Department of Psychology, University of Sheffield, Sheffield S10 2TP, United Kingdom
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ABSTRACT |
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Dean, Paul, John Porrill, and Paul A. Warren. Optimality of position commands to horizontal eye muscles: a test of the minimum-norm rule. Six muscles control the position of the eye, which has three degrees of freedom. Daunicht proposed an optimization rule for solving this redundancy problem, whereby small changes in eye position are maintained by the minimum possible change in motor commands to the eye (the minimum-norm rule). The present study sought to test this proposal for the simplified one-dimensional case of small changes in conjugate eye position in the horizontal plane. Assuming such changes involve only the horizontal recti, Daunicht's hypothesis predicts reciprocal innervation with the size of the change in command matched to the strength of the recipient muscle at every starting position of the eye. If the motor command to a muscle is interpreted as the summed firing rate of its oculomotor neuron (OMN) pool, the minimum-norm prediction can be tested by comparing OMN firing rates with forces in the horizontal recti. The comparison showed 1) for the OMN firing rates given by Van Gisbergen and Van Opstal and the muscle forces given by Robinson, there was good agreement between the minimum-norm prediction and experimental observation over about a ±30° range of eye positions. This fit was robust with respect to variations in muscle stiffness and in methods of calculating muscle innervation. 2) Other data sets gave different estimates for the range of eye-positions within which the minimum-norm prediction held. The main sources of variation appeared to be disagreement about the proportion of OMNs with very low firing-rate thresholds (i.e., less than ~35° in the OFF direction) and uncertainty about eye-muscle behavior for extreme (>30°) positions of the eye. 3) For all data sets, the range of eye positions over which the minimum-norm rule applied was determined by the pattern of motor-unit recruitment inferred for those data. It corresponded to the range of eye positions over which the size principle of recruitment was obeyed by both agonist and antagonist muscles. It is argued that the current best estimate of the oculomotor range over which minimum-norm control could be used for conjugate horizontal eye position is approximately ±30°. The uncertainty associated with this estimate would be reduced by obtaining unbiased samples of OMN firing rates. Minimum-norm control may result from reduction of the image movement produced by noise in OMN firing rates.
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INTRODUCTION |
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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.
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METHODS |
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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.
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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
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(1) |
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
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(2) |
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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)
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(3) |
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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
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(4) |
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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
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(5) |
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
(FRi). The sum of these changes in firing rate would then correspond to
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(6) |
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(8) |
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(9) |
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(10) |
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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)![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2) Slope is plotted against threshold for 78 OMNs in
Fig. 2, A and B, of the study by King et
al. (1981)![]() |
3) Slope is plotted against threshold for 81 OMNs in
Fig. 5 of the study by Fuchs et al. (1988)![]() |
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) ...
(FRj)] (Eq. 9). The altered firing rates in
turn change the forces delivered by the motor units
(
f1 ...
fj),
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
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(11) |
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)
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(14) |
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
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(15) |
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
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(16) |
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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.
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![]() ![]()
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3) Simonsz and Spekreise (1996)![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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RESULTS |
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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 position
see
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.
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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.
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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 measure would be the sum of squared errors, where error
means the difference between observed and predicted values, and the squaring is introduced so that positive and negative errors do not
cancel. In this case, the values are of ratios, so the difference between the logarithms of the observed and predicted values becomes the
appropriate measure, thus avoiding the relative underweighting of
errors for low expected values. The plot, in Fig. 8B, which omits the error score of 0 at the primary position, shows clearly how
the error increases sharply outside the range ±30°. The sum of the
squared errors (for 5° steps) within this range was 0.156 in contrast
to the value of 5.92 for the range ±30-45°.
