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2018年度 修 士 論 文
バンドパスフィルタを用いた振動子集団の同期の解析
2019年 1月
お茶の水女子大学大学院人間文化創成科学研究科 情報科学コース
学籍番号: g1740646
稲垣 志帆
指導教員: 工藤 和恵 准教授
Analysis of synchronization in oscillators using a
band-pass filter
by
Shiho Inagaki
Master’s Thesis
Department of Information Sciences
Ochanomizu University
Supervised by Prof. Kazue Kudo
January 2019
概 要異なる固有振動数を持つ振動子集団が作用すると, 同期とよばれる秩序化現象が起こる. 脳では,
異なる周波数を持つ複数の重要な脳波が様々な領域で生成されている. 短期記憶課題など特定の学習を行うと, 脳の部位間に相互作用が生じ, 脳波に同期が起きることが知られている. 時系列データから同期を検出し, 相互作用の向きや強さを推定することで, 各部位の機能や主従関係に関する重要な知見が得られると期待される.
しかし脳波の時系列データは非常に多くのノイズを含むため, そのまま解析するのは困難である. ノイズを含むデータに対しては, 時系列にバンドパスフィルタとよばれる, ある周波数成分のみを残し, 他の周波数成分をカットするフィルタを適用して解析することがある.
また第 1章で紹介する先行研究の多くは, 各振動子の時系列を用いて解析を行っている. しかし,
系によっては, 系を構成する個々の振動子の振る舞いを記録するのは難しく, 振動子集団の総和のみが観測可能な場合もある.
本研究では, 振動子集団の平均的な振る舞いのみを観測できる場合に, バンドパスフィルタを用いることで, 個々の振動子の振る舞いを取り出し, 周期外力との同期を検出できるかを第 2章で示す手順で検証する. 電極によって測定される脳波は, 様々な振動成分を持つ 1次元の時系列と考えられる. そこで, 周期外力の加わる系で, 様々な振動成分を持つ 1次元の時系列データを簡単な数理モデルで作る. この時系列データから観測量のデータとして, 平均的な振る舞いの時系列データを出す. さらに観測量のデータをバンドパスフィルタに通すことで観測位相を算出する. この観測位相の時系列を用いて, 振動数解析および周期外力との同期の検出を行う.
第 3章は単一振動子の場合を扱う. 周期外力と結合していない場合は, フィルタが通す振動数帯に関係なく, 観測位相の実効的な振動数は個々の振動子の固有振動数と一致する. 周期外力と同期している場合は, フィルタが通す振動数帯に関係なく, 観測位相の実効的な振動数は周期外力の振動数と一致する. 周期外力と結合しているが, 同期はしていない場合は, 観測位相の実効的な振動数はフィルタが通す振動数帯に依存する. このことを定性的に説明するために, 解析計算を行う.
第 4章は多振動子の場合を扱う. フィルタの通過帯域幅を適切に設定すると周期外力との同期を検出できることを確かめる. また, フィルタの適切な通過帯域幅は系を構成する振動子の固有振動数の分布に依存する.
これらの結果から, 振動子集団の平均的な振る舞いの時系列データをフィルタに通すと振動数解析を行うことができ, 同期の検出が可能であることがわかった.
キーワード: 同期現象，バンドパスフィルタ, 振動数解析
i
Abstract
Oscillators synchronize in the presence of interaction even if they have different natural frequen-
cies. In particular, that property applies to brainwaves as well. In the brain, several important
brainwaves are generated. When a specific learning task such as a short-term memory task is
performed, interactions among brain regions occur, and brainwaves synchronize. It is expected
that important information on the functions of the brain regions and on the direction of signal
transmissions can be obtained by detecting the synchronization and inferring the direction and
the strength of the interaction using time-series data.
In the data analysis of brainwaves, various types of band-pass filters are used to extract certain
frequency components and to smooth noisy data. In the theoritical study of coupled oscillator
systems, it is often assumed that time-series of all the constituent oscillators can be observed.
However, in most of realistic situations, only global behavior of the assembly of oscillators can
be observed.
In this study, I aim to verify whether synchronization can be detected using a band-pass filter
using a procedure introduced in Chapter 2 even when only the average dynamics of constituent
oscillators can be observed. I generated time-series data using simple mathematical models
describing oscillators with interaction and periodic external forcing. Here I assume that I can
only observe its average behavior. By filtering the observed data, the observation phase is
calculated. The frequency analysis and the detection of synchronization is carried out using the
time-series data of the observation phase.
In Chapter 3, I consider a system consisting of an oscillator subjected to periodic external
force. I first confirm that regardless of the frequency allowed by the band-pass filter, the actual
oscillation frequency of the observation phase equals that of the natural frequency of the oscillator
in the absence of periodic external force. In the presence of external forces, the actual oscillation
frequency of the observation phase always equals to the frequency of the periodic external force
when the oscillator synchronizes well with the external force. However, when the oscillator does
not perfectly synchronize with the periodic external force, the actual oscillation frequency of the
observation phase depends on the frequency allowed by the band-pass filter.
