J Neurophysiol 93: 2614-2624, 2005.
First published January 12, 2005; doi:10.1152/jn.00960.2004
0022-3077/05 $8.00
Muscle Activity Determined by Cosine Tuning With a Nontrivial Preferred Direction During Isometric Force Exertion by Lower Limb
Daichi Nozaki,
Kimitaka Nakazawa and
Masami Akai
Department of Rehabilitation for Movement Functions, Research Institute of National Rehabilitation Center for Persons with Disabilities, Tokorozawa, Japan
Submitted 15 September 2004;
accepted in final form 6 January 2005
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ABSTRACT
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We investigated how the CNS selects a unique muscle activation pattern under a redundant situation resulting from the existence of bi-articular muscles. Surface electromyographic (EMG) activity was recorded from eight lower limb muscles while 11 subjects were exerting isometric knee and hip joint torque simultaneously (Tk and Th, respectively. Extension torque was defined as positive). The knee joint was kept at either 90 or 60°. Various combinations of torque were imposed on both joints by pulling a cable attached to an ankle brace with approximately three levels of isometric force in 16 directions. The distribution of the data in the three-dimensional plot (muscle activation level quantified by the root mean squared value of EMG vs. Tk and Th) demonstrates that the muscle activation level M can be approximated by a single model as M = 
aTk + bTh
where x = max (x,0) and a and b are constants. The percentage of variance explained by this model averaged over all muscles was 82.3 ± 14.0% (mean ± SD), indicating that the degree of fit of the data to the plane was high. This model suggests that the CNS uses a cosine tuning function on the torque plane (Tk, Th) to recruit muscles. Interestingly, the muscle's preferred direction (PD) defined as the direction where it is maximally active on the torque plane deviated from its own mechanical pulling direction (MD). This deviation was apparent in the mono-articular knee extensor (MD = 0°, whereas PD = 14.1 ± 3.7° for vastus lateralis) and in the mono-articular hip extensor (MD = 90°, whereas PD = 53.4 ± 6.4° for gluteus maximus). Such misalignment between MD and PD indicates that the mono-articular muscle's activation level depends on the torque of the joint that it does not span. Practical implications of this observation for the motor control studies were discussed. We also demonstrated that the observed shift from the MD to the PD is plausible in the configuration of our musculo-skeletal system and that the experimental results are likely to be explained by the CNS process to minimize the variability of the endpoint force vector under the existence of signal-dependent noise.
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INTRODUCTION
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In the human musculo-skeletal system, multiple muscles are involved in the torque production around a single joint. Therefore infinite combinations of muscle activation can produce a given joint torque. It remains unknown how the CNS deals with such redundant conditions (Bernstein 1967
). Intuitively, the muscle activity is supposed to increase with the torque of the joint that it spans. One of the concepts in this category is a "muscle equivalent" model proposed by Bouisset (1973), where the muscles spanning a joint are considered to work together. Although such synergistic action contributes to reduce the redundancy, this concept is too simple to explain actual situations. For example, both the biceps brachii and the brachialis are elbow flexor muscles, but the former also works as a supinator of the forearm. Hence, their activities during elbow flexion torque exertion do not necessarily covary (Buchanan et al. 1989; Gielen and van Zuylen 1986). Namely, muscles which have a common function in one degree of freedom may have different functions in another degree of freedom, suggesting that muscles cannot be lumped together just because they span the same joint.
Such torque interaction among different degrees of freedom is not restricted to the case of single joints the muscles of which generated torques at multiple degrees of freedom. For example, the elbow extension/flexion and the shoulder extension/flexion seem completely independent. However, joint torque interaction is inevitable even for the isometric condition because of the existence of bi-articular muscles spanning both joints. To see the influence of bi-articular muscles, consider a task that requires torque productions at two neighboring joints such as pushing a wall isometrically with an arm using elbow and shoulder extension/flexion torque. If the musculo-skeletal system is composed of only mono-articular muscles (Fig. 1A), the joint torque vector T on a joint torque plane required for the task can be easily decomposed into each joint torque component as |T|cos
and |T|sin (i.e., cosine tuning). Hence, the muscle activity level is dependent only on the torque which it spans, that is, the 2 degrees of freedom are independent. In this case, even if multiple muscles exist within each degree of freedom, the problem of redundancy can be handled individually (Fig. 1B).
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FIG. 1. Construction of torque vector T. A: if only mono-articular muscles existed, the T could be easily decomposed into each axis component (cosine tuning). B: in the case of A, the problem of how the T is constructed by multiple muscles can be reduced to the problem in each degree of freedom. C: on the other hand, in the actual situation, the situation is much more complicated because of the existence of the bi-articular muscles. D: there is a myriad of possible combinations to realize a certain T.
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In contrast, actual situations are more complicated (Fig. 1C): the muscle activation pattern cannot be uniquely determined due to the torque interaction between both joints through bi-articular muscles (Fig. 1D). In this case, we have to consider both degrees of freedom simultaneously to deal with the problem of how to construct a torque vector (Binding et al. 2000). The problem is what strategy the CNS uses in this case. Does the CNS still use the cosine tuning function as in the case when bi-articular muscles are absent? Alternatively, does the CNS use a more complicated muscle recruitment strategy? Up to now, our understanding of the strategy taken by the CNS is insufficient.
