We introduce a novel, transient model for the electroencephalogram (EEG) as the noisy addition of linear filters responding to trains of delta functions. We set the synthesis part as a parameter-tuning problem and obtain synthetic EEG-like data that visually resembles brain activity in the time and frequency domains. For the analysis counterpart, we use sparse approximation to decompose the signal in relevant events via Matching Pursuit. We improve this algorithm by incorporating the Gini Index as a stopping criteria; in this way, we promote sparse sources while, at the same time, eliminating one of the free parameters of Matching Pursuit. Results are presented using synthetic EEG and BCI competition data. Statistics of the model parameters are more informative and posses finer temporal resolution than classical methods such as Power Spectral Density (PSD) estimation.