  Within the neurosciences, to observe variability across time in the dynamics
of an underlying brain process is neither new nor unexpected. Wavelets are
essential in analyzing 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). As neurologists consider localised spectra of brain
signals to be most informative, here we develop a novel wavelet-based tool
capable to formally represent process nonstationarities across both time and
replicate dimensions. Specifically, we propose the Replicate Locally Stationary
Wavelet (RLSW) process, that captures the potential nonstationary behaviour
within and across trials. Estimation using wavelets gives a natural desired
time- and replicate-localisation of the process dynamics. We develop the
associated spectral estimation framework and establish 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, demonstrate the applicability of our
proposed methodology, as well as the new insights it provides.
