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The complexity of spontaneous brain activity changes in schizophrenia, bipolar disorder, and ADHD was examined using different variations of entropy

Abstract Adult attention deficit/hyperactivity disorder (ADHD), schizophrenia (SCHZ), and bipolar disorder (BP) have common symptoms and differences, and the underlying neural mechanisms are still unclear. This article will thoroughly discuss the differences between ADHD, BP, and SCHZ (31 healthy control and 31 ADHD; 34 healthy control and 34 BP; 42 healthy control and 42 SCHZ) relative to healthy subjects in combination with three atlases (et al., the Brainnetome atlas, the Dosenbach atlas, the Power atlas) and seven entropies (et al., approximate entropy (ApEn), sample entropy (SaEn), permutation entropy (PeEn), fuzzy entropy (FuEn), differential entropy (DiffEn), range entropy (RaEn), and dispersion entropy (DispEn)), as well as the prominent significant brain regions, in the hope of giving information that is more suitable for analyzing different diseases' entropy. First, the reliability (et al., intraclass correlation coefficient [ICC]) of seven kinds of entropy is calculated and analyzed by using the MSC dataset (10 subjects and 100 sessions in total) and simulation data; then, seven types of entropy and multiscale entropy expanded based on seven kinds of entropy are used to explore the differences and brain regions of ADHD, BP, and SCHZ relative to healthy subjects; and finally, by verifying the classification performance of the seven information entropies on ADHD, BP, and SCHZ, the effectiveness of the seven entropy methods is evaluated through these three methods. The core brain regions that affect the classification are given, and DiffEn performed best on ADHD, SaEn for BP, and RaEn for SCHZ.


