Matrix perturbation analysis for weakly coupled oscillators

The central pattern generator for controlling animal locomotion is generally not a single oscillator but a group of interconnected oscillators. Although the whole group can be analyzed as a bulky single oscillator, it is difficult with such an approach to reveal the essential mechanism of phase coordination among oscillators. In this paper, a new approach based on the multivariable harmonic balance is used to analyze a group of identical oscillators that are weakly coupled to each other. Specifically, a lower dimensional matrix representing the coupling structure is abstracted from the overall neuronal connectivity matrix of large dimension, so that the coupling matrix captures the essential mechanism for phase coordination of the oscillators. Moreover, this framework provides an effective way to design the neuronal interconnections to achieve prescribed coordination among the oscillators.