Multiscale spectral modelling for nonstationary time series within an ordered multiple-trial experiment

Within the neurosciences it is natural to observe variability across time in the dynamics of an underlying brain process. Wavelets are essential in analysing brain signals because, even within a single trial, brain signals exhibit nonstationary behaviour. However, neurological signals generated within an experiment may also potentially exhibit evolution across trials (replicates), even for identical stimuli. As neurologists consider localised spectra of brain signals to be most informative, we propose the MULtiple-Trials Locally Stationary Wavelet process (MULT-LSW) that fills the gap in the literature by directly giving a stochastic wavelet representation of the time series of ordered replicates itself. MULT-LSW yields a natural desired time- and trial-localisation of the process dynamics, capturing nonstationary behaviour both within and across trials. While current techniques are restricted by the assumption of uncorrelated replicates, here we account for between-trial correlation. We rigorously develop the associated wavelet spectral estimation framework along with its asymptotic properties. By means of thorough simulation studies, we demonstrate the theoretical estimator properties hold in practice. A real data investigation into the evolutionary dynamics of the hippocampus and nucleus accumbens, during an associative learning experiment, demonstrates the applicability of our proposed methodology as well as the new insights it provides. Our model is general and facilitates wider experimental data analysis than the current literature allows.