Revisiting Wiener's principle of causality -- interaction-delay reconstruction using transfer entropy and multivariate analysis on delay-weighted graphs.

To understand the function of networks we have to identify the structure of their interactions, but also interaction timing, as compromised timing of interactions may disrupt network function. We demonstrate how both questions can be addressed using a modified estimator of transfer entropy. Transfer entropy is an implementation of Wiener's principle of observational causality based on information theory, and detects arbitrary linear and non-linear interactions. Using a modified estimator that uses delayed states of the driving system and independently optimized delayed states of the receiving system, we show that transfer entropy values peak if the delay of the state of the driving system equals the true interaction delay. In addition, we show how reconstructed delays from a bivariate transfer entropy analysis of a network can be used to label spurious interactions arising from cascade effects and apply this approach to local field potential (LFP) and magnetoencephalography (MEG) data.