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The method for deriving the fit shown in Fig. 8 is in some respects
less intuitive than thinking of the minimum-norm rule in terms of
iso-position lines, as described in the INTRODUCTION. The
purpose of Fig. 9 is to illustrate the
equivalence of the two approaches by showing that the data underlying
Fig. 8 also produce the appropriate pattern when plotted with relation
to iso-position lines. This graph plots the iso-position curves at 10° intervals from 40 to +40°. The muscle model used is a slight variant of that derived from Robinson (1975)
in which
the horizontal recti are equal (see METHODS). It can be
seen that the line connecting the actual firing-rate commands crosses
the iso-position lines at ~90° within the ±30° range of eye
positions (cf. Fig. 5B), but outside this range crosses the
40 and +40° iso-position lines at an angle >90°. Thus this
representation also shows that the data are consistent with the
minimum-norm prediction only within about ±30°.
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One advantage of iso-position line diagrams is that they show clearly
the effects of different representations of motor command. Thus Fig.
10A shows what happens when
the motor commands are not summed firing rates but are linearly related
to eye position . It can be seen from Fig. 10A that even
if some level of the oculomotor system uses such commands, it does not
minimize the norm of their changes.
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Similarly, Fig. 10B plots the iso-position lines in terms of the innervation parameters e1 and e2 of the two muscles. The natural operating region of an EOM (Fig. 5A) is in those parts of the length-tension curves where the slope (i.e., stiffness) is approximately constant. In this region, the isometric-force gradient at a given position of the eye is the product of the muscle's (fixed) stiffness and the gradient of its innervation parameter at that eye-position (APPENDIX, Eq. A20). Thus in Fig. 10B, the iso-position lines are for the most part straight and parallel. They are crossed, obviously not at right angles, by the curve showing the "reciprocal innervation" relation between the two innervation parameters (APPENDIX, Eq. A17). Comparison of Figs. 9 and 10B helps clarify a potential source of confusion created by use of the term "innervation parameter": it does not refer to the motor command sent to the muscle (interpreted here as summed OMN firing rates) but to the combined effects of the motor command with muscle strength at that eye position (cf. METHODS, Eq. 11). Although a given change in innervation parameter could be achieved solely by change in motor command (by keeping muscle strength fixed), in general the effect of recruitment is to change both motor command and muscle strength simultaneously. Figure 10B can be taken to illustrate the results of keeping muscle strength fixed, so that the innervation parameter does indeed simply reflect the motor command. Minimum-norm control is not achieved in these circumstances.
Data sets for OMN firing rates
The fit illustrated in Fig. 8 between the predictions from OMN
firing rates and the data for isometric-force gradients of the
horizontal recti was obtained with firing rates taken from the data set
of Van Gisbergen and Van Opstal (1989). It is important to know how robust is the fit in Fig. 8 to variations in firing-rate data. For this purpose, four additional data sets were investigated, as
explained in METHODS. The summed firing rates for three of these sets are plotted in Fig.
11A, together with the
original data from Van Gisbergen and Van Opstal's review. The
remaining data set (from the review of Keller 1981
) is
not shown here on account of its close similarity with the data from
Van Gisbergen and Van Opstal. In Fig. 11A, the summed firing
rates have been normalized, that is, divided by the number of OMNs in
that particular set of data. For eye positions less than about
15°,
the summed firing rates for two of the distributions (King et
al. 1981
; Van Gisbergen and Van Opstal 1989
)
differ noticeably from the firing rates for the remaining two
(Fuchs et al. 1988
; Gamlin and Mays 1992
). This difference between the two pairs of distributions is seen more clearly in Fig. 11B, which shows the percentage
of OMNs recruited at different positions of the eye. Fuchs et
al. (1988)
found no units (of 81) with thresholds less than
40°, and Gamlin and Mays (1992)
found one (of 74).
In contrast, in Van Gisbergen and Van Opstal's (1989)
review, there are 11/87 units with thresholds less than
40°, and
King et al. (1981)
report 17/78 units [Keller's
(1981)
review has 39/160 units]. There is thus a >10-fold
difference in the percentage of low threshold units between the two
sets of distributions.
|
This difference is also apparent in the summed, normalized firing-rate
gradients (Fig. 12A), and
correspondingly in the squared ratio of those gradients (Fig.
