A STUDY OF STOCHASTIC RESONANCE AS A

MATHEMATICAL MODEL OF ELECTROGASTROGRAM

Matsuura Yasuyuki, Miyao Masaru

Graduate School of Information Science, Nagoya University, Furo-cho, Nagoya, Japan

Yokoyama Kiyoko

Graduate School of Design and Architecture, Nagoya City University, 2-1-10 Kitachikusa, Nagoya, Japan

Takada Hiroki

Graduate School of Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui, Japan

Keywords: Electrogastrography, Numerical solution, Wayland algorithm, Lyapunov exponent, Stochastic resonance.

Abstract: An electrogastrogram (EGG) is a recording of the electric activity of the stomach as measured on the

abdominal surface. In this study, our goal is to obtain a mathematical model of an EGG obtained for a

subject in the supine position. Initially, we applied the translation error in the Wayland algorithm to the

EGG in order to measure the degree of determinism. However, we could not determine whether or not the

mathematical model of the EGG could be defined on the basis of a chaotic process. The waveform of the

electric potential in the interstitial cells of Cajal is similar to the graphs of the numerical solutions to the Van

der Pol equation (VPE). We therefore added the VPE to a periodic function and random white noise was

used to represent the intestinal motility and other biosignals, respectively. The EGG and numerical solutions

were compared and evaluated on the basis of the translation error and the maximum Lyapunov exponent.

The EGG was well described by the stochastic resonance in the stochastic differential equations.

1 INTRODUCTION

Percutaneous electrogastrography is a useful method

for examining human gastric electrical activity

without invasion. Human gastric electrical activity

cannot be measured by any other methods such as

magnetic resonance imaging (MRI) or gastro-

fiberscopy. An electrogastrogram (EGG) is

evaluated by comparing the mean frequency and

power values obtained for it to those derived from

the spectrum analysis of previous EGG studies.

However, the amount of information that can be

obtained from such an analysis is limited. Moreover,

EGGs are used less often compared to ECGs, EEGs,

and other biosignals. However, using a mathematical

model for an EGG makes it possible to obtain

additional information.

In 1921, Walter C. Alvarez reported performing

EGG for the first time in humans (Alvarez, 1922). In

EGG, the electrical activity of the stomach is

recorded by placing electrodes on the surface of the

abdominal wall. In the stomach, a pacemaker placed

on the side of the greater curvature generates

electrical activity at a rate of 3 cycles per minute (3

cpm); the electrical signal is then transferred to the

pyloric side (Couturier et al, 1972).

Gastric electrical potential is generated by the

interstitial cells of Cajal (ICCs) (Kenneth and Robert,

2004). ICCs are pacemaker cells that spontaneously

depolarize and repolarize at the rate of 3 cpm.

The waveform of the electric potential in ICCs is

similar to the graphs of the numerical solutions to

the Van der Pol equation. We thus added the Van

der Pol equation to a periodic function and random

white noise was used to represent intestinal motility

and other biosignals.

)()()(

twtsxgradfyx

μα

++−=



(1.1)

)(

twxy

−



(1.2)

The function s(t)=sinωt and white noise w

(t)

respectively represent the weak and random

450

Yasuyuki M., Masaru M., Kiyoko Y. and Hiroki T..

A STUDY OF STOCHASTIC RESONANCE AS A MATHEMATICAL MODEL OF ELECTROGASTROGRAM.

DOI: 10.5220/0003159504500453

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2011), pages 450-453

ISBN: 978-989-8425-35-5

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

intestinal movements and other biosignals (i=1,2).

The double-well potential, f(x), generates

depolarization and repolarization in ICCs.

In this study, the gastrointestinal motility was

measured with the aim of obtaining a

mathematical model of EGG and speculating

factors to describe the diseases resulting from

constipation and erosive gastritis.

Some studies have discussed solutions to the

forward and inverse problems associated with the

dynamics generating the gastric electrical potential.

These studies suggest that it is convenient to use

current dipoles in an ellipsoid and to use computer

simulations to generate a mathematical model for the

stomach. However, results available on non-linear

analyses of the EGG are insufficient. In order to

examine whether or not a mathematical model

describes EGG data appropriately, we have proposed

a projection of time series on a two-dimensional

plane,

trans

-λ, estimated by using the Wayland

and Rosenstein algorithms (Matsuura et al.,

2008).

The Wayland algorithm has been developed in

order to evaluate the degree of determinism for

dynamics that generate a time series (Wayland et al.,

1993). This algorithm can estimate a parameter

called translation error E

trans

to measure the

smoothness of flow in an attractor, which is assumed

to generate the time-series data.

