Motor Control, 2000,4, 97-1 16 O 2000 Human Kinetics Publishers, Inc.

Muscle Coordination: The Discussion Continues Boris I. Prilutsky In this response, the major criticisms of the target article are addressed. Terminology from the target article that may have caused some confusion is clarified. In particular, the tasks that have the basic features of muscle coordination, as identified in the target article, have been limited in scope. Anew metabolic optimization criterion suggested by Alexander (2000) is examined for its ability to predict muscle coordination in walking. Issues concerning the validation of muscle force predictions, the rules of muscle coordination, and the role of directional constraints in coordination of two-joint muscles are discussed. It is shown in particular that even in one-joint systems, the forces predicted by the criterion of Crowninshield and Brand (1981) depend upon the muscle moment arms and the physiological cross-sectional areas in much more complex ways than either previously assumed in the target article, or incorrectly derived by Herzog and Ait-Haddou (2000). It is concluded that the criterion of Crowninshield and Brand qualitatively predicts the basic coordination features of the major one- and two-joint muscles in a number of highly skilled, repetitive motor tasks performed by humans under predictable conditions and little demands on stability and accuracy. A possible functional significance of such muscle coordination may be the minimization of perceived effort, muscle fatigue, andlor energy expenditure. Key Words: muscle redundancy, muscle activation, optimization

I was approached by the editor of this journal to present in a target article my own view on the problem of muscle redundancy and its relationship to the possible strategies and functional consequences of intermuscular coordination. This was a timely offer, because I had been intrigued for a long time by the fact that in a number of skilled static and dynamic tasks, patterns of muscle activation show several similar stereotypical features. At the same time, many qualitative aspects of these features seemed to be predicted by methods of static optimization. For the sake of discussion, therefore, I thought I would attempt a description and explanation of the stereotypical features of muscle activation, utilizing results of static optimization. It was anticipated that such an approach would only be capable of addressing very specific aspects of motor control and would thus provoke critical commentaries. But, after all, no scientific theory is ultimately correct. The goal of a scientific theory is to derive testable predictions and subsequently to develop

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better theories. The constructive exchange of opinion and criticism, found in the commentaries regarding the target article, will serve as a framework for achieving this goal. Several points of the target article were criticized by the commentators. In this response, I address the major criticism. First, the terminology I used is clarified. Second, I define tasks with the identified stereotypical features of muscle coordination and discuss optimization criteria that can be responsible for the selection of this coordination. Third, 1respond to criticism concerning validation of theoretically predicted activation patterns and rules of muscle coordination. Finally, directional constraints and coordination of two-joint muscles are addressed.

Terminology Muscle coordination in the target article was defined as "a distribution of muscle activation or force among individual muscles to produce a given combination of joint moments." All other notions of motor coordination (e.g., Bemstein, 1967; Turvey, 1990; von Holst, 1973) were not considered. Coordination in motor control is a very broad and complex phenomenon, the study of which requires multidisciplinary approaches. It is unlikely that such a complex phenomenon can be completely understood based on very specific definitions of coordination, such as muscle coordination (Prilutsky, 2000) or movement coordination (Kautz et al., 2000). In my opinion, the study of muscle coordination as defined in the target article provides-a useful tool for obtaining new insight into certain aspects of motor control (cf. Kautz et al., 2000). Raikova (2000) noted that the terminology used in the target article was insufficient for a full and unambiguous description of muscle function and, further, emphasized the need for new-terminology (see Raikova, 2000, for examples of this newly-proposed terminology). To avoid possible misunderstandings, some of the terms suggested by Raikova will be used in this response. Since the biomechanical definitions of agonistic and antagonistic muscles (Andrew & Hay, 1983) do not necessarily coincide with the corresponding anatomical definitions, those muscles that have anatomically-opposed functions (such as the long heads of triceps brachii and biceps brachii), or anatomically-similar functions (such as brachioradialis and brachialis; Figure 1 in Raikova, 2000) will be called anatomical antagonists and anatomical agonists,respectively. These terms will be used in addition to antagonists and agonists as defined in the target article, whose moment direction is opposite to, or coincident with, the direction of the joint moment (Andrews & Hay, 1983), respectively. Agonistic and antagonistic muscles correspond to performers and resistors, respectively, in Raikova's terminology. Loeb (2000) raised another point concerning terminology: Is the musculoskeletal system really redundant, and do we possess more muscles than in fact we need? Although it is true that we need at least two muscles to control each degree of freedom (since muscles cannot push) and that the total number of muscles does not greatly exceed the redoubled number of degrees of freedom, we do nevertheless have an infinite number of muscle force patterns available to produce a given combination of joint moments. Even for the simplest joint, containing only one degree of freedom and controlled by only two anatomical antagonists, there remains an unlimited choice of muscle force combinations to produce a given resultant moment. The fact that different subjects in a number-of skilled tasks

