TARGET ARTICLE Motor Control, 2000,4, 1-44 Q 2000 Human Kinetics Publishers, Inc.

Coordination of Two- and One-Joint Muscles: Functional Consequences and Implications for Motor Control Boris I. Prilutsky The purpose of this paper is three-fold: (a) to summarize available data on coordination of major two- and one-joint muscles in multijoint tasks and identify basic features of muscle coordination, (b) to demonstrate that there may exist an optimization criterion that predicts essential features of electromyographic activity of individual muscles in a variety of tasks, and (c) to address the functional consequences of the observed muscle coordination and underlying mechanisms of its control. The analysis of the literature revealed that basic features of muscle coordination are similar among different voluntary motor tasks and reflex responses. It is demonstrated that these basic features of coordination of one- and two-joint muscles in two-dimensional tasks are qualitativelypredicted by minimizing the sum of muscle stresses cubed. Functional consequences of the observed coordination of one- and two-joint muscles are (a) reduction of muscle force as well as stress, mechanical and metabolic energy expenditure, muscle fatigue, and perceived effort; (b) a spring-like behavior of a multi-joint limb during maintenance of an equilibrium posture; and (c) energy transfer between joints via two-joint muscles. A conceptual scheme of connections between motoneuron pools of one- and two-joint muscles, which accounts for the observed muscle coordination, is proposed. An important part of this scheme is the force-dependentinhibition and excitation from two-joint to one-joint synergists and antagonists, respectively.

Key Words: muscle redundancy, muscle activation, optimization

1. Definitions In this paper, muscle coordination is defined as a distribution of muscle activation or force among individual muscles to produce a given combination of joint moments. Other aspects of movement coordination, for example, dynamical interactions among body segments and how they are affected by one- and two-joint muscles The author is with the Center for Human Movement Studies and the Department of Health and Performance Sciences at Georgia Institute of Technology, Atlanta, GA 30332. This article is based in part on the presentation given by the author at the Satellite Symposium of the 20th Annual Meeting of the American Society of Biomechanics: "Biarticular Muscles: Biomechanics and Neural Control," 1996,Atlanta, GA. 1

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(Zajac, 1993; Zajac & Gordon, 1989), will not be considered. Two-joint muscle is defined as a muscle that can produce moments about two joints. For example, the gastrocnemius, knee flexor and ankle extensor, is a two-joint muscle. One-joint muscle is a muscle that can produce moments only about one joint, for example, the soleus. In most of the examples in this paper, motor tasks in one plane (sagittal or horizontal) will be considered. However, one- and two-joint muscles typically produce moments about more than one or two degrees of freedom, respectively (Lawrence et al., 1993; Kuo, 1994). These muscles will be referred to here as multi-degrees-of-freedomor multi-functional muscles. A muscle, whose moment at a joint (the product of muscle force and muscle moment arm) is in the same direction with the resultant joint moment (the total moment produced by all muscles crossing the joint), has the agonistic action at this joint. If the direction of moment produced by an individual muscle is opposite to the direction of the resultant joint moment, the muscle has the antagonistic action at the joint (Andrews & Hay, 1983). Extension moments will be considered positive and flexion moments negative.

2. Introduction The number of skeletal muscles in animal and human limbs exceeds the number of kinematic degrees of freedom (DOF) in the joints. Therefore, a given combination of joint moments can be produced using an infinite number of muscle activation strategies. Nevertheless, activation of major individual limb muscles in slulled motor tasks appears to have stereotyped features. This stereotypical muscle activation in the highly redundant motor system has motivated a number of researches to find an optimization criterion that could predict basic features of stereotypical muscle activation and explain why this specific activation has been selected through evolution and learning (for reviews, see An et al., 1995; Crowninshield & Brand, 1981b;Herzog, 1996;Zatsiorslicy & Prilutsky, 1992). Since the publication of studies first addressing this issue (Barbenel, 1972; Seireg & Arvikar, 1973), research has demonstrated that some optimization criteria are able qualitatively to predict activation patterns of major muscles in selected tasks. However, it has not been established whether different static and dynamic skilled tasks share similar features of muscle coordination, whether these features could be predicted by one optirnization criterion, and if they could, what would be their functional consequences and implications for motor control. The purpose of this paper is (a) to summarize available data on coordination of major two- and one-joint muscles in multijoint tasks and identify basic features of muscle coordination, (b) to demonstrate that there may exist an optimization criterion that accounts for and predicts essential features of electromyographic activity of individual muscles in a variety of tasks, and (c) to address the functional consequences of the observed muscle coordination and underlying mechanisms of its control. The main body of the paper consists of four sections. In the next section, I review muscle activation data for two- and one-joint (or two- and one-DOF) muscles in static and dynamic tasks. In the end of that section, basic features of muscle coordination are identified. In Section 4, I demonstrate that the optimization criterion of Crowninshield and Brand (1981a) qualitatively predicts the basic features

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muscle coordination are formulated. Section 5 addresses functional consequences of the observed muscle coordination. In the final section, possible neural mechanisms underlyingbasic features of muscle coordination are identified and discussed.

3. Basic Features of Muscle CoordinationExperimental Evidence Typically,two-joint or multi-functional muscles are considered to exhibit distinctly different activation patterns compared to one-joint muscles (Loeb, 1985; Perret & Cabelguen, 1980; van Ingen Schenau et al., 1994; Zajac, 1993). Therefore, in this section, activation of two- and one-joint muscles is considered separately. Later in this paper, it is demonstrated that basic features of activation of two- and one-joint muscles in two-dimensional tasks can be qualitatively predicted by one optimization criterion and explained by the same rules of muscle coordination.

3.1. Activation of Two-Joint Muscles 3. I . 1. Static Tasks. During the control of the direction of a force exerted by a distal segment on the environment (Buchanan et al., 1986, 1989; Flanders & Soechting, 1990; Jacobs & van Ingen Schenau, 1992; Keshner et al., 1989; Kumamoto, 1984; Wells & Evans, 1987) or during resisting external moments applied to two joints or about two DOF of the same joint (Caldwell et al., 1993; Fujiwara & Basmajan, 1975; Jamison & Caldwell, 1993; Wells & Evans, 1987), activation of two-joint or two-functional muscles seems to depend strongly on the direction of moments about both joints or DOF (Figure 1). A two-joint muscle shows the highest activation when it has agonistic action at both joints or DOF, the lowest activation when it has antagonistic action at both joints, and an intermediate activation when the two-joint muscle acts as agonist at one joint and antagonist at the other. For example, the highest activation of the gastrocnemius (GA, ankle extensor and knee flexor) in experiments of Wells and Evans (1987) occurred at ankle extension and knee flexion moments, whereas this muscle was only slightly active at ankle flexion and knee extension moments (Figure 1). The two-joint antagonists hamstrings (HA, knee flexor and hip extensor) and rectus femoris (RF, knee extensor and hip flexor) demonstrated reciprocal activation:At knee flexion1 hip extension combinations of joint moments, HA had the highest activation and RF the lowest; at knee extension and hip flexion moments, the two muscles demonstrated the opposite activation (Figure 1). When RF and HA act as agonists at one joint and antagonists at the other during extension moments at the knee and hip (force direction 12;Figure I), these muscles show slight coactivation (Lombard's paradox, Lombard, 1903). Consistent with the above results are reports that the recruitment threshold of the majority of studied motor units in two-joint or twoDOF muscles decreases when the magnitude of a linear combination of the moments about both joints or DOF is increased (Ballantyne et al. 1993; ter Haar Romeny et al. 1984; van Zuylen et al. 1988). 3.7.2. Dynamic Tasks. A description of muscle coordination (distribution of relative muscle activation at given joint moments) in dynamic tasks is more complicated because: (a) the excitation-contraction coupling and muscle-tendon complex dynamics cause a time shift between EMG and joint moment patterns, and (b) changes in muscle length and the rate of its change modify muscle contractile

Force direction

Hip moment Nm

Knee moment

Nm

Ankle moment Nm

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abilities and, in turn, can modify EMG-force and EMG-moment relationships.The time shift between EMG and joint moments can be found by cross-correlating the EMG linear envelopes with joint moments or with a rhythrmcally produced external force (van Ingen Schenau et al., 1995; Vos et al., 1991). The influence of the muscle force-length and force-velocity properties on the muscle force production can be taken into account by modeling muscle mechanical properties (Hof, 1984; Zajac, 1989). In some skilled dynamic movements, the change in the muscle length and velocity do not always substantially affect the EMG-force and EMG-moment relationship. For example, 91% of variation in instantaneous values of the soleus force measured during trotting in the cat can be explained by muscle activation alone, as evident from the coefficient of determination (1-2 = 0.91) calculated between the measured soleus force and the force ~redictedfrom the soleus EMG envelope without taken into account the muscle force-length and force-velocity properties (Dowling, 1997; Norman et al., 1988). In cycling at 60 revolution per minute and in lifting a load from the ground with straight legs, at least 80% of variation in the ankle, knee, and hip moments can be explained by the joint moments estimated from EMG of the major muscles crossing these joints without taken into account the muscle force-length-velocity properties (Prilutsky, Gregor, & Albrecht, submitted). Doorenbosch and van Ingen Schenau (1995) found no significant difference in the EMG-moment relationship for two-joint muscles between similar static and slow dynamic tasks. Thus, in some skilled tasks, dynamic conditions do not markedly affect EMG-force and EMG-moment relationships. This assumption will be implied in this paper. 3.1.3. Dynamic Control Tasks and Reaching Movements. In dynamic tasks that require control of external force exerted by a distal segment, activation of two-joint muscles is strongly correlated with moments at both joints. In particular, a strong correlation (r > 0.9) was reported for the relationship (RFEMG - HA EMG) versus (knee moment - hip moment) in both static and dynamic tasks of Figure 1(opposite) -EMG amplitude of major one- and two-joint muscles as a function of the direction of the exerted external force (or different combinationsof joint moments) (from Wells & Evans, 1987; reprinted with permission). A: A schematic representation of the experimental set-up (view from above). Five subjects were placed in a side-lying posture and were asked to exert constant isometric forces of equal magnitude (64 N) in 12 different directions corresponding to the numerals on a clock face. The exerted forces were measured by a force platform mounted vertically. The subject's foot was connected to the force platform by a Nordic ski boot and binding that allowed normal and shear forces to be exerted on the force platform. Subjectsreceived a visual feedback on the magnitude and direction of the exerted force. Surface EMG was recorded from major one- and two-joint muscles; EMG was rectified, low pass filtered (6 Hz), and averaged over 1 s. EMG magnitudes were normalized to the maximum activity recorded for each muscle. B: The calculated joint moments averaged over 5 subjects. Positive moments are extension; negative moments are flexion. C: Averaged EMG values of 10 muscles (filled bars) as a function of the 12 directions of the exerted force or 12 combinations of joint moments. Note that EMG of two-joint muscles is maximum when they have the agonistic action at both joints, and minimum when they have the antagonistic action at both joints.

