Partial phase synchronization, also reported as neuron cooperation, is a pivotal behavior of the brain and related to its main features, such as memory. We report here how individual characteristics of the local dynamics of neurons coupled in a network may be a fundamental player in the phase synchronization process. Global and small-world topologies are considered for a Hindmarsh-Rose-neuron network. For both coupling schema, the effects of the local dynamics are clear, inducing early or retarding the occurrence of partial phase-synchronization of the network when the coupling strength is varied. The excess or even the lack of phase synchronization are associated to brain disorders like Parkinson’s disease and epilepsy. These diseases may be related to malfunctioning of the synchronization process of the neurons, triggered by alterations of the ion-channel conductance properties. In fact, it is common the use of drugs to block or activate specific channels, changing the conductances and bringing the synchronization process to some desired behavior. In this scenario, we discus about the effect of local dynamics of the neuron, showing that it may of fundamental importance to understand and control the process of the network phase synchronization. The study brings useful information to the general understanding of network phase synchronization process.