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select similar activation patterns without instructions to activate specific muscles is not trivial even for very simple tasks (e.g., Gottlieb et al., 1989; Zatsiorsky et al., 1998). Yes, in oversimplified experimental tasks, subjects could make their own inferences about unspecified conditions (Caldwell & Li, 2000; Loeb, 2000), which should lead to an increasedbetween-subject variability of activation patterns. However, activation patterns are rather stereotypical in skilled simple tasks (Buchanan et al., 1986; Flanders & Soechting, 1990; Jacobs '& van Ingen Schenau, 1992; Wells & Evans, 1987), which suggests that the subjects may select specific muscle activation based on optimizing the same physiological parameter(s) and, if so, the simple tasks may potentially help us to find out what the brain is trying to optimize. In the following discussion, I maintain that the musculoskeletal system is redundant in the sense that different activation strategies can be used to execute a given movement or a given combination of joint moments. In the target article and in this response, qualitative agreement (see comment by Herzog & Ait-Haddou, 2000) between the predicted patterns of muscle activation or force and the measured pattern of muscle activation is defined as agreement between the shapes and phases of the curves, but not their magnitudes. Qualitative agreement was measured by the Pearson correlation coefficient (see Figures 1 & 2 in this response and Figure 6 in Prilutsky, 2000).

Tasks With the Identified Features of Muscle Coordination A number of commentators (Caldwell & Li, 2000; Gielen, 2000; Gottlieb, 2000; Herzog & Ait-Haddou, 2000; Kautz et al., 2000; Loeb, 2000; McNitt-Gray, 2000; Raikova, 2000) noted that for many motor tasks that were not mentioned in the target article (and for some tasks that were), some reported experimental facts do not always exhibit the basic features of muscle coordination, as identified in Section 3.3. Basic Features of Muscle Coordination in the target article. I realize that perhaps the criteria for the selection of motor tasks for the analysis in the target article should have been more clearly defined. The target article was concerned with highly skilled repetitive motor tasks that are characterized by stereotypical muscle activation. Most of these tasks (including "cyclical steady state movements involving no or little impact" [McNitt-Gray, 20001 in predictable environment, contact control tasks, responses to unexpected postural perturbations) are sometimes called automatic and performed with minimum cohscious control. Muscle activation in tasks like holding a gun in the hand (Brown & Loeb, 1998; Loeb, 2000) clearly depends on the subject's intentions, is variable, and therefore was not discussed in the target article. Muscle activation in nearly all highly skilled automatic tasks can be voluntarily modified, for example, by setting different levels of coactivation of anatomical antagonists. It was assumed that such voluntary interventions were minimal in the examples discussed in the article-an assumption supported by the consistency of muscle coordination in cited examples (see Sections 3.1. Activation of Two-Joint Muscles and 3.2. Activation of One-Joint Muscles). Tasks in which joint moment demands are difficult to anticipate-active maintenance of an equilibrium posture against perturbations in random directions (as opposed to reflex responses to perturbations; Figure 9 in Prilutsky, 2000), landing movements (McNitt-Gray, 2000), tracking tasks, and so on-were not considered in the target article. These types of tasks do not typically exhibit stereotypical

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muscle activation and are often characterized by coactivation of anatomical antagonists (Raikova, 2000). Other examples of motor tasks with uncertain joint moment demands are novel complex tasks in which kinematics, kinetics, and muscle activation are highly variable. As noted by several commentators (Gottlieb, 2000; Loeb, 2000; McNitt-Gray, 2000; Raikova, 2000), tasks concerned primarily with stability and accuracy, and characterized by substantial coactivation of anatomical antagonists, were not considered in the target article. Thus, the fact that in many motor tasks (landing, McNitt-Gray, 2000; holding a gun, Brown & Loeb, 1998; exertion of static forces, Raikova, 2000; etc.) the muscle activation patterns do not exhibit the specific features described in the target article could mean that these tasks are not sufficiently skilled, the joint moment demands of the tasks are difficult to anticipate, and/or there is substantial voluntary intervention in the control of individual muscles. In these circumstances, each individual may use unique, task-dependent muscle activation patterns to perform the required task. The concept of task-dependent muscle activation (Loeb, 1985) emphasizes the uniqueness of each task and the corresponding activation patterns and, therefore, can be applied to any movements including tasks with uncertain moment demands. The notion of moment-demand-dependent activation (Prilutsky, 2000; see also Kautz et al., 2000), on the other hand, can be applied only to highly skilled tasks in which activation of major individual muscles is stereotypical and closely related to joint moments. According to the notion of moment-demand-dependent activation, different tasks can have the same features of muscle activation, given similar joint moment demands.

