Biological Cybernetics

Biol. Cybernetics 27, 9 4 0 (1977)

9 by Springer-Verlag 1977

Energy-Optimal Controls in the Mammalian Neuromuscular System H. Hatze and J. D. Buys National Research Institute for Mathematical Sciences, CSIR, Pretoria, South Africa

Abstract. The hypothesis is advanced that the specific

patterns of motor unit recruitment and stimulation frequencies observed in mammalian skeletal muscle under static isometric contractions are determined by a minimum-energy principle. By performing a constrained energy optimization based on a control model o f skeletal muscle comprising three different fibre types, and appropriate expressions for the energy rates, it is indeed possible to obtain detailed predictions of recruitment and stimulation frequency patterns which agree well with the experimentally observed functions, thereby providing strong support for the minimumenergy hypothesis. Since the orderly recruitment sequence determined by the size principle is also, independently, predicted by the minimum-energy principle, it is concluded that there exists a relationship between motor unit size and the myoenergetic properties of the recruited unit. It is suggested that this relationship, together with the possibility of adjusting the relative proportions of the fibre types present in a muscle, constitutes an optimal adaptation of the neuromuscular system for practically all types of muscular performances normally encountered. For various types of muscles, the energy rates as functions of the force output are also discussed.

Introduction

The specific relationship existing in mammalian skeletal muscle between the number, size, type, and sequential order of the motor units recruited, and the stimulation frequencies of the recruited units, for different modes of contraction, is still a matter of controversy. The most consistent results have been obtained for static isometric contractions in animals and in man where it has been found that motor units are recruited 'according to their size in an orderly fashion (MilnerBrown et al., 1973a), and that their firing rates vary over

a rather restricted range only (Gydikov and Kosarov, 1973). This sequential order in the recruitment pattern has recently been shown to represent the realization, by the mammalian neuromuscular system, of a general teleological principle of optimal recruitment (grading) sensitivity over the whole range of force production (Hatze, 1977b). Discharge and recruitment patterns of motor units in non-static isometric contractions were found to differ substantially from the patterns observed in static isometric contractions (Bigland and Lippold, 1954; Clamann, 1970; Gydikov and Kosarov, 1973 ; Grimby and Hannerz, 1968 ; Milner-Brown et al., 1973b ; Person and Kudina, 1972). Starting discharge frequencies were higher (depending on the time rate of force rise) and the firing range was extended (Gillies, 1972). However, the sequential recruitment pattern of the motor units according to their size and histochemical composition was generally still observed. When non-isometric contractions were investigated (Adrian and Bronk, 1929; Gydikov and Kosarov, 1973), discharge frequencies were found to be comparatively high but it was not always possible to establish a strict order, according to unit size, in the sequence of recruitment. A model recently presented by Wani and Guha (1975) supposedly provides complete information regarding the number of motor units active and their detailed characteristics such as twitch tensions, instantaneous firing frequencies, threshold forces, contraction times, etc. for a certain gradation of the tension. The model is based on the assumptions that (i) the tension developed by a motor unit is equal to the product of its averaged twitch tension and its firing frequency for all possible frequencies occurring; (ii) the frequencies of all motor units increase linearly with the total force produced; and (iii) the same motor units (having certain twitch tensions and contraction times) are recruited at certain threshold forces under static as well as dynamic conditions.

10

These assumptions appear to be rather restrictive, since it is well known that: (i) sequences of twitches do not add up linearly when overlapping of the individual responses occurs (Rack and Westbury, 1969 ; Eberstein and Goodgold, 1968) but approach a maximum force level asymptotically as the frequency increases ; (ii) the frequency increase of the motor units recruited is, in general, nonlinear, as clearly demonstrated by MilnerBrown et al. (1973b, Fig. 4), from whom Wani and Guha took their data, and (iii) it would be rather difficult to substantiate the assumption (iii) above. These deficiencies were discovered when impossible predictions with this model were obtained; as a result, the model structure was investigated. The consistent basic pattern of recruitment and rate coding observed in static isometric contractions suggests the presence of rather general principles which determine these patterns. The fact that the small motor units recruited at low and intermediate force levels consist mainly of slow-fatigueing muscle fibres with a histochemical profile which indicates low energy consumption (the S-type fibres of Burke et al., 1971) points to the possibility of the presence of a minimum-energy principle, in addition to the principle of optimal force grading sensitivity which explains the size principle (Hatze, 1977b). On these grounds we advance the hypothesis that the special relationship between motor unit recruitment and rate coding observed in static isometric contractions, as well as the selection, by the evolutionary process, of fibre types with specific histochemical compositions for different levels of force production, constitutes the realization of a principle of minimumenergy expenditure. This hypothesis will now be tested. In order to do this we shall need an expression, which will now be derived, for the rate of energy production of the contraction process.

