Estimating Phase Amplitude Coupling Between Neural Oscillations Based on Permutation and Entropy

entropy

Article Estimating Phase Amplitude Coupling between Neural Oscillations Based on Permutation and Entropy

Liyong Yin 1, Fan Tian 2, Rui Hu 2, Zhaohui Li 2,3 and Fuzai Yin 1,*

1 Department of Internal Medicine, Hebei Medical University, Shijiazhuang 050011, China; [email protected] 2 School of Information Science and Engineering (School of Software), Yanshan University, Qinhuangdao 066004, China; [email protected] (F.T.); [email protected] (R.H.); [email protected] (Z.L.) 3 Hebei Key Laboratory of Information Transmission and Signal Processing, Yanshan University, Qinhuangdao 066004, China * Correspondence: [email protected]

Abstract: Cross-frequency phase–amplitude coupling (PAC) plays an important role in neuronal oscillations network, reﬂecting the interaction between the phase of low-frequency oscillation (LFO) and amplitude of the high-frequency oscillations (HFO). Thus, we applied four methods based on permutation analysis to measure PAC, including multiscale permutation mutual information (MPMI), permutation conditional mutual information (PCMI), symbolic joint entropy (SJE), and weighted-permutation mutual information (WPMI). To verify the ability of these four algorithms, a performance test including the effects of coupling strength, signal-to-noise ratios (SNRs), and data length was evaluated by using simulation data. It was shown that the performance of SJE was similar to that of other approaches when measuring PAC strength, but the computational efﬁciency of SJE was the highest among all these four methods. Moreover, SJE can also accurately identify the PAC Citation: Yin, L.; Tian, F.; Hu, R.; Li, frequency range under the interference of spike noise. All in all, the results demonstrate that SJE is Z.; Yin, F. Estimating Phase better for evaluating PAC between neural oscillations. Amplitude Coupling between Neural Oscillations Based on Permutation and Entropy. Entropy 2021, 23, 1070. Keywords: neuronal oscillations; phase–amplitude coupling; permutation; mutual information; https://doi.org/10.3390/e23081070 entropy

Academic Editors: Osvaldo Anibal Rosso and Fernando Montani 1. Introduction Received: 27 July 2021 Neuronal oscillations play a significant role in memory and learning [1,2]. In the study Accepted: 13 August 2021 of cognitive processes, phase–amplitude coupling is a promising measurement [3]. Contin- Published: 18 August 2021 uous electrophysiological signals which are recorded at macroscopic and mesoscopic levels, such as electroencephalogram (EEG) and local field potential (LFP), present rhythmical Publisher’s Note: MDPI stays neutral characteristics called neuronal oscillations [4]. Generally, they can be divided into five fre- with regard to jurisdictional claims in quency bands, including delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–12 Hz), beta (12–30 Hz), published maps and institutional affil- and gamma (above 30 Hz) oscillations [1,5,6]. The interaction between neuronal oscilla- iations. tions of different frequency bands is implemented by a communication mechanism, which is known as cross-frequency coupling (CFC) [7]. In previous experiments and studies, three types of CFC have been observed and reported, including PAC [8–11], phase–phase coupling (PPC) [12,13], and amplitude–amplitude coupling (AAC) [14–16]. Copyright: © 2021 by the authors. Particularly, more reports focused on PAC because of its ubiquity in electrophysiology Licensee MDPI, Basel, Switzerland. significance in brain functions. The PAC is generally defined as that the amplitude of a This article is an open access article higher-frequency oscillation is modulated by the phase of a lower-frequency oscillation [17]. distributed under the terms and Moreover, the PAC has been studied in many functions, such as memory, attention selec- conditions of the Creative Commons tion, and sensory information detection [18,19]. It is widely observed in the hippocampus; Attribution (CC BY) license (https:// neocortex; and basal ganglia from rats, mice, monkeys to humans [20–22]. Several quan- creativecommons.org/licenses/by/ titative measures have been proposed to characterize PAC, such as phase-locking value 4.0/).

Entropy 2021, 23, 1070. https://doi.org/10.3390/e23081070 https://www.mdpi.com/journal/entropy Entropy 2021, 23, 1070 2 of 16

(PLV) [23], mean vector length (MVL) [24], Kullback–Liebler (KL) based modulation index (KLmi) [9], and generalized linear modeling (GLM) [25]. Overall, these different methods were proposed based on different principles and focused on different purposes, which can be found in comparative studies [9,26–29]. Recently, based on permutation entropy [30] and mutual information theory, a new method called permutation mutual information (PMI) was applied to measure PAC [31]. Although the advantage of the PMI approach has been described, it failed to account for the multiple time scales inherent in the physiological systems. To solve this problem, Aziz et al. improved the permutation entropy (PE) and called this modified procedure multiscale permutation entropy (MPE). By comparing the analysis results of MPE and PE on physiological signals, it is proved that MPE is more robust [32]. Therefore, in the present study, we proposed a novel approach named MPMI. It measures the PAC by evaluating the mutual information of multiscale permutations between two neural oscillations, which is developed from the MPE on the basis of mutual information theory. It is well known that the classical conditional mutual information (CMI) is initially proposed to characterize the weak interaction between two phase signals with similar rhythms [33]. Cheng et al. improved it and successfully applied it in measuring PAC [34]. Afterward, Li et al. presented a new method called PCMI for assessing the directionality between two signals generated by different neuronal populations [35]. The simulations show that this method is superior to the CMI method for identifying the coupling direction. Therefore, we will use PCMI to estimate PAC. Joint entropy is a nonlinear measure used to estimate the uncertainty between one time series and another [36]. In order to weaken or eliminate the influence generated by the probability distribution of the original series, we proposed a symbolized time series to replace the original time series to calculate the probability distribution, called SJE; it can be used to measure PAC. Cui et al. proposed a novel approach to quantify the synchronization strength of EEG, which was named as normalized weighted-permutation mutual information (WPMI) [37]. It overcomes the shortcomings of the PMI algorithm. Specifically, in the extraction of the ordinal patterns, only the information of the order structure is reserved, and a large amount of amplitude information is lost. The simulation model shows that WPMI can reflect the amplitude characteristics of the time series while estimating the synchronization strength of the time series. Therefore, we will use the WPMI method to measure PAC. Finally, we will evaluate the performance of these four algorithms (MPMI, PCMI, SJE, and WPMI) through simulation data and hope that these methods can help researchers understand the characteristics of PAC in neural oscillations and further explore the structure and function of the brain.

2. Materials and Methods 2.1. Multiscale Permutation Mutual Information The MPMI is developed based on MPE [32] and mutual information theory. For a time series {x(t); t = 1, 2, ··· , N}, we first average a successively increasing number of data points in non-overlapping windows to construct a continuous coarse-grained time series [38]. s The coarse-grained time series Xk is calculated according to the following formula:

1 ks Xs = x(t), (1) k s ∑ t=(k−1)s+1

where s denotes the scale factor and 1 ≤ k ≤ N/s. The length of each coarse-grained time series is equal to the length of the original time series divided by the scale factor s. When s = 1, the coarse-grained time series is the original time series, and the calculated entropy value is the permutation entropy value. An example of a coarse-graining procedure is presented in Figure1. Entropy 2021, 23, x FOR PEER REVIEW 3 of 17

Entropy 2021, 23, x FOR PEER REVIEW 3 of 17 value is the permutation entropy value. An example of a coarse-graining procedure is presented in Figure 1.