In Chapter 4, I consider a system consisting of multiple oscillators. I found that synchroniza-
tion can be detected when the width of the frequency range of the band-pass filter is appropriate.
The appropriate width of the frequency range of the band-pass filter depends on the distribution
of the natural frequency of the oscillators.
From these results, I confirmed that frequency analysis can be performed by filtering the
observed data and synchronization can be detected.
keywords: Synchronization, Band-Pass Filter, Frequency Analysis
i
目 次
1 Introduction 1
1.1 Synchronization detection using a band-pass filter . . . . . . . . . . . . . . . . . . 1
1.2 Design of the band-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Purpose of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Model 6
2.1 Dynamics of the oscillators with external force . . . . . . . . . . . . . . . . . . . 6
2.2 Condition for synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Definition of observation phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Single Oscillator 11
3.1 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Analytical calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Comparison between synchronization and asynchronization . . . . . . . . . . . . 16
3.4 Strong coupling with a periodic external force . . . . . . . . . . . . . . . . . . . . 19
3.5 Relationship between phase lag and the frequency a band-pass filter passes . . . 20
4 Multiple Oscillators 21
5 Summary and Future Work 23
ii
Chapter 1
Introduction
1.1 Synchronization detection using a band-pass filter
Each oscillator has a natural frequency. When a pair of oscillators with different natural frequen-
cies interact with each other, actual oscillation frequencies vary and eventually become identical.
This phenomena is called synchronization. An experimental study reveals that neuronal activ-
ities of mice synchronize when they perform a working memory task [1]. In the experiment,
neural activities and local field potentials in the medial prefrontal cortex, the vental tegmental
area, and the hippocampus are recorded. The activity in the medial prefrontal cortex and in
the vental tegmental area synchronize. Additionally, 4Hz oscillations and hippocampal theta
oscillations show 1:2 phase locking. A 4Hz oscillation is dominant in the prefrontal cortex-vental
tegmental area circuits and can be observed when subjects perform a short term memory task.
By detecting the synchronization and inferring the direction and the strength of the interaction
using time-series data, it is expected that important information on the function of the brain
regions and on the direction of signal transmissions can be obtained.
Methods for inferring the network structure of interactions of the system have been proposed.
In a system without noise, the dynamics are estimated accurately [2, 3]. The partial recurrence
based synchronization analysis makes it possible to distinguish between direct and indirect in-
teractions in networks of oscillatory systems with multiple time scales [2]. A previous study
considered a system of N weakly coupled nearly identical limit cycle oscillators, and found
that the multiple-shooting method combined with the cross-validation technique is most effi-
cient when data are taken from the nonsynchronized regime [3]. Even with noise, parameters
in mathematical models can be inferred in some systems [4, 5]. By using Baysian inference, the
parameters of the Lorentz system and the system of five copled oscillators are inferred accu-
rately [4]. Time-evolving parameters can also be inferred with Baysian inference. If the system
synchronize, the accuracy of the inference gets poor. In order to demonstrate the method on
real biological data, cardiorespiratory measurements from resting human subjects whose paced
respiration was ramped down with decreasing frequency was analyzed. By inferring coupling
parameters, analyses confirm that respiration-to-heart is dominant [5]. The method based on
Baysian inference can successfully estimate the coupling function between theta oscillation and
the speech sound envelope. The theta oscillation plays an important role in speech processing.
The theta activity does not influence speech sound. The estimated dynamics shows that the
coupling function from the theta oscillation to speech sound has smaller amplitude than the
coupling function in the opposite direction [6].
1
Filters are often used in order to deal with the noisy data [7-9]. When subjects are shown
’Mooney’ faces, gamma waves synchronize in long-distance if they perceive a ’Mooney’ face as a
face [9]. A gamma wave is a pattern of brain waves which can be observed when subjects perceive
something, and its typical frequency is 40 Hz. The electroencephalogram data are band-pass
filtered. The filter pass the frequency range f0±3 Hz, where f0 is the gamma wave frequency
of each subject. ϕi(f0, t, k), which is a complex number of unit magnitude, is an instantaneous
phase value at the electronode i, for a chosen frequency f0, at time t, and trial k ∈ 1, . . . , N.The phase-locking value is computed as
ϕij(f0, t) =
∣∣∣∑Nk=1 (ϕi(f0, t, k)− ϕj(f0, t, k))
∣∣∣N
. (1.1)
If the instantaneous phase difference between electronode i and electronode j is constant, the
phase-locking value equals 1. If the instantaneous phase difference between electronode i and
electronode j is random, the phase-locking value equals 0.
Figure 1.1(a) is recognized as a face when presented in upright orientation. Figure 1.1(b)
is usually seen as a meaningless black and white shape when presented upside-down. The
’perception’ refers to the condition that the subject perceives Figure 1.1(a) as a face. The ’no-
perception’ refers to the condition that the subject perceives Figure 1.1(b) as a meaningless
shape. Figure 1.1(c) shows average scalp distribution of gamma activity and phase synchrony.
