## 2 Materials and Methods
### Effective Connectivity Analysis  
#### 2.4.1 Spectral Dynamic Causal Modeling  
Brain regions that showed significant functional connectivity were taken as ROIs in the spectral DCM analysis. The activity of each ROI was summarized with its weighted (using T value) mean, and a Full model (i.e., the brain regions were reciprocally interconnected, and the brain regions had inhibitory self-connection) structure was set. Spectral DCM was then applied using SPM12.  
The principle of spectral DCM is the same as that of classical DCM, which is to establish a Bayesian generative model to describe how various neural systems are coupled through effective connectivity and finally generate observation data (such as BOLD signal). However, the difference is that the spectral DCM no longer estimates stochastic time series, but instead uses a deterministic model to generate the cross spectral densities (CSD) of time series.  
Equation (1) denotes the neuronal model of the spectral DCM:  
\[x(t)=Ax(t)+Cu(t)+v(t) \tag{1}\]  
where \(x(t)\) is the neuronal state variable, \(A\) is the neuronal parameter (i.e., the effective connectivity), \(u(t)\) is the experimental stimulus input (\(u(t)=0\) for resting-state data) and \(v(t)\) is the state noise.  
The below hemodynamic state equation further describe how the hidden neural activity \(x(t)\) becomes the observed BOLD time series \(y(t)\):  
\[y(t)=h(x(t),\varphi)+e(t) \tag{2}\]  
where \(\varphi\) is the hemodynamic parameter and \(e(t)\) is the observation noise.  
Therefore, the parameters to be estimated in the above generative model are the neuronal parameter \(A\), the hemodynamic parameter \(\varphi\), the state noise \(v(t)\) and the observation noise \(e(t)\). However, the estimation of the latter time-varying signals \(v(t)\) and \(e(t)\) is the solution of stochastic differential equations, which is computationally inefficient. To avoid this, we further calculate the CSD of these signals, which can be expressed as follows:  
\[\begin{split} g_{v}(\omega,\theta)&=\alpha_{v} \omega^{-\beta_{v}}\\ g_{e}(\omega,\theta)&=\alpha_{e}\omega^{-\beta_{e} }\end{split} \tag{3}\]  
Figure 1: Brain template of supplementary motor area (SMA) sub-regions. The area marked in red refers to the pre-SMA, and the blue area is the SMA proper. These two sub-regions can be effectively divided by vertical commisure anterior (VCA) line (Y = 0).  
where \(g_{x}(\omega)=X(\omega)X(\omega)^{\dagger}\) is the CSD, \(X(\omega)\) is the Fourier transform of the time-domain signal \(x(t)\), \(\{\alpha,\beta\}\subset\theta\) is the parameter to be estimated for controlling the amplitude and exponent of the spectral density and \(\omega=2\pi f\) is the angular frequency. As a result, the original time-varying time series is replaced by its second-order statistics with time-invariant characteristics (i.e., the CSD). This is performed so that the parameter estimation of DCM can be turned into the solution of deterministic differential equations, which greatly improves the calculation efficiency.  
In this study, the CSD was calculated by the 4 th order autoregressive model, and the frequency range was set to [1/128, 0.1] Hz (for more detail please see [20; 26; 27]). After model inversion, the model evidence and the posterior mean and posterior covariance of each connection for each subject were obtained.