Goal-dependent modulation of effective connectivity detected by Granger causality with signal-dependent noise

Here we present a detailed approach on how to analyze fMRI data, using Granger causality with signal dependent noise. The model stability of signal dependent noise is discussed which serves as a constraint when we fitted the model parameters. We then introduced a strict procedure to regress out all artifacts due to head movements. The processed data enables us to clearly demonstrate that noise depends on signal. With the data, we then analyzed effective connectivity circuits of seven brain regions, including the three regions in the main text as a subset. 1 Granger Causality with Signal-Dependent Noise This novel approach to causality is a marriage of two approaches: Granger causality (C. W. J. Granger, 1969), and the Baba-Engle-Kraft-Kroner (BEKK) model of time series with time-varying volatility (R. F. Engle and K. F. Kroner, 1995). We have applied this approach to EEG data (Q. Luo et al., 2011), but this is the first application to fMRI data. Signal dependent noise is widely observed in physiological experiments at the spike level. As a consequence, we should expect that the derived measures such as LFP, EEG and the BOLD signal should exhibit similar features (see Discussion). However, most publications related to fMRI data do not take this key feature into account. Here we describe the stability conditions of this model, and provide a constrained minimization optimization for the model fitting. A Matlab package of the methods used in this paper is available on request. † Contributed to this work equally * Correspondence should be addressed to Jianfeng Feng, Fudan University, Shanghai 200433, P.R. China. E-mail: [email protected]. Supplementary Materials 2 Model with Signal-Dependent Noise 2.1 Model Suppose that we have two time series t X and t Y ( 1, , t T = ). Let ( , ) t t t Z X Y = , p and q the model orders, i A ( 1, , i p = ), , xy j B , , yx j B ( 1, , j q = ), xy C , yx C the model coefficient matrices, and , xy t u , , yx t v Gaussian white noise processes. The time-series model considered is given below , , 1 ( , ) ' p t i t i xy t yx t i Z A Z r r − = = + ∑ , (S1)