Coexistence of regular and irregular dynamics in complex networks of   pulse-coupled oscillators
  For general networks of pulse-coupled oscillators, including regular, random, and more complex networks, we develop an exact stability analysis of synchronous states. As opposed to conventional stability analysis, here stability is determined by a multitude of linear operators. We treat this multi-operator problem analytically and show that for inhibitory interactions the synchronous state is stable, independent of the parameters and the network connectivity. In randomly connected networks with strong interactions this synchronous state, displaying \textit{regular} dynamics, coexists with a balanced state that exhibits \textit{irregular} dynamics such that external signals may switch the network between qualitatively distinct states. 
Marc Timme, Fred Wolf, and Theo Geisel