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Common multifractality in the heart rate variability and brain activity ofCommon multifractality in the heart rate variability and brain activity of

healthy humanshealthy humans

D. C. Lin and A. Sharif

Citation: Chaos 2020, 023121 (2010); doi: 10.1063/1.3427639

View online: http://dx.doi.org/10.1063/1.3427639

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Common multifractality in the heart rate variability and brain activity

of healthy humans

D. C. Lina?and A. Sharif

Department of Mechanical and Industrial Engineering, Ryerson University, Toronto,

Ontario M5B 2K3, Canada

?Received 15 January 2010; accepted 17 April 2010; published online 17 June 2010?

The influence from the central nervous system on the human multifractal heart rate variability

?HRV? is examined under the autonomic nervous system perturbation induced by the head-up-tilt

body maneuver. We conducted the multifractal factorization analysis to factor out the common

multifractal factor in the joint fluctuation of the beat-to-beat heart rate and electroencephalography

data. Evidence of a central link in the multifractal HRV was found, where the transition towards

increased ?decreased? HRV multifractal complexity is associated with a stronger ?weaker? multi-

fractal correlation between the central and autonomic nervous systems. © 2010 American Institute

of Physics. ?doi:10.1063/1.3427639?

The autonomic control of the human heart rate is known

to bring forth multifractal heart rate variability (HRV)

that has both fundamental and clinical significance. De-

spite the extensive integration of the autonomic and cen-

tral nervous systems in the higher brain centers, the role

of the central influence in the multifractal HRV remains

unclear. In this work, we factor out the common multi-

fractal element in the beat-to-beat heart rate and surface

scalp potential (electroencephalography) fluctuations.

Our main result is the evidence of a direct central link to

the multifractality of HRV, where a stronger central and

autonomic correlation implies increased HRV multifrac-

tal complexity in the head-up-tilt maneuver. In the

broader context, our finding implies the importance of

the information processing in the central nervous system

for multifractal HRV generation and constitutes the first

step toward building a HRV paradigm as a dynamical

system driven by the central and autonomic nervous sys-

tems. We further speculate that the current finding could

have relevant implications to the more complex HRV pat-

tern witnessed in healthy subjects and the diminishing

HRV in certain heart disease conditions.

I. INTRODUCTION

Research in recent decades suggests the intriguing link

between the proper functioning of the biological system and

the underlying variability.1–3Human cardiovascular dynami-

cal system represents one such examples of natural impor-

tance. In particular, the fluctuation in human heart rate,

known as the heart rate variability ?HRV?,4,5is highly inter-

mittent and exhibits multifractal scale-free characteristics.

HRV is believed to be important since the health deteriora-

tion in certain heart disease conditions was found to accom-

pany by a diminishing HRV and a less pronounced

multifractality.1,2,4,6Hence, the fluctuation in HRV contains

information that could potentially lead to effective treatment

and better understanding of the complex fluctuation in bio-

logical systems in general.

The sympathetic ?SNS? and parasympathetic ?PNS? divi-

sions of the autonomic nervous system ?ANS? provide the

main regulation to the pace maker cells of the heart.3They

are also closely related to the scale-free property of HRV. For

example, the administration of atropine, a compound that can

block the neurotransmission in the PNS, can effectively di-

minish the multifractality in HRV.7However, when com-

pared to other cardiovascular variables, such as the blood

pressure, HRV shows qualitatively different dynamical

pattern,8suggesting additional factors are in play as far as the

generating mechanism of HRV is concerned.

In general, ANS is known to have sufficient coupling to

the central nervous system ?CNS?. The traffic of this cou-

pling is mainly carried out by the ganglionic nerve cells out-

side the CNS and the preganglionic nerve cells in the spinal

cord and brainstem. These latter connections further provide

the networking to the other parts of the brain.9Given this

extensive integration, a HRV model including the CNS could

provide further insights. Indeed, the changing HRV property

witnessed in stress,9different sleep stages,10and mental

exercises11support the view of a central influence. Concur-

rently, studies also found multifractal characteristics in the

electroencephalography ?EEG? of the surface scalp potential

fluctuation.12It is thus plausible that the CNS could play a

role in the multifractal HRV. The objective of this study is to

examine the potential of such a central influence. As the first

step, we ask how much of the multifractal HRV is correlated

with that of the EEG? Note that multifractality is a property

of the moment of all orders. The multifractal correlation thus

captures more than the second order statistics as in the tra-

ditional correlation measure.

To answer this question, the beat-to-beat heart rate and

EEG from healthy young adults were recorded under the

ANS perturbation induced by the head-up-tilt ?HUT?

maneuver. Due to gravity, the postural change in the HUT

a?Electronic mail: derlin@ryerson.ca.

CHAOS 20, 023121 ?2010?

