This article investigates the synchronization problem of multiple memristive neural networks (MMNNs) in the case of switching communication topologies and parameter mismatch. First, the distributed event-triggered control under continuous sampling conditions is studied. Then, a periodic event-triggered control (PETC) model is proposed to substantially reduce control consumption. Using the Lyapunov method, the properties of M-matrix, and some inequalities, the sufficient criteria of synchronous control are derived. The results can be used in the analysis of other multiagent nonlinear systems. A norm-based threshold function is given to determine the update time of the controller, and it is proved that the trigger condition excludes the Zeno behavior. Subject to parameter mismatch, a quasisynchronous control strategy is proposed, which can be extended to complete synchronization provided that the system mismatch or disturbance disappears. It is worth mentioning that this article introduces the signal function into the controller, so that the theoretical error can be limited to an arbitrarily small range. Furthermore, this new controller is used in the PETC strategy which automatically avoids the Zeno behavior. Finally, one example is given to illustrate our results.