Frequency Shifting and Induced Stability in Systems of Asymmetrically Coupled Oscillators*

The Haken-Kelso-Bunz (HKB) equations describe bistable rhythmic coordination phenomena, which are ubiquitous in biophysical motor, neural, and social systems. Although originally formulated for pairs of coupled oscillators, the HKB model has recently been generalized to larger systems of oscillators with diverse natural frequencies. Existing work on the generalized HKB model has been mostly restricted to the case of symmetric coupling, where any two coupled oscillators equally influence each other's dynamics. However, in natural systems such as social systems and the brain, influences between components are often unequal. In the present work, we generalize our previous work by allowing each oscillator's coupling strength to be asymmetrical, where a given oscillator may be more sensitive to the influence of its partners than vice versa. We find that asymmetric coupling changes the collective dynamics of the oscillator system, shifting the frequency at which the system coordinates. We also show that the phenomenon of induced stability, whereby subsystems of oscillators can be sustained in coordination patterns that would not be stable in isolation, is robust in the presence of asymmetric coupling. Finally, we discuss some applications of this theoretical work in gerontology and neurostimulation.