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This paper presents an approach to detect epileptic seizure segments in the neonatal electroencephalogram (EEG) by characterizing the spectral features of the EEG waveform using a rule-based algorithm cascaded with a neural network. A rule-based algorithm screens out short segments of pseudosinusoidal EEG patterns as epileptic based on features in the power spectrum. The output of the rule-based algorithm is used to train and compare the performance of conventional feedforward neural networks and quantum neural networks. The results indicate that the trained neural networks, cascaded with the rule-based algorithm, improved the performance of the rule-based algorithm acting by itself. The evaluation of the proposed cascaded scheme for the detection of pseudosinusoidal seizure segments reveals its potential as a building block of the automated seizure detection system under development.