"Doing a multiple regression with them all, I assume that we look at regions where BOLD signal significantly correlates with each given variable after taking into account the contribution of other covariates."
Yes; this is a standard fact about the way regression works.
A commonly cited paper in the context of neuroimaging is "Ambiguous Results in Functional Neuroimaging Data Analysis Due to Covariate Correlation," Andrade et al., NeuroImage 10, 483–486 (1999).
For a general statistical reference, see e.g. Neter et al., _Applied Linear Statistical Models_, 4th ed., p. 294, "Effects of simultaneous tests of beta_k."
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