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Torque Interaction among Adjacent Joints due to the Action of Biarticular Muscles


Medicine & Science in Sports & Exercise: January 2009 - Volume 41 - Issue 1 - pp 205-209
doi: 10.1249/MSS.0b013e318183c103
Applied Sciences

In our musculoskeletal system, multiple muscles are involved in the generation of torque about a joint. Hence, an infinite number of combinations of muscle activities may exist in producing a certain amount of joint torque. Many studies have assumed that the level of muscle activity level depends on torque about the joint it spans. However, this assumption does not hold true in the presence of biarticular muscles. Here we show that the muscle activity level is modulated according to a scheme called "cosine tuning" based on our experiment on the isometric force exerted by the lower limb, that is, the muscle activity about the knee and the hip joints can be represented by a linear summation of the knee and hip joint torques. This way of modulation is very simple, but it is unintuitive on the point that the activation level of the monoarticular muscles explicitly depends on the torque of the adjacent joint that they do not span. It is demonstrated that such nontrivial cosine tuning can be mathematically explained by assuming that the muscle activity pattern is determined so as to minimize the variability of the end point force. Finally, the implications of the cosine tuning for muscle strength training and motor control are discussed.

Physical Education Laboratory, Graduate School of Education, The University of Tokyo, Hongo, Bunkyo-ku; and National Institute of Information and Communication Technology, Hikaridai, Keihanna Science City, Kyoto, JAPAN

Address for correspondence: Daichi Nozaki, Ph.D., Graduate School of Education, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku 113-0033, Tokyo, Japan; E-mail:

Submitted for publication December 2007.

Accepted for publication January 2008.

Our musculoskeletal system is a redundant system in which multiple muscles are involved in the torque generation about a single joint (1). Hence, an infinite number of combinations of muscle activities can produce a certain amount of joint torque. A natural question arises as to how the joint torque is distributed among the synergistic muscles. Alternatively, a question that can be answered more easily is how the muscle activity level is modulated along with the joint torque. Intuitively, the muscle activity appears to increase with the torque about the joint that it spans. This notion is consistent with the classical notion of the "muscle equivalent model" proposed by Bouisset (4), which assumes that the muscles spanning a joint work together as a synergy.

However, such a concept is too simple to explain an actual situation. Let us assume that the activation level of the knee and the hip joint muscles is determined according to the magnitudes of the knee and the hip joint torques, respectively. There exists a major problem due to the presence of biarticular muscles in our musculoskeletal system (12). For example, the rectus femoris (RF) acts as a hip flexor as well as knee extensor. Hence, the activation level is determined by two different methods-from the knee joint torque and from the hip joint torque. However, obviously, both of them rarely coincide, indicating that this problem can never be treated as a single degree-of-freedom problem due to the presence of biarticular muscles (2). Therefore, we are required to clarify how the muscle activity level is modulated with not only the torque about the joint it spans but also the torque about the adjacent joint(s).

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Relationship between joint torque and muscle activity.

In a previous study (9), we evaluated the muscle activity level (mean EMG activity) of the major lower limb muscles when subjects isometrically exerted various combinations of hip and knee joint torques. We used a force exertion task where the participants were required to exert a force in various directions (Fig. 1). In this task, the knee (Tk) and the hip joint torques (Th) are linearly related to the force vector F = (Fx, Fy) applied at the end point as TT = AFT, where T is (Tk, Th) and A is the Jacobian matrix (superscript T denotes transpose).

Figure 2A shows the relationship between the mean EMG of muscle activity and each of the hip and knee joint torque. For example, the muscle activity of the vastus medialis (VM) increases with the knee joint torque and there is no clear relationship between the VM activity level and the hip joint torque. This appears quite natural considering that the VM is a monoarticular knee joint muscle that contributes to the generation of only the knee extension torque generation. A similar pattern is observed in the case of other monoarticular muscles. On the other hand, for biarticular muscles, the activity level is dependent on both the joint torques. For example, the activity level of the RF increases not only with the knee extension torque but also with the hip flexion torque. This is reasonable considering the muscle's mechanical function.

