Biological
Cybernetics
Biol. Cybern.44, 129-133 (1982)
9 Springer-Verlag 1982
Simulation of Post-Tetanic Potentiation
and Fatigue in Muscle Using a Visco-Elastic Model
A. Ducati 1, F. Parmiggiani 2, and M. Schieppati 3
1 Istituto di Neurochirurgia,Universit/tdi Milano, 1-20122 Milano, Italy
2 Istituto di Fisiologiadei Centri Nervosi, CNR, 1-20131 Milano,Italy
3 Istituto di FisiologiaUmana II, Universit~tdi Milano, 1-20133 Milano, Italy
Abstract. Post-tetanic potentiation (PTP) in single
motor units was simulated using a simple visco-elastic
model. Single isometric twitches and unfused tetani
were obtained using a wide range of physiological
input rates. Values of model parameters were chosen
to simulate contraction times close to those of fast and
slow muscle fibers. PTP has been attributed either to i)
an augmented plateau level of active state or ii) an
increase in time constant of active state decay. Our
results show that a prolonged decay time of active state
can account for most of the experimental data obtained in amphibian and mammalian preparations. In
particular, potentiation is more marked in unfused
tetani than in single twitches. Moreover the model
accounts for PTP even in the case of a reduction of
active state plateau due to fatigue.
Introduction
Post-tetanic potentiation (PTP) of the isometric twitch
has been the subject of several investigations but the
opinions are still divided as to its origin and mechanisms (see the reviews of Colomo and Rocchi, 1965;
Close, 1972).
Two hypotheses have been advanced: i) the twitch
evoked immediately after a conditioning tetanus
would yield a more complete activation of the contractile material, whereas a non-conditioned twitch
would induce only a partial activation (Close and Hoh,
1968). The increase in peak twitch tension (PT) without
a parallel increase in contraction time (CT) seems to
support this hypothesis (Hanson, 1974) ; ii) PTP might
be due to a decrease in the rate of decay of the active
state (AS) with a consequent increase of the period
during which series elastic elements enter into tension
with the development of force. In this second case CT
is necessarily increased, as reported by several authors
(Kernell et al., 1975 ; Burke et al., 1976; Close and Hoh,
1968).
Interpretation of the data in the literature and their
comparison present some difficulty, since they were
obtained both from single motor units (Kernell et al.,
1975 ; Burke et al., 1976) and whole muscles (Close and
Hoh, 1968; Ranatunga, 1977, 1978). In the latter case
anatomical and functional features may conceal an
initial increase of CT, since the whole muscle is
composed of mixed types of motor units with different
CT and possibly different visco-elastic properties.
Interactions among these different types do occur
whether the contraction is maximal or not.
Recently Ranatunga (1977, 1978) has reported that
mammalian muscle does not undergo PTP at a temperature of 20~
In interpreting these results he
suggested that, like those of Buller et al. (1968) and
Close and Hoh (1968), they are in keeping with the first
of the two hypotheses stated above. However, it is not
intuitive how - in fast mammalian muscles - the
activation of the contractile material (available in the
appropriate conditions) would be enhanced at low
temperature. Indeed, not only maximal tetanic tension
is reduced by about 18 % (Ranatunga, 1980), but the
rate of rise and the time to peak of the twitch tension
are markedly reduced (Ranatunga, 1977).
Note that a decrease in temperature can alter the
release of Ca 2 + from the cisternae of the sarcoplasmic
reticulum (Connolly et al., 1971), and this in turn could
explain the decrease in tetanic tension. PTP would also
be affected as the time course of the decay of AS would
increase. A similar mechanism, namely an initial "fatigue" in Ca 2 + reuptake, with a consequent prolonged
decay of AS, could also account for the potentiation of
post-tetanic twitches, as suggested some time ago by
Ritchie and Wilkie (1955) and proved true in fast frog
muscles by Connolly et al. (1971). Finally a close
scrutiny of the tension-frequency curves for unfatigued
mammalian fast motor units published by Kernell et
0340-1200/82/0044/0129/$01.00
130
-i
- KiA(t )
i
~b
.
.
.
.
.
.
.
.
.
.
.
.
.
K
e
(6)
2 Jr-c~x = B(Ki + Ke ) .
I
l--Xl~l-x2~,
.
one obtains
I
A(t)
J
Fig, 1. Schematic diagram of the muscle model utilized for the
simulations. For the meaning of the various parameters refers to the
text
When force is measured by a transducer in series with
an external spring, the force generated in the spring
will simply be"
f= -K~x
under isometric conditions (when K e ~
and
al. (1975) does suggest that the same mechanism could
be involved also in this case.
