A new directionality index based on high-resolution joint symbolic dynamics to assess information transfer in multivariate networks

The detection and quantification of coupling strength and direction are important aspects for achieving a deeper understanding of physiological regulatory processes in the field of network physiology. Due to the limitations of established approaches, we developed directionality indices based on simple mathematical symbolization principles and simple computational procedures that allow a quick and comprehensive understanding of the underlying couplings. We introduced a new directionality index (<i>D</i> _HRJSD) derived from the pattern family density matrix of the High-Resolution Joint Symbolic Dynamics (HRJSD) approach and its multivariate version (mHRJSD) to determine coupling direction and driver-response relationships. The mHRJSD approach contains the multivariate directionality index <i>D</i> _mHRJSD (<i>D</i> _mHRJSD(<i>x</i>,<i>y</i>|<i>z</i>), <i>D</i> _mHRJSD(<i>x</i>,<i>z</i>|<i>y</i>), and <i>D</i> _mHRJSD(<i>y</i>,<i>z</i>|<i>x</i>)), allowing us to determine the primary driver <i>**D</i> _mHRJSD, the secondary driver <i>*D</i> _mHRJSD, and the dominant responder ^- <i>D</i> _mHRJSD in multivariate systems that are at least weakly coupled. Different linear and non-linear bi- and multivariate coupled systems (Gaussian autoregressive models) with different mutual influences were generated to validate these indices. The simulation results showed that <i>D</i> _HRJSD was able to correctly detect the dominant coupling direction in linear bivariate coupled systems but was partly able to detect the dominant coupling direction in non-linear bivariate coupled systems. The proposed directionality index <i>D</i> _mHRJSD derived from the mHRJSD approach is able to correctly detect the driver-responder relationships in linear coupled systems. The main advantages of the newly introduced directionality indices include their insensitivity to non-stationary time series, their ability to capture couplings through a simple, fast, and easy-to-implement symbolization procedure, and their scale invariance. Additionally, they are independent of time series length, model order selection, and the procedure for determining their significance level.