Measuring complex causality and high-order interactions: An extended transfer entropy spectrum approach

Classical causal networks between complex systems are constructed based on causal interactions between pairs of systems. However, complex causal relationships involving three or more systems (e.g., indirect causality, common causality, and synergistic causal effects) and between specific system components (e.g., low-frequency components and high-frequency components) tend to have important effects on the systems. In this paper, we propose Fourier-domain conditional transfer entropy spectrum and Fourier-domain high-order transfer entropy spectrum to measure complex causality at different frequencies and different time periods. The former can distinguish indirect causality and common causality, and the latter can identify synergistic causal effects. Using this method, we construct a direct causal structure matrix that enables the observation of the causal structure of multiple systems across various frequency bands and time periods. This approach is validated through experiments using synthetic data. The application of our method to physiological networks described by polysomnography highlights its ability to detect latent causality within specific frequency bands, providing insights into the complex causal relationships inherent in sleep stages. Notably, our approach is effective in identifying direct causality within intricate causal networks, showcasing its practical applicability in understanding physiological interactions. Furthermore, the study of brain activity during motor imagery or motor execution demonstrates that, when extended to multivariate systems, the method can identify synergistic effects among different brain regions, emphasizing its potential in neuroimaging applications. These findings highlight the potential of this approach for clinical applications, particularly in understanding the biological significance of causal interactions across different frequency bands.