the connective tissue structure between muscle fibers is far more complex than the text book view of independent single fibers connected to a tendon. Muscle fibers are contained within a collagenous meshwork that cross-links the fibers (24). These links are capable of transmitting the full force of a fiber to its neighbors (22). Ounjian et al. (13) showed that in cat tibialis anterior (TA) the fast fibers generally do not run the length of a muscle, so the force from these short fibers must be transferred indirectly to the tendon. The meshwork may also serve to transmit force around local damage to a fiber (14). Because the diameter of fibers vary along their length (5), the collagen network may help to distribute the force to prevent localized strain. In summary, if the force transmission between fibers is significant, then force measured from different parts of the muscle may sum nonlinearly (see Fig. 1A).

The aponeurosis and tendon may also show nonuniform strain when different portions of the muscle are active. The aponeurosis often forms a broad sheet so that activation of a single motor unit can result in localized strain. Proske and Morgan (16) showed that single-motor units can cause nonuniform strain in the tendon. The tendon and aponeurosis may also show different stress-strain properties (9, 11). Treatment of the aponeurosis and tendon as a single elastic element may lead to errors in understanding how motor units interact.

Nonlinear summation of force (F_nl) has been demonstrated between single-motor units (4, 6, 15, 21,23). However, measurement of this interaction alone may not be adequate to determine whether F_nl is significant in whole muscle (6). When one motor unit is active, the number of active fibers is small compared with the large number of inactive muscle fibers and connective tissue. Thus force from the active fibers is easily distorted by links to the passive tissue that exhibits thixotropic behavior, possibly from weakly bound cross bridges (17, 18). As more motor units become active, the passive tissue is overwhelmed, making its behavior of little relevance to whole muscle behavior.

When the interaction between large portions of the muscle is studied, it is necessary to account for stretch of the tendon and other elastic elements shared in common (2, 15, 20). This is because a compliant common elasticity by itself can produceF_nl. Contraction in part of a muscle will stretch the common elasticity, changing the length and velocity of neighboring fibers, thus altering their force.

Despite the complex connective structure of muscle, in cat soleus (Sol), its behavior is well described by a simple model: independent fibers connected to a common elasticity (20). Thus the complex model of Fig. 1A can be reduced mathematically to Fig. 1B. In this study, the Sol was divided into two large pseudo-motor units by splitting the ventral roots into two bundles, thereby allowing each portion of the muscle to be stimulated independently. No assumption was made about the exact location of the elasticity; it may reside in any combination of tendon, aponeurosis, and intramuscular collagen matrix. The extent to which one part of the muscle stretched the elasticity of the other part was measured. Results indicate approximately one half of the total elasticity should be viewed as common, i.e., stretched by both parts of the muscle. Under all conditions, F_nl was small [<5% of maximal tetanic tension (P_o)], and most could be accounted for by stretch of the common elasticity.

F_nl may be more important in cat TA and other muscles with serial fibers. In vivo, if two isolated muscle fibers are connected in series, the steady-state force from the two fibers will be the same as the force from a single fiber (when length-tension effects are ignored). In muscles with intrafascicular terminations, force transmission is not fully understood. However, if some of the force is transmitted serially, analogous to the two-fiber example above, the serial fibers will contribute less to total muscle force than the linear sum. Glycogen depletion studies have shown that intrafascicularly terminating fibers are not connected to fibers within the same motor unit. Thus, when additional motor units are activated, some of the fibers will be in series with already active units and less than linear summation may occur. Sheard et al. (21) have examined motor units in guinea pig sternomastoid and shown, on average, the opposite effect: greater than linear summation of force between motor units. However, when only two motor units are active, the random distribution of the fibers makes the probability of two active fibers in series relatively small, possibly underplayingF_nl due to fibers in series. For this reason, summation of force needs to be studied with a larger portion of the muscle active.

