INTRODUCTION

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CONTROL OF A MUSCLE DEPENDS in part on the recruitment and derecruitment of motor units. Because of the complex connective tissue matrix within a muscle (14, 20, 26, 33, 35), the effective mechanical properties of newly recruited motor units may depend on how much of the muscle is already active and to what degree these active fibers strain neighboring passive fibers and associated connective tissue and tendon. Little is known about the mechanical interaction of muscle fibers, and a quantitative measure of this interaction is essential for understanding how whole muscle properties arise from its constituent motor units.

Fibers within a muscle may mechanically interact in several ways. Even with the simplest possible model of muscle, that is, independent parallel fibers attached to an external tendon, activation of one fiber will stretch the tendon and affect the length and velocity of neighboring fibers (see Fig. 1A). Recent research has shown that the connective tissue structure of muscle is far more complex (Fig. 1B) (26, 33, 35). Proteins within the muscle fibers are capable of transmitting lateral forces between myofibrils and to the extracellular connective tissue matrix pervasive throughout the muscle. The links between neighboring fibers may prevent sarcomere length heterogeneity observed in isolated fibers, thus preventing damage and altering muscle force (8, 21). In some muscles, fibers do not run the length of the muscle but rather rely on this extracellular matrix and other muscle fibers to transmit force (18, 24). Thus neighboring fibers are not mechanically independent. Additional sources of interaction between muscle fibers occur in the aponeurosis and tendon, where cross-links likely exist between tendon fibrils (33). Furthermore, in pennate muscle, when part of the muscle contracts, the angle of pennation of the remaining muscle fibers is likely to be altered (10, 36). It is not presently known which of these factors are functionally important.

View larger version (20K): [in this window] [in a new window]   Fig. 1.   Muscle and tendon models. A: simplified model of the common elasticity. Part A and part B refer to the muscle fibers in groups A and B, respectively. The elasticity of part A is attributed to the cross bridges within the muscle fibers, as well as elastic links within the fibers and between fibers and tendon, that act independently from part B. The common elasticity, K, refers to any elastic component that is stretched by the fibers in both parts A and B. B: detailed model with elastic links between fibers and tendons.

One measure of the mechanical interaction between parts of a muscle, and the measure that will be used in this study, is the change of force produced by one part when another part of the muscle is active. This is commonly referred to as nonlinear summation. In this paper, nonlinear summation of force (F_nl) will be defined as

(1)

F_AB(t) is the force measured when both parts of the muscle are activated together. F_A(t) and F_B(t) are the force waveforms when part A and part B, respectively, are stimulated alone. Although the measure of F_nl is only an indirect measure of the strain throughout a muscle, it has the important advantage in that force is the key variable of interest in a systems' approach to motor control studies. Thus F_nl is a good functional representation of the total mechanical effects between parts of a muscle.

Nonlinear summation has been previously studied between parts of whole muscle during isometric (fixed end) contractions (4, 15) and between individual motor units during both isometric and slow ramp movements (7, 9, 27). In whole muscle studies, forces summed linearly during static isometric contractions. Linearity is surprising, considering the many ways fibers within a muscle can mechanically interact. This study expands on those studies by examining nonlinear summation during dynamic movements. In contrast to whole muscle, motor unit summation has been shown to be quite nonlinear (7, 9, 27). However, single motor units represent a small portion of a muscle; thus sources of nonlinearity, such as compression of passive muscle, may be insignificant when larger parts of the muscle are active. For this reason, it seemed important to study F_nl using larger active portions of the muscle.