However, the observed relationship is not very precise. For instance, in the RF, the muscle activity level corresponding to a certain magnitude of the knee joint torque varies considerably, indicating that there is no clear one-to-one correspondence between them. Note that this variability was not introduced by measurement error. As observed from Figure 2B, there exists a best view direction of the 3D plot along which the variability of muscle activity level was minimal, indicating that the distribution of the data was fitted well by a single plane. Hence, the muscle activity level (M) can be represented as a linear summation of the knee and hip joint torques as M = [aTk + bTh], where [×] = max(0, x) (extension torque is defined as positive). This equation can be rewritten as M = [PTT] = [|P||T|cosθ], where P = (a,b), T = ([Tk, Th,]) and θ denotes the angle between P and T. Hence, this way of modulation is termed "cosine tuning."

We quantified the direction of the vector P as the preferred direction (PD) toward which the muscle activity level most steeply increases (Fig. 3A). Figure 3B shows the PD of each muscle (averaged for 12 subjects). The PD of biarticular muscles points toward the direction expected from the mechanical action. For example, the RF is both a knee extensor and a hip flexor, which is in agreement with the PD of approximately 315°.

However, the result of the PD of some monoarticular muscles was slightly different from the expected results. For example, the PD of the VM and the vastus lateralis (VL) was approximately 14° (Fig. 3B), indicating that the muscle activity was dependent on not only the knee joint torque but also hip joint torque. Considering the function to be represented by the mechanical pulling direction (MD), the monoarticular muscles should generate the torque only about the joint that they span. Similarly, the PD of the gluteus maximus (GM) rotated clockwise from the MD by approximately 36°, indicating that the activation level was explicitly dependent on the knee joint torque.

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Origin of the muscle activity modulation with joint torque.

As demonstrated in the previous section, the way of modulation of muscle activity level is not complicated even in the presence of biarticular muscles; it obeys the simple cosine tuning. However, the PD unintuitively shifted from the MD for several monoarticular muscles. In this section, the reason behind the modulation in such a simple yet unintuitive fashion is discussed.

On the joint toque plane, each muscle activity generates a torque vector pointing toward the MD. Hence, the problem can be mathematically formulated as follows: how a given torque vector T = (Tk, Th) is constructed by a combination of the muscle's basic vector ei = (cos μi, sin μi) as

where mi is the MD of the ith muscle, ki is the activation level, and gi represents factors such as muscle physiological cross-sectional area, moment arm, etc.

This type of problem has been rigorously investigated in the field of biomechanics (2,5,13,15), and it is usually solved using the concept of the optimization-the muscle activity is determined so as to optimize (or minimize) a cost function. According to a previous study (15), the solution of minimizing sum of muscle force (or stress or activation) squared provides the best fit to the experimental data of the upper limb. Hence, we used the same cost function to obtain the value of ki.

This problem can easily be solved by assuming that each muscle has an antagonist that has an opposite MD (although this assumption is slightly different from realistic situations, it is confirmed numerically that the conclusion derived from the following discussion is effective). The solution can be represented by the following equation:

Furthermore, from equation 1, we can obtain the map from the MD into the PD by

where Φi = (cos Φi, sin Φi), Φi is the PD, and

This equation implies three important points. First, it predicts that the muscle activity level is represented by a linear summation of the knee and the hip joint torques (i.e., cosine tuning). Second, the matrix Λ determines the PD of all the muscles, indicating that the PD of each muscle is not determined individually but is a resultant of the configuration of the entire musculoskeletal system. For example, the PD of all muscles might change if the characteristic of just one of muscles (g or μ) changes. Third, the PD shift from the MD is inevitable in our musculoskeletal system where the MD exists only in the second and fourth quadrants. Due to this configuration, the term is always less than zero; consequently, the elements (2,1) and (1,2) of the matrix Λ are always greater than zero. As a result, the clockwise and counterclockwise rotations from the MD toward the PD will be observed for the monoarticular knee and hip joint muscles, respectively. This is consistent with our experimental data (Fig. 3B).

