Higher order definition of causality by optimally conditioned transfer entropy.

The description of the dynamics of complex systems, in particular the capture of the interaction structure and causal relationships between elements of the system, is one of the central questions of interdisciplinary research. While the characterization of pairwise causal interactions is a relatively ripe field with established theoretical concepts and the current focus is on technical issues of their efficient estimation, it turns out that the standard concepts such as Granger causality or transfer entropy may not faithfully reflect possible synergies or interactions of higher orders, phenomena highly relevant for many real-world complex systems. In this paper, we propose a generalization and refinement of the information-theoretic approach to causal inference, enabling the description of truly multivariate, rather than multiple pairwise, causal interactions, and moving thus from causal networks to causal hypernetworks. In particular, while keeping the ability to control for mediating variables or common causes, in case of purely synergistic interactions such as the exclusive disjunction, it ascribes the causal role to the multivariate causal set but not to individual inputs, distinguishing it thus from the case of, e.g., two additive univariate causes. We demonstrate this concept by application to illustrative theoretical examples as well as a biophysically realistic simulation of biological neuronal dynamics recently reported to employ synergistic computations.