  One of the most well-established tools for modeling the brain as a complex
system is the functional connectivity network, which examines the correlations
between pairs of interacting brain regions. While powerful, the network model
is limited by the restriction that only pairwise dependencies are visible and
potentially higher-order structures are missed. In this work, we explore how
multivariate information theory can reveal higher-order, synergistic
dependencies in the human brain. Using the O-information, a measure of whether
the structure of a system is redundancy- or synergy-dominated, we show that
synergistic subsystems are widespread in the human brain. We provide a
mathematical analysis of the O-information to locate it within a larger
taxonomy of multivariate complexity measures. We also show the O-information is
related to a previously established measure, the Tononi-Sporns-Edelman
complexity, and can be understood as an expected difference in integration
between system scales. Highly synergistic subsystems typically sit between
canonical functional networks, and may serve to integrate those networks. We
then use simulated annealing to find maximally synergistic subsystems, finding
that such systems typically comprise $\approx$10 brain regions, also recruited
from multiple canonical brain systems. Though ubiquitous, highly synergistic
subsystems are invisible when considering pairwise functional connectivity,
suggesting that higher-order dependencies form a kind of ``shadow structure"
that has been unrecognized by established network-based analyses. We assert
that higher-order interactions in the brain represent a vast and under-explored
space that, made accessible with tools of multivariate information theory, may
offer novel scientific insights.