Entropy 2021, 23, 1070 value is the permutation entropy value. An example of a coarse-graining procedure 3is of 16 presented in Figure 1.

Scale 2 x1 x2 x3 x4 x5 x6 xt xt +1

() X1 X 2 X 3 Xk=+ x t x t +1 2 Scale 2 x x1 x x2 x x3 x x4 x x5 x x6 xt xt +1 Scale 3 1 2 3 4 5 6 xt xt +1 xt +2

() X1 X 2 X 3 Xk=+ x t x t +1 2 X X X=() x + x + x 3 x 1 x x x 2 x x k t t++12 t Scale 3 1 2 3 4 5 6 xt xt +1 xt +2 Figure 1. Schematic illustration of the coarse-graining procedure for scales s = 2 and s = 3.

X X X=() x + x + x 3 1 2 k t t++12 t {}s Secondly, when the scale factor is s, calculate the coarse-grained time series X k em- FigureFigure 1. Schematic 1. Schematic illustration illustration ofm,, τthes of coarse the{,,,} coarse-graining s-graining s procedure s procedure for scale for,,,() scaless s = 2s and= 2 ands = 3.s = 3. bedding delay expression ZXXX11,()l= l l +τ ld + − τ , l=1 2 3 M − m − 1 τ , m,,τ s τ Z s s where m isSecondly, the embedding when the dimension, scale factor and is s, is calculate the time the delay. coarse-grained Then, 1,l timeis arranged series{}X in em- Secondly, when the scale factor is s, calculate the coarse-grained time series X k em-k m,τ,s n s s s m,,τ s o m,,τ s= s s··· s = ··· − ( − ) increasingbedding order delay of ma expressiongnitudes.Z When theX ltwo, Xl +valueτ, s ,ofX +(Z1,l− ) are,l equal,1, 2, they 3, areM sortedm 1 τ, bedding delay expression ZXXX1,,()l = {,,,}l d 1, τl=1,,,() 2 3 M − m − 1 τ , 11l l l +τ ld + − τ m,τ,s accordingwhere tom theis thesize embedding of the subscript. dimension, Therefore, and τeachis the of the time sub delay.-vector Then,s inm Z,,mτ s dimensionalis arranged in where m is the embedding dimension, and τ is the time delay. Then, Z 1,lis arranged in π m,τ ,,,!im= 12Zm,τ,s 1,l spaceincreasing is mapped order into of magnitudes. an ordinal When pattern the two valuesi of,,1,l are. equal, Each they of are the sorted increasing order of magnitudes. When the two values of Z m τ s are equal, they are sorted T=according M −() m − to1 τ the sub size-vectors of the subscript.corresponds Therefore, to one of eachm! ofpossible1 the,l sub-vectors arrangements. in m dimensional For in- according to the size of the subscript. Therefore,m,τ ieach= of ···them sub-vectors in Tm =dimensionalM − (m − ) stancespace, if the is mappedembedding into dimension an ordinal patternis set asπ 3,i it, will1, take 2, 6 motifs!. Each for of theeach time series, 1 τ sub-vectors corresponds to one of m! possible arrangements.π m,τ ,,,!im= 12 For instance, if the embedding whichspace is illustrated is mapped in Figure into an2b. Figure ordinal 2a shows pattern an examplei of some permutation. Each of pat- the dimension is set as 3, it will take 6 motifs for each time series, which is illustrated in ternsT= that M may −() m appear − 1 τ subin the-vectors signal. corresponds Then, we can to one calculate of m! the possible number arrangements. of occurrences For of in- Figure2b . Figure2am shows,τ an example of some permutation patterns that may appear in eachstance ordinal, if the pattern embedding π dimension which is indicatedis set as 3, byit willC .take Consequently, 6 motifs for theeach probabilit time series,y m,τ the signal. Then, wei can calculate the number of occurrencesi of each ordinal pattern π which is illustrated in Figure 2b. Figure 2a shows an example of some permutation pat- i PX()which=  m, is indicatedof the occurrence by C . Consequently, of each symbol the probability is obtained P from(X = theπm , reconstructedτ) of the occurrence se- of terns thati may appear in thei signal. Then, we can calculate the numberi of occurrences of each symbol is obtained from the reconstructed sequence: quence: π m,τ each ordinal pattern i which is indicated by Ci . Consequently, the probability m, Ci PX()=  m, m,τ Ci i of the occurrenceP( XP (= ofX = eachπ ) = symbol) = , i =, isi 1,2, = obtained1, 2, ,··· m ! from, m! the reconstructed( 2 se-) (2) i i T quence: T C (a) P( X= m, ) =i , i = 1,2, , m ! (2) i T

(a) M =2 τ=1 M =1 τ=2 M =5 τ=2 M =4 τ=1

M =2 τ=1 M =1 τ=2 M =5 τ=2 M =4 τ=1 (b) M =1 M =2 M =3 M =4 M =5 M =6

(b) M =1 M =2 M =3 M =4 M =5 M =6 M=Motif FigureFigure 2. Example 2. Example of sorting of sorting modes. modes. (a) A segment (a) A segment of real of EEG real signal EEG signal (upper (upper) and some) and someof the ofmotifs the motifs (m = 3,( mand= 3,τ = and 1 orτ 2)= 1(bottom or 2) (bottom). (b) All). (ofb) the All motifs of the motifsare of the are oforder the of order 3 (3!). of 3M (3!). represents M represents the motif the motif of the ofordinal the ordinal patterns patterns.. M=Motif Figure 2. Example of sorting modes. (a) A segment of real EEG signal (upper) and some of the motifs (m = 3, andFollowing, τ = 1 or 2) ( webottom deﬁne). (b the) All probability of the motifs distribution are of the order of {ofx (3t (3!).)} as Mp representsx, and then, the themotif MPE of theof ordinal{x(t)} patternsis deﬁned. as m! MPE(X) = −∑ px log(px) (3) l=1 Consider two-time series {x(t)} and {y(t)}; let MPE(X) and MPE(Y) denote their m,τ m,τ MPE. The joint probability pxy = p(π , π ) of each symbol occurrence in the signal i j Entropy 2021, 23, 1070 4 of 16

be calculated [35]. Thus, the multiscale joint permutation entropy MPE(X, Y) can be denoted by m! m! MPE(X, Y) = −∑ ∑ pxylog(pxy) (4) i=1 j=1 Afterwards, on the basis of MPE and mutual information theory, we can deﬁne MPMI as follows: MPMI(X; Y) = MPE(X) + MPE(Y) − MPE(X, Y) (5)

Finally, the normalized multiscale permutation mutual information (MPMINor) can be deﬁned as MPMI(X; Y) MPMI = (6) Nor min(MPE(X), MPE(Y))

where the MPMINor ranges from 0 to 1. The greater the value of MPMINor, the stronger the relationship between X and Y.