The purpose of this study was to measure the nonlinear interaction between two parts of cat soleus during static and dynamic conditions. The soleus was experimentally divided into two large pseudo-motor units by splitting the ventral roots into two bundles. F_nl was measured during equal-duration isometric tetani to each part of the muscle, different-duration tetani to each part of the muscle, and during stretches and releases of the muscle. Finally, the lumped parameter model of Fig. 1A was fit to the data to determine to what extent the model of Fig. 1A can account for the nonlinearity. Each part of the muscle was assumed to have its own independent-series elastic element (representing the elasticity of the cross bridges plus the independent portion of the tendon), which in turn connect to a common-series elasticity. The physical location of the common elasticity is not restricted to the external tendon but rather represents any compliance stretched by both parts of the muscle. The model was shown to provide a good enough fit so more complex models may not be needed for most applications.

	METHODS

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The data were obtained from six cats (male and female). All surgical and experimental procedures conformed with the policies of Northwestern University and the National Institutes of Health. The surgical procedures are only briefly summarized here because a detailed account was recently presented (30). The cats were anesthetized with isoflurane during the surgical procedures and switched to pentobarbital sodium for data collection. The left hind legs were partially denervated and mounted in a rigid frame. The nerve and blood supplies to the soleus were preserved. The calcaneus was cut and attached to a servomechanism (custom device with a compliance of 0.01 mm/N), allowing the soleus to be moved by computer while simultaneously measuring muscle force. Before the tendon was cut, the foot was manually dorsiflexed and a thread was tied to the tendons of both the soleus and medial gastrocnemius. After the soleus was attached to the puller, realignment of the threads was defined as maximum physiological length (0 mm). L_o is defined in this paper as the length at which peak isometric tetanic tension (P_o) occurs. (This corresponds to L_o in Eq. 2.) The ventral roots were exposed via laminectomy and divided into two bundles, each part innervating roughly half of the soleus, and placed on separate hook electrodes so that each part could be stimulated independently. Stimulus isolation units were used, and the roots were stimulated using suprathreshold pulses 0.1 ms in duration and 2-10 V in amplitude. Stimulus trains were 100 Hz. The muscle force and length signals were generally sampled at 1 kHz, although for some quick stretches and releases sampling was at 10 kHz. Passive tension was always measured and subtracted from active tension.

Nonlinear summation was measured under three experimental conditions: 1) isometric contractions with equal-duration tetani to both parts of the muscle, 2) isometric contractions with unequal-duration tetani, simulating recruitment of new motor units to an already partially active muscle, and 3) quick stretches and releases of the muscle, simulating a changing external load. Experiment 1 was selected because of its simplicity. Experiments 2 and 3 were selected because they were expected to produce the largest nonlinear summation. All three experiments share a common protocol. First, the whole muscle was stimulated by activating the ventral roots for part A and part B simultaneously. Next, the ventral roots for part A and part B were stimulated individually, and muscle force was measured. The length of the muscle-tendon complex was the same in each of the three trials. Nonlinear summation error, F_nl, was defined by Eq. 1. It was essential that there was no cross-stimulation of the ventral roots. This possibility was ruled out because tetanic tension from the two parts of the muscle summed linearly to 0.5% when the muscle was stimulated near maximum physiological length, even when the stimulating voltage was increased by a factor of two. It is also critical that the muscle did not fatigue during the times when parts A, B, and AB were measured. This was not a problem in cat soleus because the muscle is resistant to fatigue and because 1-min rest intervals were allowed between trials. In every group of trials (A, B, and AB), at least one waveform was remeasured to test for fatigue. In some experiments, the measurements were made several times and the order of stimulation was varied. Waveforms were generally repeatable with errors <0.5% of maximum tetanic tension.

Some of the data were analyzed by constructing length-tension curves during whole or partial muscle stimulation. The data were fit to Otten's equation (3, 23)

(2)

where f is force (in N), x is muscle length (in mm), L_o is the peak of the length-tension curve (in mm), and W is the width of the length tension curve (in mm). This empirical equation fit the soleus length tension data quite well by optimizing only three parameters: L_o, W, and P_o. This procedure could detect small shifts in L_o.

