LETTERS

Journal of the Physical Society of Japan 81 (2012) 073801
DOI: 10.1143/JPSJ.81.073801

Bekki–Nozaki Hole in Traveling Excited Waves
on Human Cardiac Interventricular Septum
Naoaki BEKKI, Yoshifumi HARADA1 , and Hiroshi KANAI2
College of Engineering, Nihon University, Koriyama, Fukushima 963-8642, Japan
1
Emeritus Professor of Fukui University, Fukui 910-8507, Japan2
Department of Electronic Engineering, Tohoku University, Sendai 980-8579, Japan(Received May 8, 2012; accepted May 29, 2012; published online June 19, 2012)

We observe some phase singularities in traveling excited waves noninvasively measured by a novel ultrasonicmethod, on a human cardiac interventricular septum (IVS) for a healthy young male. We present a possible physicalmodel explaining a part of one-dimensional cardiac dynamics of the observed phase defects on the IVS. We show thatat least one of the observed phase singularities in the excited waves on the IVS can be explained by the Bekki–Nozaki
hole solution in the complex Ginzburg–Landau equation, although the creation and annihilation of phase singularities
on the IVS give birth to a variety of complex patterns.
KEYWORDS: Bekki–Nozaki hole, cardiac dynamics, traveling excited waves, phase singularity

The eﬀects of external electric ﬁelds on cardiac tissue of
rabbits and the propagation of excited waves by producing
standing waves of membrane depolarization have beenstudied both experimentally and numerically.1,2) Using amethod of identifying phase singularities, Gray et al.
elucidated the mechanisms for the formation and terminationof these phase singularities.3)
Also, a variety of complex patterns which include spiralwaves4) and the spontaneous response of the myocardium
to electrical excitation have been observed in human heartby developing an ultrasonic noninvasive novel imaging
modality with high temporal and spatial resolutions.5)
Visualizing the propagation of the myocardial response of
the electric excitation in human hearts during systole, as isshown in Fig. 1, Kanai6) observed the velocity components
toward the ultrasonic probe as waveforms and theirinstantaneous phases of 40 Hz components. A velocity
component corresponding to the contraction was generated
on the septum at a time of T-wave of ECG (end-systole), andpropagated slowly (40 mm/ms) in clockwise direction along
the left ventricle circumferential direction. Thus, the behavior of phase singularities in a human healthy heart is one of
the most interesting subjects in physics and biophysics. In
order to explain its behavior in a human healthy heart, acertain model of explanation is therefore needed on the basisof the direct observation of phase defects. We present here
a possible physical model explaining a part of cardiac
dynamics of these phase singularities on the interventricular
septum (IVS).
In this Letter, we show that at least one of the phasesingularities in the excited waves on a human cardiac IVS
can be explained by the Bekki–Nozaki (BN) hole solution7)in the complex Ginzburg–Landau equation (CGLE).
The CGLE8–15) is well known for one of the simplest
models that account for the behaviors of nonlinear waves
and the spontaneously formed complicated patterns in thespatially extended non-equilibrium systems: the ionization
waves in the glow discharge,16) the chemical oscillations
and turbulence,17) the hydrothermal nonlinear waves in alaterally heated layer,18) and so on. A wide class of nonlinearE-mail: bekki@ge.ce.nihon-u.ac.jp

Fig. 1. (Color online) A snapshot of two-dimensional spatial pattern ofphase ðx; y; tÞ in the excited waves on the IVS for a healthy young male, byusing the novel ultrasonic measurement technique for myocardial motions
in vivo found by Kanai. A typical phase singularity of excited waves on IVS
is shown by a large circle. The cross-sectional image of the color-coded
phase values just before the time of aortic-valve closure at end-systole isshown here. A phase value at a local point on IVS, for example, changes
from cyan (+180 ) near a certain phase-defect, through green (+90 ), andto red (0 ) at the diﬀerent point and time. The left inset shows the scanning
range of the ultrasonic beams in this measurement: LV, left ventricle; LA,left atrium; RV, right ventricle; RA, right atrium; US prove, ultrasonicprove; IVS, interventricular septum; Ao, aorta; ECG, electrocardiogram;
PCG, phonocardiogram (heart sound).

waves for such strong dispersive systems can be describedby the one-dimensional (1D) nonlinear partial diﬀerential
equation which is called CGLE,i@@2¼ þ p 2 ¼ þ qj¼j2 ¼ ¼ i¼;
@t
@x

ð1Þ

where ¼ is a complex function of scaled time t and space x,
and with the two complex coeﬃcients (p ¼ pr þ ipi ,q ¼ qr þ iqi ) and a real positive constant . It is noted thatCGLE is represented by the full coeﬃcients without
rescaling in order to make a direct comparison between
the observed data and the exact solutions of CGLE.
One of the exact solutions of CGLE connects two
diﬀerent patterns speciﬁed by the asymptotic wavenumbers

