A Nonlinear Effective Connectivity Measure Based on Granger Causality and Volterra Series

Estimating effective connectivity, especially in brain networks, is an important topic to find out the brain functions. Various effective connectivity measures are presented, but they have drawbacks, including bivariate structure, the problem in detecting nonlinear interactions, and high computational cost. In this paper, we have proposed a novel multivariate effective connectivity measure based on a hierarchical realization of the Volterra series model and Granger causality concept, namely hierarchical Volterra Granger causality (HVGC). HVGC is a multivariate connectivity measure that can detect linear and nonlinear causal effects. The performance of HVGC is compared with Granger causality index (GCI), conditional Granger causality index (CGCI), transfer entropy (TE), phase transfer entropy (Phase TE), and partial transfer entropy (Partial TE) in simulated and physiological datasets. In addition to accuracy, specificity, and sensitivity, the Matthews correlation coefficient (MCC) is used to evaluate the connectivity estimation in simulated datasets. Furthermore influence of different SNRs is investigated on the estimated connectivity. The obtained results show that HVGC with a minimum MCC of 0.76 performs well in the detection of both linear and nonlinear interactions in simulated data. HVGC is also applied to a physiological dataset that was cardiorespiratory interaction signals recorded during sleep from a patient suffering from sleep apnea. The results of this dataset also demonstrate the capability of the proposed method in the detection of causal interactions. Applying HVGC on the simulated fMRI dataset led to a high MCC of 0.78. Moreover, the results indicate that HVGC has slight changes in different SNRs. The results indicate that HVGC can estimate the causal effects of a linear and nonlinear system with a low computational cost and it is slightly affected by noise.