12B). Comparison of those ratios with the ratio of
isometric-force gradients derived from Robinson (1975)
indicates that the region of reasonable fit (errors
0.01) was
similar to that of Van Gisbergen and Van Opstal for the distributions with appreciable numbers of low threshold units: ±37° (King
et al. 1981
); ±32° (Keller 1981
). However,
for the distribution of Gamlin and Mays (1992)
the
region of fit was about ±15°, and there was no fit at all with the
distribution of Fuchs et al. (1988)
.
|
The fit between data and minimum-norm prediction illustrated in Fig. 8
thus is influenced strongly by which set of experimental results are
taken for the comparison. A major factor influencing the outcome is the
proportion of low-threshold OMNs in the sample, though the difference
in fit between the data set of Gamlin and Mays (1992)
and that of Fuchs et al. (1988)
indicates that the pattern of recruitment at higher thresholds is also influential.
Variation of muscle parameters
The length-tension curves for extraocular muscle can be
approximated by Eq. 1 of Robinson (1975), which contains the
three parameters k, a, and e. The fit
between prediction and data illustrated in Fig. 8 was obtained using a
particular set of values for these parameters for medial and lateral
recti (Robinson 1975
). However, because a number of
other values subsequently have been proposed, it becomes important to
know the robustness of the fit in Fig. 8 to variations in parameter
values. This section first considers variations in individual
parameters (see METHODS).
PARAMETER k.
This parameter controls the slope of the length-tension curves when the
muscle is stretched (Fig. 1). Doubling or halving its value in both
muscles changed the error scores (cf. Fig. 8B) by <10% over the
eye-position range ±30° (not shown). This is because k
affects the isometric-force gradient in each of the two horizontal
recti, so that the ratio between the two gradients is little altered
(APPENDIX: Isometric force gradient with respect to
fixation command). When the stiffnesses of the lateral and medial
recti were made identical (Clement 1987; Simonsz
and Spekreise 1996
) there is a 40% reduction in the already
low error scores within the ±30° range of eye positions (not shown).
This is perhaps not surprising in view of the assumption that the
firing rates of the two OMNs were identical. However, the range of eye
positions over which the errors remained small (
0.01) stayed at about
±32°.
PARAMETER a.
This parameter controls the curvature of the length-tension curves
(Fig. 1). Doubling or halving the value of a in both muscles alters the error score by <10% over the eye-position range ±30° (not shown). The reason for the relative insensitivity of the isometric-force gradient to the curvature parameter can be deduced from
Fig. 5A: the natural operating region of the horizontal
recti is where the length-tension curves are fairly linear (cf.
Collins 1971).
PARAMETER e.
This parameter controls the spacing of the length-tension curves that
correspond to different values of the fixation command. A variety of
relations between the parameter e and the fixation command
have been proposed (see METHODS). However, these appear to have little effect on the goodness of fit between isometric-force and firing-rate gradients. For example, one curious feature of the
original model is the use of "secondary innervation." If values for e are taken from the original length-tension
measurements (so-called primary innervation), the difference in force
produced by the two muscles acting in concert is close to zero
(APPENDIX, Fig. A1). The extra secondary innervation is
needed to provide sufficient force to offset the passive torque of the
orbital tissues, as described in INTRODUCTION. However,
substituting the original estimates of the parameter e has
little effect on the ratio of isometric-force gradients, slightly
reducing the range of eye positions giving a good fit to about ±25°
(not shown). This is true for primary innervation values calculated
either from Robinson (1975)
or from Robinson
(1981)
. A second example comes from Clement (1985)
, who suggested a slightly different equation for
secondary innervation (see METHODS). The effects of this
alteration on the isometric-force gradient for an individual muscle is
shown in Fig. 13A, and on
the ratio of the gradients for the two horizontal recti in Fig.
