Summation of Motor Unit Force in Passive and Active Muscle Thomas G

ARTICLE

Summation of Motor Unit Force in Passive and Active Muscle Thomas G. Sandercock Department of Physiology, Northwestern University Medical School, Chicago, IL

SANDERCOCK, T.G. Summation of motor unit force in passive and active muscle. Exerc. Sport Sci. Rev., Vol. 33, No. 2, pp. 76–83, 2005. Nonlinear summation of force has been observed between motor units. The complex structure of muscle suggests many reasons why this could happen. When large portions of the muscle are active, however, the nonlinearities are small, and generally explained by stretch of the common elasticity. This suggests that the type of nonlinearity observed between motor units may not be physiologically significant. Key Words: motor unit, summation, model, architecture, nonlinear

INTRODUCTION the width of the length–tension curve will remain the same. This simple example suggests that the force from separate Because of the complex structure of muscle, the transmission motor units might sum linearly in a muscle with parallel of force from a single fiber to the action of the whole muscle fibers. 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 Force Transmission Within a Muscle two motor units when they are activated alone. There are many reasons why the force of motor units may not sum The actual structure of even the simplest muscle is more linearly. This review examines the theoretical expectations complex than the serial or parallel arrangements described and the experimental literature on the summation of motor above. As a consequence, there are many pathways for force unit forces. transmission (4). For example, although specialized adapta- tions at the ends of the muscle fiber allow force transmission to the microtendons, not all force is transferred along this MUSCLE STRUCTURE AND SUMMATION OF FORCE path. Experiments on isolated frog muscle show that when the attachment at the end of a fiber is cut, full force can still The arrangement of muscle fibers has a profound effect on be recorded from the fiber, demonstrating that force can be the combined mechanical output. For example, consider the transmitted laterally to the neighboring fiber. Huijing et al. effects of in-series and in-parallel arrangements of two iden- (4) elaborated on these studies, showing that one tendon of tical, but independent, muscle fibers. When the two fibers are the rat extensor digitorum longus muscle could be cut with connected in series, the total force will be the same as a single little effect on muscle force other than a broadening of the fiber (Fig. 1); however, the maximal velocity of shortening length–tension properties. (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 Nonlinear Summation From Serial Fibers parallel, the force will be twice as large (Fig. 1), but Vmax and Force transmission is complex in serial-fibered muscles and in muscles where some of the fibers terminate within the Address for correspondence: Thomas G. Sandercock, Ph.D., Department of Physiol- fascicle. Cat sartorius is composed of serial fibers with trans- ogy, M211, Ward 5–295, Northwestern University School of Medicine, 303 E. Chicago Ave, Chicago, IL 60611 (E-mail: [email protected]). verse bands innervated by different nerve branches (11). Accepted for publication: November 9, 2004. Thus, one part of the muscle could be activated while its serial neighbor remained passive, which would increase the 0091-6331/3302/76–83 Exercise and Sport Sciences Reviews series compliance and broaden the length–tension curve. In Copyright © 2005 by the American College of Sports Medicine cat tibialis anterior, the fast fibers do not run the full length

76 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 Figure 1. Force from identical muscle fibers connected in: (A) serial and produces dimpling on the surface, presumably caused by the (B) parallel. contraction of single fibers. of the fascicle, and hence must transmit force to the longer neighboring slow fibers (10). Further complications arise Change in Muscle Fiber Length or Velocity Caused with variation in muscle fiber diameter along its length. The by Common Elastic Elements larger diameter sections produce more force, which must be transmitted to neighboring fibers to avoid heterogeneity When parts of a muscle (or motor units) are connected to along the fiber. Thus, even parallel-fibered muscles may a common elasticity, the stretch of the elastic element will transmit some force serially to neighbor fibers, which would contribute to nonlinear behavior. Figure 3 shows an example result in subadditive summation. 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. Differential Strain–Shear in Aponeurosis and Thus, the muscle fibers will shorten to different lengths with Tendon variation in the amount of activation. Depending on the position on the length–tension curve, more or less tension The transmission of force within a muscle is further com- will be produced during a steady contraction. Also, at the plicated by the nonuniformity of the aponeurosis and tendon. onset or offset of the contraction, when force is changing and The direction of the collagen fibrils generates an anisotropic the elasticity is stretching, different shortening velocities will material that is much stiffer along the longitudinal axis be encountered, resulting in different forces. Note that when compared with the transverse axis. When only part of a a motor unit produces 1% of whole muscle force, the stretch muscle is active, the stretch will be greater in the part of the is proportionally smaller (Fig. 3). This concept is simple

Figure 2. Nonuniform strain within the muscle, aponeurosis, and tendon. (A) All fibers are relaxed. (B) One fiber is active. The degree of shortening is exaggerated.