Therefore, our data can be well explained by the model of minimizing the sum of the squared force (or activation or stress). However, the reason why the cost function takes a quadratic form remains unknown. It appears difficult to understand that the CNS calculates such a quadratic sum of forces across all the muscles. A recently suggested plausible solution to overcome this problem is that minimizing the quadratic cost function is equivalent to minimizing the variability of the end point force vector in the presence of signal-dependent noise in a motor command (6). This concept was first proposed by van Bolhuis and Gielen (15), then Todorov (14) mathematically proved the equivalence and pointed out that the cosine tuning minimizes the error in the end point force. This concept is fascinating because the information of the error in the end point force is directly available for the CNS.

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Implications for strength training.

A single-joint torque exertion task such as knee extension/flexion has been considered to be least influenced by a person's skill. Due to this simplicity, it is considered as a suitable and efficient task to control the activation level of muscles spanning the corresponding joint. In more practical situations like strength training, this type of torque exertion might be the first choice. Notably, in this case, the outputx of the muscular force is controlled by specifying the magnitude of the torque about the relevant joint. For example, only the knee extension torque is specified (e.g., 80% of one repetition maximum) when an individual attempts to increase the strength of the knee extensors.

However, as described in the previous sections, such negligence of adjacent hip joint torque might be problematic because the activity of even the monoarticular knee joint muscles is explicitly dependent on the hip joint torque. One might consider this negligence to be a minor problem from a practical viewpoint; however, this is not the case. In our previous study (8), participants were asked to exert various levels of knee joint torque. A computer displayed the visual information of the knee joint torque. However, we recorded the hip joint torque as well as knee joint torque without notifying the participants.

Figure 4 shows the relationship between the knee and the hip joint torques, each panel of which represents the data obtained from an individual. It is apparent that they adopted various strategies to exert knee joint torque, particularly during the knee extension toque exertion. The increase in the knee extension torque is associated with the increase in hip extension (Fig. 4A), increase in hip flexion (Figs. 4D-F), or negligible hip joint torque (Fig. 4B). In several participants, the relationship was not a straight line but was highly curved (Figs. 4A and C) or variable (Figs. 4E and F). On taking the PD of the knee extensors into account, these results indicate that the contribution of each muscle to the knee extension torque might differ from subject to subject, from trial to trial, or with the magnitude of knee extension torque.

Therefore, the adjacent joint torque should be controlled in order to activate the knee joint muscles in a reproducible fashion. This can be easily accomplished by monitoring the force direction because a torque vector uniquely corresponds to the vector of the end point of the limb (Fig. 1). Importantly, this notion leads to the concept of training the mono- and biarticular muscles separately. The difference in the activation of mono- and biarticular muscles has been reported in various movements such as cycling (16), walking and running (7), and jumping (3) [There is an excellent review on functional differences between mono- and biarticular muscles (12).] By modulating the force direction with reference to the PD, the strength training can be freely designed to satisfy the requirements of the movements.

Ohshima et al. have suggested a very interesting concept in context to this study that enables us to evaluate the monoarticular and biarticular muscles' strength separately (11). Their concept is briefly described as follows (our explanation is slightly different from theirs). Consider a person whose maximal torques of knee extension and hip extension are already known (they are supposed to be measured separately). The problem is whether or not he or she can exert the torques around both the joints simultaneously. Obviously, the answer is no because the RF that contributes to the maximal knee extension inevitably reduces the hip extension torque. Therefore, by examining the degree of reduction, the contribution of the RF can be estimated. By combining the training method with evaluation methods, a more efficient training method specific to the requirement can be realized.

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Implication for motor control.

van Ingen Schenau et al. (16) have proposed an influential idea that bi- and monoarticular muscles might be involved in different aspects of motor control, that is, biarticular and monoarticular muscles contribute to controlling the force and the movement directions, respectively, of the end point of the limb. The control of force direction is discussed from the viewpoint described in this article.

The transformation of the joint torque vector into a force vector at the end point is linear, indicating that the muscle activity level can also be represented as the cosine tuning on the force vector plane. In this context about the isometric force exertion task, both the mono- and the biarticular muscles contribute equally to the control of the direction of the end point (8,9). This finding does not support the assertion by van Ingen Schenau's group.