13B. Clement's suggestion improves the errors within the
eye-position range ±30° by ~60% but has very little effect on the
range over which the good fit is obtained.
|
2) The worst fit between data and prediction was
obtained from the measurements of Collins et al. (1975)![]() ![]() |
3) The final muscle variant to be investigated was based
on the data of Simonsz (e.g., Simonsz and Spekreise
1996![]() ![]() ![]() ![]() ![]() ![]() ![]() |
|
In summary, variants in the values of muscle parameters that are based
on the original measurements used by Robinson (1975)![]() ![]() ![]() |
|
Estimation of muscle strength z
Muscle strength z, as defined by Daunicht and
interpreted here, is the increase in isometric force produced by unit
increase in motor command m. Its value for a particular
position of the eye depends on the average strength of the motor units
that are recruited in that position (METHODS, Eq. 14). As such, z is not simply an intrinsic property of
the muscle itself but also is influenced by the order in which motor
units are recruited as the fixation command increases in magnitude.
Altering the structure of the fixation command thus could alter the
value of z. Consequently z cannot be measured
directly by electrical stimulation of the motor nerve, because such
unphysiological stimulation lacks the organization of the natural
fixation command. However, it is possible to estimate z
indirectly from direct measurements, by dividing the isometric-force
gradient at a given eye-position by the summed firing-rate gradient at
that position (METHODS, Eq. 11). These estimates
are shown in Fig. 15, for four
different combinations of force and firing-rate measurement.
|
The two curves in Fig. 15A illustrate how z
varies with eye position for two data-set combinations that produced a
good fit between data and minimum-norm prediction. One curve is for a
variant of the "basic" combination, of firing rates from the
review of Van Gisbergen and Van Opstal (1989) with
isometric force from the model of Robinson (1975)
with
identical horizontal recti (Clement 1987
). This
combination produces a reasonable fit with the minimum-norm prediction
over the eye-position range ±30°. For eye positions greater than
30°, z increases monotonically with eye position. At
30°, however, z increases as eye position decreases,
producing an overall U-shaped curve. Thus the region of good fit to the minimum-norm prediction coincides with the region in which z
increases with eye position in the ON direction for both
muscles. This is also the case for the second curve shown in Fig.
15A, which is for the same firing-rate data (Van
Gisbergen and Van Opstal 1989
) together with the EOM
length-tension relations illustrated in Fig. 14A. By design,
this combination produces a good fit over the entire oculomotor range
(Fig. 14B). As Fig. 15A shows, it also produces
values for z that increase as eye position increases over
the full ±45° range of eye positions.
In contrast, the two curves of Fig. 15B show how
z varies with eye position for two data-set combinations
that produced a poor fit between data and minimum-norm prediction. One
curve is for the same firing-rate data shown in Fig. 15A
(Van Gisbergen and Van Opstal 1989), together with
isometric force estimates from the data given by Collins et al.
(1975)
. The curve shows z decreasing with eye
position up to about
10°, then increasing slightly. The region of
fit between data and minimum-norm prediction for this combination of
data were about ±10° (Fig. 13B). Finally, the remaining
curve in Fig. 15B is for firing-rate data from Fuchs et al. (1988)
together with the isometric force from the model of Robinson (1975)
. This combination produced no region
of fit between data and prediction, and correspondingly in Fig.
15B z starts to increase with eye position only for
positions greater than about +5°. There is no range of eye positions
for which z increases with ON direction for both
muscles.
These results show how implementation of minimum-norm control requires
the appropriate recruitment of EOM motor units. In particular, it
requires that the muscle strength z should increase as eye
position moves into the ON direction of the muscle. Given the interpretation of muscle strength adopted here, this in turn means
that motor units have to be recruited in order of increasing strength.
Thus an important and possibly unexpected corollary of minimum-norm
control in the oculomotor system is that it appears to require the
"size principle" of motor unit recruitment (e.g., Henneman
and Mendell 1981; Henneman et al. 1965
).
