Biol. Cybern. 75, 409—417 (1996)

From twitch to tetanus: performance of excitation dynamics optimized for a twitch in predicting tetanic muscle forces Jan Peter van Zandwijk, Maarten F. Bobbert, Guus C. Baan, Peter A. Huijing Institute for Fundamental and Clinical Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands Received: 4 January 1996/Accepted in revised form: 30 April 1996

Abstract. In models of the excitation of muscles it is often assumed that excitation during a tetanic contraction can be obtained by the linear summation of responses to individual stimuli from which the active state of the muscle is calculated. The purpose of this study was to investigate whether such a model adequately describes the process of excitation of muscle. Parameters describing the contraction dynamics of the muscle model used were derived from physiological and morphological measurements made on the gastrocnemius medialis muscle of three adult Wistar rats. Parameters pertaining to the excitation dynamics were optimized such that the muscle model correctly predicted force histories recorded during an isometric twitch. When a relationship between intracellular calcium and active state from literature on rat muscle was used, the muscle model was capable of generating force histories at stimulation frequencies of 20, 40, 60 and 80 Hz and other muscle-tendon complex lengths which closely matched those measured experimentally — albeit forces were underestimated slightly in all cases. Differences in responses to higher stimulation frequencies between animals could be traced back to differences in twitch dynamics between the animals and adequate predictions of muscle forces were obtained for all animals. These results suggest that the linear summation of responses to individual stimuli indeed gives an adequate description of the excitation of muscle.

1 Introduction In simulation studies addressing questions in the field of multi-segment movement control, muscle models are used as actuators which drive models of the skeleton. In these studies it is obviously important that properties of the model closely resemble the properties of the real musculo-skeletal system. For muscle models which have been used in simulating the push-off phase in vertical

Correspondence to: J. P. van Zandwijk

jumping (Pandy et al. 1990; Pandy and Zajac 1991; van Soest and Bobbert 1993; Bobbert and van Zandwijk 1994), it was found that the rate of force development is too large when compared with experimental data (see e.g. fig. 2 of Pandy and Zajac 1991). This fact renewed our interest in the dynamics of force development. The process of generating muscle force involves excitation of muscle by its motoneurons as well as contraction of muscle fibres. Usually these processes are incorporated into muscle models by two separate sets of equations, referred to as excitation dynamics and contraction dynamics respectively. The too rapid development of force in our model may be due to the excitation dynamics of the muscle model being too fast, the contraction dynamics of the muscle model being too fast, or stimulation of the muscle model rising too rapidly. In our studies, first-order dynamics as described by Hatze (1977, 1981) was used to model muscle excitation. As explained in Hatze (1977, 1981) the first-order dynamics are a simplification of a more complex model of the excitation dynamics of human muscle and give a description of the averaged behaviour of this model. The more complex model assumes that excitation of muscle during a tetanic contraction can be described by the linear summation of responses to individual stimuli, from which the active state of the muscle is calculated by means of a non-linear transformation, the so-called active state—calcium relationship. Unfortunately no experimental data are provided in Hatze (1977, 1981) to support the assumption that such an extrapolation of excitation from twitches to tetani adequately describes the process of muscle excitation. As a first step in the search for an explanation of the excessively fast rise of muscle force in our model we set out to investigate whether a model of the excitation of muscle that sums responses to individual stimuli linearly can adequately describe the process of excitation of real muscle. In this study we will address this issue for an animal model since it is easily accessible to experimentation. To do so, we will first ensure that the contraction dynamics of our muscle model accurately reflect the contractile properties of the muscle under investigation

410

by deriving the values of all parameters describing the contraction dynamics from measurements done on the same animal. Next, given an active state—calcium relationship from literature on rat muscle, we will optimize parameters pertaining to the excitation dynamics such that the muscle model correctly predicts force histories recorded during an isometric twitch. Finally we will evaluate whether the optimized excitation dynamics provide an adequate description of the excitation of muscle by comparing force histories of the model with experimentally measured force histories at other stimulation frequencies and other muscle-tendon complex (MTC) lengths.

2 Methods In this section we will describe the muscle model as used in this study, the procedure adopted in the animal experiment and finally the way of obtaining parameter values in both the excitation and the contraction dynamics of the model using the results of the experimental study.