Color coding indicates gamma power. Gamma activity is similar between conditions. Black
and green lines correspond to a significant increase or decrease in synchrony. When the subject
does not perceive the face, phase synchrony between electronode pair in long-distance does not
increase. When the subject perceives the face, phase synchrony between electronode pair in
long-distance increases [9].
Figure 1.1: (a)(b) Examples of ’Mooney’ faces. (a) is an upright picture and can be recognized as a
human face easily. (b) is an upside-down picture and is difficult to be recognized as a human face. (c)
Black and green lines correspond to a significant increase or decrease in synchrony [9].
2
1.2 Design of the band-pass filter
BiQuad filter is often used for time-series analysis [10]. Ik is the input data at a time-step k,
Ok is the output data at a time-step k (Figure 1.2). The output ON is calculated as
ON =b0a0IN +
b1a0IN−1 +
b2a0IN−2 −
a1a0ON−1 −
a2a0ON−2. (1.2)
The calculation of O0 requires the value of I−2, I−1, O−2, O−1 is needed. For convenience,
I−2, I−1, O−2, O−1 are assumed to be 0.
The frequency characteristic depends on the coefficients. For making band-pass filters, the
parameters are set as follows:
α = sin(ωf∆t)sinh(log 2) sin(ωf∆t)
2βωf∆t,
a0 = 1 + α,
a1 = −2 cos(ωf∆t),
a2 = 1− α,
b0 = α,
b1 = 0,
b2 = −α,
(1.3)
where ωf is the center frequency of the filter, ∆t is the reciprocal of the sampling rate, β describes
the width of frequency range of the band-pass filter in octave. When the frequency ω satisfies
ωf2−β
2 < ω < ωf2β2 , the output signal is larger than −3 dB, where dB (decibel) is defined as
20 log10amplitude of output signal
amplitude of input signal. (1.4)
For simplicity, the frequency range ωf2−β
2 < ω < ωf2β2 is called the frequency range of the
band-pass filter.
Figure 1.3 shows the frequency characteristic of the band-pass filter. The input data F (t) are
defined as
F (t) = sinωt, (1.5)
where ω is natural frequency of a given oscillator.
The output data F (t) are given by
F (t) = a sin(ωt+ C), (1.6)
where C is introduced for phase lag which can occur by filtering. The numerical results shown
in Figure 1.3 can be obtained by changing frequency ω.
The purple line in Figure 1.3 indicates the −3 dB level. The blue line indicates the frequency
characteristic of the band-pass filter when the width of frequency range of the band-pass filter
β = 0.01. The purple line and the blue line cross when log2ω
ωf= ±0.005 = ±β
2. The red
line indicates the frequency characteristic of the band-pass filter when the width of frequency
range of the band-pass filter β = 0.22. Similarly, the purple line and the red line cross when
3
log2ω
ωf= ±0.11 = ±β
2. It is confirmed that when the frequency ω falls in the frequency range
of the band-pass filter ωf2−β
2 < ω < ωf2β2 , the output signal is larger than −3 dB, and when ω
does not satisfy frequency range of the band-pass filter ωf2−β
2 < ω < ωf2β2 , the output signal is
smaller than −3 dB.
Table 1.1 is an example of frequency ranges for the band-pass filter. In Chapter 3 and Chapter
4, I focus on a width of a frequency range β = 0.01 and β = 0.22. In the previous study [9], the
width of frequency range of the band-pass filter was β = 0.22.
The component of the frequency which equals 2 cannot be seen clearly in the original data in
Figure 1.4(a), but it can be seen clearly in the filtered data in Figure 1.4(b)
Figure 1.2: Example of BiQuad filter. (a)input data. (b)output data.
-40
-35
-30
-25
-20
-15
-10
-5
0
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
dB
log2(ω/ωf)
Figure 1.3: frequency characteristics of a band-pass filter. blue:
β = 0.01 red: β = 0.22
ωf β ωf2−β
2 ωf2β2
0.1 0.01 0.0996540 0.100347
0.1 0.22 0.0926588 0.107923
0.5 0.01 0.498270 0.501736
0.5 0.22 0.463294 0.539614
0.7 0.01 0.697578 0.702430
0.7 0.22 0.648612 0.755460
Table 1.1: an example of fre-
quency range of the band-pass fil-
ter.