![]() |
DISCUSSION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The question addressed in this study was whether the control
system for horizontal eye position used the minimum-norm principle, as
formulated by Daunicht (1988, 1991
), to determine the
forces exerted by the horizontal recti. To answer the question, data on
the summed firing-rate gradients of OMN pools in primate were compared
with data on the isometric-force gradients of EOMs in people. The
results of the comparison can be summarized as follows.
There was good agreement between the minimum-norm prediction and experimental observation over about a ±30° range of eye positions for the pair of data sets selected as representative. This fit was robust with respect to variations in muscle stiffness and in methods of calculating muscle innervation.
Other pairs of data sets gave different estimates for the range of eye positions within which the minimum-norm prediction held. These estimates varied from 0° (that is, no fit at all) to ±45° (essentially the full oculomotor range).
For each pair of data sets, the range of eye positions over which the minimum-norm rule applied was determined by the pattern of motor unit recruitment inferred for that pair. It corresponded to the range of eye positions over which the size principle of recruitment was obeyed by both agonist and antagonist muscles.
These findings raise the following issues for discussion: the origin of the differences between the data sets, a possible rationale for the implementation of minimum-norm control, and the implications of its implementation.
Differences between data sets
MEASUREMENTS OF OMN FIRING RATES.
The success of the minimum-norm rule in predicting experimental results
was affected markedly by the origin of the data for the summed OMN
firing rates. When compared with isometric-force gradients derived from
the length-tension measurements incorporated in the original Robinson
model (Robinson 1975), three of the OMN samples gave
reasonable fits over at least a ±30° range of eye positions, one
gave a fit over ±15°, and one gave no fit at all (Fig.
12B). Goodness of fit was related to the proportion of OMNs in a sample with thresholds less than
40° (Fig. 11B).
The two populations of OMNs containing
1% of these units gave poor
fits to the minimum-norm prediction, whereas the three populations containing >10% of such units gave better fits. The important issue,
therefore, is the origin of these differences in the proportion of
low-threshold units.
MEASUREMENTS OF EOM LENGTH-TENSION CURVES.
The success of the minimum-norm rule in predicting experimental results
[using firing-rate data from Van Gisbergen and Van Opstal
(1989)] was in some respects very robust to variations in EOM
length-tension curves. As explained in METHODS,
length-tension curves usually are specified by three parameters,
related to stiffness, "curvature," and innervation. A fit to the
minimum-norm prediction over about a ±30° range of eye positions was
obtained with muscle parameters derived from the length-tension
measurements summarized in 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
), and are the basis for the models of
Robinson (1975
, 1981
), Miller and Robinson
(1984)
, and Miller and Shamaeva (1995)
. The fit
to minimum-norm prediction was little affected either by substantial
variations in the muscle stiffness and curvature parameters including
those related to symmetry or by the changes in innervation parameter
produced by different estimates of its relation to fixation command or
by implementation of a full three-dimensional EOM model (see
RESULTS). The relative insensitivity of the fit to these
variations is important given the possible problems associated with
data obtained from muscles in strabismic patients.
Rationale for minimum-norm control
The purpose of the present study was simply to compare the
predictions of the minimum-norm rule with relevant data: the possible theoretical basis of the rule itself was the subject of a companion modeling study (Dean and Porrill 1998). Nonetheless,
interpretation of the comparison's outcome raises a theoretical
question that requires consideration here.
The question is whether there is a compelling rationale for
minimum-effort control or whether there are alternative minimization rules that would fit the data just as well. Simple minimization of the
energy spent in either neural firing or muscular contraction seems not
to be a plausible alternative given the activity observed in what might
have been a resting state with the eye, namely the primary position
(see INTRODUCTION). About 80% of OMNs fire in the primary
position (e.g., Keller 1981), and many EOM fibers demonstrate specialized metabolic adaptations for continuous activity (e.g., Porter et al. 1995
).
