How synaptic weights determine stability of synchrony in networks of pulse-coupled excitatory and inhibitory oscillators

Under which conditions can a network of pulse-coupled oscillators sustain stable collective activity states? Previously, it was shown that stability of the simplest pattern conceivable, i.e., global synchrony, in networks of symmetrically pulse-coupled oscillators can be decided in a rigorous mathematical fashion, if interactions either all advance or all retard oscillation phases (“mono-interaction network”). Yet, many real-world networks—for example neuronal circuits—are asymmetric and moreover crucially feature both types of interactions. Here, we study complex networks of excitatory (phase-advancing) and inhibitory (phase-retarding) leaky integrate-and-fire (LIF) oscillators. We show that for small coupling strength, previous results for mono-interaction networks also apply here: pulse time perturbations eventually decay if they are smaller than a transmission delay and if all eigenvalues of the linear stability operator have absolute value smaller or equal to one. In this case, the level of inhibition must typically be significantly stronger than that of excitation to ensure local stability of synchrony. For stronger coupling, however, network synchrony eventually becomes unstable to any finite perturbation, even if inhibition is strong and all eigenvalues of the stability operator are at most unity. This new type of instability occurs when any oscillator, inspite of receiving inhibitory input from the network on average, can by chance receive sufficient excitatory input to fire a pulse before all other pulses in the system are delivered, thus breaking the near-synchronous perturbation pattern.