interval v-1. Multiplying this function by the stimulation rate v we obtain the activation heat rate 0 for a muscle having a mass of G kg o f which a proportion u is active as 0 = u g l y [ 1 - e x p ( - ~:8 - tcT/v)]/

[~ - f e x p ( - ~:8 - ~7/9)],

where ~ (in W/kg muscle mass) is the muscle-specific activation heat rate constant, ~ is the maximum stimulation frequency occurring, and the constants ~:7 and ~:8 have values of 18.2 and 0.25 respectively. Wendt and Gibbs (1973) have also determined values of the ratio q5 defined by q~= ~/(~ + ~-),

The total energy rate/~ of a muscular contraction can be expressed (Mommaerts, 1969) as the sum of the activation heat rate 0, the maintenance heat rate/~, the shortening heat rate ~, the work rate w, and the rate i" of heat dissipated in the parallel elastic structures. Hence E=0+~i+~+w+~.

(I)

The activation heat is thought to be connected to the release and re-uptake of Ca by the sarcoplasmic reticulum and is that portion of the total internal heat production which is not connected to the actomyosin system (Woledge, 1971). The activation heat per stimulus has been measured in slow and fast rat muscle (Gibbs and Gibson, 1972 ; Wendt and Gibbs, 1973), and was found to be an exponential function of the stimulus

(3)

where ~ is the muscle-specific maintenance heat rate constant. The values for ~b were found to be about 0.35 for fast mammalian fibres and 0.45 for slow mammalian fibres (see Table 2 of Wendt and Gibbs, 1973), and agree with observations by Edwards et al. (1975) who found an average value of 0.34 (since 66 Tooof the heat rate was due to glycolytic reactions) for the human quadriceps muscle. Data for (0 + h) for fast and slow human muscle fibres have also been collected (Bolstad and Ersland, 1975). The respective values are 150 W kg- 1 and 24.4 W kg- 1. These values as well as their ratio agree remarkably well with the data observed in rat muscle at 27 ~ C (Gibbs and Gibson, 1972 ; Wendt and Gibbs, 1973). They constitute, however, only average values of a population of a certain fibre type and fairly large deviations from average properties within such populations have been documented (Burke et al., 1971). Hill (1938) has demonstrated that the combined activation and maintenance heat rates can be expressed as

0 + h = ab = (a/Po)2Po V o ,

Energy Rates of the Contraction Process

(2)

(4)

where it has been found (Ritchie and Wilkie, 1958; Jewell and Wilkie, 1960 ; Bahler et al., 1968) that Hill's "constants" a, Po, V0 are, in fact, complicated functions of the relative length { and the active state q of the contractile element. The significance of these experimentally observed functions has been discussed in (Hatze, 1977a) where the following expressions were derived

Po(~, q) = k(Oq(v, ~, t)P,

(5)

Vo( ~, q) = ( - -2o)[- a 2 -

a71 arctanh {blb2 e x p [ - 2.6(~- 1 ) ] / q k ( ~ ) - 1}].

(6)

The symbols have the following meaning : ~ = 2/2 is the relative length o f the contractile element C E with actual length 2 and optimum length 2 (see Hatze, 1977a), ff is

It

the tetanic force produced when the CE is at 2 and fully stimulated with rate ,gmax; "~0, at, a2, bl, and b 2 are muscle-specific constants, the length-tension relation /c(~) for a fibre is given by k(~)= 0.32 +0.71 e x p { - 1.112(4- 1)} 9sin{3.722(~- 0.656)},

- (a/FYFJ7o = (~+ ~)C,

0.58 < ~__=K* the minimizing vector (u*, v*)r of the reduced problem is the solution of the unconstrained minimization problem rain L(u, v; A*), (u.v)

(A3)

where A* is the Lagrange multiplier associated with the reduced problem and can be found by the following iterative technique. Choose a value A (~ solve (A3) with A* = A (~ and let (u(1~,v(1~)r be the corresponding solution vector. Compute A ~) from A ~) = A (~ + 2K~,o(u (~), v I~)) and repeat the whole procedure using A t~)

instead of A (~ etc. It may be shown that the sequence {A(~1} so obtained converges to A* and {(u(~, v(J~)r} converges to (u*, v*)r. The iterations were stopped when IW(u,v)] < t 0 - i0. The choice of K is generally not critical. The rate of convergence can be increased by increasing K but too large a value might create problems in computing (A3). In the present case, a starting value of K = 100 was chosen and the condition

was checked at each iteration. If (A4) was not satisfied the current value K ~j)was replaced by 5K(Jk The upper limit of K was set at 101 o. The subroutine VA09A (Harwell Subroutine Library) which is based on Fletcher's (1970) variable metric algorithm was used to perform the unconstrained minimizations (A3). All computations were carried out on a CDC Cyber 174 digital computer.

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