What Does CNS Optimize in Tasks With Stereotypical Activation? The skilled tasks that were analyzed in the target article are characterized by stereotypical activation patterns, despite the muscle redundancy. One possible explanation for the selection of these specific activation patterns in the process of evolution and/or learning is that the nervous system could be selecting an activation pattern that optimizes a particular physiological criterion. Even if the quantitative validation of the predicted individual muscle forces in humans could have been possible (see below), it remains a difficult problem to ascertain what specific physiological criterion is optimized. The reason for this uncertainty is that some optimization criterion that show good performance in the prediction of muscle coordination may be "a correlate of the criterion that actually drives muscle choice" (Alexander, 2000; see also, Gielen, 2000). Alexander suggested that in many of the tasks discussed in the target article, the minimization of metabolic cost could be this criterion. In response to this suggestion, I calculated muscle activation patterns in walking by minimizing metabolic rate P, as defined by Alexander (2000) and Minetti and Alexander (1997), with the use of static optimization and the model employed in the target article: Minimize P = &a, - F-,,

- v m , ,. @[vilv,,,]

subject to equality constraints Mj - &ai - do- 9[vjvm,,1 = 0 and inequality constraints 0 I ai5 1

(2) (3)

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where function @ is the metabolic cost function (Minetti & Alexander, 1997): @[VJV-,~]= r0.054 + 0.506 - (v~/v-,~)+ 2.46 - (~ilv,,~)']/[l- 1.13 . (vi/vm,J + 12.8 function Y2 is the force produced by the muscle at ( V J V ~ ,-, )1.64 ~ - (v~v-,~)~]; fuI1 activation and given muscle velocity vi (Minetti & Alexander, 1997): ?[v) v,,] =F,,i- i1.8-0.8 -[(v~,i+vi)l(v~,i-7.56~vi)]},ifv, 2.16 Vm (mls) 0.686

185 2.7 0.2 0.54 0.172

GA

VA

RF

BFS

HA

IL GLM

88.6 196.2 4.3 6.8 0.5 0.5

40.2 6.6 0.48

12.2 13.3 0.35

91 11.1 0.33

62 48.6 9.8 13.6 0.5 0.5

3.17 1.007

4.66 1.480

2.15 0.683

3.40 1.08

3.66 4.90 6.80 1.163 1.557 2.16

Note. TA, tibialis anterior; SO, soleus; GA, gastrocnemius; VA, vastii; RF, rectus femoris; BFS, short head of biceps femoris; HA, two-joint hamstrings; IL,iliacus; GLM, gluteus

maximus; PCSA is physiological cross-sectional area (determined by multiplying PCSA from Table 1 in Prilutsky & Gregor, 1997, by factor 2);Lfis optimal fiber length (estimated from Yarnaguchi et al., 1990);SF is the fraction of slow-twitch fibers in the muscle (taken fromTable 1in Prilutsky & Gregor, 1997);(1 -SF> is the fraction of fast-twitch fibers in the muscle; LJ1 -SF) is velocity factor, which was assumed to be proportional to the maximum shortening velocity (vmJ The vmUof VA was estimated from the maximum isokinetic knee extension velocity (18 radls; Tihanyi et al., 1982) and an approximate moment arm of the knee extensors(=0.06 m). Assuming that vmUis proportionalto the velocity factor LA1 -SF), V- values for muscles other than VA were calculated as vmm = 1.08LJ1- Sfl13.40, where coefficients 1.08 and 3.40 are the VA maximum shortening velocity and velocity factor, respectively.

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ear EMG envelopes and predicted activation patterns decreased for TA (from 0.832 to 0.405), SO (from 0.856 to 0.673), and HA (from 0.866 to 0.694), and increased for RF (from 0.508 to 0.583) and GLM (from 0.804 to 0.828). The most substantial difference between the two solutions is the difference between the predicted HA activation and force at the early stance. Alexander's criterion predicts zero HA activation between the 9-th and 14-th percent of the cycle. This difference could be related to the relatively high normalized shortening velocity of the HA compared to GLM, due to simultaneous knee flexion and hip extension during this period. However, because of uncertainty in the model parameters and simplifying assumptions of the two models (Alexander, 2000; Prilutsky, 2000) and because other optimization criteria can also predict similar muscle coordination (Gielen, 2000), it is difficult to decide which criterion can be responsible for the observed muscle coordination. 1n this response, 1have found it necessary to repeat some of the statements that were clearly stated in the target article concerning the performance of Crowninshield and Brand's criterion (see Section 4.2. Validation of Crowninshield and Brand's Criterion and Rules of Muscle Coordination), as it seems that some of the commentators did not appear to notice them. The criterion of Crowninshield and Brand (1981) qualitatively predicts activation patterns of the major leg muscles operating in the sagittal plane during a number of skilled tasks. This criterion, however, fails to predict satisfactory activation of smaller muscles and those muscles acting outside the sagittal plane (for references, see Prilutsky, 2000). Additionally, the criterion of minimum fatigue in formulation of Dul et al. (1984b), which takes into account the percentage of slow-twitch fibers within the muscle, seems to predict individual forces in cats better than the Crowninshield and Brand's criterion (Prilutsky et al., 1997b).