These data suggest that energy is being killed for some other functional consideration. In the case of muscles specialized for controlling the position of the eye, an obvious possibility is stability of the visual image. Is there any connection between minimum-norm control and the reduction of image movements that would disturb visual processing? The connection suggested by the companion modeling study is as follows.
OMN firing rates are assumed to be noisy. The effects of this noise on eye position will depend on the strength of the motor unit that the OMN controls: the stronger the unit, the greater the image movement induced by OMN noise.
To minimize image movement, the control system must therefore learn to
recruit weak motor units before strong ones, so implementing the size
principle (cf. Senn et al. 1997).
Because OMNs with higher thresholds have greater firing-rate slopes
(see Interpretation and measurement of m), stronger motor units are in effect given larger changes in motor command: this particular implementation of the size principle gives a close approximation to minimum-norm control (Dean and Porrill
1998
).
There is evidence in favor of each of the links in this chain connecting image stability to minimum-noise control.
Variability in OMN firing-rates has been observed in cat
(Gómez et al. 1986) and primate (Goldstein
and Robinson 1986
).
The maintenance of accurate position commands to OMNs generally is
recognized to require some form of calibration, especially in the face
of naturally occurring alterations to crucial variables such as overall
muscle strength or vestibular input (Berthoz and Melvill Jones
1985; Grossberg and Kuperstein 1989
). Inaccurate position commands give rise to postsaccadic drift of the eye (e.g., Optican and Robinson 1980
), and artificially induced
drift produces changes in position command (Optican and Miles
1985
). Moreover, it appears that different position commands
can be sent to motor units of differing strength. A simulation study of
the recruitment pattern of medial rectus motoneurons (Dean
1997
) indicated that it was unlikely to be the result of a
common motor command acting equally on motoneurons which varied in
threshold. Rather, the recruitment pattern appeared to arise, at least
in part, from different OMNs receiving different synaptic inputs.
It can be seen from the results plotted in Fig. 15 of the present study that, whether a particular combination of data sets gives a good or a poor fit to the minimum-norm prediction, there is a close relationship between the range of eye positions giving the fit, and the range over which muscle strength z increases monotonically with ON direction eye position for both horizontal recti. The relationship follows from the fundamental requirement of minimum-norm control that change in motor command be matched to muscle strength (Fig. 3B). Because the gradient of summed OMN firing rates increases with eye position, one way of achieving the match is to have muscle strength increase also. Given the simplifying assumptions described in METHODS (Interpretation and measurement of z), the only way a muscle can increase its strength is to recruit stronger motor units. Thus there appears to be a fundamental connection between the size principle of motor unit recruitment and minimum-norm control.
In addition, the optimization of image stability may be consistent with
the apparent restriction of minimum-norm control to a central region of
the oculomotor range if it is the case that the first motor units to be
recruited consist of multiply innervated fibers (Dean
1996). It has been suggested that multiply innervated fibers
are a functional specialization of EOMs designed to minimize fluctuations in muscle force (Robinson 1978
). If so, the
contribution to noise-induced image drift of a unit containing these
fibers would be less than that of a conventional unit of identical
strength. The learning mechanism for calibrating the position command
therefore would allow such units to be stronger than subsequently
recruited conventional units, giving rise to a U-shaped recruitment
pattern (Fig. 15) and the corresponding restriction on minimum-norm
control.
There is thus a variety of evidence apparently consistent with the
suggestion that the approximation to minimum-norm control observed in
the present study arises from optimization of retinal-image stability,
using a distributed system of noisy motor units of differing strengths.
However, crucial evidence concerning the relationship of OMN
firing-rate threshold to motor unit strength, and the precise
functional nature of different types of EOM fiber, are not currently
available (issues briefly reviewed in Dean 1996).
