Effects of Sleep Deprivation on the Brain Electrical Activity in Mice

Abstract

Sleep plays a crucial role in maintaining brain health. Insufficient sleep leads to an enhanced permeability of the blood–brain barrier and the development of diseases of small cerebral vessels. In this study, we discuss the possibility of detecting changes in the electrical activity of the brain associated with sleep deficit, using an extended detrended fluctuation analysis (EDFA). We apply this approach to electroencephalograms (EEG) in mice to identify signs of changes that can be caused by short-term sleep deprivation (SD). Although the SD effect is usually subject-dependent, analysis of a group of animals shows the appearance of a pronounced decrease in EDFA scaling exponents, describing power-law correlations and the impact of nonstationarity as a fairly typical response. Using EDFA, we revealed an SD effect in 9 out of 10 mice (Mann–Whitney test, p<0.05) that outperforms the DFA results (7 out of 10 mice). This tool may be a promising method for quantifying SD-induced pathological changes in the brain.

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applied
sciences
Article
Effects of Sleep Deprivation on the Brain Electrical Activity
in Mice
Alexey N. Pavlov 1,* , Alexander I. Dubrovskii 2, Olga N. Pavlova 2
and Oxana V. Semyachkina-Glushkovskaya 3


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
Publisher’s Note: MDPI stays neu-
tral with regard to jurisdictional clai-
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Copyright: © 2021 by the authors. Li-
censee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and con-
ditions of the Creative Commons At-
tribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Faculty of Nonlinear Processes, Saratov State University, Astrakhanskaya Str. 83, 410012 Saratov, Russia
2Physics Department, Saratov State University, Astrakhanskaya Str. 83, 410012 Saratov, Russia;
paskalkamal@mail.ru (A.I.D.); pavlov_lesha@yahoo.com (O.N.P.)
3Biology Department, Saratov State University, Astrakhanskaya Str. 83, 410012 Saratov, Russia;
glushkovskaya@mail.ru
*Correspondence: pavlov.alexeyn@gmail.com
Abstract:
Sleep plays a crucial role in maintaining brain health. Insufficient sleep leads to an enhanced
permeability of the blood–brain barrier and the development of diseases of small cerebral vessels.
In this study, we discuss the possibility of detecting changes in the electrical activity of the brain
associated with sleep deficit, using an extended detrended fluctuation analysis (EDFA). We apply
this approach to electroencephalograms (EEG) in mice to identify signs of changes that can be caused
by short-term sleep deprivation (SD). Although the SD effect is usually subject-dependent, analysis
of a group of animals shows the appearance of a pronounced decrease in EDFA scaling exponents,
describing power-law correlations and the impact of nonstationarity as a fairly typical response.
Using EDFA, we revealed an SD effect in 9 out of 10 mice (Mann–Whitney test,
p<
0.05) that
outperforms the DFA results (7 out of 10 mice). This tool may be a promising method for quantifying
SD-induced pathological changes in the brain.
Keywords:
detrended fluctuation analysis; long-range correlations; electroencephalogram; sleep
deprivation; nonstationarity
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
Appl. Sci. 2021,11, 1182. https://doi.org/10.3390/app11031182 https://www.mdpi.com/journal/applsci

Appl. Sci. 2021,11, 1182 2 of 10
complex dynamics of natural systems [
16
–
22
]. The traditional correlation function
C(τ)
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
t1
,
t2
. Only for stationary processes there is a dependence on their difference
τ=t2−t1
, 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 power-
law 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 4summarizes the main findings of the study.