One of the approaches has been taken by van Ingen Schenau's group (Jacobs and van Ingen Schenau 1992; van Bolhuis et al. 1998; van Ingen Schenau et al. 1992). They have hypothesized that the bi-articular muscles contribute to control the force directions of an endpoint of a limb (e.g., hand or foot), whereas the mono-articular muscles control the movement directions. In fact, these researchers showed that the hypothesis is partially correct: the bi-articular muscle activity correlates more strongly with the endpoint force directions than its movement directions, although the mono-articular muscle activation does not necessarily obey the hypothesis (Doorenbosch et al. 1997; van Bolhuis et al. 1998). However, because both the movement and force directions are realized by combinations of joint torques, it seems inappropriate to relate the muscle activity to those spatial parameters easily without considering the possible relationship between the joints torque and those parameters.
Another well-known approach to this problem is the use of optimization concept. For example, the muscle activation pattern has been suggested to be determined so as to minimize the sum of the muscle force (Seireg and Arvikar 1973), squared (van Bolhuis and Gielen 1999) or cubed (Crowninshield and Brand 1981; Prilutsky 2000). van Bolhuis and Gielen (1999) investigated what models can best capture the actual arm muscle activity patterns during isometric force exertion and found that the squared sum of the muscular force (or stress or activation) gave the best fit. However, the actual data were not so well characterized, making the comparison ambiguous. The characterization of the data should be performed before assuming the cost function a priori.
Therefore in this study, we first tried to obtain a precise landscape of the muscle activity level of a lower limb when various joint extension/flexion torques were simultaneously imposed on the hip and knee joints. Then we compared this landscape achieved by the CNS with the mechanical functions of the muscles. To impose torque at both joints, the ankle position was pulled in various directions with various force levels, and subjects were asked to maintain their lower limb posture with minimal effort. This task can be accomplished more easily than a task in which the subjects have to control the force magnitude and direction by themselves. In addition, the knee joint muscles, except gastrocnemius muscle, do not span the ankle joint. Hence, it is easier to limit the degree of freedom only to the hip and knee joints than to perform the task using the shoulder and elbow joint because a lot of elbow muscles span the wrist or finger joint. Furthermore, due to the larger muscle size, the electromyography (EMG) used to evaluate the level of lower limb muscle activity is expected to be less contaminated by the electrical cross-talk from adjacent muscles. These aspects are suitable for obtaining the exact mapping from the joint torque to the muscle activation level.
Our results show that a very clear cosine tuning is working on the joint torque plane, as in the case without bi-articular muscles shown in Fig. 1A. One remarkable and unintuitive difference was that the muscle's preferred direction (PD) where it is most active deviated from its mechanical pulling direction (MD) (Cabel et al. 2001; Fagg et al. 2002; Hoffman and Strick 1999). By use of a simple analysis, we demonstrate that the observed shift of the PD from the MD satisfies a necessary condition that the CNS uses a cosine tuning function to recruit muscles. In addition, based on the cosine tuning, we reconsider the hypothesis regarding the contribution of bi-articular muscles on the control of the endpoint force direction. Finally, we discuss the possible relationship between our results and approaches using optimization theory. These results have been partly published in abstract form (Nozaki and Nakazawa 2001).
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METHODS
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Procedure imposing joint torques
In this study, to apply the various levels of torque to both knee and hip joints simultaneously, subjects maintained the lower limb posture against the force applied to the ankle position (Fig. 2A). The endpoint force vector F = (Fx, Fy) was transformed into the joint torque vector T = (Tk, Th) (Tk and Th are knee and hip joint torque, respectively) as
 | (1) |
where T indicates transposition and
 | (2) |
The definition of parameters is shown in Fig. 2A. In this study, extension torque is defined to be positive. This is a linear mapping to transform a circle (i.e., constant magnitude of force) and a line (i.e., constant direction of force) on the force plane (Fx, Fy; Fig. 2B) into an ellipse and a line on the torque plane (Tk, Th), respectively (Fig. 2, C and D). This mapping indicates that various combinations of both joints' torque are systematically imposed by changing the force vectors of the endpoint (Jacobs and van Ingen Schenau 1992).
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FIG. 2. Transformation from the force coordinate to the torque coordinate. A: definition of parameters. The y axis is defined as the direction from the ankle to the knee joint, and the x axis is orthogonal to the y axis. B: circles and lines indicate, respectively, the constant force magnitude and the force direction trajectory in the force coordinate. C and D: the transformation matrix (Eq. 1) transforms the circle on the force coordinate into the oblique ellipse on the joint torque plane (Tk knee joint torque, Th hip joint torque). The lines separated by the equal force angle on the force coordinate are also transformed into lines separated unequally. C and D indicate the cases when the knee angle ( ) is 90 and 60° (full flexion corresponds to 0°), respectively.
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Experimental procedures
Eleven male subjects (22–39 yr of age) participated in the experiments with informed consent. The experiments were conducted according to the Declaration of Helsinki. The experimental procedure was approved by the ethical committee of our institute. Subjects laid on their left side on a bed with the right leg supported horizontally by a sling (Fig. 3A). The experimenter pulled the cable attached to the subject's ankle brace immobilizing the right ankle joint from various directions with various force magnitudes (Fig. 3A). Subjects were requested to maintain the right lower limb posture as constantly as possible for 
4 s. They were also instructed not to stiffen the leg excessively and to minimize muscle co-contraction. The knee joint angle was maintained at 90° in experiment 1 (n = 11) and at 60° in experiment 2 (n = 6). The hip joint was maintained at 
90° in both experiments.
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FIG. 3. Experimental setup. A: the right leg of the subject was supported horizontally by a sling. The cable attached to the ankle was pulled in various force magnitudes and directions by the experimenter. The ankle joint was immobilized by an ankle brace. The force magnitude was measured by a load cell. B: definition of force angle.