| INTRODUCTION
The human brain is a complex system characterized by large-scale distributed neural networks of inherently fluctuating signals, revealed by accumulating functional magnetic resonance imaging (fMRI) studies (Biswal et al., 2010;Deco et al., 2011;Gordon et al., 2017). Restingstate fMRI (rs-fMRI) can reflect the spontaneous neural activity of the human brain and can be used to study its intrinsic function. In the past decades, resting-state functional connectivity (rsFC) has been increasingly used in mapping the intrinsic functional networks of the human brain, with a focus on the inherently fluctuating signals and their interactions between different brain regions (Biswal et al., 1995;Fox & Greicius, 2010;Greicius et al., 2003Greicius et al., , 2009). There are numerous complexity methods of rs-fMRI brain signals, such as the Lyapunov exponent, correlation dimension, Lempel-Ziv complexity, Hurst exponent, and entropy (Ali et al., 2018;Guan et al., 2020Guan et al., , 2022Ju et al., 2019;Lin et al., 2022;Smith et al., 2013;Yakovleva et al., 2020). Among these methods, due to its simple algorithm, the small amount of data required, and the strong anti-noise ability in calculation, entropy is one of the most widely used complexity methods for evaluating the dynamic characteristics of brain signals. In physiological signal analysis, entropy methods have been widely used, including approximate entropy (ApEn; Yentes et al., 2012), sample entropy (SaEn; Richman et al., 2004), fuzzy entropy (FuEn; Chen et al., 2007), and permutation entropy (PeEn; Bandt & Pompe, 2002). SaEn, as proposed by Richman and Moorman (2000), is an improved version of Pincus's ApEn and aims to overcome the limitations of ApEn and reduce statistical bias.
FuEn is an improvement over the ApEn and SaEn algorithms. PeEn, another widely used entropy measure, is a novel method developed by Bandt to characterize the complexity of time series (Bandt & Pompe, 2002). PeEn is suitable for capturing the complex dynamics and rich time structure embedded in biological systems.
Besides, there are some scholars have also proposed new entropy methods, such as Range entropy (RaEn; Omidvarnia et al., 2018), Dispersion entropy (DispEn; Rostaghi & Azami, 2016), and Differential entropy (DiffEn; Duan et al., 2013). RaEn is a proposed modification to ApEn and SaEn, which are highly sensitive to signal amplitude changes and less affected by variation in the magnitude of signals.
DispEn originates from SaEn and PeEn, which can detect the noise bandwidth and simultaneous frequency and amplitude change. It does not lead to undefined results in short signals, is less sensitive to noise, and is considerably faster than SaEn. DiffEn is used to measure the complexity of a continuous random variable, is the entropy of a continuous random variable, and is also related to minimum description length. However, relatively few studies have employed these entropies simultaneously approach for fMRI (Lin et al., 2019;Sokunbi et al., 2011Sokunbi et al., , 2013Sokunbi et al., , 2014Sokunbi et al., , 2015. Specifically, Sokunbi et al. (2011) conducted multiple fMRI signal studies using different entropy measures.
Using ApEn to study individual differences in cognitive performance in an elderly population, the results showed that higher regional signal entropy was associated with better cognitive performance. SaEn was used to study the complexity of fMRI signals in the brain of schizophrenia (SCHZ) patients. It was found that the complexity of the signals from these patients was higher than that of HC at the global and local levels (Sokunbi et al., 2014). They also used SaEn to analyze the patients' fMRI signals from attention-deficit/hyperactivity disorder (ADHD). They found that the entropy values of the fMRI signal from the whole brain of these patients were lower than those of the controls (Sokunbi et al., 2013). The analysis of the complexity of fMRI signals from 41 HC adults (41 males, 19-85 years old) showed that brain entropy (BEN) values at the global and local levels were negatively correlated with age (Sokunbi et al., 2015). Lin et al. (2019) used SaEn to explore changes in the entropy of fMRI signals from the brains of patients with depression. Compared with HC, patients with depression showed reduced entropy in the medial orbitofrontal cortex and cingulate cortex but increased brain signal complexity in the motor cortex. These studies showed that entropy measures could analyze the temporal changes in fMRI signals and locate the relevant brain space. Studies on fMRI showed that the change in entropy value can act as a biomarker for disease states and can also be used to study the internal mechanism of the HC human brain.
Furthermore, a few studies have systematically analyzed the test-retest reliability of these entropy methods applied to fMRI signals to determine the entropy method that most reliably reflects the complexity of biological systems (Liu et al., 2020;Niu et al., 2020). For example, Niu and colleagues (Niu et al., 2020) investigated the distribution and test-retest reliability of four entropy measures in three independent data sets. They showed that analyzing fMRI signals with entropy showed strong tissue sensitivity, with PeEn and FuEn superior to ApEn and SaEn at all three levels. ApEn and SaEn are two of the most commonly used measures in contemporary science among the signal mentioned above entropy measures. SaEn stems from ApEn after addressing some limitations, including inconsistency and strong dependency on the input signal length. However, both measures still suffer from sensitivity to signal amplitude changes. Although PeEn is conceptually simple and computationally fast, the method does not consider the mean value of amplitudes and the differences between amplitude values (Rostaghi & Azami, 2016). Liu et al. (2020) (Finn et al., 2015) applied to whole-cortex entropy profiles. In addition, Zhang and colleagues (Zhang et al., 2021) used multiscale sample entropy of rs-fMRI from 15 SCHZ, 49 BP, and 49 HC to assess differences in rs-fMRI signal complexity. Niu et al. (2020) compared the test-retest reliability of ApEn, SaEn, FuEn, and PeEn, results showed that the highest testretest reliability was achieved with PeEn and FuEn were superior to ApEn and SaEn at all three levels, that is, based on voxels, brain regions, and functional networks. However, brain rs-fMRI signals are very susceptible to noise during the acquisition process, so anti-noise performance of entropy is important (Niu et al., 2019). DiffEn, DispEn, and RaEn are not currently used in rs-fMRI, so whether DiffEn, Dis-pEn, and RaEn are likely to have better anti-noise performance is a new topic that warrants exploration.
Based on three different atlases, this article will focus on finding a reliable and effective entropy method to reveal differences in functional connectivity between ADHD, BP, and SCHZ. So, the distribution and test-retest performance of the seven entropy methods (ApEn, SaEn, PeEn, FuEn, DiffEn, DispEn, and RaEn) in this aim were also compared. The similarities and differences between ADHD, BP, and SCHZ as revealed by entropy methods, are also explored.
Additionally, multiscale entropy expanded based on the seven kinds of entropy is used to analyze the differences and brain regions of ADHD, BP, and SCHZ relative to healthy subjects. The entropy methods that are most effective in these tasks will be determined using classification analysis.  (Gordon et al., 2017).
Therefore, the rs-fMRI data of 9 subjects with 10 sessions were analyzed in this study.

