Because of the complex structure of muscle, the transmission of force from a single fiber to the action of the whole muscle is poorly understood. The force produced by the activation of two motor units, for example, can be more (superadditive) or less (subadditive) than the force exerted by the sum of the two motor units when they are activated alone. There are many reasons why the force of motor units may not sum linearly. This review examines the theoretical expectations and the experimental literature on the summation of motor unit forces.
MUSCLE STRUCTURE AND SUMMATION OF FORCE
The arrangement of muscle fibers has a profound effect on the combined mechanical output. For example, consider the effects of in-series and in-parallel arrangements of two identical, but independent, muscle fibers. When the two fibers are connected in series, the total force will be the same as a single fiber (Fig. 1); however, the maximal velocity of shortening (Vmax) will be twice as large as a single fiber, and the length–tension curve will be twice as broad (expansion along the length axis). When the two fibers are connected in parallel, the force will be twice as large (Fig. 1), but Vmax and the width of the length–tension curve will remain the same. This simple example suggests that the force from separate motor units might sum linearly in a muscle with parallel fibers.
Force Transmission Within a Muscle
The actual structure of even the simplest muscle is more complex than the serial or parallel arrangements described above. As a consequence, there are many pathways for force transmission (4). For example, although specialized adaptations at the ends of the muscle fiber allow force transmission to the microtendons, not all force is transferred along this path. Experiments on isolated frog muscle show that when the attachment at the end of a fiber is cut, full force can still be recorded from the fiber, demonstrating that force can be transmitted laterally to the neighboring fiber. Huijing et al. (4) elaborated on these studies, showing that one tendon of the rat extensor digitorum longus muscle could be cut with little effect on muscle force other than a broadening of the length–tension properties.
Nonlinear Summation From Serial Fibers
Force transmission is complex in serial-fibered muscles and in muscles where some of the fibers terminate within the fascicle. Cat sartorius is composed of serial fibers with transverse bands innervated by different nerve branches (11). Thus, one part of the muscle could be activated while its serial neighbor remained passive, which would increase the series compliance and broaden the length–tension curve. In cat tibialis anterior, the fast fibers do not run the full length of the fascicle, and hence must transmit force to the longer neighboring slow fibers (10). Further complications arise with variation in muscle fiber diameter along its length. The larger diameter sections produce more force, which must be transmitted to neighboring fibers to avoid heterogeneity along the fiber. Thus, even parallel-fibered muscles may transmit some force serially to neighbor fibers, which would result in subadditive summation.
Differential Strain–Shear in Aponeurosis and Tendon
The transmission of force within a muscle is further complicated by the nonuniformity of the aponeurosis and tendon. The direction of the collagen fibrils generates an anisotropic material that is much stiffer along the longitudinal axis compared with the transverse axis. When only part of a muscle is active, the stretch will be greater in the part of the aponeurosis near the active fibers, leading to a nonuniform strain across the aponeurosis (Fig. 2). A similar effect is observed in tendon. When several parts of the muscle are active, there may be nonuniform strain at one end of the tendon. The extent to which nonuniform stress applied at one end of the tendon has evened out by the other end of the tendon is not clear. Proske and Morgan (7) tackled this problem in cat gastrocnemius, and concluded that the force was uniform by the end of the tendon. Yet the aponeurosis does not always show uniform strain. For example, in muscles with broad thin aponeuroses on the surface, such as cat medial gastrocnemius, activation of a single motor unit often produces dimpling on the surface, presumably caused by the contraction of single fibers.
Change in Muscle Fiber Length or Velocity Caused by Common Elastic Elements
When parts of a muscle (or motor units) are connected to a common elasticity, the stretch of the elastic element will contribute to nonlinear behavior. Figure 3 shows an example of a muscle connected to a linear spring with different levels of recruitment. When the whole muscle is active, the spring will stretch twice as far as when half the muscle is active. Thus, the muscle fibers will shorten to different lengths with variation in the amount of activation. Depending on the position on the length–tension curve, more or less tension will be produced during a steady contraction. Also, at the onset or offset of the contraction, when force is changing and the elasticity is stretching, different shortening velocities will be encountered, resulting in different forces. Note that when a motor unit produces 1% of whole muscle force, the stretch is proportionally smaller (Fig. 3). This concept is simple when the location and stiffness of the elastic element is known. Because the location of the fibers in a motor unit and the material properties of the cross-links are unknown, however, the stiffness of the common elasticity may differ from the tendon.
