Nonlinear Dyn (2012) 69:263–273DOI 10.1007/s11071-011-0262-5

O R I G I NA L PA P E R

Multifractal Detrended Cross-Correlation Analysis of BVPmodel time series

Chongfeng Xue · Pengjian Shang · Wang Jing

Received: 26 July 2011 / Accepted: 31 October 2011 / Published online: 24 November 2011© Springer Science+Business Media B.V. 2011

Abstract The Bonhoeffer–van der Pol (BVP) oscilla-tor has widely application in the modeling of biolog-ical processes, and it also has rich nonlinear behav-ior including different topological properties betweenthe continuous-time BVP oscillator and the discreteone. Multifractal Detrended Cross-Correlation Anal-ysis (MF-DCCA) has proved useful in describing thecorrelations and detecting the scaling exponent of timeseries. We investigate the autocorrelations and cross-correlations of the time series in BVP model by usingthe MF-DCCA method. Our results show that there ex-ist long range autocorrelations and cross-correlationsin the BVP model time series, and that multifractalfeatures are significant in the analyzed BVP model.

Keywords Autocorrelation · Cross-correlation ·Multifractal Detrended Cross-Correlation Analysis(MF-DCCA) · Bonhoeffer–van der Pol (BVP)model · Multifractality

1 Introduction

Map-based neuron models have received much atten-tion over the past decade, especially in the large-scalenumerical simulation of collective behavior of neuron

C. Xue (�) · P. Shang · W. JingSchool of Science, Beijing Jiaotong University, Beijing100044, P.R. Chinae-mail: xue_right@163.com

networks [1, 2]. As an advantageous simplification,map-based models have shown to be comparable tocontinuous neuron models in reproducing characteris-tic behavior of biological neurons [3–12].

As a simplified version to the Hodgin–Huxleynerve equations [13], the continuous BVP oscillatorcan be transformed into a map-based BVP model byusing the forward Euler scheme. As a representativeof a wide class of nonlinear excitable oscillator, BVPoscillator is taken into account not only because theBVP oscillator has wide applications in the model-ing of biological processes, but also has rich nonlin-ear behaviors including different topological proper-ties between the continuous-time BVP oscillator andthe discrete one. We investigate the classical continu-ous BVP oscillator, and it can be transformed into atwo-dimensional map by using the forward Euler dis-crete scheme. As to the discrete BVP model, we usethe numerical method to get two series changing overtime. Furthermore, we generate two new BVP serieswhich is called NBVP time series, whose initial val-ues are random variables. Then we use the multifractaldetrended cross-correlation analysis (MF-DCCA) forinvestigating the cross-correlations between two char-acteristic quantities.

In recent years, the detrended fluctuation anal-ysis (DFA) method [14, 15] has become a widelyused technique for the determination of fractal scal-ing properties and long-range correlations in non-stationary time series [16]. It has been successfullyapplied to diverse fields such as DNA sequences

264 C. Xue et al.

[14, 15, 17], heart rate dynamics [18, 19], neuronspiking, and so on [20–24]. For two time series,Detrended Cross-Correlation Analysis (DCCA), ageneralization of MF-DFA method, is a newly pro-posed method to investigate the cross-correlations be-tween two characteristic quantities [25]. There are anumber of situations where different signals exhibitcross-correlations. In seismology, the degree of cross-correlation among noise signals taken at different an-tennas of detector arrays is an alert signaling earth-quakes and volcanic eruptions [25]. In finance, risk isestimated on the basis of cross-correlation matrices fordifferent assets and investment portfolios [26].

Previous studies that deal with cross-correlationare based on the dubious assumption that both ofthe analyzed time series are stationary. But unfortu-nately, in many cases, the approaches in the currentliterature may lead to a spurious detection of auto-correlation and cross-correlation. Jun et al. proposeda detrended cross-correlation approach to quantifythe correlations between positive and negative fluc-tuations in a single time series [24]. Based on theprevious study, Podobnik and Stanley proposed De-trended Cross-Correlation Analysis (DCCA) for in-vestigating power-law cross-correlations between twosimultaneously recorded time series in the presenceof nonstationarity [25]. To unveil multifractal featuresof two cross-correlated nonstationary signals, Zhouproposed Multifractal Detrended Cross-CorrelationAnalysis (MF-DCCA) by combining Multifractal De-trended Fluctuation Analysis (MF-DFA) and DCCAapproaches [26].