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Force was applied from 16 directions (0, 30, 60,...330, 45, 75, 225, 255°) at three force levels (the target was roughly determined to be 30, 60, and 90 N). The force direction was controlled by the experimenter monitoring the pulling angle by the circular protractor attached to the ankle position (see Fig. 3B for the definition of the angle). The force magnitude was measured using a load cell (Model 1269F, Takei Scientific Instrument). We verified that the position of the ankle joint and the direction of the cable were, respectively, within a 3-cm distance and ±2° from the predetermined value. The force magnitude 90 N was generally <30% of the maximal voluntary contraction level, although it depends on the force direction (Nozaki, unpublished observation). The measurement was undertaken at least three times for each force magnitude and direction with the total number of trials per subjects being 144 (16 directions x 3 force levels x 3 repetitions). Sufficient rest was taken between trials to minimize muscle fatigue.
Surface EMG signals were obtained from major leg muscles including gluteus maximus (GM), biceps femoris long head (BFL), semitendinosus (ST), rectus femoris (RF), biceps femoris short head (BFS), vastus lateralis (VL), vastus medialis (VM), and medial gastrocnemius (GAS). The type and function of these muscles are summarized in Table 1. The EMG signal was amplified with band-pass filtering between 20 and 500 Hz (Amplifier; The Bagnoli 8 EMG System, DELSYS). The data concerning muscular EMG activity and force were digitized at 1 kHz for off-line analysis.
Data analysis
We chose a 1-s period where the fluctuation of force was smallest. The root mean squared (RMS) value of the EMG activity during this period was used to evaluate the muscle activation level. Our results were not substantially influenced when using the integrated value of the rectified EMG signal instead of the RMS. The torque of both joints was estimated using Eq. 1 and the relationship of how the muscle activation level was modulated with the torque at each joint was first investigated. Following this ordinary analysis, we constructed the three-dimensional (3D) plot of the muscle activation level (z axis) as a function of both joints' torque (x,y axes). As shown in Fig. 4A, it seems that the data distribution can be approximated by a single plane on this 3D plot. The plane was identified using a multiple regression analysis: The model used was M = aTk + bTh, where M is the muscle activity (the RMS of EMG), x = max (x,0), and a and b are constants. The constants a and b were calculated using the Levenberg-Marquardt nonlinear least-squares optimization method (Press et al. 1992). To evaluate the goodness of the fit of this model to the data, we calculated the coefficient of determination R2 value, which represents the ratio of the variance explained by the model to the total variance (Zar 1999). Furthermore, the PD along which the muscle activation level increased most steeply was obtained in degrees as the angle from the Tk axis to (a,b) (Fig. 4, A and B).
In addition to the data representation on the 3D plot (Fig. 4A), we constructed a polar plot on the force plane that has been conventionally adopted in previous studies (Buchanan et al. 1989; Fagg et al. 2002; Hoffman and Strick 1999; van Bolhuis et al. 1998; van Bolhuis and Gielen 1999). In this plot, the muscle activity level corresponding to a certain force level is represented as a radius for each force direction. In contrast to the previous studies recording the EMG activity in response to a constant force magnitude, we did not control the force magnitude so strictly. Hence, to make such a plot, we first applied a linear regression analysis to data of the EMG versus the force magnitude for each force direction (Fig. 4C). Then the slope of the regression line, which represents a gain of EMG to a unit force magnitude, was used as the radius of the polar plot (Fig. 4D). As a reference, we drew the PD into the polar plot. Substituting Eq. 1 into M = aTk + bTh obtained in the preceding text gives M = a'Fx + b'Fy where (a',b') = (a,b)A. This implies that the PD on the force plane (i.e., on the polar plot) is directed to (a',b'). It should be noted that the PD on the torque plane is not mapped directly into the PD on the force plane using Eq. 1: i.e., (a',b') 
(a,b)(A–1)T. This is because the A is not an orthogonal matrix (Strang 1976).
Mechanical pulling direction of muscle
The increase in the muscle's activation can be represented as a vector on the torque plane: for example, the GM activation increases the vector in a direction of 90° [i.e., (0,1)]. Here we call this direction the mechanical pulling direction (MD). For the mono-articular muscle, the MD is easily obtained as in the case of the GM; 0° (VM and VL) and 180° (BFS and GAS). On the other hand, information about the moment arms around both joints is needed for the bi-articular muscles because the muscular force of these muscles generates a torque of both joints the ratio of which is proportional to the moment arm ratio between both joints. We estimated the moment arms of BFL, ST, and RF using the method and the data provided in Delp (1990). The MD of each muscle is shown in Table 1.
The MD on the force plane was also obtained for the polar plot data representation. It is defined as a force direction generated at an endpoint by a muscle. In the case of the VL, the direction ran from the hip joint toward the ankle position (Hof 2001). Mathematically, this direction can be calculated by transforming the MD on the torque plane using the inverse transformation of Eq. 1 as (cos µ, sin µ)(A–1)T, where µ is the MD on the torque plane shown in Table 1. In addition to this conventional definition of the MD, we employed another definition. Assuming naively that muscle activation is proportional to the orthogonal projection from T to its MD as in the case of Fig. 1A, M 
cos µ Tk + sin µ Th. In this case, the MD vector (cos µ, sin µ) is transformed into the MD on the force plane as (cos µ, sin µ)A (The method of the transformation is identical to that of the PD described in the previous section). This corresponds to a force direction that maximizes the torque of the joint that the muscle spans. In the case of the VL, it is orthogonal to the shank segment because a force in this direction maximizes the knee joint torque. Both types of MD were drawn in the polar plot.