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external force control (Figure 2; Doorenbosch & van Ingen Schenau, 1995). A negative sign of the difference (knee moment - hip moment) corresponds, in general, to a knee flexionhip extension combination of joint moments at which RF EMG is low and HA EMG is high (or the difference [RF EMG - HA EMG] < 0). When the difference (knee moment - hip moment) is positive-that is, knee moment is extension and hip moment is flexion-the difference (RF EMG - HA EMG) is also positive, which means that RF EMG is high and HA EMG is low. Thus, the two relationships between EMG of two-joint muscles and the joint moments shown in Figures 1 and 2 illustrate the same result: Activation of two-joint muscles is strongly related to the moments at both joints. Relationships similar to those in Figures 1 and 2 were also reported for cycling (Prilutsky, Gregor, & Albrecht, submitted; Prilutsky, Gregor, Albrecht, & Ryan, 1997a, 1997b).A strong correlation was also found between GA EMG and the difference (ankle moment knee moment) in load lifting tasks (Prilutsky, Isaka, Albrecht, & Gregor, 1998; Prilutsky, Isaka, Albrecht, Ryan, & Gregor, 1997; Toussent at al., 1992), which is a reflection of the dependence of GA EMG on the ankle and knee moments (Figure 1, GA). Activation of two-joint or two-DOF muscles, comparable with that in contact control tasks (Figure l), was also reported for multijoint unrestrained arm movements. When two-joint or two-DOF muscles contribute to the desired angular displacement about both DOF, they have a greater EMG activity compared to the case where they contribute to the displacement about only one or none of the two DOF (Flanders et al., 1994; Karst & Hasan, 1991a, 1991b; Sergio & Ostry, 1994, 1995). Since joint angular velocity and displacement are determined to a large extent by the joint moment, the coordination of two-joint muscles in the above studies seems to be consistent with the coordination demonstrated in Figures 1 and 2. 3.7.4. Responses to Postural Perturbations. When external forces are unexpectedly applied to the trunk (Foix & Tevenard, 1923; Kobayashi et al., 1974), or the supporting surface is translated (Brookhart et al., 1970; Dietz et al., 1992; Macpherson, 1988a, 1988b; Nashner, 1986), the corrective activation of two-joint muscles appears to have features similar to those described previously. When the force is applied to the trunk of standing humans in the forward direction (or the supporting surface is shifted backward), two-joint long head of biceps femoris, semimembranosus, semitendinosus, and gastrocnemius presumably have the agonistic actions at both joints they cross (at hip and knee, or knee and ankle), and their EMG activity is the highest, while rectus femoris has presumably the antagonistic action at the knee and the hip and is not active. The opposite seems to be true when an external force is applied to the trunk in the backward direction (or the supporting surface is shifted forward): The activity of rectus femoris is the highest, and the posterior two-joint muscles are not active.

Figure 2 (opposite) -Experimental set-up and relationship (RF EMG - HA EMG) vs. (knee moment - hip moment) (from Doorenbosch & van Ingen Schenau, 1995; reprinted with permission).A. Three positions of the dynamometer: 0°, IS0,and -15'. Ten subjects were seated on a bike saddle and their right foot was placed on the force platform. After training, subjects were asked to exert a force of 300 N in seven different directions from 45' to 13S0 with intervals of 15' using visual feedback. In each position of the

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I

X

-300

-100

0

100 Mknee-Mhip (Nrn)

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dynamometer and for each force direction, subjects exerted forces isometrically and isokinetically. During isokinetic trials, the force plate of the dynamometer moved with a constant velocity of 2 cm/s along the direction of the exerted force. Surface EMG was recorded, rectified, low pass filtered (2 Hz), and normalized to the EMG level at a standard maximum isometric contraction (SIC); joint moments were calculated using a standard inverse dynamics analysis. B: Relationship (RF EMG - HA EMG) versus (knee moment - hip moment) obtained in isometric (circlesand solid line) and isokinetic (crosses and dotted line) conditions. Positive moments are extension; negative moments are flexion. The presented relationship means, in general, that the EMG magnitude of RF is large and that of HA is low at the knee extensionlhipflexion joint moment combinations (when the difference M,,, - M,, has high positive values). When the difference M,,, M,, is negative and large (large knee flexion and hip extension moments), HA EMG is high and RF EMG is low.

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Similar results were reported for two-joint muscles in the dog (Brookhart et al., 1970) and cat (Jacobs & Macpherson, 1996; Macpherson, 1988a, 1988b).The posterior two-joint hindlimb muscles seem to act as agonists at both joints when the supporting surface moves forward, and their activity is high, whereas the antagonist RF is not active. When the supporting surface is shifted backward, rectus femoris has the agonistic action at the hip and the knee and demonstrates the highest EMG, whereas the two-joint HA have minimum activity. This reciprocal activation of RF and HA results in a strong correlation between the differences (RF EMG - HA EMG) and (knee moment - hip moment) (Jacobs & Macpherson, 1996, their Figure 8). 3.7.5. Locomotion. During human walking and running, there are at least two periods of the cycle where two-joint muscles act as agonists at both joints they cross. In the end of the swing phase, there is a hip extensionlkneeflexion combination of joint moments, and the HA has its activationpeak around this period, whereas its anatomical antagonist RF is typically silent. In the end of stance and the beginning of swing,joint moments tend to flex the hip and extend the knee. Around this phase, the RF has a burst of activity, and activity of HA is low or absent (Kadaba et al., 1989; Nilsson et al., 1985; Prilutsky, Gregor, & Ryan, 1998;Prilutsky, Petrova, & Raitsin, 1996; Wells & Evans, 1987; Winter, 1983, 1991). The reciprocal activation of RF and HA during the swing phase of walking and running in humans is manifested in a high correlation (r > 0.9) between the differences (RF EMG - HA EMG) and (knee moment - hip moment) (Prilutsky, Gregor, & Ryan, 1998). During cat locomotion, coordination of two-joint muscles is similar to that of human locomotion (Grillner 1981; Prochazka et al. 1989). The cat two-joint hip extensors and knee flexors (biceps femoris posterior, semitendinosus, and semimembranosus posterior) typically have their peak activity at the end of the swing phase acting as agonist at both joints (Chanaud et al., 1991; English & Weeks, 1987; Manter, 1938; Perell et al., 1993). The two-joint hip and knee flexor medial compartment of sartorius has its EMG peak and contributes to knee and hip flexion moments at the beginning of the swing in walking and trotting (Manter, 1938; Pratt & Loeb, 1991; Pratt et al., 1996). Hip flexion and knee extension moment demands increase during the onset of the swing in fast galloping, and RF (knee extensor and hip flexor) demonstrates an increased EMG activity (Engberg & Lundberg, 1969; Rassmussen et al., 1978; Smith et al., 1993). During the stance phase of backward walking in the cat, the joint moments tend to extend the ankle and to flex the knee (Perell et al., 1996). During this time, GA (ankle extensor and knee flexor) appears to have a higher EMG activity and force than during the stance phase of normal walking (Perell et al., 1996), where GA acts as agonist at the ankle and antagonist at the knee. Mutability of activation patterns of two-joint muscles in animal locomotion is higher than that of one-joint muscles (e.g., Loeb, 1985; Perret & Cabelguen, 1980; Pratt et al., 1996; van Ingen Schenau et al., 1994). This observation is consistent with the notion that activation of two-joint muscles depends on moment demands at the two joints, whereas activation of one-joint muscles primarily depends on moment demands at the joint the muscle crosses (see below). The notion of task-dependent muscle activation (Loeb, 1985)may then be refined as momentdemand-dependent activation. From the preceding review, it appears that in many skilled tasks, activation oftwo-joint (or-two-DOF) muscles strongly depends on moment demands at both

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joints: When a two-joint muscle has the agonistic action at both joints, its activation is higher than when the muscle has the agonistic action at one joint and the antagonistic action at the other joint; activation of a two-joint muscle is minimal when the muscle acts as an antagonist at both joints. Similar findings have been reported for two-DOF muscles.

3.2. Activation of One-Joint Muscles Typically, activation of one-joint muscles increases with the joint moment to which the muscles can contribute as agonists. This relationship was reported for tasks of exerting a prescribed end-point force (Buchanan et al., 1986; Doorenbosch & van Ingen Schenau, 1995; Flanders & Soechting, 1990; Jacobs & van Ingen Schenau, 1992;Wells &Evans, 1987;vanzuylen et al., 1988),for cycling (Pr~lutsky,Gregor, & Albrecht, submitted; van Ingen Schenau et al., 1995), for postural responses to perturbations (Jacobs & Macpherson, 1996), and for reaching movements (Hong et al., 1994). Sometimes, one-joint muscles are active when they act as antagonists (thus producing a moment opposite to the resultantjoint moment; Buchanan et al., 1989;Flanders & Soechting, 1990;Jacobs & vanIngen Schenau, 1992). This "paradoxical" behavior of one-joint muscles has been reported to occur when their twojoint antagonists are also active. In running and cycling, activation of one-joint muscles is typically correlated not only with the corresponding joint moment, but also with the rate of muscle shortening (Prilutsky, Gregor, & Albrecht, submitted; van Ingen Schenau et al., 1995). Several authors have reported that activation of one-joint muscles is proportional to their moment arm at thejoint (Buchanan et al., 1986;Flanders & Soechting, 1990; Jacobs & van Ingen Schenau, 1992; Legrand et al., 1996).A consequence of this relationship is a reciprocal activation of one-joint muscles crossing a joint from opposite sides, which typically occurs in skilled tasks.

