Emergence of oscillations in a model of weakly coupled two Bonhoeffer-van der Pol equations

Bifurcations of periodic solutions in a model of weakly coupled two Bonhoeffer-van der Pol equations are studied. The model realizes a half-center model with reciprocal inhibition, a typical model used in the field of neural motor control to account for the generation of alternating rhythmic bursts observed in motoneurons and spinal neural networks. Several oscillatory solutions such as in-phase, anti-phase as well as out-of-phase solutions emerge from the model's equilibrium as one of the parameters of the model changes. Among the variety of bifurcations exhibited by the model, we analyze Hopf bifurcations, by which several periodic solutions emerge, and illustrate generation mechanisms of alternating oscillations in the model.