Optimization-based models of muscle coordination

Abstract

Optimization-based models may provide reasonably accurate estimates of activation and force patterns of individual muscles in selected well-learned tasks with submaximal efforts. Such optimization criteria as minimum energy expenditure, minimum muscle fatigue, and minimum sense of effort seem most promising.

Figure 1

A two-dimensional model of the human leg. It has three DoF and is controlled by nine muscles. Therefore, the moments (M) at the joints can be produced by an infinite number of muscle force combinations. TA is tibialis anterior, SO is soleus, GA is gastrocnemius, VA is vastii, RF is rectus femoris, BFS is short head of biceps femoris, HA is two-joint hamstrings, GLM is gluteus maximus, and IL is iliacus. For review of model parameters (segment inertial parameters, muscle moment arms, PCSA, muscle composition, F_max, V_max) and their values (10).

Figure 2

Recorded EMG linear envelopes (EMG) and muscle forces and activation predicted by optimizing minimum fatigue criterion Z₂, min/max criterion Z₃, and minimum metabolic cost criteria (Z₄, p = 1 and p = 2) during cycle of walking. EMG was obtained from 10 subjects walking on a treadmill with a speed of 1.82 m·s⁻¹, and muscle forces and activation were calculated from kinematics and ground reaction forces of one typical subject during over ground walking at a similar speed (10). The EMG linear envelopes were normalized to the EMG peak in the cycle and shifted in time by 40 ms to account for the delay between the EMG and the joint moments. The Pearson correlation coefficients (r) calculated between the EMG and predicted force/activation patterns are typically between 0.7 and 0.9 (exception: the RF muscle). The best performance is demonstrated by the minimum fatigue criterion Z₂; the linear version of the metabolic cost criterion (Z₄, p = 1) has typically the worst performance. [Adapted from Prilutsky, B.I., “Coordination of two- and one-joint muscles: functional consequences and implications for motor control,” Motor Control 4:18, 2000, and from Prilutsky, B.I., “Muscle coordination: the discussion continues,” Motor Control, 4:103, 2000. Copyright © 2000 Human Kinetics Publishers, Inc. Used with permission.]

Figure 3

Recorded EMG linear envelopes (EMG) and muscle forces and activation predicted by optimizing minimum fatigue criterion Z₂, min·max criterion Z₃, and minimum metabolic cost criteria (Z₄, p = 1 and p = 2) during cycle of pedaling at a cadence of 60 rpm and a power of 200 W. The EMG was recorded, and the muscle forces and activation were calculated from kinematics and pedal reaction forces of one typical subject (11). The EMG linear envelopes were normalized to the peak EMG values recorded in maximum isometric contractions. The EMG envelopes were shifted in time to account for the delay between the EMG and the joint moments. The time shift was found for each muscle by cross-correlating the EMG envelopes and the joint moments (11). The Pearson correlation coefficients (r) calculated between the EMG and predicted force/activation patterns are typically between 0.6 and 0.9. The best performance is demonstrated by the minimum fatigue criterion Z₂; the linear version of the metabolic cost criterion (Z₄, p = 1) has typically the worst performance.

Figure 4

Measured forces (solid lines) and predicted forces (using criterion Z₁; dotted lines) of the cat soleus (SO), plantaris (PL), and gastrocnemius (GA) muscles during seven consecutive step cycles of trotting of the cat at a speed of 1.5 m·s⁻¹. Because the percentage of slow-twitch fibers engaged in the force development during this trial was unknown, different physiologically reasonable values were examined (see Table 2 in (12)). The percentage of slow-twitch fibers in SO, GA, and PL used to calculate forces presented in this figure were 100%, 20%, and 35%, respectively. [Reproduced from Prilutsky, B.I., W. Herzog, and T.L. Allinger. Forces of individual cat ankle extensor muscles during locomotion predicted using static optimization. J. Biomech. 30:1029, 1997. Copyright © 1997 Elsevier Science Ltd. Used with permission.]