3.3. Basic Features o f Muscle Coordination The following features of muscle coordination can be identified based on the preceding review: 1. Activation of one-joint muscles is typically correlated with the joint moment and depends on their moment arms. As a consequence, one-joint muscles crossing a joint from opposite sides have reciprocal activation. 2. Two-joint muscles have the highest activation when they act as agonists at both joints, the lowest activation when they act as antagonists at both joints, and an intermediate activation when the muscles have agonistic action at one joint and antagonistic action at the other joint. As a consequence, two-joint anatomical antagonists RF and HA have reciprocal activation at hip flexionhee extension or hip extensionknee flexion combinations of joint moments. The RF and HA may be coactive when simultaneous production of extension moments at the knee and hip is required (Lombard's paradox; Lombard, 1903), such as in jumping (Bobbert & van Ingen Schenau, 1988; Pandy & Zajac, 1991), cycling (Jorge & Hull 1986; Prilutsky, Gregor, & Albrecht, submitted; Prilutsky, Gregor, Albrecht, & Ryan, 1997a, 1997b;Ryan & Gregor, 1992;van Ingen Schenau et al., 1992), walking and running (Nillson et al., 1985; Prilutsky, Petrova, & Raitsin, 1996; Winter, 1991), and rising from a chair (Doorenbosch et al., 1994).

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3. One- and two-joint muscles crossing a joint from opposite sides are often coactivated. This coactivation may occur when the one-joint muscles have the antagonistic action at the joint. 4. One- and two-joint synergists (muscles with the same action at a joint) are often active simultaneously (e.g., the vastii [VA] and RF in the beginning of the stance phase in walking and running). Sometimes,when a two-joint muscle acts as agonist in the both joints, its one-joint synergist is not active (e.g., the VA and RF in the end of the stance phase and in the beginning of the swing phase in walking and running; Nillson et al., 1985;Prilutsky, Petrova, & Raitsin, 1996;Winter, 1991).

4. Prediction and Explanation of Muscle Coordination This section demonstrates that the basic features of muscle coordination in planar motor tasks are predicted by the optimization criterion of Crowninshield and Brand (1981a). Three rules of muscle coordination are derived from this criterion.

4.1. Criterion of Crowninshield and Brand and Corresponding Rules of Muscle Coordination Crowninshield and Brand (1981a), Dul, Johnson, Shiavi, and Townsend (1984), and Dul, Townsend, Shiavi, and Johnson (1984) demonstrated that certain optimization criteria reasonably predict activation patterns of major individual muscles in selected tasks. These authors (and many others, see reviews An et a]., 1995; Crowninshield & Brand, 1981b;Herzog, 1996;Zatsiorsky & Prilutsky, 1992) predicted forces of individual muscles by solving a static optimization problem similar to the following: for each time instant of movement: minimize objective function 'Ci(F,IPCSAj)", i = 1,2, . . ., m; n = 2,3,4,5, (1) subject to the equality constraints Mi-Z,d,F,=O;j= 1 , 2 , . . . , k ; i = 1 , 2,..., m,

(2)

and the inequality constraints Fi > 0; i = 1,2, . . ., m,

(3)

where m is the number of muscles in the model, k is the number of joints, F, is the unknown force of the i-th muscle, PCSA, is physiological cross-sectional area of the i-th muscle, M, is the resultant moment at the j-th joint, and d,, is the moment arm of the i-th muscle relative to thej-th joint. Moments M, are typically determined using an inverse dynamics analysis and experimentally recorded kinematics and external forces applied to the body (Elftman, 1939). Moment arms d,, are often estimated fromregression equations (Prilutsky & Gregor, 1997;Visser et al., 1990) that relate joint angles with muscle moment arms and are developed based on experimental measurements of muscle moment arms (Nemeth & Ohlsen, 1985; Smidt, 1973; Visser et al., 1990). The equality constraint (2) requires the sum of moments produced by individual muscles to be equal to given joint moments. The inequality constraint (3) requires muscle forces to produce force only in the pulling direction. The objective function (1) with power n = 3 was proposed by Crowninshield and Brand (198 la) based on the empirical relationship found in experimental studies on humans between endurance time (T, time during which a muscle can produce a

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given force) and muscle stress (FIPCSA): T = (FIPCSA)". The inverse of the endurance time lIT= (FIPCSA)?can be defined as muscle fatigue (equation 1, n = 3; Crowninshield & Brand, 198la). According to Crowninshield and Brand (1981a), power n in the equation T = (FIPCSA)" was found to vary slightly; also, the solutions to the optimization problem (1)-(3) corresponding to values n = 2 to 5 were found to be similar. The power n = 1 in the objective function (1) gives a dramatically different solution: Typically, the number of predicted nonzero forces is equal to the number of DOF in the model, and these force-producing muscles are muscles with the largest product of their moment arm and PCSA (Crowninshield & Brand, 1981a; Dul, Townsend et al., 1984). With increasing power n, more equal distribution of muscle stresses is predicted. In a one-DOF case, this more equal stress distribution is typically achieved by predicting forces in all agonists, with forces being higher in muscles with a long moment arm and a large PCSA. For example in a one-DOF case, minimizing the function (1) with n > 1 predicts the distribution of forces between any two agonists according to the following equation (Dul, Townsend et al., 1984): FJF, = (dJd,JJ1("-') (PCSAJPCSA,)"'("-I), where v and w are indices of two agonists. It can be seen that all muscles are predicted to produce nonzero force, with a relatively greater force predicted for muscles with a long moment arm and a large PCSA. When n increases and approaches infinity, the power terms ll(n - 1) and nl(n - 1) approach zero and unity, respectively, and predicted stresses of any two agonists become equal: FJ(PCSA,) = F,I(PCSA,). For more realistic multijoint systems, a general force distribution equation has not been obtained. However, it is evident from numerical solutions of the problem (1)-(3) that the forces of individual muscles in multijoint systems, predicted by solving the problem (1)-(3), correspond to the following rules: Rule 1: Relatively more force is allocated to muscles with a long moment arm. Rule 2: Relatively more force is allocated to muscles with a large PCSA. Rule 3: Muscles exhibit synergistic action (or more equal distribution of stresses among muscles).As a result, the number of predicted active muscles exceeds the number of DOE These three rules will be called rules of muscle coordination.

4.2. Validation of Crowninshield and Brand's Criterion and Rules of Muscle Coordination The goal of this section is to compare the recorded EMG envelopes with muscle forces predicted by solving optimization problem (1)-(3) (which satisfy Rules 13) in several static and dynamic tasks. This comparison was done using the following model of the lower extremity and the methods of EMG processing and determination of joint moments (for details, see Prilutsky, Isaka, Albrecht, & Gregor, 1998; Prilutsky, Petrova, & Raitsin, 1996; Prilutsky, Vasilyev, Raitsin, & Aktov, 1989; Prilutsky & Zatsiorsky, 1994). The human leg was modeled as a system of four rigid segments (foot, shank, thigh, and pelvis) interconnected by three frictionless hinge joints (ankle, knee, and hip). The model is two-dimensional and acts in the sagittalplane. Coordinates

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of the fifth metatarsophalangeal, ankle, knee, and hip joints, and the iliac crest were recorded using a high speed stereophotogrammetric(Aleshinsky & Zatsiorsky, 1978)or video (Peak Performance Technologies) systems. Linear and angular displacements and accelerations of the leg segments and the recorded ground reaction forces allowed for calculating the resultant joint moments (Aleshinsky & Zatsiorsky, 1978; Crowninshield & Brand, 1981b; Elftman, 1939). The resultant joint moment was assumed to be produced by muscles crossing the joint, and the influence of passive joint structures was neglected. It was also assumed that the joint moments are produced by nine major leg muscles: tibialis anterior (TA, ankle flexor), soleus (SO, ankle extensor), gastrocnemius (GA, ankle extensor, knee flexor), vastii (VA, knee extensor), rectus femoris (RF, knee extensor, hip flexor), short head of biceps femoris (BFS, knee flexor), two-joint hamstring (HA, knee flexor, hip extensor), iliacus (IL, hip flexor), and gluteus maximus (GLM, hip extensor). Each of these muscles represented the action of all synergists with similar actions. The PCSA of the muscles was estimated from Yamaguchi et al. (1990) by adding PCSA values of the synergists (and can be found in Prilutsky & Gregor, 1997); length changes and moment arms of the muscles were calculated from joint angles using the empirical regression equations (Prilutsky & Gregor, 1997). The calculated muscle forces were normalized to the maximum force estimated as F,,, = uPCSA,, where u = 40 N/cm2 (Hatze, 1981). Simultaneously with recordings of kinematics and ground reaction forces, surface EMG of major leg muscles was collected at a sampling frequency of 1000 Hz (Prilutsky, Gregor, Albrecht, &Ryan, 1997a, 1997b; Prilutsky, Gregor, &Ryan, 1998; Prilutsky, Isaka, Albrecht, & Gregor, 1998;Prilutsky, Isaka, Albrecht, Ryan, & Gregor, 1997) or 2083 Hz (Prilutsky, Petrova, & Raitsin, 1996;Prilutsky, Raitsin, & Poltorapavlov, 1991). EMGs were band pass filtered, rectified, and then low pass filtered (Butterworth, zero lag, fourth order, cut off frequency of 6 to 10 Hz) to obtain linear envelopes. Among studied muscles were TA, SO, medial or lateral heads of GA, medial or lateral heads of VA, RF, long head of biceps femoris, semimembranosus, or semitendinosus (HA), and GLM. Due to the excitation-contractioncoupling and muscle-tendon dynamics, there is the electromechanical delay between an EMG envelope and the corresponding joint moment in dynamic tasks (Hof, 1984; Olney & Winter, 1985; Vos et al., 1991). To compare experimentally obtained EMG-moment relationships (Figures 1 & 2) with the corresponding muscle force-moment relationships predicted by solving the problem (1)-(3) and to compare patterns of the EMG envelopes with the predicted muscle forces, the EMG linear envelopes should be aligned in time with the joint moments. The delay between muscle activation and joint moments was found by cross-correlating EMG envelopes, sampled at the frequency of the video recording, with joint moments (Prilutsky, Gregor, & Albrecht, submitted; Prilutsky, Isaka, Albrecht, & Gregor, 1998; van Ingen Schenau et al., 1995; Vos et al., 1991). Electromechanical delay for one-joint muscles was defined as a time shift corresponding to the highest correlation between the EMG linear envelope and the corresponding joint moment. The electromechanical delay of two-joint muscles was defined as the mean of two delays found between EMG and the moments at the two joints the muscle crosses. Several tasks were investigated: static exertion of forces on the ground, walking, running, cycling, and load lifting using a back lift technique (similar to the