| Data preprocessing
For each subject and each session, those two rs-fMRI datasets were preprocessed by using SPM12 (Ashburner, 2012) and DPABI (Yan et al., 2016), with the following steps: (1) discarding the first 10 volumes for the equilibrium of magnetization; (2) slice timing correction to account for temporal shifts during data acquisition; (3) motion contaminated volumes were then identified by frame-by-frame displacement [FD, described in Power et al. (2011)], calculated as the sum of absolute values of the differentials of the three translational motion parameters (including one filtered parameter) and three rotational motion parameters. Frames with FD >0.2 mm were flagged as motion-contaminated; (4) regressing out nuisance variables including the Friston-24 head motion time-series (Friston et al., 1996) as well as the average time-series of white matter, cerebrospinal fluid, and global signal (Fox & Raichle, 2007); (5) normalizing functional images into the standard Montreal Neurological Institute space by using the EPI atlas with the resampled voxel size of 3 mm; (6) smoothing by using a 6mm full width at half maximum (FWHM) Gaussian kernel; (7) band-pass filtering (0.009-0.08 Hz); and (8) Scrubbing settings: FD type refer Power, FD threshold for bad time is 0.2 mm, and scrubbing time points before and after bad time is one and two, respectively. The scrubbing method was a linear interpolation, and Linear interpolation refers to a method that uses a straight line connecting two known quantities to determine an unknown value between the two known quantities. The interpolation function in MATLAB2016b is "interp1.m." The analytical pipeline is illustrated in Figure 1.

| Entropy calculation
To compare the test-retest reliability of entropy methods, we applied entropy to investigate the complexity of rs-fMRI signals. Entropy is the rate of new information generation, which measures the probability of generating a new pattern in the signal. The greater the probability of generating a new pattern, the greater the signal complexity. This value was calculated with the following formula: In the formula, p i represents the probability of the i-th discrete state.
Here, seven entropy methods were applied as complexity estimators of rs-fMRI signals:

| Approximate entropy
ApEn (Sokunbi et al., 2015) is roughly equivalent to the mean value of the logarithmic conditional probability of the new state appearing when the dimension changes. It has a certain value in measuring the complexity of the time-series. Its calculation formula can be expressed as follows.
Step 1: Given a time series with N data points g , and a priori determination of two unknown parameters, m and r. The parameter m determines the length of the sequences to be compared, and its selection can be estimated by calculating the false nearest neighbor. The second parameter, r, is the tolerance threshold for accepting similar patterns between two segments and has been recommended to be within 0.1-0.2 times the standard deviation of the data.
Step 2: By reconstructing the original data, subsequences can be Step 3: Calculate the distance d m ¼ X i ð Þ, X j ð Þ ½ , the distance d m is determined by the maximum difference between the corresponding position elements in the two vectors. This includes the distance i ¼ j.
Step 4: Then count the number of vectors that satisfy the following conditions, and find the ratio between them and the total number of statistics: This process is called the atlas matching process of X i ð Þ, C m i r ð Þ represents the probability of matching between any X j ð Þ and the atlas.
Step 5: Define the average similarity rate when the number of subsequences is m: Step 6: According to the above 1-5, calculate the average similarity rate Φ mþ1 r ð Þ when the number of divided subsequences is m þ 1.

| Sample entropy
SaEn (Richman & Moorman, 2000) is an improvement relative to the ApEn algorithm in that SaEn calculates the logarithm of the sum and aims to reduce the ApEn's error to is more closely consistent with the known random part. Its calculation formula can be expressed as follows.
F I G U R E 1 The flowchart of the complexity of spontaneous brain activity by using multi-type entropies. The raw rs-fMRI images of the MSC dataset and UCLA dataset underwent preprocessing and time-series extraction and then were used to compute entropy values using seven entropy methods. The entropy analysis was conducted based on whole-brain gray matter voxels. Test-retest reliability was finally examined using the intraclass correlation coefficient (ICC). Besides, altered complexity of spontaneous brain activity and classification results were explored Step 1: Given a time series with N data points x 1 ð Þ, x 2 ð Þ, …, x N ð Þ f g , and a priori determination of two unknown parameters, m and r. The parameter m determines the length of the sequences to be compared, and its selection can be estimated by calculating the false nearest neighbor. The second parameter, r, is the tolerance threshold for accepting similar patterns between two segments and has been recommended to be within 0.1-0.2 times the standard deviation of the data.
Step 2: By reconstructing the original data, subsequences can be Step 3: Calculate the distance d m ¼ X i ð Þ, X j ð Þ ½ , the distance d m is determined by the maximum difference between the corresponding position elements in the two vectors. This includes the distance i ≠ j.
Step 4: Then count the number of vectors that satisfy the following conditions, and find the ratio between them and the total number of statistics: This process is called the atlas matching process of X i ð Þ, B m i r ð Þ represents the probability of matching between any X j ð Þ and the atlas.
Step 5: Define the average similarity rate when the number of subsequences is m: Step 6: According to the above 1-5, calculate the average similarity rate Φ mþ1 r ð Þ when the number of divided subsequences is m þ 1.