Change in Stiffness of Series Elasticity
Isolated tendon is known to have exponential length–tension properties. The tendon is quite compliant at low forces, but becomes stiffer as force increases. Thus, a motor unit recruited in an active muscle, where the tendon is already stretched, might experience a stiffer tendon and thus exhibit different contractile properties, such as steeper rise and relaxation times. Alternately, the distributed strain in the aponeurosis and tendon may well make this relation more linear than predicted from isolated tendon.
Change in the Angle of Pennation
Few muscles have fibers parallel to a line between the origin and insertion of the muscle; rather, they are in pennate or bipennate, or comprise more complicated structures. The angle of pennation increases during a contraction, which may decrease the force. Newly recruited motor units, therefore, may contribute less force because of the disadvantageous position.
Passive force is generally elastic in nature, but exhibits many complex behaviors, including hysteresis, a nonlinear stress–strain relation, and creep. The structures that contribute to passive force include titin within the sarcomeres, and the endomysium, epimysium, and perimysium outside the fibers (4). These elements are distributed, and not necessarily parallel to, the contractile elements. Although passive tension is subtracted from total tension to estimate active tension in many experiments, this is not totally correct, because localized strain near active fibers will change the length, and hence the force exerted by, the passive elements.
In addition to elastic-type behavior, passive muscle shows a complex thixotropic, or sticky friction, behavior (5,6,10). The muscle resists small changes in length, but once the initial friction is overcome, the fibers stretch or compress smoothly. This behavior may result from weakly bound crossbridges. Stretching the muscle seems to break the crossbridges, and it takes some time for them to reform. Friction between neighboring fibers may also contribute. Because muscle fibers are bound together, with every fiber connected to some elastic tissue, active fibers shorten even during an isometric (fixed end) contraction, and drag the passive fibers along. When the passive fibers resist compression with a thixotropic behavior, or when they resist compression and the friction between fibers is thixotropic, this can result in superadditive summation of force between motor units.
Fibers Within a Motor Unit
The arrangement of fibers within a motor unit adds further complexity to force transmission. Fibers within a motor unit are distributed randomly over a fairly broad area of a muscle. The degree to which neighboring passive fibers are shortened may depend on the exact position of the active fibers. Thus, force and nonlinear summation may have a random component. In serial-fibered muscles, fibers do not generally terminate on others belonging to the same motor unit. Thus, when two or more motor units are active, some fibers are likely to be in series, which will result in a less than linear summation of force.
Summation of Parts of Whole Muscle
Few studies have examined the summation of force in whole muscle. Early studies took subadditive summation of twitches as evidence of polyneuronal innervation of the muscle fibers (6). Brown and Mathews (1) correctly argued that the stretch of the tendon was responsible for the nonlinear effects.
Sandercock (9) recently reexamined the issue of force summation between parts of the cat soleus muscle. The study was undertaken to determine the extent to which a simple lumped parameter model (Fig. 4) can account for the nonlinear summation of force. Such a parameter is required to develop a whole muscle model based on motor unit contractile properties and firing rates. No assumptions were made about the location of the common elasticity. The parameter was estimated by measuring the interaction between two separate parts of the muscle. Nonlinear summation errors were generally small, but greatest with changes in force caused either by the onset or offset of the stimulus, or by changes in muscle velocity (Figs. 5 and 6). Under all conditions, the errors were less than 6% of maximal tetanic tension. When force was steady, the error was superadditive at long lengths and subadditive at short muscle lengths, consistent with the position on the length–tension curve and stretch of the common elasticity. The magnitude of the common elasticity was estimated in three ways: (1) the shift in the length–tension relation; (2) by using the puller to mimic the extra stretch of the elasticity when both parts of the muscle were active (Fig. 6); and (3) as an algebraic calculation based on whole muscle stiffness measurements when one or both parts of the muscle were active. The lumped parameter model was shown to provide a good fit to the measurements, and could account for 70% of the measured nonlinear error. When smaller portions of the muscle were activated, the errors were smaller and consistent with the same size of the common elasticity. Thus, the common elasticity measured by the interaction of the parts is more linear than expected from the isolated tendon. The good fit of this simple model suggests it can be used to model motor unit recruitment, in which motor units are assumed to be parallel and independent force generators, connected to a common elasticity.
Similar experiments were repeated in the cat tibialis anterior muscle, which has fibers that terminate within fascicles and a long tendon (10). Nonlinear errors were small in this muscle as well, less than 8% of maximal tetanic tension for the muscle under all circumstances. As with cat soleus, the errors were greatest during dynamic conditions in which force was rapidly changing, such as the onset or offset of stimulation or rapid changes in length. Surprisingly, the common tendon model did not account for much of the nonlinearity, perhaps because of the high stiffness of the tendon (10). Subadditive summation was observed during the force plateau, which is consistent with the serial transmission of some force.
