Abstract
Dynamical models of two coupled biological oscillators interpret the detuning term as an arithmetic difference between the uncoupled frequencies, Δω =(ω1 − ω2) . This Δω interpretation of detuning was addressed in four experiments in which human subjects oscillated pendulums in their right and left hands in 1∶1 frequency locking in antiphase (Experiments 1–3) or inphase (Experiment 4). Differences between the uncoupled frequencies were manipulated through differences in the equivalent simple pendulum lengths, and the effects of this manipulation on the detuning of relative phase from π or 0 and the standard deviation of relative phase SDφ were measured. In Experiment 1, the same values of ω i were satisfied by several different physical configurations. The experiment confirmed that the detuning term is related strictly to the uncoupled frequencies rather than to other physical characteristics of the oscillators. Experiments 2, 3 and 4 showed, however, that the particular dependency of fixed point drift and SDφ on Δω depends on the particulars of ω 1 and ω 2. With variations in Δω brought about by different ω 1 and ω 2 that always formed a constant ratio, fixed point drift related inversely to Δω, and SDφ varied with Δω in ways that depended on the magnitude of the constant ratio. These outcomes do not conform to expectations from models of coordination dynamics that interpret detuning as (ω 1)−ω 2).
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References
Arnold VI (1965) Small denominators. I. Mapping of the circumference onto itself. Am Math Soc Transl 46:213–284
Cohen AH, Holmes PJ, Rand RH (1982) The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: a mathematical model. J Math Biol 13:345–369
Gilmore R (1981) Catastrophe theory for scientists and engineers. Wiley, New York
Haken H (1977) Synergetics. Springer, Berlin Heidelberg New York
Haken H (1983) Advanced synergetics. Springer, Berlin Heidelberg New York
Haken H, Kelso JAS, Bunz H (1985) A theoretical model of phase transitions in human hand movements. Biol Cybern 51:347–356
Holst E von (1973) The behavioral physiology of animal and man. Coral Gables, FL: University of Miami Press. (Original work published 1939)
Jackson EA (1989) Perspectives of nonlinear dynamics. Cambridge University Press, Cambridge, UK
Jeka JJ, Kelso JAS, Kiemel T (1993) Pattern switching in human multilimb coordination dynamics. Bull Math Biol 55:829–845
Kadar EE, Schmidt RC, Turvey MT (1993) Constants underlying frequency changes in biological rhythmic movements. Biol Cybern 68:421–430
Kay BA, Kelso JAS, Saltzman ES, Schöner G (1987) Space-time behavior of single and bimanual rhythmical movements. J Exp Psychol Hum Percept Perform 13:178–192
Kay BA, Saltzman ES, Kelso JAS (1991) Steady state and perturbed rhythmical movements: a dynamical analysis. J Exp Psychol Hum Percept Perform 17:183–197
Keith WL, Rand RH (1984) 1∶1 and 2∶1 phase entrainment in a system of two coupled limit cycle oscillators. J of Math Biology 20:133–152
Kelso JAS (1984) Phase transitions and critical behavior in human bimanual coordination. Am J Phys Reg Integr Comp 15:R1000-R1004
Kelso JAS, Ding M (1993) Fluctuations, intermittency and controllable chaos in biological coordination. In: Newell KM, Corcos DM (eds) Variability and motor control. Human Kinetics, Champaigne,pp 291–316
Kelso JAS, Jeka JJ (1992) Symmetry breaking dynamics of human multilimb coordination. J Exp Psychol Hum Percept Perform 18:645–668
Kelso JAS, Schöner G, Scholz JP, Haken H (1987) Phase-locked modes, phase transitions and component oscillators in biological motion. Physica Scripta 35:79–87
Kelso JAS, Delcolle JD, Schöner G (1990) Action-perception as a pattern formation process. In: Jeannerod M (eds) Attention and performance XIII. Erlbaum, Hillsdale, pp 139–169
Kopell N (1988) Toward a theory of modelling central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 369–413
Kopell N, Ermentrout GB (1988) Coupled oscillators and the design of central pattern generators. Math Biosci 90:87–109
Kugler PN, Turvey MT (1987) Information, natural law and the self-assembly of rhythmic movement. Erlbaum, Hillsdale
McClellan AD, Sigvardt KA (1988) Features of entrainment of spinal pattern generators for locomotor activity in the lamprey spinal cord. J Neuroscience 8:133–145
Murray JD (1990) Mathematical biology. Springer, Berlin Heidelberg New York
Rand RH, Holmes PJ (1980) Bifurcations of periodic motions in two weakly coupled van der Pol oscillators. Int J Nonlinear Mechanics 15:387–399
Rand RH, Cohen AH, Holmes PJ (1988) Systems of coupled oscillators as models of central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 333–367
Schmidt RC, Turvey MT (1994) Phase-entrainment dynamics of visually coupled rhythmic movements. Biol Cybern 70:369–376
Schmidt RC, Carello C, Turvey MT (1990) Phase transitions and critical fluctuations in the visual coordination of rhythmic movements between people. J Exp Psychol Hum Percept Perform 16:227–247
Schmidt RC, Beek PJ, Treffner PJ, Turvey MT (1991) Dynamical substructure of coordinated rhythmic movements. J Exp Psychol Hum Percept Perform 17:635–651
Schmidt RC, Shaw BS, Turvey MT (1993) Coupling dynamics in interlimb coordination. J Exp Psychol Hum Percept Perform 19:397–415
Schöner G, Haken H, Kelso JAS (1986) A stochastic theory of phase transitions in human hand movement. Biol Cybern 53:442–452
Sternad D, Turvey MT, Schmidt RC (1992) Average phase difference theory and 1∶1 phase entrainment in interlimb coordination. Biol Cybern 67:223–231
Treffner PJ, Turvey MT (in press) Handedness and the asymmetric dynamics of bimanual rhythmic coordination. J Exp Psychol Hum Percept and Perform
Turvey MT, Rosenblum L, Schmidt RC, Kugler PN (1986) Fluctuations and phase symmetry in coordinated rhythmic movements. J Exp Psychol Hum Percept Perform 12:564–583
Turvey MT, Schmidt RC, Beek PJ (1993) Fluctuations in interlimb rhythmic coordinations. In: Newell K, Corcos D (eds) Variability in motor control. Human Kinetics, Champaigne, pp 381–411
Williams TL, Sigvardt KA, Kopell N, Ermentrout GB, Remler M (1990) Forcing of coupled nonlinear oscillators: studies of intersegmental coordination in the lamprey locomotor central pattern generator. J Neurophysiology 64:862–871
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Sternad, D., Collins, D. & Turvey, M.T. The detuning factor in the dynamics of interlimb rhythmic coordination. Biol. Cybern. 73, 27–35 (1995). https://doi.org/10.1007/BF00199053
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DOI: https://doi.org/10.1007/BF00199053