Statistical analysis
The paired t-test was used to compare the value of PD between knee angle conditions and between muscles. The t-value was also calculated to test the null hypothesis that the PD was equivalent to the MD or a predetermined value. The probability of P < 0.05 was accepted as a significant level.
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RESULTS
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The subjects were able to keep the force output stable as evaluated by the value of the coefficient of variation (<1%). Furthermore, using a different period (within 4 s) for calculating force magnitude and the RMS of EMG activity did not affect our results substantially, indicating that our results shown in the following text were robust.
Dependence of muscle activation on each joint torque
Figure 5 shows the relationship between each joint torque and the muscle activation level. The activation level of mono-articular muscle increased with the torque of the joint which it spans. For example, the activation level of GM (hip extensor muscle) increased as the hip extension torque became larger, whereas there was no apparent relationship between the GM activation level and the knee joint torque (Fig. 5, GM). Likewise, the activation level of VM and VL (knee extensor muscle) increased with the knee extension torque but had no clear relationship with the hip joint torque (Fig. 5, VM and VL). On the other hand, the activation level of the bi-articular muscle was dependent on the torque of both joints. For example, the RF, which is a hip flexor as well as a knee extensor, was more activated when the hip flexion or the knee extension torque became larger (Fig. 5, RF). The opposite pattern was observed for BFL and ST, which are hip-extensor/knee-flexors (Fig. 5, BFL and ST).
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FIG. 5. The relationship between EMG activity and each joint torque. These typical data were obtained from 1 subject. The EMG level (vertical axis) increases with the torque of the joint over which the muscle spans. For example, for the gluteus maximus (GM) and vastus medialis (VM), the EMG level increases with the hip joint torque (Th) and the knee joint torque (Tk), respectively. On the other hand, the apparent relationship is not observed between the EMG level and the joint torque over which the muscle does not span. The muscle abbreviations are summarized in Table 1.
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Dependence of muscle activation on torque at both joints
The experimental result that the muscle activation level increased with the torque of the joint that it spans was not unexpected. However, it should be noted that the relationship was not so strict. For example, considerable variability existed in the activation level corresponding to a certain level of joint torque (Fig. 5); namely, there was no clear one-to-one correspondence between them. However, changing the view of the direction of the data considerably dissolved this ambiguity. The 3D plot between muscle activation level and the torque of both joints (Fig. 6A) demonstrates that there existed a best view direction from which the variability of muscle activation level seemed minimal. The existence of such view points (Fig. 6, A–C) indicates that the activation level of the bi-articular muscle can be described as a linear sum of the torque of both joints, implying that the data span a single plane in the 3D plot. We quantified to what extent the EMG level can be described by this plane using a multiple regression analysis (Fig. 4A). Overall, the EMG recorded from all muscles fitted well with the plane. This can be ascertained by the large value of the variance explained by the model (R2 in Table 2) except for the GM and GAS. In the case of the GAS, reliable parameters could not be calculated for four subjects. On the other hand, the relatively low R2 value of the GM might be partly due to the fact that the thick fatty tissue of the buttocks weakens its EMG signal. This result guarantees the validity of calculating the PD for each muscle. The PD of RF was 315.1 ± 15.1° when the knee angle was 90°, demonstrating that the activity of this muscle changes most sensitively when the ratio of knee extension torque to hip flexion torque is 99.7% [=|tan(315.1°)|]. The PD of a bi-articular muscle is not counterintuitive because the value of the PD was located in the same quadrant as the MD on the torque plane. For example, the anatomical function of BFL is to extend the hip joint as well as to flex the knee joint, and its PD (159.3 ± 11.3°) actually coincided with this direction.
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FIG. 6. The 3-dimensional (3D) plot showing the relationship between EMG and torque of both joints. A: rectus femoris (RF) is a hip flexor as well as a knee extensor, hence the EMG level increases with the torque of each joint (top 2 panels). Tk and Th indicate the knee and hip joint torque, respectively. However, by changing the view direction of the 3D plot (bottom left), we can obtain the view direction from which the variation of the EMG data are smaller (bottom right). The situation is same for other bi-articular muscles [BFL, biceps femoris long head (B); ST, semitendinousus (C)]. D: the VM is a mono-articular knee joint extending muscle, hence the EMG level increases with knee extension torque (top left) but seems independent of the hip joint torque (top right). However, as is in the case of RF, by changing the view direction (bottom left), the better view direction can be obtained (bottom right). This tendency is also true for other mono-articular muscles [VL, vastus lateralis (E); GM, gluteus maximus (F)].
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More interestingly, even in the case of mono-articular muscles, a slight change of view point of the 3D plot reduced the variability of the EMG level (VM in Fig. 6D). That is, although these muscles generate only one joint torque, their activation levels are also dependent on the torque of the joint which they do not span (Fig. 6, D–F). The PD of the VL was 14.1 ± 3.7°, indicating that the VL is activated most efficiently when the knee extension torque is accompanied by the hip extension torque whose magnitude is 25.1% [=tan(14.1°)] of the knee extension torque. The result for the PD is summarized in Fig. 7 and Table 2. The PD significantly differed from the MD for GM, VL, and VM (P < 0.001). The shift from the MD to the PD was clockwise for VL and VM and counterclockwise for GM. No significant difference was observed for the PD between VL and VM. The difference between the PD and the MD in bi-articular muscles was observed only for ST (P < 0.001); however, the results were not conclusive because the exact MD of each subject was unknown for bi-articular muscles. The PD of BFL was significantly smaller than that of ST (P < 0.01).