Volume 33 ⅐ Number 2 ⅐ April 2005 Summation of Motor Unit Force 77 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

Figure 3. Idealized lumped parameter model showing a muscle con- Passive force is generally elastic in nature, but exhibits nected to a linear spring representing a tendon. (A) The muscle is relaxed. many complex behaviors, including hysteresis, a nonlinear (B) The activity of a single motor unit. (C) A muscle that is half active. (D) stress–strain relation, and creep. The structures that contrib- A muscle that is fully active. Stretch of the linear spring is proportional to ute to passive force include titin within the sarcomeres, and force. Thus, a motor unit can barely stretch the spring. In effect, the series elasticity appears stiffer for the motor unit, potentially altering its contrac- the endomysium, epimysium, and perimysium outside the tile properties. 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 ten- when the location and stiffness of the elastic element is sion in many experiments, this is not totally correct, because known. Because the location of the fibers in a motor unit and localized strain near active fibers will change the length, and the material properties of the cross-links are unknown, how- hence the force exerted by, the passive elements. ever, the stiffness of the common elasticity may differ from In addition to elastic-type behavior, passive muscle shows the tendon. 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 Change in Stiffness of Series Elasticity smoothly. This behavior may result from weakly bound crossbridges. Stretching the muscle seems to break the cross- Isolated tendon is known to have exponential length– bridges, and it takes some time for them to reform. Friction tension properties. The tendon is quite compliant at low between neighboring fibers may also contribute. Because forces, but becomes stiffer as force increases. Thus, a motor muscle fibers are bound together, with every fiber connected unit recruited in an active muscle, where the tendon is to some elastic tissue, active fibers shorten even during an already stretched, might experience a stiffer tendon and thus isometric (fixed end) contraction, and drag the passive fibers exhibit different contractile properties, such as steeper rise along. When the passive fibers resist compression with a and relaxation times. Alternately, the distributed strain in thixotropic behavior, or when they resist compression and the aponeurosis and tendon may well make this relation more the friction between fibers is thixotropic, this can result in linear than predicted from isolated tendon. superadditive summation of force between motor units.

Figure 4. Models of common elasticity. (Left) Detailed model with elastic links between fibers and tendons. (Right) Simplified model of the common elasticity. (A and B) Muscle fibers in groups A and B, respectively, based on the grouping of the axons in the ventral root bundles. The simplified model (right) depicts an independent elasticity associated with each group of fibers, attributed to the crossbridges within the muscle fibers, as well as elastic links within the fibers and between fibers and tendon. The common elasticity, K, refers to any elastic component that is stretched by both the fibers in part A and part B. (Reprinted from Sandercock, T.G. Nonlinear summation of force in cat tibialis anterior: A muscle with intrafascicularly terminating fibers. J. Applied Physiol. 94:1955–1963, 2003. Copyright © 2003 American Physiological Society. Used with permission.)

78 Exercise and Sport Sciences Reviews www.acsm-essr.org Fibers Within a Motor Unit Similar experiments were repeated in the cat tibialis anterior muscle, which has fibers that terminate within fascicles and a The arrangement of fibers within a motor unit adds further long tendon (10). Nonlinear errors were small in this muscle as complexity to force transmission. Fibers within a motor unit well, less than 8% of maximal tetanic tension for the muscle are distributed randomly over a fairly broad area of a muscle. under all circumstances. As with cat soleus, the errors were The degree to which neighboring passive fibers are shortened greatest during dynamic conditions in which force was rapidly may depend on the exact position of the active fibers. Thus, changing, such as the onset or offset of stimulation or rapid force and nonlinear summation may have a random compo- changes in length. Surprisingly, the common tendon model did nent. In serial-fibered muscles, fibers do not generally termi- not account for much of the nonlinearity, perhaps because of nate on others belonging to the same motor unit. Thus, when the high stiffness of the tendon (10). Subadditive summation 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.