| Fuzzy entropy
FuEn (Chen et al., 2007) adds a fuzzy function to the comparison of continuous point vectors and generalizes these vectors to reduce the impact of measurement drift. Its calculation formula can be expressed as follows.
Step 1: Given a time series with N data points x 1 ð Þ, x 2 ð Þ, …, x N ð Þ f g , and a priori determination of two unknown parameters, m and r. The parameter m determines the length of the sequences to be compared, and its selection can be estimated by calculating the false nearest neighbor. The second parameter, r, is the tolerance threshold for accepting similar patterns between two segments and has been recommended to be within 0.1-0.2 times the standard deviation of the data.
Step 2: By reconstructing the original data, subsequences can be Step 3. Calculate the distance d m ¼ X i ð Þ, X j ð Þ ½ , the distance d m is determined by the maximum difference between the corresponding position elements in the two vectors. This includes the distance i ≠ j.
Step 4: The fuzzy membership function: This process is called the atlas matching process of X i ð Þ, B m i r ð Þ represents the probability of matching between any X j ð Þ and the atlas.
Step 5: Define the average similarity rate when the number of subsequences is m: Step 6: According to the above 1-5, calculate the average similarity rate Φ mþ1 r ð Þ when the number of divided subsequences is m þ 1.

| Permutation entropy
PeEn (Bandt & Pompe, 2002) uses the shape of neighboring points to evaluate complexity based on permutation patterns, and it has good computing performance for analyses of any real-world data. Its calculation formula can be expressed as follows.
Step 1: Given a time series with N data points and a priori determination of two unknown parameters, m and L. The parameter m determines the length of the sequences to be compared, and its selection can be estimated by calculating the false nearest neighbor. The second parameter, L, is the delay time.
Step 2: By reconstructing the original data, subsequences can be Step 3: Then do an increasing sort internally for each X i ð Þ, that is If two values are equal, sort them according to the subscript i of j i . In this case, an X i ð Þ is mapped to, which is exactly one of m! permutations. That is, every m dimensional subsequence X i ð Þ is mapped to one of m! permutations.
Step 4: Through the above Step 1-Step 3, the continuous m dimensional subspace is represented by such a sequence of symbols, where the number of these symbols is m!. Denote the probability distribution of all symbols in terms of P 1 ,P 2 ,…,P K , where K ≤ m!.

| Dispersion entropy
DispEn is calculated by first conducting permutations based on the original time series (Rostaghi & Azami, 2016). DispEn can detect the noise bandwidth and simultaneous frequency and amplitude change.
Also, it originates from SaEn and PeEn. Its calculation formula can be expressed as follows.
Step 1: Given a time series with N data points g , and a priori determination of two unknown parameters, m and c. The parameter m determines the length of the sequences to be compared, and its selection can be estimated by calculating the false nearest neighbor. The second parameter, c, is the number of classes that can be members of the time series. We first employ the normal cumulative distribution function to map g from 0 to 1. and a priori determination of two unknown parameters, m and L. The parameter m determines the length of the sequences to be compared, and its selection can be estimated by calculating the false nearest neighbor. The second parameter, L, is the delay time.
Step 2: We use a linear algorithm to assign each y k ð Þ to an integer from 1 to c. To do so, for each member of the mapped signal, we use where z c k shows the kth member of the classified time series and rounding involves increasing or decreasing a number to the next digit.
Step 3: Each embedding vector z m,c k with embedding dimension m and time delay L is created according to The number of possible dispersion patterns that can be assigned to each time series z m,c k is equal to c m , since the signal has m members and each member can be one of the integers from 1 to c.
Step 4: For each of c m potential dispersion patterns, relative frequency is obtained as follows: Step 5: Get dispersion entropy:

| Differential entropy
DiffEn is equivalent to the logarithm energy spectrum in a specific frequency band and was employed to construct features in the frequency bands mentioned above (Duan et al., 2013). Its calculation formula can be expressed as where X is a random variable, f x ð Þ is the probability density function of X. For the time series X obeying the Gauss distribution N 0, σ 2 À Á , its DiffEncan be defined as, where σ 2 denotes the DiffEnof the corresponding rs-fMRI signal variance.