4
-1.5
-1
-0.5
0
0.5
1
1.5
9000 9010 9020 9030 9040 9050
(a)
F(t
)
t-0.15
-0.1
-0.05
0
0.05
0.1
0.15
9000 9010 9020 9030 9040 9050
(b)
F~(t
)
tFigure 1.4: (a)input data F (t) = sin t+
1
10cos 2t. (b)output data. ωf = 2.0, β = 0.01
1.3 Purpose of this study
In the previous studies introduced above [2-6], it is assumed that time-series of all the constituent
oscillators can be observed. However, in most realistic situations, only the global behavior of
the assembly of oscillators can be observed. For example, there are millions of neurons in the
detection target range of an electronode, and the electronode receives the sum of the action
potential emitted by the vast number of neurons. The approach for inferring the network
structure of interactions of the system using time-series data of the global behavior of the
assembly of the oscillators is required. Also, synchronization appears ubiquitously. For example,
the neurons fire simultaneously. As another example, circadian rhythm is affected by the external
stimuli such as light. According to the previous study [5], the accuracy of the inference gets
poor when the system synchronize. There is still room for improvement in order to infer the
network structure of interactions of the system even when the system synchronize accurately.
Filters may be useful in order to improve the approach for inferring the network structure of
interactions of the system because we can get some information on frequencies by filtering the
time-series data. In fact, filters are often used when we analyze time-series data. In the data
analysis of brainwaves, various type of band-pass filters are used to extract certain frequency
components and to smooth noisy data[7-9]. However, it is not clear what kind of properties of
the oscillator dynamics behind the brainwaves are extracted by the filter. Also, an important
dynamical properties may be lost by filtering the time-series data.
In the previous study [9], the filter pass the frequency range f0 ± 3 Hz, f0 is the gamma wave
frequence of each subject. The center frequency of the filter ωf should equal to the subject
frequency. In some case, we cannot know the subject frequency, but if we calculate with many
kinds of the center frequency of the filter ωf, we spend far too much time trying to analyze the
time-series data. However, the appropriate width of frequency range of the band-pass filter is
not clear. In the first place, analytical approaches on filters are inadequate.
In this study, I aim to verify whether synchronization can be detected using a band-pass
filter even when only the average dynamics of constituent oscillators can be observed and the
filter characteristics. First, I consider a system consisting of an oscillator subjected to periodic
external force. I did numerical analysis and analytical calculation. Second, I consider a system
consisting of multiple oscillators. I examined the relationship between the width of the frequency
range of the band-pass filter and the distribution of the natural frequency.
5
Chapter 2
Model
2.1 Dynamics of the oscillators with external force
Phase oscillators under periodic external force can be described as [11]
ϕk = ωk + κ sin(Ωt− ϕk), (2.1)
where ϕk, ωk are the phase and the natural frequency of oscillator k ∈ 1, . . . , N, respectively,Ω is the frequency of the periodic external force, κ(> 0) is the intensity of coupling with periodic
external force.
There are no interactions between oscillators. The second term of Equation (2.1) is the effect
of synchronizing with periodic external force. If the phase of the oscillator ϕk is a bit larger than
the phase of periodic external force Ωt, ϕk decrease because of sin(Ωt − ϕk) < 0. If the phase
of the oscillator ϕk is a bit smaller than the phase of periodic external force Ωt, ϕk increase
because of sin(Ωt− ϕk) > 0.
2.2 Condition for synchronization
The condition for synchronization is determined as follows [12]. The dynamics of periodic
external force is
ϕex = Ω, (2.2)
where ϕex is the phase of the periodic external force. By subtracting Equation (2.2) from
Equation (2.1), I have
ϕex − ϕk = Ω− ωk − κ sin(Ωt− ϕk). (2.3)
By putting the phase difference between the phase of the oscillator and the one of periodic
external force ψ = Ωt− ϕ, I have
ψ = (Ω− ω)− κ sinψ. (2.4)
When ψ = 0, phase locking occurs between the oscillator and the periodic external force. From
the condition for the existence of the solution to ψ = 0, one finds that synchronization occurs
when |Ω− ω| ≤ κ. The red rigion in Figure 2.1 shows the parameter region of synchronization.
This region is called Arnold tongue.
6
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
κ
Ω-ω
Figure 2.1: The red region is the parameter sets which the system synchronize. This is called Arnold
tongue.
2.3 Definition of observation phase
I assumed that only the data of the average behavior of the oscillators are available. In general,
the observed data F (t) is calculated as
F (t) =1
N
N∑k=1
f(ϕk), (2.5)
where N is the number of oscillators, f(ϕk) is an output function. In this study, the output
function is set to f(ϕk) = cosϕk.
The observation phase is obtained by following procedure:
1. The time-series data F (t) is band-pass filtered data of the observed data F (t).
2. H(t) is the Hilbert transformation of filtered time-series data F (t).
3. z ∈ C is an analytic signal. The argument of z(t) = F (t) + iH(t) is Φωf(t). Hereinafter,
Φωf(t) is called observation phase.
The Hilbert transformation delays the phase byπ
2for every frequency component [13]. Figure
2.2 shows the flowchart of this procedure.