Fig. 2a–d. Relations describing the behaviour of the three elements in the muscle model. a Force-length relation of SEE. b Force-length relation of PEE. c Force-length relation of CE. d Force-velocity relation of CE. The meaning of the parameters is explained in the main text

2.1 Muscle model Simulations of muscle contractions were performed using a model of a MTC based on the classical structural model of Hill (1938). The model consists of a contractile element (CE), a series elastic element (SEE) and a parallel elastic element (PEE) and has already been described elsewhere (van Soest and Bobbert 1993; van Soest et al. 1995). Figure 1 shows schematically the arrangement of the elements with respect to each other as well as the definition of the different lengths used in this study. 2.1.1 Contraction dynamics. The contraction dynamics of the muscle is determined by the properties of CE, SEE and PEE, which are specified by the setting of a number of parameters. Figure 2 shows schematically the relations describing the behaviour of the three elements in the contraction dynamics together with the required parameters. CE. The behaviour of CE is governed by its force-length relationship, its force-velocity relationship and its active state. The CE forcelength relationship is described as a parabola and is determined by optimum CE length l (m), maximal isometric force at CE optimum #%,015 length F (N), and the dimensionless shape parameter w, which speci.!9 fies the width of the CE force-length relationship. The CE force-velocity relationship is described by the classical hyperbolic Hill equation. Its shape is determined by the dimensionless parameter a and the parameter b (Hz) [being a/F and b/l respectively, 3%where a (N) and .!9 #%,015 b (m/s) 3%are the usual parameters in the Hill equation]. In the eccentric

Fig. 1. Schematic view of the arrangement of the contractile element (CE), series elastic element (SEE) and parallel elastic element (PEE) with respect to each other. The length measures used in this study are indicated. Note that in all cases PEE length equals CE length

Fig. 3. Active state—calcium relationships used in this study. Calcium concentration is expressed as pCa, active state as relative force. Data points are reproduced from Stephenson and Williams (1982) with permission. Continuous line, relation as modelled in Hatze (1977, 1981); dashed line, relation used in the modified model in this study. Note that both active—state calcium relations depend on CE length. For clarity only the curves pertaining to CE optimum length (sarcomere length) (2.4 lm) are shown. Sarcomere lengths: open circles, 2.65 lm; filled circles, 3.14 lm; crosses, 3.60 lm. The thin lines are guides for the eye

part of the force-velocity relationship force approaches 1.5 F as .!9 eccentric velocity goes to infinity. The active state of CE is determined by the excitation dynamics, which will be described below. SEE. SEE force depends quadratically on SEE extension. Parameters are the spring constant k (N/m2) and the slack length 4%% l (m). 4%%,0 PEE. PEE force also depends quadratically on PEE extension. Parameters are the spring constant k (N/m2) and the slack length 1%% l (m). 1%%,0 2.1.2 Excitation dynamics. Excitation dynamics is modeled as described by Hatze (1977, 1981). It consists of two parts. The first part describes the calcium dynamics within the muscle and consists of two steps. First the neural excitation a(t) of the muscle fibres spreads inward along the T-tubuli leading to a depolarization b(t) of the T-tubuli. Due to this depolarization Ca2` is released from the sarcoplasmic reticulum leading to an increase in the intracellular Ca2` concentration c(t). The second part of the model of the excitation of muscle relates the intracellular Ca2` concentration c(t) to the active state q of the muscle, which is defined as the relative amount of Ca2` bound to troponin (Ebashi and Endo 1968). This part will henceforth be referred to as the active state—calcium relation. On theoretical grounds Hatze (1977, 1981) derived an expression for the active state—calcium relation for human muscle. Experimental data of Stephenson and Williams (1982) on skinned fibres of rat extensor digitorum longus (EDL) muscle at

411 35 °C revealed for that muscle a relationship between intracellular Ca2` concentration and active state somewhat different from that derived by Hatze (1977, 1981). Since it is presently unclear which active state—calcium relationship can best be used for rat gastrocnemius medialis (GM) muscle, the muscle investigated in this study, it was decided to perform simulations using both these models. Henceforth the model of the excitation dynamics based on the active state—calcium relation derived by Hatze (1977, 1981) will be referred to as the standard model, while the model of the excitation dynamics using an active state—calcium relation closely matching the experimental data on rat EDL of Stephenson and Williams (1982) will be referred to as the modified model. Figure 3 shows both active state—calcium relations as used in this study. Appendix A provides a detailed description of the excitation dynamics of both models. The MTC model is described by a set of five coupled differential equations, which are solved by numerical integration using a variableorder variable-stepsize integrator (Shampine and Gordon 1975). Input to the model is MTC length and stimulation rate of the muscle; output is, among other things, force exerted by the MTC, which may be calculated from state variables.

