Proceedings of the International Conference on Man-Machine Systems (ICoMMS)
11 – 13 October 2009, Batu Ferringhi, Penang, MALAYSIA

On the Chaotic Nature of Biological Signals Using
Nonlinear Data Analysis Methodology
Azian Azamimi Abdullah,Yoshifumi Nishio
School of Mechatronic Engineering,Universiti Malaysia Perlis,Perlis, MALAYSIA
Department of Electrical and Electronic Engineering,The University of Tokushima,Tokushima, JAPAN

Abstract- In this study, we analyze the characteristic of biologicalsignals using nonlinear data analysis methodology. Biologicalsignals are not linear so to get a more accurate portrait ofnonlinear signals, we must analyze them with nonlinear analysis
methods.
The nonlinear analysis method is emerging as
relatively new and rapidly growing in biomedical field. One of
the most useful techniques in nonlinear data analysis is theconcept of Lyapunov exponent. As we may know, Lyapunov
exponent is often used to define whether a dynamical system ischaotic or not. If the system exhibits at least one positive
Lyapunov exponent and is purely deterministic, then it is chaotic.
In this work, we measure the finger pulse signal for twenty
minutes in two different situations. Then, we analyze the finger
pulse signal using nonlinear data analysis method. We extractand evaluate Lyapunov exponent parameters from the finger
pulse signal. We finally find the positive value of Lyapunov
exponent and confirm the existence of chaotic nature in
biological systems.

I. INTRODUCTION
The analysis of chaotic behavior in biological signals has
attracted great interest in recent years due to its importantdemands for clinical and biomedical application. Biologicalsignals can be defined as a collective electrical signal acquired
from any organ in human body that represents a physicalvariable of interest. The example of biological signals are
EEG signal which measures the electrical activity of a brain,
EMG signal represents the electrical activity generated frommuscle fibers and also finger pulse signal measures the bloodflow in our body. Biological signals are decidedly nonlinear[1] and usually exhibit complex behavior with nonlinear
dynamic properties. Regarding this matter, the nonlinear
dynamic theory may be a better approach than conventional
linear methods in characterizing the complex nature ofbiological signals. The study of nonlinear dynamic and chaostheory can give the opportunities to develop new approaches
that are needed to understand and control the complex system
in biology and medicine. The literature study on theapplication of the nonlinear dynamics theory to analyze
physiological signals shows that nonlinear methods were used
for analysis of heart rate, nerve activity, arterial pressure, EEG
and respiratory signals [2, 3].
In this study, the finger pulse signal is measured for a certain
time in different situations. In the beginning of experiment,
the finger pulse is measured in relax condition while in theend of experiment, the finger pulse is measured in concentrate
situation which the subject must solve some mathematical
calculation. Then, the time series of this signal is evaluatedand analyzed using nonlinear method. We also make acomparison with the traditional linear method in term ofcharacterizing the complex nature of biological signals. We
choose the Lyapunov Exponent technique as nonlinear
analysis method and Fast Fourier Transform as linear method.
II. EXPERIMENTAL SETUP
The finger pulse signal is measured by using pulse oximetry.
Pulse oximetry is a non-invasive method, which measures thearterial oxygen saturation (SaO2) and allows monitoring of theoxygenating of a subject’s hemoglobin. This method involves
placing a sensor on a thin part of the subject’s anatomy suchas fingertip or earlobe. A light containing both red andinfrared wavelengths is passed from one side to the other.
These light waveforms are transmitted by a semiconductor
light emitting diode and detected by a photo detector. Pulse
oximetry determines SaO2 by analyzing two wavelengths oflight that are absorbed differently by specific hemoglobin
structures in the blood. The relative absorption of thesewavelengths reflects the ratio of oxygenated to totalhemoglobin and this ratio is expressed as a percentage. A
SaO2 of 97% to 99% is considered normal for a healthy
human adult [4].
In this study, the finger pulse signal of one healthy person ismeasured by using pulse oximetry for duration of twenty
minutes. The pulse rate signal is first recorded for the subject
under resting normal condition (relax). In second state, thepulse rate signal is measured with subject solving somemathematical calculation. This state requires subject to be inconcentrate condition because it will involve the thinking
process on how to solve that mathematical calculation. Thenthe data is digitally filtered and analyzed using linear andnonlinear analysis methods.

