Delay-induced multistable synchronization of biological oscillators

We analyze the dynamics of pulse coupled oscillators depending on strength and delay of the interaction. For two oscillators, we derive return maps for subsequent phase differences, and construct phase diagrams for a broad range of parameters. In-phase synchronization proves stable for inhibitory coupling and unstable for excitatory coupling if the delay is not zero. If the coupling strength is high, additional regimes with marginally stable synchronization are found. Simulations with $N\gg 2$ oscillators reveal a complex dynamics including spontaneous synchronization and desynchronization with excitatory coupling, and multistable phase clustering with inhibitory coupling. We simulate a continuous description of the system for $\overrightarrow{N}\infty$ oscillators and demonstrate that these phenomena are independent of the size of the system. Phase clustering is shown to relate to stability and basins of attraction of fixed points in the return map of two oscillators. Our findings are generic in the sense that they qualitatively are robust with respect to modeling details. We demonstrate this using also pulses of finite rise time and the more realistic model by Hodgkin and Huxley which exhibits multistable synchronization as predicted from our analysis as well.