Epilepsy is one of the most common brain disorders worldwide, affecting millions of people every year. Although significant effort has been put into better understanding it and mitigating its effects, the conventional treatments are not fully effective. Advances in computational neuroscience, using mathematical dynamic models that represent brain activities at different scales, have enabled addressing epilepsy from a more theoretical standpoint. In particular, the recently proposed Epileptor model stands out among these models, because it represents well the main features of seizures, and the results from its simulations have been consistent with experimental observations. In addition, there has been an increasing interest in designing control techniques for Epileptor that might lead to possible realistic feedback controllers in the future. However, such approaches rely on knowing all of the states of the model, which is not the case in practice. The work explored in this letter aims to develop a state observer to estimate Epileptor's unmeasurable variables, as well as reconstruct the respective so-called bursters. Furthermore, an alternative modeling is presented for enhancing the convergence speed of an observer. The results show that the proposed approach is efficient under two main conditions: when the brain is undergoing a seizure and when a transition from the healthy to the epileptiform activity occurs.