| Range entropy
RaEn is a proposed modification to ApEn and SaEn, and the idea here is to estimate the (logarithmic) likelihood that runs of patterns that are close remain close following incremental comparisons (Omidvarnia et al., 2018). Its calculation formula can be expressed as follows.
Step 1: Given a time series with N data points g , and a priori determination of two unknown parameters, m and r. The parameter m determines the length of the sequences to be compared, and its selection can be estimated by calculating the false nearest neighbor. The second parameter, r, is the tolerance threshold for accepting similar patterns between two segments and has been recommended to be within 0.1-0.2 times the standard deviation of the data.
Step 2: By reconstructing the original data, subsequences can be Step 3: In the spacial case of two-dimensional reconstructed phase space Step 4. Then count the number of vectors that satisfy the following conditions, and find the ratio between them and the total number of statistics: This process is called the atlas matching process of X i ð Þ, B m i r ð Þ represents the probability of matching between any X j ð Þ and the atlas.
Step 5: Define the average similarity rate when the number of subsequences is m: Step 6: According to the above 1-5, calculate the average similarity rate Φ mþ1 r ð Þ when the number of divided subsequences is m þ 1.

| Parameter selection
This study uses three fMRI datasets to compare the test-retest performance of the seven entropy methods. ApEn, SaEn, RaEn, and FuEn need to set three parameters: N, m, and r. N is the number of time points, m specifies the dimension of the phase space, and r is the similarity tolerance. Many studies have discussed the setting of these parameters (Schultz et al., 2008;Tocado et al., 2009). The parameter m is taken as 2, and the similarity tolerance r is taken as 0.25 times the standard deviation of the original data. The PeEn algorithm also involves the setting of three parameters: N, pm, and τ, where N is the time-series, pm is the embedding dimension, and τ is the delay time.
The pm value setting is based on the following considerations: when pm < 3, the process is meaningless because there are too few permutations and combinations; the larger the pm is, the more the algorithm time complexity will increase as larger m corresponds to more permutations. Bandt and Pompe suggested that pm should be 3-7 (Bandt & Pompe, 2002); Li et al. (2014), to ensure sensitivity to the transient characteristics of the system, suggested that the pm should be a small value to reduce the time complexity of the algorithm. In this study, pm was set as 4. For the time delay in the sorting and symbolization process, the value used in this study was 1. In this setting, more information can be captured in brain signals. The DispEn involves the setting of 4 parameters: pm, nc, ma, and τ, where pm is the embedding dimension, τ is the delay time, nc is the number of classes which usually equal to a number between 3 and 9, and ma is the mapping approach, chosen from "LM" (linear mapping), "NCDF" (normal cumulative distribution function), "TANSIG" (tangent sigmoid), "LOGSIG" (logarithm sigmoid), and "SORT" (sorting method; Azami & Escudero, 2018). There are fewer number parameters in DiffEn: range, and level, where the range is assumed to be a rectangle and the level is the depth in the tree with a default is 0.

| Multiscale entropy calculations
The multiscale entropy analysis was introduced to estimate the entropy on multiple time scales. The multiscale entropy calculation can be summarized as follows.
Step 1: a coarse-grained time series according to a range of scale factors was constructed; Step 2: the entropy of each coarse-grained time series was quantified; Step 3: the multiscale entropy profile was examined using a range of scales.
The length of each coarse-grained time series was equal to the length of the original time series divided by the scale factor. For scale 1, the time series was simply the original time series. So, in this article, all seven entropy methods mentioned in this article are extended to multi-scale entropy, and the results of those seven multi-scale entropies are further analyzed.

