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 nonlinear behaviour, hence undermining the validity of MVAR-based GC (MVAR-GC). Most nonlinear GC estimators only cater for additive nonlinearities or, alternatively, are based on recurrent neural networks or long short-term memory networks, which present considerable training difficulties and tailoring needs. We reformulate the GC framework in terms of echo-state networks-based models for arbitrarily complex networks, and characterize its ability to capture nonlinear causal relations in a network of noisy Duffing oscillators, showing a net advantage of echo state GC (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. In addition, we examine joint brain-heart signals in 15 subjects where we explore directed interaction between brain networks and central vagal cardiac control in order to investigate the so-called central autonomic network in a causal manner. This article is part of the theme issue 'Advanced computation in cardiovascular physiology: new challenges and opportunities'.
Keywords: Granger causality; brain connectivity; brain-heart interaction; echo state network.