1C5-1Proceedings of the International Conference on Man-Machine Systems (ICoMMS)
11 – 13 October 2009, Batu Ferringhi, Penang, MALAYSIA

x(t0), x(t0 + t), ..., x(t0 + (m- 1)t)(2)

And denote the distance between these two points as L(t0).
At a later time t1, initial length will evolve to length L0(t1).
In implementation of this program, the following set ofnumerical parameters has to be chosen:

P = (m, t, T, Smax, Smax, thmax)
Fig. 1. Pulse oximetry

III.

ANALYSIS METHODS

A. Fast Fourier Transform (FFT)Fast Fourier Transform (FFT) is an efficient algorithm tocompute the discrete Fourier transform and its inverse [5].
FFT has a great importance to wide variety of applications,from digital signal processing and solving partial differential
equations to algorithms for quickly multiplying large integers.
In biomedical engineering application, FFT are also widelyused for biological signal processing especially EEG signals.
PCs with FFT implemented already in EEG-data acquisition
software were quickly accepted in clinical environment
because FFT gives physicians what they have been using andfast computational time. The only weakness of FFT is that,
this method is a linear method and it can only describe thespectral characteristic of biological signals. The linear method
may be not appropriate to model and analyze living systems or
biological signals because the living systems itself arecomplex, nonlinear and operate far from equilibrium.

(3)

where m is the embedding dimension, t is time delay, T
is an evaluation time and Smax, Smin are the maximum andminimum separations of replacement point respectively.
Therefore, thmax is the maximum orientation error.
IV. SIMULATION RESULTS
Figure 2 shows the time series data of finger pulse signal for
one healthy person. This data is measured for twenty minutes
in two different situations. In the beginning of experiment, thefinger pulse is measured in relax condition while in the end ofexperiment, the finger pulse is measured in concentratecondition where the subject must solve some mathematical
calculation. Time series data for each relax and concentratecondition can be shown as in Figs. 3(a) and (b).

B. Lyapunov Exponent
The algorithm calculating Lyapunov Exponent λfrom timeseries was originally proposed by Wolf et al [6]. Lyapunov
Exponent provide a direct measure of the sensitive
dependence on the initial conditions by quantifying theexponential rates at which neighboring orbits on an attractor
diverge as the system evolves in time. The existence of apositive value of exponent for almost all initial conditions in abounded dynamical system is used to define the deterministicchaos. To discriminate between chaotic dynamics and periodic
signals, Lyapunov exponent is often used. It is a measure of
the rate at which the trajectories separate one from other.
Lyapunoc exponent can characterize the average rate ofdivergence of these neighboring trajectories. A negative
exponent implies that the orbits approach a common fixed
point. A zero exponent means the orbit are on a stable attractor
and finally, a positive exponent implies that the orbits are on achaotic attractor.
This algorithm proposed by Wolf is used to measure theLargest Lyapunov exponent from a time series. For given timeseries x(t) for m dimensional phase space with delay
coordinate t, a point on the attractor is given byFig. 2. Time series data of finger pulse signal for one healthy
person (twenty minutes).

(a)x(t), x(t + t), ..., x(t + (m- 1)t)(1)Then, the nearest neighbor must be located to initial point

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Proceedings of the International Conference on Man-Machine Systems (ICoMMS)
11 – 13 October 2009, Batu Ferringhi, Penang, MALAYSIA(b)
Fig. 3. Time series data of finger pulse signal.
(b) Concentrate.

(a) Relax.

After we measure the finger pulse signal, the time series data
is filtered and analyzed using linear and nonlinear method.
We make a comparison in terms of characterizing thebiological signals by using these two different methods. We
want to confirm which method is the most appropriate foranalyzing and modeling the biological signals. Figure 4 and 5
show the analysis result for FFT method and Lyapunov
Exponent method both for relax and concentrate condition
respectively.

(b)Fig. 4. The analysis result of finger pulse signal using FFT
method. (a) Relax. (b) Concentrate.