1054-1500/2010/20?2?/023121/7/$30.00© 2010 American Institute of Physics

20, 023121-1

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maneuver is known to cause a reduction in the afferent

baroreflex traffic13and perturb the ANS toward SNS activa-

tion and PNS withdrawal.14These effects provide a natural

setting to systematically examine a potential CNS link in the

multifractal HRV generation. In this study, we further the

previously introduced multifractal factorization approach15

to define and characterize a common multifractal factor

?CMF? in the heart rate and EEG fluctuations. Using the

proposed method, we are able to extract CMF from the heart

rate and EEG data and measure their multifractal correlation.

We will show that this correlation has a specific meaning in

the multifractal HRV that suggests a direct CNS link.

In Sec. II, the idea of multifractal factorization is intro-

duced and the CMF is defined and characterized. In particu-

lar, our previous approach is extended to measure the multi-

fractal correlation in jointly fluctuating multifractal time

series. In Sec. III, the experimental methods and data analy-

sis are described and the results are shown. In Sec. IV, the

discussion is presented. The conclusion is given in Sec. V.

II. MULTIFRACTAL FACTORIZATION

The idea of multifractal factorization is based on the

joint partition function approach ?JPFA? proposed by

Meneveau et al.16For completeness, we will first summarize

the main points of JPFA and its connection to the notion of

relative multifractality ?RMA?.17,18The main goal of this

section is to introduce the CMF and the estimation of multi-

fractal correlation.

A. Background

Multifractality, in general, refers to the specific scale-

free property contributed by interwoven fractal subsets of

different singularity strength ? ?Hölder exponent?. For time

series, a small ? implies a strongly intermittent fluctuation.

The multifractal analysis provides a decomposition of

such ?-subsets using the parametrization of the ?free? mo-

ment parameter q. The multifractality is normally character-

ized by the dimension of these fractal subsets, f???, as a

function of ?.19Multifractality covering a large ? interval,

W0=?min−?max, f??min?=f??max?=0, means more complex-

ity, with a wide range of different “calm” and “violent” fluc-

tuations coexisting in the data.

JPFA proposed by Meneveau et al. represents an exten-

sion of the above framework. Without loss of generosity,

consider multifractal ?probability? measures, ? and ? in

?0,1?, and a generic partition H, satisfying ??I??a??,

??I??a??, where a=?I?, I?H is the length of the interval.

JPFA provides a joint analysis based on the scaling relation-

ship of the joint partition function,

Z?a;q,p? =?

I?H

??I?q??I?p? a??q,p?.

?1?

RMA was designed to accomplish a similar goal by using

one multifractal measure to gauge the singularity of the

other.17,18Recall, the partition functions of ?,? read

???I?q??I????q?, ???I?q??I????q?, where ?·? denotes the

Lebesgue measure. The main idea of RMA is to replace ?·?.

For example, the multifractality of ? relative to ? is to

replace ?·? by ? and to study the partition function of the

form

???I?q??I?−t?q?= O??I??,

where O denotes the “big O,” O??I??→const. as ?I?→0.

Cole20proved that a unique upper bound for t exists. Let

??/?=sup?t?q??. It characterizes the set with the power law

property ?????/??q?. Let L?c?=??q,p?,??q,p?=c?. A direct

comparison of Eqs. ?1? and ?2? indicates that ??/?can be

obtained from ?q,p??L?0?, where

p = − ??/??q?.

Note that, the same L?0? also describes the RMF of ? with

respect to ?,

?2?

?3?

q = − ??/??p?.

?4?

B. CMF and multifractal correlation

The multifractal factorization means that ??q,p? can be

written as the sum of two terms,

? = ?1+ ?2.

This is achieved by identifying ?1with L?0?. Specifically, for

?q,p??L?c?, one can find ?Qc?q,p?,Pc?q,p???L?0?, such

that ?1?q,p?=??Qc,Pc?=0. With ?1so defined, the “remain-

der term” contributed by the individual multifractality is cap-

tured by ?2=???qc

details. As a result, the factorization in terms of the joint

partition function Z?a;q,p? reads

Z?a;q,p? =??q?p? a?1+?2???q?−??/??q?

??????pc

?5?

??=c or ?2=???pc

??=c; see Ref. 15, for more

??=??q?−??/??q???????qc

??.

?6?

It is interesting to note that this expression shares the same

form as the energy partition function of a statistical system

consisting of two independent components. Hereafter, we

will call ?A, A=?/?, ?/? the CMF.

In the traditional multifractal framework, it is possible to

further the above by considering the Hölder exponent ?Aand

multifractal spectrum f??A?, A=?/?,?/?, associated with

the CMF, i.e.,

?A?q? = d?A?q?/dq,

f??A? = q?A− ?A.

?7?

Let us now study a few important scenarios.