2.2. Permutation Conditional Mutual Information T T Presumably, two-time series X = (X1, X2, ··· , Xn) and Y = (Y1, Y2, ··· , Yn) , based on the Shannon information theory and the permutation method introduced in Section 2.1, the PE and the joint permutation entropy of the time series can be calculated by the following formulas: m! PE(X) = −∑ pxln(px) (7) l=1 m! PE(Y) = −∑ pyln(py) (8) l=1 m! m! PE(X, Y) = −∑ ∑ pxylog(pxy) (9) i=1 j=1 Mathematically, the PCMI can be deﬁned by

δ PCMIX→Y = PCMI(X; Yδ Y) = PE(X, Y) + PE(Yδ, Y) − PE(Y) − PE(X, Yδ, Y) (10)

where Yδ is an observable derived from the state of the process Y’s δ steps delay, i.e., Yδ : yt+δ = yt. In general, the value of δ ranges from 3 to 15 [39]. At some later time points, the information that is transmitted from one process X to another process Y can be deﬁned as N 1 δ PCMIX→Y = ∑ PCMIX→Y (11) N δ=1

where the N is the maximal later points. PCMIX→Y means that the coupling strength of X to Y.

2.3. Symbolic Joint Entropy Given the time series as X = [x(1), x(2), ... , x(N)], sequence reconstruction is to convert X into non-overlapping sub-vectors Xk. The conversion of the new vector is shown as  X1= [x(1), x(2), ··· , x(m)]   X2= [x(m + 1), x(m + 2), ··· , x(2 ∗ m)] (12) .  .  Xk= [x((k − 1) ∗ m + 1), ··· , x(k ∗ m)]

where m represents the embed dimension, k = 1, 2, ··· , M = [N/m]. After generating Xk, we can symbolize Xk in accordance with certain rules to reduce the requirements for the original sequence. The basic principle of the permutation is to replace the original value Entropy 2021, 23, 1070 5 of 16

with anon-negative integer. For a given window, it can be denoted by a symbol according to the permutation rule in the Table1. We take m = 3 as an example, as demonstrated in Table1.

Table 1. The permutation rule in symbolic joint entropy.

Permutation Symbol

xi ≤ xi+1 ≤ xi+2 012 xi ≤ xi+2 ≤ xi+1 021 xi+1 ≤ xi ≤ xi+2 102 xi+1 ≤ xi+2 ≤ xi 120 xi+2 ≤ xi+1 ≤ xi 210 xi+2 ≤ xi ≤ xi+1 201

Therefore, we can get a symbolized time series {sx(t); t = 1, 2, ··· , M ∗ m}. Simi- larly, symbolized time series of the time series {y(t); t = 1, 2, ··· , N} can be expressed as sy(t); t = 1, 2, ··· , M ∗ m . After that, calculate the probability of occurrence of the corresponding time πl in the two symbolized time series: #π p = l l = 1, 2, ··· , m2 (13) l M ∗ m

where #πl denotes its frequency of occurrence in the time series. πl can be expressed as

0 0 0 1 1 1 m − 1 m − 1 π = , , ··· , , , , ··· , , ··· , , ··· (14) l 0 1 m − 1 0 1 m − 1 0 m − 1

Subsequently, the symbolic joint entropy is deﬁned as

m2 ESJ = − ∑ pllog2 pl (15) l=1 pl 6=0

For two series without correlation, after symbolizing, the occurrence probability of all 1 potential symbol vectors is approximately equal, i.e., m2 . In this case, the symbolic entropy achieves the theoretical maximum. It can be deﬁned as

2 m 1 1 = − ( ) = 2 ESJ ∑ 2 log2 2 log2m (16) l=1 m m

On the contrary, the minimum symbol joint entropy (log2m) is obtained when the series are completely linearly correlated. In order to minimize the influence of the parameter m, we further defined a normalized coupling coefficient (CSJ) as

E − log m E = − SJ 2 = − SJ CSJ 1 2 2 (17) log2m − log2m log2m

The index of CSJ is positively correlated with the coupling strength between the two series. The value 1 indicated the perfect inter-series coupling whereas the value 0 suggested no inter-series correlation. Entropy 2021, 23, 1070 6 of 16

2.4. Weighted-Permutation Mutual Information Given two time series X and Y which are embedded into m dimensional space, each of the sub-vectors is assigned to an ordinal pattern πi. Speciﬁcally, the weighted probability pω for each ordinal pattern of X can be calculated as

∑ Iπi (Xi)ωi pω(πi) = (18) ∑ IΠ(Xi)ωi

where Π is a collection of ordinal patterns πi, i = 1, 2, ··· , m!. Iπi (Xi) = 1, if the sub-vector

Xi can be mapped to ordinal pattern πi, otherwise Iπi (Xi) = 0. ωi is a weight which is determined by a speciﬁc feature of each sub-vector Xi. Different features are selected according to the needs, but the relationship ∑i pw(πi) = 1 is always present. The weight ωi is calculated by m 1 2 ωi = ∑ (xi+(k−1)τ − Xi) (19) m k=1

where Xi is mean of sub-vector Xi obtained by the following formula:

1 m Xi = ∑ xi+(k−1)τ (20) m k=1

Therefore, the probability distribution pωx of weighted ordinal pattern in time series X is obtained. pωy can be calculated in the same way. For these two-time series, πi and πj are used to indicate the ordinal patterns of Xk and Yk, respectively. Thus, m! ∗ m! joint ordinal patterns are available. A new method for calculating the probability of each joint ordinal pattern is proposed as follows:

∑ Iπij (Xi; Yj)ωij pω(πij) = (21) ∑ IΠ(Xi; Yj)ωij

where Π is a collection of ordinal patterns πij.Iπij (Xi; Yj) = 1, if the sub-vector Xi and Yj can

be mapped into ordinal pattern πij, otherwise Iπij (Xi; Yj) = 0. Weight ωij = ωiωj, ωi and ωj denote the weight of the sub-vector Xi and Yj, respectively. Thus, the weighted probability distribution of joint ordinal patterns pwxy is obtained. According to the information theory, the WPE of the time series X and Y can be calculated as

m! WPE(X) = −∑ pωxlog(pωx) (22) i=1

m! WPE(Y) = −∑ pωylog(pωy) (23) j=1

m! m! WPE(X, Y) = −∑ ∑ pωxy log(pωxy) (24) i=1 j=1 Then, WPMI is deﬁned as

WPMI(X; Y) = WPE(X) + WPE(Y) − WPE(X, Y). (25)

Afterward, the WPMI can be normalized as

WPMI(X; Y) WPMI = , (26) Nor max{WPMI(X; X), WPMI(Y; Y)} Entropy 2021, 23, 1070 7 of 16

where WPMINor varies from 0 to 1. If X and Y are two perfect synchronous time series, WPMINor equals 1; if there is completely no synchronization between X and Y, WPMINor is close to 0. The stronger the synchronization between X and Y, the higher the WPMINor.

2.5. Acquisition of Simulation Data First, multiple sine waves with different frequencies, amplitudes, and phases are added together to generate the simulation data [40,41]. The low-frequency oscillation (LFO) and high-frequency oscillation (HFO) are denoted by low(t) and high(t), respectively. The low(t) is a 4–8 Hz theta oscillation and the high(t) is a 60–70 Hz gamma oscillation, and the sampling frequency is 1000 Hz. The amplitude of the components was inversely proportional to their frequencies. Both oscillations are normalized to a range of −1 to 1; the purpose is to eliminate the interference caused by amplitude fluctuation and maintain the fluctuation of its instantaneous frequency. The phases were randomly selected from [−π, π]. After that, the Von Mises coupling method is used to generate the amplitude series of HFO [42]: c A (t) = exp[λ × low(t)]. (27) high exp λ The parameter c controls the maximum amplitude of HFO. λ is a concentration parameter. The λ with bigger value generates the larger amplitude of HFO around the preferred phase. Specifically, the λ with a zero value caused the equal amplitudes of the HFO at all phases. In the following simulation analysis, parameters are c = 1, λ = 1. Consequently, the raw data with PAC can be generated as below:

Entropy 2021, 23, x FOR PEER REVIEW Raw = low(t) + A (t) × high(t).8 of (28) 17 high A construction example of the simulation data is shown in Figure3.

