Steffen Schulz ^1* Andy Schumann ² Karl-Jürgen Bär ² Jens Haueisen ³ Georg Seifert ¹ Andreas Voss ^1,3

• ¹ Charité Competence Center for Traditional and Integrative Medicine (CCCTIM), Charité University Medicine Berlin, Berlin, Germany
• ² Lab for Autonomic Neuroscience, Imaging and Cognition (LANIC), Department for Psychosomatic Medicine and Psychotherapy, University Hospital Jena, Jena, Thuringia, Germany
• ³ University of Technology Ilmenau, Institute of Biomedical Engineering and Informatics, Ilmenau, Germany

The detection and quantification of coupling strength and direction are important aspects for a more detailed understanding of physiological regulatory processes in the field of Network Physiology. Due to the limitations of established approaches, we have developed directionality indices based on simple mathematical symbolization principles and simple computational procedures that allow a comprehensive understanding of the underlying couplings in a quick and easy way. We introduce a new directionality index (DHRJSD) 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 multivariate directionality index DmHRJSD (DmHRJSD(x,y|z), DmHRJSD(x,z|y), and DmHRJSD(y,z|x)) allowing to determine the primary driver **DmHRJSD, the secondary driver *DmHRJSD and dominant responder ‾DmHRJSD in multivariate systems that at least are weakly coupled. To validate these indices, different linear and non-linear bi-and multivariate coupled systems (Gaussian autoregressive models) with different mutual influences were generated. The simulation results showed that DHRJSD 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 DmHRJSD derived from the mHRJSD approach is able to correctly detect the driver-responder relationships in linear coupled systems. The main advantages of the new introduced directionality indices are that they are insensitive to non-stationary time series; they are able to capture couplings through a simple, fast and easy to implement symbolization procedure; they are scale invariant; they are independent of time series length, model order selection and significance level determination procedure.