Echo State Network models for nonlinear Granger causality
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.