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J. Phys. Soc. Jpn. 81 (2012) 073801

L ETTERS

Fig. 2. (Color online) Observed local phase proﬁle ðx; tÞ (deg) on the
IVS with a phase-jump at xh ¼ 10:5 mm in the line direction for1415:7 < t < 1424:7 ms and 3:19 < x < 16:95 mm. We can clearlyobserve a proﬁle of traveling phase-jump.

and a phase-jump between two patterns, which is called the
BN hole.7) However, very few experimental investigationsof BN hole have been reported up to now.18,19) The
estimation of the rescaled coeﬃcients of CGLE from theexperimental data has been a diﬃcult task, because somelocalized amplitude holes have been observed in thehydrothermal nonlinear waves, and not adequately comparedwith BN hole solution in CGLE.18)
Let us ﬁrst demonstrate a typical observation related to thephase singularities in the excited waves on the IVS for ahealthy young male, as is shown in Figs. 1–3, by developing
an ultrasonic noninvasive novel imaging modality with hightemporal and spatial resolutions5) which shows that the
propagation of the mechanical wave-front occurs at the endof the cardiac systole by simultaneous measurement of thevibrations at many points (10,000 points) set in the IVS. We
obtained 2D patterns of phase and amplitude of the excited~ y; tÞj,waves on the IVS: ðx; y; tÞ and Aðx; y; tÞ ¼ jðx;where x-axis denotes the line (beam) direction and y-axis
does the circumferential (depth) direction. Indeed, theobserved data of phases and amplitudes demonstrate avariety of complex patterns, which include a zigzag pattern
and the target waves on the IVS.20) Our interest is, however,focussed on 1D cardiac dynamics of phase singularities on
the IVS, which may be viewed as a 1D generalization ofa core of 2D target patterns.6,20) For the ﬁxed y (depth
direction for long-axis view), therefore, let us deﬁne 1D dataof measurements of excited waves as follows:~ tÞ ¼ Aðx; tÞ exp½iðx; tÞ;
ðx;

ð2Þ

where t (ms) denotes an observable time of phasesingularities (T1 < t < T2 ).
As is shown in Fig. 2, we can obtain a pair of asymptotic
local wavenumbers k~j ( j ¼ 1; 2) deﬁned byZ T2
1ðx2 ; tÞ  ðx1 ; tÞ
k~j ¼
dt;
ð3Þ
T2  T1 T1
x2  x1
where k~1 for x1 < x2 < xh and k~2 for xh < x1 < x2 during
T1 < t < T2 , respectively. A position of phase singularity(hole) is denoted by xh , and a life-time of hole in our case is

N. BEKKI et al.

Fig. 3. (Color online) Observed amplitude proﬁle Aðx; tÞ near the phasejump at xh ¼ 10:5 mm in the line direction for 1415:7 < t < 1424:7 ms and3:19 < x < 16:95 mm. We can clearly observe an amplitude-hole withphase singularity and obtain the velocity of the amplitude-hole [see eq. (5)]:c~h ¼ 0:08 mm/ms.

about 10 ms for a very slow speed of hole. We also deﬁne aphase-jump ~ ob ,~ ob ¼ lim sup jðx  xh  ; tÞ  ðx  xh þ ; tÞj; ð4Þ!þ0 x2R

where the phase ðx; tÞ is linearly extrapolated at a ﬁxed
time. Figure 2 shows a typical one-dimensional phase ðx; tÞ(deg) at a certain small region (3:19 < x < 16:95 mm and1415:7 < t < 1424:7 ms). From Fig. 2 and eq. (4), we canobserve a phase-jump ~ ob ¼ 2:9 rad.
Next, let us deﬁne a position x0 of minimum amplitude attime t1 and a position x00 of minimum amplitude at time t2 ,then, in a uniform linear motion of phase singularities, wehave its velocity c~h (mm/ms) deﬁned byc~h ¼

x00  x0:
t2  t1

ð5Þ

As is shown in Fig. 3, we can observe a propagation of thephase singularity on the IVS and we have c~h ¼ 0:08
mm/ms from eq. (5).
Finally, from eqs. (3) and (4), as is shown in Fig. 4, wehave obtained the following fundamental physical quantities
related to BN hole: (i) a pair of asymptotic local
wavenumbers k~1 ¼ 0:13  0:01 mm1 and k~2 ¼ 0:05 
0:005 mm1 , (ii) the phase-jump ~ ob ¼ 2:9  0:1 rad, andthe curvature deﬁned by eq. (11) near the hole j~j ¼
0:39  0:02 mm1 . Here the curvature does not mean thereciprocal of its radius. On the other hand, from eq. (5),(iii) we have also obtained the velocity of phase singularityc~h ¼ 0:08  0:01 mm/ms for 1415:7 < t < 1424:7 ms, as
is shown in Fig. 3. A set of observations obtained from thedata (2) is represented byK~ ob ¼ fk~1 ; k~2 ; c~h ; ~ ob ; j~jg 2 R5 :

