Electroencephalogram (EEG) is used routinely for diagnosis of diseases occurring in the brain. It is a very useful clinical tool in the classification of epileptic seizures and the diagnosis of epilepsy. In this study, epilepsy diagnosis has been investigated using EEG records. For this purpose, an artificial neural network (ANN), widely used and known as an active classification technique, is applied. The particle swarm optimization (PSO) method, which does not need gradient calculation, derivative information, or any solution of differential equations, is preferred as the training algorithm for the ANN. A PSO-based neural network (PSONN) model is diversified according to PSO versions, and 7 PSO-based neural network models are described. Among these models, PSONN3 and PSONN4 are determined to be appropriate models for epilepsy diagnosis due to having better classification accuracy. The training methods-based PSO versions are compared with the backpropagation algorithm, which is a traditional method. In addition, different numbers of neurons, iterations/generations, and swarm sizes have been considered and tried. Results obtained from the models are evaluated, interpreted, and compared with the results of earlier works done with the same dataset in the literature.