To determine whether the common-elastic lumped parameter model of Fig. 1A provided a reasonable account of the data, the puller was used to mimic stretch of the common elasticity. The common elasticity was assumed to be a linear Hookian element (K). When both halves of the muscle are active, the common elasticity will stretch by a distance (L_AB) proportional to F_AB

(3)

When part A is stimulated alone, the muscle fibers will not shorten as much, since now the common elasticity will only be stretched by the force from part A

(4)

Thus, for the muscle fibers of part A to shorten by the same amount when part A is stimulated alone, compared with when both parts of the muscle are active, the servomechanism needs to move by the difference between L_AB(t) and L_A(t). This movement can be approximated by

(5)

The force waveform measured during activation of part B was divided by the estimated stiffness of the common elasticity. The resulting waveform is an approximation to the elastic stretch attributed to part B. It was used to drive the puller during activation of part A. Conversely, the force from part A was used to estimate the tendon stretch attributed to part A and used to drive the puller during activation of part B (Eq. 4). The summation of these two new waveforms was compared with the force from whole muscle stimulation to determine whether this could account for the nonlinearity. This procedure was repeated at both one-half and two times the initial estimate of K.

Quick stretches (0.5 mm in 5 ms) were used to measure muscle stiffness. Within this distance and time, the muscle acts as a linear spring, so the change in the force waveform is equal to the change in the length waveform multiplied by the stiffness (s) (22). A computer program calculated the s that minimized the difference between the force and scaled length waveforms for the 5-ms period after the initiation of the stretch. This procedure provided a more accurate estimate of s than that obtained using a single-point measure 5 ms after stretch initiation.

The three-element model in Fig. 1A, coupled with the experimental stiffness measurements, can be used to determine the stiffness of the common elasticity. An algebraic solution was obtained by assuming that the common elasticity was to be linear over the range of forces measured (F_A to F_AB)

(6)

where K is the stiffness of the common-elasticity and s_AB, s_A, and s_B are the experimentally measured stiffness of the muscle when parts AB, A, and B, respectively, are stimulated with the same stimulus train.

	RESULTS

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Isometric contractions with equal-duration tetani. Figure 2 shows nonlinear summation during an isometric contraction near L_o. Tetani of 100 Hz, from 200 to 600 ms, were delivered to parts A, B, and AB together. The resulting force waveforms are shown in Fig. 2, top. The mathematical subtraction of these waveforms (F_nl) is shown in Fig. 2, middle. Overall F_nl(t) is quite small. The largest F_nl (0.65 N) is ~3% of P_o (25 N) and occurs during tension development (200-350 ms) and tension relaxation (650-850 ms). Note that F_nl during the force plateau (500-600 ms) was near zero.

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Top: typical example of nonlinear summation during an isometric contraction. The heavy line AB, thinner line A, and dotted line B denote the forces from the whole muscle when ventral roots bundle A and B together, A alone, and B alone were stimulated, respectively (100 Hz, 0.2-0.6 s). Middle depicts nonlinear summation of force (Fnl) as defined by Eq. 1 in the text. Bottom shows the length of the muscle-tendon complex. In this muscle, length where peak isometric tetanic tension occurs (Lo) was 6.5 mm.

View larger version (15K): [in this window] [in a new window]   Fig. 3.   Servo movement to compensate for stretch of common elasticity during equal duration tetani. Heavy line is Fnl redrawn from Fig. 2. Light lines show puller compensation for 3 different values of K (10, 20, and 40 N/mm).

View larger version (17K): [in this window] [in a new window]   Fig. 4.   Example of nonlinear summation during isometric contractions with equal duration tetani at different muscle lengths. Same duration tetani as in Fig. 2 were used.

