Control of desynchronization transitions in delay-coupled networks of type-I and type-II excitable systems

We discuss synchronization and desynchronization transitions in networks of delay-coupled excitable systems. These transitions arise in response to varying the balance of excitatory and inhibitory couplings in a small-world topology. To describe the local dynamics, we use generic models for type-I excitability, which arises close to a saddle-node bifurcation on an invariant cycle (SNIC or SNIPER), and for type-II excitability, which occurs close to a Hopf bifurcation (FitzHugh-Nagumo model). For large delay times both type-I and type-II systems behave in a similar way. This is different for small delay times, where in case of type-I excitability we find novel multiple transitions between synchronization and desynchronization, when the fraction of inhibitory links is increased. In contrast, only a single desynchronization transition occurs for the FitzHugh-Nagumo model (type-II excitability) for all values of the delay time.