Validation of Muscle Force Prediction The comparison of two criteria proposed by Alexander (2000) and Crowninshield and Brand (1981) demonstrates the difficulties in validating predicted muscle forces and activation in humans. One major problem originates from the relationship between EMG and muscle force (e.g., Gielen, 1999,2000). In dynamic tasks, the EMG-force relationship is more complex due to electromechanical delay (EMD) between EMG and force (Caldwell & Li, 2000). If the "stretching of the series elastic component, to a point where muscle force can be detected, is the primary determinant of the EMD phenomenon" (Norman & Komi, 1979, p. 241; see also Cavanagh & Komi, 1979; Vos et al., 1991), it could be expected that muscle-tendon complexes with different compliant properties (Zajac, 1989) would have different EMDs. Therefore, we estimated EMD for each one-joint muscle by cross-correlating its EMG linear envelope with the corresponding joint moment. The EMD for the two-joint muscles was obtained as the mean of the time-delays that were found between the EMG and the joint moments at both of the joints involved (for details, see the target paper; Prilutsky et al., 1998; Prilutsky et al., submitted). This approach has limitations (Caldwell & Li, 2000) because the underlying assumptions are difficult to verify. For example, Caldwell and Li argue that our estimation of the EMD for two-joint muscles, which were calculated by averaging the two

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Figure 1 Muscle EMG linear envelopes (thin lines) and activation predicted by minimizing metabolic cost function P (equation 1;Alexander, 2000, thick lines) during a walking cycle. The EMG envelopes, muscle moment arms, and joint moments are the same as those used to obtain Figure 6 in the target article. TA is tibialis anterior, SO is soleus, GA is gastrocnemius, VA is vastii, RF is rectus femoris, HA is two-joint hamstrings, and GLM is gluteus maximus. Parameters of the musculoskeletal model are listed in Table 1.Note that values of PCSAare two times greater than those used in Figure 6 of the target article. The Pearson correlation coefficients (r) were used to estimate similarity between the EMG envelopes and predicted activation patterns.

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time-delays, decreases the strength of the EMG-moment relationship. A sensitivity analysis of the correlation between EMG of two-joint muscles and the difference of the corresponding joint moments in load lifting (Prilutsky et al., 1998a) and in cycling (Prilutsky et al., submitted) by alterations in the EMD, revealed that the EMD obtained as described in the target article gave the highest correlation. Caldwell and Li also suggested that EMD should be estimated independently by cross-correlating EMG linear envelopes with the rate of muscle length change (van Ingen Schenau et al., 1995). In our opinion, the rate of muscle length change is not an independent variable. It is related to the joint moment. The correlation coefficients calculated between these two variables for one-joint muscles in cycling were between 0.59 and 0.74 (Prilutsky et al., submitted). Although the strength of the EMG-moment relationships in our study of pedaling on a stationary bike (Prilutsky et al., 1997a, submitted) is substantially higher than that obtained by Caldwell and Li for uphill seated and standing cycling (2000, Figures 1 & 2), our data support the conclusion of Caldwell and Li that the EMD is an important factor in determining the relationship between EMG and joint moments in dynamic tasks. The difference in the strength of the EMG-moment relationships between the two studies could be associated with the fact that riding a stationary bike is a better skilled task, and therefore it may have more stereotypical muscle activation compared to uphill cycling. I agree with the commentators who noted that EMG should be used with great caution for validation of force predictions (Caldwell & Li, 2000; Gielen, 2000; Herzog & Ait-Haddpu, 2000). In our studies (Prilutsky et al., 1997a, 1998, submitted), we justified the comparison of EMG envelope patterns with predicted muscle force patterns by the fact that in the tasks we studied, joint moments that were estimated from EMG explained approximately 80% of the variation of the resultantjoint moments obtained from inverse dynamics. Even so, only qualitative comparisons between patterns of EMG and predicted muscle forces can be made. The magnitudes of the predicted muscle forces is difficult to validate by EMG. The uncertainty in the relationship between EMG and joint moments in dynamic tasks allows for a different interpretation of data presented by Gotllieb (2000, Figure 1). In his commentary, Gottlieb suggested "that in some intervals, the same joint torque will be produced by different patterns of muscle activation, depending upon properties of the task other than the values of the kinematic variables at the current instant." Although I agree that, depending on the task, the same moments can be produced by different activation patterns (see the section entitled Tasks With the Identified Features of Muscle Coordination in this response), in my opinion, Gottlieb's data do not convincingly prove this point. If the phase lags between the EMG envelopes of the elbow flexors and extensors, and the angular acceleration (or joint moment), are assumed to be 41 ms and 26 ms, respectively (as reported by Norman & Komi, 1979, for elbow flexion and extension movements), then the changes in EMG in Gottlieb's Figure 1can, in general, be used to explain the difference in the angular accelerationbetween the pointing and reversal movements. The decrease in flexion acceleration in the pointing movement, compared to the reversal movement (if one considers this significant), starts at 0.160 s-that is, 25 ms after the difference in activation of the lateral head of triceps between the two tasks becomes apparent. The decrease in extension acceleration in the pointing movement, with respect to the acceleration of the reversal movement, starts at about 0.200 s-that is, approximately 40 ms after the flexor EMG for pointing