(7)

Isometric contractions with unequal-duration tetani. The data above show that the maximum nonlinearities occurred during tension development and relaxation. Similar measurements were made during isometric contractions, except with unequal-duration tetani to each part of the muscle. The idea behind these experiments is that, if one part of a muscle contracts isometrically, stimulation of the second part will further stretch the common elasticity, transiently shortening the already active fibers. The force-velocity properties of muscle suggest this should decrease force from the active part. In contrast, during relaxation, stretch of the active fibers will lead to a positive velocity and increased force. Part A was stimulated at 100 Hz from 0.05 to 1.0 s, and part B was stimulated at 100 Hz from 0.20 to 0.50 s. Parts A and B were again stimulated together and then by themselves. Figure 5 shows a typical result. F_nl(t) is quite small, with the peak <5% of P_o. The peak F_nl occurred during the rise and fall of tension in part B. Note that, unlike the case with equal-duration tetani, here F_nl is positive during relaxation. The average peak F_nl(t) in six cats was 3.6 ± 1.8% of P_o.

View larger version (13K): [in this window] [in a new window]   Fig. 5.   Typical example of nonlinear summation during isometric contractions with different duration tetani. Part A was stimulated at 100 Hz from 0.05 to 1.0 s, and part B was stimulated at 100-Hz train from 0.2 to 0.5 s.

View larger version (16K): [in this window] [in a new window]   Fig. 6.   Demonstration of servo movement to compensate for stretch of common elasticity during different duration tetani. Same muscle and stimulus patterns as in Fig. 5 were used. See text for details.

View larger version (16K): [in this window] [in a new window]   Fig. 7.   Servo movement to compensate for stretch of common elasticity during different duration tetani. Same muscle as in Figs. 5 and 6 was used.

Nonlinear summation during muscle stretch or release. Nonlinear summation was also measured during imposed increases or decreases in muscle length during stimulation at 100 Hz. Initial tests showed that moderate ramps (±40 mm/s or less) did not produce F_nl larger than isometric contractions with unequal-duration tetani. However, at speeds of 100 mm/s or greater, larger values of F_nl were produced. A typical example is shown in Fig. 8 in which the muscle was shortened by a ramp of 2 mm in 20 ms. Note the largest F_nl (6% of P_o) occurred during the ramp. This is consistent with the common-elasticity model because more of the stretch will occur in the muscle fibers when only part of the muscle is active (the common elasticity is proportionally stiffer). Figure 9 shows a similar result when the muscle was stretched by 2 mm. The average peak nonlinearity in six cats, for a positive or negative 2-mm ramp, was 6.5 ± 1.2% of P_o. Attempts to use the servomechanism to compensate for stretch of the common elasticity (as above in Fig. 6) were not successful due to limitations of the servomechanism.

View larger version (14K): [in this window] [in a new window]   Fig. 8.   Nonlinear summation during muscle movement. Similar to Fig. 2 except the stimulus is 100 Hz, 0.2-0.6 s, and the muscle is released with a ramp covering 2 mm in 20 ms.

View larger version (13K): [in this window] [in a new window]   Fig. 9.   Nonlinear summation during muscle movement. Experiment was similar to that in Fig. 8, except that the muscle was stretched 2 mm in 20 ms.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Cat soleus was divided into two functional parts by splitting the ventral roots into two bundles, each producing approximately the same force. Nonlinear summation between these parts was measured under three different experimental conditions. In all experiments, nonlinear summation was small, <7% of P_o. The largest value of F_nl occurred during changes in muscle force, either due to activation or deactivation of part of the muscle or by imposed movements on the whole muscle.

There are surprisingly few studies that have examined the summation of force in muscle. This may result, on one hand, from viewing muscle fibers as parallel independent force generators, and, on the other hand, from knowing that the actual anatomy is so complex that a completely accurate anatomic model is out of reach. Early studies concerned with polyneuronal innervation of muscle fibers examined summation when the muscle was stimulated by the ventral roots (4, 15). These investigators found twitch summation less that linear. Brown and Mathews (4) correctly attributed this to stretch of the series elasticity. Rack and Westbury (29) 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 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. They speculated that the asynchronous stimulation reduced internal shortening of the fibers, preventing the breaking and reforming of the cross bridges, thus producing more tension. These studies are consistent with the data from this study.