Rack and Westbury (8) used distributed stimulation to study the mechanical properties of cat soleus at low stimulation rates. They divided the ventral roots into five bundles, and asynchronously stimulated each bundle so that total muscle force was quite smooth compared with the unfused tetanus of each bundle. They noted that the average muscle force from the asynchronous stimulation was greater than the sum of each bundle stimulated separately (superadditive). They speculated that the asynchronous stimulation reduced internal shortening of the fibers, preventing the breaking and reforming of the crossbridges, thus producing more tension. Sandercock repeated Rack and Westbury’s experiments using the puller to mimic stretch of the common elastic, and demonstrated that minimizing movement could account for the extra force.
Summarizing the results for whole muscle, errors caused by nonlinear summation were small, and could be explained by the common elasticity in the cat soleus, which has a relatively simple architecture. The errors caused by nonlinear summation were also small in tibialis anterior, which comprises fibers that terminate within fascicles, but this effect was consistent with serial transmission of the force. Further study in needed in other muscles with more complicated architectures.
Measuring Force From Single Motor Units in Passive Muscle
The contractile properties of motor units are measured even when the number of active fibers in the motor unit is approximately 0.1% of the number of inactive muscle fibers in the muscle. The experimental protocols generally call for stretching a muscle to a length slightly longer than slack length. The measured contractile properties, such as time to peak twitch tension and force–frequency relations, show clear differences between fast and slow motor units that parallel histochemical measurements, at least in the muscles of experimental animals. However, uncertainty remains concerning the accuracy of the measurements. It is difficult to measure the twitch tension of the smallest motor units. A further problem is the shift in passive tension after measurement of motor unit tension; that is, after a motor unit contracts and relaxes, the passive tension often changes. Both problems might be explained by the thixotropic behavior of the passive fibers. Because small motor units cannot compress neighboring fibers, force is not detected by the force transducer. The shifting baseline may be a result of the passive fibers’ yielding to establish a new length, and then not moving once the motor unit has relaxed.
The problem of measuring motor unit force becomes more substantial during movement. Successful measurement of motor unit force during dynamic conditions is shown in Figure 7. The movement and stimulus pattern simulate a slow walk for the cat soleus muscle. The left column shows results from stimulating part of the muscle, whereas the right column shows results for a single motor unit. Note that during movement, the passive force is much greater than the active force produced by the motor unit. The similarity of the active force for the whole muscle and the motor suggests that the measurement is an accurate representation of the contractile properties of the motor unit. In this example, the motor unit generated a tetanic force of about 1% of the whole muscle force.
Motor unit force, however, often does not duplicate the change in force for the whole muscle. For example, smaller motor units do not produce forces that parallel the whole muscle force, and a stiffer tendon cannot account for the difference. Furthermore, there is a substantial difference between the two forces when the muscle approaches its slack length (Fig. 8). Similarly, errors are encountered in measuring motor unit force during a slow ramp stretch and releases (Fig. 9). The changes in motor unit force during the ramps are opposite to those measured for a whole muscle or single fiber, probably because of the thixotropic behavior of the passive fibers. For example, when a muscle is stretched beyond its slack length, the passive elements in the muscle compress the fibers, and they are prevented from shortening because the end of the muscle is held. A slow release enables compression of the passive fibers to overcome the friction (probably weakly bound crossbridges) more quickly, which allow the fibers of the motor unit to shorten and produce more tension. The cause of the decrease in force measured during a slow stretch, however, is difficult to explain. Stretching a muscle can force the passive fibers to lengthen, possibly overriding the compressive force from the motor unit. Nonetheless, this behavior was not observed during faster ramps over larger distances, and stretching and shortening the muscle beforehand minimized the problem, but did not eliminate it.
Linear Summation Between Motor Units
Recent studies have demonstrated the existence of nonlinear summation of force from individual motor units during isometric contractions (2,3,6,12,13). Clamann and Schelhorn found both superadditive and subadditive errors, but superadditive errors were most common with a mean change of force of 12% in medial gastrocnemius and 5% in soleus. Activation of a second motor unit often increased the force from the first, even after stimulation was stopped in the second unit (2). Powers and Binder found an average superadditive error of 20% in the tibialis posterior muscle. They showed the nonlinearities diminished when the muscle was moved before the units were stimulated (6). They also tried to shorten the tendon and found it had no effect on the magnitude of the nonlinearities.