7
Figure 2.2: The procedure for calculating observation phase Φωf(t)
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
-0.006 -0.003 0 0.003 0.006
(a)
H~(t
)
F~
(t)
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
-0.01 -0.005 0 0.005 0.01
(b)
H~(t
)
F~
(t)
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
-0.01 -0.005 0 0.005 0.01
(c)
H~(t
)
F~
(t)
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
(d)
H~(t
)
F~
(t)
Figure 2.3: Examples of phase planes. N = 1,Ω = 0.7, κ = 0.02, ωf = 0.4, (a), (c) ω1 = 0.1, (b), (d)
ω1 = 0.68, (a), (b) β = 0.01, (c), (d) β = 0.22.
8
-1
-0.5
0
0.5
1
9000 9100 9200 9300 9400 9500
(a)
F(t
)
t-1
-0.5
0
0.5
1
9000 9020 9040 9060 9080 9100
(b)
F(t
)t
-0.004
-0.002
0
0.002
0.004
9000 9100 9200 9300 9400 9500
(c)
F~(t
)
H~(t
)
t-0.006
-0.004
-0.002
0
0.002
0.004
0.006
9000 9020 9040 9060 9080 9100
(d)
F~(t
) H~
(t)
t
-0.1
-0.05
0
0.05
0.1
9000 9100 9200 9300 9400 9500
(e)
F~(t
) H~
(t)
t-0.15
-0.1
-0.05
0
0.05
0.1
0.15
9000 9020 9040 9060 9080 9100
(f)
F~(t
) H~
(t)
tFigure 2.4: (a), (b) The observed data F (t). (c), (d), (e), (f) Blue and red are the time-series data of F (t)
and H(t), respectively. N = 1,Ω = 0.7, κ = 0.02, ωf = 0.2, (a), (c), (e) ω1 = 0.1, (b), (d), (f) ω1 = 0.68,
(c), (d) β = 0.01, (e), (f) β = 0.22
9
The Hilbert transformation is calculated as follows. N is the number of real number data and
assumed to be even. xk(k = 0, . . . , N − 1) is calculated as
xk =X0
N+
2
N
N2−1∑
l=0
Re
Xl exp
[−i
2πlk
N
]+XN
2
Ncos kπ, (2.6)
where Xk = F(xk)(Fourier Transform).
The Hilbert transformation hk delays the phaseπ
2for every frequency component, so h(k) is
calculated as
hk =X0
N+
2
N
N2−1∑
l=0
Re
Xl exp
[−i
2πlk
N− i
π
2
]+XN
2
Ncos kπ (2.7)
I can rewrite Equation (2.4) as
hk =H0
N+
2
N
N2−1∑
l=0
Re
Hl exp
[−i
2πlk
N
]+HN
2
Ncos kπ, (2.8)
where F(hk) = Hk, the following equations are established:
H0 = X0,
Hl = −iXl
(l = 1, . . . ,
N
2− 1
),
HN2= XN
2.
(2.9)
Generally, the inverse Fourier transform is written as
hk =1
N
N−1∑l=0
Hl exp
[i2πlk
N
]. (2.10)
So, Hl
(l =
N
2+ 1, . . . , N − 1
)is also needed. hk is a real number, so Hl = iXl.
Figure 2.3 shows examples of phase plane. Figure 2.3 (a) and (c) are examples of asynchro-
nization with a periodic external force. Figure 2.3 (b) and (d) are examples of synchronization
with a periodic external force.
Figure 2.4 shows examples of time-series data of the filtered data F (t) and the Hilbert trans-
formation data H(t). Figure 2.3 (a), (c), (e) are examples of asynchronization with periodic
external force. Figure 2.3 (b), (d), (f) are examples of asynchronization with periodic external
force.
The amplitude of the filtered data F (t) is smaller than the amplitude of the observed data
F (t). When the oscillator does not synchronize with the periodic external force, the amplitude
of the filtered data F (t) and the Hilbert transform data H(t) is smaller than when the oscillator
synchronizes with the periodic external force.
10
Chapter 3
Single Oscillator
3.1 Numerical analysis
In this chapter, I focus on a single oscillator, as defined in Chapter 2. The frequency of periodic
external force is Ω = 0.7 and the intensity of coupling with periodic external force is κ = 0.02. As
described in Chapter 2.2, if the natural frequency ω1 satisfies the condition Ω−κ ≤ ω1 ≤ Ω+κ,
i.e. 0.68 ≤ ω1 ≤ 0.72, in this case, the oscillator synchronizes with the periodic external force. In
order to examine whether the synchronization can be detected using the time-series data of the
observation phase, a numerical analysis is performed in the case where the oscillator synchronizes
with periodic external force and the case where it does not. As an example of synchronization,
I set the natural frequency ω1 to ω1 = 0.68, 0.7, 0.72. As an example of asynchronization, I set
the natural frequency ω1 to ω1 = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 0.9, 1.0. Synchronization can be
detected by calculating the actual oscillation frequency of the observation phase⟨Φωf
⟩⟨Φωf
⟩=
Φωf(T2)− Φωf
(T1)
T2 − T1, (3.1)
where T1 and T2 are set as 9000 and 10000, respectively.