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movement begins increasing markedly in comparison to the flexor EMG for the reversal movement. Of course, in this analysis I assumed that activation patterns of the long head of biceps and the lateral head of triceps are representative of the activation patterns of all elbow flexor and extensor muscles, that the joint force at the elbow does not affect the joint acceleration (e.g., Aleshinsky, 1986), and that a time delay between EMG and angular acceleration does exist. According to Corcos et al. (1992), the latter assumption may be incorrect. If however, my analysis is generally correct, the results suggest that in the example of Gottlieb, different muscle activation pattenls produce different angular acceleration or joint moment patterns. The second point of Gottlieb's criticism is concerned with the inability of Crowninshield and Brand's criterion to predict coactivation of antagonists. Gottlieb is correct that Crowninshield and Brand's criterion does not predict coactivation of one-joint anatomical antagonists (e.g., short head of biceps vs. lateral head of triceps). As proposed in the target article, reciprocal activation of one-joint anatomical antagonists is one of the common features of muscle coordination in many automatic tasks, and this feature is predicted by Crowninshield and Brand's criterion (see Section 4.2 Validation of Crowninshield and Brand's Criterion and Rules of Muscle Coordination). The criterion by Crowninshield and Brand predicts, however, simultaneous force productions by one-joint muscles and their two-joint anatomical antagonists. (Note that this case is discussed in the Gottlieb's examplecoactivation of the one-joint lateral head of triceps and the two-joint long head of biceps.) These predictions are clearly seen in Figure 6 of the target article: Onejoint tibialis anterior and the two-joint gastrocnemius produce force simultaneously in late-stance and early-swing phases, and the one-joint vastii and the two-joint hamstrings are predicted to produce force simultaneously in the first half of stance. In some situations, the criterion of Crowninshield and Brand predicts the force production in a one-joint muscle when it has antagonistic action at the joint (see Figure 3 in Prilutsky, 2000: VA force and activation at flexion knee moments). Coactivation of one-joint muscles with their two-joint anatomical antagonists is one of the common features of highly skilled tasks (Bobbert & van Ingen Schenau, 1988; Bobbert & van Soest, 2000; Buchanan et al., 1989; Flanders & Table 2 Optimized Values of Electromechanical Delay and Physiological Cross-Sectional Area Used to Calculate Joint Moments From EMG in Cycling (see Equations (4)-(6) and Figure 2)

Muscles Parameters

TA

SO

GA

VA

RF

HA

GLM

PCSA (cm2) EMD (ms)

30 100

43

-41

35 116

0 0

90 -116

28 75

133

42

Note. TA, tibialis anterior; SO, soleus; GA, gastrocnemius; VA, vastii; RF, rectus femoris;

HA, two-joint hamstrings; GLM, gluteus maximus; PCSA is physiological cross-sectional area EMD is electromechanicaldelay. Minus sign correspondsto a time-shift of EMG along the time axis to the left.

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Figure 2 -Joint moments in cycling obtained from inverse dynamics (thin lines) and calculated from the linear EMG envelopes of seven major leg muscles (thick lines; see equation 4) (from Prilutsky & Gregor, unpublished results; see also Prilutsky et a]., 1997a).The electromechanicaldelay and PCSA of each muscle necessary for calculating the moments from EMG (see equation 4) were found by minimizing the sum of the squared differences between experimentallv obtained and calculated moments (see equations 5 & 6). The optimizedparameters are presented in Table 2. The similarity between measured and calculated moments was estimated using the Pearson correlation coefficient (r).Extension moments are defined as positive. Data from one subject: pedaling rate 60 rpm, average power 200 W. Raw EMG was full-wave rectified and low-pass filtered using a zero lag, fourth-order Butterworth filter with a cut-off frequency of 10 Hz.

