Disclaimer: I'm afraid I haven't done a very good job reviewing the entirety of this paper because the stats/processing part was pretty much opaque to me. I've done my best to glean what I can out of it, and then I've focused as much as I can on the acquisition, since that is one part where I can penetrate the text and offer some useful commentary. Perhaps someone with better knowledge of stats/ICA/processing will review those sections elsewhere.
The last paper I reviewed used a bias field map to attempt to correct for some of the effects of subject motion in time series EPI. A different approach is taken by Prantik Kundu et al. in another recently published study. In their paper, Differentiating BOLD from non-BOLD signals in fMRI time series using multi-echo EPI, Kundu et al. set out to differentiate between signal changes that have a plausible neurally-driven BOLD origin from those that are likely to have been modulated by something other than neuronal activity. In the latter category we have cardiac and respiratory fluctuations and, of course, subject motion.
The method involves sorting BOLD-like from spurious changes using an independent component analysis (ICA) and to then "de-noise" the time series before applying connectivity analysis. For resting state fMRI in particular, the lack of any sort of ground truth and an absence of independent knowledge that one has with task-based fMRI makes disambiguating neurally driven signal changes from artifacts a major problem. Kundu et al. use a relatively simple philosophical approach to the separation:
"We hypothesized that if TE-dependence could be used to differentiate BOLD and non-BOLD signals, non-BOLD signal could be removed to denoise data without conventional noise modeling. To test this hypothesis, whole brain multi-echo data were acquired at 3 TEs and decomposed with Independent Components Analysis (ICA) after spatially concatenating data across space and TE. Components were analyzed for the degree to which their signal changes fit models for R2* and S0 change, and summary scores were developed to characterize each component as BOLD-like or not BOLD-like."
And, noting again the caveat that there is an absence of ground truth, the approach seems to work:
"These scores clearly differentiated BOLD-like “functional network” components from non BOLD-like components related to motion, pulsatility, and other nuisance effects. Using non BOLD-like component time courses as noise regressors dramatically improved seed-based correlation mapping by reducing the effects of high and low frequency non-BOLD fluctuations."