Detecting event-related time-dependent directional couplings

Nonlinear interdependence measures can be used to detect directional couplings between stationary dynamical systems from a pair of signals measured from them. For many dynamics, however, intermittent directional couplings arise in causal relation to distinct events on timescales that are often too short to be resolved by nonlinear interdependence measures. On the other hand, in many experimental settings signals are measured for multiple instances of such events. We demonstrate how these multiple realizations can be exploited to reliably detect event-related time-dependent directional couplings. For this purpose, we propose the general concept of time-resolved causal statistics derived from embeddings across multiple realizations of time-dependent dynamics. Surrogates constructed by permuting the order of realizations can be used to test specified null hypotheses. We adapt a conventional nonlinear interdependence measure to serve as a time-resolved causal statistic and apply it to exemplary coupled Lorenz dynamics. This approach allows detecting event-related time-dependent directional couplings based on only a few tens of realizations. Changes of the coupling direction can be detected within one oscillation of the dynamics. Beyond this particular application, any metric bivariate or univariate measure can be adapted to serve as time-resolved causal statistics to characterize various aspects of event-related time-dependent dynamics.