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FIG. 7. The mean value of the preferred direction of each muscle calculated from experimental data (n = 11, knee angle was 90°). The abbreviations of the muscle names are shown in Table 1.
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Effect of knee joint angle
Table 2 also demonstrates the R2 and the PD for six subjects when the knee joint angle was 60°. For reference, the data of these subjects when the knee joint was 90° were also shown. The goodness of the fit evaluated by the R2 was high as in the case when the knee joint angle was 90°. The deviation of the PD from the MD was also observed for several muscles (GM, VL, VM) in this knee angle condition (P < 0.01). The direction of the shift was the same as that when the knee joint angle was 90°. The PD was significantly larger for the GM (P < 0.01) and smaller for the GAS (P < 0.05) when the knee joint angle was 60° (paired t-test). There was no significant difference in the PD for other muscles between both knee joint angle conditions. However, this lack of significant difference should be interpreted as meaning that there was no consistent shift of the PD independent of the subjects. Indeed, although the PDs of other muscles agreed well between both knee joint angle conditions, those of some subjects showed slight deviation (Fig. 8). Therefore there still remains a possibility that the PD shifted with the knee joint angle in a direction which was specific to each subject.
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FIG. 8. The relationship of the PD between both knee joint angle conditions. B–F, Each character represents the data measured from each subject. The abbreviations of the muscle names are shown in Table 1.
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Force plane representation
Figure 9 shows a typical example of the dependence of the muscle activity on the force direction. As easily inferred from the result that the data distribution in the 3D plot was approximated by a single plane (Fig. 6), the shape of the polar plot resembles a circle, indicating again that the cosine tuning works. In addition, the misalignment between the PD and MD is demonstrated for both types of the MD (see METHODS). That is, the muscle was neither maximally active when the force was in alignment with its mechanical pulling direction (a gray line in Fig. 9) nor when the force produced a maximal torque at the joint that the muscle spans (a broken line in Fig. 9.
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FIG. 9. The dependence of the EMG level on the force direction. A: coordinate system adopted. Note that the force is defined here as that applied by the experimenter. B–I: the radius of the polar plot indicates the sensitivity of the EMG level to a unit force. The thick straight line indicates the preferred direction of each muscle on the force plane. The gray line is the muscle's mechanical pulling direction on the force plane. The broken line is the force direction when the muscle is assumed to be proportional to the orthogonal projection from the torque vector to the mechanical pulling direction on the torque plane (see METHODS). Each type of MD for the VL is shown in A. The abbreviations of the muscle names are shown in Table 1.
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DISCUSSION
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In our experiment, subjects passively maintained the posture of the lower limb against the external force controlled by the experimenter. This task was more easily accomplished than the task in which the subjects must precisely self-control the force direction and magnitude. In fact, subjects have difficulty in controlling the force vector by themselves, and this case accompanied stronger co-contraction among the antagonists until the subjects became used to the task (Nozaki, unpublished observation). Hence, the task in the present study was less influenced by the task difficulty or the subjects' skill. In addition, the knee joint muscles do not span the ankle joint with the exception of the GAS, which contributes to restraining the degree of freedom of torque exertion only to the hip and knee joints (In this sense, the reliability of our data were relatively low when the GAS was active, because a part of the knee flexion torque might have leaked to the ankle joint through the GAS activation). These aspects may make the task suitable for examining genuine muscle activation patterns to generate the torque at two different degrees of freedom.
CNS adopts cosine tuning
We have observed that the muscle activity when various torques are imposed on the knee and hip joints is determined in a very simple and consistent fashion. The EMG data span a plane in the 3D plot as shown in Fig. 6, indicating that the muscle activity level M can be approximated as a function of knee (Tk) and hip joint torque (Th): M = aTk + bTh. This equation can be rewritten as M = PTT = |P| |T|cos, where P = (a,b), T = (Tk,Th), and denotes the angle between P and T. This means that the M obeys the cosine (more precisely, half cosine) tuning function the PD of which is pointing to P. Therefore the CNS apparently adopts the cosine tuning function (Hoffmann and Strick 1999) even under the redundant condition that the bi-articular muscles exist (Fig. 1, C and D).
In the present study, we have been dealing with the problem on the torque plane (Tk,Th). However, it can be also dealt with on the force plane (Fx,Fy) as has been adopted in several previous studies (Hoffman and Strick 1999; van Bolhuis et al. 1999). Because the mapping from T to F is linear (Eq. 1) and the muscle activity is a linear sum of Tk and Th, the muscle activity can also be expressed as a linear sum of Fx and Fy. In fact, the polar plot representation (Fig. 9) indicates that the tuning function can be approximated by a half cosine function. Accordingly, the cosine tuning holds on the force plane, too. Hence, the present results do not contradict the previous studies demonstrating that a muscle activity is cosinely tuned with isometric force directions (Hermann and Flanders 1998) or with movement directions (Hoffmann and Strick 1999).
Misalignment between PD and MD
The CNS adopts the cosine tuning as in the case where there are no bi-articular muscles (Fig. 1A). However, differently from this fictitious case, the actual PD was not aligned with the MD as clearly demonstrated in the case of mono-articular muscles (Fig. 7, Table 2). Furthermore, in the force plane (Fig. 9), the PD (thick lines) was aligned neither with the MD on the force plane (gray lines) nor with the direction obtained from the naive assumption that the muscle activity level is proportional to the torque of the joint that it spans (broken lines). Hence, the activity level of the mono-articular muscles is not determined selfishly based on its mechanical advantage. On the other hand, we did not reach a definite conclusion regarding the misalignment of the bi-articular muscles, because their exact MDs are unknown. However, the PD of bi-articular muscles was likely to differ from their own MD, because the difference in the PD between BFL and ST seemed too large to be explained by the difference in the MD between them (Table 2). Such a misalignment between the MD and PD has been recently reported in several studies (Cabel et al. 2001; Flanders and Soechting 1990; Hoffman and Strick 1999).