Therefore, we apply MF-DCCA for investigatingthe cross-correlation between two characteristic quan-tities, namely, electric potential across the cell mem-brane and recovery force in the BVP model. We arriveat the following interesting conclusions: first, nonlin-ear dependency and power-law cross-correlation arefound in the electric potential across the cell mem-brane series and recovery force series; second, thecross-correlation relationship in BVP model is foundto be monofractal; finally, the cross-correlation rela-tionship in NBVP model is found to be multifractal.

The paper is structured as follows. In Sect. 2, adescription of BVP model, MF-DFA and MF-DCCAis provided. In Sect. 3, the results of MF-DFA, MF-DCCA, and multifractal nature of the cross-correlationrelationships are provided. In Sect. 4, the conclusionsare summarized.

2 Proposed methods

2.1 The single map-based BVP model

The classical continuous BVP oscillator may be writ-ten as

x = y − 1

3x3 + x + μ,

y = ρ(a − x − by),

(1)

and it can be transformed into a two-dimensional mapby using the forward Euler discrete s scheme

xn+1 = xn + δ

(yn − 1

3x3n + xn + μ

),

yn+1 = yn + δρ(a − xn − byn),

(2)

where 0 < ρ � 1, 0 < a < 1, 0 < b < 1, μ is a stim-ulus intensity, and 0 < δ < 1 is the step size. Thestate variable x can be thought of the electric poten-tial across the cell membrane, and the other state vari-able y stands for a recovery force. Due to the fact that0 < δρ � 1 is small enough, the evolution of y (or yn)is much slower than that of x (or xn). Thus, we referto xn as the fast variable and yn as the slow variable.

Given a set of variables values and initial values x0

and y0, we can get time series {xi}, {yi} and a relatedoscillograph trace (Fig. 1, Fig. 2) by numerical calcu-lation.

2.2 DCCA method

Cross-correlation is a well-known statistical methodused to establish the degree of correlation between twodifferent time series. This is done considering that sta-tionary characterizes both time series under investiga-tion. Unfortunately, real time series are hardly station-ary and to cure that, as a rule, short intervals are con-sidered for analysis [27–33]. This is not always a validchoice, especially when the time series need to be seenand analyzed as a whole. For this reason, a methodthat deals with non-stationary time series, named de-trended cross-correlation analysis (DCCA) was intro-duced by Podobnik and Stanley. We shall recall themain points of the algorithm here in brief.

The DCCA procedure consists of five steps. Con-sider two series {xi} and {x′

i}, where i = 1,2, . . . ,N ,and N is the maximum number of sample.

Multifractal Detrended Cross-Correlation Analysis of BVP model time series 265

Fig. 1 The picture of theBVP model in (x − t) placefor δ = 0.1, ρ = 0.001,b = 0.5, a = 0.2, μ = 0.1,x0 = 0, y0 = 0.01

Fig. 2 The picture of BVPmodel in (x − y) place forδ = 0.1, ρ = 0.001,b = 0.5, a = 0.2, μ = 0.1,x0 = 0, y0 = 0.01

Step 1: Determine the “profile”

Yi =i∑

k=1[xk − 〈x〉],

Y ′i =

i∑k=1

[x′k − 〈x′〉], i = 1,2, . . . ,N.

(3)

Step 2: The integrated series Yi and Y ′i are divided into

boxes of equal length s.

Step 3: In each box of length s, we fit Yi and Y ′i , using

a polynomial function of order l which represents thetrend in that box (here linear polynomial is used in thefitting procedure).

Step 4: Calculate the covariance of the residuals ineach box:

f 2DCCA(s, v) = 1

s

s∑k=1

(Yk − Yk,v

)(Y ′

k − Y ′k,v

), (4)

266 C. Xue et al.

where Yk,v and Yk,v are the fitting polynomials in seg-ment v, respectively. Then average over all segmentsto obtain the fluctuation function:

F 2DCCA(s) = 1

2Ns

2NS∑v=1

f 2DCCA(s, v). (5)

When {xi} = {x′i}, the detrended covariance F 2

DCCA(s)

reduced to the detrended variance F 2DFA(s) used in the

DFA method.

