This paper investigates the problem of event-triggered control for the synchronization of networks of nonlinear dynamical agents; distributed model-based approaches able to guarantee the synchronization of the overall system are derived. In these control schemes all the agents use a model of their neighbourhood in order to generate triggering instants in which the local controller is updated and, if needed, local information based on the adopted control input is broadcasted to neighbouring agents, Synchronization of the network is proved and the existence of Zeno behaviour is excluded; an event triggered strategy able to guarantee the existence of a minimum lower bound between inter-event times for broadcasted information and for control signal updating is proposed, thus allowing applications where both the communication bandwidth and the maximum updating frequency of actuators are critical. This idea is further extended in an asynchronous periodic event-triggered schemes where the agents check a trigger condition via a periodic distributed communication without requiring a model based computation.