Does the PD not agree with the MD under the condition that the cosine tuning is working? Consider a situation where the torque vector T = (Tk,Th) is constructed using the muscles' basic vectors ei = (cos µi, sin µi) as 
, where µi and ki are, respectively, the MD and the contribution of the ith muscle to the joint torque. Cosine tuning implies that the muscle activity level is proportional to the orthogonal projection from T to pi = (cos 
i, sin i)
 | (3) |
where Ci is a positive constant and i is the PD and for simplicity we assume that –90° 
i 90° and that a negative value of ki represents the activity of antagonist muscles. Substituting Eq. 3 into gives
 | (4) |
 | (5) |
The necessary conditions for these equations to hold for all Tk and Th are
 | (6) |
 | (7) |
If the PD agrees with the MD for all muscles (i.e., i = µi), the term sin µi cos µi is never larger than zero because the MDs are distributed only in the second and fourth quadrants (e.g., we do not have bi-articular muscle generating extension torque around both knee and hip joints). It should be noted that Eqs. 6 and 7 are satisfied if there is no bi-articular muscle because the term sin µi cos µi is always zero. In other words, Eqs. 6 and 7 never hold under the existence of bi-articular muscles, indicating that the PD cannot agree with the MD for all muscles simultaneously as long as the CNS adopts cosine tuning.
In the present results, the PDs of bi-articular muscles are directed to the same quadrants as their MDs, indicating that the term sin i cos µi is smaller than 0 for these muscles. Hence, considering that the cos µi is 1 and 0, respectively, for mono-articular knee and hip joint muscles, for Eq. 6 to hold, 
has to be larger than 0 ("MK" denotes mono-articular knee joint muscles). Our result that the PD of mono-articular knee extensor muscles such as VM and VL rotates from the Tk axis in an anticlockwise direction (Fig. 7) satisfies this necessary condition because sin i is larger than 0 for these muscles. Similarly, the PD of the mono-articular hip extensor muscle (GM) rotating in a clockwise direction from the MD (= 0°) satisfies the necessary condition (Eq. 7). Therefore the misalignment between the MD and PD observed in this study satisfies the necessary conditions that the cosine tuning works on the joint torque plane.
Implications of cosine tuning
Descriptions of how muscles are recruited have remained rough since the classic study of Fujiwara and Basmajian (1975). For example, even in a recent review article of Prilutsky (2000), it is stated that a mono-articular muscle activity increases with the joint torque that it spans and that a bi-articular muscle activity is highest and lowest, respectively, when it acts as an agonist and an antagonist at both joints and is intermediate when it has agonistic action at one joint and antagonistic action at the other joint. Now, we have obtained more accurate descriptions to enable us to predict the muscle activity when the torques at the knee and hip joints are needed.
The most unintuitive point deduced from the present results is the difference between the PD and MD in mono-articular muscles because it indicates that the muscle activation level is dependent on the torque of the joint that it does not span. From the discussion described in the preceding text, we can conclude that this observation comes from the interaction between the torque of the neighboring joints through bi-articular muscles. A similar discussion can be found in a study which demonstrates that the muscle spindle of mono-articular muscle is involved in the computation of its nonspanning joint angle (Scott and Loeb 1994).
The misalignment between the PD and MD has some important practical implications. First, it explains the recruitment of antagonist muscles that is often observed in a simple single joint torque exertion task (Kimura et al. 2003). The muscle is active on the torque plane ranging from –90 to 90° around its PD. For example, the VL is active in the range from –75.9°(= 14.1 –90) to 104.1°(=14.1 +90; 14.1° is the PD of VL) on the joint torque plane, suggesting that the VL is active even when the knee flexion torque is needed (i.e., in the range from 90 to 104.1°). Furthermore, the range where the VL is active overlapped that where knee flexor muscles such as BFL and ST are active; therefore the co-contraction occurs around the knee joint. This co-contraction comes not from stiffening the knee joint excessively but from an inevitable consequence of the cosine tuning.
Second, our results imply that the degrees of freedom of the knee and hip joints cannot be separated. That is, the concept "single joint torque exertion" is not viable in principle under the existence of bi-articular muscles. Single joint torque exertion tasks, due to their simplicity, have been widely used to control the muscle activation level in motor control studies or in muscular strength training. However, now we have to question the simplicity of such tasks. For example, it is quite usual for those who want to train the knee joint muscles to set only the magnitude of the knee joint torque as the training intensity. However, specifying only the knee joint torque is insufficient to uniquely determine the activation level of the knee joint muscles. This is true even for mono-articular knee joint muscles that span only the knee joint. In fact, during an isometric knee extension or flexion task, the profile of unintentionally generated hip joint torque is different from subject to subject, and this difference affects the relative contribution between mono- and bi-articular muscles and would bias the muscles to be strengthened (unpublished data). Many previous studies employing the single-joint torque exertion task have a similar weakness in that the torque of the neighboring joint was not controlled; therefore in some cases, the experimental results should be re-examined under the condition that the neighboring joint torque is controlled.