ð6ÞSimilarly, we have observed another traveling amplitude
holes at many points in y-axis (circumferential direction) as
well as in x-axis (beam direction) and two-dimensionaltarget waves on the IVS.20)
Solving the reality condition in the bilinear form of
CGLE,7,12) we have an important parameter  mentioned
previously,

073801-2#2012 The Physical Society of JapanL ETTERS

J. Phys. Soc. Jpn. 81 (2012) 073801

N. BEKKI et al.

1

150
~

Bekki-Nozaki hole solution

0.8

Amplitude

Phase [deg]

(a)

k1

100

σ~ ob = 2.9

50

0
−50

−100

0.6

0.4

Observed hole

0.2

~

(b)

k2
0

4

6

8

10

12

14

16

4

6

8

x [mm]

10

12

14

16

x [mm]

6
0.4

(b)

(a)

5

0.2

k1 = 0.133
0k2 = −0.051

σ [rad]Asymptotic wavenumbers

Fig. 4. (a) A snapshot of observed phase-jump near the local point xh ¼ 10:5 mm at a ﬁxed time t ¼ 1421:1. Circle denotes the phase of excited waves
ðx; tÞ (deg). We obtain the asymptotic wave-numbers k~1 ¼ 0:13 mm1 , k~2 ¼ 0:05 mm1 , and the phase-jump ~ ob ¼ 2:9 rad. (b) A snapshot of observed
amplitude hole and the BN hole with the coeﬃcients Ch . Triangle denotes the amplitude hole Aðx; tÞ for the ﬁxed time t ¼ 1421:1. The curvature near theobserved hole deﬁned by eq. (11) is j~j ¼ 0:39 mm1 and the curvature near BN hole is jj ¼ 0:4031.

σ = 2.983

3
2

−0.2
−0.4

4

1

ch = −0.0793−0.8 −0.6 −0.4 −0.2 0

ch = −0.0793

0−0.8 −0.6 −0.4 −0.2 0

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

Velocity of BN hole

Velocity of BN holeFig. 5. (a) Asymptotic wavenumbers k1 and k2 versus the velocity (ch ) of BN hole [0:5 < k1 ðor k2 Þ < 0:5 and 0:8 < ch < 0:8]. From the analytic formof BN hole solution, we obtain k1 ¼ 0:133, k2 ¼ 0:051, and ch ¼ 0:0793, respectively. We have þ ð¼ k1 þ k2 Þ ¼ 0 in case of ch ¼ 0. (b) Phase-jump() versus the velocity (ch ) of BN-hole (0 5  5 2 and 0:8 < ch < 0:8). From eqs. (7) to (13) with the coeﬃcients Ch , we can obtain the value of thephase-jump  ¼ 2:983 rad for ch ¼ 0:0793. We have  ¼  in case of ch ¼ 0.

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 
3 pr qr þ pi qi
2 ¼    þ 2
:2 pr qi  pi qr

ð7Þ

A set of all the coeﬃcients in CGLE is written by
C ¼ fpr ; pi ; qr ; qi ; g 2 R5 ;

ð8Þwhere pi < 0 and qi > 0. Algebraic condition in CGLE
uniquely determines k1 , k2 , ch , argðb2 =b1 Þ and jb2 =b1 j after
algebraic manipulations.
The velocity of propagating BN hole is given bypr qi  pi qr
ðk1 þ k2 Þ:
qi

ch ¼

connects discontinuously two diﬀerent patterns speciﬁedby wavenumbers k1 and k2 .
The curvature near the hole is given byð9Þ

We have a phase-jump (0    2) b2
  arg
b1
9
8 

>
>
p
1
>
>
i
>
>
>
=
< 2 q ðpr qi  pi qr Þ þ  ðpr qr þ pi qi Þ þ  >i;
¼ arctan

 2

>
>
pi
1
>
>
2 2
2 2
>
>
>
>
jqj þ  1 þ 2 jpj 
;
:

q

¼


:
2

We also obtain analytically the ratio  22 1=2
 b2   ¼ ðs1 þ s2 Þ þ ðt1  t2 Þ;b 
ðs1  s2 Þ2 þ ðt1 þ t2 Þ2
1
where
p
r
 p i  ;s1 ¼ pi þ ;
s2 ¼


qr
pi
t1 ¼ pi þ ; t2 ¼ pr þ
 :
qi

Asymptotic wavenumbers k1 and k2 satisfyið10Þ
where  ¼ k1  k2 . It is noted that this phase-jump
073801-3ðþ Þ2 ð Þ2
þ 2 ¼ 1;
a21
a2
a21 ¼