Step 5: Determine the scaling behavior of the fluctu-ation functions by analyzing log – log plots F 2

DCCA(s)

versus s. For two cross-correlated time series {xi} and{x′

i}, there is a power-law relationship between thefluctuation function F 2

DCCA(s) and the scale s:

F 2DCCA(s) ∼ sλ. (6)

The value of λ represents the degree of the cross-correlation between the two time series {xi} and {x′

i}.

2.3 Multifractal Detrended Cross-CorrelationAnalysis (MF-DCCA)

The Multifractal Detrended Cross-Correlation Analy-sis (MF-DCCA) procedure consists of five steps. Thefirst three steps are essentially identical to the conven-tional DCCA procedure. Consider two series {xi} and{x′

i}, where i = 1,2, . . . ,N , and N is the maximumnumber of sample.

Step 1: Determine the “profile”

Yi =i∑

k=1[xk − 〈x〉],

Y ′i =

i∑k=1

[x′k − 〈x′〉], i = 1,2, . . . ,N.

(7)

Step 2: Divide the profile Yi and Y ′i into Ns =

int(N/s) nonoverlapping segments of equal length s.Since the length N of the series is often not a multipleof the considered time scale s, the same procedure canbe repeated starting from the opposite end. Thereby,2Ns segments are obtained altogether.

Step 3: In each box of length s, we fit Yi and Y ′i , using

a polynomial function of order l which represents thetrend in that box (here linear polynomial is used in thefitting procedure).

Step 4: Calculate the covariance of the residuals ineach box:

f 2DCCA(s, v) = 1

s

s∑k=1

(Yk − Yk,v

)(Y ′

k − Y ′k,v

), (8)

where Yk,v and Yk,v are the fitting polynomials in seg-ment v, respectively. Then average over all segmentsto obtain the q-th order fluctuation function:

Fq(s, v) ={

1

2Ns

2NS∑v=1

(f 2

DCCA(s, v))q/2

}1/q

. (9)

This is the generalization of (5). For q = 2, the stan-dard DCCA procedure is retrieved. One is interestedin how the generalizedqdependent fluctuation func-tion Fq,v(s) depend on the time scale s. It is appar-ent that Fq,v(s) will increase with increasing s. Ofcourse, Fq,v(s) depends on the order m. By construc-tion, Fq,v(s) is only defined for s ≥ m + 2.

Step 5: Determine the scaling behavior of the fluctua-tion functions by analyzing log – log plots Fq,v(s) ver-sus s for each value of q . For two cross-correlated timeseries {xi} and {x′

i}, there is a power-law relationshipbetween the fluctuation function Fq,v(s) and the scales:

Fq(s, v) ∼ sh(q). (10)

For very large scales, s > N/4, Fq,v(s) becomes sta-tistically unreliable because the number of segmentsNs for the averaging procedure in step 4 becomesvery small. Thus, scales s > N4 should be excludedfrom the fitting procedure determining h(q). Besidesthat, systematic deviations from the scaling behaviorin (10), which can be corrected, occur for small scaless ≈ 10. In general, the exponent h(q) in (10) may de-pend on q . For stationary time series, h(2) is identicalto the well-known Hurst exponent H . Thus, we willcall the function h(q) generalized Hurst exponent.

The value of h(0), which corresponds to the limith(q) for q → 0, cannot be determined directly usingthe averaging procedure in (4) because of the divergingexponent. Instead, a logarithmic averaging procedurehas to be employed,

F0(s, v) = exp

{1

4Ns

2Ns∑v=1

ln[F 2(v, s)

]} ∼ sh(0). (11)

Note that h(0) cannot be defined for time series withfractal support, where h(q) diverges for q → 0.

Multifractal Detrended Cross-Correlation Analysis of BVP model time series 267

2.4 Relation to standard multifractal analysis

For stationary, normalized series defining a measurewith compact support the multifractal scaling expo-nents h(q) defined in (10) are directly related, asshown below, to the scaling exponents τ(q) definedby the standard partition function-based multifractalformalism.

