In this paper, a tripartite synapse network is constructed to examine external and internal triggering factors of epilepsy transition and propagation in neurons with the Epileptor-2 model. We first explore the external stimuli in the environment that induce epileptic activities and transition behaviors among Ictal Discharges (IDs) and Interictal Discharges (IIDs) states. The higher the strength and abruptness of the stimuli, the more severe is the occurrence of epilepsy within a reasonable range of parameters. Then for the internal triggering factors, the results of the tripartite synapse network, which is improved by combining the Epileptor-2 model with astrocyte by means of ion exchange and new connections, show that astrocytes can transmit normal physiological activity information and filter out abnormal discharge information of neurons. One of the causes for epileptic seizures is the abnormal release of glial neurotransmitters in astrocytes. The excessive release of glutamate causes the discharge state of neurons to transit from nonepileptic to IIDs, IDs and tonic, while adenosine triphosphate can alleviate epilepsy. Meanwhile, the synapse dysfunction of an astrocyte-free network can also lead to seizures, and the epilepsy propagation ability of a tripartite synapse network becomes weaker than that of an astrocyte-free network. Our research is expected to provide some theoretical basis for the therapeutic approach to curing epilepsy in the intracellular and extracellular contexts.