Many evidences have shown that synchronization takes a key role in brain functions, but most of its theoretical studies are so far only focused on the global brain network. However, it is well-known that all the cognitive tasks are performed by individual local cognitive subnetworks but not the global brain network, thus it is necessary to study the synchronization of local brain networks. We here focus on this problem by considering the synchronization transition of all the 63 local subnetworks of a real brain network and interestingly find that these local networks of cortical regions may show a completely different synchronization transition from that of global brain network, i.e., non-monotonic order parameter with folding. Moreover, we find that the forward and backward processes show different synchronization transitions, such as one being the first-order synchronization transition and the other being the second-order synchronization transition, and even amplitude death, depending on the different topologies of local networks. By extensive numerical simulations, we show that the amplitude death is prefer to occur in the local networks with larger size but not prefer to occur in the local networks with smaller size. A brief theoretical analysis is presented to explain the mechanism of amplitude death.