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-003-

-300 -600 -400 -200 0 200 400 Knee Hip moment, Nm

-

-200 -100 0 100 200 Knee moment, Nm Figure 3 -Measured and predicted relationships between EMG and joint moments in isometric exertions of externalforces in different directions (from Prilutsky & Gregor, 1997). Positive moments are extension; negative moments are flexion. RF is rectus femoris, HA is two-joint hamstrings, and VA is vastii. EMG was taken from the study of Jacobs and van Ingen Schenau (1992) in which subjects seated in a chair in three diierent leg positions exerted forces of 300 and 600 N on a force platform in five directions from 60° to 120° with a step of 154 In the study of Jacobs and van Ingen Schenau, surface EMG of major one- and two-joint muscles was recorded, rectified, low pass filtered, and then normalized to the filtered EMG in standard isometric maximum contractions. The muscle forces were computed by minimizing the objective function of Crowninshield and Brand (see equations 1-3, power n = 3) and normalized to the estimated maximum force of each muscle. The input for the calculations were joint moments (calculated from the known leg positions, ground reaction force, and mass and inertia characteristics of the segments; Zatsiorsky et al., 1990),muscle moment arms (estimatedfrom the regression equations; Prilutsky & Gregor, 1997),and muscle physiological cross-sectional area (estimated from Yamaguchi et al., 1990). Note a qualitative agreement between the EMG-moment and muscle forcemoment relationships.

Prilutsky

-150

-100 -50 0 50 Knee-Hip moment, Nm

100 X

Run, 2.7 m/s

A Walk, 2.7 m/s

0 Walk, 1.8 m/s

-150

-100 -50 0 50 Knee-Hip moment, Nm

100

Figure 4 -Measured relationships between (RF EMG - HA EMG) and (knee moment

- hip moment) (top) and predicted relationships between (RF force - HA force) and

(knee moment - hip moment) (bottom) in the swing phase of walking and running at different speeds (from Prilutsky et al., 1998b). Surface EMG of RF and HA were measured from 4 subjects during walking and running at four constant speeds on a treadmill. The EMG was rectified, low pass filtered, shifted in time to account for the delay between the EMG linear envelope and the joint moments, normalized to the EMG peak in a given trial, and averaged at each percent of the swing over three swing phases and 4 subjectsfor a given speed. The muscle forces were computed by minimizing the objective function of Crowninshield and Brand (see equations 1-3, power n = 3) and normalized to the estimated maximum force of each muscle. The EMG, muscle forces, and joint moments were taken every 5% of the swing phase. Both EMG-moment and predicted force-moment relationships were highly

Coordination of Two- and One-Joint Muscles

15

In every task, the activation features of two-joint muscles discussed above (Figures 1& 2) were qualitatively predicted by minimizing the objective function (1) with n = 3. For example, both the experimentally obtained and theoretically predicted relationships between EMG or force of two-joint muscles and the difference in joint moments were strongly correlated (Figures 3 & 4). The coefficient of determination for the relationships (EMG, - EMGGA)versus (knee - hip moment), (force, - force,) versus (knee - hip moment), EMGGAversus (ankle knee moment), and force,, versus (ankle - knee moment) were typically close to 0.9 or higher for each of the studied tasks. Both the EMG and predicted force of two-joint muscles with the agonistic action at the two joints the muscles cross were strongly correlated with the moments at both joints (Figure 5). As a consequence, EMG and force of each two-joint muscle were highest when the muscle had the agonistic action at both joints, and EMG and force were lowest when the muscle acted as an antagonist at both joints (Figure 5). This feature of muscle coordination was observed and predicted in all studied tasks (Prilutsky, 1995; Prilutsky & Gregor, 1997; Prilutsky, Gregor, & Albrecht, submitted; Prilutsky, Gregor, Albrecht, & Ryan, 1997a, 199%; Prilutsky, Gregor, & Ryan, 1998;Prilutsky, Isaka, Albrecht, & Gregor, 1998;Prilutsky, Isaka,Albrecht, Ryan, & Gregor, 1997; Prilutsky, Raitsin, & Poltorapavlov, 1991). The EMG-moment relationships obtained for one-joint muscles were also qualitatively predicted by the above criterion (Figure 3, VA). Note that the model predicts nonzero VA forces at flexion knee moments in accordance with the experimental observations. Predicted patterns of forces of major two- and one-joint muscles in walking and running (Prilutsky, Gregor, &Ryan, 1998a;Prilutsky, Raitsin, & Poltorapavlov, 1991; see also Collins, 1995; Crowninshield &Brand, 1981a;Pedotti et al., 1978), cycling (Prilutsky, Gregor, & Albrecht, submitted; Prilutsky, Gregory,Albrecht, & Ryan, 1997a, 1997b), and load lifting (Prilutsky et al., 1997a, 1998a) were also qualitatively similar to patterns of the shifted EMG envelopes. Figure 6 shows an example of measured EMG and predicted muscle force patterns during a walking cycle at a speed of 1.82 rnls. The similarity of patterns between EMG and force was estimated by calculating the Pearson coefficient of correlation. The correlation coefficients ranged between 0.804 and 0.958 with one exception (0.508; Figure 6, RF). In cycling, forces computed by minimizing the function 1 with n = 3 (or according to Rules 1-3 of muscle coordination) correctly predict a phase shift and time differences in peak occurrence between activation of synergistic one- and two-joint muscles (SO vs. GA, VA vs. RF, and GLM vs. HA; Prilutsky, Gregor, & Albrecht, submitted). Reciprocal activation of one-joint muscles (Figure 6, TA and SO) and a strong correlation between their activation and the joint moment (Prilutsky, Gregor, & Albrecht, submitted; Prilutsky, Gregor, Albrecht, & Ryan, 1997a, 1997b) is also predicted. Muscle force patterns computed for cycling by minimizing the objective function (1) with n = 2,4, and 6 had generally similar high correlations with the EMG envelopes but slightly greater errors of the prediction of relative muscle activation compared to the function with n = 3 (Prilutsky, Gregor, & Albrecht, submitted). These forces were characterized by slightly smaller (for n = 2) or larger (n = 4,6) synergistic action (or lesslmore equal stress distribution) among muscles. Minimization of the objective function (1) with n = 1 and infinity (for details see Crowninshield & Brand, 1981a) or minimization of functions of the general form

16

Prilutsky

Figure 5 - EMG (top) and predicted forces (bottom) of two-joint muscles of a representative subject as functions of the moments at two joints crossed by the muscles during load lifting (from Prilutsky, Isaka, Albrecht, & Gregor, 1998). The subject lifted barbell weights of 18kg using a back lift technique in which his knees and elbows were extended. EMG, force, and moment data taken every 1% of the movement time are presented. The movement time was defined as the time between the smallest hip flexion angle at the start position and the point at which the hip angle reached a plateau at its largest value. Surface EMG was rectified, low pass filtered, normalized to the peak filtered EMG in a 85%-maximum snatch, and then shifted in time to account for the delay between muscle activation and joint moments. The muscle forces were computed by minimizing the objective function of Crowninshield and Brand (1981a; see equations 1-3, power n = 3) and normalized to the estimated maximum force of each muscle. GA is gastrocnemius, GAM is medial gastrocnemius, BFL is long head of biceps femoris, and RF is rectus femoris. When two-joint muscles had the agonistic action at both joints (GA and BFL), their EMG and predicted force increased with increasingmoments at each joint. When a two-joint muscle had the antagonistic action at both joints (RF), its EMG and predicted force were low.

pr.iixr*

-

Z,(F,)" with power n = 1, 2, 3, 4, 6, and infinity, predicted the relative muscle activation with a statistically greater error compared to the function of Crowninshield and Brand, C,(F,/PCSA,)"( p < .05; Prilutsky, Gregor, & Albrecht, submitted). The comparison between normalized shifted EMG envelopes and predicted normalized muscle forces has several limitations. First, magnitude of the estimated maximum EMG and force is sensitive to subject motivation and estimated model parameters, respectively. Therefore, the comparison of the magnitudes of the normalized-EMG- d - p d i c t e d 'forces is-questionable. Second, irn- d p & c tasks,