It was therefore tempting to test the idea, although
indirectly, by means of a mathematical model of
muscle (Bawa et al., 1976; Stein and O~uzt6reli, 1976)
in which the relevant parameters can be easy varied.
(7)
, Ke>>Ki)
j~+ aisof = KiA(t)/B,
(8)
where
~iso= lim c~= (K~ + Kp)/B.
Ke--+ oo
Since the time constant of the visco-elastic system is :
% = 1/ai~o = B/(K i + Kp)
Methods
by substituting this expression in expression (8) one
obtains
The Model
The linear model used is based on the classical Hill's
(1938) view of muscle properties and has been reproposed in recent years by Bawa et al. (1976) and Stein
and O~uzt6reli (1976). Although it has recently been
criticized (Hatze, 1977) the limits of its usefulness are
well within the scope of our aims.
It contains (see Fig. 1) an internal series elastic
element (with stiffness Ki), a parallel elastic component
(with stiffness Kv), a linear dashpot (of viscosity B) and
an active state element (A(t)). The latter produces the
contractile force every time a stimulus is applied. Also
included in Fig. 1 is an external elastic element (with
stiffness K~) which represents the elastic load against
which contraction takes place.
In Fig. 1, let x 1 and x 2 represent the length changes
of the parallel and series elastic elements, respectively,
during a twitch. Then, the equations of motion at
nodes a and b are:
KpX 1 -I- B2 t + A(t) = K i x 2 ,
(1)
Kix 2 -t- Kex = 0,
(2)
where x = x a + x 2 is the total displacement of the
muscle. Substituting in (1) for
X i =
X --
X 2 =
x(1 + Ke/Ki)
(3)
and rearranging
2B(K~ + Kr + x(KvK i + KpKr + KiKe) = - K~A(t).
(4)
Defining a rate constant
=
K p K i -I- KpKe + K i K e
B(K~ + Ke)
(5)
%j'+ f =rA(t),
(9)
where
r = Ki/(K i + Kp).
In the case of fully activated muscle K i ~>Kp and r = 1.
Values of the Parameters Used for the Simulations
r : it was made equal to 1 since during isometric twitch
the parallel elastic component is negligible with respect
to the series elastic component. In this situation CT is
equal to the value of AS at that instant (Richtie and
Wilkie, 1955).
%: measurements of the time constant of the
viscoelastic system for the whole muscle (Stein and
Parmiggiani, unpublished observations) have led to
values ranging from 40 to lOOms. A % = 8 0 m s was
chosen in our simulations to obtain twitches time
courses comparable with those we wanted to simulate.
A(t): a duration of 5ms for the fast muscle and
15 ms for the slow muscle were assigned to the early
phase of the plateau because a fused tetanus (without
ripple) is evoked at frequencies above 200Hz and
66 Hz in the two types of muscle (Cooper and Eccles,
1930; Bullet and Lewis, 1965; see also Brust, 1966).
Time constants of the decay of the active state, vas,
were 20ms and 50ms respectively, for normal
conditions.
With the above parameters the contraction times,
half-twitch durations and twitch-tetanus ratios were in
accordance with experimental data present in the
literature.
131
A
B
100
C
100
%
100
%
0
ms 5 0 0
%
0
ms 5 0 0
0
ms 5 0 0
Fig. 2. A Time course of active state (AS) and twitch tension in a fast muscle in response to a single shock. Values of parameters : r = 1,
zv= 80 ms, z,s = 20 ms, AS plateau = 5 ms. In this and the following graphs, tension is expressed as percentage of active state. B Active state
and twitch tension in a fast muscle for increasing values of zas. Values of other parameters as in A. Note the increase in CT and PT. C Same as
in B but with a reduction of AS plateau at 75 %, due to fatigue
Results
Single Twitch
Fast Muscles. Figure 2A shows the simulation by the
model of the twitch of a fast muscle. Contraction time
is 37 ms and PT is at 19 % of the plateau of the active
state (AS). These values are in fair accordance with the
behaviour of fast motor units (Kernell et al., 1975;
Brust, 1976). To simulate the changes in PT and twitch
time course taking place during cooling or fatigue - the
latter as a result of a long period of stimulation at
tetanic frequency (Close and Hoh, 1968) - Zas was then
increased. The simulated twitches presented in Fig. 2B
are for three different values of %s. Note the increase of
CT (50,80, and 97ms) and PT (27.8, 38.9, and 46.1%) as
Za~ is increased to 40, 80, and 120ms. Changes of the
same order of magnitude have been reported by
Kernell et al. (1975, see their Table 1) for twitches
evoked immediately after a conditioning fused tetanus,
in spite of the absence of an increase in maximal
tetanic tension or even when the latter was decreased.