(a) ) 1 1

(

d u

t 0 0

p m

A 1 1 (b) 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2

) 0.2 1

(

d u

t 0 0

p m

A 0.2 1 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 (c)

) 1 2

(

d u

t 0 0

p m

A 1 2 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 Time(s) Time(s)

FigureFigure 3. 3. AnAn example example for generating thethe simulationsimulation data. data. (a ()a A) A nonstationary nonstationary slow slow oscillation oscillation (4–8Hz, (4– 8redHz, curve,red curve,left), left normalized), normalized to −1 to 1−1 (right to 1 ),(right aiming), aiming to remove to remove amplitude amplitude ﬂuctuations fluctuations and retain and the retainﬂuctuation the fluctuation of its instantaneous of its instantaneous frequency. frequency. (b) A nonstationary (b) A nonstationary fast oscillation fast oscillation (60–70Hz, (6 blue0–70 curve,Hz, blueleft ),curve, normalized left), norm to −alized1 to 1 (toright −1 to). 1 (c ()right The). amplitude (c) The amplitude envelope envelope of the fast of oscillation the fast oscillation (red, left ).(red, The left). The raw data (magenta, right) were the sum of the normalized slow oscillation and fast oscil- raw data (magenta, right) were the sum of the normalized slow oscillation and fast oscillation signal. lation signal. Afterward, to regulate the PAC strength, the raw data are combined with an interfer- entialAfterward, signal (IFS) to regulate and normalize the PAC in strength, the same the way. raw Therefore, data are combined the original with signal an interfe- can be rentialreconstructed signal (IFS) as and normalize in the same way. Therefore, the original signal can be reconstructed as Raw0 = k × low(t) + (1 − k) × IFS(t) + A (t) × high(t), (29) ()()()()()high Raw = k  low t +1 − k  IFS t + Ahigh t  high t , (29)

The parameter k represents the coupling strength, ranging from 0 to 1. Figure 4 shows the simulation results under different coupling strengths.

2 2 2 )

V 1 1 1

μ ( 0 0 0

1 1 1 Amplitude

2 2 2 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 Time(s) Time(s) Time(s) (a) (b) (c)

Figure 4. Examples of the simulation data with different PAC strengths. (a) k = 0.3. (b) k = 0.5. (c) k = 0.8.

2.6. Calculating Phase–Amplitude Coupling () () The coupling strength between the amplitude Athigh of HFO and phase θlow t of LFO can be measured by MPMI, PCMI, SJE, and WPMI methods. However, the phase series of the low() t is a periodic function with monotonically increase in each cy- cle. Thus, the ordinal patterns of would be very simple, and it is inappropriate for analyzing the coupling between and .Conversely, if there is a prominent ( ( )) PAC between them, the magnitude orders of cos θlow t will be similar to that of

Entropy 2021, 23, x FOR PEER REVIEW 8 of 17

(a) ) 1 1

(

d u

t 0 0

p m

A 1 1 (b) 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2

) 0.2 1

(

d u

t 0 0

p m

A 0.2 1 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 (c)

) 1 2

(

d u

t 0 0

p m

A 1 2 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 Time(s) Time(s)

Figure 3. An example for generating the simulation data. (a) A nonstationary slow oscillation (4– 8Hz, red curve, left), normalized to −1 to 1 (right), aiming to remove amplitude fluctuations and retain the fluctuation of its instantaneous frequency. (b) A nonstationary fast oscillation (60–70Hz, blue curve, left), normalized to −1 to 1 (right). (c) The amplitude envelope of the fast oscillation (red, left). The raw data (magenta, right) were the sum of the normalized slow oscillation and fast oscillation signal.

Afterward, to regulate the PAC strength, the raw data are combined with an interfe- rential signal (IFS) and normalize in the same way. Therefore, the original signal can be reconstructed as Entropy 2021, 23, 1070 8 of 16 ()()()()() Raw = k  low t +1 − k  IFS t + Ahigh t  high t , (29)

The parameter k represents the coupling strength, ranging from 0 to 1. Figure 4 shows the The parameter k represents the coupling strength, ranging from 0 to 1. Figure4 shows the simulation results under different coupling strengths. simulation results under different coupling strengths.

2 2 2 )

V 1 1 1

μ ( 0 0 0

1 1 1 Amplitude

2 2 2 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 Time(s) Time(s) Time(s) (a) (b) (c)

FigureFigure 4. Examples 4. Examples of of the the simulation simulation datadata with with different different PAC PAC strengths. strengths. (a) k (=a) 0.3. k = (0.3.b) k (=b 0.5.) k = ( c0.5.) k = ( 0.8.c) k = 0.8.

2.6.2.6. Calculating Calculating Phase Phase–Amplitude–Amplitude Coupling Coupling The coupling strength between the amplitude A (t) of HFO and phase θ (t) of highAt() low () LFOThe can coupling be measured strength by MPMI, between PCMI, the SJE, amplitude and WPMI high methods. of HFO However, and phase the phase θlow t of LFOseries canθlow be(t )measuredof the low (byt) isMPMI, a periodic PCMI, function SJE, withand monotonicallyWPMI methods. increase However, in each the cy- phase cle. Thus, the ordinal patterns of θ (t) would be very simple, and it is inappropriate series of the low() t is a periodiclow function with monotonically increase in each cy- for analyzing the coupling between θlow(t) and Ahigh(t).Conversely, if there is a promi- clenent. Thus, PAC the between ordinal them, patterns the magnitude of orderswould of becos very(θlow simple,(t)) will and be similar it is inappropriate to that of for A (t). Correspondingly, it would be beneﬁcial for calculating PAC between cos(θ (t)) analyzinghigh the coupling between and .Conversely, if there is alow prominent and Ahigh(t). First of all, by using Hilbert transformation: ( ( )) PAC between them, the magnitude orders of cos θlow t will be similar to that of

iθlow(t) low(t) = Re(Alow(t) ∗ e ), (30)

where θ (t) and A (t) are the phase and amplitude series of the slow oscillation, low low respectively. Thus, cos(θlow(t)) can be generated by dividing low(t) by its instantaneous amplitude Alow(t): low(t) iθlow(t) = Re(e ) = cos(θlow(t)), (31) Alow(t)

Therefore, the amplitude of cos(θlow(t)) is only dependent on the phase time series of the slow oscillation, which is more conducive to the measurement of PAC. Finally, after generating the phase time series cos(θlow(t)), we further use the Hilbert transform to produce the amplitude Ahigh(t) of the fast wave oscillation high(t). Then, the PAC of two time series can be obtained by cos(θlow(t)) and Ahigh(t).

