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).
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