Reconsideration of force direction control by bi-articular muscles
If the CNS did not use the cosine tuning, the PD would point to the same direction as the MD. For example, the RF and the VL could be active, respectively, only in the fourth quadrant or only in the first and fourth quadrants on the torque plane. In this case, the tuning function should be more complicated and the tuning width is expected to be different between mono- and bi-articular muscles. However, the CNS does not use such strategy and instead adopts a simpler and more universal tuning function. Accordingly, we did not find any difference in the tuning pattern itself between mono- and bi-articular muscles. However, on the other hand, problems regarding the functional significance of bi-articular muscles have attracted researchers' interest (Prilutsky 2000). One influential idea that has been proposed by van Ingen Schenau and his colleagues holds that the bi-articular muscle plays a key role in controlling of the force direction of the endpoint of a limb (Doorenbosch et al. 1997; Jacobs and van Ingen Schenau 1992; van Bolhuis et al. 1998; van Ingen Schenau et al. 1992).
As described in METHODS, Eq. 1 transforms a circle and a line on the force plane (Fig. 10A) into an oblique ellipse and a line on the torque plane (Fig. 10B), respectively. To change the force direction from c to d in Fig. 10A, the joint torque must move from c' to d' in Fig. 10B. The direction c'd' is closer to the PD of RF, indicating that the RF is more sensitive to the change in force direction cd, as van Ingen Schenau's hypothesis suggested. However, when the force direction is changed from e to f, the activity of VM or GM plays a key role because the vector e'f' is closer to the PD of VM than that of RF. Likewise, the modulation of force magnitude is more influenced by the GM or VM than the RF (direction ab) or by the RF than the GM or VM (direction gf). Hence, the type of muscle that contributes more to controlling the force direction and magnitude is not strictly fixed but depends on the situation.
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FIG. 10. Mapping of the iso-force-direction and iso-force-magnitude trajectory to those on the torque plane. A: the trajectory of force when controlling the force magnitude with the direction kept constant (ab and gf) and when controlling force direction with the magnitude kept constant (cd and ef). B: the vectors in A are transformed by Eq. 1 on the torque plane (e.g., ab into a'b'). The PDs of the GM, VM, and RF are also shown for reference.
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The dependence of muscle activity on force direction and magnitude can be easily understood by considering the muscle activity level M is expressed as M|F| cos ( – ) where |F| is the magnitude of force and and are, respectively, the angle of the endpoint force and the PD on the force plane. By taking its derivative with or |F|, the sensitivity of M to and |F| can be obtained as dM/d–|F| sin ( – ) and dM/d|F|cos ( – ), respectively. The shape of these functions (cosine function) does not differ between mono- and bi-articular muscles, which is clearly in contrast with the assertion by van Ingen Schenau et al. The reason they reached their conclusion was that they did not explore the entire range of force direction but examined the narrow range where the the bi-articular muscles activities were more sensitive to the change in the force direction (e.g., b'c' in Fig. 10B) (Jacobs and van Ingen Schenau 1992). Therefore from the viewpoint of cosine tuning in this study, we conclude that both mono- and bi-articular muscles equally contribute to control both the force direction and magnitude. One remaining possibility to support their assertion might come from a bias of distribution of the location of the torque vector required for the movement in our daily life. If the required torque vector is more often located in the range between b' and c' in Fig. 10B, we could say that the contribution of the bi-articular muscles to control force direction is higher than that of mono-articular muscle. It is possible because the movements such as locomotion, standing, and reaching do not require all possible combinations of torque at multiple muscles equally but preferential combinations (Scott et al. 2001).
Origin of cosine tuning
The next fundamental question is what the origin of cosine tuning is. Conventionally, the problem of how a joint torque is distributed among multiple muscles has been dealt with from the viewpoint of optimizing cost functions (Binding et al. 2000; Crowninshield and Brand 1981; Seireg and Arvikar 1973; van Bolhuis and Gielen 1999; Yeo 1976). In isometric force exertion of an upper limb, for example, van Bolhuis and Gielen (1999) have reported that the solution minimizing sum of muscle force (or stress or activation) squared provides the best fit to the experimental data of muscle activation.
To examine to what extent such cost functions can reproduce the cosine tuning and the PD observed in this study, we again consider the problem of how T = (Tk,Th) is constructed using ei = (cos µi, sin µi) as 
, where we introduce the variable gi as a gain that includes such factors as muscle physiological cross-sectional area (PCSA), moment arm, etc. We can handle various types of cost functions by assigning the value of gi for each muscle. Minimizing the sum of squared ki (muscular force, stress, or activation) can be accomplished using the Lagrange method (Luenberger 1984). We can construct the Lagrangian as
 | (8) |
where 
1 and 2 are Lagrange multipliers. The necessary condition that 
ki2 is minimized is 
L/ki = 0 (i = 1, 2,..., N). The indices i = 1 and i = 2 are assigned to one of the mono-articular knee extensors and hip extensors, respectively (i.e., cos µ1 = 1, sin µ1= 0, cos µ2 = 0, and sin µ2 = 1). It should be noted that this assignment does not mean that there are no other examples of these types of muscles. Strictly speaking, the inequality constraint ki 0 should be taken into account because the muscle cannot push. For the sake of simplicity, here we considered a symmetric system in which each muscle has an antagonist the MD of which is opposite but the other characteristics of which are the same. In such models, we can interpret the negative ki as the antagonist muscle's activation. The general case in which the inequality constraint was considered was thoroughly dealt with in the study of Todorov (2002). Summarizing up the equations after eliminating 1 and 2 gives
 | (9) |
where k = (k1, k2,..., kN)T, 
, and N–2
 | (10) |
Equation 9 means that the value of ki is a linear summation of Tk and Th, hence, the tuning function of each muscle is a half cosine. Therefore criteria minimizing the summation of squared muscular force (or stress or activation) generate cosine tuning (Todorov 2002). Some calculation yields
 | (11) |
Equation 11 indicates that the MD (µi) is mapped into the PD (i) by
 | (12) |
where pi = (cos i, sin i) and
 | (13) |
The matrix 
has two different eigen-vectors that are orthogonal to each other because it is a symmetry matrix (Strang 1976). On the joint torque plane, one of the eigen-vectors [v1 = (cos 
1, sin 2)] is directed toward the first (or 3rd) quadrant and another [v2 = (cos 2, sin 2)] is directed toward the fourth (or 2nd) quadrant (Fig. 11A). These eigen-vectors correspond to the "fixed direction" where the MD is mapped into the same direction (i.e., i = µi). The value of gi2 sin µi cos µi is never larger than 0 in the configuration of our musculo-skeletal system. In this condition, the MD is mapped in the clockwise direction into the PD (i.e., i µi) either when µi 1 or µi 2, while the MD is mapped in the anti-clockwise direction into the PD (i.e., i µi) when 2 µi 1 (Fig. 11B) (unpublished data). The shift from the MD to the PD observed in GM (clockwise direction), VM, and VL (anti-clockwise direction) agrees with this prediction.