2.7. Parameter Choices in the Algorithm In each algorithm, two key parameters embed dimension m and time lag τ are included. The order m represents the number of data points involved in the motif. Bandt et al. suggested that the value range of m is 3–7 [30], we set m = 3. The lag is referred to as the number of sample points spanned by each motif. Regarding the choice of τ, we refer to the mutual information method proposed by Li et al. [35]. In the simulation data, by comparing the mutual information derived from the different τ values, the lag τ corresponding to the maximal value of mutual information is an optimum parameter. As shown in Figure5 , when τ = 10, the mutual information reaches the maximum value between cos(θlow(t)) and Ahigh(t) with coupling strength k = 0.3, k = 0.5, and k = 0.8. Entropy 2021, 23, x FOR PEER REVIEW 9 of 17

( ( )) . Correspondingly, it would be beneficial for calculating PAC between cos θlow t and () Athigh . First of all, by using Hilbert transformation:

iθ ()t ()(())low low t=Re Alow t e , (30) () () where θlow t and Atlow are the phase and amplitude series of the slow oscillation, re- ( ( )) spectively. Thus, cos θlow t can be generated by dividing low() t by its instantaneous () amplitude Atlow :

low() t iθ ()t ==Re(e low ) cos (θ (t )), () low (31) Atlow ( ( )) Therefore, the amplitude of cos θlow t is only dependent on the phase time series of the slow oscillation, which is more conducive to the measurement of PAC. ( ( )) Finally, after generating the phase time series cos θlow t , we further use the Hilbert () transform to produce the amplitude Athigh of the fast wave oscillation high() t . Then, ( ( )) () the PAC of two time series can be obtained by cos θlow t and Athigh .

2.7. Parameter Choices in the Algorithm In each algorithm, two key parameters embed dimension m and time lag τ are included. The order m represents the number of data points involved in the motif. Bandt et al. suggested that the value range of m is 3–7 [30], we set m = 3. The lag is referred to as the number of sample points spanned by each motif. Regarding the choice of τ, we refer to the mutual information method proposed by Li et al. [35]. In the simulation data, by comparing the mutual information derived from the different τ values, the lag τ corresponding to the maximal value of mutual information is an optimum parameter. As

Entropy 2021, 23, 1070 shown in Figure 5, when τ = 10, the mutual information reaches the maximum value9 of be- 16 ( ( )) () tween cos θlow t and Athigh with coupling strength k = 0.3, k = 0.5, and k = 0.8.

Figure 5. With the coupling strengthstrength kk= = 0.3,0.3, kk == 0.5, kk == 0.8, and τ ranging fromfrom 11 toto 18,18, thethe curvescurves of mutualof mutual information information are are between between the the amplitude amplitude of of the the HFO HFO and and the the phase phase of of the the LFO. LFO. There There were 100were realizations 100 realizations for each for couplingeach coupling strength. strength.

2.8. Parameter Choice in MPMI When analyzing the MPE, the coarse-grainedcoarse-grained process isis veryvery important.important. The speciﬁcspecific Entropy 2021, 23, x FOR PEER REVIEWoperation is to truncate the raw time series into several small sequences with the10 sameof 17 operation is to truncate the raw time series into several small sequences with the same length and average each segment to obtain aa newnew timetime series.series. Therefore, thethe analysisanalysis ofof signal complexity needs t too consider the choice of the scale factor ss inin thethe coarse-grainingcoarse-graining process. The adjacent elements of the original time series contain sequence information. If beprocess. extracted The toadjacent the greatest elements extent, of the so originalthe analysis time effectseries iscontain not obvious. sequence However, information. when If the value of the scale factor s is too small, the information of the relevant fragments cannot thethe complexityvalue of the differencescale factor between s is too small, the signals the information is small, theof the scale relevant factor fragments s should notcannot be be extracted to the greatest extent, so the analysis effect is not obvious. However, when the selected too large, otherwise, the difference between the signals may be eliminated in the complexity difference between the signals is small, the scale factor s should not be selected averaging process. In this study, the MPMI value varies with coupling strength under too large, otherwise, the difference between the signals may be eliminated in the averaging different scale factors s is set. As shown in Figure 6, we can observe that MPMI is propor- process. In this study, the MPMI value varies with coupling strength under different scale tional to k. When s increases, the MPMI curve moves upward as a whole, but the overall factors s is set. As shown in Figure6, we can observe that MPMI is proportional to k. change trend remains unchanged. When k = 0, to avoid a higher MPMI value, s = 3 is When s increases, the MPMI curve moves upward as a whole, but the overall change trend selected. remains unchanged. When k = 0, to avoid a higher MPMI value, s = 3 is selected.

FigureFigure 6 6.. TheThe curve curve of of the the change change of of MPMI MPMI value value with with the the coupling coupling strength strength kk underunder different different scale factorsfactors ss.. There There were were 100 100 realizations realizations for for e eachach coupling coupling strength. strength.

2.9.2.9. Parameter Parameter Choice Choice in in PCMI PCMI WhenWhen applying the PCMIPCMI method,method, it it is is necessary necessary to to consider consider the the selection selection of of parameter param- eterδ because δ because of its of inﬂuence its influence on the on PCMI the PCMI estimation. estimation. The algorithm The algorithm itself determines itself determines that the thatvalue the of valueδ cannot of δ becannot less thanbe less the than order them orderin PCMI m in [ 33PCMI]. In [33]. the simulation In the simulation data, we data, plotted we plottedthe variation the variation curve of curve PCMI of withPCMI the with coupling the coupling strength strengthk when kδ whentakes δ from takes 3 fr toom 10. 3 Asto 10.shown As shown in Figure in Figure7, we can 7, we see can that see with that the with increase the increase of δ, the of changingδ, the chang trending oftrend the of PCMI the PCMIcurve remainscurve remains constant, constant, and increases and increases with k. However,with k. However, when δ = when 10, k =δ 0.7= 10, to k 1, = the 0.7 PCMI to 1, thevalue PCMI dropped, value causingdropped, inaccurate causing inaccurate measurement measurement results. That results. is, when Thatδ takesis, when from δ 3takes to 9, δ fromthe PCMI 3 to 9 estimation, the PCMI is estimation stable. Therefore, is stable. in Therefore, this paper, in we this suggest paper, that we the suggest choice that of the= 5 is an appropriate option. choice of δ = 5 is an appropriate option.

Figure 7. When δ takes different values, the PCMI curve varies with the coupling strength k (100 realizations for each strength).

3. Results 3.1. Dependence on Data Length In order to analyze the influence of data length on four kinds of algorithms, simulated time series with data length from 5 s to 30 s (5000 to 30,000 sample points) are generated by using the simulation data. We conducted 100 experiments on each epoch set of data length when k = 0.3, 0.5, 0.8. It can be seen from Figure 8a,c,d that MPMI, SJE, and WPMI

Entropy 2021, 23, x FOR PEER REVIEW 10 of 17

be extracted to the greatest extent, so the analysis effect is not obvious. However, when the complexity difference between the signals is small, the scale factor s should not be selected too large, otherwise, the difference between the signals may be eliminated in the averaging process. In this study, the MPMI value varies with coupling strength under different scale factors s is set. As shown in Figure 6, we can observe that MPMI is proportional to k. When s increases, the MPMI curve moves upward as a whole, but the overall change trend remains unchanged. When k = 0, to avoid a higher MPMI value, s = 3 is selected.

Figure 6. The curve of the change of MPMI value with the coupling strength k under different scale factors s. There were 100 realizations for each coupling strength.

2.9. Parameter Choice in PCMI When applying the PCMI method, it is necessary to consider the selection of parameter δ because of its influence on the PCMI estimation. The algorithm itself determines that the value of δ cannot be less than the order m in PCMI [33]. In the simulation data, we plotted the variation curve of PCMI with the coupling strength k when δ takes from 3 to 10. As shown in Figure 7, we can see that with the increase of δ, the changing trend of the PCMI curve remains constant, and increases with k. However, when δ = 10, k = 0.7 to 1, the PCMI value dropped, causing inaccurate measurement results. That is, when δ takes Entropy 2021, 23, 1070 from 3 to 9, the PCMI estimation is stable. Therefore, in this paper, we suggest that10 of the 16 choice of δ = 5 is an appropriate option.