Soechting, 1990; van Ingen Schenau et al., 1992; Wells & Evans, 1987) and is predicted by Crowninshield and Brand's criterion (Prilutsky, 2000). Muscle coordination predicted by Crowninshield and Brand's criterion may provide a necessary stiffness of the limb and resistance to external perturbations (see Figure 9 in Prilutsky, 2000)-important requirements for many motor tasks (Gottlieb, 2000; Loeb, 2000).

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Another problem associated with modeling and validating the output of individual muscles in human movements is related to the simplifying assumptions upon which the model is based. At present, there are a number of model parameters and assumptions concerning the internal geometry and functional properties of the human musculoskeletal system that simply cannot be verified. The fact that the parameters of the model can be optimized in such a way that the model predicts the experimentally recorded kinematics, kinetics, and timing of muscle activity (Kautz et al., 2000; Neptune & Hull, 1999; Raash et al., 1997) does not guarantee that these parameters are correct. Figure 2 illustrates this point. It shows joint moments in cycling, as obtained from inverse dypamics (M, thin lines), and the joint moments as calculated from linear EMG envelopes of seven muscles (EM, thick lines), assuming a linear relationship between EMG and muscle force. The moment EM was calculated using the following formula: where EMGijtis linear EMG envelope of the i-th muscle (i = 1, .. . ,7; see Table 2) at the t-th time instant crossing the j-th joint (j= 1,2, 3), EMD, is the electromechanical delay of the i-th muscle, EMGim is EMG of the i-th muscle recorded in the maximum isometric contraction, the product u . PCSA, is the maximum force of the i-th muscle ( a = 40 N/cm2and PCSA is muscle physiological cross-sectional area), dijris the moment arm of the of the i-th muscle at the t-th time instant crossing the j-th joint. Parameters EMD, and PCSAi of expression (4) were found by solving the following optimization problem: minimize xsj(M,.f - EMjf12

(5)

subject to PCSAi < 0

(6)

,,

Values of parameters EMD and PCSA, found for each muscle by solving the problem (5 - 6) (Table 2), provide a reasonable match between joint moments obtained from inverse dynamics and from EMG (Figure 2), given a relatively small number of unknown parameters that can be adjusted compared to more sophisticated models (e.g., Raash et al., 1997). As evident from Table 2 (e.g., VA PCSA = O), the good agreement between the experimentally obtained and predicted joint moments does not guarantee that the estimated EMD and PCSA are correct. Since the true values of many parameters of the human musculoskeletal system cannot be found, the predictions of any model used to simulate human movements should be considered with caution. In general, results obtained by using sophisticated models are more uncertain than the results obtained by using simpler models, because the former require a greater number of assumptions and unknown parameters in order to perform calculations. Therefore, I am not convinced by the results of Kautz et al. (2000) and Neptune and Hull (1999), which suggest that muscle coordination in cycling does not minimize fatigue function as defined by Crowninshield and Brand (1981). As mentioned in the target article (section 5.4. Reduction of Perceived Effort), muscle fatigue is closely related to the sense of perceived effort, which appears to be minimized when subjects select the preferred pedaling rate in cycling (Coast et al., 1986) or the preferred gate in treadmill locomotion (Noble et al., 1973). However, I agree with Kautz et al. that since static optimization utilizes joint moments as input, this method cannot be used to explain "why the nervous system produces a

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particular combination of joint moments for a given dynamic task" (Kautz et al., 2000) in tasks where the joint moments are freely chosen by subjects. In static tasks where subjects are required to produce a specific external force (Jacobs & van Ingen Schenau, 1992), joint moments (which are determined by the direction and magnitude of the exerted force) appear to qualitatively satisfy the minimum fatigue criterion by Crowninshield and Brand (Prilutsky & Gregor, 1997).

Rules of Muscle Coordination I would like to thank Herzog and Ait-Haddou (2000) for raising the issue of rules of muscle coordination. However, it appears that their own mathematical derivation of these rules is incorrect (at least for one-joint systems; see below). At the time of preparing the target article, no analytical solution of the optimization problem (1)-(3) of the target article was available for any reasonably realistic multijoint musculoskeletal model. Herzog and Ait-Haddou argue that such analytical solutions were obtained in studies by Herzog and Binding (1992, 1993). The authors of these studies assumed, however, for the sake of simplicity,that all of the muscles in the model had the same moment arm and the same PCSA. Apart from the fact that these assumptions are highly unrealistic, they make it impossible to analyze the relationship between predicted forces of individual muscles and their moment arms and PCSAs. Since, at the time, there was no analytical solution available for the model discussed in the target article, I used the previous results of numerical optimization (Prilutsky & Gregor, 1997; Prilutsky et al., 1997a, 1998, submitted) and analytical solutions for one-joint systems (Dul et al., 1984a) to infer the relationships between predicted forces of individual muscles and their moment arms and PCSAs. My conclusions about the relationship between predicted muscle forces, and moment arms and PCSAs were inaccurate, as correctly pointed out by Herzog and Ait-Haddou. Their comments motivated me to find explicit expressions for muscle forces as functions of the moment arms, PCSAs, and joint moments by solving analytically problem (1)-(3) of the target article. In this response, these expressions are presented for criterion rnin @ = Ci(F;IPCSAi)2,which were obtained using Lagrange multipliers method (Dul et al., 1984a; Raikova, 1996) in which the optimization problem is reduced to an unconstraint optimization of the Lagrange function L:

where F,, F2, . . . ,F, are forces of nine muscles of the model (see Table 1; 1 = TA,

2=SO;3=GA;4=VA,5=RF,6=BFS;7=HA,8=L;9=GLM);h,,&,A3are Lagrange multipliers; M,, M2, and M, are the resultant joint moments at the ankle, knee, and hip, respectively; A,, A,, . . . ,A, are PCSAs of nine muscles; muscle moment arms (6) with one subscript are moment arms of one-joint muscles 1,2,4, 6,8, and 9; moment arms with two subscripts are moment arms of two-joint muscles (3,5, and 7) at the two joints (a = ankle, k = knee, h = hip). The forces satisfying the necessary conditions for existence of an extremum of function L are:

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Since the Hessian matrix of the objective function Q> = Ci(F,IPCSAi)2is positive definite, the obtained equations can be used to calculate muscle forces that minimize function @, subject to inequality constraints FiI0 (i = 1,2, . . . ,9). As seen from the equations obtained, force of any particular muscle is a very complicated nonlinear function of moment arms and PCSA of all muscles in the

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80 120 160 VA PCSA. N/cmA2

i

Figure 3-Forces of selected muscles as a function of moment arms and physiological cross-sectional area of soleus (SO) and vastii (VA). The forces were predicted by minimizing function Z,(FJPCSA,)3(see equations 1-3 in Prilutsky, 2000) for a threejoint, nine-muscle leg model (see Prilutsky & Gregor, 1997). The leg position corresponded to Position I11 in Figure 1in Prilutsky and Gregor (1997); the ankle, knee, and hip-joint moments were extension moment of 26 Nm, extension moment of 70 Nm, and flexion moment of 36 Nm, respectively. During calculations, the moment arm or PCSA of either SO or VA was changed, keeping moment arms and PCSA of all other muscles without change. (Nominal values of moment arms and PCSA were taken from Prilutsky & Gregor, 1997.) GA is gastrocnemius, RF is rectus femoris, and IL is iliacus.

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model, and of the resultant moments at each of the three joints. It is therefore difficult to formulate simple rules of muscle coordination that would express predicted forces through model parameters. Even in a system equipped with one-joint muscles exclusively, a predicted muscle force is not just a function of the product d,P: as was suggested in the target article, or in the commentary by Herzog and Ait-Haddou (2000). The force of the i-th one-joint muscle (F,) at the j-th joint depends on its muscle moment arm (dv)and PCSA (Ai) in a more complex way:

where m is the number of one-joint muscles crossing the j-th joint. The relationship between muscle force and correspondingmoment arm and physiological crosssectional area can differ, depending on the moment arms and PCSAs of other muscles crossing the same joint. For example, if n = 2 and the moment arms of all of the other muscles, except muscle i, are negligibly small, then the muscle force is essentially independentof PCSA of the muscle and is inversely related to the muscle moment arm. Figure 3 demonstrates how forces of selected muscles predicted by minimizing Crowninshield and Brand's cost function for the three-joint, nine-muscle system (see equations [I]-[3] in Prilutsky, 2000) depend on the moment arm and PCSA of the SO and VA. These relationships were obtained using numerical optimization. The leg position corresponded to Position 111in Figure 1 in Prilutsky and Gregor (1997), and the ankle, knee, and hip joint moments were extension moment of 26 Nm, extension moment of 70 Nm, and flexion moment of 36 Nm, respectively. The changes in SO and VAmoment arm and PCSA affect not only the predicted force of SO and VA, but also affect the force of other muscles (e.g., GA, RE and IL). For example, an increase in SO PCSA from 10 to 70 N/cm2results in an increase in SO force and in a decrease in GA and VA forces. These numerical results, together with the analytical solutions described above, underline the difficulty associated with establishing a simple set of rules of muscle coordination. These results may also suggest a potential explanation for the between-subject variability of muscle activation patterns at similar joint moments, as recorded for some motor tasks (e.g., McNitt-Gray, 2000): Substantial differences in moment arms andor PCSA between subjects would result in different predictions of muscle force patterns for similar joint moment demands. In conclusion of this section, it should be emphasized that all optimization results that were demonstrated in the target article (Figures 3 4 9 , & 11in Prilutsky, 2000) are correct. In particular, the criterion of Crowninshield and Brand (1981) qualitatively predicts basic coordination features of major one- and two-joint muscles in the tasks analyzed in the target article. Correspondingly, the conclusions about the possible consequences and functional significance of the stereotypical patterns of muscle activation, as discussed in the target article, are justifiable. However, the inaccuracy that did exist in the target article (Prilutsky, 2000) was in the expression of the numerical optimization results in the simple terms of three coordination rules. The commentary of Herzog and Ait-Haddou (2000) and tlus response demonstrate that such a simplification is incorrect.

