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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 72 (1995), S. 511-518 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract.  An order parameter equation for correlated limb movements was applied to rhythmic coordination between the limbs of two people. The interlimb coordination was established and maintained through vision. Manipulations of frequency competition, coupled frequency, and intended mode (in-phase or anti-phase) produced equilibria and fluctuations in relative phase predicted by the order parameter equation and confirmed originally in within-person coordination. It was concluded that there is an elementary coordination dynamics governing the rhythmic coordination between organisms as well as between components of a single organism.
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 73 (1995), S. 27-35 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: 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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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 67 (1992), S. 223-231 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract The dynamics of coupled biological oscillators can be modeled by averaging the effects of coupling over each oscillatory cycle so that the coupling depends on the phase difference φ between the two oscillators and not on their specific states. Average phase difference theory claims that mode locking phenomena can be predicted by the average effects of the coupling influences. As a starting point for both empirical and theoretical investigations, Rand et al. (1988) have proposed dφ/dt=Δω — K sin φ), with phase-locked solutions φ=arcsin(Δω /K), where Δω is the difference between the uncoupled frequencies and K is the coupling strength. Phase-locking was evaluated in three experiments using an interlimb coordination paradigm in which a person oscillates hand-held pendulums.Δω was controlled through length differences in the left and right pendulums. The coupled frequency ωc was varied by a metronome, and scaled to the eigenfrequency ωv of the coupled system K was assumed to vary inversely with ωc. The results indicate that: (1) Δω and K contribute multiplicatively to φ (2) φ =0 or φ = π regardless of K when Δω=0; (3) φ ≈ 0 or φ ≈ π regardless of Δω when K is large (relative to Δω); (4) results (1) to (3) hold identically for both in phase and antiphase coordination. The results also indicate that the relevant frequency is ωc/ωv rather than ωc. Discussion high-lighted the significance of confirming φ=arcsin(Δω/K) for more general treatments of phase-locking, such as circle map dynamics, and for the 1∶1 phase-entrainment which characterizes biological movement systems.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 73 (1995), S. 27-35 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: 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 ω1 and ω2. With variations in Δω brought about by different ω1 and ω2 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 (ω1−ω2).
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 73 (1995), S. 499-507 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract Biological rhythmic movements can be viewed as instances of self-sustained oscillators. Auto-oscillatory phenomena must involve a nonlinear friction function, and usually involve a nonlinear elastic function. With respect to rhythmic movements, the question is: What kinds of nonlinear friction and elastic functions are involved? The nonlinear friction functions of the kind identified by Rayleigh (involving terms such as $$\dot \theta ^3 $$ ) and van der Pol (involving terms such as $$\theta ^2 \dot \theta $$ ), and the nonlinear elastic functions identified by Duffing (involving terms such as $$\theta ^3 $$ ), constitute elementary nonlinear components for the assembling of self-sustained oscillators. Recently, additional elementary nonlinear friction and stiffness functions expressed, respectively, through terms such as $$\theta ^2 \dot \theta ^3 $$ and $$\theta \dot \theta ^2 $$ , and a methodology for evaluating the contribution of the elementary components to any given cyclic activity have been identified. The methodology uses a quantification of the continuous deviation of oscillatory motion from ideal (harmonic) motion. Multiple regression of this quantity on the elementary linear and nonlinear terms reveals the individual contribution of each term to the oscillator's non-harmonic behavior. In the present article the methodology was applied to the data from three experiments in which human subjects produced pendular rhythmic movements under manipulations of rotational inertia (experiment 1), rotational inertia and frequency (experiment 2), and rotational inertia and amplitude (experiment 3). The analysis revealed that the pendular oscillators assembled in the three experiments were compositionally rich, braiding linear and nonlinear friction and elastic functions in a manner that depended on the nature of the task.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 74 (1996), S. 107-115 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract It is hypothesized that metabolic and mechanical changes in human locomotion associated with changes in speed v are constrained by two attractive strategies: $$Q_{{\text{metab}}} = 1{\text{ and }}\Delta Q_{{\text{metab}}} /\Delta v = {\text{a}}$$ positive definite constant. $$Q_{{\text{metab}}} = \Delta {\rm E}_{\text{k}} {\text{s}}^{{\text{ - 1}}} /{\text{ml O}}_{\text{2}} {\text{s}}^{{\text{ - 1}}} $$ where ΔEs−1 is the summed increments and decrements per unit time in the translational and rotational kinetic energies of the body's segments and ml O2s−1 is the rate at which chemical energy is dissipated. The expected constancy of ΔQ metab/Δv metab was derived from an extension of Ehrenfest's adiabatic hypothesis by which transformations (increases, decreases) in locomotion v can be considered as adiabatic, even though the biological conditions are nonconservative and non-rate-limited. The expected significance of Q metab=1 was derived from stability considerations of the symmetry per stride of stored and dissipated energy. An experimental evaluation was provided by collecting metabolic and mechanical measures on walking (10 subjects) and running (9 subjects) at progressively greater treadmill speeds but within the aerobic limit. Results revealed that walking was restricted to ometab ⩽ 1 with a nonlinear trajectory in v×Q metab coordinates shaped by Q metab=1 (primarily) and the constancy of ΔQ metab/Δv. Running satisfied Q metab 〉 1, with a linear trajectory in v×Q metab coordinates conforming to ΔQ metab/Δv=a constant, with the constant predicted from invariants in the mechanical space v×ΔE ks−1. Results also suggested that the metabolic costs of running might be predictable from measures made in the v×ΔE ks−1 space.