FigureFigure 7.7. WhenWhenδ δtakes takes different different values, values, the the PCMI PCMI curve curve varies varies with with the couplingthe coupling strength strengthk (100 k real-(100 izationsrealizations for each for each strength). strength).

3.3. Results 3.1.3.1. Dependence on Data LengthLength In order to analyze the influence of data length on four kinds of algorithms, simulated Entropy 2021, 23, x FOR PEER REVIEW In order to analyze the influence of data length on four kinds of algorithms, simulated11 of 17 time series with data length from 5 s to 30 s (5000 to 30,000 sample points) are generated time series with data length from 5 s to 30 s (5000 to 30,000 sample points) are generated by using the simulation data. We conducted 100 experiments on each epoch set of data by using the simulation data. We conducted 100 experiments on each epoch set of data length when k = 0.3, 0.5, 0.8. It can be seen from Figure8a,c,d that MPMI, SJE, and WPMI length when k = 0.3, 0.5, 0.8. It can be seen from Figure 8a,c,d that MPMI, SJE, and WPMI all present three small fluctuationfluctuation horizontal curves, indicating that these three algorithmsalgorithms are not sensitivesensitive to changeschanges in thethe datadata length.length. In addition,addition, it cancan alsoalso bebe explainedexplained thatthat when the PAC intensityintensity isis constantconstant withinwithin aa period,period, thethe datadata lengthlength hashas almostalmost no effecteffect on the performanceperformance of thesethese threethree algorithms.algorithms. When kk == 0.8,0.8, comparedcompared withwith MPMIMPMI andand SJE, the WPMI curve is more stable,stable, andand thethe performanceperformance isis thethe best.best. As demonstrated in Figure8 8b,b , when k = 0.3, 0.3, 0.5, 0.5, 0.8 0.8,, t thehe three three curves curves show show small fluctu fluctuations,ations, which can reflectreflect the PACPAC strength strength effectively. effectively In. In short, short, these these four four algorithms algorithms can reflectcan reflect the PAC the valuePAC undervalue differentunder differentk within k within 5 s to 30 5 s.s to 30 s.

(a) (b)

(c) (d) Figure 8. Effect of data length on the four methods at different coupling strengths (100 realizations Figure 8. Effect of data length on the four methods at different coupling strengths (100 realizations for each strength).strength). (a)) MPMI. ((b)) PCMI.PCMI. ((c)) SJE.SJE. ((d)) WPMI.WPMI. 3.2. The Effects of Coupling Coefficient on Methods 3.2. The Effects of Coupling Coefficient on Methods To analyze MPMI, PCMI, SJE, and WPMI characteristics with the changing of coupling strength,To analyze coupling MPMI, coefficient PCMI, kSJE,gradually and WPMI increased characteristics from 0 towith 1 with the changing the step ofof 0.05.cou- Figurepling strength,9 plots the coupling relationship coefficient between k gradually MPMI/PCMI/SJE/WPMI increased from 0 to and1 with coupling the step strength of 0.05. kFigure, where 9 plots the PAC the relationship estimators allbetween gradually MPMI/PCMI/SJE/WPMI increase as the k increases. and coupling However, strength when k, 0where≤ k ≤ the0.2 PACor 0.8 estimators≤ k ≤ 1, the all PAC gradually estimators increase increased as the comparatively k increases. slowly.However, It shows when that0k when 0.2 theor coupling0.8k 1 is, the weak PAC or estimators strong, it is increased difficult tocomparatively distinguish anyslowly difference. It shows of PACthat when intensities. the coupling is weak or strong, it is difficult to distinguish any difference of PAC intensities.

(a) (b)

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all present three small fluctuation horizontal curves, indicating that these three algorithms are not sensitive to changes in the data length. In addition, it can also be explained that when the PAC intensity is constant within a period, the data length has almost no effect on the performance of these three algorithms. When k = 0.8, compared with MPMI and SJE, the WPMI curve is more stable, and the performance is the best. As demonstrated in Figure 8b, when k = 0.3, 0.5, 0.8, the three curves show small fluctuations, which can reflect the PAC strength effectively. In short, these four algorithms can reflect the PAC value under different k within 5 s to 30 s.

(a) (b)

Figure 8. Effect of data length on the four methods at different coupling strengths (100 realizations for each strength). (a) MPMI. (b) PCMI. (c) SJE. (d) WPMI.

3.2. The Effects of Coupling Coefficient on Methods To analyze MPMI, PCMI, SJE, and WPMI characteristics with the changing of coupling strength, coupling coefficient k gradually increased from 0 to 1 with the step of 0.05. Figure 9 plots the relationship between MPMI/PCMI/SJE/WPMI and coupling strength k, where the PAC estimators all gradually increase as the k increases. However, when 0k 0.2 or 0.8k 1, the PAC estimators increased comparatively slowly. It shows Entropy 2021, 23, 1070 11 of 16 that when the coupling is weak or strong, it is difficult to distinguish any difference of PAC intensities.

(a) (b)

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(c) (d) Figure 9. Variation of the four methods with the coupling strength (100 realizations). (a) MPMI. Figure(b) PCMI. 9. Variation (c) SJE. (d of) WPMI the four. methods with the coupling strength (100 realizations). (a) MPMI. (b) PCMI. (c) SJE. (d) WPMI. 3.3. The Effects of Noise on Methods 3.3. The Effects of Noise on Methods To analyze the effects of noise on four methods, white Gaussian noise with an SNR varyingTo analyze from −5 thedB to effects 20 dB of in noise steps on of four1 dB is methods, added to white the coupled Gaussian signal. noise The with data an length SNR − varyingis 10,000. from In the 5case dB toof 20k = dB0.3, in 0.5, steps 0.8, ofrespectively, 1 dB is added Figure to the10 plots coupled the signal.variation The of dataPAC k lengthwith white is 10,000. Gaussian In the noise case and of coupling= 0.3, 0.5, strengths 0.8, respectively, of k = 0.3 Figure, 0.5, 0.8 10. In plots order the to variation compare of PAC with white Gaussian noise and coupling strengths of k = 0.3, 0.5, 0.8. In order the four methods in the same coordinate system, the normalization curve is realized by to compare the four methods in the same coordinate system, the normalization curve is dividing the coupling value with noise by the coupling value without noise.

realized by dividing the coupling value with noise by the coupling value without noise.

) )

1 1 ) 1

d d

e e

z z

i i

l l

a a

m m

r r

o o

0.5 SJE 0.5 SJE o 0.5 WPMI

n n

( (

WPMI WPMI ( SJE

C C

A A

MPMI MPMI A MPMI

P P PCMI PCMI P PCMI 0 0 0 5 0 5 10 15 20 5 0 5 10 15 20 5 0 5 10 15 20 SNR (dB) SNR (dB) SNR (dB) (a) (b) (c) Figure 10. Effect of white Gaussian noise (100 realizations for each strength). (a) k = 0.3. (b) k = 0.5. (c) k = 0.8. Figure 10. Effect of white Gaussian noise (100 realizations for each strength). (a) k = 0.3. (b) k = 0.5. (c) k = 0.8.

