On Neuroimaging Analysis
But being highlighted means being active (at least more active than other regions)
This is not the case in functional connectivity analysis, or indeed not necessarily the case in any sort of functional neuroimaging analysis. The colormap highlighting in such images as you referred to (see the primary reference for that particular image) reflect statistic scores, commonly z- or t-statistics. Correlation simply means that time-courses behave similarly over time, and the color coding tells you how strongly they correlate with the “common profile” of the cluster. This makes no direct statement regarding the total “activation level“ over time, as represented by e.g. the area under the curve of the timecourse. In fact, such a metric would be meaningless, since fMRI signals are de-meaned so that they fluctuate around zero (as seen e.g. in the timecourses depicted here). Functional connectivity networks thus do not imply in whatever fashion a higher level of activity in highlighted networks. For further visual clarification you can consult Figure 1. of this article showing that functional connectivity networks can be and often are active at the same amplitudes and have similar levels of “total activity“ as visually inferred from the integral of the power spectra. This integral analysis is not presented, however, as it is meaningless.
On Inhibitory Functional Networks
Whether functional connectivity networks are implied to be excitatory depends on the exact mode of analysis. If the analysis is truly based only on detecting large correlation scores, identified networks will not resolve inhibitory projections, as such connections would cause anticorrelation rather than correlation. It could still be that a functional connectivity network is based on multiple areas receiving strong inhibitory input from an upstream node, yet that node would not be part of the network. Consequently, proficient functional connectivity analysis will identify networks via methods such as independent component analysis (ICA), which are able to group both strong correlation and anticorrelation together, as seen here.
Yes it can, it often is, and it's generally highlighted in blue.