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