Here, we report that explosive synchronization can be observed in a network when the coupling between nodes is biased by their degrees. Moreover, we observe that there is a transition from explosive (first-order) to classical (second-order) synchronization in a branched acyclic system once a cycle emerges in the system. This transition represents a potential mechanism with which a neuronal system can naturally control a switching system between states of explosive and classical synchronization such that it can avoid pathological states like epilepsy or chronic pain.