If the actual oscillation frequency of the observation phase⟨Φωf
⟩equals to the frequency of
the periodic external force, the oscillator synchronizes with the periodic external force.
In the absence of a periodic external force, i.e. when the intensity of coupling κ = 0, regardless
of the value of the width of frequency range of the band-pass filter β and the center frequency
of the filter ωf, the actual oscillation frequency of the observation phase equals to the natural
frequency of the oscillator (Figure 3.1 (a), (b)). These are valid results.
When the oscillator synchronizes with periodic external force, i.e. the intensity of coupling
κ = 0.02, regardless of the value of the width of frequency range of the band-pass filter β and the
center frequency of the band-pass filter ωf, the actual oscillation frequency of the observation
phase equals to the frequency of periodic external force (Figure 3.1(c),(d) blue line). These are
also valid results. From these results, it can be said that the synchronization can be detected
using the time-series data of the observation phase.
When the oscillator does not perfectly synchronize with periodic external force, i.e. κ <
|Ω− ωk|, the detected frequencies depend on the width of the frequency range of the band-pass
filter β (Figure 3.1(c),(d)). When a width of frequency range of the band-pass filter β = 0.01,
the values of the plateaus are ω1, |Ω−2ω1|,Ω. When a width of frequency range of the band-pass
filter β = 0.22, the values of the plateau is only ω1.
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
(a)
κ=0.0 β=0.01
frequency
frequency 0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
(b)
κ=0.0 β=0.22
frequency
frequency
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
(c)
κ=0.02 β=0.01
frequency
frequency 0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
(d)
κ=0.02 β=0.22
frequency
frequency
Figure 3.1: Each line represents the actual oscillation frequency of the observation phase which is calcu-
lated using a time-series data of an oscillator whose natural frequency is ω1. horizontal axis: ωf, vertical
axis:⟨Φωf
⟩
natural frequency the values of the plateaus
0.1 0.1, 0.5, 0.7
0.2 0.2, 0.3, 0.7
0.3 0.3, 0.1, 0.7
0.4 0.4, 0.1, 0.7
0.5 0.5, 0.3, 0.7
0.6 0.6, 0.5, 0.7
natural frequency the values of the plateaus
0.68 0.7
0.7 0.7
0.72 0.7
0.8 0.8, 0.9, 0.7
0.9 0.9, 0.7
1.0 1.0, 0.7
Table 3.1: The relationship between the natural frequency of an oscillator and the values of the plateaus
when the intensity of coupling with periodic external force κ equals 0.02 and the width of frequency range
of the band-pass filter β equals 0.01.
12
3.2 Analytical calculation
Pertubation theory is used in order to explain why the values of the plateaus depend on the
width of frequency range of the band-pass filter β. The following calculation can be applied
when κ is sufficiently small. When the osillator does not synchronize with periodic external
force, the solution of Equation (2.1) is given by
ϕ1(t) = ω1t+ C +∞∑n=1
κnfn(t), (3.2)
where Ck is a constant value and fn(t) does not contain κ. By differentiating Equation (3.2)
with respect to t, I have
ϕ1(t) = ω1 +∞∑n=1
κnfn(t) (3.3)
By substituting Equation (3.2) into Equation (2.1), I have
ϕ1 = ω1 + κ sin
(Ω− ω1)t− C −
∞∑n=1
κnfn(t)
= ω1 + κ
[sin (Ω− ω1)t− C+
∞∑m=1
[1
m
(−
∞∑n=1
κnfn(t)
)m]]
= ω1 + κ
[sin (Ω− ω1)t− C+
∞∑m=1
[1
m!(Ω− ω1)
m sin(Ω− ω1)t− C +
m
2π(
−∞∑n=1
κnfn(t)
)m]]= ω1 + κ sin (Ω− ω1)t− C − κ2f1(t)(Ω− ω1) sin
(Ω− ω1)t− C +
π
2
+O(κ3)
(3.4)
By comparing Equation (3.3) and Equation (3.4), I have
f1(t) = sin (Ω− ω1)t− C
f1(t) =1
ω1 − Ωcos (Ω− ω1)t− C (3.5)
f2(t) = −(Ω− ω1)
(ω1 − Ω)cos (Ω− ω1)t− C sin
(Ω− ω1)t− C +
π
2
=
1
2
[sin2(Ω− ω1)t− 2C +
π
2
+ sin
π
2
]=
1
2sin2(Ω− ω1)t− 2C +
π
2
+
1
2
f2(t) = − 1
4(Ω− ω1)cos2(Ω− ω1)t− 2C +
π
2
+
1
2t (3.6)
By substituting Equation (3.5) and Equation (3.6) into Equation (3.2), I have
ϕ1(t) =
(ω1 +
κ2
2
)t+C+
κ
ω1 − Ωcos (Ω− ω1)t− C− κ2
4(Ω− ω1)cos2(Ω− ω1)t− 2C +
π
2
+O(κ3)
(3.7)
13
Equation (3.7) is substituted for Equation (2.5).