2.2 Animal experiments 2.2.1 Animals and experimental protocol. Measurements were performed on three adult Wistar rats. Guidelines and regulations according to Dutch law were followed to ensure animal welfare. The animals were anaesthesized with sodium pentobarbitone (80 mg/kg) injected intraperitoneally. The GM muscle was exposed and freed from surrounding structures, leaving its blood supply intact. Details concerning the surgical procedure can be found in Roszek et al. (1994). The calcaneus was cut and the Achilles tendon was looped together with part of the calcaneus around a piece of steel wire and fixed by means of a ligature. The steel wire was connected to a force transducer. To prevent drying, the muscle was covered with a layer of paraffin oil during the experiments. In all experiments muscle temperature was kept at 35 °C using a feedback control system which consisted of a heating lamp coupled to a thermosensor positioned close to the muscle. The GM muscle was stimulated supramaximally through its severed nerve by means of cuff electrodes using 3 mA square current pulses of 0.1 ms duration. Both isometric contractions and isotonic releases were performed on a general-purpose muscle ergometer (Woittiez et al. 1987). Each contraction was preceded by a single twitch to let the muscle adapt to its new length. Contraction started 350 ms after the onset of the twitch and 150 ms after each contraction a second twitch was elicited. Between subsequent contractions the muscle was allowed to recover for 3 min at slack length. During contractions, force exerted by the muscle, length of the GM MTC and stimulation synchronization pulses were recorded using an AD-converter (sample rate 2000 Hz) and stored on a personal computer. A special-purpose microcomputer was used to control the timing of the equipment. 2.2.2 Physiological measurements. Isometric contractions at different MTC lengths were elicited by stimulating the muscle at a frequency of 80 Hz for 500 ms. This stimulation frequency yielded a virtually maximal tetanic contraction. First the length at which the GM MTC exerted the largest force was determined. Henceforth this length will be referred to as optimum MTC length. Starting from optimum MTC length, the length of the GM MTC was decreased in steps of 1 mm and at each MTC length force histories were recorded. Next the GM MTC was brought to optimum length again and the same procedure was repeated for a few MTC lengths greater than optimum length. Finally, keeping the GM muscle at its MTC optimum length, it was stimulated at frequencies of 20, 40, 60 and 80 Hz for 500 ms and force histories were recorded. In the isotonic experiments the muscle was stimulated at a frequency of 80 Hz for 350 ms isometrically and was subsequently allowed to shorten against loads ranging between approximately 10% and 90% of isometric force, starting with the highest loads and subsequently reducing the value of the load. All isotonic contractions were started at a MTC length 2.0 mm greater than MTC optimum length. 2.2.3 Morphological measurements. After the experiments the animals were killed and the GM muscle was removed from the body and

preserved in a solution consisting of 4% formaldehyde, 15% absolute alcohol and 1.5 mg/l thymol. Next the muscle was prepared to allow for collection of single muscle fibres using the method described by Huijing (1985). In this method the muscle is first exposed for 4 h to a 26% nitric acid solution to soften connective tissue and is subsequently preserved in a 50% glycerol solution. Four fibres from the most distal part of the GM muscle were isolated and in each the number of sarcomeres in series was determined using a semi-automatic counting system (IBAS, Kontron Elektronik, Echting, Germany).

2.3 Determination of parameter values 2.3.1 Contraction dynamics. CE. To estimate CE optimum length, l , it was assumed that all fibres in parallel to each other contain the #%,015 same number of sarcomeres. In this case, CE optimum length is simply the average number of sarcomeres in a fibre multiplied by sarcomere optimum length. In this study a value of rat sarcomere optimum length of 2.4 lm was used, based on the work of Zuurbier et al. (1995) who measured for rat GM muscle average sarcomere force-length relations. Effects of inhomogeneties of sarcomere lengths in a fibre are implicitly accounted for by using this value of sarcomere optimum length. Maximal isometric force at CE optimum length F was obtained from the .!9 isometric recordings at MTC optimum length. The shape parameter w of the CE force-length relation was obtained from the normalized sarcomere force-length relation and was taken to be 0.47 (Zuurbier et al. 1995). Finally the parameters a and b defining the shape of the 3%3%hyperbolic force-velocity relation were obtained by fitting the Hill equation to isotonic force versus MTC velocity data. SEE. The value of k of the SEE in rat GM muscle was determined 4%% from the size of the rapid change in MTC length at the start of the isotonic releases (Wilkie 1956). Assuming that this rapid MTC length change is taken up entirely by SEE, it can be derived that DF"!k