The purpose of this study was to 1) measureF_nl in cat TA; 2) test the hypothesis that in cat TA the degree of F_nl between two parts of the muscle can be accounted for by a simple common elasticity model; and 3) test the hypothesis that the intrafascicularly terminating fibers in cat TA will produce less than linear summation due to force transmission in series. Essentially, the study in cat Sol (20) was repeated in TA (a muscle with fibers that do not run the length of the muscle). The results showed thatF_nl in TA was quite small. However, unlike cat Sol, the nonlinearity is not accounted for by a stretch of a common elasticity. The less-than-linear summation may be explained by fibers effectively connected in series.

METHODS

The methods have recently been described in detail (20) and will be summarized here. The data were obtained from four cats (male and female). All surgical and experimental procedures conformed with the policies of Northwestern University and the National Institutes of Health. The cats were anesthetized with isoflurane during the surgical procedures and switched to pentobarbital sodium (intravenous) for data collection. The left hind legs were partially denervated and mounted in rigid frame. The nerve and blood supply to the TA was preserved. The complete TA tendon was freed. At its insertion on the medial side of the foot, a piece of the bone was removed by using a dental drill. The bone chip was attached to a servomechanism (custom device with a compliance of 0.01 mm/N) that allowed the TA to be moved by computer while muscle force was simultaneously measured. The ventral roots were exposed via laminectomy and divided into two bundles. Each part innervated roughly half of the TA. They were placed on separate hook electrodes so that each part could be stimulated independently. The muscle force and length signals were generally sampled at 1 kHz. Passive tension was always measured and subtracted from active tension. L_o is defined in this paper as the length where P_o occurs during stimulation at 100 Hz.

F_nl is defined as

where F_ab is the force when both parts are stimulated together, F_a is the force when part A is stimulated, F_b is the force when part B is stimulated, andt is time. F_nl was measured under two experimental conditions: 1) isometric contractions with equal duration tetany to both parts of the muscle and2) isometric contractions with unequal duration tetany simulating recruitment of new motor units to an already partially active muscle. These conditions were shown to produce the largest nonlinearities in cat Sol (20). Similar experiments were performed to estimate the common elasticity by using quick stretches to measure stiffness. All experiments share a common protocol. First, the whole muscle was stimulated by activating the ventral roots for parts A and B simultaneously. Then, then the ventral roots for part A and part B were stimulated individually, and muscle force measured. Because fatigue is a problem in TA, whole muscle force was measured again, allowing fatigue to be monitored. Fatigue was partially compensated for by averaging whole muscle force before and after stimulation of the individual parts. The length of the muscle-tendon complex was the same in each of the four trials.

In three experiments, F_nl was measured between large and small parts of the muscle. These measurements were performed to simulate the recruitment of a large motor unit to an already active muscle. The large part had a force of approximately half the muscle. The ventral roots were subdivided so that the small piece produced a force from 2.0 to 0.7 N.

To determine whether the common-elastic lumped parameter model of Fig.1B 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 element. When both halves of the muscle are active, the common elasticity will stretch by a distance (L_ab) proportional to F_ab

where K is the stiffness of the common elasticity. When part A is stimulated alone, the muscle fibers will not shorten as much because now the common elasticity will only be stretched by the force from part AThus, 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) andL_a(t). This movement can be approximated byThe force waveform measured during activation of part B was divided by the estimated stiffness of the common elasticity. The resulting waveform is an approximation of the elastic stretch attributed to part B. It was used to drive the puller during activation of part A, and the resulting force is referred to as F_aM. Conversely, the force from part A was used to estimate the tendon stretch and drive the puller during activation of part B. The resulting force is referred to as F_bM. These new waveforms can be used to determine the nonlinearity that would result from a linear common elastic element of magnitude of Kwhere F_nlmodel is theF_nl of this model. In a previous paper (20), F_aM and F_bM were subtracted from F_ab. That method worked well in cat Sol where, with the proper selection of K, the resulting waveform approached zero, indicating that the model fit the data. That did not happen in TA with any value for K. The waveform described by Eq.5 is easier to interpret.

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 (12). A computer program calculated the stiffness 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 stiffness than that obtained by using a single point measured 5 ms after stretch initiation. The three-element model in Fig.1B, coupled with the experimental stiffness measurements, can be used to determine K. An algebraic solution was obtained by assuming the common elasticity was linear over the range of forces measured (F_a to F_ab)

where s_AB, s_A, ands_B are the experimentally measured stiffness of the muscle when parts AB, A, and B are respectively stimulated with the same stimulus train.