The possibility that an "improvement" of the electromechanical coupling might be responsible for this
effect (Ranatunga, 1977) is quite unlikely since, according to the model response, to obtain such amount of
potentiation without changing the value of Za~ it was
necessary to increase the plateau level of AS by more
than 200%. Actually, from experiments on barnacle
muscle, only an increase of the order of 20 %, with a
maximum of 50 %, has been reported by Desmedt and
Hainault (1978), who measured Ca 2+ activity in a
potentiated twitch as indication of AS time course.
Another way to approximate, with the model, the
condition of fatigue is by reducing the AS plateau to
75 % of control value (Fig. 2C). This procedure, which
is in keeping with the proposal by Kernell et al. (1975),
led to the following values of peak twitch tension: 15.7,
21.8, and 25.9 % of AS plateau, for values ofz~ equal to
40, 80, and 120ms, respectively.
lOO
0
B
A
0
100~,
ms 500
0
ms 500
Fig. 3. A Time course of active state (AS) and twitch tension in a
slow muscle. Values of the parameters : r = 1, % = 80 ms, %~= 50 ms,
AS plateau =15 ms. B Active state and twitch tension in a slow
muscle for increasing values of z,s, respectively 80ms, 120ms, and
160ms. Values of other parameters as in A
Slow Muscles. Figure 3A shows the simulation by the
model of one twitch of a slow muscle. Parameters
chosen were: 15 ms for the duration of AS plateau,
control Zas equal to 50 ms (see Methods). The resulting
CT is 65 ms and the PT is 26.8 % of AS plateau. When
"Caswas increased, CT and PT increased (Fig. 3B). The
former attained values of 81, 98, and l l 2 m s and the
latter reached 43.5, 49.8, and 54.5 % of maximal tetanic
tension (never exceeding 64% as in Ramsey and
Streets, 1941), for values of as of 80, 120, and 160ms,
respectively.
As fatigue is known to be negligible in slow muscle
fibres (Burke et al., 1976 ; Parmiggiani and Stein, 1981)
no attempt was made to reproduce changes induced by
a decrease of AS plateau.
Unfused Tetani
Fast Muscles. In order to examine PTP under conditions which closely approximate the experimental
situation, we simulated the effect of changing AS
values on the relation between frequency of activation
and tension development. Figure 4 A illustrates the
frequency-tension relation in unfatigued, unpoten-
132
B
A
100,
lOO
%
%
c
.o
c
o0
•
E
E
i
i
i
~
ms
i
500
0
ms 500
Fig. 4. A Plot of max tetanic tension attained during unfused tetani
versus interstimulus interval in a fast muscle. Curve ( x ) refers to
unfatigued, unpotentiated conditions, i.e. ~,~ = 20 ms and AS = 100 %.
Curve ( + ) refers to same muscle when AS plateau is reduced to 75 %
(fatigue) and za~ is increased to 120 ms (potentiation)9 B Plot of max
tetanic tension attained during unfused tetani versus interstimulus
interval in a slow muscle. Curve ( x ) refers to unpotentiated
conditions, i.e. Zas= 50 ms while curve ( + ) refers to the same muscle
when Za~ is increased to 120ms (potentiation)
tiated conditions ( x ) and in conditions simulating
fatigue (+), that is when the magnitude of AS plateau
was reduced to 75 % of control and "Ca~was increased to
120 ms. It is evident that between 4 and 20 Hz, that is,
in physiological range (Kernell, 1965; Kernell and
Sj6holm, 1973; Baldissera and Parmiggiani, 1979), a
striking increase in absolute value of tension in unfused tetani is obtained in fatigued conditions; this
increase is even larger than the increase for potentiated
twitches (see Fig. 2C).
