**3. Results and Discussion**

According to earlier studies [31,32], the slopes of the dependences (3) and (5) on the lg–lg plot vary with the segment length, which characterizes the range of power-law correlations. Often, significant differences between the properties of long-range and short-range correlations are observed in many types of physiological signals, which was demonstrated in pioneering works [14,15]. Knowledge of the features of power-law dependences (3) and (5) is important for establishing informative markers that quantify changes caused by transitions between different physiological states. In the case of the rat EEG, noticeable changes in correlation properties during sleep were found in the region of slow-wave dynamics associated with the lg *n* >3.3 range, which refers to frequencies below 1 Hz. To specify this range for the current study, we performed a preliminary visual analysis of the EDFA results. By analogy with the works in [31,32], transitions between distinct physiological states (dynamics before or after SD) can be observed in the area of long-range correlations, but changes between slopes, quantitatively determined by the exponents *α*

and *β*, are usually subject-dependent and may also vary throughout a single recording for the same animal. In some cases, the range lg *n* >3.3 is appropriate to illustrate the effect of sleep deprivation. In other cases, lower frequencies associated with larger values of lg *n* seem preferable. Thus, Figure 3 shows two examples of the dependence lg *F* vs. lg *n* for individual 5-min EEG segments measured in awake mice before and after SD. They illustrate the most pronounced differences between the states for lg *n* >3.9. As lg *n* decreases, the distinctions in slopes are still observed, but they become weaker. Insert show the results of statistical analysis performed to select an appropriate range of scales (the values of lg *n* related to changes in the slopes of lg *F* vs. lg *n*).

**Figure 3.** Examples of the dependence (3) in the lg–lg plot for 5-min EEG segments acquired in an awake mice before and after SD. Insert shows the results of statistical analysis over different EEG segments.

For the *β*-exponent, the effect can be more pronounced, as this exponent can change its sign upon transition to another physiological state. This is illustrated in Figure 4 for the same measurements as in Figure 3. Again, the range lg *n* >3.9 is better suited for quantifying the distinctions caused by SD. Analogous visual analysis performed on other animals or data segments allowed us to capture this range of scales for further statistical analysis of the groups.

**Figure 4.** Examples of the dependence (6) in the lg–lg plot for 5-min EEG segments acquired in an awake mice before and after SD. Insert shows the results of statistical analysis over different EEG segments.

When conducting statistical analysis, several important points should be mentioned. First, there is a significant variability in the measures *α*, *β* within each state, due to the individual peculiarities of animals. The latter complicates comparison of the states based on absolute values of the scaling exponents, and accounting for differences between exponents *α*1, *β*<sup>1</sup> related to state 1 (awake mice before SD) and *α*2, *β*<sup>2</sup> associated with state 2 (awake mice after SD) seems to be a more promising approach. Thus, we introduce two measures

$$
\Delta \mathfrak{a} = \mathfrak{a}\_1 - \mathfrak{a}\_2, \quad \Delta \mathfrak{f} = \mathfrak{f}\_1 - \mathfrak{f}\_2 \tag{7}
$$

for a quantitative description of SD effects.

Another circumstance is the significant variability in scaling exponents between different parts of each recording. On the one hand, we can take longer datasets (e.g., several hours), estimate the corresponding values of *α* and *β*, and then compare these quantities for the two states under consideration. However, this way is accompanied by time-varying dynamics and several types of nonstationary behavior that can change the expected values of the scaling exponents. On the other hand, we can select fairly homogeneous (more stationary) segments, analyze them with EDFA, and then average the results for each animal and each condition. Our preliminary analysis of the simulated datasets [27] showed that this way gives more stable and reliable estimations, and we use it here for EEG processing.

The established distinctions caused by SD for the entire group of animals are given in Figure 5, where different symbols indicate distinct responses, and in Table 1. For six out of 10 mice, a pronounced effect of SD is observed, characterized by a decrease in the *α* and *β* exponents, i.e., positive values of Δ*α* and Δ*β* (Figure 5, circles). Several animals (three out of 10 mice) demonstrated relatively subtle signs of SD-induced changes (Figure 5, triangles), although these changes are significant according to the Mann–Whitney test (*p* < 0.05). In this study, only one day of sleep deprivation was used. Longer SD periods are expected to elicit stronger responses; however, our goal was to examine the effects of short-term SD when sleep deprivation is not associated with neurodegenerative disorders. According to Figure 5, one mouse showed a different reaction (square), but this behavior can be treated atypical compared to other animals. Our results indicate that short-term effects of SD can be detected in EEG recordings, although the strength of the response is subject-dependent. Moreover, accounting for the *β* exponent of the proposed EDFA can surpass the *α* exponent of the standard DFA—the Δ*β* range is about twice as large as the Δ*α* range (0.12 ± 0.04 versus 0.05 ± 0.02). Consequently, the changes in the features of nonstationarity caused by SD are more pronounced than the changes in the properties of long-range correlations associated with them.

**Figure 5.** Individual responses of mice from the entire group to one-day SD quantified with the differences between the scaling exponents (7) before and after SD (Δ*α* and Δ*β*). Maroon circles show pronounced effects of SD, blue triangles indicate relatively subtle signs of SD-induced changes, and an orange square marks atypical response.

Thus, in this study we show that EDFA sensitively reflects the changes induced by SD. The sleep is a natural factor of activation of the lymphatic clearing and drainage functions of the brain [1,13]. The SD causes significant suppression of the clearance of toxins from the brain [1]. There are animal data suggesting that sleep deficit leads to sterile inflammation [7–9], an increase in the BBB permeability [8,10], and long SD is accompanied by hallucination and various cognitive deficiencies [12]. We hypothesized that sleep is a biomarker of the BBB permeability and EEG is an important informative platform for the analysis of BBB leakage and the cerebral lymphatic functions [20]. We show that EDFA may be applied to study changes in the electrical brain activity after SD.

**Table 1.** Characterization of SD effects with measures (7). The results are given as mean values ± SE. Asterisks indicate statistically significant changes according to the Mann–Whitney test (*p* < 0.05). EDFA shows significant changes for 9 out of 10 animals (Δ*β*), while DFA provides significant distinctions for 7 out of 10 mice (Δ*α*). The last column indicates that changes in Δ*β* are stronger in 8 mice (|Δ*β*|>|Δ*α*|).