RESULTS

Stimulation with unequal length tetany.

Stimulation with unequal length tetany is analogous to the recruitment of a very large motor unit in a partially active muscle. Typical results are shown in Fig. 2. First, consider the waveforms on the left side of Fig. 2. They show the experimental determination of F_nl. Thetop panel shows the force when parts A, B, and AB were stimulated together. Part A was stimulated with a 100-Hz train beginning at 0.15 s and ending at 0.8 s. Part B was stimulated with a 100-Hz train beginning at 0.3 s and ending at 0.5 s. In this example, part B is about twice as large as part A. The middle panel shows F_nl as calculated by Eq. 1. Note that F_nl is fairly small. During the plateau (from 350 to 500 ms),F_nl is negative, indicating less-than-linear summation. F_nl is largest during the relaxation of part B, where it reaches +2.2 N (6.4% of P_o). This shape was typical of all four experiments.

The right side of Fig. 2 depicts the experimental determination ofF_nlmodel (Eq. 5). The model in Fig.1B was assumed to be correct, and the puller was used to reproduce estimated stretch of the common elasticity. A value ofK = 20 N/mm was used in Eqs. 3 and 4 to calculate the puller movements (see the bottom right panel of Fig. 2). Note that F_nlmodelis substantially different than F_nl. It shows an initial negative spike, and the plateau region is much smaller. The results using K = 20 N/mm are shown because they provide the best match between F_nlmodel andF_nl.

Figure 3 shows the results from another experiment using three different values for K. F_nlmodel was calculated by usingK = 10, 20, and 40 N/mm. The experimental protocol was the same as shown in Fig. 2 except the stimulus trains were longer. Here, part A was stimulated from 100 to 800 ms at 100 Hz, and part B was stimulated from 200 to 500 ms at 100 Hz. No values of Kproduce a F_nlmodel that accurately matchesF_nl. As K became more compliant, the initial negative spike and terminal positive spike ofF_nlmodel became larger, but the overall shape is a poor fit.

Similar results were seen in all four experiments that used the unequal length isometric tetany. F_nl was never large, its maximum amplitude ranged from 5.1 to 6.8% of P_o (Table1). All showed a negative plateau region in F_nl that was not matched byF_nlmodel. The size of the plateau ranged from about −1 to −2 N. The positive peak in F_nl, resulting from the relaxation of part B, was seen in all four experiments but varied widely in both amplitude and duration. In two experiments, F_nl was best matched byF_nlmodel using K = 20 N/mm and in two others by K = 40 N/mm.

Stimulation with equal length tetany: length-tension curves.

Next the protocol was altered so both parts of the muscle were stimulated with equal length tetany. Equal length tetany allows the construction of length-tension curves during partial and whole muscle stimulation. Figure 4 shows a typical example. Stimulation to both parts was applied from 0.4 to 0.6 s at 100 Hz while the muscle was held isometrically atL_o. F_nl is initially negative when both parts are active and then becomes positive during relaxation. F_nlmodel was measured by usingK = 20 N/mm and is plotted along withF_nl. It is clearly a poor match toF_nl. Figure 5shows F_nl at different lengths. The data are from the same muscle, and the stimulation parameters are the same as in Fig. 4. F_nl decreased at shorter isometric lengths, becoming slightly positive at −12 mm with respect toL_o.

The decrease in F_nl with length means that the length-tension functions of the parts may have a different shape than the length-tension function of the muscle as a whole. Thetop plot in Fig. 6 shows data from the same muscle presented in Figs. 4 and 5. Rather thanF_nl, the digital summation of parts A and B are shown. Note that the sum is larger at L_o but becomes smaller by −15 mm. Similar results were seen in all four muscles. The mean difference at L_o was 4.1% of P_o. In one experiment, the puller was adjusted to measure the length-tension data at longer lengths (Fig. 6, bottomplot). Here, F_nl became positive at a length of 7 mm. These data are inconsistent with a simple shift in the length-tension curve.