Suppose that the series xk, yk of the equal lengthN is a stationary, positive, and normalized sequence,xk ≥ 0, yk ≥ 0 and

∑Nk=1 xk = 1,

∑Nk=1 yk = 1. Then

the detrending procedure in step 3 of the MF-DCCAmethod is not required, since no trend has to be elim-inated. Thus, the DCCA can be replaced by the stan-dard cross-correlation analysis, which is identical tothe DCCA except for a simplified definition of thevariance for each segment v, v = 1, . . . ,Ns , in step 3.

F 2DCCA(v, s) = [

Y(vs) − Y((v − 1)s

)]× [

X(vs) − X((v − 1)s

)]. (12)

Inserting this simplified definition into (9) and using(10), we obtain{

1

2Ns

2Ns∑v=1

∣∣(Y(vs) − Y((v − 1)s

))

× (X(vs) − X

((v − 1)s

))∣∣q/2

}1/q

∼ sh(q). (13)

For simplicity, we can assume that the length N of theseries is an integer multiple of the scale s, obtainingNs = N/S and, therefore,

N/s∑v=1

∣∣(Y(vs) − Y((v − 1)s

))

× (X(vs) − X

((v − 1)s

))∣∣q/2 ∼ sqh(q)−1. (14)

This already corresponds to the multifractal formalismused. In fact, a hierarchy of exponents Hq similar toour h(q) has been introduced based on (14).

In order to relate the MF-DCCA also to the stan-dard textbook box counting formalism, we employ thedefinition of the profile in (3). It is evident that theterm Y(vs) − Y((v − 1)s) and X(vs) − X((v − 1)s)

in (11) is identical to the sum of the numbers yk, xk

within each segment v of size s. This sum is known asthe box probability ps(v) in the standard multifractalformalism for normalized series xk , yk ,

ps(v) =vs∑

k=(v−1)s+1

xk · yk

= (X(vs) − X

((v − 1)s

))× (

Y(vs) − Y((v − 1)s

)). (15)

The scaling exponent τ(q) is usually defined via thepartition function Zq(s),

Zq(s) =N/s∑v=1

∣∣ps(v)∣∣q/2 ∼ sτ(q), (16)

where q is real parameter as in the MF-DCCA above.Sometimes τ(q) is defined with the opposite sign.

Using (15), we see that (16) is identical to (14), andobtain analytically the relation between the two sets ofmultifractal scaling exponents,

τ(q) = qh(q) − 1. (17)

Thus, we have shown that h(q) defined in (15) for theMF-DFA is directly related to the classical multifractalscaling exponents τ(q).

Another way to characterize a multifractal series isthe singularity spectrum f (α), that is, related to τ(q)

via a Legendre transform,

α = τ ′(q) and f (α) = qα − τ(q). (18)

Here, α is the singularity strength or Hölder exponent,while f (α) denotes the dimension of the subset of theseries that is characterized by α. Using (17), we candirectly relate α and f (α) to h(q),

α = h(q) + qh′(q) and f (α) = q[α − h(q)

] + 1.

(19)

3 Analysis and results

3.1 Autocorrelation and cross-correlation of the BVPseries

In order to study the dynamics of the BVP serieschange over time, we first consider two series gener-ated by the BVP Model. We apply the DCCA methoddescribed in the previous section to study the autocor-relation of the time series {xi}, {yi}, and the cross-correlation between the two time series.

In Fig. 3, we can know each of the two series of ab-solute values of the successive differences of {xi}, {yi}exhibits power-law autocorrelations with the scaling

268 C. Xue et al.

Fig. 3 The DCCA fluctuation functions F 2DCCA(n) are shown

versus the scale s in log–log plots for: (a) the blue color isthe autocorrelations with scaling exponents hxx = 1.5822; (b)the red color is the autocorrelations with scaling exponenthyy = 1.10196; (c) the green color is the cross-correlations

with scaling exponent hxy = 1.8494. The time series {xi}, {yi}are generated in the BVP model with a set of variables valuesδ = 0.1, ρ = 0.001, b = 0.5, a = 0.2, μ = 0.1 and initial valuesx0 = 0, y0 = 0.01

