine (Simonsz et al. 1988). This difference between innervated and passive muscle means that the length-tension curves are compressed along the tension axis (e.g., Fig. 14A), with consequent alteration of the isometric-force gradients. The precise effects of the compression cannot be directly explored with the model of Simonsz and Spekreise (1996), because in that model innervation is taken to be zero for a fixation command of 30°. It was thought important to explore them, however, because they appeared likely to improve the fit of the minimum-norm prediction for large deviations of the eye (more than ~30°). An indirect method therefore was chosen that consisted of asking what the length-tension curves of the horizontal recti would look like if the minimum-norm rule were obeyed exactly. Details of this procedure are given in the APPENDIX (Minimum-norm equations for muscle).

	RESULTS

Top Abstract Introduction Methods Results Discussion Appendix References

If the horizontal eye-position control system were using the minimum-norm rule as described by Daunicht (1988, 1991), then there would be a particular relationship between the strengths of the horizontal recti muscles and the motor commands sent to them to produce a small change in the position of the eye. The command to the agonist muscle should increase and that to the antagonist muscle decrease (reciprocal innervation): and the size of the change should be proportional to the strength of the muscle (METHODS, Eq. 5).

The motor command to a muscle is interpreted here as meaning the summed firing rates of the parent OMNs, and so the strength of the muscle is measured as the change in isometric force produced by unit change in summed firing rate. Because muscle strength so defined is crucially dependent on recruitment strategy and therefore difficult to measure directly, a more convenient way of testing the minimum-norm rule is to use the equivalent relationship depicted by Eq. 15 of METHODS. This relationship links two quantities: the ratio of the firing-rate gradients of the two OMN pools and the ratio of the isometric-force gradients of the two muscles. The minimum-norm rule predicts that the latter equals the square of the former.

Testing this prediction therefore requires comparison of data for OMN firing rates with data for EOM isometric-force gradients. The comparison is carried out as follows. First, the basic data sets described in METHODS are compared. Second, the effects on this comparison are assessed for variations in each data set in turn. Finally, the data are used to derive an indirect measurement of muscle strength so that the relation between motor unit recruitment and the minimum-norm rule can be demonstrated.

Comparison of basic data sets

The basic data set for the firing rates of OMNs was taken from the review of Van Gisbergen and Van Opstal (1989). The summed firing rates of the OMNs in that data set are shown as a function of fixation command (and hence eye positionsee METHODS) in Fig. 6A. Recruitment begins at about 60° and stops at about +25°. This is easier to see in the plot of the gradient of summed firing rate that is shown in Fig. 6B. The gradient changes slowly below approximately 40°, then increases almost linearly until it levels out fairly abruptly about +25°. It is this gradient that is taken here as corresponding to the small change in motor command m that produces a small change in eye position (e.g., Fig. 3). The method of testing the minimum-norm rule described in METHODS requires the ratio of m₁ and m₂ for the two horizontal recti. The initial assumption here is that the OMN pools for the two muscles behave identically with respect to their own ON directions so that the firing-rate gradient shown for one of the OMN pools in Fig. 6B can be reflected simply around the line representing the primary position ( = 0) for the other muscle. The ratio of these two gradients is plotted on a logarithmic scale in Fig. 6C. The plot is approximately linear in the range 30 to +30°, which means that the logarithm of the ratio changes by equal amounts for equal changes in eye position. Outside this range, the firing-rate gradient of the off-direction muscle starts to decline more rapidly: the ratio of the gradients therefore increases (more than +30°) or decreases (less than 30°) more sharply than it does within the 30° range. Overall, the ratio of the two gradients, representing the ratio of the changes in motor commands on the left hand side of Eq. 15, varies by >100-fold over the oculomotor range.

View larger version (25K): [in this window] [in a new window]   Fig. 6. Properties of a sample of OMNs (n = 87) described in Fig. 1 of Van Gisbergen and Van Opstal (1989) plotted in each case against eye position. A: summed firing rate. B: gradient of summed firing rate with respect to eye position, referred to as the agonist gradient. Reflection of this curve about the vertical line "eye position = 0" gives the antagonist gradient. C: ratio of the 2 gradients from B, plotted on a logarithmic scale. Ratio varies by >100-fold over the oculomotor range.

The basic data set for isometric-force gradient is taken from the model described in Robinson (1975), and the gradients for lateral and medial recti plotted as functions of eye position in Fig. 7A. For convenience, this figure ignores the fact that the medial rectus acts in the opposite direction to the lateral rectus so that its force has the opposite sign (a convention used in subsequent plots). It can be seen that although the muscles are of unequal length and cross-sectional area in the Robinson (1975) model, their isometric-force gradients are fairly symmetrical. The force gradient for each muscle increases steadily as the eye moves into its ON direction, apart from the region beyond ~40-45° in the OFF direction: in this region the gradient declines slightly. The ratio of the gradients is shown on a logarithmic scale in Fig. 7B. The plot is approximately linear over the range 35 to +35°, but then starts to reverse its direction because of the behavior of the isometric-force gradient of the off-direction muscle.

View larger version (25K): [in this window] [in a new window]   Fig. 7. A: isometric force gradients for the horizontal recti, using the equation and parameter values from Robinson (1975). Curve for the lateral rectus shows, for each position of the eye, how the force exerted by the muscle increases with fixation command when muscle length is held fixed at that position. Curve for the medial rectus muscle is reversed in sign. B: ratio of the isometric gradients shown in A, plotted on a logarithmic scale.

The minimum-norm rule requires that, ignoring sign, the isometric-force gradient ratio should equal the square of the firing-rate gradient ratio in size (METHODS, Eq. 15). The relevant comparison of the two data sets is shown Fig. 8A. The fit appears quite close in the range 30 to +30° but then becomes poor. An error measure was devised to quantify closeness of fit (Fig. 8B). A standard measu