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 68 (1993), S. 421-430 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract When an animal increases or decreases the frequency of its limb motions, how should the transformation in timing be characterized? It has been hypothesized that the transformation is adiabatic, even though the biological conditions are nonconservative and non-rate-limited (Kugler and Turvey 1987). An adiabatic transformation requires that the rhythmic system's action (energy/frequency) and entropy production remain time-invariant throughout the transformation. The non-conservative adiabatic hypothesis was evaluated through an experiment on human rhythmic hand movements. On each trial, a subject began at a prescribed frequency and then, over a 30 s interval, increased (or decreased) the frequency continuously at will. For each subject, on each increasing and decreasing trial, cycle kinetic energy was a linear function of cycle frequency with a negative energy intercept. By the adiabatic hypothesis, the slope of the function defines the constant action and the intercept defines the constant dissipation — changes in cycle frequency incur no changes in energy dissipated per cycle. Slopes and intercepts were correlated suggesting a common basis for the two constants, and the variety of cycle amplitude-cycle duration relations were in agreement with the nonmonotonic, nonlinear space-time function predicted by the hypothesis. The possibilities of addressing aspects of the data through (a) muscle modeled as a continuum of Kelvin bodies with a continuous relaxation spectrum, and (b) various classes of autonomous differential equations, were discussed. Most importantly, the discussion focused on the puzzling independence of energy cost and speed exhibited by locomoting animals differing in morphology, physiology, size, and taxa. It was suggested that the independence may reflect a very general principle — adiabatic transformability of biological movement systems.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 72 (1995), S. 511-518 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract An order parameter equation for correlated limb movements was applied to rhythmic coordination between the limbs of two people. The interlimb coordination was established and maintained through vision. Manipulations of frequency competition, coupled frequency, and intended mode (in-phase or anti-phase) produced equilibria and fluctuations in relative phase predicted by the order parameter equation and confirmed originally in within-person coordination. It was concluded that there is an elementary coordination dynamics governing the rhythmic coordination between organisms as well as between components of a single organism.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 70 (1994), S. 369-376 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract Do interlimb rhythmic coordinations between individuals exhibit the same relations among the same observable quantities as interlimb rhythmic coordination within an individual? The 1∶1 frequency locking between the limbs of two people was investigated using a paradigm in which each person oscillated a hand-held pendulum, achieving and maintaining the mutual entrainment through vision. The intended coordination was antiphase, φ=π, and the difference between the uncoupled eigenfrequencies, Δω, was manipulated through differences in the lengths of the two pendulums. The mean phase relation and its variance for visually coupled coordinations differing in Δω were predicted by an order parameter equation developed by Haken et al. (1985) and Schöner et al. (1986) for the relative phase of correlated movements of limb segments. Specifically, the experiment revealed that: (1) the deviation of φ from π increased with increasing deviation of Δω from 0; and (2) fluctuations in φ increased with increasing deviation of Δω from 0. With deviations of Δω from 0, new peaks were added at higher harmonics in φ's power spectrum. These results were in agreement with previous research on the stable states of interlimb coordination within a person, mediated by mechanoreceptive rather than photoreceptive mechanisms. Additionally, they were in agreement with previous research on phase transitions in interlimb coordination which have been shown to conform to the same order parameter dynamics whether the coupling be mechanoreceptively or photoreceptively based. It was suggested that phase entrainment in biological movement systems may abide by dynamical principles that are indifferent to the details of the coupling.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Biological cybernetics 70 (1994), S. 369-376 
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract. Do interlimb rhythmic coordinations between individuals exhibit the same relations among the same observable quantities as interlimb rhythmic coordination within an individual? The 1:1 frequency locking between the limbs of two people was investigated using a paradigm in which each person oscillated a hand-held pendulum, achieving and maintaining the mutual entrainment through vision. The intended coordination was antiphase, φ=π, and the difference between the uncoupled eigenfrequencies, Δω, was manipulated through differences in the lengths of the two pendulums. The mean phase relation and its variance for visually coupled coordinations differing in Δω were predicted by an order parameter equation developed by Haken et al. (1985) and Schöner et al. (1986) for the relative phase of correlated movements of limb segments. Specifically, the experiment revealed that: (1) the deviation of φ from π increased with increasing deviation of Δω from 0; and (2) fluctuations in φ increased with increasing deviation of Δω from 0. With deviations of Δω from 0, new peaks were added at higher harmonics in φ’s power spectrum. These results were in agreement with previous research on the stable states of interlimb coordination within a person, mediated by mechanoreceptive rather than photoreceptive mechanisms. Additionally, they were in agreement with previous research on phase transitions in interlimb coordination which have been shown to conform to the same order parameter dynamics whether the coupling be mechanoreceptively or photoreceptively based. It was suggested that phase entrainment in biological movement systems may abide by dynamical principles that are indifferent to the details of the coupling.
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