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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 phgr 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 dphgr/dt=DeltaohgrK sin phgr), with phase-locked solutions phgr=arcsin(Deltaohgr /K), where Deltaohgr 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.Deltaohgr was controlled through length differences in the left and right pendulums. The coupled frequency ohgrc was varied by a metronome, and scaled to the eigenfrequency ohgrv of the coupled system K was assumed to vary inversely with ohgrc. The results indicate that: (1) Deltaohgr and K contribute multiplicatively to phgr (2) phgr =0 or phgr = pgr regardless of K when Deltaohgr=0; (3) phgr ap 0 or phgr ap pgr regardless of Deltaohgr when K is large (relative to Deltaohgr); (4) results (1) to (3) hold identically for both in phase and antiphase coordination. The results also indicate that the relevant frequency is ohgrc/ohgrv rather than ohgrc. Discussion high-lighted the significance of confirming phgr=arcsin(Deltaohgr/K) for more general treatments of phase-locking, such as circle map dynamics, and for the 1ratio1 phase-entrainment which characterizes biological movement systems.

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