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While the standard network description of complex systems is based onquantifying links between pairs of system units, higher-order interactions(HOIs) involving three or more units play a major role in governing thecollective network behavior. This work introduces an approach to quantifypairwise and HOIs for multivariate rhythmic processes interacting acrossmultiple time scales. We define the so-called O-information rate (OIR) as a newmetric to assess HOIs for multivariate time series, and propose a framework todecompose it into measures quantifying Granger-causal and instantaneousinfluences, as well as to expand it in the frequency domain. The frameworkexploits the spectral representation of vector autoregressive and state-spacemodels to assess synergistic and redundant interactions among groups ofprocesses, both in specific bands and in the time domain after whole-bandintegration. Validation on simulated networks illustrates how the spectral OIRcan highlight redundant and synergistic HOIs emerging at specific frequenciesbut not using time-domain measures. The application to physiological networksdescribed by heart period, arterial pressure and respiration measured inhealthy subjects during paced breathing, and to brain networks described byECoG signals acquired in an animal experiment during anesthesia, document thecapability of our approach to identify informational circuits relevant towell-defined cardiovascular oscillations and brain rhythms and related tospecific physiological mechanisms of autonomic control and alteredconsciousness. The proposed framework allows a hierarchically-organizedevaluation of time- and frequency-domain interactions in networks mapped bymultivariate time series, and its high flexibility and scalability make itsuitable to investigate networks beyond pairwise interactions in neuroscience,physiology and other fields.

A Framework for the Time- and Frequency-Domain Assessment of High-Order Interactions in Brain and Physiological Networks