Stiffness measurements.

Quick stretches were used to measure stiffness when parts A, B, and both A and B were stimulated together. The data from one experiment are shown in Fig. 7. The dotted line is simply a visual reference and was drawn from the origin to the largest data point. Note that the data points from part A and B are slightly above the reference line. This indicates that their normalized stiffness (stiffness over force) is just slightly greater than that from whole muscle. When the model of Fig. 1 is applied, Eq.6 can be used to estimate K. On the basis of four cats, whole muscle stiffness was measured at 13.1 N/mm and K was measured at 41.9 N/mm (Table 1). By these calculations, about one-fourth of the whole muscle compliance can be attributed to the common elasticity.

F_nl between smaller pieces of the muscle.

In three muscles, F_nl was also measured with smaller pieces of the muscle. A typical example is shown in Fig.8. The experiment was similar to the unequal tetany protocol shown in Fig. 2. Because this experiment was very sensitive to fatigue, the complete protocol was repeated twice in each muscle. The larger piece of muscle (generating from 1/3 to 1/2 the total muscle force; see Table 2) was stimulated from 200 to 800 ms at 100 Hz. The smaller piece (generating 2–7% of the total muscle) was stimulated from 300 to 500 ms at 100 Hz, as shown in Fig. 8. The results are summarized in Table 2.F_nl is expressed as a percentage of the smaller part of the muscle (Fig. 8B). The averageF_nl for the four trials in the two different muscles was 21.8%. The whole muscle data was expressed as a percentage of whole muscle P_o. For comparison, the whole muscle data should be normalized by one-half of P_o, which gives a meanF_nl of 8.2%. Thus it appears that nonlinearity increases with small pieces of muscle.

DISCUSSION

This study examined F_nl in cat TA. The muscle was divided into two parts by splitting the ventral roots. Division into two large pieces resulted in F_nlthat was quite small (<7% of P_o).F_nl was negative except at the offset of stimulation of part of the muscle. When smaller pieces of the muscle were considered, F_nl was somewhat larger (∼20% of the force from the small piece). In an attempt to account for the source of F_nl, common elasticity was measured in two ways: 1) by mimicking its effects by using the muscle puller and measuring the predictedF_nl; and 2) measurement of the change in short-term stiffness when part or all of the muscle was active. Both measures indicated the contribution of the common elasticity was small.F_nl is, at best, only partially explained by stretch of the common elasticity. The data are consistent with the hypothesis that less than linear summation is produced in muscles with intrafascicularly terminating fibers.

Common elasticity can result in F_nl because of the length-tension and force-velocity properties of a muscle. Consider Fig. 2 where part A of the muscle is active and part B is recruited. The additional force from part B will stretch the common elasticity, placing part A at a shorter length on its length-tension curve. This could produce more or less force depending on the start length. However, because TA has such a broad length-tension curve (Fig. 6) and the common elasticity is quite stiff, this effect is small. During the development of force in part B, the stretch of the common elasticity will transiently produce a shortening velocity in part A, leading to a reduction of force on the basis of the force-velocity properties. The opposite occurs during the relaxation of part B, where part A is transiently stretched, increasing force. These effects are apparent inF_nlmodel in Fig. 2, where stretch of the common elasticity was simulated. They are not readily apparent inF_nl in Fig. 2. It is possible that these effects are present and masked by an additional nonlinearity. The initial decrease is similar, but F_nl remains negative, and the length-tension properties do not account for this. The spike during relaxation is similar. The predicted effects of common elasticity are slightly more complicated when both parts of the muscle relax at the same time. The increased velocity of stretch when both parts are active can lead to a faster relaxation of the muscle, producing a negative F_nl (3). This effect is apparent in F_nlmodel in Fig. 4. Again, the effect is not observed in the measured F_nl.