Implications
The horizontal recti in people exert ~10 g of force in the
primary position. The question posed in the Introduction was how this
value was determined by the oculomotor control system. The present
results suggest the possible explanation that it arises from a
combination of the minimum-norm rule and the size of the oculomotor
range. The minimum-norm rule itself requires only that the
motor-command values lie on a line at right angles to the iso-position
lines (Fig. 3B), not specifying any particular crossing point. But once the ends of the motor-command line are fixed by the
oculomotor range (cf. Fig. 9), then the whole line is fixed. The value
of 10 g is thus the least force that is compatible with minimum-norm control and an oculomotor range of about ±45°. One implication of this suggestion is that variations in the size of
oculomotor range between species (e.g., Barmack 1982),
or within species at different stages of development, should correlate
with the variations in active muscle force at the primary position.
A more significant implication of the minimum-norm rule concerns the
control of eye-position in three dimensions. The model described by
Daunicht (1988, 1991
) is for three-dimensional control (cf. APPENDIX: Pseudoinverse control) and
addresses a problem not considered in the present study, namely the
coordination of muscles of markedly different strength. Initial results
(Warren et al. 1998
) suggest that it is compatible with
current mechanical models of the six EOMs (e.g., Miller and
Shamaeva 1995
). Such compatibility may be of clinical
relevance. A control system using the minimum-norm rule will adjust its
command signals in response to a change in muscle strength. The effects
of, for example, strabismus surgery therefore will include not only
direct changes in muscle mechanics but also the consequent changes in
motor command. An understanding of minimum-norm control therefore might
assist in the prediction of surgical outcomes.
Finally, it has been observed that the position-related firing rates of
OMNs vary with vergence state (e.g., Gamlin and Mays 1992; Gamlin et al. 1989
; King et al.
1994
; Mays and Porter 1984
). Although recent
data obtained with implanted force transducers indicate that
cocontraction does not occur during vergence displacements (Miller 1998
), the possibility of minimum-norm violation
associated with vergence is an important issue for further
investigation.
![]() |
APPENDIX |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Derivation of minimum-norm rule for horizontal eye position
The first step is to determine the relationship between the
motor commands m1 and m2
sent to the horizontal recti and the resultant position of the eye.
This can be regarded as the forward control equation for horizontal eye
position. Because muscle force is a nonlinear function of length and
motor command, the problem is linearized by dealing with the change in
eye position
that results from small changes in motor command
m1 and
m2. If we take an individual
muscle with command m
and length
p
, corresponding to eye position
,
Taylor's expansion gives
![]() |
(A1) |
![]() |
![]() |
(A2) |
![]() |
![]() |
![]() |
![]() |
(A3) |
In the case where two motor commands, m1 and
m2, produce force changes
f1 and
f2, the
resultant change in angle
is
![]() |
(A4) |
![]() |
(A5) |
![]() |
(A6) |
The inverse equation for control of horizontal eye position would give
the commands m1 and
m2 that were needed to produce a desired
horizontal displacement
. As Eq. A6 shows, there is no
unique solution to this problem, unless some additional constraint linking
m1 and
m2
is introduced. The constraint used here is that of minimum norm, where
the norm N is a measure of the magnitude of the change in
the two motor commands expressed as the vector (dm1, dm2) (cf. next
section)
![]() |
(A7) |
![]() |
(A8) |
![]() |
![]() |
(A9) |
![]() |
![]() |
Finally, the relation between m1 and
m2 that is imposed by the minimum-norm
constraint can be used with the forward Eq. A6 to produce
the inverse equations
![]() |
![]() |
(A10) |
Pseudoinverse control
The forward Eq. A6 can be rewritten in matrix form as
follows
![]() |
(A11) |
![]() |
(A12) |
The corresponding form of the inverse Eq. A10 is now
![]() |
(A13) |
Secondary innervation and the parameter e
The values of e from Fig. 3 of Robinson
(1975), which are derived directly from measurement of
length-tension curves, are plotted as a function of fixation command
in Fig. A1A. This figure shows both the data points and the best fit cubic (···) to them
![]() |
(A14) |
|
The parameter values for k, a, and e
from Robinson (1975) (see METHODS) then were
incorporated into a computer simulation of the two horizontal recti.