| Test-retest reliability
The test-retest reliability evaluates the statistical stability of the index at different measurement times (Spitzer, 1992). It comprehensively considers the changes within the individual and among different individuals, reflecting the stability and consistency of the index across time (Rongsawad et al., 2019). Test-retest reliability is essential in various fields, including sociology, behavior, physics, biology, and medicine (Chambers, 1985). Due to the interference of various factors in the actual measurement, it is critical to choose a reliable index. The intraclass correlation coefficient (ICC) is a commonly used reliability coefficient index to measure test-retest reliability (Spitzer, 1992). The ICC of intrinsic functional networks was computed as shown below.
Equation (20) estimates the correlation of the subject signal intensities between sessions, modeled by a two-way ANOVA, with random subject effects and fixed session effects (Shrout & Fleiss, 1979

| Classification analysis
To classify subjects into HC and patients using ROI-based BENs, a linear support vector machines (SVM) approach was implemented (Dosenbach et al., 2010). SVMs have supervised learning models which can be used for classification tasks (Cortes & Vapnik, 1995).
Given labeled training data with n features, the algorithm outputs an optimal hyperplane that can separate the classes with a maximal mar-

| Node definition
Considering that different atlases will affect the analysis results, three atlases are used in this paper, namely the Brainnetome atlas (Fan et al., 2016), the Dosenbach atlas (Dosenbach et al., 2010), and the Power atlas (Power et al., 2011), shown in Figure 2. By comparing the differences between different atlases and finding no significant difference, the results obtained in this paper can be said to be atlas independent.

| Reliability comparison of different BENs
In this part, we use rs-fMRI data with 10 sessions from the MSC dataset to explore the reliability of different BENs, in Figures 3 and 4.
Additionally, we simulated and generated three distributed random sequences, calculated their entropy, and then explored the reliability of the different entropies, in Figure 5.

In
Step 1, extracted ROIs within DMN in the Dosenbach atlas (Dosenbach et al., 2010) and calculated the BENs of the extracted rs-fMRI data using the seven entropy methods mentioned above. Each session of rs-fMRI was calculated to obtain session BENs of nine subjects in total. In Step 2, we calculated the ICC of the repeated sessions with the data obtained from Step 1. In Step 3, we select k ¼ 3, 6, 10 ½ (k means how many rs-fMRI sessions were used) to test BEN's reproducing reliability, and Figure 3 shows the distribution of ICC values.
In Figure 4, we also performed a two-sample t-test on different entropy methods when k ¼ 10 to see whether there is a significant difference between methods. This proves that the ICC values vary when calculated by different entropy methods. The results displayed on the three templates are consistent; that is, the reliability of DiffEn was better than those of the other entropy methods, regardless of the template used. From Figure 3, DiffEn is more stable when the number of sessions increases and has the highest average ICC value. Based on Figure 4, the two-sample t-test showed a significant difference among entropy methods. Furthermore, the performance is consistent with no significant difference during these three atlases, namely the Brainnetome atlas (Fan et al., 2016), the Dosenbach atlas (Dosenbach et al., 2010), and the Power atlas (Power et al., 2011).

| Altered complexity of spontaneous brain activity
In this part, we classified ADHD, BP, and SCHZ using BEN obtained from the seven entropy methods mentioned above. Before the classification task, we first performed a two-sample t-test between HC and patients on ROI levels. Figure 6 exhibits the results of the two-sample t-test between HC and different diseases. Based on Figure 6, Tables   (Csermely et al., 2013) greatly, and DMN, occipital, and cingulo-opercular changed more so than the cerebellum.
From F I G U R E 6 Slice images of the brain regions with significant differences between HC and patient. The number of specific brain regions with significant differences is shown in Tables 1-4 Similar meaningful results are seen for each condition based on the described above. These three disorders share brain regions that > frontoparietal. Furthermore, Figure 6 is slice images of the brain regions with significant differences between HC and patient. It is worth noting that a similar pattern is obtained on the Brainnetome atlas and the Power atlas based on the Dosenbach atlas. That is to say, the conclusion is not affected by the atlas.

| Multiscale entropy analysis
For each atlas, we extracted the average multiscale entropy to ROIs.
For each ROI, the differences in multiscale entropy values among the three groups were compared using an independent samples t-test, and p values are listed in Table 4. The average multiscale entropy values in all ROIs over multiple time scales for the four groups (ADHD, BP, SCHZ, and HC) are shown in Figure 7. Regardless of the atlas used, the mean multiscale entropy values for the ADHD, BP, and SCHZ were significantly lower than for the HCs (both p-value <.05).
More detailed results are presented in Table 4. Besides, it is also found that the performance of different multiscale entropy algorithms on different diseases is also different, such as DiffEn, RaEn, and SaEn performed better in ADHD, SCHZ, and BP, respectively.