Fig. 4 The scaling exponents are shown versus different ini-tial value μ: (a) the change of the scaling exponent of auto-correlation of time series {xi} when the initial value μ changesfrom 0.1 to 1; (b) the change of the scaling exponent of auto-

correlation of time series {yi} when the initial value μ changesfrom 0.1 to 1; and (c) the change of the scaling exponent ofcross-correlation between {xi} and {yi} when the initial value μ

changes from 0.1 to 1

exponents hxx = 1.5822, hyy = 1.10196. Figure 3 alsoshows that the strong cross-correlation between {xi},{yi} exists and can be fit a power law sh with exponenthxy = 1.8494. It is worth for us to noticing the factthat, according to the definition of cross-correlation,each of the variables at any time depends not only onits own past values but also on past values of the othervariable. This fact could well explain the way we gen-erate the two series {xi} and {yi}.

To further study the cross-correlation between thetwo series generated in the BVP model, we examinewhether the two series generated with different initialvalue exist the same cross-correlations. Figure 4 showsthe two series still exhibit strong cross-correlationswith the scaling exponent decrease from 1.5931 to

1.2142 while the scaling exponent of the electric po-tential decreases from 1.3771 to 1.1412 and then in-creases from 1.1412 to 1.1930 when the initial value μ

change from 0.1 to 1.0.

3.2 Multifractal Detrended Cross-CorrelationAnalysis (MF-DCCA) of BVP time series

In Fig. 5(a), the relationship is displayed betweencross-correlation exponent hxy(q) and q (greencurves). To make a comparison, we also estimatethe generalize Hurst exponents hxx(q) of time se-ries {xi} and hyy(q) of time series {yi} by meansof MF-DFA (red color stands for electric poten-tial {xi} and blue color stands for a recovery force

Multifractal Detrended Cross-Correlation Analysis of BVP model time series 269

Fig. 5 (a) The q

dependence of thegeneralized Hurst exponenth(q) determined by fits isshown for MF-DCCA anddifferent parameters.(b) τ(q) = qH − 1 isshown. Series of lengthN = 8000 were analyzed

Fig. 6 The multifractalspectra f (α) are shownversus related α.(a), (b), and (c) illustraterelationships between f (α)

and α in the electricpotential, recovery force,and their cross-correlation

{yi} in Fig. 5(a)). As we know, if the exponent isa constant, the time series is monofractal, other-wise it is multifractal. From this plot in Fig. 5, wecan easily find that the relationships are monofrac-tal because for different q , there are same expo-nents.

By means of (18), the monofractal exponentsτxx(q), τyy(q) and τxy(q) are estimated. FromFig. 5(b), the slopes of tangents of τ(q) can be seen.We can find that τ(q) is linearly dependent on q , andthis is another piece of evidence of monofractality.

Then all the slopes for different q and the multifrac-tal spectra are estimated by means of (19) (see Fig. 6).It is widely known that the multifractal spectrum ofmonofractality is a point or a straight line, namely, thewidth of multifractal spectrum is zero if the system un-der study is monofractal. In Fig. 6, we can find that allthe three curves exhibit monofractal nature.

3.3 The cross-correlation of NBVP series

In the previous section, we study the cross-correlationwith constant initial values. Next, we will generatetwo more complex BVP series which are called NBVPtime series, whose initial value μ is a random vari-able, satisfying μ ∼ Norm(0.2,12) and other initialvalues are constant values, with ρ = 0.001, b = 0.5,a = 0.2, x0 = 0, y0 = 0.01. Then we apply the DCCAand Multi-DCCA method to the NBVP series to studythe cross-correlation and multifractal nature.