Chaos processes generate complexity in the

attractor, which can be reconstructed from a time

series (Takens, 1981). These processes have a

sensitive dependence on the initial conditions and

can be quantified using the Lyapunov exponent

(Sato et al., 1987; Rosenstein et al., 1993). If the

Lyapunov exponent has a positive value, the

dynamics are considered to be a chaos process. In

this study, Rosenstein’s algorithm (Sato et al., 1987;

Rosenstein et al., 1993) was used to calculate the

maximum Lyapunov exponent (MLE), λ.

According to the analysis of the degree of

determinism for the time series dynamics, the EGG

data obtained 30 min after a subject’s postural

change were significantly different from the initial

EGG data (Matsuura et al, 2008), which were also

regarded as a stationary time series in this study.

During the latter period, the autonomic nervous

system could be represented by a stationary process

because it controls the gastric electrical activity,

which can be measured by an EGG.

In this study, the gastrointestinal motility was

measured with the aim of obtaining a mathematical

model of the stationary EGG, and we examined

whether numerical solutions to the stochastic

resonance (SR) would fit the EGG data.

2 MATERIALS AND METHODS

The subjects were 14 healthy people (7 males and 7

females) aged between 21 and 25 years. A sufficient

explanation of the experiment was provided to all

the subjects, and a written consent was obtained

from them.

2.1 Experimental Procedure

EGGs were obtained at 1 KHz for 150 min for a

subject in the supine position by using an A/D

converter (AD16-16U (PCI) EH; CONTEC, Japan).

The EGGs were amplified using a bio-amplifier

(MT11; NEC Medical, Japan) and recorded using a

tape recorder (PC216Ax; Sony Precision

Technology, Japan).

In this experiment, EGGs are obtained with

electrodes arranged for monopolar recordings

(Vitrode Bs; Nihon Kohden Inc., Tokyo, Japan).

A reference electrode is positioned on the midline

of the patient’s abdomen near the umbilicus. An

active electrode should be placed approximately

10 cm cephalad from the umbilicus and 6 cm to

the patient’s left. It is the position closest to the

pacemaker of gastrointestinal motility.

To remove the noise from the time series of the

EGG data

{

}

1,2,1,0 −= Njy

obtained at 1 kHz,

resampling was performed at 1 Hz. For the analysis,

we obtained a resampled time series

{

}

1)1000/(,2,1,0 −= Nix

as follows:

∑

999

1000

∑

×=

1999

10001

1000

, … ,

∑

+×

×=

9991000

1000

The following delay coordinates were used:

{

}

)1(1

(

−++

mtttt

xxxx "

Here, m represents the embedding dimension. These

delay coordinates could be used to reconstruct a

continuous trajectory without intersections in an

embedding space having a large m. The embedding

delay, τ, is defined as the minimum delay (

0≥

)

when the auto-correlation coefficient is zero. In this

study, we assumed that there was no correlation

when The auto-correlation function initially

decreased to a value below 1/e (

0≥t

2.2 Calculation Procedure

In this study, we numerically solved Equations (1.1)

and (1.2) and verified the SR in the Stochastic

differential equations (SDEs). We converted

A STUDY OF STOCHASTIC RESONANCE AS A MATHEMATICAL MODEL OF ELECTROGASTROGRAM

451

Equations (1.1) and (1.2) into difference equations

and obtained numerical solutions using the Runge–

Kutta–Gill formula for the numerical calculations.

The initial values were set to (0, 0.5). Pseudorandom

numbers were substituted for

)(tw

()

2,1=i

. These

pseudorandom numbers were generated by using the

Mersenne Twister (Matsumoto and Nishimura,

1998). These numerical calculations were performed

for N = 24000 time steps. Each time step was 0.05

units.

The values of the numerical solutions were

recorded after every 20 time steps, which is

equivalent to a signal sampling rate of 1 Hz. For

each value of μ, we obtained 20 numerical solutions

to Equations (1.1) and (1.2).

1) Using Wayland and Rosenstein’s algorithms,

estimate the translation errors (E

trans

) and MLEs (λ)

in the attractors generating EGG data, except for 30

min after the postural change. Then, project the

stationary EGG onto the E

trans

–λ plane.

2) Calculate the mean values (m(i)) of E

trans

and λ for

all of the projections obtained in (1). According to

statistical theory, 95.5% of the EGGs would project

onto the region

{

}

)(2)()(2)(

λσλσ

±×±ℜ mEEm

transtranss

3) Calculate the standard deviations (σ(i)) of E

trans

and λ

for all of the projections obtained in (1).

4) Project the numerical solutions of Equation (1) onto

the E

trans

–λ plane obtained in (1).

5) Count the number of numerical solutions projected

onto region

ℜ

of the E

trans

–λ plane.

6) Calculate the conformity ratio of the number

counted in step (5) to 20, i.e., the number of

numerical solutions for each value of μ.