Coordination of Two- and One-JointMuscles

17

changes in muscle length and velocity modify muscle contractile abilities and, as a consequence, EMG-force and EMG-joint moment relationships might not be linear, which would complicate the comparison of patterns of EMG and predicted forces. The criterion of Crowninshield and Brand and the corresponding three rules of muscle coordination seem to predict essential features of activation of one- and two-joint muscles and can be used to explain the observed muscle activation. According to Rule 1, more force is allocated to muscles with a long moment arm (i.e., to muscles that can produce desired moments with a smaller muscle force). When a two-joint muscle has the agonistic action at both joints, this muscle is the most advantageous for producing moments at the two joints with less force compared to one-joint synergists or two-joint muscles that have the antagonistic action at one or both joints (see also Herzog & Binding, 1994). Thus, Rule 1 of muscle coordination can potentially explain the relationships demonstrated in Figures 1-5. When a one-joint muscle and its two-joint synergist have the agonistic action at one joint and, at the same time, the two-joint muscle has the antagonistic action at the other joint, the one-joint muscle may be more advantageous than the two-joint muscle to produce the desired combination of joint moments. Then, according to Rule 1, the one-joint muscle may be activated more extensively. Potential examples of such coordination between one- and two-joint synergists can be seen in Figure 6: EMG and force peaks of one-joint VA and GLM occur when their two-joint synergists (RF and HA, respectively) have the agonistic action at the knee and hip, respectively, and antagonistic action at the other joint. EMG and predicted force peaks of two-joint muscles, on the other hand, occur when they have the agonistic action at both joints (Figures 1 & 5). Rule 2 of muscle coordination appears also to be important in explaining the observed muscle coordination. This rule seems to allow for a more accurate prediction of muscle forces as evident from the results of Prilutsky, Gregor, and Albrecht (submitted): The error in the relative activation predicted by minimizing functions 2,F; with n = 2,3,4,5,6 (which do not allocate more force to muscles with a large PCSA; Dul, Townsend et al., 1984) were statistically higher than that of functions Z,(F,IPCSA,)"with n = 2,3,4,5,6 which allocate relatively more force to muscles with a larger PCSA (Prilutsky, Gregor, & Albrecht, submitted; see also Prilutslq & Gregor, 1997). Rule 3 of muscle coordination (synergistic action) also seems necessary to correctly predict force patterns. The degree of the synergistic muscle action is determined by the power n in the objective function (1). When Rule 3 is ignored (n is assumed to be I), nonzero forces are typically predicted in only three muscles, which does not agree with the observed EMG (see, for example, Figure 6). Values of n between 2 and 6 seem best at predicting muscle activation patterns in humans (Crowninshield & Brand, 198la; Hughes et al., 1994;Pedersen et al., 1987; Pedotti et al., 1978; Prilutslq & Gregor, 1997; Prilutsky, Gregor, & Albrecht, submitted). Although Rules 1-3 (or objective functions Z,[F,/PCSAJn with n = 2,3,4,5, 6 ) predict essential features of muscle activation in several skilled tasks in humans, these rules appear to be insufficient for an adequate prediction of muscle forces in cat locomotion, and some additional rules (such as allocation of more force to muscles with a large percentage of slow twitch fibers) should be introduced to explain the force distribution among ankle extensors in cats (Prilutsky, Herzog, & Allinger, 1997).

Force, %

0

a Force, %

I

g

4

Force, %

o

(0

Coordination of Two- and One-JointMuscles

19

It should be noted that when complex three-dimensional musculoskeletal models of human lower and upper exh-emitieswith a substantiallyhigher number of muscles, including relatively small muscles located close to joint centers, are employed, the agreement between the measured activation and the predicted muscle forces is typically not satisfactory (Buchanan & Shreeve, 1996; Crowninshield & Brand, 1981a; Glitsch & Baumann, 1997; Karlsson & Peterson, 1992; van der Helm, 1994). It could be that big muscles with relatively long moment arms have different roles (and different principles of control) compared to muscles of a small size and short moment arm (like "spurt" and "shunt" muscles; see MacConaill, 1967). Some small muscles around the cat ankle, for example, do not have stereotypical activation patterns in locomotion (Loeb, 1993), which makes it impossible to predict their behavior based on the above or similar rules. In addition, the position of small muscles with respect to the joints (and, therefore, the moment arms about different DOF) is more difficult to evaluate accurately, which may cause incorrect muscle force predictions (e.g., Herzog, 1992).

5. Functional Consequences of the Observed Coordination of One- and Two-JointMuscles In the preceding section, it was demonstrated that the criterion of Crowninshield and Brand (1981a) predicts basic features of muscle coordination in planar static and dynamic tasks. In this section, some functional consequences of the observed muscle coordination are addressed. Many of them are related to, or can be derived from, the criterion of Crowninshield and Brand.

5.1. Reduction of Total Muscle Force and Stress One obvious functional consequence of the increased force production of a twojoint muscle when it has the agonistic action at both joints can be illustrated by the Figure 6 (opposite) - EMG linear envelopes (thin lines) and predicted forces (thick lines) of one- and two-joint muscles during a walking cycle (from Prilutsky, Raitsin, & Poltorapavlov, 1991). EMG was recorded from 10 subjects during walking on a treadmill at a speed of 1.82 mls. EMG was rectified, integrated over 40 ms, interpolated by an interpolation cubic spline, normalized to the EMG peak in a cycle, and then shifted in time to account for the delay between the integrated EMG and the joint moments. Muscle forces were calculatedfor one subject with a typical activation pattern. The subject walked with a similar speed over a force platform while coordinates of markers attached to the joints were recorded (Prilutsky, Petrova, & Raitsin, 1996; Prilutsky, Raitsin, & Poltorapavlov, 1991). The muscle forces were computed by minimizing the objective function of Crowninshield and Brand (1981a; see equations 1-3, power n = 3) and normalized to the estimated maximum force of each muscle. TA is tibialis anterior, SO is soleus, GA is gastrocnemius, VA is vastii, RF is rectus fernoris, HA is two-joint hamstrings, and GLM is gluteus maximus. The mean EMG patterns and the joint moments of the subject agreed well with the typical data from the literature (Nisson et al., 1985;Winter, 1991). Note a qualitative agreement between the EMG and predicted force patterns. The Pearson correlation coefficients (r) calculated between the EMG and force patterns were typically higher than 0.8.

20

Prilutsky

following example. Let a motor task be the simultaneous production of an extension moment of 20 N at joint 1 and a flexion moment of -20 N at joint 2 (Figure 7A). The most efficient strategy to reduce total muscle force and stress in this example (assuming that moment arms of all muscles at joints 1 and 2 equal 0.05 m and PCSA of all muscles are equal) is to produce force of 400 N by two-joint muscle 2, which has a two-times larger mechanical advantage for this task than muscles 1 and 3. A much less efficient strategy to produce this combination of joint moments would be not to activate the most advantageous muscle (muscle 2) but to produce a force of 400 N by two one-joint muscles 1 and 3. As demonstrated in the preceding sections, in similar situations, the nervous system appears preferentially to activate muscle 2 and, thus, allows for the reduction of the total muscle force and stress. Activating antagonistic muscles in the example shown in Figure 7A would be the most disadvantageous strategy in terms of the total muscle force and stress production.

5.2. Economy of Mechanical Energy Expenditure Another functional consequence of the increased activity of two-joint muscles when they have the agonistic action at both joints is demonstrated in Figure 7B. Again, the moments tend to extend joint 1 and flex joint 2. Also, there are equal changes in the joint angles. At joint 1, the directions of the moment and angular movement coincide, and at joint 2 they are opposite. The most efficient strategy to reduce the total mechanical energy expenditure (MEE; for definitions, see Aleshinsky, 1986; Zatsiorsky, 1986), or positive and negative work done by all muscles, is to produce force by two-joint muscle 2 because the length change of this muscle is smaller than in muscles 1 and 3. (In this particular example, muscle 2 does not change its length.) The positive and negative work done by all muscles in this case is zero. A much less efficient strategy to produce this combination of joint moments (from a standpoint of reducing total positive and negative work production) would be not to activate muscle 2, but to produce forces by each one-joint muscle 1 and 3. In this case, muscle 1 would do positive work F, . (At$, d,,) and muscle 3 would do negative work -F, - (A& . d32)(where the product At$ . d is the change in muscle length). The simultaneous production of positive and negative work by two-joint muscles at the adjacent joints occurs in walking and running (Elftman, 1940,1941; Momson, 1970; Prilutsky, Petrova, & Raitsin, 1996; Prilutsky & Zatsiorsky, 1992; Wells, 1988), in jumping (Bobbert & van Ingen Schenau, 1988), in cycling (Broker & Gregor, 1994; van Ingen Schenau et al., 1992), and in generating an endpoint force by the arm (Gielen & van Ingen Schenau, 1992; Gielen et al., 1990). The economy of MEE or mechanical work by two-joint muscles estimated in the above cited studies ranges from 3 to 100%.

5.3. Reduction of Muscle Fatigue Another functional consequence of the observed coordination between two- and one-joint muscles is a reduction of muscle fatigue. As demonstrated above, the criterion of Crowninshield and Brand (1981a) predicts forces of major muscles of the leg that are qualitatively similar to EMG envelope patterns in several static and

Coordinationof Two- and One-JointMuscles

Figure 7 - Consequences of preferential activation of two-joint muscles when they have the agonistic action at both joints. Six muscles crossingjoints 1and 2 are assumed to have the same moment arm length at both joints (0.05 m) and the same PCSA. The muscles produce extension (M,) and flexion (M,) moments of 20 N at joints 1and 2, respectively. A: An example illustrating a reduction of total muscle force and stress. A combination of extension and flexion moments at joints 1and 2, respectively, is produced with the lowest total muscle force and stress by muscle 2 producing force of 400 N. If muscle 2 is not activated, the next most eff~cientstrategy to produce the desired combination of joint moments with a smaller total muscle force and stress would be producing a force of 400 N by each of one-joint agonists 1and 3. If one of antagonist muscles 4, 5, or 6 is activated in this task, the total muscle force and stress would increase. B: Illustration of a reduction in total positive and negative work done by the muscles. The same combination of moments as in example A is required. Also, joints 1 and 2 are extending by angles Af, and hf,, respectively, and Af, = Af,. The lowest (zero) total muscle positive work and negative work are achieved if two-joint muscle 2 produces 400 N. If muscle 2 is not active and one-joint agonists 1and 3 produce the required joint moments, muscles 1and 2 would produce nonzero positive and negative work, respectively.

was derived from a nonlinear, inverse relationship between muscle endurance time and muscle stress obtained in static and dynamic tasks (Edwards, 1981; Enoka & Stuart, 1992; Mannion & Dolan, 1996), which can be approximated by the function T = (FIPCSA)" (Crowninshield & Brand, 1981a). The inverse of this function IIT= (F1PCSA)kan be defined as muscle fatigue (see equation 1). Muscle forces predicted by the minimum fatigue criterion suggest that muscle fatigue may be minimized by allocating relatively more force to muscles with a long moment arm and a large PCSA and by more equal stress distribution among muscles. It seems that the criterion by Crowninshield and Brand (1981a) and the corresponding Rules 1-3 of allocating forces to individual muscles do not predict satisfactory forces of the cat ankle extensors during locomotion. Another minimum fatigue criterion proposed by Dul, Johnson et al. (1984) demonstrates a better performance in that case (Prilutsky, Herzog, & Allinger, 1997). This fact suggests that there may be some differences in factor(s) determining fatigue between human and cat leg muscles. One such factor may be the percentage of fatigue resistant, slow twitch fibers in the muscles (Dul, Johnson et al., 1984; Petrofsky & Lind, 1979).There is a big difference in the percentage of slow twitch fibers among cat hindlimb muscles: from 98-100% in SO and vastus intermedius to 13-18% in vastus medialis and GA lateralis (Ariano et al., 1973). The distribution of slow twitch fibers in human leg muscles does not have such large extremes (Johnson et al., 1973).