3.3.1 Autocorrelation and cross-correlation of theNBVP series

We apply the DCCA method to the NBVP model timeseries to study the autocorrelation of the new series

270 C. Xue et al.

Fig. 7 The DCCA fluctuation functions F 2DCCA(n) on NBVP

model time series are shown versus the scale s in log–log plotsfor (a) the blue color is the autocorrelations of time series{xi} with scaling exponents hxx = 0.6029 (b) the red color is

the autocorrelations of time series {yi} with scaling exponenthyy = 0.9371; and (c) the green color is the cross-correlationsbetween {xi} and {yi} with scaling exponent hxy = 0.7329

Fig. 8 For NBVP modeltime series (a) The q

dependence of thegeneralized Hurst exponenth(q) determined by fits isshown for MF-DCCA.(b) τ(q) = qH − 1 isshown. Series of lengthN = 8000 are analyzed

{xi}, {yi} and the cross-correlation between the two

series. The effect of the method applied on the time se-

ries in NBVP model is illustrated in Fig. 7, where the

autocorrelation of time series {xi} with scaling expo-

nent hxx = 0.6029, the autocorrelation of time series

{yi} with scaling exponent hyy = 0.9371, and the scal-

ing exponent of cross-correlation between time series

{xi} and {yi} hxy = 0.7329. The values of the scal-

ing exponent indicate that the series {xi} {yi} and the

cross-correlation between the two series show long-

term correlation.

3.3.2 Multifractal Detrended Cross-CorrelationAnalysis (MF-DCCA) of NBVP series

Now we apply the MF-DCCA method to the new com-plex time series in NBVP model generated with a ran-dom variable as initial value to investigate the multi-fractal nature of the cross-correlation between the twotime series.

In Fig. 8(a), the relationship between cross-corre-lation exponent hxy(q) and q (green curves) is dis-played. To make a comparison, we also estimate thegeneralized Hurst exponents hxx(q) of time series {xi}and hyy(q) of time series {yi} by means of MF-DFA

Multifractal Detrended Cross-Correlation Analysis of BVP model time series 271

Fig. 9 The multifractalspectra f (α) are shownversus related α. (a), (b),and (c) illustraterelationships between f (α)

and α in the electricpotential, recovery force,and their cross-correlationwith �αx = 0.0614,�αy = 0.1020,�αxy = 0.0872

(red color stands for electric potential {xi} and bluecolor stands for a recovery force {yi} in Fig. 5(a)) inthe NBVP model. From this plot, we can find that therelationships are multifractal because for different q ,there are different exponents; that is, for different q ,there are different power-law cross-correlations.

By means of (18), the multifractal exponentsτxx(q), τyy(q) and τxy(q) are estimated. FromFig. 8(b), we can find that all the three curves τ(q)

are nonlinearly dependent on q and multifractalityexists in electric potential, recovery power, and thecross-correlation between electric potential and recov-ery power. This is another piece of evidence of multi-fractality.

Then all the slopes for different q and the multifrac-tal spectra are estimated by means of (19) (see Fig. 9).Actually, the width of multifractal spectrum can be re-garded as an estimate of multifractal strength. The nu-merical results of the widths are listed as followed:�αx = 0.0614, �αy = 0.1020, �αxy = 0.0872. Es-pecially, for the electric potential, recovery power andtheir cross-correlation, the width of cross-correlationbetween the electric potential and the recovery powermultifractal spectrum is narrower than the separatelyanalyzed recovery power but wider than the electricpotential. The width of the cross-correlation multifrac-tal spectrum is significantly nonzero, which imply thatthere are clear departures from random process for allthe cases.

Based on the above-mentioned results, nonlineardependency and multifractality can be clearly foundbetween the electric potential and the recovery force.Therefore, a researcher or a technical analyst can com-plement her understandings on BVP model by ob-taining more comprehensive knowledge from the inte-

grated point of view rather than from separated quan-tities; and as the dependency exists, one cannot simplyjump to a conclusion as to which variable is the causeof the volatility of the other, since they mutually inter-act.

4 Conclusions

In this paper, we study the autocorrelations, cross-correlations and their multifractality by using MF-DFA and MF-DCCA technique. The technique hasbeen implemented on the time series of absolute val-ues of the successive differences of the original BVPmodel variables. Concretely, our main results are asfollows:

First of all, in the BVP model, there exists powerlaw cross-correlation between the electric potentialand recovery force, which suggests that a large incre-ment of electric potential across the cell membrane ismore likely to be followed by a large increment of re-covery force.

Secondly, in the BVP model monofractality is sig-nificant in cross-correlation between electric potentialacross the cell membrane and the recovery force.

Thirdly, in the NBVP model, there still existspower-law cross-correlation between the electric po-tential and the recovery force. Furthermore, multifrac-tality is significant in the cross-correlation relation-ship.