Appl. Sci. 2021,11, 1182 3 of 10
2. Methods and Experiments
2.1. DFA
DFA is a variant of the correlation analysis of experimental datasets originally pro-
posed by Peng et al. [
14
,
15
]. It is based on RMS analysis of signal profile and includes the
following steps:
(1) Transition from signal x(i),i=1, . . . , Nto its profile y(k),k=1, . . . , N
y(k) =
k
∑
i=1
[x(i)− hxi],hxi=
N
∑
i=1
x(i). (1)
(2) Segmentation of the profile y(k)into parts of length n(n<N).
(3) Computation of the local trend
yn(k)
for each segment using a least-squares straight-
line fit.
(4) Estimation of the standard deviation,
F(n) = v
u
u
t
1
N
N
∑
k=1
[y(k)−yn(k)]2. (2)
(5) Implementation steps 2–4 over a wide range of n.
(6) Computation of the scaling exponent α,
F(n)∼nα. (3)
Power-law dependence (3) is observed for various random processes, but many real-
world datasets with an inhomogeneous structure often exhibit different local slopes of
lg F
vs.
lg n
, and
α
may differ for short-range and long-range correlations. DFA is usually
applied to reveal the features of complex dynamics related to the region of long-range
correlations. Specific values of
α
, associated with
α<
0.5,
α=
0.5, and 0.5
<α<
1, describe,
respectively, anti-correlated behavior (alternation of large and small values of
x(i)
), lack of
correlations (e.g., white noise), and positive power-law correlations (large values of
x(i)
tend to follow large values, and vice versa) [
15
]. Positive correlations, which may differ
from power-law behavior, are associated with α>1.
2.2. EDFA
Signal properties can vary strongly between different parts of a dataset. This is
observed, e.g., for transients from one state to another, when well-pronounced variations
in the local mean value affect the scaling exponent. Considering datasets with and without
such transients can lead to distinct results of DFA. In a recent study [
27
], we illustrated the
effects of nonstationarity for several cases: low-frequency trend, intermittent dynamics,
and nonstationarity in energy. Besides the case when time-varying dynamics occurs
throughout the signal, the
α
exponent is also influenced by short-term failures of the
recording equipment or artifacts. Such data segments provide distinct local standard
deviations (2) compared to the averaged quantities. In order to characterize the differences
in nonstationarity across the entire signal, we have proposed the following modification of
the method, called EDFA [31,32]. Within this approach, a new measure
dF(n) = max[Floc(n)]−min[Floc(n)],Floc (n) = s1
n
n
∑
k=1
[y(k)−yn(k)]2(4)
is introduced, where
Floc (n)
are the local standard deviations of the profile from the trend,
which are estimated for each segment. The difference
dF(n)
contains information about the
impact of signal inhomogeneity. If the properties of the signal vary insignificantly,
dF(n)

Appl. Sci. 2021,11, 1182 4 of 10
takes values close to zero. Otherwise, a wide distribution of
Floc (n)
appears, and
dF(n)
varies with n, exhibiting power-law behavior characterized by the scaling exponent β
dF(n)∼nβ. (5)
In this definition,
β
becomes highly sensitive to artifacts in experimental recordings.
The existence of a single artifact can lead to a large
Floc (n)
associated with
max[Floc(n)]
,
and the latter reduces the stability of the EDFA method. In particular, the
dF(n)
dependence
can show strong fluctuations with
n
. A more stable algorithm is based on the statistical
analysis of
Floc (n)
, and the use of the standard deviation
σ
(
Floc (n)
) as a measure of the
signal inhomogeneity. Thus, here we propose to consider the dependence
σ(Floc (n)) ∼nβ. (6)
Figure 1shows both dependences (5) and (6) in a lg–lg plot for the case of 1/
f
-noise
used as a simple example of a homogeneous process with power-law correlations. This
figure confirms that the latter definition provides reduced variability in the estimated
values. Thus, standard error of the
β
estimates decreases from 0.0038 for the definiton (5)
to 0.0023 for the definiton (6).
For physiological datasets, differences are usually larger. Although the exponents
β
in
Equations (5) and (6) may differ, we use the same designation (
β
) to quantify the impact of
nonstationarity and a more stable algorithm based on
σ
(
Floc (n)
) for its evaluation. The
β
exponent can take as positive, as negative values [
30
]. Both
α
and
β
exponents describe
different signal properties and are independent quantities.
1.5 1.8 2.1 2.4 2.7 3.0
lg n
−1.0
−0.6
−0.2
0.2
0.6
1.0
lg dF, lg σ(Floc)
Eq. (5)
Eq. (6)
Figure 1.
Dependences described by Equations (5) and (6) in the lg–lg plot for 1/
f
-noise. The
β
-
exponent is estimated with the standard errors 0.0038 and 0.0023, respectively.
2.3. Subjects and Experiments
Experiments were carried out on ten C57BL/6 male mice (20–25 g) in accordance with
the Guide for the Care and Use of Laboratory Animals (8th ed., The National Academies
Press, Washington, 2011). The protocols were approved by the Local Bioethics Commission
of the Saratov State University. The mice were kept in a light/dark environment with
the lights on from 8:00 to 20:00 and fed ad libitum with standard rodent food and water.
The ambient temperature and humidity were maintained at 24.5
±
0.5
◦
C and 40–60%,
respectively.
A two-channel cortical EEG (Pinnacle Technology, Taiwan) was recorded (
Figure 2
)
using two silver electrodes (tip diameter 2–3
µ
m) located at a depth of 150
µ
m in coordinates
(L: 2.5 mm and D: 2 mm) from Bregma on either sides of the midline under inhalation
anesthesia with 2% isoflurane at 1 L/min
N2O
/
O2
—70:30. The head plate was mounted
and small burr holes were drilled. Thereafter, wire EEG leads were inserted into burr
holes on one side of the midline between the skull and the underlying dura mater. EEG
leads were fixed with dental acrylic. Ibuprofen (15 mg/kg) for relief of postoperative pain
was provided in water supply for two to three days before surgery and for three or more