F = cos
[(ω1 +
κ2
2
)t+ C +
κ
ω1 − Ωcos (Ω− ω1)t− C − κ2
4(Ω− ω1)cos2(Ω− ω1)t− 2C +
π
2
+O(κ3)
]= cos
(ω1 +
κ2
2
)t+ C
−[
κ
ω1 − Ωcos (Ω− ω1)t− C+O(κ2)
](ω1 +
κ2
2
)sin
(ω1 +
κ2
2
)t+ C
= cos
(ω1 +
κ2
2
)t+ C
− κ
2(ω1 − Ω)
(ω1 +
κ2
2
)sin
(2ω1 +
κ2
2− Ω
)t+ 2C
− κ
2(ω1 − Ω)
(ω1 +
κ2
2
)sin
(κ2
2+ Ω
)t
+O(κ2)
(3.8)
O(κ2) can be neglected when κ equals to 0.01. If O(κ2) is neglected in Equation (3.8), I have
F = cos(ω1t+ C)− κω1
2(ω1 − Ω)sin (2ω1 − Ω) t+ 2C − ω1κ
2(ω1 − Ω)sinΩt (3.9)
By Equation(3.6), the observed data F (t) obtains the frequency elements which equals to
ω1, |2ω1 − Ω|,Ω.
F = a1 cos(ω1t+ C)− a2 sin (2ω1 − Ω) t+ 2C − a3 sinΩt (3.10)
, where a1 = 1, a2 = a3 =κω1
2(ω1 − Ω). If the approximated F (t) defined as Equation (3.9) are
filtered, F (t) is given by
F (t) = a1 cos(ω1t+ C1) + a2 sin(Ω− 2ω1)t+ C2
+ a3 sin(Ωt+ C3), (3.11)
where C1, C2, C3 are introduced for phase lag which can occur by filtering.
Figure 3.2(a) shows that the coefficient of the |2ω1 − Ω| frequency component |a2| is larger
than the coefficient of the ω1 frequency component |a1| if the center frequency of the band-pass
filter ωf is around 0.5, and the coefficient of the Ω frequency component |a3| is larger than the
coefficient of the ω1 frequency component |a1| if the center frequency of the band-pass filter ωfis around 0.7. Figure 3.2(c) and (d) are enlarged views of Figure 3.2(a). Figure 3.2(c) and (d)
show that the range of the center frequency of the band-pass filter ωf which the coefficient of the
|2ω1 −Ω| frequency component |a2| is larger than the coefficient of the ω1 frequency component
|a1| is wider than the one which the coefficient of the Ω frequency component |a3| is larger thanthe coefficient of the ω1 frequency component |a1|. Figure 3.2(b) shows that the coefficient of
the ω1 frequency component |a1| is always larger than the coefficient of the |2ω1 −Ω| frequencycomponent |a2| and the coefficient of the Ω frequency component |a3|. Figure 3.2 explains Figure
3.1(c), (d) qualitatively.
When the intensity of coupling with periodic external force κ = 0, the coefficient of the
|2ω1−Ω| frequency component |a2| and the coefficient of the Ω frequency component |a3| equals0. So, the coefficient of the ω1 frequency component |a1| is always larger than the coefficient of
the |2ω1 − Ω| frequency component |a2| and the coefficient of the Ω frequency component |a3|.This is the qualitative explaination of Figure 3.1(a), (b).
14
0
0.001
0.002
0.003
0.004
0.005
0 0.2 0.4 0.6 0.8 1
(a)
κ=0.02
β=0.01
|a~
1|,
|a~
2|,
|a~
3|
ωf
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.2 0.4 0.6 0.8 1
(b)
κ=0.02
β=0.22
|a~
1|, |a~
2|, |a~
3|
ωf
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.48 0.5 0.52
(c)
κ=0.02
β=0.01
|a~
1|,
|a~
2|,
|a~
3|
ωf
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.68 0.7 0.72
(d)
κ=0.02
β=0.01
|a~
1|, |a~
2|, |a~
3|
ωfFigure 3.2: ω1 = 0.1,Ω = 0.7, κ = 0.02. red line: the coefficient of the ω1 frequency component |a1|,blue line: the coefficient of the |2ω1 − Ω| frequency component |a2|, purple line: the coefficient of the Ω
frequency component |a3|
condition the range of the center frequency of the band-pass filter ωf|a2| > |a1| 0.499 < ωf < 0.501
|a3| > |a1| 0.697 < ωf < 0.703
the value of the plateau equals to Ω− 2ω1 0.483 < ωf < 0.517
the value of the plateau equals to Ω 0.669 < ωf < 0.737
Table 3.2: Correspodence table when ω1 is 0.1, κ is 0.02 and β is 0.01.
15
3.3 Comparison between synchronization and asynchronization
When the center frequency of the filter is ωf = 0.7, the actual oscillation frequency is 0.7
regardless of the natural frequency ω1. Although the actual oscillation frequency is the same
value in both cases, the behavior in phase plane is different.
