Journal of Mathematical Biology

, Volume 4, Issue 3, pp 203–236 | Cite as

A model of optimal voluntary muscular control

  • R. FitzHugh
Article

Summary

In the absence of detailed knowledge of how the CNS controls a muscle through its motor fibers, a reasonable hypothesis is that of optimal control. This hypothesis is studied using a simplified mathematical model of a single muscle, based on A. V. Hill's equations, with series elastic element omitted, and with the motor signal represented by a single input variable.

Two cost functions were used. The first was total energy expended by the muscle (work plus heat). If the load is a constant force, with no inertia, Hill's optimal velocity of shortening results. If the load includes a mass, analysis by optimal control theory shows that the motor signal to the muscle consists of three phases: (1) maximal stimulation to accelerate the mass to the optimal velocity as quickly as possible, (2) an intermediate level of stimulation to hold the velocity at its optimal value, once reached, and (3) zero stimulation, to permit the mass to slow down, as quickly as possible, to zero velocity at the specified distance shortened. If the latter distance is too small, or the mass too large, the optimal velocity is not reached, and phase (2) is absent. For lengthening, there is no optimal velocity; there are only two phases, zero stimulation followed by maximal stimulation.

The second cost function was total time. The optimal control for shortening consists of only phases (1) and (3) above, and is identical to the minimal energy control whenever phase (2) is absent from the latter.

Generalization of this model to include viscous loads and a series elastic element are discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abbott, B. C., Bigland, B., Ritchie, J. M.: The physiological cost of negative work. J. Physiol. 117, 380–390 (1952).Google Scholar
  2. Athans, M., Falb, P. L.: Optimal Control. New York: McGraw-Hill 1966.Google Scholar
  3. Ayoub, M. A., Ayoub, M. M., Walkevar, A. G.: A biomechanical model for the upper extremity using optimization techniques. Human Factors 16, 585–594 (1974).Google Scholar
  4. Basmajian, J. V.: Muscles Alive. Their Functions Revealed by Electromyography. Baltimore: Williams and Wilkins 1962.Google Scholar
  5. Beckett, R., Chang, K.: An evaluation of the kinematics of gait by minimum energy, pp. 15–26 in: Biomechanics (Bootzin, D., Muffley, H. C., eds.). New York: Plenum 1969.Google Scholar
  6. Bigland, B., Lippold, O. C. J.: The relation between force, velocity and integrated electrical activity in human muscles. J. Physiol. 123, 214–224 (1954).Google Scholar
  7. Chow, C. K., Jacobson, D. H.: Studies of human locomotion via optimal programming. Math. Biosci. 10, 239–306 (1971).Google Scholar
  8. Close, J. R.: Motor Functions in the Lower Extremities. Analyses by Electronic Instrumentation. Springfield: Ch. C Thomas 1964.Google Scholar
  9. Cohn, D. L.: Optimal systems: I. The vascular system. Bull. Math. Biophys. 16, 59–74 (1954).Google Scholar
  10. Halze, H.: Optimization of human motions, pp 138–143 in Medicine and Sport (Cerquilini, S., Venerando, A., Wartenweiler, J., eds.). Vol. 8. Biomechaanics III. Basel: Karger 1973.Google Scholar
  11. Hill, A. V.: The heat of shortening and the dynamic constants of muscle. Proc. Roy. Soc. B 126, 262–274 (1938).Google Scholar
  12. Hill, A. V.: The mechanical efficiency of frog's muscle. Proc. Roy. Soc. B 127, 434–451 (1939).Google Scholar
  13. Hill, A. V.: The dynamic constants of human muscle. Proc. Roy. Soc. B 128, 262–274 (1940).Google Scholar
  14. Hill, A. V.: Production and absorption of work by muscle. Science 131, 897–903 (1960).Google Scholar
  15. Hill, A. V.: The effect of load on the heat of shortening of muscle. Proc. Roy. Soc. B 159, 297–318 (1964a).Google Scholar
  16. Hill, A. V.: The efficiency of mechanical power development during muscular shortening and its relation to load. Proc. Roy. Soc. B 159, 319–324 (1964b).Google Scholar
  17. Hill, A. V., Howarth, J. V.: The reversal of chemical reactions in contracting muscle during an applied stretch. Proc. Roy. Soc. B 151, 169–193 (1959).Google Scholar
  18. Hopf, H. C., Handwerker, H., Hausmanns, J.: Die rasche Willkürbewegung des Menschen. Deutsche Zeitschr. f. Nervenheilkunde 191, 186–209 (1967).Google Scholar
  19. Huxley, A. F., Simmons, R. M.: Proposed mechanism of force generation in striated muscle. Nature 233, 533–538 (1971).Google Scholar
  20. Joyce, G. C., Rack, P. M. H., Westbury, D. R.: Some mechanical properties of the cat soleus muscle at various stimulus rates. J. Physiol. 197, 23P-25P (1968).Google Scholar
  21. Katz, B.: The relation between force and speed in muscular contraction. J. Physiol. 96, 45–64 (1939).Google Scholar
  22. Leitman, G.: An Introduction to Optimal Control. New York: McGraw-Hill 1966.Google Scholar
  23. Matthews, P. B. C.: Mammalian Muscle Receptors and Their Central Actions. Baltimore: Williams and Wilkins 1972.Google Scholar
  24. Merriam, C. W. III.: Optimization Theory and the Design of Feedback Control Systems. New York: McGraw-Hill 1964.Google Scholar
  25. Nubar, Y., Contini, R.: A minimal principle in biomechanics. Bull. Math. Biophys. 23, 377–391 (1961).Google Scholar
  26. Pfeiffer, A.: Ein kybernetisches Gütemaß für die Koordinierung von Bewegungen. Z. f. Psychol. 171, 395–399 (1965).Google Scholar
  27. Pini, A.: Relazione forza-velocita nel muscolo umano in vivo nel lavoro motore e resistente. Boll. Soc. Ital. Biol. Sper. 41, 1548–1550 (1965).Google Scholar
  28. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., Mishchenko, E. F.: The Mathematical Theory of Optimal Processes. New York: Interscience 1962.Google Scholar
  29. Ralston, H. J.: Recent advances in neuromuscular physiology. Am J. Physical Medicine 36, 94–120 (1957).Google Scholar
  30. Rashevsky, N.: Mathematical Biophysics, 3rd ed., Vol. 2, p. 292. New York: Dover 1960.Google Scholar
  31. Rosen, R.: Optimality Principles in Biology. New York: Plenum 1967.Google Scholar
  32. Wacholder, K.: Willkürliche Haltung und Bewegung, insbesondere im Lichte elektrophysiologischer Untersuchungen. Ergebnisse der Physiologie 26, 568–775 (1928).Google Scholar
  33. Wacholder, K.: Die Arbeitsfähigkeit des Menschen in ihrer Abhängigkeit von der Funktionsweise des Muskel- und Nervensystems, pp. 587–642, in: Handbuch der normalen und pathologischen Physiologie (Bethe, A., et al., eds.), Band 15, 1. Hälfte. 1930.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • R. FitzHugh
    • 1
  1. 1.National Institutes of HealthBethesdaUSA

Personalised recommendations