EXPERIMENTAL STUDIES

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 Figure 5. Example of nonlinear summation between two parts of the long lengths and subadditive at short muscle lengths, con- cat soleus muscle during an isometric contraction. The heavy line AB, sistent with the position on the length–tension curve and thinner line A, and dotted line B denote the force from the whole muscle stretch of the common elasticity. The magnitude of the when ventral root bundles A and B were activated (100 Hz, 0.2–0.6 seconds) together, bundle A alone, and bundle B alone, respectively. The common elasticity was estimated in three ways: (1) the shift middle plot depicts nonlinear summation (Fnl). The bottom plot shows the in the length–tension relation; (2) by using the puller to length of the muscle–tendon complex. Resting length (Lo) was Ϫ6.5 mm. mimic the extra stretch of the elasticity when both parts of The largest error (0.65 N) was approximately 3% of Po (25 N), and oc- the muscle were active (Fig. 6); and (3) as an algebraic curred during tension development (200–350 ms) and tension relaxation calculation based on whole muscle stiffness measurements (650–850 ms). Note that Fnl during the force plateau (500–600 ms) was near zero. These results are consistent with the common elasticity model when one or both parts of the muscle were active. The of Figure 4. The common elasticity stretched more during the rise in force lumped parameter model was shown to provide a good fit to when both parts of the muscle were active, which resulted in a higher the measurements, and could account for 70% of the mea- velocity of shortening, and hence a lower force. Fiber velocity was near sured nonlinear error. When smaller portions of the muscle zero during the force plateau, so Fnl depended primarily on altered fiber were activated, the errors were smaller and consistent with lengths and the resulting shift along the length–tension curve. Because this contraction occurred near Lo, Fnl was small during the force plateau. the same size of the common elasticity. Thus, the common Fnl during the force plateau increases at shorter or longer muscle lengths. elasticity measured by the interaction of the parts is more Fnl became negative again during the decrease in force, probably caused linear than expected from the isolated tendon. The good fit by a greater stretch velocity when both parts of the muscle relaxed to- of this simple model suggests it can be used to model motor gether. Stretch during relaxation has been shown to cause muscle force to fall more rapidly (9). (Reprinted from Sandercock, T.G. Nonlinear summa- unit recruitment, in which motor units are assumed to be tion of force in cat soleus muscle results primarily from stretch of the parallel and independent force generators, connected to a common elastic elements. J. Appl. Physiol. 89:2206–2214, 2000. Copy- common elasticity. right © 2000 American Physiological Society. Used with permission.)

