Phase models and oscillators with time delayed coupling

We consider two identical oscillators with time delayed coupling, modelled by a system of delay differential equations. We reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. By analyzing the phase model, we show how the time delay affects the stability of phase-locked periodic solutions and causes stability switching of in-phase and anti-phase solutions as the delay is increased. In particular, we show how the phase model can predict when the phase-flip bifurcation will occur in the original delay differential equation model. The results of the phase model analysis are applied to pairs of Morris-Lecar oscillators with diffusive or synaptic coupling and compared with numerical studies of the full system of delay differential equations.