(Dl )2#2JFk Dl (1) 4%% 4%% where DF is the magnitude of the drop in force at the start of the isotonic release, Dl the rapid MTC length change and F the force at the onset of the isotonic release. The spring constant k of SEE was 4%% obtained by fitting (1) to the experimentally measured DF versus Dl data. The slack length l of SEE was determined by noting that at 4%%,0 MTC optimum length CE length equals l . Given the value of the #%,015 spring constant k of SEE and the value of MTC optimum length 4%% l , slack length l of SEE was calculated from .5#,015 4%%,0

S

F .!9 (2) k 4%% PEE. The spring constant k of PEE was obtained from data of 1%% passive force at different MTC lengths, from which the effective spring constant k of the passive MTC was determined. It can be shown that %&& k equals %&& k k 4%% 1%% k " (3) %&& k #k #2Jk k 4%% 1%% 4%% 1%% The spring constant k of PEE can be obtained from (3) given the 1%% value of k . The slack length of PEE, l , was adjusted to match the 4%% 1%%,0 passive force at optimum MTC length. l "l !l ! 4%%,0 .5#,015 #%,015

2.3.2 Excitation dynamics. Parameters in the excitation dynamics were estimated by minimization of the least squares objective function U(h)"+ (F (t )!F (t , h))2 (4) %91 i #!- i i with respect to the vector of parameters h describing the excitation dynamics. F (t ) is the experimentally measured muscle force at time %91 i t and F (t , h) the simulated muscle force at time t . Since integrator i #!- i i step size was not constant, state variables were interpolated to the times t and simulated muscle force was calculated from these interpolated i state variables. To calculate the vector of parameters minimizing the objective function U(h) it is beneficial to impose upper and lower bounds on the parameters being optimized. This was done using the sequential unconstrained minimization technique (SUMT) (Bard 1974). In this technique, for each bound imposed on the parameters a penalty

412 function of the form f (h),a /(h !h ) (5) k k i k,"06/$ is added to the objective function (4), yielding the modified objective function U*(h)"U(h)#+ f (h) (6) k k where h is the boundary value imposed on the i th parameter and k,"06/$ k a is a weighting coefficient. In SUMT a number of consecutive optik mizations is performed in which the modified objective function U*(h) is minimized with respect to the vector of parameters h. In subsequent runs the value of the weighting coefficients a of the penalty functions is k reduced and the optimization is restarted using as initial guess the best vector of parameters found so far. The actual minimizations of the modified objective function U*(h) were performed using the LevenbergMarquardt method. This method uses both the value of the modified objective function and its derivative with respect to the parameters to find iteratively the parameters minimizing U*(h). This derivative is the sum of the derivative of the penalty functions f (h) with respect to the k parameters and the derivative of the original objective function U(h) with respect to the parameters. The derivative of the penalty functions f (h) with respect to the parameters can be calculated directly from (5). k To calculate the derivative of the original objective function U(h) with respect to the parameters, the method of sensitivity equations (Bard 1974) was used. In this method, extra differential equations, the socalled sensitivity equations (SEQs), are integrated in parallel to the original model equations and the desired derivative is calculated from these SEQs. Appendix B gives a short overview of this elegant method because we feel it deserves more attention.

3 Results 3.1 Animal experiment Figure 4 gives a typical example of the force histories obtained in both the isometric and isotonic experiments. Figure 4a shows force histories recorded during the isometric experiments at different stimulation frequencies at optimum MTC length. Note that the twitch following the tetanic contraction is potentiated, the amount of potentiation increasing with stimulation frequency. Figure 4b shows superimposed force histories obtained during the isotonic experiments for different values of the isotonic load. Note that passive force is higher and maximal tetanic force is lower than in Fig. 4a because MTC length is 2 mm above optimum MTC length in the isometric part of the contraction. Figure 5 shows for the same animal the force-length relation of the GM MTC as obtained from the isometric experiments. The results of the measurements of the spring constant k of SEE are 4%% shown in Fig. 6, which shows the rapid drop in force DF at the onset of the isotonic contraction as a function of the rapid change in MTC length Dl. Typically MTC velocities ranged between 150 and 300 mm/s during this phase. The regression line shown is the fit of (1) to the experimental data from which the value of k was de4%% rived. Finally, the force-velocity relation of CE as calculated from the isotonic release experiments is shown in Fig. 7 together with the fit of the Hill equation. The deviation of the data points in the low-velocity region from the Hill equation might be due to fatigue, since for these data points it takes more time to reach the CE length at which the velocity is determined. Omitting these points from the regression analysis affected the

