## 3 Results  
The proposed method for generating signals is illustrated using the \(EEG\) Motor Movement/Imagery Dataset [25]. Specifically, the S001R01 baseline recording with eyes open is utilized. These data were sampled at 160 Hz (\(f_{s}\)) and spanned a duration of 61 s. The selected order for autoregressive models, \(M\), was 19, which is equivalent to 120 ms, a time horizon relevant for \(EEG\) connectivity analysis. This selection was based on prior studies indicating its relevance for capturing \(EEG\) connectivity [26; 27; 28]. We also experimented with other regression orders but found that 19 provided a good balance between model accuracy and computational efficiency. The \(GC\) connectivity matrix was computed (see Figure 1), revealing a high \(GC\) connectivity of 0.92 between electrodes FC1 (\(T\)) and F1 (\(S\)). To generate signals, subsections from electrodes FC1 and F1 were selected: from 0 to 14.4 s for \(T^{\prime}\) (recorded by FC1) and from 28.8 to 43.2 s for \(S^{\prime}\) (recorded by F1). The \(GC\) value between these unrelated signal segments was calculated to be 0.008. The temporal connectivity parameter \(K\) was set to 1 during the first interval of \(R^{\prime}\) (0 to 4.8 s) and the third interval (9.61 to 14.4 s), while it was set to 0 in the second interval (4.81 to 9.6 s). Using the proposed method (Equation (6)) and the previously obtainedautoregressive model parameters, a synthetic signal was generated. This synthetic signal \(R^{\prime}\) and the source signal \(S^{\prime}\) were then used for functional connectivity analysis using GC (see Figure 2). The GC values obtained for the three defined intervals were 0.77, 0.032, and 0.77, respectively (referred to below as reference functional connectivity (\(RFC\)) values). As anticipated, the \(RFC\) value for the second interval is low, whereas those for the first and third intervals are high. However, these values are still lower than the GC of the original signals (calculated for the entire period of the signal). Figure 1: Connectivity matrix GC of order 19 for subject S001R01 [25], from the baseline eyes-open experiment [29]. Figure 2: Synthetically generated signal and reference GC values (in three intervals) [29]. To demonstrate the applicability of the signal generation method for analyzing dynamic connectivity estimation methods, dynamic GC [28; 30; 31] with a sliding window approach was employed. Here, _GC_ was estimated for the generated signal, which mimics the modified FC1 signal relative to the source signal F1. Figures 3a and 3b depict the results obtained using window sizes of 2 s and 400 ms, respectively. It is evident that the estimated connectivity varies over time, showing expected deviations from the _RFC_, particularly during transitions. Figure 3, shows that there are different limitations in the estimation of the dynamic _GC_ for both wide and narrow window sizes. For wider windows, there is a good agreement between _RFC_ and the estimated values when there is constant reference connectivity. When an abrupt change occurs in _RFC_, the estimated dynamic _GC_ slowly transitions to a new value, as expected. The wide window limits the rate of change, as observed in Figure 3a. For narrow windows, a noisier estimation of dynamic _GC_ is expected, along with a faster response to abrupt changes in _RFC_. This underscores the limited capability of _GC_ measures in identifying the precise times of connectivity changes and highlights the necessity for further research in this direction. To further investigate the aforementioned limitations, dynamic Granger causality (_GC_) was calculated for various window sizes, ranging from a minimal window size to 3000 ms with a step of 25 ms, and for different regression orders (during estimation), \(M_{GC}\), ranging from 5 to 35 with a step of 1. The minimal window size equals \((M_{GC}+1)/f_{s}\). During signal generation, a regression order of 19 was used, and the temporal connectivity parameter \(K\) was set to 1.