3 RESULTS

3.1 Subjective Evaluation

We analyzed the EGG data. Wayland and

Rosenstein’s algorithms were applied to the

attractors in the case of all 252 (14 subjects × 18 ×

10 min-EGGs = 252 EGGs) EGG data items.

The attractors of the EGGs were reconstructed in

accordance with Takens’ embedding method. The

form of the attractors could be evaluated by E

trans

and λ. The embedding delays and embedding

dimensions were distributed from 2 (s) to 4 (s) and

from 2 to 7, respectively.

The translation errors were distributed from 0.23

to 0.61. The average ± standard deviation in the

trans

was found to be 0.45 ± 0.10.

The MLEs were distributed from 0.67 to 0.81. All

of the MLEs were greater than 0. The average ±

standard deviation in the MLEs derived from the

EGG data was found to be 0.75 ± 0.024.

3.2 Simulation Evaluation

In the 24000 time steps, there was no exception

wherein the numerical solutions did not diverge

for

20,,2,1 "

; the value of τ derived from the

first component of the numerical solution was not

different from that derived from the second

component. We compared this numerical solution

with the EGG data.

The cross-correlation coefficient between the

observed signal, x(t), and the periodic function, s(t),

was calculated as a substitute for the SNR used in

previous studies in which the occurrence of the SR

was investigated. The cross-correlation coefficient

between the numerical solutions,



, and the periodic

function, s(t), in Equation (1.1). The cross-

correlation coefficient was maximized for a

moderate value of noise intensity, 11 < μ

≤ 12.

Thus, the SR could be generated using Equations

(1.1) and (1.2) with 11 < μ

≤ 12. Numerical

solutions were projected onto the E

trans

–λ plane

(Figure 1)

0.4

0.65

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

trans

Figure 1:

trans

-λ plane (simulation results).

With respect to the EGG data taken 30 min after

the postural change, the amount of EGG data

projected onto region

ℜ

was less than the statistical

standard. In contrast, 100% of the stationary EGG

data was projected on the following region.

{

}{ }

)(2)()(2)(

±×

mEEm

transtrans

We quantitatively examined the conformity of the

numerical solutions in region

ℜ

of the E

trans

-λ

plane. The conformity ratio for μ = 11.6 was the

highest. Equations (1.1) and (1.2) for μ = 11.6 could

be regarded as a mathematical model of the

stationary EGG. Therefore, the SR appropriately

describes the stationary EGG data.

BIOSIGNALS 2011 - International Conference on Bio-inspired Systems and Signal Processing

452

4 DISCUSSION

In this study, we analyzed EGG time-series data

using complex dynamical methods. E

trans

and λ were

calculated from the EGG data.

SDEs were proposed as a mathematical model of

an EGG by Matsuura et al. (2008). SR occurs for an

appropriate coefficient, μ. Some biosystems are

based on the nonlinear phenomenon of SR, in which

the detection of small afferent signals can be

enhanced by the addition of an appropriate amount

of noise (Benzi et al., 1981). We examined whether

or not the SR generated using Equations (1) could

describe an EGG time series. Stationary EGGs are

well described by the SDEs in the case of μ = 11.6,

which might represent the SR (11 < μ

≤ 12). We

herein claim that SR can be regarded as a

mathematical model of an EGG. Moreover,

distribution of the numerical solutions in the SR fits

the distribution of the EGGs, which can be

correlated as shown in Figure 2 (R

= 0.972).

12500

25000

-3.01

-0.84

1.33

3.5

x(t)

[

arb. unit.

]

Times

1250

2500

EGG

Numerical solutions

Figure 2: Distributions of EGGs and numerical solutions

(μ = 11).

The diseases resulting from constipation and

erosive gastritis (an illness in which the inside of the

stomach becomes swollen and painful) are

accompanied by anomalous autonomic nervous

activity. A decline in the electrical activity of the

stomach should change the degree of determinism

trans

) and the complexity (λ) in the attractor

reconstructed from the EGG data. By using the

mathematical model of an EGG, electrogastrography

will be of assistance in the diagnosis of diseases of

the alimentary canal and autonomic nervous system.

5 CONCLUSIONS

As a mathematical model of an EGG, we added the

van der Pol equation to a periodic function and

random white noises that represented the intestinal

motility and other biosignals, respectively. By

projecting the data from a stationary EGG obtained

for a subject in the supine position, along with the

numerical solutions, onto the E

trans

-λ plane, we

qualitatively evaluated the affinity between them.

The SR was statistically the most appropriate with

regard to the mathematical model of the stationary

EGG. It is necessary to further investigate the

reliability of a simplified measurement method by

increasing the number of EGGs studied. The next

step will also involve the suggestion of a

mathematical model for an EGG, derived from data

from elderly subjects and meal tolerance tests.

ACKNOWLEDGEMENTS

This work was supported in part by a Grant-in-Aid

for JSPS Fellows and the Hori Information Science

Promotion Foundation.

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