Volume 33 ⅐ Number 2 ⅐ April 2005 Summation of Motor Unit Force 79 Figure 6. Example of nonlinear summation during isometric contractions with tetani of different durations. The left column is similar to Figure 5, except that the duration of the activation differs. Bundle A was stimulated at 100 Hz from 0.05 to 1.0 s, and bundle B was stimulated at 100 Hz train from 0.2 to 0.5 s. Fnl is relatively small, with the peak less than 5% of Po. The peak Fnl occurred during the rise and fall of tension because of activation of bundle B. Note that unlike the case with tetani of equal duration, Fnl was positive during relaxation. The right column demonstrates servo movement to compensate for stretch of common elasticity during different duration tetani. The puller movements based on the estimated tendon stretch (K ϭ 20 N·mmϪ1) are shown in the lower plot. Force waveforms resulting from these movements are shown in the upper plot. The error was recalculated using these two waveforms, and appears in the middle plot along with the original Fnl. Note that the error was substantially reduced with the inclusion of the common elasticity. The mean square error was reduced by 81%. (Reprinted from Sandercock, T.G. Nonlinear summation of force in cat soleus muscle results primarily from stretch of the common elastic elements. J. Appl. Physiol. 89:2206–2214, 2000. Copyright © 2000 American Physiological Society. Used with permission.) was observed during the force plateau, which is consistent with approximately 0.1% of the number of inactive muscle fibers the serial transmission of some force. in the muscle. The experimental protocols generally call for Rack and Westbury (8) used distributed stimulation to study stretching a muscle to a length slightly longer than slack the mechanical properties of cat soleus at low stimulation rates. length. The measured contractile properties, such as time to They divided the ventral roots into five bundles, and asynchro- peak twitch tension and force–frequency relations, show nously stimulated each bundle so that total muscle force was clear differences between fast and slow motor units that quite smooth compared with the unfused tetanus of each bun- parallel histochemical measurements, at least in the muscles dle. They noted that the average muscle force from the asyn- of experimental animals. However, uncertainty remains con- chronous stimulation was greater than the sum of each bundle cerning the accuracy of the measurements. It is difficult to stimulated separately (superadditive). They speculated that the measure the twitch tension of the smallest motor units. A asynchronous stimulation reduced internal shortening of the further problem is the shift in passive tension after measure- fibers, preventing the breaking and reforming of the cross- ment of motor unit tension; that is, after a motor unit bridges, thus producing more tension. Sandercock repeated contracts and relaxes, the passive tension often changes. Rack and Westbury’s experiments using the puller to mimic Both problems might be explained by the thixotropic behav- stretch of the common elastic, and demonstrated that minimiz- ior of the passive fibers. Because small motor units cannot ing movement could account for the extra force. compress neighboring fibers, force is not detected by the force Summarizing the results for whole muscle, errors caused by transducer. The shifting baseline may be a result of the nonlinear summation were small, and could be explained by passive fibers’ yielding to establish a new length, and then the common elasticity in the cat soleus, which has a rela- not moving once the motor unit has relaxed. tively simple architecture. The errors caused by nonlinear The problem of measuring motor unit force becomes more summation were also small in tibialis anterior, which com- substantial during movement. Successful measurement of prises fibers that terminate within fascicles, but this effect motor unit force during dynamic conditions is shown in was consistent with serial transmission of the force. Figure 7. The movement and stimulus pattern simulate a slow Further study in needed in other muscles with more com- walk for the cat soleus muscle. The left column shows results plicated architectures. from stimulating part of the muscle, whereas the right column shows results for a single motor unit. Note that during Measuring Force From Single Motor Units in movement, the passive force is much greater than the active Passive Muscle force produced by the motor unit. The similarity of the active The contractile properties of motor units are measured force for the whole muscle and the motor suggests that the even when the number of active fibers in the motor unit is measurement is an accurate representation of the contractile