Fig. 4. Typical examples of force histories obtained in isometric and isotonic experiments. Above: Superimposed force histories obtained at optimum muscle-tendon complex (MTC) length at different stimulation frequencies. The uppermost trace pertains to a stimulation frequency of 80 Hz, subsequent traces to 60 Hz, 40 Hz and 20 Hz, respectively. Note the potentiation of the twitch following the tetanic contraction, the amount of which increases with the stimulation frequency used. Below: Superimposed traces of four isotonic release contractions. In all cases stimulation frequency was 80 Hz. MTC length was 2.0 mm above MTC optimum length before the start of the isotonic part of the contraction

value of the parameters a and b only marginally. 3%3%Table 1 summarizes the parameters specifying the contraction dynamics for all animals as computed from the isometric and isotonic release experiments as well as from the morphological measurements. 3.2 Numerical experiment Using the contraction dynamics defined by the parameter values in Table 1, the excitation dynamics of the model was optimized for an isometric twitch by minimization of the objective function (4). Figure 8 shows, for the same animal as in Figs. 4—7, measured twitch forces histories as well as calculated force histories generated by the standard and the modified model after optimization of the parameters pertaining to the excitation dynamics. The values of the parameters in the excitation dynamics

413 Table 1. Parameters describing the contraction dynamics

k 4%% l 4%%,0 k 1%% l 1%%,0 l #%,015 F .!9 a 3%b 3%-

(N/m2) (m) (N/m2) (m) (m) (N) (Hz)

Animal 1

Animal 2

Animal 3

4.22]106 2.83]10~2 2.13]105 1.39]10~2 1.32]10~2 13.39 0.20 3.15

3.64]106 3.03]10~2 1.65]105 1.32]10~2 1.23]10~2 13.81 0.13 2.02

3.47]106 2.65]10~2 5.11]105 1.34]10~2 1.12]10~2 12.28 0.21 3.73

Parameters values describing the contraction dynamics were obtained from the isometric and isotonic release experiments as well as from morphological measurements. Abbreviations of parameters are the same as used in Fig. 2

Fig. 5. Typical example of a force-length relationship of the gastrocnemius medialis (GM) MTC for the same animal as in Fig. 4. Stimulation frequency was 80 Hz in all cases. Optimum MTC length is indicated by the continuous line. The dashed line is a guide for the eye

Fig. 6. Results of measurements of the spring constant k from the 4%% isotonic release experiments. Abscissa, rapid change in MTC length Dl at the onset of the isotonic release; ordinate, rapid change in force DF at the onset of the isotonic release. The continuous line is the fit of (1) to the experimental data, from which the spring constant k is derived 4%%

Fig. 7. Force-velocity relation of CE as obtained from the isotonic release experiments. All data points pertain to the same CE length. Stimulation frequency was 80 Hz in all cases. Continuous line is the fit of the hyperbolic Hill equation

Fig. 8. Force histories of both simulated and experimentally measured twitches at GM MTC optimum length. The continuous curve pertains to the experimentally measured twitch force, the dashed curve to the force generated by the standard model and the dash-dotted curve to the modified model. The simulated twitches have been obtained by optimization of the parameters pertaining to the excitation dynamics

corresponding to the simulations shown in Fig. 8 are given in Table 2. Subsequently performance of this optimized excitation dynamics in predicting force histories at other stimulation frequencies and MTC lengths was evaluated by comparing experimentally measured force histories with those generated by the model. Figure 9 shows for both the standard and the modified model predictions of muscle force at other stimulation frequencies at MTC optimum length, as well as experimentally measured muscle forces. Figure 10 shows predicted isometric forces at a fixed stimulation frequency of 80 Hz at different MTC lengths. From Figs. 9 and 10 it is apparent that the modified model is better able to predict muscle forces than the standard model, although both models underestimate tetanic forces at a stimulation frequency of 80 Hz in all cases. It is important to point out that the results presented in Figs. 8—10 do not depend on the shape and width of the pulses used for the neural excitation a(t). These only affect the value of the parameters describing the excitation dynamics as shown in Table 2, but the behaviour of the model after optimization was

414 Table 2. Parameter values of the excitation dynamics Parameter

Standard model

Modified model

h 1 h 2 h 3 h 4 h 5

7.7]104 (5.3]104—1.1]105) 6.9]106 (5.1]106—9.7]106) 1.3]104 (6.5]103—3.1]104) 1.2]106 (7.0]105—3.2]106) 4.2]1014 (1.8]1014—2.4]1015)

3.7]104 6.5]106 1.8]104 7.3]105 3.2]108

(3.2]104—3.8]104) (5.7]106—6.7]106) (1.0]104—2.4]104) (3.8]105—9.9]105) (1.7]108—4.4]108)