| Classification comparison of different BENs
When a two-sample t-test was done, we did a classification task based on entropy values calculated using different ROI levels and the whole brain based on three different atlases. The result is presented in Tables 5 and 6.   Table 5 for details. Table 6 is based on the best results of a wholebrain classification in Table 5.
The weights of the corresponding linear kernel SVM classifiers are analyzed to explore the critical brain regions that affect ADHD, BP, and SCHZ. Among them, DiffEn achieves an accuracy of 0.8548.  the three templates are consistent; that is, the reliability of DiffEn was better than those of the other entropy methods. Then, we used seven kinds of entropy and multiscale entropy expanded based on seven kinds of entropy to explore the differences between brain regions.

| The entropy method comparison
The multiscale entropy is obtained only by scale expansion on the single-scale entropy, and then the complexity of sequence or topology can be studied from different scales (Costa et al., 2005;Ferraz & Kihara, 2022;Marinazzo et al., 2020). The core of different multiscale entropy is the single-scale entropy, so it is necessary to discuss the difference in single-scale entropy. In physiological signal analysis, ApEn and SaEn are two of the most commonly used measures. SaEn is an improved version of ApEn and aims to overcome the limitations of ApEn and reduce statistical bias (Richman & Moorman, 2000).
However, both measures still suffer from sensitivity to signal T A B L E 3 Counting the ROIs which have a significant difference between HC and SCHZ (num_of_diff_area refers to the number of ROIs which significantly differ from HC and SCHZ)                amplitude changes, so, Chen et al. (2007) proposed the FuEn, which is an improvement over the ApEn and SaEn algorithms. As another widely used entropy measure, PeEn is a novel method developed by Bandt to characterize the complexity of time series (Bandt & Pompe, 2002). Although PeEn is conceptually simple and computationally fast, PeEn does not consider the mean value of amplitudes and the differences between amplitude values (Rostaghi & Azami, 2016). RaEn (Omidvarnia et al., 2018) is a proposed modification to ApEn and SaEn, which are highly sensitive to signal amplitude changes and less affected by variation in the magnitude of signals.
DispEn (Rostaghi & Azami, 2016) originates from SaEn and PeEn, which can detect the noise bandwidth and simultaneous frequency and amplitude change. It does not lead to undefined results in short signals, is less sensitive to noise, and is considerably faster than SaEn.
DiffEn (Duan et al., 2013) measures the complexity of a continuous random variable, is the entropy of a continuous random variable, and is also related to minimum description length. Besides, rs-fMRI signals maybe include multi-type noises, complex nonlinear (Guan et al., 2020), and fractal structures (Guan et al., 2022). Therefore, if some entropy methods can be less sensitive to noise or highly sensitive to signal amplitude changes and less affected by variation in the magnitude of signals, then these entropy methods should be able to show good performance. The results of the ICC of seven kinds of entropy using the MSC dataset and simulation data displayed on the three templates are consistent, and the reliability of DiffEn was better.
The core brain regions that affect the classification are given, and Dif-fEn performed best on ADHD, SaEn for BP, and RaEn for SCHZ.