Appl. Sci. 2021,11, 1182 5 of 10
days after surgery. Before starting the experiment, the mice were given 10 days to recover
from surgery.
Figure 2.
Experiment design: (
a
) implantation of a two–channel cortical EEG, (
b
) SD by presenting new objects to the mouse,
and (c) EEG recording in an awake mouse. Insert shows experimental EEG signals—voltages (µV) vs. time (seconds).
As standard sleep staging rules for mice are not currently available, we referred
to the visual assessment criteria from the studies [
1
,
38
]. Sleep deprivation was carried
out according to the method described in [
39
], with adaptation to the vivarium regime.
The mice were deprived of sleep from 8:00 pm to 8:00 am and were immediately used for
the experiment. Sleep deprivation was maintained by bringing new objects and sounds
into the experiment room [
40
]. The mice were constantly monitored to make sure they
were actually studying objects.
Signals were measured in awake and sleeping mice (day 1, 10-h recording), and after
SD (day 2, 4-h recording). This study compares two states: (1) awake mice, background
EEG activity, and (2) awake mice after SD. All recordings were done with a sampling rate
of 2 kHz. At the stage of preprocessing, twelve 5-min segments with a quite homogeneous
structure and less distorted by artifacts were selected for each EEG channel. Every 5-min
segment was analyzed with EDFA to evaluate
σ
(
Floc (n)
) and the
β
exponent. The results
for each mouse state were averaged over all selected segments and both channels.
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 corre-
lations. 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

Appl. Sci. 2021,11, 1182 6 of 10
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 3shows 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 nrelated to changes in the slopes of lg Fvs. lg n).
3.2 3.5 3.8 4.1 4.4 4.7
lg n
3.1
3.3
3.5
3.7
3.9
lg F
before SD
after SD
3.0
3.5
4.0
4.5
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 4for
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.
3.2 3.5 3.8 4.1 4.4 4.7
lg n
2.6
2.8
3.0
3.2
3.4
lg σ(Floc)
before SD
after SD
3.0
3.5
4.0
4.5
lg n
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
α1
,
β1
related to state 1 (awake mice before SD) and
α2
,
β2
associated with state 2 (awake
mice after SD) seems to be a more promising approach. Thus, we introduce two measures
∆α=α1−α2,∆β=β1−β2(7)

Appl. Sci. 2021,11, 1182 7 of 10
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
∆α
and
∆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
∆β
range is about twice as large as the
∆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.
−0.1 0.0 0.1 0.2
∆α
−0.1
0.0
0.1
0.2
0.3
0.4
∆β
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 (
∆α
and
∆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.