5.4. Reduction of Perceived Effort The relationship between the inverse of the endurance time (that is, fatigue) and muscle stresslforce described by a power function IIT = (FIPCSA)" (n = 3; Crowninshield & Brand, 1981a; see also equation 1) is similar to a general relationship between perceived magnitude of physical stimulus and stimulus intensity (Stevens' power low; Stevens, 1975):

+

+

where is the sensation magnitude, is the stimulus magnitude, K and P are constants, and K depends on the units of measurements. According to Stevens (1975, his Table I), the perceived muscle effort as a function of the grip force measured by a hand dynamometer is described by equation (4) with the power P = 1.7. In his review of the relationships between the perceived effort and the actual physical level of exertion in different activities, Banister (1979) reported values of P between 1.3 and 3.1. Given reported P values and setting to be the exerted muscle force and K = IIPCSAP, the similarity between the two functions of muscle fatigue-stress and the perceived muscle effort-stress becomes evident. Two important points concerning this similarity should be made. First, the criterion of Crowninshield and Brand, which is often called the minimum fatigue criterion, might minimize the perceived muscle effort as well. In this case, the features of muscle coordination observed in skilled tasks (sections 3.3), which are predicted by the above criterion (section 4), would lead to the reduction of the perceived muscle effort. Second, the notion of achieving the minimum perceived effort in skilled tasks suggests that minimization of perceived effort may be a "learning rule" used by the motor network to select specific, stereotyped activation patterns during skill acquisition.

+

Coordination of Two- and One-Joint Muscles

23

5.5. Economy o f Metabolic Energy Expenditure Reducing the total muscle force, stress, positive work, and fatigue by activating muscles according to the criterion of Crowninshield and Brand and the corresponding three rules of muscle coordination (see above) is likely to lead to economy of metabolic energy expenditure: Metabolic cost increases as a function of muscle stress, positive work, and fatigue (Edwards, 1981;Hill, 1964; Ma & Zahalak, 1991; Woledge et al., 1985). Thus, another consequence of the observed coordination of two- and one-joint muscles may be the reduction of metabolic energy expenditure. Some modeling studies (e.g., Alexander, 1997; Davy & Audu, 1987; de Lussanet &Alexander, 1997)seem to support this hypothesis. For example,Alexander (1997) compared arm trajectories in fast pointing movements calculated by minimizing metabolic cost using two models: with and without two-joint muscles. The model with two-joint muscles was generally more successful in predicting the empirical arm trajectories obtained by Atkeson and Hollerbach (1985) and in reducing movement cost. Apossible explanation for this result may be that the model with two-joint muscles was able to reduce metabolic cost by using two-joint muscles in the phases of movement where they had the agonistic action at the shoulder and elbow joints.

5.6. Transfer o f Mechanical Energy Between Joints As demonstrated above, one of the features of muscle coordination at extension moments at the hip, knee, and ankle is coactivation of one- and two-joint antagonists (for examples, GLM vs. RF and VA vs. GA) and of the two-joint RF and GA. A functional consequence of this coordination is the transfer of mechanical energy between the joints. A possible significance of this transfer of energy may be suggested based on the fact that the muscles located on the proximal segments of the leg have larger volumes (Alexander & Ker, 1990; Gambaryan, 1974), and thus a greater potential to do mechanical work than muscles of the distal segments. This muscle design reduces the moment of inertia of the leg about the hip and, possibly, metabolic energy expenditure for locomotion (Myers & Steudel, 1985). Because of the limited amount of work the distal muscles can do, dissipation and generation of mechanical energy by them is limited. Two-joint muscles may offset this limitation by distributing mechanical energy between distal and proximal joints (Prilutslq, Herzog, & Allinger, 1996;Prilutsky & Zatsiorsky, 1994).Also, the above mentioned coordination of one- and two-joint muscles appears to improve the performance in explosive leg extensions (Bobbert et al., 1986; Bobbert & van Ingen Schenau, 1988; Jacobs et al., 1996; van Ingen Schenau, 1989; van Soest et al., 1993; however, see Pandy & Zajac, 1991). An increase in the height of vertical jumping due to the coactivation of the one-joint VA and two-joint GA was demonstrated using a physical model (van Ingen Schenau, 1989) and computer simulations (Bobbert & Zandwijk, 1994; van Soest et al., 1993). Possible mechanisms of energy transfer from distal to proximal and from proximal to distal joints can be demonstrated using a physical model of the leg (Prilutsky, 1991) that consists of four rigid segments: pelvis, thigh, shank, and foot, and three muscles: one-joint hip extensor represented by a spring, and twojoint RF and GA represented by nonstretchable threads (Figure 8). During a yield phase of movement (e.g., landing), the ankle and knee moments do negative work (joints are flexing against the joint moments that equal the product of the muscle

24

Prilutsky

forces and moment arms), but muscles crossing these joints do not do work because their length is constant. The negative work done by the ankle and knee moments is absorbed by the hip extensor (its length is increasing), and one can say that mechanical energy is transferred from the distal to the proximal joints through GA and RE During a push-off phase of movement, part of the energy generated by the hip extensor appears as the positive work done by the moments at the knee and ankle: It can be seen that the knee and ankle joints are extending, but the work done by the muscles crossing these joints is again zero because their length is constant. The energy generated at the ankle and knee joints in this example is equal to the energy released by the hip extensor and transported to the distal joints through two-joint muscles. The estimated values of energy transported by two-joint muscles from distal to proximal segments during the yield phase of locomotion range from 22 to 39% of negative work at the ankle in cat locomotion (Prilutsky, Herzog, & Allinger, 1996) and in human landing (Prilutsky & Zatsiorsky, 1994), respectively. In the leg extension phase of locomotion, the contribution of energy transferred from the knee to the ankle to positive work at the ankle was reported to be 7-14% in cat locomotion (Prilutsky, Herzog, & Allinger, 1996), 7% in human jogging (Prilutsky & Zatsiorsky, 1994), 28% in human sprinting (Jacobs et al., 1996), and 23% (Prilutsky & Zatsiorsky, 1994) and 25% (Bobbert et al., 1986; Jacobs et al., 1996) in human vertical jumping. The energy transferred from the hip to the knee is 6% in human jogging (Prilutsky & Zatsiorsky, 1994), and 21% and 3 1% in human jumping and sprinting, respectively (Jacobs et al., 1996). Thus, coactivation of one- and two-joint antagonists (GA and VA, RF and GLM) and coactivation of GA and RF when the extension moments are produced at the ankle, knee, and hip during the stance phase of locomotion and in landing and jumping allows bigger proximal muscles to compensate a limited ability of distal muscles to do mechanical work.

5.7. Maintenance of an Equilibrium Posture A limb slightly displaced from an equilibrium posture tends to return to the original position. Mussa-Ivaldi et al. (1985) measured the restoring forces at the end point of the human arm after small perturbations of the end-point position and before any voluntary response. They showed that the perturbed arm demonstrated a spring-likebehavior: The measured end-point stiffness was essentially symmetrical, and the field of forces generated at the end point was close to conservative. In this case, definitions of end-point stiffness, joint stiffness, and muscle stiffness (which are often ambiguously defined in the literature; for review, see Latash & Zatsiorsky, 1993) can be derived from each other (Hogan, 1985; Flash & Gurevich, 1997; Flash & Mussa-Ivaldi, 1990; Kumamoto et al., 1994; Mussa-Ivaldi et al., 1985;Tsuji, 1997). For example, for a two-DOF arm driven in the horizontal plane by six major elbow and shoulder muscles (Figure 9A), the relations between endpoint stiffness, joint stiffness, and muscle stiffness are: where K,, K,, and K,,, are matrices of joint, end-point, and muscle stiffness,respectively; superscript t denotes the transpose operation; J is the arm Jacobian, the matrix of transformation between small joint (do) and end-point (dx) coordinate

Coordination of Two- and One-JointMuscles

LANWNG

JUMPING

Figure 8 -Physical model illustrating a transfer of mechanical energy between joints via two-joint muscles (from Prilutsky, 1991). The model consists of four rigid segments (foot, shank, thigh, and pelvis) connected by three hinge joints (ankle, knee, and hip). The model has three muscles: one-joint hip extensor represented by a spring, and twojoint RF and GA represented by nonstretchable threads. When the model held on the hip joint is pressed against the ground, all three joints will be flexed (left two postures), and negative work will be done at three joints. However, muscles crossing the ankle and knee (GA and RF) do not do work because their length is constant. The negative work done by the ankle and knee joint moments is absorbed by the one-joint hip extensor whose length is increasing, and one can say that mechanical energy is transferred from the distal to the proximal joints. If the model is released from its lowest position (right two postures), all three joints will extend. Since muscles crossing the ankle and knee do not do work, the positive work done at the ankle and knee equals the energy that is released by the one-joint hip extensor during its shortening and transported to the knee and ankle by two-joint RF and GA.

changes: dx = Jd0; D is the transformation matrix between the vectors of small muscle length (dL) and joint angle changes respectively: dL = Dd0. The elements of the matrix D are moment arms of six muscles with respect to the elbow and the shoulder joints:

where subscripts s and e denote the shoulder and elbow joints, respectively. The 6 X 6 muscle stiffness matrix is defined as

K, = diag.[-dF,ldl,, -dF,ldl,, . . ., -dFddl,]

(7)

