## 2 Materials and Methods
### Extended Detrended Fluctuation Analysis of EEG Structure  
There are several reasons that complicate the quantification of long-range power-law correlations in physiological data sets, namely, the nonstationarity and the fast rate of the decay of the correlation function. The latter produces values close to zero, where computational errors can strongly affect the results. Due to problems, the analysis of very slow physiological processes in terms of the correlation function is ineffective, and alternative approaches are necessary. One such approach is the detrended fluctuation analysis (DFA), which was proposed by [PERSON] et al. [36; 37] to improve the correlation analysis of natural processes. This method can be treated as a version of the root mean square study of a signal profile.  
The data set \(x(i)\), \(i=1\),..., \(N\) is analyzed according to the following algorithm:  
(1) The creation of a random walk \(y(k)\) associated with the original dataset  
\[y(k)=\sum_{i=1}^{k}[x(i)-\langle x\rangle], \tag{1}\]  
where \(\langle x\rangle\) is the mean value of \(x(i)\).  
(2) Segmentation of \(y(k)\) into fragments of size \(n\) and estimation of the local trend \(y\_n}\) (\(k\)) in every segment using linear fitting.  
(3) Computation of the root mean square fluctuation  
\[F(n)=\sqrt{\frac{1}{N}\sum_{k=1}^{N}[y(k)-y_{n}(k)]^{2}}\sim n^{\alpha}. \tag{2}\]  
Such power-law behavior (Equation (2)) is typical for many natural systems. Consideration of this dependence on a double logarithmic plot provides an ability to estimate the scaling exponent \(\alpha\), whose value is nearly constant for homogeneous datasets but can vary with \(n\) for inhomogeneous processes.  
Using \(\alpha\), the correlation analysis of nonstationary processes can be provided, where \(\alpha<0.5\) quantifies the alternation of large and small values (anti-correlations), \(\alpha=0.5\) indicates the absence of correlations, and \(\alpha>0.5\) marks different types of positive correlations, which are related to power-law behavior (\(\alpha<1\)), and a behavior that may differ from power-law statistics (\(\alpha>1\)).  
The above standard DFA algorithm does not account for possible changes in nonstationary behavior across the entire dataset, which are clearly pronounced for transients, intermittency, or abrupt changes in energy as a typical in EEG data. To adapt the DFA tool to the case of variable nonstationary properties, we have proposed an extended DFA (EDFA) [38]. Its main idea is to account for local distinctions in the RMS deviation (2) between different parts of the dataset, i.e., to consider the inhomogeneous structure of the signals. For this purpose, a new measure is introduced:  
\[dF(n)=max[F_{loc}(n)]-min[F_{loc}(n)]. \tag{3}\]  
It characterizes the impact of nonstationarity and determines the difference between the extreme values of local RMS deviations. If nonstationarity does not show significant changes between different parts of the data, measure (3) does not take large values. However, for clearly inhomogeneous datasets, \(dF(n)\) can show strong distinctions. Typically, the dependence (3) shows a power-law behavior, although the related scaling exponent is different:  
\[dF(n)\sim n^{\beta} \tag{4}\]  
The given approach has been successfully applied in several our studies [28; 31].