We would like to stress that although we gener-ate NBVP model with a random variable as initialvalue, there still exist power-law cross-correlations be-tween electric potential across the cell membrane and

272 C. Xue et al.

the recovery force. Furthermore, we can obtain morecomprehensive knowledge from the integrated point ofview rather than from separated quantitie because ofthe significant multifractality of the cross-correlation.

Our present report mainly considers power-law au-tocorrelations and cross-correlations in the time se-ries of the absolute values of given variables. In fact,we have carefully investigated the long-range cross-correlations between coupled BVP model time series{xi} and {yi}, but we do not find long-range cross-correlations between them, although there are power-law autocorrelations themselves.

The findings presented in this paper also suggestthe need for further research. The proposed technol-ogy, though promising, must be regarded as prelimi-nary because of the limitations of our technique usedfor model tests. More techniques such as DMA [27]and HXA [28] are necessary to generalize to MF-DMA and MF-DXA. Finally, the findings presentedhere encourage us to think that this model can be ex-tended to be applied by other technique such as MF-DMA and MF-DXA. We can compare the accuracyand the efficiency of the three techniques. In fact, someeffort is already underway in this area.

Acknowledgements The financial support from the funds ofthe State Key Laboratory of Rail Traffic Control and Safety(RCS2010ZT006), the China National Science (60772036,61071142), and the National High Technology Research De-velopment Program of China (863 Program) (2007AA11Z212)are gratefully acknowledged.

References

1. Rulkov, N.F., Timofeev, I., Bazhenov, M.: Oscillations inlarge-scale cortical networks: map-based model. J. Com-put. Neurosci. 17, 203–233 (2004)

2. Izhikevich, E.M.: Dynamical Systems in Neuroscience:The Geometry of Excitability and Bursting. MIT Press,Cambridge (2007)

3. Rulkov, N.F.: Regularization of synchronized chaoticbursts. Phys. Rev. Lett. 86, 183–186 (2001)

4. De Vries, G.: Bursting as an emergent phenomenon in cou-pled chaotic maps. Phys. Rev. E 64, 051914 (2001)

5. Shilnikov, A.L., Rulkov, N.F.: Subthreshold oscillations ina map-based neuron model. Phys. Lett. A 328, 177–189(2004)

6. Casado, J.M.: Transient activation in a network of coupledmap neurons. Phys. Rev. Lett. 91, 208108 (2003)

7. Casado, J.M., Ibarz, B., Sanjuán, M.A.F.: Winnerless com-petition in networks of coupled neurons. Mod. Phys. Lett.B 18, 1347–1366 (2004)

8. Ibarz, B., Cao, H.J., Sanjuán, M.A.F.: Map-based neuronnetworks. In: Garrido, P.L., Marro, J. (eds.), 9th GranadaSeminar on Computational and Statistical Physics. AIPProceedings, vol. 887, Melville, NY, pp. 69–76 (2007)

9. Ibarz, B., Cao, H.J., Sanjuán, M.A.F.: Patterns in inhibitorynetworks of simple map neurons. Phys. Rev. E 75, 041911(2007)

10. Ibarz, B., Tanaka, H.J., Sanjuán, M.A.F., Aihara, K.:Sensitivity versus resonance in tow-dimensional spiking-bursting neuron models. Phys. Rev. E 75, 041902 (2007)

11. Ibarz, B., Cao, H.J., Sanjuán, M.A.F.: Bursting regimes inmap-based neuron models coupled through fast thresholdmodulation. Phys. Rev. E 77, 05918 (2007)

12. Tanaka, G., Ibarz, B., Sanjuán, M.A.F., Aihara, K.: Syn-chronization and propagation of bursts in a ring of coupledmap neurons. Chaos 16, 013113 (2006)

13. Hodgin, A.L., Huxley, A.F.: A quantitative description ofmembrane current and its application to conduction and ex-citation on nerve. J. Physiol. 117, 500–544 (1952)

14. Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stan-ley, H.E., Goldberger, A.L.: Mosaic organization of DNAnucleotides. Phys. Rev. E 49, 1685 (1994)