The underlying cause of the nonlinearity in the summation of motor unit force is not fully understood, but probably relates more to the large mass of passive tissue in relation to the small number of active fibers, rather than the stretch of the common elastic elements. Certainly, the type of nonlinearities observed between motor units shows little resemblance to that observed between the two large pieces of cat soleus (9). Superlinear summation between motor units can be explained by compression of the passive fibers. When a motor unit contracts and produces a force that can be measured by the force transducer, its fibers must shorten slightly. Because the active fibers are linked to passive neighbors through the endomysium, and possibly by friction as well, the active fibers will compress the neighboring fibers. Any resistance to this compression will result in a decrease in force measured (see earlier section of thixotropic properties). When enough force is applied, the fiber will yield. This mechanism would not play a major role when large portions of the muscle are active. Sublinear summation between motor units is more difficult to explain. Serial transmission of force is possible, but these muscles are not noted for serial fibers. Stretch of the common elasticity—such that when both units are active the muscle is on an unfavorable position on the length–tension curve—is possible, but the motor units are not strong enough to stretch the whole tendon (Fig. 2). This could happen only when the fibers from the active units happened to be near each other and there was substantial localized strain of the aponeurosis. A decreased angle of pennation would also produce sublinear summation, but this seems unlikely, given the small force produced by the motor units.
Sheard (12) studied nonlinear summation in serial- and parallel-fibered muscles during isometric contractions, and found superadditive errors of 20 and 9%, respectively. Greater superadditive error is predicted for serial-fibered muscles. Troiani et al. (13) measured nonlinear summation between motor units in cat peroneus longus muscle, and found systematic differences between motor unit types. In general, they found greater superadditive summation between type S and FR units, but subadditive summation between FF units. They noted that type S and FR motor units produce maximal force at a shorter lengths than type FF motor units, and thus they attributed the observed differences to steady-state changes in fiber length during single and multiple motor unit activation. Their results highlight the potentially complicated interactions between motor units, particularly for different locations of resting length on the length–tension relation.
Motor Unit in Partially Active Muscle
Few studies have examined the summation of force for multiple motor units or for a motor unit and a larger piece of muscle. These are technically difficult experiments, in part, because they challenge the limits of the force transducer, which must be sensitive enough to record motor unit force, yet not saturate when recording larger forces.
Emonet-Denand et al. (3) examined the nonlinear error between single motor units and larger numbers of motor units. They found that the magnitude of the error decreased as the size of the piece of muscle increased. This result is consistent with nonlinear summation between motor units caused by the effect of passive tissue.
Perreault et al. (5) examined the linear summation of up to 10 single motor units and larger pieces of muscle in the cat soleus. These protocols allowed motor unit force summation to be examined from approximately 0 to 25% of tetanic force. Nonlinear summation was assessed by comparing the actual forces to the algebraic sum of individual units and bundles stimulated in isolation. As in other experiments, nonlinear summation in the soleus was modest, but a clear transition from predominately superadditive to predominantly subadditive summation occurred in the range of 6–8% of tetanic force. The largest nonlinearities were transient, and appeared at the onset of recruitment and derecruitment of groups of motor units. These results are consistent with the explanation that summation errors between small motor units in parallel-fibered muscles are primarily superadditive because of compression of the passive fibers, whereas summation errors in larger pieces of muscle result from stretching of the common elastic elements.
There are numerous reasons why force from motor units within a muscle might not sum linearly. Most motor unit studies have observed substantial nonlinearity—some subadditive, but mainly superadditive. Because there are few active fibers compared with the number of passive fibers, the force from the motor unit is likely to be distorted by compression of the neighboring passive fibers or by friction with the neighboring fibers. These errors can be minimized by: 1) moving the muscle before making the measurements, and 2) keeping the muscle at lengths longer than slack length. This type of summation error seems unimportant when larger parts of the muscle are activated. Therefore, most studies of nonlinear summation of force in motor units are probably irrelevant to understanding muscle function. As larger parts of the muscle are activated, stretch of the common elasticity contributes to nonlinear summation of force. Because of nonuniform strain in the aponeurosis and tendon, common elasticity may not directly correspond to the stiffness of the aponeurosis and tendon. Further study is needed to measure localized strain within a muscle. Nonlinear summation may be substantial in serial-fibered muscles, but the contractile properties of such muscles have not been studied widely. There are other potential sources of summation error, but experimental data are lacking, and the theoretical issues remain highly speculative because of the absence of quantitative models of the force transmission within muscles.
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Keywords:©2005 The American College of Sports Medicine
motor unit; summation; model; architecture; nonlinear