Common elasticity was measured two ways. Each method is different and selected to determine how large parts of the muscle interact. It was hoped that a simple relationship would suggest how to mathematically represent recruitment in a muscle model. If the model in Fig.1B was correct and the common elasticity was near linear, the two methods would have provided the same estimate of K. This was not the case. Unlike the direct measurement of tendon or aponeurosis stretch, the methods make no assumption about the location of the nonlinearity; rather, they measure the interaction between different parts of the muscle by using their length-tension or force-velocity properties. Each method had its own limitations. Quick stretches (Eq. 6) provide an estimate of common elasticity, provided the elasticity is linear over the range of forces studied. Thus the elasticity must be linear starting from the force produced by the smaller piece of muscle and extending to the force when both parts of the muscle are active (12, 20). An exponential stress-strain relationship over this range would lead to an underestimation of the common elasticity. Although isolated tendon is known to have a nonlinear stress-strain relationship, at high forces levels it is often considered to be linear (1, 26). The mean K was estimated to be very stiff at 42 N/mm, which by itself would produce small nonlinearities (see Fig. 3;K = 40 N/mm). The second method (Fig. 2) used the puller to mimic expected changes in the length of the common elasticity. Different values of K were used because the value of K estimated from the quick stretches was very stiff and was unable to match F_nl. More compliant values of K did not fit the data either. Substitution of a nonlinear common elasticity was not tried since a range of physiological values was already tried and found lacking. In cat Sol, this method was successful because puller movements could account for most of F_nl, providing confidence in the common elasticity model as well as the value for K. Failure to match F_nl in the TA provides little justification for the common elasticity model, which makes the value ofK determined with this method suspect.

Cat TA has a long tendon (∼4 cm). Despite the length of the tendon, the measured common elasticity was very stiff, almost twice that of cat Sol. Although surprising, this is consistent with previous results in cat TA (19, 25). Roeleveld et al. (19) studied the mechanical properties of cat TA with and without its tendon. They found that removal of the tendon made little difference in the overall contractile properties of the muscle.

Few studies have examined F_nl between large pieces of a muscle. F_nl was initially examined to investigate polyneuronal innervation of muscle fibers (8). The small degree of F_nl was attributed to the stretch of the tendon (2). Sandercock (20), using the same techniques used in this study, measured F_nl in cat Sol, finding the common elasticity explained a major part of F_nl. The shape of the F_nl waveforms in the Sol study is very similar to F_nlmodel measured in this study.

The common elasticity probably plays a minor role in the TAF_nl for several reasons. First, the tendon in the TA is stiff, so large forces are necessary before there are substantial changes in fiber lengths. A change in fiber length will have two effects. The steady-state, or constant, length changes will affect the operating point on the length-tension curve. Faster changes in length will affect force through the force-velocity properties of muscle. The common elasticity in TA is ∼40 N/mm compared with 20 N/mm in cat Sol. Second, the TA has a very flat length-tension curve (Fig.6), so steady-state changes in fiber length need to be large before they will have much effect on force. Its length-tension properties are comparable to cat Sol. Finally, it has a relatively fast maximum velocity of shortening. The TA's maximum velocity of shortening is 320 mm/s, which is almost twice as fast as cat Sol. Thus, when only the common elasticity is considered, TA should have half the steady-stateF_nl and a quarter of the dynamic effects onF_nl observed in cat Sol.

If common elasticity cannot account for F_nl, what can? In cat TA, type F muscle fibers do not run the full length of the TA. These short fibers terminate short of the tendon and must somehow transmit their force to the tendon. The endomysial connective tissue matrix outside the muscle fibers (7, 14, 24) probably serves to transmit the force (13). The intrafascicularly terminating fibers are likely to affectF_nl in two ways: 1) increased compliance and 2) the serial transmission of force.

Serial fibers within a muscle may increase the compliance. Glycogen depletion studies on muscles with serial fibers have shown that the short fibers terminate near fibers of different motor units (10,13). Thus the intrafascicularly terminating fibers probably transmit force to fibers that are not within the same motor unit. Because these fibers may be not be active, the compliance seen by the motor unit may increase significantly. When a single motor unit is activated, its fibers must shorten farther. This may alter the operating point on the length-tension curve. Increased compliance, and hence shortening, may increase the nonlinearities due to friction between or the breaking of weakly bound crossbridges of neighboring fibers (see discussion of motor unit nonlinearities below). Sheard et al. (21) used this argument to explain greater-than-linear summation in the serial-fibered guinea pig sternomastoid muscle. There was little evidence of increased compliance is this study. Increased compliance would be expected to shift the peak of the length-tension curve to longer lengths when part of the muscle was active compared with the whole muscle. This was not observed.

