Average phase difference theory and 1∶1 phase entrainment in interlimb coordination

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.