Recent studies have demonstrated the nonlinear summation of force from individual motor units (7, 9, 11, 27). The underlying cause of the nonlinearity is not fully understood. It is likely caused by friction or elastic links between the relatively small number of active fibers in the motor units and the surrounding mass of passive fibers or other tissue. In this study, the ventral roots were divided into two bundles to form two large pseudo-motor units. Because the territory of a single motor unit can span almost the whole soleus (5), it can be assumed that the fibers for part A were randomly distributed between the fibers for part B. Thus nonlinear summation similar to that shown in motor units might have been observed in this study. However, the nature of the nonlinearities observed was much different. In the motor unit studies, activation of a second motor unit often increased the force from the first, even after stimulation stopped in the second unit (7). Powers and Binder (27) showed that the nonlinearities diminished if the muscle was moved. Stretch and relaxation effects similar to those shown in Fig. 5 were not observed. Therefore, it seems that the cause of the nonlinearities in motor units relates more to the large mass of passive tissue and not to the stretch of the common elastic elements. Thus nonlinearities measured between single motor units probably do not have relevance to this study.

The angle of pennation of the muscle fibers may contribute to nonlinear summation, but it does not appear to be a major factor in this study. In a pennate muscle, such as the soleus, as more of the muscle contracts, the tendon stretches and the angle of pennation increases. The resulting force from the muscle is believed to decrease by the cosine of the angle of pennation. Therefore, pennation effects would be expected to produce a negative F_nl even during constant force contractions. In this study, the steady-state F_nl was small (plateau in Figs. 2, 4, 5, 8, and 9) and at least partially accounted for by the shift in the length-tension curve. Therefore, pennation effects do not account for substantial nonlinearity in cat soleus.

The mechanical links between fibers within a muscle may contribute to nonlinear summation of force. Recent research has shown that the force transmission in muscle is complex and incompletely understood (14, 26, 33, 35). It is generally agreed that cross-bridge connections between actin and myosin form the basis for active force generation (16). These cross bridges are an important part of the total muscle compliance. Titin also appears to be a significant load-bearing structure. It connects the myosin filaments to the Z disk and is probably responsible for most of the passive tension within a muscle (12, 13, 19). Other cytoskeletal proteins probably play roles in maintaining lattice structure (26). Intermediate filaments, desmin and skelemin, probably cross-link the myofibrils, keeping them in register (17). Regularly spaced costameres are probably the structures that transmit force between the myofibrils and the extracellular space (25). Thus, within the sarcolemma, the actin-myosin cross bridges and titin support longitudinal force transmission and other proteins are involved in transverse transmission of force. Outside the muscle fibers, there is a well-developed endomysial connective tissue matrix capable of transmitting muscle force (33, 34). Muscle fibers taper at their ending. Although there are specialized structures where they insert to the tendon plate, because the fibers always taper, the connective tissue matrix is likely to transmit some of the force. Furthermore, recent evidence shows that many muscles have fibers that do not run the length of the muscle (18, 20, 24). These fibers by necessity transmit their force to this endomysial connective tissue matrix. Street (32) recently demonstrated this transmission in frog fibers, which have a weaker endomysial matrix than mammalian fibers. Thus force transmission in skeletal muscle is complex, far more complex than the common view of independent parallel fibers. It appears that muscle fibers are tightly linked to each other, which may affect the way force from parts of a muscle sum together. Despite the links between muscle fibers, the simple common-elasticity model of Fig. 1A fits the data reasonably well. A portion of the common elasticity measured in this study may be attributable to the links between muscle fibers.