The second possible effect of intrafascicularly terminating fibers is the serial transmission of force. Two fibers connected in series will theoretically produce the same force whether one or both of the fibers is active. Thus, when both parts of TA muscle are active, some fibers from each part will be effectively in series, so they will not contribute additional force, resulting in less-than-linear summation. This may explain the less-than-linear F_nlobserved in whole muscle.

Fatigue was a problem with these experiments. Its significance varied between cats. The protocol required stimulating the muscle at 1-min intervals with 100-Hz trains lasting 700 ms. When trials were repeated, it was noticed that force fell anywhere from 0 to ∼2%. Its effects were minimized by calculating F_nl by using a small group of waveforms measured together with constant timing, the stimulus of part AB, A, B, and AB again. The repetition of part AB allows an assessment of fatigue. It also allows the AB waveforms to be averaged, thus mitigating the fatigue effect. Some of the experiments allowed additional controls. During stimulation with unequal tetany (Fig. 2), F_nl should be zero before part B is active. Any deviation provides an index of the fatigue of part A. Figures 2, 3, and 8 show small deviations that are less than the measured F_nl. Thus, when unequal parts of the muscle were stimulated (Fig. 8), it is unlikely thatF_nl results from the fatigue of part A. Because part B is small, its fatigue cannot contribute substantially toF_nl.

Substantial F_nl has been demonstrated between single motor units (4, 15, 21). When a single motor unit is activated, its fibers must shorten slightly before force can be measured. Even in an isometric (fixed end) contraction, shortening cannot be avoided because there is some compliance in the force transducer as well as compliance in the tendon and aponeurosis. The endomysial connective tissue matrix suggests that fibers are coupled to their neighbors. Thus, for a fiber to shorten, it must drag its neighbors along. If the neighboring fibers resist shortening because of weakly bound cross bridges or other types of friction, then some of the potential force from the motor unit is lost and is not measured with the force transducer because it is used in the compression of neighboring fibers. A similar argument can be made if the muscle fiber must slide by its neighbors. If there is some friction preventing easy movement, force will be lost in the compression of its neighbors (17). This would explain the increase in force seen when pairs of motor units were stimulated together (positiveF_nl). Emonet-Denand et al. (6) showed that, when F_nl was measured between groups of motor units, it was smaller than between single motor units. Thus this mechanism would not play a major role when large portions of the muscle are active, such as in this study. Sheard et al. (21) studied F_nl in serial- and parallel-fibered muscles and found greater-than-linear increases in force of 20 and 9%, respectively. The opposite was observed in this study. Troiani et al. (23) measuredF_nl between motor units in cat peroneus longus muscle. They found systematic differences between different motor unit types. In general, they found greater-than-linear summation between type S and FR units but less-than-linear summation between FF units. They noted that type S and FR motor units produce maximum force at shorter lengths than type FF motor units, 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. In this study, force from a small piece of muscle decreased when it was stimulated along with a larger piece. The reason for the greater negative F_nl with a small piece of TA is not clear. It does not appear to be stretch of the common elasticity.

In summary, the interaction (F_nl) was measured between two parts of cat TA. This is a muscle with an unusual architecture in that some fibers do not run the full length of the muscle. The measured nonlinearities were generally small, but they could not be fully explained by the common elasticity. For motor control studies, the magnitude of the error is small enough that they probably do not need to be considered. However, although small,F_nl is still of interest in understanding the structure of muscle.

FOOTNOTES

• Address for reprint requests and other correspondence: T. G. Sandercock, Dept. of Physiology, M211, Ward 5-295, Northwestern Univ. School of Medicine, 303 E. Chicago Ave., Chicago, IL 60611 (E-mail:[email protected]edu).

• The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

• First published January 10, 2003;10.1152/japplphysiol.00718.2001