When the 2 is located between the MDs of two bi-articular muscles, the difference in these MDs is amplified by the map of Eq. 12. The relatively large difference in the PD between BFL and ST (Fig. 12A) as compared with the possible small difference in the MD (Table 1) might result from such an amplification mechanism. Equation 13 also indicates that the PD of the ith muscle does not directly depend on the value of gi. That is, the PD is not affected by each muscle's moment arm, PCSA, and so on. Hence, the muscles with the same MD should have the same PD. This can be ascertained from the PDs of VL and VM: although their PCSA is considerably different (Wickiewicz et al. 1983), their PDs agreed quite well (Fig. 12B). As can be understood from the matrix (Eq. 13), the PD is determined by the configuration of the entire musculo-skeletal system. That is, the change in parameter gi in a certain muscle can affect the way that the PD is shifted from the MD of other entire muscles. This effect was exemplified in our experimental result showing that the PD of the GM spanning only the "hip" joint was significantly affected by the "knee" joint angle (Table 2). In other muscles except the GM and GAS, no consistent shift of the PD with knee joint angle independent of the subject was observed (Table 2). However, considering a possible inter-subject difference in the structure of the matrix , there is a possibility that the direction of the PD shift shown in Fig. 8 depends on the subject.
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FIG. 12. A: the PD of semitendinousus (ST) and biceps femorirs long head (BFL). The PD of ST was significantly larger than that of BFL. In addition, the variability of BFL was larger. B: the PD of vastus lateralis (VL) and vastus medialis (VM). Both PDs agree well.
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Therefore our data can be well explained by the model minimizing the sum of the squared force (or activation or stress) as in case of the upper limbs (van Bolhuis and Gielen 1999). One interpretation of this cost function is to consider that it is related to the energy consumption (Happee and van der Helm 1995). However, the question of why the cost function takes a quadratic form remains unsolved. An alternative idea bridging the gap between the cost function and the behavioral consequence has been recently proposed (Todorov 2002) with the basic assumption being that the force output of the muscle is not constant but fluctuates (McAuley et al. 1997; Nozaki et al. 1995), and the amount of fluctuation is generally proportional to the magnitude of the output itself (Jones et al. 2001; Todorov 2002). Under the existence of such signal-dependent noise, the solution minimizing the error of final hand position arrived at via a reaching movement can well reproduce the actual movement characteristics such as thetrajectory or shape of the velocity profile of the endpoint (Harris and Wolpert 1998). Similarly in the isometric force exertion task, first van Bolhuis and Gielen (1999) have pointed out that minimizing the sum of the squared muscular force is equivalent to minimizing the variance of the endpoint force vector under the existence of signal dependent noise in the muscular force output. Then Todorov (2002) has theoretically proven this equivalence and proposed that the cosine tuning comes from the process of minimizing the variance of the endpoint force vector. Such a behavioral goal can be achieved through a feedback control process by the CNS involving the motor cortex (Scott 2004) and cerebellum (Wolpert et al. 1998). Considering that the PD of the motor units within a muscle are distributed over a relatively wide range (Hermann and Flanders 1998; van Zuylen et al. 1988) (However, note that these studies also reported examples of motor units which did not obey cosine tuning) and that the neuron within the primary cortex exhibits a muscle-like tuning pattern (Kakei et al. 1999), such a process might work at different levels of the CNS.
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GRANTS
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This work was supported by the Descente and Ishimoto Memorial Foundation, by the Combi Wellness Academy, and by the Japanese Ministry of Health, Labor, and Welfare.
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ACKNOWLEDGMENTS
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We thank Dr. Stephen Scott for helpful discussion and suggestions. We also thank T. Mita for preparation of the experimental device and T. Kato for experimental help.
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FOOTNOTES
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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.
Address for reprint requests and other correspondence: D. Nozaki, Dept. of Rehabilitation for Movement Functions, Research Institute NRCD, 4-1 Namiki Tokorozawa, Saitama 359-8555, Japan (E-mail: dnozaki{at}rehab.go.jp)
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REFERENCES
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Bernstein NA. The Coordination and Regulation of Movement. Oxford, UK: Pergamon, 1967.
Binding P, Jinha A, and Herzog W. Analytic analysis of the force sharing among synergistic muscles in one- and two-degree-of-freedom models. J Biomech 33: 1423–1432, 2000.
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ABSTRACT
INTRODUCTION