80 Exercise and Sport Sciences Reviews www.acsm-essr.org Figure 7. Force from whole muscle and a motor unit in cat soleus during dynamic changes in length and activation that simulated a slow walk. The left column shows measurements made when 25% of the muscle was activated. The right column shows measurements for a single motor unit. The top panel in both columns shows passive force and total force. Note that in the case of the motor unit, passive force is more than 20 times (it saturated at 2 N) the motor unit force. The second row of panels shows the digital subtraction of passive force from total force. Note that the force from a single motor unit is similar to that for the whole muscle. The third row of panels indicates the change in muscle length. The bottom panels in both columns show the timing of the stimulus pulses. (Unpublished data from Sandercock and Heckman. See (9) for methods.) properties of the motor unit. In this example, the motor unit Linear Summation Between Motor Units generated a tetanic force of about 1% of the whole muscle Recent studies have demonstrated the existence of non- force. linear summation of force from individual motor units during Motor unit force, however, often does not duplicate the isometric contractions (2,3,6,12,13). Clamann and Schel- change in force for the whole muscle. For example, smaller horn found both superadditive and subadditive errors, but motor units do not produce forces that parallel the whole superadditive errors were most common with a mean change muscle force, and a stiffer tendon cannot account for the of force of 12% in medial gastrocnemius and 5% in soleus. difference. Furthermore, there is a substantial difference be- Activation of a second motor unit often increased the force tween the two forces when the muscle approaches its slack from the first, even after stimulation was stopped in the length (Fig. 8). Similarly, errors are encountered in measur- second unit (2). Powers and Binder found an average su- ing motor unit force during a slow ramp stretch and releases peradditive error of 20% in the tibialis posterior muscle. They (Fig. 9). The changes in motor unit force during the ramps showed the nonlinearities diminished when the muscle was are opposite to those measured for a whole muscle or single moved before the units were stimulated (6). They also tried fiber, probably because of the thixotropic behavior of the to shorten the tendon and found it had no effect on the passive fibers. For example, when a muscle is stretched be- magnitude of the nonlinearities. yond its slack length, the passive elements in the muscle The underlying cause of the nonlinearity in the summa- compress the fibers, and they are prevented from shortening tion of motor unit force is not fully understood, but probably because the end of the muscle is held. A slow release enables relates more to the large mass of passive tissue in relation to compression of the passive fibers to overcome the friction the small number of active fibers, rather than the stretch of (probably weakly bound crossbridges) more quickly, which the common elastic elements. Certainly, the type of nonlin- allow the fibers of the motor unit to shorten and produce earities observed between motor units shows little resem- more tension. The cause of the decrease in force measured blance to that observed between the two large pieces of cat during a slow stretch, however, is difficult to explain. soleus (9). Superlinear summation between motor units can Stretching a muscle can force the passive fibers to lengthen, be explained by compression of the passive fibers. When a possibly overriding the compressive force from the motor motor unit contracts and produces a force that can be mea- unit. Nonetheless, this behavior was not observed during sured by the force transducer, its fibers must shorten slightly. faster ramps over larger distances, and stretching and short- Because the active fibers are linked to passive neighbors ening the muscle beforehand minimized the problem, but did through the endomysium, and possibly by friction as well, not eliminate it. the active fibers will compress the neighboring fibers. Any

Volume 33 ⅐ Number 2 ⅐ April 2005 Summation of Motor Unit Force 81 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 Figure 8. Error in measurement of motor unit force as the soleus they attributed the observed differences to steady-state muscle neared its slack length. The motor unit was stimulated at 100 Hz from 250–900 ms. Figure panels (top to bottom): passive and total motor changes in fiber length during single and multiple motor unit unit force; active motor unit force; whole muscle force (measured at a activation. Their results highlight the potentially compli- later time in the same muscle); and muscle length. Note that when passive cated interactions between motor units, particularly for dif- force reached 0.1 N (see vertical line), active force in the motor unit began ferent locations of resting length on the length–tension to decline. Whole muscle force (same stimulus and movement) remained relation. constant throughout the ramp. (Unpublished data from Sandercock and Heckman. See (9) for methods.) Motor Unit in Partially Active Muscle Few studies have examined the summation of force for motor units. These results are consistent with the explana- multiple motor units or for a motor unit and a larger piece of tion that summation errors between small motor units in muscle. These are technically difficult experiments, in part, parallel-fibered muscles are primarily superadditive because because they challenge the limits of the force transducer, of compression of the passive fibers, whereas summation which must be sensitive enough to record motor unit force, errors in larger pieces of muscle result from stretching of the yet not saturate when recording larger forces. common elastic elements. 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 CONCLUSION as the size of the piece of muscle increased. This result is consistent with nonlinear summation between motor units There are numerous reasons why force from motor units caused by the effect of passive tissue. within a muscle might not sum linearly. Most motor unit Perreault et al. (5) examined the linear summation of up to studies have observed substantial nonlinearity—some subad- 10 single motor units and larger pieces of muscle in the cat ditive, but mainly superadditive. Because there are few active soleus. These protocols allowed motor unit force summation fibers compared with the number of passive fibers, the force to be examined from approximately 0 to 25% of tetanic force. from the motor unit is likely to be distorted by compression Nonlinear summation was assessed by comparing the actual of the neighboring passive fibers or by friction with the forces to the algebraic sum of individual units and bundles neighboring fibers. These errors can be minimized by: 1) stimulated in isolation. As in other experiments, nonlinear moving the muscle before making the measurements, and 2) summation in the soleus was modest, but a clear transition keeping the muscle at lengths longer than slack length. This from predominately superadditive to predominantly subaddi- type of summation error seems unimportant when larger parts tive summation occurred in the range of 6–8% of tetanic of the muscle are activated. Therefore, most studies of non- force. The largest nonlinearities were transient, and appeared linear summation of force in motor units are probably irrel- at the onset of recruitment and derecruitment of groups of evant to understanding muscle function. As larger parts of

