Almost all currently available Granger Causality(GC)-based approaches to the estimation of information flow between brain regions are based on linear bivariate/multivariate autoregressive (MVAR) models. The fMRI signal is an articulate results of convolving neural activity with a locally and time-varying haemodynamic response function (HRF), and a linear MVAR approach appears only partially suitable in reconstructing the multiple nonlinearities/time-scales which concur to the BOLD effect. Additionally, employing a linear model to reconstruct complex dynamics necessarily leads to an increase in model order often incompatible with typical fMRI acquisitions. We employ a Volterra-Wiener decomposition using orthogonal Laguerre polynomials as base functions in order to build parsimonious MVAR models which more accurately model BOLD dynamics, and use this framework to estimate Laguerre-based Granger Causality (LGC) in the context of directed functional brain connectivity.

Estimating globally conditioned GC (GCGC) from variable Y_t to the variable X_t (Y→X) amounts to testing the null-hypothesis that knowledge of the past of Y_t does not improve the prediction of the future of X_t. To this end, in the GCGC approach two models are employed: first, the "restricted" AR model for X_t, which includes the past values of X_t itself and Z_t, which accounts for all other variables except Y_t. Second, the "unrestricted" AR model, which includes all variables X_t, Y_t, and Z_t [1]. We consider the restricted and unrestricted system as a MVAR system defined over the components that comes from a Volterra-Wiener expansion with Laguerre polynomials: $$x_t={\mathbf{A}}\big(\,\mathcal{L}^{(m)}(x)\oplus\mathcal{L}^{(m)}(z)\,\big)+{\varepsilon_t}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(restricted model)}$$ $$x_t={\mathbf{A'}}\big(\,\mathcal{L}^{(m)}(x)\oplus\mathcal{L}^{(m)}(y)\oplus\mathcal{L}^{(m)}(z)\,\big)+{\varepsilon_t'}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(unrestricted model)}$$ where the discrete-time Volterra-Wiener decomposition with Laguerre polynomials $$$\mathcal{L}^{(m)}(\cdot)$$$ over the discrete time signal $$${x_t}$$$ is defined as: $$\mathcal{L}^{(m)}(x) = \sum_{n=1}^N \phi_m(n) ( x_{N-n} - x_{N-n-1}) $$ The $$$m^{\text{th}}$$$-order, discrete time, orthonormal Laguerre polynomial $$$\phi_m(n)$$$ is defined as $$\phi_m(n) = \alpha^\frac{n-m}{2} (1-\alpha)^\frac{1}{2}\sum_{j=0}^m (-1)^{j}\binom{n}{j}\binom{m}{j}\alpha^{m-j}(1-\alpha)^j$$ where parameter $$$\alpha\,(0<\alpha<1)$$$ determines the rate of exponential asymptotic decline of $$$phi_m(n)$$$ [2].

The Null-hypothesis that Yt does not cause Xt., conditioned to Zt, is rejected if the $$$f_\text{ratio}$$$ of the residual sum of squares (RSS) $$f_\text{ratio}=\frac{RSS_\text{r}-RSS_\text{ur}}{RSS_\text{ur}}\frac{N_\text{obs}-2m}{m}$$ is extreme with respect of its parent distribution ($$$\chi2$$$-distribution with $$$N_\text{obs}-2m$$$ and $$$m$$$ degrees of freedom [3].

We first characterize the ability of LGC to capture nonlinear/hidden causal relations by simulating multivariate coupling in a network of noisy, driven Duffing oscillators (Figure 1) which interact through integral relationships with different decay constants. The ability of detecting true causal links while rejecting false causal links is quantified as the area under the ROC curve (AUC) as a function of the threshold in causality strength (Figure 2). Successively, we explore the structure of LGC-based networks in the human brain in functional MRI (fMRI) data from 440 healthy subjects scanned at rest at 3T (4 sessions,1200 volumes/subject, TR=0.72S) within the "HCP 500-Subjects PTN Release" by employing the subject-specific timecourses of 25 components resulting from spatiotemporal group independent component analysis (ICA).Figure 1 shows the synthetic network employed to validate the LCG approach as well as the nonlinear noisy oscillators at each node. Figure 2 shows the difference in AUC between employing linear MVAR conditioned GC approach and employing the Laguerre-based MVAR approach. Figure 3 is a graphical summary of the 440 subject directed, between-component brain connectivity network.Synthetic validation showed a clear advantage of the LGC approach in detecting nonlinear, causal links across different timescales at low model order. This is in line with the idea that Laguerre polynomials are smooth basis functions, able to capture damped multiple-frequency oscillations with fewer parameters when compared to linear MVAR models. In turn, overparameterising the MVAR model leads to a degradation of the performance in detecting causality. The within-ICA component LCG analysis consistently showed that posterior occipital-inferior parietal networks strongly drive the activity in the right dorsolateral prefrontal cortex (PFC)-superior parietal circuits which is consistent with the view of a "bottom-up" information processing flow from the visual systems to the right-lateralized attentional executive system [4]. We also found evidence for a “top-down” information processing flow from the superior parietal networks to the visual systems as well as from the “cognitive" cerebellar regions to the visual networks. This is particularly intriguing in view of the emerging evidence showing the critical role of the posterior lobules of the cerebellum in cognitive control and “coordination” of sensory information processing. While this observation warrants additional validation through specific task-based investigation, in general we have shown that the LGC method is able to detect in vivo functional interactions and causal dynamics across multiple neural networks while delivering superior performance as compared to classical, linear MVAR-based Granger causality methods.