Prilutsky

26

where dF and dl are small changes in muscle force and length, respectively, caused by a small displacement of the end point (muscle elongation is defined as negative); and diag. [ ] stands for the diagonal matrix. From ( 3 , (6), and (7) it follows that the joint stiffness matrix Kj is

where K,,,= d2,, dF,ldll + d2,, . dF21d12+ 8, . dF,ldl, + 8, . dFddl, K,,, = Kj2,= d,, - d,, - dF,ldl, + d6, . d6e- dFdd16 qZ2 = d23e. dF31d13+ dZ4,- dF4/d14+ dZ5, . dF51d15+ 626, - dFdd16 a

The elements of the joint stiffness matrix K,,, and Kj2,relate the joint angle change to the joint moment change at the shoulder and elbow joints, respectively. The and K,,, depend on the action of two-joint muscles and relate the elements q.21 angular change in one joint with the moment change at the second joint. It can be seen from equation (8) that coordination (force distribution) between one- and two-joint muscles can influence joint stiffness and thus the response of the arm to a perturbation. For example, if two-joint muscles (muscles 5 and 6 in the model, Figure 9A) do not change their force (i.e., coefficients K,,, and Kj2,in the joint stiffnessmatrix equal zero), it will be impossible for a restoring force at the end point to be generated in exactly the opposite direction to the end-point displacement and to be proportional in magnitude to the displacement(Hogan, 1985). It was demonstrated (Prilutsky, 1998) that the optimization criterion of Crowninshield and Brand and the corresponding three rules of muscle coordination predict reasonably well the shape and orientation of the end-point stiffness ellipses measured by Flash and Mussa-Ivaldi (1990), which characterize the elastic force fields restoring the initial posture after a small perturbation (Figure 9B). On the other hand, criteria that do not predict the synergistic muscle action (min CiFjPCSAi)or do not allocate relatively more force to muscles with a large PCSA (min CiFi3)are less successful in predicting the measured end-point stiffness at least at some arm orientations (Figure 9C). The optimization criterion of Crowninshield and Brand always predicted the simultaneous production of nonzero forces in at least three muscles: two one-joint muscles and one two-joint muscle. Also, it sometimes predicted coactivation of one-joint and two-joint antagonists during the maintenance of the equilibrium arm posture (Prilutsky, 1998). These results suggest that the strategy of muscle coordination in reflex maintenance of posture and voluntary skilled movements (locomotion, static force production, etc.; see above) may be similar.

6. Implications for Motor Control

z .-

a

The features of muscle coordination predicted by the criterion of Crowninshield and Brand seem to occur in a variety of motor tasks: in reflex responses during the maintenance of posture (Figure 9; Aruin & Latash, 1995; Jacobs & Macpherson, 1996; Macpherson, 1988b; Nashner, 1986), in normal locomotion in cats and humans (Figure 6), in locomotor movements elicited in animals with a transacted 1996; Smith, 1986), and in skilled motor spinal.'-cord (Grillner,_l981;_Rossignol, , -----

Coordination of Two- and One-Joint Muscles

27

Figure 9 -Predictions of the field of restoring forces after small perturbations of the arm posture (from Prilntsky, 1998). A: The arm was represented as a two-degree-offreedom model with six muscles acting about the shoulder and elbow joints in the horizontal plane. The length of the upper arm and forearm was assumed to be 0.31 m. Muscle moment arms at the shoulder and elbow were calculated from the muscle origin and insertion coordinates, the coordinates of joint centers, and the orientation of flexion1 extension revolute axes published by Wood et at. (1989a, 1989b), assuming that muscles produce force along the straight line between points of their origin and insertion (see, also, Raikova, 1996). The PCSA (in cm2)of each muscle was estimated from Lemay and Crago (1996) and van der Helm (1994): PCSA, = 20.3, PCSA, = 16.6, PCSA, = 10.4, PCSA, = 12.8, PCSA, = 2.5, and PCSA, = 6.3. Muscles 1and 2 are one-joint shoulder flexor and extensor, respectively; muscles 3 and 4 are one-joint elbow flexor and extensor, respectively; and muscles 5 and 6 are two-joint long heads of biceps and triceps, respectively. B: Measured (solid lines; Flash & Mussa-Ivaldi, 1990) and predicted (dotted lines) stiffness ellipses at the end point of the arm. The arm coordinates are measured in meters (abscissa and ordinate axes), and stiffness is measured in N/m (scale is shown in the circle). The arm postures in the horizontal plane and joint stiffness values reported by Flash and Mussa-Ivaldi (1990) and 1cm displacements of the end point of the arm in different directions in the horizontal plane were the input for the calculations of the changes in joint moments that tend to restore the initial arm position. The muscle forces producing the moment changes were calculated by minimizing the objective function of Crowninshield and Brand (1981a; see equations 1-3). The muscle (continued)

28

Prilutsky

tasks like cycling (Gregor, Broker, & Ryan, 1991; Prilutsky, Gregor, & Albrecht, submitted; Prilutsky, Gregor, Albrecht, & Ryan, 1997a, 1997b; van Ingen Shenau et al., 1992,1995), load lifting (de Looze et al., 1993; Prilustky, Isaka, Albrecht, & Gregor, 1998; Prilutsky, Isaka, Albrecht, Ryan, & Gregor, 1997; Toussaint et al., 1992), reaching movements (Flanders et al., 1994; Karst & Hasan, 1991a, 1991b; Sergio & Ostry, 1994,1995), and in the static tasks of producing end-point forces (Figure 3; Buchanan et al., 1986,1989; Caldwell et al., 1993;Flanders & Soechting, 1990; Jacobs & van Ingen Schenau, 1992; Jamison & Caldwell, 1993; Wells & Evans, 1987). It seems plausible that common features of muscle coordination in different movements originate from common neural control mechanisms.

6.1. Possible Neural Mechanisms Underlying Coordination o f Two- a n d One-joint Muscles Reciprocal coordination of one-joint antagonists (e.g., SO vs. TA) and two-joint antagonists (e.g., RF vs. HA) may be controlled through pathways originating from Ia afferents and involving Ia interneurons mediating reciprocal inhibition (Baldissera et al., 1981;Jankowska, 1992). Since Ia afferents are sensitive to muscle fiber length changes, their effect on antagonistic muscles may potentially be influenced by the type of muscle contraction (eccentric, concentric, and isometric). It does not appear from available experimental data that the direction of muscle length change substantially affects the reciprocal coordination. First, the Ia afferents appear to be active in phase with muscle activation (not the muscle length change) during cat locomotion, paw shaking, and scratching (Loeb, Hoffer, & Marks, 1985; Loeb, Hoffer, & Pratt, 1985; Perret & Cabelguen, 1980; Severin, 1970) and during isometric contractions in humans (Vallbo, 1971). Second, similar reciprocal coordination of antagonists occurs in isometric contractions (Buchanan et al., 1986; Jacobs & van Ingen Schenau, 1992; Wells & Evans, 1987); in concentric contractions during the control of external forces (Doorenbosch & van Ingen Schenau, 1995), load lifting (Prilutsky, Isaka, Albrecht, & Gregor, 1998; Prilutsky, Isaka, Albrecht, Ryan, & Gregor, 1997; Tounsend et al., 1992), cycling (Gregor et al., 1991; Jorge & Hull, 1986; Prilutsky, Gregor, & Albrecht, submitted; Prilutsky, Gregor, Albrecht, & Ryan, 1997a, 1997b; van Ingen Schenau et al., 1992, 1995), Figure 9 (continued) stiffness matrix (see equation 7) was calculated from the predicted changes in muscle forces and from the correspondingchanges in muscle length caused by the displacement of the end point. The end-point stiffness was calculated from the calculated muscle stiffness (equation 5). There was an agreement between the magnitude, shape, and orientation of the calculated and measured endpoint stiffness ellipses. C: Comparison of the end-point stiffness ellipses in one arm position predicted by minimizing different objective functions. The thin solid line represents the measured stiffness ellipse (Flash & Mussa-Ivaldi, 1990); the thin dotted line represents the ellipse predicted by minimizing function 2,i(FJPCSAi)3(Crowninshield & Brand, 1981a); the thick solid line represents the ellipse predicted by minimizingfunction C,(FJ3;and the thick dashed line represents the ellipse predicted by minimizing function 8pJPCSAi.The criterion of Crowninshieldand Brand (1981a) demonstrates the best performance in predicting

Coordination of Two- and One-Joint Muscles

29

and upslope walking in the cat (Carlson-Kuhta et al., 1998; Gregor, Smith,Albrecht, Prilutsky, & Smith, 1998; Smith & Carlson-Kuhta, 1995); and in eccentric contractions during the maintenance of posture (Gurfinkel et al., 1979; Jacobs & Macpherson, 1996; Macpherson, 1988b; Prilutsky, 1998), load lowering (de Looz et al., 1993), the swing phase of human locomotion (Prilutsky, Gregor, & Ryan, 1998), and downslope walking in the cat (Smith & Carlson-Kuhta, 1995; Smith et al., 1998). The rhythmical activity of Ia inhibitory interneurons in cat locomotion (Feldman & Orlovsky, 1975), which can be responsible for reciprocal inhibition of antagonists (Baldissera et al., 1981; Jankowska, 1992), does not appear to be related to another afferent input to Ia inhibitory neurons from Renshaw cells but is likely to be controlled by an excitatory command from a central pattern generator (Rossignol, 1996). Synergistic coactivation of muscles with similar functions at a joint can be mediated by length-dependent pathways originating from Ia afferents (Eccles et al., 1957;Eccles & Lundberg, 1958;Laporte & Lloyd, 1952;Nichols, 1994)whose activity during automatic rhythmic movements and voluntary tasks is modulated to a large extent by the fusirnotor system (Loeb, Hoffer, & Marks, 1985; Loeb, Hoffer, and Pratt, 1985; Prochazka et al., 1989;Vallbo, 1971).Also, Ib inputs originating from the Golgi tendon organs appear to excite motoneurons of extensor synergists during the stance phase in cat locomotion (Gossard et al., 1994; Guertin et al., 1995; Whelan & Pearson, 1997). As demonstrated above (see section 4.2), activation of one- and two-joint muscles with a similar synergistic action at one joint (e-g., SO and GA, VA and RF, and HA and GLM) is somewhat different (Figure 6). Apparently, this occurs because activation of two-joint synergists depends on joint moment demands at two adjacent joints (Figures 1 & 5), whereas the activity of one-joint synergists is influenced primarily by the moment demand at one joint. Such differential activation of one- and two-joint synergists may be controlled in part through the forcedependent inhibitory connections described by Nichols (1989,1994) for one- and two-joint members of the cat triceps surae and quadriceps muscle groups. These force-dependent inhibitory connections are likely mediated through pathways originating from Golgi tendon organs (Nichols, 1989, 1994). These pathways may also regulate the force-dependent excitation from twojoint muscles (e.g., cat GA and PL) to their one-joint antagonists (e.g., TA; Nichols, 1989, 1994). The consequence of this force-dependent excitation may be coactivation of one- and two-joint muscles crossing the jointfiom opposite sides, which often occurs in skilled tasks (see sections 3.2 and 3.3). The dependence of activation of two-joint muscles on moments at bothjoints they cross may originate from two independent central commands to motoneurons of the two-joint muscles (Perret & Cabelguen, 1980; Smith et al., 1998; Vidal et al., 1979). These two central commands may be related to the moment demands at the two joints.

