Background: Quantifying interactions among many neurons is fundamental to understanding system-level phenomena such as attention, learning and even conscious experience. Causal influences in the brain, quantified as integrated information, are thought to support subjective conscious experience. Recent empirical work has shown that the spectral decomposition of causal influences, for example using Granger causality, can reveal frequency-specific influences that are not observed in the time domain. However, a spectral decomposition of integrated information has not been put forward, limiting its adoption for analyzing neural data. New method: We present a general and flexible framework for deriving the spectral decomposition of causal influences in autoregressive processes. Results: We use the framework to derive a spectral decomposition of integrated information. We show that other well-known measures, including Granger causality, can be derived using the same framework. Using simulations, we demonstrate a complex interplay between the spectral decomposition of integrated information and other measures that is not observed in the time domain. Comparison with existing methods: This paper provides a spectral decomposition of integrated information for the first time. Although a spectral decomposition of Granger causality has been derived, that approach is only applicable to uni-directional causal influences, not multi-directional causal influences as required for integrated information. Conclusions: Our novel framework can be used to derive the spectral decomposition of uni- and multi-directional measures of causal influences. We use this framework to derive a spectral decomposition of integrated information, paving the way for better understanding how frequency-specific causal influences in the brain relate to cognition.
- Granger causality
- Integrated information
- Integrated information theory
- Spectral decomposition
- Transfer entropy