| ADHD and HC
Several studies have shown a downward trend in brain complexity in visual brain regions in ADHD compared with HC (Akdeniz, 2017;Sokunbi et al., 2013). A meta-analytical study suggests that ADHDrelated dysfunction is associated with multiple neuronal systems involved in higher-order cognitive functions and sensorimotor processes, including the visual system and the DMN (Cortese et al., 2012). Cortese and colleagues found ADHD-related hyperactivation in the visual network and the DMN (Cortese et al., 2012). Furthermore, Sutcubasi and colleagues (Sutcubasi et al., 2020) found that ADHD was associated with a connectivity disruption within the DMN than HC. Sun and colleagues  found that regions of the DMN and sensorimotor network were altered in ADHD compared with HC by comparing local consistency in ADHD and HC. In addition,  symptoms. Bollmann and colleagues (Poil et al., 2014) studied spatial working memory in adults with ADHD. They found that the left lateral occipital cortex, middle frontal gyrus, and supramarginal gyrus positively affected working memory load.
In contrast, the frontal pole, middle temporal gyrus, and occipital pole cingulate gyrus negatively affected working memory load. Tang et al. (2018) used multicenter data to analyze local and global differences between individuals with ADHD and typically developing.
They found that the ALFF, the fraction ALFF, and the ReHo in the ADHD group were significantly higher in the medial orbital compared with the typically developing group. There were abnormalities in the frontal cortex, anterior cingulate cortex, postcentral gyrus, thalamus, precuneus, and cerebellum. Meanwhile, from the perspective of brain development, brain network development was delayed in the ADHD group, especially in the DMN. The cingulate gyrus is involved in attentional processing and executive function (Makris et al., 2005(Makris et al., , 2007 et al., 2012) found decreased fALFF in the medial prefrontal and orbitofrontal cortices and DMNs in bilateral putamen with increased rsFC in the left insula and bilateral dorsal prefrontal cortex. In SCHZ, ALFF was reduced in the bilateral ventral frontal cortex (Lui et al., 2010).
SCHZ exhibits high connectivity between subcortical and auditory networks (Lottman et al., 2019). It can be seen from the above studies that the difference between SCHZ and HC is mainly concentrated in DMN.
Meanwhile, this experiment also found significant differences in SMN and VN. SCHZ is associated with dysfunction in visual perception, and the right lateral occipital cortex is a mid-level visual area critical for object recognition. Previous fMRI studies have reported structural and functional abnormalities in the lateral occipital cortex of SCHZ (Cant & Goodale, 2007). Li et al. (2020) investigated the spatiotemporal interactions of the lateral occipital cortex with other brain networks through functional connectivity. They found that the temporal instability of lateral occipital cortical connections was increased in both resting-state and task-switching situations, and in the resting state, compared to controls, the patients' lateral occipital cortex interacted with the FPN and thalamus effect increased. At the same time, the interaction of the lateral occipital cortex with the DMN increased during task switching. The study also found that temporal instability of lateral occipital cortical connectivity was positively associated with patients' switching costs during task performance and the severity of hallucinations. These results suggest that reduced stability of lateral occipital cortical connections may be an important factor in neurocognitive dysfunction and symptom severity in SCHZ. The thalamus regulates input from multiple brain regions, is considered a major relay hub, and may play a specific role in SCHZ pathophysiology (Shenton et al., 2001). Gifford and colleagues (Gifford et al., 2020) compared flexibility in 55 SCHZ and 72 HC. The study found that nodes covering the left thalamus had significantly higher flexibility scores at the node level.
The temporal plane, located posterior to the superior temporal lobe, is a vital brain area associated with language and speech production (Galaburda, 1978) and is also thought to be involved in the pathophysiology of SCHZ (Shenton et al., 1992). Kwon et al. (1999) used magnetic resonance imaging to measure subtemporal gray matter in 16 SCHZ, and 16 HC matched for age, sex, handedness, and parental economic status volume. The study found that gray matter volume in the left temporal plane was significantly reduced (28.2%) in the SCHZ group compared to the HC group. The left temporal plane was more asymmetric than the right temporal plane in the SCHZ group in the HC group. Found the opposite result, which is thought to be the basis of language processing impairment and SCHZ-specific paranoia. Zhou et al. (2010) analyzed the functional connectivity of 19 early-stage SCHZ and 19 HC and found that the functional connectivity strength increased in the left inferior temporal gyrus. An fMRI study using meta-analysis (Taylor et al., 2012) found that the HC showed stronger activation in the left occipital pole than the SCHZ group in terms of affective experience.

| CONCLUSIONS
This article will thoroughly discuss the differences between ADHD, BP, and SCHZ (

| Limitations
Although this article considers the performance of seven different entropies on ADHD, BP, and SCHZ and then extends these to multiscale and three different atlases, this article's amount of rs-fMRI data is not large enough, which is also the main disadvantage. The next project will explore this topic by collecting multiple sites' rs-fMRI data.
Besides, three parameters (N is the number of time points, m specifies the dimension of the phase space, and r is the similarity tolerance) selection for each entropy method is limited. Although already some journals have discussed the setting of these parameters ( et al., 2008;Tocado et al., 2009), how to get a more suitable parameter was still not sure. So, based on our previous works (Guan et al., 2020(Guan et al., , 2022, the nonlinear complexity of rs-fMRI will further explore the more suitable parameters.

DATA AVAILABILITY STATEMENT
The MSC dataset is publicly available at https://openneuro.org/ datasets/ds000224, and the UCLA dataset is publicly available at the OpenfMRI database with the accession number ds000030 (https:// www.openfmri.org/dataset/ds000030/). Data were processed using publicly available software (SPM12 http://www.fil.ion.ucl.ac.uk/spm/ software/spm12/ and DPAPI http://rfmri.org/DPABI). The simulation time-series and seven entropy methods were performed using MATLAB code available from the authors upon request.