For identical ?=?, it is intuitive that the referenced

measure is viewed completely “uniform” by the referencing

measure. As a result, one expects a simple monofractal type

of scaling relationship. Indeed, in this case, the joint partition

function is simply a reparametrization,

Z?a;q,p? =?

I?H

??I?q??I?p=?

I?H

??I?q+p? a???q+p?,

?8?

i.e., ??q,p?=???q+p?. By definition, L?0?=??q,p?,??q,p?

=0?=??q,p?,???q+p?=0?=?q+p=1?. Hence, the L?0? is

given by the straight line q=1−p and ?A=1. Applying the

Legendre transform, one has a degenerate or a monofractal

description: f??A?=1 for ?A=1 and f??A?=0 elsewhere. In

023121-2D. C. Lin and A. Sharif Chaos 20, 023121 ?2010?

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general, ?Ais bounded. For example, ??/?lies between ??

when ?=?·? and 1/??when ?=?·?. The former follows from

the definition of the multifractal spectrum of ? and the latter

follows from Eqs. ?3?, ?4?, and ?7?.

The estimation of ??q,p? can be effectively carried out

by extending the idea of wavelet transform modulus

maxima21?WTMM? method in one dimension to a joint

WTMM ?JWTMM? approach.22To this end, the partition

function ?1? is written using the wavelet modulus maxima

that are paired together from the nearest available maxima

lines in the time-scale plane ?Appendix A?. Once the joint

scaling exponent ??q,p? is estimated, the CMF is defined

and the calculation of its Hölder exponent is carried out by

Eq. ?7?.

The calculation of the multifractal spectrum of CMF

serves the important purpose to measure the degree of mul-

tifractal correlation. From the monofractal scenario for ?

=? to the more general case discussed above, it is evident

that the stronger the correlation between ?,? is, the nar-

rower f??A? becomes, and vice versa. Since CMF is derived

from the same L?0?, this statement holds regardless of the

“point of view,” A=?/? or ?/?. Thus, one can use the

width of f??A?, to measure the multifractal correlation in

jointly fluctuating scale-free processes. In what follows, this

program will be employed to analyze the potential CNS link

in the multifractal HRV.

III. CMF of EEG and heart rate fluctuations

A. Experimental method and data analysis

We now turn to the experimental data and extract the

CMF in heart rate and EEG fluctuations. Eleven subjects,

eight males and three females ?age: 25.72?4.3 year old;

weight: 69.48?12.2 kg; height: 173.83?8.2 cm?, without

known cardiovascular, pulmonary, and neurological condi-

tions participated in the study. Each subject was fully ex-

plained about the goal and detail of the test reviewed and

approved by the University Ethic Board and signed an in-

formed consent form. The experiment consists of two

?40 min sessions of supine ?control? and upright tests on

two separate days. The tests were arranged to be in approxi-

mately the same time of the day. Prior to each test, subjects

were reminded not to deviate from normal daily activity

and routine, refrain from heavy exercise, and have sufficient

sleep. The test was conducted in a temperature-controlled

and shielded room of slightly dim lighting condition

??200 lx?.

During the test, the subjects were in eyes open and asked

to stay calm and remain steady on a tilt table with foot rest.

In HUT, subjects were first in the supine position ?SUP? for

?10 min before tilted up to a 75° upright position ?UPR?.

There were no syncope event in the tilt test. Electrocardio-

gram ?ECG? ?5-lead? and EEG bipolar measurements ?Inter-

national 10–20 system? were taken simultaneously via a

16-bit ADC ambulatory recorder at a 256 Hz sampling rate

?g.MobiLab, GTEC Inc., Austria?. For the EEG, the elec-

trodes were placed on the frontal ?FP1-FC3, FP2-FC4? and

occipital sites ?CP3-O1, CP4-O2?. Neighboring frontal sites

?AF3-F3, AF6-F4? were also recorded and showed similar

results. The recorder has a, hardcoded, built-in filter that

effects a ?0.01,100? Hz passband on the raw ECG signal and

a ?0.01,30? Hz passband on the raw EEG signal.

In the data analysis, the beat-to-beat R-wave interval se-

quence ?RRi? is extracted from the ECG data. Let RRi

and its time stamp be r?n? and t?n?=?l=1

nr?l?, n=1,2,...,

TABLE I. Spectral characteristics of RRi and EEG: AZ?sympathovagal index?, Z=UPR,SUP, and the ratio of

the EEG spectral powers R?=S?

beta-frequency band spectral powers, respectively ?Appendix B?. All entries are dimensionless quantities.

UPR/S?