Values of the parameters describing the excitation dynamics for one animal for both the standard and the modified model after optimization of the excitation dynamics on the basis of an isometric twitch. The meaning of the parameters is explained in Appendix A. Note that h has a different meaning in the modified model compared 5 with the standard model. The 95% confidence limits on the parameter estimates are given in parethenses

Fig. 9. Force histories at MTC optimum length at different stimulation frequencies. In each panel, continuous lines pertain to experimentally measured forces and dashed curves to forces generated in model calculations. For both the experimental and calculated force histories the uppermost force trace corresponds to a stimulation 80 Hz and subsequent traces to frequencies of 60 Hz, 40 Hz and 20 Hz respectively. Above: Predictions made by the standard model. Below: Predictions made by the modified model. For both models the value of the parameters pertaining to the excitation dynamics is the same as found by optimization of model behaviour to an isometric twitch for the model, i.e. the data shown in Fig. 8

Fig. 10. Force histories at different MTC lengths. In each panel, continuous lines pertain to experimentally measured forces, dashed lines to forces generated in model calculations. For all curves, stimulation frequency was 80 Hz. The uppermost trace in both panels corresponds to MTC optimum length; subsequent traces are at lower MTC lengths. The difference in MTC length between subsequent traces amounts to 2 mm. Above: Predictions made by the standard model. Below: Predictions made by the modified model. For each model the value of the parameters pertaining to the excitation dynamics is the same as that found by optimization of model behaviour to approximate an isomeric twitch, i.e. the simulation results shown in Fig. 8

415

found to be the same for different pulse shapes and pulse widths of the neural excitation a(t). For the other animals similar results to those shown in Figs. 8—10 were obtained. This is illustrated in Fig. 11, which shows for another animal measured and predicted muscle forces at MTC optimum length for different stimulation frequencies obtained after optimizing the excitation dynamics for an isometric twitch. Note that for this animal both the shape of the twitch and the unfused tetanus at 40 Hz are quite different from the data shown in Fig. 9. Nevertheless, when parameters pertaining to the excitation dynamics are optimized such that the model correctly predicts force histories for an isometric

Fig. 11. Force histories at MTC optimum length at different stimulation frequencies for an animal different from the one that produced the results depicted in Figs. 8—10. In each panel continuous lines pertain to experimentally measured force histories and dashed curves to forces obtained in the model calculations. For both experimental and calculated force histories the uppermost trace corresponds to a stimulation frequency of 80 Hz. Subsequent traces correspond to a stimulation frequency of 60 Hz, 40 Hz and 20 Hz respectively. Note that for this animal both the twitch force and the 40 Hz unfused tetanus are different from the ones shown in Fig. 9. Above: Model predictions after the parameters of the excitation dynamics have been optimized to approximate an isometric twitch using the standard model. Below: Model predictions after the parameters pertaining to the excitation dynamics have been optimized to approximate an isometric twitch using the modified model

twitch, it is capable of correctly predicting forces at other stimulation frequencies for this animal also. 4 Discussion The purpose of this study was to determine whether a model which describes excitation of muscle during a tetanic contraction by the linear summation of responses to individual stimuli provides an adequate description of the process of excitation of rat GM muscle. To do so, we first determined for rat GM MTC parameter values in the contraction dynamics on the basis of physiological and morphological measurements. Second, we optimized parameters in the excitation dynamics, given an active state—calcium relation from literature on rat muscle, such that the muscle model correctly predicted force histories in an isometric twitch. Finally, we evaluated whether this optimized model is capable of predicting muscle forces at other stimulation frequencies and MTC lengths. The results shown in Figs. 9 and 10 indicate that the model of the excitation of muscle indeed gives an adequate description of the excitation of rat GM muscle, since force histories generated by the model at other stimulation frequencies and other MTC lengths correspond closely to the ones measured experimentally. However, performance of the model depends on the active state—calcium relationship used. The standard model with the active state—calcium relation derived by Hatze (1977, 1981) for human muscle predicts forces at higher stimulation frequencies less well than the modified model based on experimental data on rat EDL by Stephenson and Williams (1982). Unfortunately the modified model also underestimates steady-state forces when compared with experimentally measured ones, especially at high stimulation frequencies. The reason for this may be that not all processes involved in the excitation of muscle have been incorporated in the model. For instance, Fig. 4a shows twitch potentiation following a tetanic contraction, a phenomenon that cannot be reproduced by our model. Nevertheless, the modified model is capable of giving encouraging predictions of muscle forces when excitation dynamics is optimized for an isometric twitch, which suggests that the model used to describe the excitation of muscle is indeed adequate. It is interesting to observe that when the method of obtaining the excitation dynamics as described in the previous sections is applied to data from another animal which display a different twitch shape, the model is still capable of correctly predicting force histories at other stimulation frequencies after the excitation dynamics has been optimized for an isometric twitch. This suggests that differences in tetanic contractions between animals can be traced back to differences in twitch shapes between the animals. Unfortunately this observation also suggests that to obtain an adequate description of the behaviour of a specific muscle it is indeed important that parameter values of both the excitation and contraction dynamics are derived on the basis of measurements done on that animal.

