We explore and illustrate the use of time series decomposition methods for evaluating and comparing latent structure in nonstationary electroencephalographic (EEG) traces obtained from depressed patients during brain seizures induced as part of electroconvulsive therapy (ECT). Analysis of the patterns of change over time in the frequency structure of such EEG data provides insight into the neurophysiological mechanisms of action of this effective but poorly understood antidepressant treatment, and allows clinicians to modify ECT treatments to optimize therapeutic benefits while minimizing associated side effects. Our work has introduced new methods of time-frequency analysis of EEG series that identify the complete pattern of time evolution of frequency structure over the course of a seizure, and usefully assist in these scientific and clinical studies. New methods of decomposition of flexible dynamic models provide time domain decompositions of individual EEG series into collections of latent components in different frequency bands. This allows us to explore ECT seizure characteristics via inferences on the time-varying parameters that characterize these latent components, and to relate differences in such characteristics across seizures to differences in the therapeutic effectiveness and cognitive side effects of those seizures. This article discusses the scientific context and problems, development of nonstationary time series models and new methods of decomposition to explore time-frequency structure, and aspects of model fitting and analysis. We include applied studies on two datasets from recent clinical ECT studies. One is an initial illustrative analysis of a single EEG trace, the second compares the EEG data recorded during two types of ECT treatment that differ in therapeutic effectiveness and cognitive side effects. The uses of these models and time series decomposition methods in extracting and contrasting key features of the seizure underlying the EEG signals are highlighted. Through the use of these models we have quantified, for the first time, decreases in the dominant frequencies of low-frequency EEG components during ECT seizures. We have also identified preliminary evidence that such decreases are enhanced under the more effective ECTs at higher electrical dosages, a finding consistent with prior reports and the hypothesis that more effective forms of ECT are more effective in eliciting neurophysiological inhibitory processes.