ItIt couldcould bebe seenseen thatthat inin FigureFigure 10 10a,a, when whenk k= = 0.3, 0.3, SNR SNR from from− −55 dBdB toto 66 dB,dB, comparedcompared withwith thethe otherother threethree methods,methods, SJESJE methodmethod isis closercloser toto thethe couplingcoupling valuevalue whenwhen therethere isis nono noise,noise, andand thethe ability ability to to suppress suppress noise noise better. better. When When the the value value of of SNR SNR ranges ranges from from 10 dB10 dB to 20to dB,20 dB, SJE, SJE, WPMI WPMI and and MPMI MPMI areall are close all close to the to noise-free the noise coupling-free coupling value, value, while thewhile PCMI the valuePCMI increases value increases with the with increase the increase of the SNR, of the until SNR SNR, until = 20 SNR dB, it= still20 dB, does it notstill reach does thenot couplingreach the value coupling under value noise-free under conditionsnoise-free value.conditions Figure value. 10b showsFigure that10b whenshowsk that= 0.5 when and SNRk = 0.5 range and fromSNR isrange−5 dBfrom to 5is dB, −5 dB the to anti-noise 5 dB, the performance anti-noise performance of SJE is slightly of SJE better is slightly than thebetter other than three the methods.other three In methods. Figure 10 Inc, Figure when k10c,= 0.8, when SNR k = ranges 0.8, SNR from ranges−5 dB from to 10 −5 dB, dB WPMIto 10 dB, against WPMI noise against is the noise best, is exceeding the best, SJE.exceeding This is becauseSJE. This under is because the same under data the length, same thedata higher length, the the coupling higher strength, the coupling the better strength, the performance the better the of theperformance WPMI. Moreover, of the WPMI. when SNRMoreover, is from when 7 dB SNR to 20 is dB, from SJE 7 and dB MPMIto 20 dB, have SJE similar and MPMI performance. have similar performance.

3.4.3.4. DetectionDetection ofof thethe FrequencyFrequency PairsPairs withwith PACPAC InIn thethe processprocess ofof collectingcollecting EEGEEG data,data, there will inevitably be some noise in interference.terference. Therefore,Therefore,in in thisthis paper,paper, whitewhite GaussianGaussian noisenoise isis usedused toto simulatesimulate measurementmeasurement noisenoise [[43].43]. In addition, when analyzing EEG signals collected by humans or animals under different behaviors and cognitive situations, we do not know the magnitude of the PAC strength value and the coupling frequency range of each stage in advance. Therefore, it is necessary to measure the PAC to obtain the best possible frequency estimation of the phase and amplitude coupling components. The PAC value can also be used to evaluate the degree of PAC affected under different behaviors or physiological and pathological mechanisms. At present, several methods of exploring PAC have been adopted. Among them, the most popular analysis is the comodulogram, in which a “coupling palette” is computed and used to indicate the strength of the modulation of amplitude by phase within wide ranges of phase-giving and amplitude-giving frequencies, [21,44,45]. Its advantage is that the experimenter can directly observe the frequency range showing the phase and amplitude of strong coupling. However, before the analysis, the possible coupling frequency range must be chosen. We used a comodulogram to test whether these four algorithms can detect PAC frequency pairs in a given frequency range. In order to verify the accuracy of the algorithm, the frequency range of the simulation data was expanded to 1–100 Hz, and the actual PAC coupling frequency range was 4–8 Hz and 60–70 Hz. Usually, the

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In addition, when analyzing EEG signals collected by humans or animals under different behaviors and cognitive situations, we do not know the magnitude of the PAC strength value and the coupling frequency range of each stage in advance. Therefore, it is necessary to measure the PAC to obtain the best possible frequency estimation of the phase and amplitude coupling components. The PAC value can also be used to evaluate the degree of PAC affected under different behaviors or physiological and pathological mechanisms. At present, several methods of exploring PAC have been adopted. Among them, the most popular analysis is the comodulogram, in which a “coupling palette” is computed and used to indicate the strength of the modulation of amplitude by phase within wide ranges of phase-giving and amplitude-giving frequencies, [21,44,45]. Its advantage is that the experimenter can directly observe the frequency range showing the phase and amplitude of strong coupling. However, before the analysis, the possible coupling frequency range must Entropy 2021, 23, x FOR PEER REVIEWbe chosen. We used a comodulogram to test whether these four algorithms can13 detect of 17 PAC

frequency pairs in a given frequency range. In order to verify the accuracy of the algorithm, the frequency range of the simulation data was expanded to 1–100 Hz, and the actual abscissaPAC coupling represents frequency the frequencies range was analyzed 4–8 Hz and as phase 60–70-modulating Hz. Usually,f the, whereas abscissa ampli- represents the frequencies analyzed as phase-modulating f , whereas amplitude-modulatedP f is tude-modulated f is represented in the ordinate axis;P that is, jet colors in a given coor- A represented in theA ordinate axis; that is, jet colors in a given coordinated (x, y) of the dinatedbi-dimensional (x, y) of the map bi indicate-dimensional that themap phase indicate of thethatx thefrequency phase of modulates the x frequency the amplitude modu- of lattheesy thefrequency. amplitude of the y frequency. InIn Figure Figure 11, 11 we, we show show an example an example of applying of applying a comodulogram a comodulogram to simulate to simulate data. The data. comodulograms were constructed by using f calculated in 0.5 Hz steps with 4 Hz The comodulograms were constructed by usingP fP calculated in 0.5 Hz steps with 4 Hz f bandwidthsbandwidths and and fAA calculatedcalculated in in 2 2Hz Hz steps steps with with 10 10Hz Hz bandwidths. bandwidths. Next, Next, in order in order to to simulatesimulate the the measurement measurement noise noise in inthe the EEG EEG signal signal more more realistically, realistically, white white Gaussian Gaussian noise noise with SNR SNR = = 5 5dB dB was was added added to tothe the original original data, data, and and the thePAC PAC comodulogram comodulogram of the of four the four methodmethodss at at kk == 0.8 0.8 was was obtained obtained as as shown shown below. below. The The comodulograms comodulograms of MPMI of MPMI and and WPMI are are relatively relatively clustered clustered and and can can correctly correctly show show the the phase phase–amplitude–amplitude frequency frequency range.range. Although Although SJE SJE is ismore more concentrated concentrated than than thethe MPMI MPMI and and WPMI, WPMI, white white Gaussian Gaussian noisenoise has has an an impact impact o onn it. it. The The comodulogram comodulogram obtained obtained by by PCMI PCMI is is relatively relatively scattered, scattered, the thefrequency frequency range range of theof the HFO HFO becomes becomes wider, wider, and and white white Gaussian Gaussian noise noise has has a greater a greater impact impacton it. Furthermore, on it. Furthermore, the running the running time oftime the of four the algorithmsfour algorithms was alsowas compared.also compared. The SJE Thealgorithm SJE algorithm has the ha fastests the runningfastest running time, WPMI time, WPMI takes thetakes longest the longest time, andtime, MPMI and MPMI is slightly isfaster slightly than faster PCMI. than PCMI.