Appl. Sci. 2021,11, 1182 8 of 10
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 (
∆β
), while DFA provides significant
distinctions for 7 out of 10 mice (∆α). The last column indicates that changes in ∆βare stronger in 8
mice (|∆β|>|∆α|).
Experiment DFA (∆α)EDFA (∆β)|∆β|/|∆α|
1 0.14 ±0.02 ∗0.34 ±0.03 ∗2.4
2 0.06 ±0.02 ∗0.22 ±0.02 ∗3.7
3−0.01 ±0.01 0.01 ±0.01 1.0
4 0.06 ±0.02 ∗0.14 ±0.02 ∗2.3
5 0.11 ±0.02 ∗0.21 ±0.02 ∗1.9
6 0.02 ±0.01 0.05 ±0.01 ∗2.5
7−0.03 ±0.01 −0.04 ±0.01 ∗1.3
8 0.09 ±0.02 ∗0.13 ±0.02 ∗1.4
9−0.08 ±0.02∗−0.06 ±0.02 ∗0.8
10 0.16 ±0.03 ∗0.25 ±0.04 ∗1.6
4. Conclusions
The scaling features of long-range power-law correlations are important markers of
the complex dynamics of many real-world systems that can be used to diagnose the state
of a system based on experimentally measured datasets. We have discussed a modified
approach to the fluctuation analysis of inhomogeneous processes, in which the nonstation-
arity varies over the entire signal. This approach, EDFA, computes two scaling exponents
that quantify power-law correlations and nonstationarity features. To provide a more
stable computational procedure and reduce the impact of localized artifacts, we consider
the standard deviation of the local RMS fluctuations of the signal profile around the trend,
rather than the difference of extreme values. The benefits of the latter procedure are
illustrated using simulated datasets.
We then applied this approach to EEG signals in mice to reveal signs of changes in
electrical activity of the brain that could be caused by a day’s sleep deprivation. These
signs can be fairly subtle, in contrast to the effects of prolonged SD, taking into account
the significant variability of characteristics during long-term EEG recordings. Using a
group of 10 mice, we found quite strong reductions in
α
and
β
scaling exponents in six
animals, with only one mouse showing a pronounced opposite effect. In these animals,
the
β
exponent provided stronger responses than the
α
exponent of the conventional DFA.
Thus, the proposed modified version of the method can be a useful prognostic tool for the
evaluation of SD-mediated suppression of the clearance of toxins from the brain that is in
accordance with the work [
1
]. Important open questions that could be further analyzed are
the role of SD duration and the factor of age.
Author Contributions:
Conceptualization, A.N.P. and O.V.S.-G.; methodology, A.N.P.; data curation,
A.I.D.; software, O.N.P.; formal analysis, A.I.D., O.N.P.; investigation, A.N.P., O.N.P.; writing—
original draft preparation, A.N.P. and O.V.S.-G.; writing—review and editing, A.N.P. and O.V.S.-G.;
visualization, A.N.P.; supervision, A.N.P. and O.V.S.-G.; project administration, O.V.S.-G.; funding
acquisition, O.V.S.-G. All authors have read and agreed to the published version of the manuscript.
Funding:
This work was supported by the grant of the Government of the Russian Federation No.
075-15-2019-1885 in the part of experimental studies and numerical analysis. A.P. acknowledges
the support by the grant of the Russian Science Foundation (Agreement 19-12-00037) in the part of
theoretical studies (the development of EDFA method).
Institutional Review Board Statement:
The study was carried out in accordance with the Guide for
the Care and Use of Laboratory Animals (8th ed., The National Academies Press, Washington, 2011),
and approved by the Local Bioethics Commission of the Saratov State University (protocol No. 12,
17.02.2020).

Appl. Sci. 2021,11, 1182 9 of 10
Informed Consent Statement: Not applicable.
Data Availability Statement:
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
DFA Detrended fluctuation analysis
EDFA Extended detrended fluctuation analysis
EEG Electroencephalogram
SD Sleep deprivation
RMS Root mean square
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