Astrocytes have an important role in the indirect regulation of neuronal excitability. The abnormalities of their ion channels cause neurons to discharge abnormally, which may induce seizures. The inwardly rectifying potassium channel 4.1 (Kir4.1 channel) and the voltage-gated calcium channel (VGCC) of an astrocyte play important roles in maintaining the homeostasis of these potassium and calcium ions, and have been found to be associated with seizures. However, the underlying mechanisms by which they induce seizures remain unclear. This paper established a neuron-astrocyte network model, which is a model consisting of a neuron and an astrocyte, to explore some mechanisms of epileptic seizures. Through a series of simulations based on this model, the results showed that low conductance of Kir4.1 channel can induce spontaneous periodic epileptic activity (SPEA) whereas higher conductance results in spontaneous periodic bursting event (SPBE) and high-frequency tonic discharges (HFTD). The abnormalities of VGCC also lead to the generation of SPEA and SPBE. As the changes of potassium concentration in the largest nearby reservoir which is analogous to a bath solution that contains a specific concentration of potassium, SPEA can undergo a process from appearance to disappearance. Thus, the research findings showed that the transitions of seizure-like discharges provide further theoretical analyses to clarify the complex mechanism of seizures.