Partial phase synchronization is an essential behavior of the brain. The lack or excess of phase synchronization is associated with brain disorders such as epilepsy, Alzheimer's, autism, Parkinson's disease, among others. These disorders may be related to the malfunctioning of the synchronization process among neurons, triggered by changes in the local dynamics of these cells. In some cases, medications are used to modify this local dynamics, adjusting the synchronization process as needed. In this work, we show that there are two distinct mechanisms that lead to phase-synchronized states. The first mechanism is strongly influenced by the individual dynamics of neurons, allowing synchronization to occur at low coupling values, regardless of the network topology. The second synchronized state is induced by network coupling, where the coupling strength promotes globally synchronized states, known as network-driven synchronization. We report that individual characteristics of the local dynamics of neurons, such as their linear stability, when coupled in a network, can play a crucial role in the phase synchronization process depending on the coupling strength. Global and small-world topologies were considered for a network of Hindmarsh–Rose neurons. In both coupling schemes, the effects of local dynamics are clear, either advancing or delaying the occurrence of partial phase synchronization of the network as the coupling strength varies. In this context, we discuss the importance of the local dynamics of neurons, showing that it can be fundamental for understanding and controlling the phase synchronization process in networks. This study also provides valuable insights for the general understanding of phase synchronization processes in networks.