ð11Þ

ð12Þ

ð13Þ

4K2m
;
3pi jqj21þ
ð1 þ 2 Þqi ðpr qi  pi qr Þ
#2012 The Physical Society of Japan

J. Phys. Soc. Jpn. 81 (2012) 073801

a22 ¼

L ETTERS

4K2m;
3qi jpj2
1þpi ðpr qi  pi qr Þ

KBN ¼ K~ ob :where a21 and a22 are positive
constants on account of the real
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
condition and Km ¼ =ðpi Þ (pi < 0).
The Bekki–Nozaki hole solution7) is given by
b1 expð Þ þ b2 expð Þ¼ðx; tÞ ¼
expð Þ þ expð Þ
 Z

i
exp
ðþ þ  tanh xÞ dx  it ; ð14Þ
2
where
¼ x  ch t and  ¼ pr K2m  ch ðk1 k2 þ K2m Þ=ðk1 þ k2 Þ. Equations (9) and (13) give the algebraic relation
between the velocity (ch ) and the asymptotic wavenumbersof BN hole solution with C. It is noted that the family of BN
hole solution can be parametrized by the velocity of BN
hole, as is shown in Fig. 5.
Also, eq. (14) gives the following inequality 2
 b2 
jb1 j2sech2 ð Þ expð Þ    expð Þ
4b1
5 j¼j2 2
 b2 
jb1 j22sech ð Þ expð Þ þ   expð Þ :
5
4
b1

ð15Þ

We have observed two diﬀerent patterns speciﬁedwavenumbers k~1 and k~2 near the phase defect, and thephase-jump between two patterns. The phase-jump occurs at
xh ¼ 10:5 mm, as is shown in Figs. 2 and 3, the amplitude
Aðx; tÞ ¼ j¼ðx; tÞj of excited waves decreases and forms a
dip shaped like a hole. In order to make a direct comparison
between the observed data and the exact solutions of CGLE,
we must ﬁnd a set of all the coeﬃcients in CGLE, taking
into account the Benjamin–Feir instability.14) Finally, we
ﬁnd all the coeﬃcients in CGLE after much trial and error7)
Ch ¼ f1:8; 2:0; 0:5; 1:2; 0:8g:

N. BEKKI et al.

ð16Þ

Let us show numerically that a set of data of BN holesolution (KBN ) is almost equivalent to that of K~ ob . Fromeqs. (7) and (15), we have  ¼ 0:228. Since a pair ofwavenumbers k1 and k2 is chosen so that eq. (13) is satisﬁed,
as is shown in Fig. 5(a), we can obtain k1 ¼ 0:133 andk2 ¼ 0:051. Therefore, from eq. (9), we obtain the velocityof BN hole ch ¼ 0:0793. It is noted that the selection ofwavenumber occurs so that the phase-jump turns into BN
hole. From eq. (11), we have the curvature near BN holejj ¼ 0:4031, as is shown in Fig. 4(b). Substituting theseparameters obtained above into eq. (10), we obtain  ¼
2:983 rad, as is shown in Fig. 5(b). From eq. (12), we havealso jb2 =b1 j ¼ 1:02. As is shown in Fig. 5, we can obtainconsistently KBN :ð17Þ

Since the observation of max jc~h j ¼ 1 mm/ms for several
faster holes is corresponding to eq. (13), as is shown in
Fig. 5, the model of BN hole solution is consistent to theobserved data (2). This suggests that BN hole solution plays
an important role in understanding of one-dimensional
cardiac dynamics of phase singularities as a criterion of ahealthy heart.
Although it is diﬃcult to estimate all the coeﬃcients in
CGLE from the observed data, we found all the corresponding coeﬃcients in our case. Substitution of these coeﬃcients
into eq. (15) gives an amplitude proﬁle of BN hole as inFig. 4(b). Diﬀerent boundary condition of observed holes
from BN hole explains the large deviation from BN hole forj j
1 since we can observe BN holes only in local ﬁnite
small regions on the IVS. Thus, we have shown that at least
one of the observed phase singularities in the excited waves
on the IVS can be explained by BN hole solution of CGLE
with the coeﬃcients Ch .
A variety of complex patterns of phase-defects on the IVS
except for observations of amplitude holes will be publishedelsewhere.20)
Acknowledgments The authors were present in Tohoku when March 11th,
2011 Great East Earthquake happened and wish to dedicate the present paper tothe memory of victims of that disaster. We would like to thank Y. Yoshikawa
and A. Harada of Hitachi Aloka Medical for stimulating discussions about ourdata in ultrasonic equipment (SSD6500).

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