We find that it is hard to extract the characteristic of finger
pulse signal using FFT method. From these signals, according
to Nyquist Theorem, one cannot recover Fourier components
with frequencies higher that 35 – 40 Hz.
On the other hand, it is easy to find the different
characteristic of two different conditions by using Lyapunov
Exponent method. As we may know, Lyapunov Exponent is anonlinear analysis method and widely used to define thedeterministic chaos. If a positive λvalue exists in a system, it
can be said that the system is in a chaotic attractor.

(a)

(a)

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Proceedings of the International Conference on Man-Machine Systems (ICoMMS)
11 – 13 October 2009, Batu Ferringhi, Penang, MALAYSIA

methods are still new in medical field. Moreover, thenonlinear analysis methods are costly and need morecomputational time compared with the traditional method.
These results show only a little information on the biologicalsystems since it is the first step for developing nonlinear
analysis methods. In the future, we will make a further
analysis method of other biological signals. We would like toanalyze the changes in EEG activity due to different mental
physiological state.
REFERENCES[1][2](b)Fig. 5. The analysis result of finger pulse signal using
Lyapunov Exponent method. (a) Relax. (b) Concentrate.

Figure 5 (a) and (b) show the analysis result of Lyapunov
Exponent method using Sunday Chaos Times software [7]. In
this analysis, we have chosen an embedding dimension of 3
and the time delay of 1. Table 1 shows the final value of
Lyapunov exponent for both relax and concentrate condition.
From this table, we can conclude that we finally find thepositive exponent of both situations, showing that theexistence of chaotic behavior in biological signals. As we canunderstand from the Lyapunov Exponent theory, the larger thevalue of λ, the more chaotic the system will be. Hence, it isclear to say that the biological signal become less complex
when the person is in relax (normal) condition. On the otherhand, the biological signal become more chaotic if a person do
difficult tasks such as solving mathematical calculation
because it will require more complicated thinking process.

[3]

[4]
[5]

[6]

[7]

TABLE 1
FINAL VALUE OF LYAPUNOV EXPONENTS

Condition
Final λvalue

Relax
0.0126

Concentrate
0.0229V. CONCLUSIONSIn this study, we make a comparison between linear andnonlinear analysis method for characterizing the biologicalsignals. We have extracted the nonlinear parameter fromfinger pulse signal and confirm the existence the chaotic
nature in biological system by using Lyapunov Exponent
method. We also observe that the chaotic nature of biologicalsignals can be reduced if one person is in relaxing condition.
From the analysis result, we can conclude that the nonlinear
analysis approach is more suitable used in modeling thebiological system due that the biological system is alreadynonlinear. But still nonlinear methods are used only inresearch and not in everyday clinical practice because these

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Betolaccini, Bussoliti and Padovini, “ A nonlinear filtering technique forthe identification of biological signals,” IEEE Trans. Of Bio. Engin. :1978; Mar25(2): 159-65. 2
Klonowski W, Olerjarczyk E, Stepien R: Nonlinear Dynamic Algorithm
for Time Series Analysis – Implementation in EEG Data Acquisition /Analysis System . In Attractors, Signals and Synagertics, Frontiers on
Nonlinear Dynamics Volume 1. Edited by: Klonowski W. Lengerich,
Berlin: Pabst Science Publishers; 2002:553-560
Indic P, Pratap P, Nampoori VP, Pradhan N : Significance of Time
Scales in Nonlinear Dynamic Analysis of Electroencephalogram Signals.
IntJNeuroscience 1999, 99 (1-4): 181 – 194.
ADInstruments http://www.adinstruments.com/solutions/research/PulseOximetry/
J. W. Cooley and J. W. Tukey. An algorithm for the machine
calculation of complex Fourier series. Mathematics of Computation,
19:297.301, 1965
A. Wolf, JB. Swift, H. L Swinney, and J.A. Vastano, “Determining
Lyapunov exponents from a time series,” Physica D: Nonlinear
Phenomena 1985; 16(3):285-317.3
Sunday Chaos Times, AIHARA Electrical Engineering Co., LTD.