Functional connectivity (FC) describes the statistical dependence between neuronal populations or brain regions in resting-state fMRI studies and is commonly estimated as the Pearson correlation of time courses. Clustering or community detection reveals densely coupled sets of regions constituting resting-state networks or functional systems. These systems manifest most clearly when FC is sampled over longer epochs but appear to fluctuate on shorter timescales. Here, we propose a new approach to reveal temporal fluctuations in neuronal time series. Unwrapping FC signal correlations yields pairwise co-fluctuation time series, one for each node pair or edge, and allows tracking of fine-scale dynamics across the network. Co-fluctuations partition the network, at each time step, into exactly two communities. Sampled over time, the overlay of these bipartitions, a binary decomposition of the original time series, very closely approximates functional connectivity. Bipartitions exhibit characteristic spatiotemporal patterns that are reproducible across participants and imaging runs, capture individual differences, and disclose fine-scale temporal expression of functional systems. Our findings document that functional systems appear transiently and intermittently, and that FC results from the overlay of many variable instances of system expression. Potential applications of this decomposition of functional connectivity into a set of binary patterns are discussed. Author Summary Numerous studies of functional connectivity have revealed densely coupled sets of brain regions corresponding to resting-state networks or functional systems. Prior work suggests that functional connectivity fluctuates over time. Here, we extend those studies by suggesting that functional connectivity can be decomposed into a set of momentary network states, with each one partitioning the network into exactly two clusters or communities. We show that these bipartitions exhibit characteristic spatiotemporal patterns that are reproducible across participants and imaging runs, and can capture individual differences. Our decomposition approach discloses fine-scale dynamics of functional systems, and reveals that functional systems coalesce and dissolve at different times and on fast timescales. Numerous applications and extensions of the approach are discussed.