## 2 Materials and Methods
### Generation of Synthetic Signals  
Once the regression models are understood, they can be employed to transform real signals into synthetic ones. This allows us to regulate the influence of source signals by introducing an additional mixing parameter, which can vary arbitrarily over time and thus govern the dynamics of connectivity. Before the analysis, the stationarity of the _EEG_ signals was verified using the augmented Dickey-Fuller (_ADF_) test. The _ADF_ test results indicated that the signals (from electrodes FC1 and F1 after offline preprocessing) were stationary (\(p\)-value for both is equal 0.0010), meeting the requirements for _GC_ analysis. However,a limitation lies in the framework's assumption of stationarity within selected intervals, which may reduce accuracy when simulating highly non-stationary or rapidly fluctuating EEG signals.  
Initially, a method for generating signals needs to be established in which the maximum connectivity equals that of the source signals, derived from independent sources. For example, consider two actual source signals, \(T\) and \(S\), and construct their regression models to estimate their respective univariate and bivariate regression errors, denoted as \(e_{T}\) and \(e_{TS}\), alongside their connectivity measure \(GC_{TS}\). In this study, two signal segments are extracted from different time periods to ensure minimal correlation, providing a suitable baseline for generating synthetic signals with controlled connectivity. These subsections, named \(T^{\prime}\) and \(S^{\prime}\), are expected to exhibit minimal correlation and low Granger causality. Using these sequences, a new sequence \(R_{1}\) is generated that resembles signal \(T^{\prime}\) but also incorporates aspects of \(S^{\prime}\), employing the following regression model:  
\[R_{1}(n)=\sum_{i=1}^{M}\omega_{Ti}T^{\prime}(n-i)+\sum_{i=1}^{M}\omega_{Si}S^{ \prime}(n-i)+e^{\prime}_{TS}(n). \tag{4}\]  
Incorporating the regression error \(e^{\prime}_{TS}\), which equals \(e_{TS}\) in the subsection of signal \(T^{\prime}\), enhances the realism of the generated sequence in terms of predictability. The resultant signal \(R_{1}\) is anticipated to exhibit Granger causality (\(GC_{RIS}\)) with respect to \(S\), similar to \(GC_{TS}\).  
However, the drawback of this model lies in its inability to dynamically adjust the actual connectivity. To obtain a signal that is unrelated to \(S\), one could employ a univariate regression model. Nevertheless, the resulting outcome would mirror the target signal \(T^{\prime}\):  
\[R_{2}(n)=\sum_{i=1}^{M}\omega_{i}T^{\prime}(n-i)+e^{\prime}_{T}(n)=T^{\prime}( n). \tag{5}\]  
Here, \(e^{\prime}_{T}\) equals \(e_{T^{\prime}}\). By combining realistic signals resembling both high and low connectivity, such as \(R_{1}\) and \(R_{2}\), the connectivity can be adjusted dynamically by mixing them together:  
\[R^{\prime}(n)=K(n)\cdot R_{1}(n)+(1-K(n))\cdot R_{2}(n) \tag{6}\]  
The parameter \(K\) represents the level of connectivity between the synthetic signal and the source. When \(K=0\), no connectivity is imposed (\(GC=0\)), whereas \(K=1\) corresponds to the maximum connectivity level derived from the original signals.  
Therefore, utilizing the pair \(\{R^{\prime},S^{\prime}\}\) will allow us to explore dynamic connectivity methods in future studies.