15. Ossadnik, S.M., Buldyrev, S.B., Goldberger, A.L., Havlin,S., Mantegna, R.N., Peng, C.-K., Simons, M., Stanley,H.E.: Correlation approach to identify coding regions inDNA sequences. Biophys. J. 67, 64 (1994)

16. Taqqu, M.S., Teverovsky, V., Willinger, W.: Correlated anduncorrelated regions in heart-rate fluctuations during sleep.Fractals 3, 785 (1995)

17. Buldyrev, S.V., Goldberger, A.L., Havlin, S., Mantegna,R.N., Matsa, M.E., Peng, C.-K., Simons, M., Stanley, H.E.:Long-range properties of coding and noncoding DNS se-quences: GenBank analysis. Phys. Rev. E 51, 5084 (1995)

18. Peng, C.-K., Havlin, S., Stanley, H.E., Goldberger, A.L.:Quantification of scaling exponents and crossover phenom-ena in nonstationary heartbeat time series. Chaos 5, 82(1995)

19. Ivanov, P.Ch., Bunde, A., Amaral, L.A.N., Havlin, S.,Fritsch-Yelle, J., Baevsky, R.M., Stanley, H.E., Goldberger,A.L.: Sleep-wake differences in scaling behavior of the hu-man heartbeat: analysis of terrestrial and long-term spaceflight data. Europhys. Lett. 48, 594 (1999)

20. Blesic, S., Milosevic, S., Stratimirovic, D., Ljubisavljevic,M.: Detecting long-range correlations in time series of dor-sal horn neuron discharges. Physica A 268, 275 (1999)

21. Bahar, S., Kantelhardt, J.W., Neiman, A., Rego, H.H.A.,Russell, D.F., Wilkens, L., Budnde, A., Moss, F.: Long-range temporal anti-correlations in paddlefish electrorecep-tors. Europhys. Lett. 56, 454 (2001)

22. Penzel, T., Kantelhart, J.W., Grote, L., Peter, T.H., Bunde,A.: Comparison of detrended fluctuation analysis and spec-tral analysis for heart rate variability in sleep and sleep ap-nea. IEEE Trans. Biomed. Eng. 10, 1143 (2003)

23. Lee, J.-M., Kim, D.-J.: Detrended fluctuation analysis ofEEG in sleep apnea using MIT/BIH polysomnography data.Comput. Biol. Med. 32(1), 37 (2002)

24. Jun, W.C., Oh, G., Kim, S.: Understanding volatility cor-relation behavior with a magnitude cross-correlation. Eur.Phys. J. B 56, 47–52 (2007)

25. Podobnik, B., Stanley, H.E.: Detrended cross-correlationanalysis: a new method for analyzing two nonstationarytime series. Phys. Rev. Lett. 100 (2008)

Multifractal Detrended Cross-Correlation Analysis of BVP model time series 273

26. Zhou, W.-X.: Multifractal detrended cross-correlation anal-ysis for two nonstationary signals. Phys. Rev. Lett. 100(2008)

27. Alessio, E., Carbone, A., Castelli, G., Frappietro, V.:Second-order moving average and scaling of stochastictime series. Eur. Phys. J. B 27, 197 (2002)

28. Barabasi, A.-L., Szepfalusy, P., Vicsek, T.: Multifractalspectre of multi-affine functions. Physica A 178, 17–28(1991)

29. Lin, A.J., Shang, P.J., Zhao, X.J.: The cross-correlationsof stock markets based on DCCA and time-delay DCCA.Nonlinear Dyn. (2011). doi:10.1007/s11071-011-9991-8

30. Xu, N., Shang, P.J., Kamae, S.: Modeling traffic flow corre-lation using DFA and DCCA. Nonlinear Dyn. 61, 207–216(2010)

31. Mora, A., Haase, M.: Quantitative methods for the char-acterization of complex surface structures. Nonlinear Dyn.44, 307–313 (2006)

32. Zhang, T., Liu, J.L.: Multi-dimensional self-affine fractalinterpolation model in tensor form. Nonlinear Dyn. 52, 83–87 (2008)

33. Ben Mabrouk, A.: On some nonlinear nonisotropic quasi-self-similar functions. Nonlinear Dyn. 51, 379–398 (2008)