# Hypergraph Analysis of Functional Brain Connectivity During Figurative Attention
## 4 Experimental Data Analysis
### Wavelet Analysis  
The wavelet analysis provides a detailed view of how spectral energy fluctuates over time, offering insights into the dynamic nature of decision-making processes as captured  
Figure 3: FFT spectra of preprocessed EEG from the O1 channel during (**a**) MI, (**b**) MRV, and (**c**) MLV tasks. The vertical dashed lines bound frequency regions of interest (\(f_{1}\) and \(f_{2}\)), representing the brain response to stimulus modulation.  
through neurophysiological data. The wavelet power spectrum serves as a quantifiable metric, enabling the precise measurement of these temporal changes [43; 65; 66; 67; 68; 69; 70].  
While the spectra depicted in Figure 3 were computed over the entire 30 s EEG time series, it is important to note that sustained attention is not maintained uniformly throughout this period. To identify specific time windows during which attention was focused, we employed wavelet analysis. Using the Morlet wavelet, we were able to pinpoint intervals where attention was predominantly directed toward one interpretation of the cube over another.  
The temporal dynamics of the cube's orientation interpretation were explored through continuous wavelet transform (CWT) analysis, implemented using the Brainstorm software (version 3.4.0.0). This method allowed us to investigate how attention shifted over time, revealing the moments when the subject focused attention on specific interpretations of the cube. By leveraging wavelet analysis, we gained a more granular understanding of the temporal patterns underlying attentional processes during the task.  
Figures 4a,b depict the variations in wavelet energy corresponding to figurative attention directed towards the left (MLV) and right (MRV) cube orientations, respectively. The wavelet energy was calculated using the following equation:  
\[W(f,t)=\sqrt{f}\int_{t-4/f}^{t+4/f}X(t)\psi^{*}(f,t)dt, \tag{12}\]  
where the asterisk (*) denotes the complex conjugate, \(X(t)\) represents the analyzed EEG signal, and \(\psi(f,t)\) is the complex-valued Morlet wavelet, chosen as the mother wavelet. The Morlet wavelet is defined as follows:  
\[\psi(f,t)=\sqrt{f}\pi^{1/4}e^{i\omega_{0}f(t-t_{0})}e^{f(t-t_{0})^{2}/2} \tag{13}\]  
with \(\omega_{0}=2\pi f_{0}\) being the central frequency of the Morlet wavelets, \(i=\sqrt{-1}\) the imaginary unit, and \(t_{0}\) representing the time shift parameter. This formulation allows for a precise analysis of the temporal and spectral characteristics of the EEG signal, capturing the dynamic shifts in attention during the task.  
It is evident that the prominence of specific components corresponds to the subject's directed attention towards either the left or right orientation of the cube. The spectral component at \(f_{1}\) is dominant when the subject perceives the cube as left-oriented (Figure 4a), while \(f_{2}\) prevails when the cube is interpreted as right-oriented (Figure 4b). As shown in Figure 4, spectral energy at \(f_{1}\) dominates during the time interval \(t_{L}\in[6.88,28.12]\) seconds in the MLV test, whereas spectral energy at \(f_{2}\) dominates during the interval \(t_{R}\in[3.94,30]\) seconds in the MRV test. In the subsequent analysis, connectivity is calculated within these identified time intervals.  
Figure 4: _Cont._