Aponeurosis and tendon characteristics must also be considered when examining nonlinear summation. The external tendons of muscles have been studied in vitro by attaching them to pullers and studying their stress-strain characteristics. Generally, their properties have proven to be complex. The stress-strain relationship shows a toe region, where the tendon is quite compliant, followed by either an exponential or piecewise linear relationship until the stress becomes high enough that the tendon fails (36). The common-elasticity model used in this study assumed a linear K that may have contributed to the error. Studies have shown that tendons, when stretched to a length and held, exhibit creep over time, stretching with at least two different time constants (1, 36). In cycles of stretch and release, tendons show marked hysteresis, with release exhibiting less stiffness. This may explain why different values of K were needed to match the different regions of F_nl in Fig. 7. The tendon may be stiffer during stretch than during release. The problem with studying excised tendons is the difficulty attaching a cut tendon to a puller and knowing what part of the tendon is held firm and what part is allowed to stretch. In a careful study by Bennett et al. (2), attempts were made to determine whether the stress-strain relationship is constant for tendons of different mammalian species. They concluded that the difference between species was small compared with the error of measurement. They estimated compliance measurements were in error by a factor of two due to problems attaching the tendon to a measuring device. Scott and Loeb (31) measured the mechanical properties of aponeurosis and tendon in cat soleus using sonomicrometers and concluded that the mechanical properties were the same. This fits with the conclusion of Bennet et al., although there is probably at least as much problem attaching sonomicrometers to the tendon as Bennet et al. encountered in securing the cut end of a tendon. Thus uncertainty remains as to tendon and aponeurosis properties.

In this study, when the common-elasticity model was fit to the data, different values for K were derived, which were dependent on whether the movements were dynamic or static. The shift in the length-tension curve and compensation for tendon stretch during the plateau regions yielded an estimate of K of ~10 N/mm. Estimates of K during tension development and quick stretches gave a K value >20 N/mm. These methods of estimating K are advantageous because they are noninvasive and because the parameter can be determined in a way muscle is normally used. The disadvantage is that these methods are not direct measures of tendon strain; rather, they use the measured nonlinearity of summation to estimate common-elastic stretch. Small errors in measurement lead to large errors in stiffness estimates. This may explain why the common-elastic stiffness estimated via a shift in the length-tension curve was one-half that of the other two methods. Alternately, as mentioned above, tendon stiffness may be greater for dynamic stretch and thus a lower value during static length-tension measurements.

Estimates of common-elastic stiffness in this study agree closely with tendon stiffness measurements made using other techniques. Morgan (22) developed the alpha method to measure all the elasticity in a muscle other than that attributed to the cross bridges. In cat soleus, the total tendon stiffness was estimated to be 16.9 N/mm (22). This method is likely to measure some elastic elements that are not shared between the fibers and, therefore, not measured in this study. Rack and Westbury (28) used muscle spindles to estimate what they called the "entire tendon" in cat soleus. At 11 N, they estimated the tendon compliance to be 25 N/mm. This is close to the value of 24.5 N/mm obtained by mimicking tendon stretch during tension development and 22.5 N/mm estimated using quick stretches and algebraic calculations.

In this study, the common elasticity was measured during tetanic stimulation. Thus it was measured at relatively high force levels, ranging from about 50% of maximal tetanic force when half the muscle was stimulated to 100% of maximal tetanic force when both halves were active. As mentioned above, tendon properties are known to be nonlinear with greater compliance at low force levels (36). The elastic elements comprising the common elasticity may have similar properties. Thus, during natural activation at low firing rates, the common elasticity will probably be more compliant. However, the force during partial activation will also be less; thus fiber shortening, and therefore nonlinear summation, will not necessarily be greater than that measured in this study. Further work examining partial activation at subtetanic rates is necessary.

In conclusion, 1) the interaction (nonlinear summation of force) between two parts of cat soleus muscle was small. During isometric contractions the F_nl was <3% of P_o. 2) The largest nonlinearities occurred during dynamic changes in muscle force, resulting from the onset or offset of a stimulus train or movement of the muscle. Still, the F_nl was <7% of P_o. 3) Most of the nonlinear interaction can be explained by a simple lumped parameter model in which the interaction occurs through the stretch of a shared elastic elements (common elasticity). The stiffness of the common elasticity at maximum tetanic force was estimated at two times the stiffness of the whole muscle.