Therefore, the functional difference between the mono- and biarticular muscles should be studied by comparing their PD with the trajectory of the joint torque vector changing during the task. In fact, our group recently found that introducing this idea aided the explanation of the difference between the behavior of the soleus and gastrocnemius muscles in human upright stance (10). Specifically, the ankle and the knee joint torque vectors required to control the anterior-posterior body sway in the upright stance are necessarily constrained along a single line on the ankle and the knee joint torque plane. The PD of the gastrocnemius and the soleus are parallel and almost orthogonal to this line, respectively, which explains the phasic and tonic activity patterns of each muscle.

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In the present article, the concept of cosine tuning with a nontrivial PD, its plausible origin, and its implication on exercise science have been introduced. The conventional belief that the activity of the monoarticular muscle depends only on the torque of the joint that it spans should be reconsidered. I believe that the scheme introduced here is a clue to understand the muscle coordinations during various movements and force exertions. For further details, the readers can refer to our previous articles on this topic (8,9,10).

The works described here were supported by the Descente and Ishimoto Memorial Foundation, by the Combi Wellness Academy, by the Japanese Ministry of Health, Labor, and Welfare, and by the Japanese Ministry of Education, Culture, Sports, Science and Technology (C, #18500456; Wakate-S, 20670008). I declare that the results of the present article do not constitute of endorsement by ACSM.

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1. Bernstein NA. The Coordination and Regulation of Movement. Oxford (UK): Pergamon; 1967. 196 p.
2. Binding P, Jinha A, Herzog W. Analytic analysis of the force sharing among synergistic muscles in one- and two-degree-of-freedom models. J Biomech. 2000;33:1423-32.
3. Bobbert MF, van Ingen Schnau GJ. Coordination in vertical jumping. J Biomech. 1988;21:249-62.
4. Bouisset S. EMG and muscle force in normal activities. In: Desmedt JE, editor. New Developments in Electromyography and Clinical Neurophysiology. Basel: Karger; 1973. pp. 547-83.
5. Crowninshield RD, Brand RA. A physiologically based criterion of muscle force prediction in locomotion. J Biomech. 1981;14:793-801.
6. Harris CM, Wolpert DM. Signal-dependent noise determines motor planning. Nature. 1998;394:780-4.
7. Nillson J, Thorstensson A, Halbersma J. Changes in leg movements and muscle activity with speed of locomotion and mode of progression in humans. Acta Phyiol Scand. 1985;123:457-75.
8. Nozaki D, Nakazawa K, Akai M. Uncertainty of knee joint muscle activity during knee joint torque exertion: the significance of controlling adjacent joint torque. J Appl Physiol. 2005;99:1093-103.
9. Nozaki D, Nakazawa K, Akai M. Muscle activity determined by cosine tuning with a nontrivial preferred direction during isometric force exertion by lower limb. J Neurophysiol. 2005;93:2614-24.
10. Nozaki D, Hirano T, Nakazawa K, Akai M. Cosine tuning can predict different activity patterns among triceps surae muscles during human quiet standing. Soc Neurosci Abstract. 2004;306:5.
11. Ohshima T, Fujikawa T, Kumamoto M. Functional evaluation of effective muscle strength based on a muscle coordinate system consisted of bi-articular and mono-articular muscles. Seimitsu Kougakkaishi. 1999;65:1772-7.
12. Prilutsky B. Coordination of two- and one-joint muscles: functional consequences and implications for motor control. Mot Control. 2000;4:1-44.
13. Seireg A, Arvikar RJ. A mathematical model for evaluation of forces in lower extremities of the musculo-skeletal system. J Biomech. 1973;6:313-26.
14. Todorov E. Cosine tuning minimized motor errors. Neural Comp. 2002;14:1233-60.
15. van Bolhuis BM, Gielen CCCM. A comparison of models explaining muscle activation patterns for isometric contractions. Biol Cybern. 1999;81:249-61.
16. van Ingen Schenau GJ, Boots PJM, de Groot G, Snackers RJ, van Woensel WWWLM. The constrained control of force and position in multi-joint movement. Neuroscience. 1992;46:197-207.

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