Abstract
A new model based on the theory of dynamical systems is proposed for the intrinsic random or pscudo-random mechanism underlying certain types of muscular tremor. The active length-tension curve of the individual sarcomere, in conjunction with the passive length-tension relation is a map from length to tension with an observed time delay between length change and resulting tension change. The passive length tension relation is assumed to instantaneously relate this tension change back to a change in length. The stability properties of this iterated interval map are investigated by means of computer simulation and computation of the Lyapunov exponent and the bifurcation tree. The resulting analysis is related to experimental tremor data in the literature in terms of period doubling, bifurcation points, and “chaotic” behavior. The model appears to have its most fruitful application in understanding the insect type and isometric mammalian types of tremor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armstrong CF, Huxley AF, Julian FJ (1966) Oscillatory responses in frog skeletal muscle fibres. J. Physiol 186:26–27P
Bawa P, Mannard A, Stein RB (1976) Effects of elastic loads on the contractions of cat muscles. Biol Cybern 22:129–137
Bloch AM, Busquet H (1904) Etude sur le tremblement physiologique. CR Seances Soc Biol Paris 56:151–153
Ebashi S, Endo M (1968) Calcium ion and muscle contraction. Prog Biophys Mol Biol 18:123–183
Ebashi S, Endo M, Ohtsuki I (1969) Control of muscle contraction. Q Rev Biophys 2:351–384
Flitney FW, Hirst DG (1978) Filament sliding and energy absorbed by the cross-bridges in active muscle subjected to cyclical length changes. J Physiol 276:467–479
Fox JR, Randall JE (1970) Relationship between forearm tremor and the biceps electromyogram. J Appl Physiol 29:103–108
Griffiths W (1888) On the rhythm of muscular response to volitional impulses in man. J Physiol 9:39–54
Halliday AM, Redfearn JWT (1956) An analysis of the frequencies of finger tremor in healthy subjects. J Physiol 134:600–611
Hamoen AJ (1962) Über das Wesen des physiologischen Tremors der Hand. Psychiatr Neurol Neurochir 65:109–116
Hatze H (1977) A myocybernetic control model of skeletal muscle. Biol Cybern 25:103–119
Heinl P, Kuhn HJ, Ruegg JC (1974) Tension responses to quick length changes of glycerinated skeletal muscle fibres from the frog and tortoise. J Physiol 237:243–258
Hill AV (1953) The mechanics of active muscle. Proc R Soc London Ser B 141:104–117
Huxley AF (1957) Muscle structure and theories of contraction. Prog Biophys Biophys Chem 7:255–318
Huxley AF (1979) Review lecture: muscle contraction. J Physiol 243:1–43
Huxley AF, Peachey LD (1961) The maximum length for contraction in vertebrate striated muscle. J Physiol 156:150–165
Huxley AF, Simmons RM (1971) Proposed mechanism of force generation in striated muscle. Nature 233:533–538
Jewell BR, Ruegg JC (1966) Oscillatory contraction of insect fibrillar muscle after glycerol extraction. Proc R Soc London B 164:428–459
Joyce GC, Rack PMH (1974) The effects of load and force on tremor at the normal human elbow joint. J Physiol 240: 375–396
Julian FJ, Morgan DL (1979) Intersacomere dynamics during fixed-end tetanic contractions of frog muscle fibres. J Physiol 293:365–378
Julian FJ, Sollins MR, Moss RL (1978) Sarcomere length nonuniformity in relation to tetanic responses of stretched skeletal muscle fibres. Proc R Soc London Ser B 200:109–116
Kantor JC (1984) A dynamical instability of spark-ignited engines. Science 224:1233–1235
Kunke E (1952) Über eine Frequenzlücke zwischen den schnellsten Willkürbewegungen und dem Versteifungstremor. Pfluegers Arch Gesamte Physiol 225:530–543
Lance JW, Schwab RS, Peterson EA (1963) Action tremor and the cogwheel phenomenon in Parkinson's disease. Brain 86:95–110
Li TY, Yorke JA (1975) Period three implies chaos. Am Math Monthly 82:985–992
Lippold OCJ (1970) Oscillation in the stretch reflex arc and the origin of the rhythmical, 8–12 c/s component of physiological tremor. J Physiol 206:359–382
Marsden CD, Meadows JC, Lange GW, Watson RS (1967) Effect of deafferentation on human physiological tremor. Lancet 2:700–702
Marshall J, Walsch EG (1956) Physiological tremor. J Neurol Neurosurg Psychiatry 19:260–267
May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–467
Poincaré H (1892) Les methodes nouvelles de la mechanique celeste. Gauthier-Villars, Paris
Pringle JWS (1949) The excitation and contraction of the flight muscles of insects. J Physiol 108:226–232
Robson JG (1959) The effect of loading upon the frequency of muscle tremor. J Physiol 149:29–30p
Schafer EA (1886) On the rhythm of muscular response to volitional impulses in man. J Physiol 7:11–117
Shaw R (1981) Strange attractors, chaotic behavior, and information flow. Z Naturforsch 36A:80–112
Stein RB, Oğuztöreli MN (1978) Reflex involvement in tremor and clonus, in physiological tremor, pathological tremor and clonus. In: Desmedt JE (ed) Prog. Clin Neurophysiol 5. Karger, Basel, pp 28–50
Thorson J, White DCS (1969) Distributed representations for actin-myosin interaction in the oscillatory contraction of muscle. Biophys J 9:360–390
Ueda Y, Akamatsu N (1981) Chaotically transitional phenomena in the forced negative-resistance oscillator. IEEE Trans CAS-28:217–224
White DGS, Thorson J (1972) Phosphate starvation and the nonlinear dynamics of insect fibrillar flight muscle. J Gen Physiol 60:307–336
Yap CB, Boshes B (1967) The frequency and pattern of normal tremor. Electroencephalogr Clin Neurophysiol 22:197–203
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Akamatsu, N., Hannaford, B. & Stark, L. An intrinsic mechanism for the oscillatory contraction of muscle. Biol. Cybern. 53, 219–227 (1986). https://doi.org/10.1007/BF00336992
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00336992