Adaptive coupling induced recurrence of phase coalescence

It is well known that there are various rhythms in human brain and the brain system can quickly switch from one rhythm to another, especially in the resting-state of brain, but its mechanism is still not completely understood. To go a further step toward this problem, here we present an adaptive model of the Kuramoto oscillators from the angle of phase coalescence. We find that the adaptive coupling can induce a recurrence of spike-like phase coalescence for the case that the non-adaptive coupling does not allow for such a phase coalescence, and the transient time T_tr for the appearance of phase coalescence depends on the coupling strength K₀ in a power law relationship. We show that this feature of spike-like phase coalescence favors the fast switching between different frequency combs. Further, we find that the spike-like recurrence of phase coalescence will become shoulder-like when the coupling is nearby the critical point, and the shoulder width T_s depends on the distance to the critical point in another power law. Then, we extend the model to the case of biharmonic coupling function and find that the transient time T_tr can be adjusted by the parameter γ and there is an optimal γ. Moreover, a rigorous theoretical analysis is presented to explain these results.