Slow Muscles. As shown in Fig. 4B, tension of unfused
tetanus is much larger when %~ is increased from 50 to
120 ms, maximal increases in amplitude in relation to
unfused, unpotentiated tetani being reached at
frequencies between 3 and 10 Hz. It is also evident that
potentiation of force during tetanus is of the same
order of magnitude than potentiation of the single
twitch, both in absolute and in relative values. From
the plots it can also be concluded that to obtain a
tension equal to 60% of maximal the required frequency is at about 10 Hz, whereas a frequency of about
4Hz is enough to develop the same tension in a
potentiated muscle (Zas=120ms). Quantitatively similar results have been obtained by NystriSm (1968) in
cat muscles. Note also that the frequency-tension
curves for the slow muscle do not cross. This is at
variance with the results obtained for fast muscles (Fig.
4A) and it means that at all frequencies of tetanization
there is potentiation in the slow muscle. In fast muscle,
on the other hand, the effect due to the decrease in AS
predominates over the advantage due to an increase in
r,~, for frequencies above 400 Hz.
Discussion
In this paper a mathematical model has been used to
simulate the effects of post-tetanic potentiation and
fatigue in slow and fast muscles, on the assumption
that PTP is due to an increase of r,s and fatigue to a
decrease in AS plateau. Therefore in the muscle model
the only parameters subjected to change were: the
duration of active state plateau, plateau level and time
constant of decay of AS. Although the model contains
values of elasticity and viscosity which were obtained
from whole muscle, where interactions between different types of motor units occur, the chosen values
permit to duplicate rather well the behaviour of single
motor units.
According to the results obtained, PTP and fatigue
in fast and slow muscles appear indeed to be related to
changes in different parts of time course of AS, respectively a decrease in the rate of decay of "casand a
reduction in AS plateau. The model accounts also for
presence of PTP in fast muscles even in the case of a
reduction of AS plateau due to fatigue.
The results appear to be in fair agreement with
those of Kernell et al. (1975), where PTP is stressed by
low frequency repetitive stimulation and fatigue by
high frequency stimulation in fast and slow muscle
units, respectively (cf. their Fig. 8). Selecting appropriate values of Vasand AS plateau, the model simulations
show in fact: i) a higher level of twitch PTP in fast
muscles than in slow ones, even when the former is
fatigued (see Figs. 2 and 3); ii) maximal potentiation in
unfused tetani of fast muscles at physiological rates,
presence of fatigue notwithstanding, and iii) less potentiation of unfused tetani in slow muscles, even without
any fatigue effect.
The results of the model simulations are also in
keeping with the hypothesis put forward by Richtie
and Wilkie (1955), according to whom PTP is due to an
increase of Za~. Although the data of these authors are
clear, the magnitude of this effect (as measured in their
experimental conditions) cannot match the results for
mammalian muscles since their experiments were performed at 0 ~ This by itself causes a marked increase
in %~ for thermodynamic reasons. Therefore the possibility to observe a further clear-cut increase of ~,s is
likely to be either prevented and/or masked. However,
a significant increase in "c,s certainly occurs at low
temperature as one can infer from Ranatunga's paper
(1980), where both contraction time and peak tension
are increased while maximal tetanic tension is
diminished9
Finally all our data run contrary to the suggestions
that PTP and the so-called post-activation potentiation are different aspects of the same phenomenon,
as proposed by Ranatunga (1977). The latter effect,
which is induced by applying a conditioning stimulus
at very short intervals (Burke et al., 1976; Ranatunga
1978), can be better explained in terms of an increase in
muscle stiffness (Parmiggiani and Stein, 1981). An
133
analogy might instead be found between PTP and
isometric twitch potentiation
induced by iodide
( R a n a t u n g a , 1979) w h i c h is a b l e t o i n h i b i t C a 2 §
binding capacity of the sarcoplasmic
reticulum
( E b a s h i , 1965).
Acknowledgements. The authors are grateful to Prof. C. Terzuolo for
helpful comments on the manuscript.
References
Baldissera, F., Parmiggiani, F. : After hyperpolarization conductance
time-course and repetitive firing in a motoneurone model with
early inactivation of the slow potassium conductance system.