SUP, where S?, ?=theta, alpha, beta, denotes the EEG theta-, alpha-, and

SubjectS1S2S3S4S5S6S7S8 S9S10 S11Mean?SD

AUPR

ASUP

AUPR/ASUP

RTheta

RAlpha

RBeta

3.61

1.17

3.09

0.79

0.51

1.04

9.24

0.62

15.82

0.56

0.41

1.64

13.00

2.95

4.40

1.05

1.14

0.90

3.62

2.27

1.59

0.64

0.70

1.40

7.87

1.03

7.67

1.07

0.92

0.77

7.27

2.58

2.82

0.67

1.93

0.73

1.46

0.97

1.50

0.90

0.71

1.62

14.74

1.23

12.02

1.36

0.76

1.13

9.94

0.59

16.93

1.04

1.07

0.95

8.61

1.11

7.74

1.00

1.25

0.77

13.41

1.18

11.33

0.65

0.81

1.33

8.43?4.3

1.43?0.80

7.63?5.47

0.88?0.25

0.93?0.42

1.12?0.33

500 (beat)10 2 (sec)

5

−5

−5

5

msec

µV

µV

msec

547

469

1016

781

RRi (UPR)

RRi (SUP)EEG (SUP)

EEG (UPR)

FIG. 1. ?Color online? Data segments from a typical subject ?S5?. Left

column: RRi in SUP ?top? and UPR ?bottom?. Right column: EEG in SUP

?top? and UPR ?bottom?.

023121-3CNS and multifractal HRV Chaos 20, 023121 ?2010?

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respectively. Denote the EEG by e?s?,s=?t,2?t,... and

?t=1/256 s. To find the CMF, e?s? is first aggregated based

on t?n?,

y?n? =? ?

t?n?

t?n+1?

e?s????t?n + 1? − t?n??.

?9?

The joint partition function Z?a;q,p?=?CR

formed based on the wavelet modulus maxima CR,CEof

R?n?=?nr?i?, E?n?=?ny?i?, respectively. To be specific, the

CMF referenced to the EEG ?E? is considered, p=−?R/E?q?,

for

?q,p??L?0?, andthe

=d log??R/E?/log?q? is calculated to estimate the multifractal

correlation.

qCE

p

is then

Hölderexponent

?R/E

B. Results

On average, there are 3849 uninterrupted RRi in SUP

?mean?SD: 0.965?0.079 s? and 5308 RRi in UPR

?mean?SD: 0.662?0.055 s?. The EEG recordings from the

frontal site electrodes are reported in this study. The record-

ing from a typical subject’s RRi and EEG are shown in Fig.

1. Traditional HRV and EEG characterizations in the fre-

quency domain are conducted ?Appendix B? and summarized

in Table I. These spectral characteristics are interesting in

their own right and may provide a classification of the sig-

nals. The multifractal properties for r?n? and y?n? were found

in both SUP, UPR, using WTMM.23The results from a typi-

cal subject are shown in Fig. 2.

The JWTMM analysis follows the steps outlined in

Sec. II B. The result from a typical subject is shown in Fig.

3. The ?R/E, f??R/E? associated with the CMF in SUP and

UPR are obtained using Eq. ?7? and shown in Fig. 4. Using

the first and second Gaussian derivative wavelets produced

similar results. But higher order Gaussian derivative wave-

lets lead to more fluctuation in Z?a;q,p? and subject the

estimation of ??q,p? to more statistical error. These observa-

tions are consistent to the numerical study reported in the

past.22At this time, we do not have the insight as to why the

0.51 1.52

log(a)

2.53 3.5

−2

0

2

4

6

8

10

12

14

log( Z(a; q) )

0.511.52

log(a)

2.53 3.5

0

2

4

6

8

10

log( Z(a; q) )

0 0.10.20.3 0.40.5

0.8

0.9

1

αr

f(αr)

1 1.52 2.53 3.5

3

5

7

9

log(a)

log( Z(a; q) )

1 1.52

log(a)

2.53 3.5

5

7

9

log( Z(a; q) )

0 0.10.20.3

0.8

0.9

1

αy

f(αy)

(b)(a)

(c)(d)

FIG. 2. ?Color online? Multifractality from the same subject shown in Fig. 1. ?a? Scaling plots of the partition function of the RRi r?n?, Z?a;q??a??q?, in SUP

?left panel? and UPR ?right panel?. ?b? Multifractal spectrum of r?n?, f??r?, in SUP ?“?”? and UPR ?“+”?. Notice the width increase in f??r? in UPR. ?c?

Scaling plots of the partition function of the aggregated EEG y?n?, Z?a;q??a??q?, in SUP ?left panel? and UPR ?right panel?. ?d? Multifractal spectrum of y?n?,

f??y?, in SUP ?“?”? and UPR ?“+”?, In ?a? and ?c?, q=−1.5,−0.5,0,0.5,1.5 ?bottom to top, separated for clarity? and the vertical long-dashed lines indicate

the scaling range.

023121-4D. C. Lin and A. SharifChaos 20, 023121 ?2010?

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