6.2. Conceptual Model o f Neural Network Controlling Two- a n d One-Joint Muscles Consider a conceptual model of connections among motoneuron pools of major two- and one-joint muscles of the human lower extremity (Figure 10). This model

is based on the basic features of coordination of one- and two-joint muscles and the corresponding possible interactions between their motoneuron pools discussed earlier in this section. In this model (which resembles several similar models of pattern generators in automatic tasks; Lundberg, 1981; Grillner, 1981; Perret & Cabelguen, 1980; Smith, 1986; Smith et al., 1998), it is assumed that excitation inputs to motoneurons of two- and one-joint muscles are proportional to the joint moment demands. Also, the distribution of excitation inputs to synergisticmuscles crossing a given joint corresponds to the following rules: More excitation is allocated to muscles with a longer moment arm and a larger PCSA (see section 4.1). Centers generating these inputs for the ankle, knee, and hip joints (large circles in Figure 10) may be a part of the spinal circuitry and/or supraspinal motor systems and are able to function independently of each other. Each joint center, which is responsible for extension or flexion in a joint, excites two- and one-joint synergists that can contribute to the required joint moment. (The corresponding connections are shown by small open knobs and solid lines between the joint centers and muscle motoneuron pools.) Thus, motoneurons of two-joint muscles can receive excitation from two centers (see also Perret & Cabelquen, 1980; Smith, 1986). Both one- and two-joint antagonist muscles inhibit each other through reciprocal inhibition (closed knobs connected by solid lines in Figure 10). When two-joint muscles produce large force, they can inhibit their one-joint synergists (force-dependent inhibition; closed knobs and dotted lines in Figure 10) and excite their one-joint antagonists (force-dependent excitation; open knobs and dotted lines in Figure 10). The output of each motoneuron pool (activation of the muscle) is assumed to be proportional to the sum of all positive (excitatory) and negative (inhibitory) inputs to the pool. This model of excitatory and inhibitory connections among motoneurons of two- and one-joint muscles has a potential to account for intermuscular coordination at different combinations of joint moments reviewed in this paper. Consider the following examples. Let us keep hip extensor and ankle extensor moment demands (or the corresponding extensor commands) constant at 20 Nm and change the knee flexion command from zero to -200 Nm. At the zero knee flexion moment, among muscles crossing the ankle, the SO will receive more excitation than the GA because of their size differences. The TA will not be active because it does not receive excitation from the ankle flexion center; it could receive a weak forcedependent excitation from GA and reciprocal inhibition from the SO and GA. With increasing the knee flexion command, the GA activity increases, which leads to a progressive decrease of the SO activation through a force-dependent inhibition from GA (Figure 10). The decrease in SO activation decreases the reciprocal inhibition of TA. The reciprocal inhibition from GA to TA increases, but at high GA forces, the force-dependent excitation of TA from GA gets more powerful, and the net effect on TA will be excitatory. Thus, the prediction from the model is that at a constant and moderate ankle extension moment and increasing knee flexion moment, GA activation increases, SO activation decreases, and TA activation increases. The optimization of the criterion of Crowninshield and Brand (see equations 1-3) with power n = 3, which qualitativelypredicts activation of individualmuscles in a number of static and dynamic tasks (Figures 3-45), can be used to evaluate

Coordination of Two- and One-Joint Muscles

Figure 10-A conceptual model of connections among motoneuron pools of one- and two-joint muscles of the human lower extremity. Motoneuron pools of one- and twojoint muscles receive excitation inputs (shown by open knobs and solid lines) from flexion and extension joint centers (large circles). The distribution of excitation inputs to synergisticmuscles crossing a given joint corresponds to the following rules: More excitation is allocated to the muscles with a longer moment arm and a larger PCSA. Two-joint muscles can receive excitation from two joint centers. Both one- and twojoint antagonist muscles inhibit each other through reciprocal inhibition (closed knobs connected by solid lines). At high forces, two-joint muscles can inhibit their one-joint synergists (force-dependentinhibition; closed knobs and dotted lines) and excite their one-joint antagonists (force-dependent excitation; open knobs and dotted lines). The activation of each motoneuron pool is assumed to be proportional to the sum of all positive (excitatory) and negative (inhibitory) inputs to the pool. This model can potentially account for the observed coordination between one- and two-joint muscles at different combinations of joint moments.

B 5000 4000

z 3000

db

'L 2000

1000 0 hklemoment. Nm

Knee moment,Nm

le momenl= -20 Nm: Knee moment = 50 Nr

-

Figure 11 Forces of selected muscles calculated by minimizing the objectivefunction of Crowninshield and Brand (1981a; equations 1-3) at different combinations of joint moments. A: Predicted forces of gastrocnemius (GA), soleus (SO), and tibialis anterior (TA) at extension hip and ankle moments of 20 Nm and the knee moment changing from -200 Nm of flexion to 100Nm of extension. There is coactivation of GA and TA at large flexion (negative)knee moments. At zero knee moment, GA and SO are coactive. At extension (positive) knee moments, only SO produces force. B: Predicted forces of GA, SO, and TA at extension hip and knee moments of 20 Nm and the ankle moment changing from -50 Nm of flexion to 100 Nm of extension. At extension (positive) ankle moments there is coactivation of GA and SO; at the ankle flexion moment, only TA is active. C: Predicted forces of rectus femoris (RF), vastii (VA), and two-joint hamstrings (HA) at a flexion ankle moment of -20 N, an extension knee moments of 50 Nm, and the hip moment changingfrom -200 Nm of flexion to 250 Nm of extension.At extension (positive) hip moments there is coactivation of VA and HA; at hip moments of zero and -50 Nm, there is coactivation of RF and VA; and at flexion hip moments between -100 and -200 Nm, only RF is active.

>. -

-s-

ankle and hip extension moments (20 Nm), the knee extension moment increases from zero to 100 Nm, the model and the calculations predict a decrease in GA force (due to the reciprocal inhibition from VA and RF), the corresponding incre * --no

Coordination of Two- and One-JointMuscles

33

If hip and knee extension moments are set at 20 Nm, and ankle extension moment increases from 0 to 200 Nm, GA and SO demonstrate the synergistic action-their activity increases with the ankle extension moment (excitation command from the ankle extension center), and the TA does not produce force (Figure 11B). At the ankle flexion moment of -50 Nm, only the TA produces force among muscles crossing the ankle. In the third example, moments at the ankle and the knee are -20 Nm and 50 Nm, respectively, and the hip moment ranges from -200 Nm of flexion to 250 Nm of extension (Figure 11C). With increasing hip flexion moments, RF force increases due to the excitation command from the hip flexion center, and VA force decreases under the influence of the force-dependent inhibition from the RF (Figure 10). When hip extension moment increases, the HA and VA increase in force together: HA, due to the hip extension command, and VA, due to the force-dependent excitation from the HA. The described conceptual model of the neural connections between muscle motoneuron pools and the joint excitation centers is of course greatly oversimplified. However, this model seems to account for the selected reflex interactions between two- and one-joint agonist and antagonist muscles (reciprocal inhibition, force-dependent excitation of one-joint antagonists, and force-dependent inhibition of one-joint agonists), may potentially predict the features of muscle coordination in a variety of tasks (Figures 3-6), and may explain the functional significance of the above-mentioned interactions between motoneuron pools of one- and two-joint muscles. Stereotypical muscle activation patterns observed in skilled tasks of locomotion, external force control, load lifting, cycling, and so on, typically require a long period of acquisition (Corcos et al., 1996; Person, 1958). This skill acquisition appears to be a trial-and-error process (Alexander et al., 1992; Bernstein, 1947) during which patterns of muscle coordination converge to the specific stereotypical muscle coordination. Several learning rules that the motor system might use during acquisition of muscle coordinationhave been proposed in the literature: the minimization of neural interaction (total afferentation; Gelfand & Tsetlin, 1966), the minimization of muscle force (McConaill, 1967), the minimization of information processing (Tomovic & Bellman, 1970), and so on. The facts that (a) the criterion of Crowninshield and Brand (198la) qualitativelypredicts the major features of muscle coordination and (b) the relationships fatigue-stress and perceived effort-stress obtained for human muscles are similar (section 5.4) may suggest that minimization of perceived effort and, in turn, muscle fatigue could be the learning rule that guides the motor system in selecting the specific muscle coordination through motor skill acquisition.

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Acknowledgments This article is dedicated to the memory of late Dr. Gerrit Jan van Ingen Schenau whose work inspired me to study muscle coordination and functions of two-joint muscles. This paper was prepared while the author worked at the Biomechanics Laboratory of the Center for Human Movement Studies at the Georgia Institute of Technology, Atlanta, GA. I thank Dr. Robert J. Gregor, the Director of the Laboratory, for his enthusiastic support and collaboration in several experimental studies. I thank Dr. Vladirnir M. Zatsiorsky for many stimulating discussions about functions of two-joint muscles. I also thank Drs. Arthur D. Kuo and Vladimir M. Zatsiorsky for reviewing the manuscript and making useful suggestions. This work was partially supported by a grant from the Office of Interdisciplinary Programs at the Georgia Institute of Technology. Manuscript submitted: October 1998