Coupled Systems Of Fast And Slow Oscillators As A Model Of Respiratory Networks

We propose a coupled system of fast and slow phase oscillators.We observe two-step transitions to quasi-periodic motions by direct numerical simulations of this coupled oscillator system.A low-dimensional equation for order parameters is derived using the Ott-Antonsen ansatz.The applicability of the ansatz is checked by the comparison of numerical results of the coupled oscillator system and the reduced low-dimensional equation.We investigate further several interesting phenomena in which mutual interactions between the fast and slow oscillators play an essential role.Fast oscillations appear intermittently as a result of excitatory interactions with slow oscillators in a certain parameter range.Slow oscillators experience an oscillator-death phenomenon owing their interaction with fast oscillators.This oscillator death is explained as a result of saddle-node bifurcation in a simple phase equation obtained using the temporal average of the fast oscillations.Finally,we show macroscopic synchronization of the order 1:m between the slow and fast oscillators.