However, when the values of e shown in Eq. A14
were used in Eq. 1 to derive the forces exerted by both lateral and medial recti, the summed directional force of the two
muscles was close to zero (Fig. A1B). It can be seen from
Eq. 2 that this summed force is needed to balance the
passive force exerted by the stretched orbital tissues: hence the
values of e in Fig. A1A correspond to a situation
where the orbital tissue forces are zero. Robinson
(1975)
termed the values of muscle force at the primary
position produced by these values of e primary innervation.
Values of e consistent with realistic values of the orbital
tissue force were derived by Robinson (1975) as follows.
The basic idea is to find two equations that link the innervation
parameters e1 and e2 (for
agonist and antagonist muscle, respectively) and then solve them. One
of the equations is for the balance of forces (Eq. 2).
Substitutions are made for each of the three terms in this equation.
The muscle forces f1 and
f2 are calculated from the basic muscle
Eq. 1
![]() |
(A15) |
![]() |
(A16) |
![]() |
(A17) |
It can be seen from Fig. A1A that the new values for
e are higher than the original values for > 0: this
difference, required to balance the passive forces of the orbital
tissue, corresponds to the term secondary innervation of
Robinson (1975)
. The best fit quartic curves to
e1 and e2 are given in
Eq. 16 in METHODS.
Isometric force gradient with respect to fixation command
The fixation command does not appear directly in the
length-tension curves of Eq. 1 but only indirectly as the
innervation parameter e. It is therefore necessary to apply
the following chain rule
![]() |
(A18) |
![]() |
(A19) |
Equation A19 can be simplified greatly if the effects of the
a2 term are negligible. In this case, the
isometric-force gradient
![]() |
(A20) |
Minimum-norm equations for muscle
It is possible to calculate length-tension curves for EOM that
fit the predictions of minimum-norm control. In the method used here,
the predicted ratio of isometric-force gradients was derived from the
summed firing rates of the OMN population described in Van
Gisbergen and Van Opstal (1989), using Eq. 15. The
muscle equations (with identical horizontal recti) (Clement
1987
) then were adjusted to give this ratio. The adjustment was
made for eye positions at 5° intervals from
10 to
50°. At each
position, a "squashing" parameter s was introduced,
which altered the tension difference between the passive fixation
command (assumed equal to
50°) and an unchanged
15° command
(derived from Simonsz and Spekreise 1996
). The relative
spacing of the intermediate fixation commands (as determined by the
parameter e) was maintained. The value of s that
gave the required isometric-force gradient then could be calculated,
using a bisection/interpolation algorithm (MATLAB). The calculated
value of s then could be used to derive the tension values
at each eye position for different values of fixation command. Finally,
these tension values were used to plot the new length-tension curves
(Fig. 14).
![]() |
ACKNOWLEDGMENTS |
---|
We thank Dr. Christian Quaia for very helpful comments on an
earlier version of the manuscript and an anonymous referee for suggesting an explanation for artificial stabilization of muscle strength in the study of Collins et al. (1975).
![]() |
FOOTNOTES |
---|
Address for reprint requests: P. Dean, Dept. of Psychology, University of Sheffield, Western Bank, Sheffield S10 2TP, UK.
The costs of publication of this article were defrayed in part by the payment of page charges. the article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
1 For consistency with previous studies, the term "position" is used throughout instead of the term "orientation," which might be considered more appropriate for rotational systems.
2 The term "size principle" is used here to refer only to muscle units being recruited systematically in order of increasing strength. This restricted sense has no implications for whether OMNs themselves are recruited in order of increasing soma and dendrite surface area (cf. correspondence in Science 281: 919, 1998).
Received 5 June 1998; accepted in final form 7 October 1998.
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REFERENCES |
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