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-phi 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 d-phi/dt = DELTA-omega - K sin phi, with phase-locked solutions phi = arcsin(DELTA-omega/K), where DELTA-omega 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. DELTA-omega was controlled through length differences in the left and right pendulums. The coupled frequency omega(c) was varied by a metronome, and scaled to the eigenfrequency omega(v) of the coupled system; K was assumed to vary inversely with omega(c). The results indicate that: (1) DELTA-omega and K contribute multiplicatively to phi; (2) phi = 0 or phi = pi regardless of K when DELTA-omega = 0; (3) phi almost-equal-to 0 or phi almost-equal-to pi regardless of DELTA-omega when K is large (relative to DELTA-omega); (4) results (1) to (3) hold identically for both in phase and antiphase coordination. The results also indicate that the relevant frequency is omega(c)/omega(v) rather than omega(c). Discussion highlighted the significance of confirming phi = arcsin(DELTA-omega/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.