### Eneko's Digital Garden

Assuming monoexponential decay, the voxelwise fMRI signal ($\mathrm{S}$) can be expressed in signal percentage change as: $\frac{\mathrm{S}-\overline{\mathrm{S}}}{\mathrm{S}}=\Delta \rho-\mathrm{TE} \cdot \Delta \mathrm{R}_{2}^{*}+\mathrm{n}$, where $\Delta \rho$ represents non-BOLD related changes in net magnetisation, $\Delta R_2^*$ represemts BOLD-related susceptibility changes, and $\mathrm{n}$ denotes the random noise (Kundu et al. (2012)).
Noise in fMRI can be reduced by using multi-echo (ME) acquisitions that sample the data at multiple successive echo times (TE). A weighted combination of the multiple echoes based on each voxel’s $T_2^*$ value (Posse et al. (1999)) or temporal signal-to-noise ratio (Poser et al. (2006)) can smear out random noise and enhance the sensitivity to the BOLD contrast. In fact, compared with single-echo data, this optimal combination can improve the mapping of neuronal activity at 3 Tesla (Fernandez et al. (2017)) and 7T (Puckett et al. (2018)), with results comparable to other preprocessing techniques requiring extra data such as RETROICOR (Atwi et al. (2018)).
As the BOLD-related signal can be expressed as a function of the TE whereas changes in the net magnetization are independent of TE, the information available in multiple echoes can be leveraged for the purpose of denoising. For example, in a dual-echo acquisition where the first TE is sufficiently short, the first echo signal mainly captures changes in $\Delta \rho$ rather than in $\Delta R_2^*$. It is then possible to remove artefactual effects, through voxelwise regression, from the second echo signal acquired at a longer TE with appropriate BOLD contrast (Bright and Murphy (2015)).
Collecting more echoes opens up the possibility of applying ICA and classifying independent components into BOLD-related (i.e. describing $\Delta R_{2}^{*}$ fluctuations with a linear TE-dependency) or noise (i.e. independent of TE, related to fluctuations in the net magnetization $\Delta \rho$), an approach known as multi-echo independent component analysis (ME-ICA) (Kundu et al. (2013); Kundu et al. (2012); Kundu et al. (2017)). *