416

The results presented in this paper provide encouragement that a similar approach to the one adopted here will be successful in evaluating whether the excitation dynamics also provides a good description of the process of excitation of human muscle. Once this has been established it will be possible to determine what causes the excessive rate of force development of models used to stimulate the push-off phase in vertical jumping.

Appendix A Details of the models of excitation dynamics Following Hatze (1977, 1981), the excitation dynamics of the contractile element of the muscle is described by two second-order differential equations: d2b db #h #h b"a(t) 1 dt 2 dt2

(A1)

d2c dc #h #h c"b(t) 3 dt 4 dt2

(A2)

Here a(t) is the neural control signal, b(t) the depolarization of the T-tubuli and c(t) the free intracellular Ca2` concentration as described in the main text. When driven by a single pulse for a(t), the system (A1)—(A2) yields an asymmetrically shaped response for the intracellular Ca2` concentration c(t), a fact which is also observed in experiments where intracellular calcium transients are recorded by means of calcium-sensitive dyes (i.e. Blinks et al. 1978). Note that the system of equations (A1)—(A2) is linear in the driving term a(t) and that the output of the system (A1)—(A2) is determined by the value of the parameters h —h and the shape of the neural control signal a(t). In 1 4 this study we used a half-sine wave with a half period of 1 ms for a single stimulation pulse. This means that when the model is stimulated at a frequency of 80 Hz, the neural control signal is given by a(t)"sin (1000n(t!t )), 0 n t " , n3N 0 80

t (t(t #0.001, 0 0 (A3)

a(t)"0 elsewhere In both models, the active state q of the contractile element is calculated from the intracellular Ca2` concentration, using the active state—calcium relationships shown in Fig. 3.

Here m "!o $Jo2!1, where o is a constant, 1,2 2 2 2 m is the length of CE relative to l and q is a constant. #%,015 0 The length dependence of the active state q is incorporated into (A4) by means of the term m[ s!1 o*(m)" (m[ /m )s!1 where s and m[ are constants. The shape of the active state—calcium relationship (A4) is determined by the value of the constants o , s, q and m[ . For details concer2 0 ning the derivation of (A4) as well as its underlying assumptions the reader is referred to Hatze (1977, 1981). Modified model In the case of the modified model, we chose to describe the sigmoid-shaped active state—calcium relationships from Stephenson and Williams (1982) by means of the equation 1 q(c)" (A5) 1#exp (A (log c!log c )) 0 Here, A and c , the Ca2` concentration at which q"0.5, 0 are constants. As shown by Stephenson and Williams (1982), the active state—calcium relationship shifts linearly along the pCa axis when fibre length is changed. This effect can be incorporated into (A5) by making c dependent on CE length, by setting 0 c (m)"b (m!b ) (A6) 0 1 2 where b and b are constants and m is the length of CE 1 2 relative to l , as before. Using (A6), (A5) can be #%,015 rewritten as 1 q(c, m)" (A7) 1#(h c/c (m))r 5 0 where r"A/ln 10. The parameter h is introduced to 5 scale up c(t) coming from (A1)—(A2) before calculating the active state q. The shape of the active state—calcium relationship of the modified model is determined by the value of the parameters b , b and r. 1 2 For both models the parameters describing the excitation dynamics are h to h . These are the parameters 1 5 which are optimized as explained in the main text. Parameter values determining the shape of the active state—calcium relationship for both models are summarized in Table A1.