Biol. Cybern. 34, 233 240 (1979)
Bawa, P., Mannard, A., Stein, R.B. : Predictions and experimental
tests of a visco-elastic muscle model using elastic and inertial
loads. Biol. Cybern. 22, 139-145 (1976)
Brust, M. : Relative resistence to dystrophy of slow skeletal muscle of
the mouse. Am. J. Physiol. 210, 445-451 (1966)
Brust, M. : Fatigue and caffeine effects in fast-twitch and slow-twitch
muscles of the mouse. Pfliigers Arch. 367, 189-220 (1976)
Buller, A.J., Lewis, D.M. : The rate of tension development in
isometric tetanic contractions of mammalian fast and slow
skeletal muscle. J. Physiol. 176, 337-354 (1965)
Buller, A.J., Ranatunga, K.W., Smith, J.M. : The influence on the
contractile characteristics of mammalian fast and slow twitch
skeletal muscles. J. Physiol. 196, 82P (1968)
Burke, R.E., Rudomin, P., Zajac III, F.E. : The effect of activation
history on tension production by individual muscle units. Brain
Res. 109, 515-529 (1976)
Close, R.I.: Dynamic properties of mammalian skeletal muscle.
Physiol. Rev. 52, 129-197 (1972)
Close, R, Hob, J.F.Y. : The after-effects of repetitive stimulation on
the isometric twitch contraction of rat fast skeletal muscle. J.
Physiol. 197, 461-477 (1968)
Connolly, R., Gough, W., Winegrad, S. : Characteristics of the
isometric twitch of skeletal muscle immediately after a tetanus.
J. Gen. Physiol. 57, 697-709 (1971)
Cooper, S., Ecctes, J.C. : The isometric responses of mammalian
muscles. J. Physiol. 69, 377-385 (1930)
Desmedt, J.E., Hainaut, K.: Excitation-contraction coupling i n
single muscle fibers and the calcium channel in sarcoplasmic
reticulum. Ann. N. Y. Acad. Sci. 307, 433 (1978)
Ebashi, S. : Excitation-contraction coupling. Ann. Rev. Physiol. 38,
293-313 (1976)
Hanson, J. : The effects of repetitive stimulation on the action
potential and the twitch of rat muscle. Acta Physiol. Scand.
90, 387-400 (1974)
Hatze, H. : A myocybernetic control model of skeletal muscle. Biol.
Cybern. 25, 103-119 (1977)
Hill, A.V. : The heat of shortening and the dynamic constants of
muscle. Proc. R. Soc. (London) B 126, 136-195 (1938)
Kernelt, D. : The limits of firing frequency in cat lumbosacral
motoneurones possessing different time course of afterhyperpolarization. Acta Physiol. Scand. 65, 87-100 (1965)
Kernetl, D., Ducati, A., Sj6holm, H. :Properties of motor units in the
first deep lumbrical muscle of cat's foot. Brain Res. 98, 37-55
(1975)
Kernell, D., Sj/Sholm, H. : Repetitive impulse firing comparisons
between neurone models based on "voltage clamp equations"
and spinal motoneurones. Acta Physiol. Scand. 87, 40-56 (1973)
Nystr/Sm, B. : Effect of direct tetanization on twitch tension in
developing cat leg muscles. Acta Physiol. Scand. 74, 319-330
(1968)
Parmiggiani, F., Stein, R.B. : Nonlinear summation of contractions
of cat muscle. II. The later facilitation and stiffness changes. J.
Gen. Physiol. 78, 295 311 (1981)
Ramsey, R.W., Streets, S.F. : Muscle function as studied in single
muscle fibres. Biol. Symp. 3, 9-34 (1941)
Ranatunga, K.W. : Influence of temperature on the characteristics of
summation of isometric mechanical responses of mammalian
skeletal muscle. Exp. Neurol. 54, 513-532 (1977)
Ranatunga, K.W.: Characteristics of tension recruitment and
mechanical activation in mammalian skeletal muscle. Exp.
Neurol. 61, 175-184 (1978)
Ranatunga, K.W. : Potentiation of the isometric twitch and mechanisms of tension recruitment in mammalian skeletal muscle.
Exp. Neurol. 63, 266-276 (1979)
Ranatunga, K.W. : Influence of temperature on isometric tension
development in mouse fast-and slow-twitch skeletal muscle.
Exp. Neurol. 70, 211-219 (1980)
Richtie, J.M., Wilkie, D.R. : The effect of previous stimulation on the
active state of muscle. J. Physiol. 130, 488-496 (1955)
Stein, R.B., O~uzt6reli, M.N. : Tremor and other oscillations in
neuromuscular systems. Biol. Cybern. 22, 145 157 (1976)
Received: November 16, 1981
Dr. F. Parmiggiani
Istituto Fisiologia Centri Nervosi, CNR
Via Mario Bianco, 9
1-20131 Milano
Italy