82 Exercise and Sport Sciences Reviews www.acsm-essr.org Figure 9. Error in the measurement of motor unit force in cat soleus. The left column shows the response to a slow ramp release in muscle length (Ϫ1 mm·sϪ1). The upper plot shows the passive, total, and active forces. Two trials are plotted on top of each other to demonstrate the reliability of the observation. The lower plot shows muscle length. Note that the active force increased during the ramp release, whereas whole muscle force decreased. The right column shows the response to a slow ramp stretch (ϩ1 mm·sϪ1). In this instance, the force declines during the start of the ramp, whereas whole muscle increases. (Unpublished data from Sandercock and Heckman. See (9) for methods.) the muscle are activated, stretch of the common elasticity 5. Perreault, E.J., S.J. Day, M. Hulliger, C.J. Heckman, and T.G. Sander- contributes to nonlinear summation of force. Because of cock. Summation of forces from multiple motor units in the cat soleus nonuniform strain in the aponeurosis and tendon, common muscle. J. Neurophysiol. 89:738–744, 2003. 6. Powers, R.K., and M.D. Binder. Summation of motor unit tensions in elasticity may not directly correspond to the stiffness of the the tibialis posterior muscle of the cat under isometric and nonisometric aponeurosis and tendon. Further study is needed to measure conditions. J. Neurophysiol. 66:1838–1846, 1991. localized strain within a muscle. Nonlinear summation may 7. Proske, U., and D.L. Morgan. Stiffness of cat soleus muscle and be substantial in serial-fibered muscles, but the contractile tendon during activation of part of muscle. J. Neurophysiol. 52:459– properties of such muscles have not been studied widely. 468, 1984. 8. Rack, P.M.H., and D.R. Westbury. The effects of length and stimulus There are other potential sources of summation error, but rate on tension in the isometric cat soleus muscle. J. Physiol. Lond. experimental data are lacking, and the theoretical issues 204:443–460, 1969. remain highly speculative because of the absence of quanti- 9. Sandercock, T.G. Nonlinear summation of force in cat soleus muscle tative models of the force transmission within muscles. results primarily from stretch of the common elastic elements. J. Appl. Physiol. 89:2206–2214, 2000. 10. Sandercock, T.G. Nonlinear summation of force in cat tibialis anterior: References A muscle with intrafascicularly terminating fibers. J. Applied Physiol. 94:1955–1963, 2003. 1. Brown, M.C., and P.B.C. Mathews. An investigation into the possible 11. Scott, S.H., D.B. Thomson, F.J. Richmond, and G.E. Loeb. Neuromus- existence of polyneuronal innervation of individual skeletal muscle fibers in cular organization of feline anterior sartorius: II. Intramuscular length certain hind-limb muscles of the cat. J. Physiol. 151:436–457, 1960. changes and complex length-tension relationships during stimulation of 2. Clamann, H.P., and T.B. Schelhorn. Nonlinear force addition of newly individual nerve branches. J. Morphol. 213:171–183, 1992. recruited motor units in the cat hindlimb. Muscle and Nerve 11:1079– 12. Sheard, P.W., P. McHannigan, and M.J. Duxson. Single and paired 1089, 1988. motor unit performance in skeletal muscles: comparison between simple 3. Emonet-Denand, F., Y. Laporte, and U. Proske. Summation of tension in motor and series-fibred muscles from the rat and the guinea pig. Basic Appl. units of the soleus muscle of the cat. Neurosci. Lett. 116:112–117, 1990. Myol. 9:79–87, 1999. 4. Huijing, P.A. Muscular force transmission necessitates a multilevel 13. Troiani, D., G.M. Filippi, and F.A. Bassi. Nonlinear tension summation integrative approach to the analysis of function of skeletal muscle. of different combinations of motor units in the anaesthetized cat pero- Exercise Sport Sci. Rev. 31:167–175, 2003. neus longus muscle. J. Neurophysiol. 81:771–780, 1999.

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