  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.
