Measuring dynamical interaction from data

Dynamical interactions, such as between brain and behavior, are ubiquitous in nature. However, measuring such coupling from data is challenging because the underlying interaction can be dynamic, i.e., depend on the states of the systems, and the observation may be incomplete, e.g., only a subset of variables are observable. A key idea to overcome such problems is to evaluate the mutual predictability of individually reconstructed phase spaces - cross-embedding. When applied to real data however, the quantification of coupling through cross-embedding is complicated by the multiple ways to implement and evaluate the mutual prediction. Here, we introduce a new approach, the mutual information between individually partitioned state spaces, with which we can describe the state-dependent coupling by the phase space density of one system conditioned by the state of the other. We apply this approach to coupled Rössler systems, where the underlying interaction process is successfully detected from incomplete observation.