**1. Introduction**

Sleep plays a critical role in maintaining the health of the central nervous system and resisting small vessel disease in the brain. Over the past decades, there has been a better understanding of the effects of sleep on the body [1–4]. Sleep is important for attention and learning and affects long-term memory, decision-making, etc. [5,6]. It is vital to maintain good overall brain health, and prolonged periods of the absence of sleep can have serious consequences. Good sleep reduces the risk of neurodegenerative disorders, and insufficient sleep leads to sterile inflammation in the absence of infection [7–9] and an enhanced permeability of the blood–brain barrier (BBB) [8,10]. Total sleep deprivation (SD) of rats resulted in their death [11]. In humans, the longest wakefulness time (11 days) is accompanied by hallucinations and various cognitive impairments [12]. Thus, it seems clear that sleep plays an important role in restoring brain function. Sleep is a biomarker of BBB permeability, and electroencephalography (EEG) is an important informative platform for analyzing BBB leakage, especially in amyloid lesions of small vessels of the brain [13]. It is interesting to note that the opening of the BBB and deep sleep are accompanied by similar activation of toxins clearance from the brain [13]. Thus, nighttime EEG patterns also hide information about lymphatic drainage and cleansing functions of the brain. Detecting such information requires techniques that deal with nonstationary signal processing, and one such tool is the detrended fluctuation analysis (DFA).

Since its appearance [14,15], DFA has attracted considerable attention in many areas of research, where correlation features of experimental datasets are used to characterize the complex dynamics of natural systems [16–22]. The traditional correlation function *C*(*τ*)

**Citation:** Pavlov, A.N.; Dubrovskii, A.I.; Pavlova, O.N.; Semyachkina-Glushkovskaya, O.V. Effects of Sleep Deprivation on the Brain Electrical Activity in Mice. *Appl. Sci.* **2021**, *11*, 1182. https://doi.org/10.3390/ app11031182

Academic Editors: Leo K. Cheng, Keun-Chang Kwak and Jing Jin Received: 28 November 2020 Accepted: 25 January 2021 Published: 28 January 2021

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has two main restrictions, the first of which is the decay of *C*(*τ*) with increasing time lag *τ*, which is fast for broadband random processes. The latter limits the ability to compute the scaling exponent describing long-range power-law correlations, because *C*(*τ*) approaches zero and becomes comparable to computational errors for noisy datasets. The second restriction arises for time-varying dynamics, when the value of *C* is determined by two time moments *t*1, *t*2. Only for stationary processes there is a dependence on their difference *τ* = *t*<sup>2</sup> − *t*1, and the correlation function is described by one variable. Many natural processes do not satisfy this requirement, and the traditional approach is used under the assumption of quasistationarity for short segments of the dataset or after excluding the trend due to data filtering. The origin of nonstationarity differs. It can be caused by recording equipment failures or by transients between various system states. Otherwise, it appears due to internal slow dynamics with time scales comparable to the duration of the available datasets. In the latter case, we interpret part of the internal dynamics of a system with time-varying components as a trend. The advantage of DFA is the inclusion of data filtering (detrending) in the signal processing algorithm [14]. Moreover, this detrending is carried out for each time scale separately, which is important for inhomogeneous datasets. Another advantage is the transformation of the decreasing correlation function into a growing dependence of the root mean square (RMS) fluctuations of the signal profile around the local trend on the time scale, and the scaling exponent describing its powerlaw features is easier to estimate, especially in the region of long-range correlations [15]. The DFA has some limitations that were discussed in earlier studies [23–26]. Despite the detrending procedure, nonstationarity influences the results, and data preprocessing is still important for analyzing complex systems using experimentally recorded signals [27].

In its original version [14,15], the DFA considers one basic type of nonstationarity, namely, slow variations in the local mean value (trend). However, natural processes can include other types of time-varying behavior, e.g., repeated regular or random switching between system states, variations in energy, etc. The application of DFA can lead to misinterpretation of scaling exponents for inhomogeneous datasets, where segments with small and large RMS fluctuations can coexist, and their number affects the results. Several attempts have been made to modify the conventional method, such as multifractal DFA, which introduces a number of quantities instead of a single scaling exponent [28,29]. Recently, we proposed another modification that takes into account local RMS fluctuations and estimates two scaling exponents describing the features of power-law correlations and the impact of nonstationarity [30]. This approach, extended DFA (EDFA) [31,32], has been applied to various types of physiological processes to improve the diagnostic capabilities of the conventional method. The main idea of EDFA is to take into account the difference between the maximum and minimum local RMS fluctuations of the signal profile (random walk) around the trend depending on the time scale. Here, we perform some modification of the EDFA to provide a more stable computation algorithm, and consider the standard deviation of the local RMS fluctuations. Such improvement allows us to avoid or at least reduce the effect of artifacts in experimental measurements, when localized artifacts or short-term instabilities strongly influence the RMS fluctuations within the conventional DFA and alter the quantitative measures of long-range correlations.

To illustrate the EDFA's ability to characterize effects of SD on the brain electrical activity, here we analyze EEGs acquired in awake mice in two different states—background electrical brain activity and activity after SD [33–37], when the animals did not sleep for a day. Unlike prolonged SD, the effects of short-term SD are less obvious. Here, we study how one-day SD alters long-range power-law correlations in electrical activity in the brain. The manuscript is organized as follows. In Section 2, we describe the subjects, experimental procedures, and data measurements used in this work. We also provide a brief description of DFA and its modified version, EDFA. The results of EEG studies in mice during background activity and after sleep deprivation are presented in Section 3. Section 4 summarizes the main findings of the study.