Table A1. Parameter values for active state—calcium relations

Standard model According to Hatze (1977, 1981) the active state q of the contractile element is, in the case of the standard model, calculated from the free intracellular Ca2` concentration c(t) by means of 1!q 0 (m em2o* (m)hsc!m em1o* (m)hsc ) (A4) q(c, m)"1! 1 2 m !m 1 2

Standard model

Modified model

q 0 o 2 m[ s

b 1 b 2 r

0.005 1.05 2.90 1.0

!1.07]10~6 2.34 !3.04

Values given are the parameters determining the shape of the active state—calcium relations for both models

417

The many structural details incorporated in both the standard and the modified model of the excitation dynamics are important for obtaining similar behaviour of the models as observed experimentally. For example, from physiological literature it is well known that optimum MTC length shifts to higher lengths when the stimulation frequency is reduced (i.e. Roszek et al. 1994). Without the length dependence of the active state—calcium relation (by means of A6), the modified model is unable to reproduce this behaviour. When studying the behaviour of isolated skeletal muscle, the approach adopted in this study provides the advantage of including many of the physiological processes involved in muscle excitation. In other types of studies i.e. forward dynamic modeling of human movement) these structural details may be of less importance. In those cases the complexity of the model of the excitation of muscle may be reduced for the sake of reducing computational effort.

Appendix B The method of sensitivity equations The behaviour of a dynamical system is completely characterized by the vector of its state variables s and a function h which determines the time evolution of s. This function h may depend on the state variables s, the parameters h and the independent inputs x: ds "h(s, h, x) dt

(B1)

The vector y of observables (in our case the force exerted by the muscle) also depends on s, h and x: y"y(s, h, x)

(B2)

Using the chain rule to calculate the derivative of the observable y with respect to the parameters h, one obtains: dy Ly NOSV Ly Ls i" i # + i k (B3) dh Lh Ls Lh j j k/1 k j where NOSV is the number of state variables. The explicit dependence of the observable y on the parameters h can be obtained directly. The quantities ds/dh, which are called sensitivity equations (SEQs), are obtained by noting that d ds L Lh NOSV Lh Ls k " h" # + (B4) dh dt Lh Lh Ls Lh j j j j k k/1 which yields after reversal of the order of differentiation

d ds Lh NOSV Lh Ls k " # + (B5) dt dh Lh Ls Lh j j k/1 k j This equation is formally identical to (B1) since all the quantities on the right-hand side can be computed from the model equations. This means that SEQs can be obtained by integration of (B5) in parallel with the model equations (B1). The desired derivative can then be calculated directly using (B3). References Bard Y (1974) Nonlinear parameter estimation. Academic Press, New York Blinks JR, Ru¨del R, Taylor SR (1978) Calcium transients in isolated amphibian muscle fibres: detection with aequorin. J Physiol (Lond) 277:291—323 Bobbert MF, Zandwijk JP Van (1994) Dependence of human maximum jump height on moment arms of the bi-articular in gastrocnemius; a simulation study. Hum Mov Sci 13:697—716 Ebashi S, Endo M (1968) Calcium ion and muscle contraction. Progr Biophys Mol Biol 18:125—183 Hatze H (1977) A myocybernetic control model of skeletal muscle. Biol Cybern 25:103—119 Hatze H (1981) Myocybernetic control models of skeletal muscle. University of South Africa, Pretoria Hill AV (1938) The heat of shortening and the dynamic constants of muscle. Proc R Soc Lond B 126:136—195 Huijing PA (1985) Architecture of the human gastrocnemius muscle and some functional consequences. Acta Anat 123:101—107 Pandy MG, Zajac FE (1991) Optimal muscular coordination strategies for jumping. J Biomech 24:1—10 Pandy MG, Zajac FE, Sim E, Levine WS (1990) An optimal control model for maximum height human jumping. J Biomech 23: 1185—1198 Roszek B, Baan GC, Huijing PA (1994) Decreasing stimulation frequency-dependent length-force characteristics of rat muscle. J Appl Physiol 77:2115—2124 Shampine LF, Gordon MK (1975) Computer solution of ordinary differential equations: the initial value problem. WH Freeman, San Fransisco Soest AJ van, Bobbert MF (1993) The contribution of muscle properties in the control of explosive movements. Biol Cybern 69: 195—204 Soest AJ van, Huijing PA, Solomonow M (1995) The effect of tendon on muscle force in dynamic isometric contractions: a simulation study. J Biomech 28:801—807 Stephenson DG, Williams DA (1982) Effects of sarcomere length on the force-pCa relation in fast- and slow-twitch skinned muscle fibres from the rat. J Physiol (Lond) 333:637—653 Wilkie DR (1956) Measurement of the series elastic component at various times during a single muscle twitch. J Physiol (Lond) 134:527—530 Woittiez RD Brand C, de Haan A, Hollander AP, Huijing PA, Van der Tak R, Rijnsburger WH (1987) A multi-purpose muscle ergometer. J Biomech 20:215—218 Zuurbier CJ, Heslinga JW, Lee-de Groot, MBE, Van der Laarse WJ (1995) Mean sarcomere length-force relationship of rat muscle fibre bundles. J Biomech 28:83—87