(a) (b)

(c) (d) FigureFigure 11. 11. TheThe phase phase–amplitude–amplitude comodulograms comodulograms of ofthe the four four methods methods when when white white Gaussian Gaussian noise noise is isadded added to to the the simulated simulated data data (50 (50 realizations). realizations). The The abscissa abscissarepresents represents thethe frequencyfrequency forfor thethe modulatedmodulated phase, and the ordinate represents the frequency for the modulated amplitude. The center phase, and the ordinate represents the frequency for the modulated amplitude. The center frequencies frequencies (LFO: 3–10 Hz with a step of 0.5 Hz; HFO: 30–100 Hz with a step of 2 Hz) of the bandpass(LFO: filter 3–10 corresponding Hz with a step to the of coordinates 0.5 Hz; HFO: in each 30–100 comodulogram. Hz with a step (a) MPMI. of 2 Hz) (b) of PCMI. the bandpass (c) SJE. ﬁlter (correspondingd) WPMI. to the coordinates in each comodulogram. (a) MPMI. (b) PCMI. (c) SJE. (d) WPMI.

3.5. Sensibility to Spurious Coupling The instantaneous sharp edges in the time domain correspond to the broadband harmonic components in the frequency domain. They affect the phase of low-frequency components and the amplitude of high-frequency components. Therefore, even if there is no real physiological interaction, spurious CFC may be generated [46–48]. In the previous simulation analysis, we only considered the interference caused by measurement noise. In fact, in real EEG recordings, noise components may be more complex and intense, such as sharp waveforms and large artifacts. Using simulated data, Kramer and colleagues believe that in some cases, sharp edges in the data may result in coupling in a wide range of frequencies-for-amplitude [49]. However, this type of coupling has not been confirmed in real the scalp or intracranial EEG data until recently, and it is unclear under what circumstances, if at all, such a situation may arise [47,50]. In this study, the Hilbert transform is used to extract the analytic signal with the instantaneous amplitude and phase

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3.5. Sensibility to Spurious Coupling The instantaneous sharp edges in the time domain correspond to the broadband harmonic components in the frequency domain. They affect the phase of low-frequency components and the amplitude of high-frequency components. Therefore, even if there is no real physiological interaction, spurious CFC may be generated [46–48]. In the previous simulation analysis, we only considered the interference caused by measurement noise. In fact, in real EEG recordings, noise components may be more complex and intense, such as sharp waveforms and large artifacts. Using simulated data, Kramer and colleagues believe that in some cases, sharp edges in the data may result in coupling in a wide range of frequencies-for-amplitude [49]. However, this type of coupling has not been Entropy 2021, 23, x FOR PEER REVIEW 14 of 17 conﬁrmed in real the scalp or intracranial EEG data until recently, and it is unclear under what circumstances, if at all, such a situation may arise [47,50]. In this study, the Hilbert transform is used to extract the analytic signal with the instantaneous amplitude and phase components.components. Therefore, it inevitably leads to to non-meaningful non-meaningful amplitude amplitude estimates estimates for for PACPAC computation.computation. Currently,Currently, existingexisting paperspapers havehave studiedstudied thethe detectiondetection ofof spuriousspurious coupling phenom- enaena in signals without without interaction interaction [49]. [49]. Thus, Thus, we we studied studied the the impact impact of spike of spike noise noise on the on thedetection detection of PAC of PAC frequency frequency bands bands by these by these four fouralgorithms algorithms in the in case the caseof strong of strong coupling. coupling.In the simulati In the simulationon data, set data, k = 0.8, set thek = data 0.8, length the data is 10 length s, and is a 10 train s, and of a aspike train (3 of standard a spike (3deviations standard height, deviations 20 ms height, width 20 at ms half width-maximum, at half-maximum, 100 ms inter 100-peak ms inter-peakinterval) are interval) super- areimposed superimposed on the background on the background signal. In signal.the presence In the of presence spurious of coupling, spurious the coupling, phase–am- the phase–amplitudeplitude comodulograms comodulograms of the four of methods the four methodsare obtained, are obtained, as shown as in shown Figure in 12. Figure 12.

(a) (b)

FigureFigure 12. The phase phase–amplitude–amplitude comodulograms comodulograms of of four four methods methods when when a train a train of ofspikes spikes are are su- su- perimposedperimposed on the background signal (50 (50 realizations). realizations). The The abscissa abscissa represents represents the the frequen frequencycy for for thethe modulatedmodulated phase,phase, andand thethe ordinateordinate representsrepresents thethe frequencyfrequency forfor thethe modulatedmodulated amplitude.amplitude. TheThe centercenter frequenciesfrequencies (LFO:(LFO: 3–103–10 Hz with a step of 0.5 Hz; HFO: 30–10030–100 Hz with a step of 2 Hz) of the bandpassbandpass ﬁlterfilter correspondingcorresponding to to the the coordinates coordinates in in each each comodulogram. comodulogram. (a )(a MPMI.) MPMI. (b )(b PCMI.) PCMI. (c) ( SJE.c) (SJE.d) WPMI. (d) WPMI.

WeWe cancan observeobserve thatthat comparedcompared withwith otherother methods,methods, thethe SJESJE hashas aa higherhigher degreedegree ofof aggregationaggregation andand can can better better reflect reflect the the true true coupling coupling frequency frequency range, range, and and spurious spurious coupling cou- haspling less has influence less influence on it. Inon theit. In MPMI the MPMI and WPMI and WPMI methods, methods, the range the ofrange high-frequency of high-fre- rhythmsquency rhythms in their comodulogramsin their comodulograms becomes becomes wider due wider to the due influence to the influence of spurious of coupling.spurious PCMIcoupling. loses PCMI the ability loses tothe detect ability the tocoupling detect the frequency coupling range frequency in the range presence in the of spikepresence noise. of spike noise.

4. Conclusions In this study, the methods based on permutation analysis were applied to measure the PAC strength. The comparison was carefully performed between the performances of SJE and that of MPMI, PCMI, and WPMI. The results demonstrated that compared with other methods, the SJE algorithm exhibited superior performance, including the lower complexity and the highest computational efficiency. Furthermore, the results also show that SJE can better resist additive white Gaussian noise except for high coupling strengths where WPMI is more effective. In addition, we also found that the SJE has the ability to detect the pairs of coupling frequency in a wide frequency range when the signal is mixed with noise of low signal-to-noise ratio. moreover, it is capable to depict the comodulogram of a signal containing spike noise. In conclusion, it suggests that SJE is possibly a better choice to evaluate the PAC under certain conditions.

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4. Conclusions In this study, the methods based on permutation analysis were applied to measure the PAC strength. The comparison was carefully performed between the performances of SJE and that of MPMI, PCMI, and WPMI. The results demonstrated that compared with other methods, the SJE algorithm exhibited superior performance, including the lower complexity and the highest computational efﬁciency. Furthermore, the results also show that SJE can better resist additive white Gaussian noise except for high coupling strengths where WPMI is more effective. In addition, we also found that the SJE has the ability to detect the pairs of coupling frequency in a wide frequency range when the signal is mixed with noise of low signal-to-noise ratio. moreover, it is capable to depict the comodulogram of a signal containing spike noise. In conclusion, it suggests that SJE is possibly a better choice to evaluate the PAC under certain conditions.

Author Contributions: Conceptualization, L.Y.; methodology, F.T.; software, R.H.; validation, F.T.; formal analysis, F.Y.; data curation, Z.L.; writing—original draft preparation, R.H.; writing—review and editing, Z.L.; funding acquisition, F.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded in part by the National Natural Science Foundation of China (61971374, 61603327). Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The authors wish to acknowledge the editor and the anonymous reviewers for their valuable suggestions which improved the quality of this work. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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