Abstract
While Granger Causality (GC) has been often employed in network neuroscience, most GC applications are based on linear multivariate autoregressive (MVAR) models. However, real-life systems like biological networks exhibit notable non-linear behavior, hence undermining the validity of MVAR-based GC (MVAR-GC). Current nonlinear GC estimators only cater for additive nonlinearities or, alternatively, are based on recurrent neural networks (RNN) or Long short-term memory (LSTM) networks, which present considerable training difficulties and tailoring needs. We define a novel approach to estimating nonlinear, directed within-network interactions through a RNN class termed echo-state networks (ESN), where training is replaced by random initialization of an internal basis based on orthonormal matrices. We reformulate the GC framework in terms of ESN-based models, our ESN-based Granger Causality (ES-GC) estimator in a network of noisy Duffing oscillators, showing a net advantage of ES-GC in detecting nonlinear, causal links. We then explore the structure of ES-GC networks in the human brain employing functional MRI data from 1003 healthy subjects drawn from the human connectome project, demonstrating the existence of previously unknown directed within-brain interactions. ES-GC performs better than commonly used and recently developed GC approaches, making it a valuable tool for the analysis of e.g. multivariate biological networks.