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Directional Constraints and Activation of Two-JointMuscles Bobbert and van Soest (2000) and Gielen (2000) advocate the unique role of twojoint muscles for the increase in maximum performance and the reduction of dissipated energy during motor tasks where directional constraints limit the amount of positive work that can be done by one-joint muscles. As suggested by van Ingen Schenau and his colleagues, this role is accomplished by coactivation of one-joint muscles and their two-joint antagonists. This coactivation appears to be a common feature of many static and dynamic tasks (Bobbert & van Soest, 2000; Prilutsky, 2000; van Ingen Schenau et al., 1992)and is predicted by Crowninshield and Brand's criterion (Prilutsky, 2000). I agree that this coordination between one- and twojoint antagonists has important consequences for reducing total positive and negative work done by muscles (see Section 5.2. Economy of Mechanical Energy Expenditure in Prilutsky, 2000), increasing task performance, and compensating for a reduced ability of distal muscles to do work (5.6. Transfer of Mechanical Energy Between Joints). Whether or not the above advantages of the stereotypical activation patterns of two-joint muscles are the primary reason for having evolved twojoint muscles and the mechanisms of their control is still open to question in my opinion. First, during maximum performance motor tasks, the coactivation of onejoint muscles and their two-joint antagonists yields minimal improvements (Bobbert & van Soest, 2000; van Soest et al., 1993). Second, the same type of coactivation between one- and two-joint antagonists occurs in submaximum tasks that involve primarily concentric muscle actions (cycling, Gregor et al., 1991;van Ingen Schenau et al., 1992, 1995; extending the arm when joint angle changes are opposite to the moment at some joints, Gielen & van Ingen Schenau, 1992; Gielen et al., 1990; van Ingen Schenau, 1989). Without activation of two-joint muscles, these tasks would have involved the dissipation of a substantial amount of energy (Bobbert & van Soest, 2000). However, since metabolic cost of performing negative work is relatively low (Abbot et al., 1952; Asmussen, 1953), this energy dissipation may not substantially affect the efficiency and/or the endurance time of the task. Third, the same type of coactivation between two- and one-joint anatomical antagonists occurs in static tasks (Buchanan et al., 1989; Flanders & Soechting, 1990; Jacobs & van Ingen Schenau, 1992; Wells & Evans, 1987) and in tasks that involve primarily eccentric actions (landing, McNitt-Gray, 2000; load lowering, de Looze et al., 1993; etc.). The argument that the coactivation of two- and one-joint anatomical antagonists allows for a reduction in the dissipation of energy cannot be used to explain muscle coordination in these tasks. Fourth, the advantage of two-joint muscles is often associated with the control of external forces applied to the multijoint extremity (Bolhuis et al., 1998; Gielen & van Ingen Schenau, 1992; Gielen et al., 1990; van Ingen Schenau, 1989). However, the specific coordination patterns of two-joint muscles seems to persist in situations where no external force is applied to the limb (as in swing of walking and running; Prilutsky et al., 199813; Figure 4 in Prilutsky, 2000). Therefore, it is possible that the benefits of two-joint muscles, as described above, may be a consequence of, rather than a primary reason for, this specific stereotypical coordination. The reason for the observed coordination of two-joint muscles in automatic and highly skilled motor tasks could be the optimization of some specific physiologicalcriterion, or set of criteria (McNittGray, 2000), in the process of evolution and/or learning.

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In conclusion, the commentaries and the results presented in the target article and this response suggest to me that the criterion of Crowninshield and Brand qualitatively predicts the basic coordination features of the major one- and twojoint muscles in a number of highly skilled, repetitive motor tasks performed by humans under predictable conditions and little demands on stability and accuracy. A possible functional significance of such muscle coordination can b e the minimization of perceived effort, muscle fatigue, and/or energy expenditure (see Prilutsky, 2000).

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Acknowledgment I would like to thank Dr. Rositsa,Raikova for her valuable advice concerning the analytical solution of the optimization problem and for independently checking the solution obtained in this response.