Figure 3.3(a), (b) show the observed data F (t) when the natural frequency ω1 = 0.1 and
ω1 = 0.68, respectively. When the oscillator does not synchronize with the periodic external
force, the observed data F (t) is a deformed sin-curve. When the oscillator synchronizes with
periodic external force, the observed data F (t) is a sin-curve.
Figure 3.3(c), (d) show the phase plane when 9000 < t < 9500. When the oscillator synchro-
nize with the periodic external force, the radius of the phase plane is large. When the oscillator
does not synchronize periodic external force, the radius of the phase plane is small.
Figure 3.3(e), (f) show the time-series data of the filtered data F (t) and the Hilbert trans-
formation H(t). When the oscillator synchronize with periodic external force, the filtered data
F (t) is the same as the observed data F (t) except for phase lag. When the oscillator does not
synchronize with periodic external force, the sin-curve winds and the amplitude gets smaller.
This occurs because the frequency ω1 and the frequency Ω coexist in the filtered data F (t). In
order to confirm this, some F (t) are defined as follows.
First definition is as Equation (3.9).
F (t) = cos(ω1t+ C)− κω1
2(ω1 − Ω)sin (2ω1 − Ω)t+ 2C − ω1κ
2(ω1 − Ω)sinΩt (3.12)
In order to fit original observed data F (t), C is set to π. Next definition contains ω1 and Ω
frequency components.
F (t) = cos(ω1t+ C)− κω1
2(ω1 − Ω)sinΩt (3.13)
The following definition contains Ω− 2ω1 and Ω frequency components.
F (t) = − κω1
2(ω1 − Ω)sin (2ω1 − Ω)t+ 2C − ω1κ
2(ω1 − Ω)sinΩt (3.14)
The final sample contains only Ω frequency component.
F (t) = − ω1κ
2(ω1 − Ω)sinΩt (3.15)
In all cases, the amplitude of F (t) gets smaller than the one of F (t). ω1 in Equation (3.12) and
(3.13) is strong enough in order to wind a sin-curve (Figure 3.4(a), (b)). When the observed
data does not contain ω1 frequency, the sin-curve does not wind (Figure 3.4(c), (d)).
Although F (t) defined in Equation(3.12) is similar to the original observation data written in
Equation(2.5) (Figure 3.4(e)), the plateau is narrower than using Equation (2.5) (Figure 3.4(f)
compare with Figure 3.1(c)).
16
-1
-0.5
0
0.5
1
9000 9020 9040 9060 9080 9100
(a)
F(t
)
t-1
-0.5
0
0.5
1
9000 9020 9040 9060 9080 9100
(b)
F(t
)t
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
-0.02 -0.01 0 0.01 0.02
(c)
H~(t
)
F~
(t)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-1.2 -0.9 -0.6 -0.3 0 0.3 0.6 0.9 1.2
(d)
H~(t
)
F~
(t)
-0.02
-0.01
0
0.01
0.02
0.03
9000 9020 9040 9060 9080 9100
(e)
F~(t
) H~
(t)
t-1
-0.5
0
0.5
1
9000 9020 9040 9060 9080 9100
(f)
F~(t
) H~
(t)
tFigure 3.3: (a), (b) The observed data F (t). (c), (d) Phase plane. (e), (f) Blue and red is the time-series
data of F (t), H(t), respectively. N = 1,Ω = 0.7, κ = 0.02, ωf = 0.7, β = 0.01, (a), (c), (e) ω1 = 0.1, (b),
(d), (f) ω1 = 0.68.
17
-0.02
-0.01
0
0.01
0.02
0.03
9000 9020 9040 9060 9080 9100
(a)
F~(t
)
t-0.02
-0.01
0
0.01
0.02
0.03
9000 9020 9040 9060 9080 9100
(b)
F~(t
)t
-0.02
-0.01
0
0.01
0.02
0.03
9000 9020 9040 9060 9080 9100
(c)
F~(t
)
t-0.02
-0.01
0
0.01
0.02
0.03
9000 9020 9040 9060 9080 9100
(d)
F~(t
)
t
-1
-0.5
0
0.5
1
9000 9100 9200 9300 9400 9500
(e)
F(t
)
t
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
(f)
κ=0.02 β=0.01
frequency
frequency
Figure 3.4: (a), (b), (c), (d) blue line: The filtered time-series data F (t) of Equation(2.5). red line:
The filtered time-series data F (t) of (a) Equation(3.12), (b) Equation(3.13), (c) Equation(3.14), (d)
Equation(3.15), respectively. (e) The observed data F (t). blue line: defined as Equation(2.2). red:
defined as Equation(3.12). (f)The actual oscillation frequency of the observation phase which is calculated
using a time-series data of Equation (3.12) when ω1 = 0.1, horizontal axis: ωf, vertical axis:⟨Φωf
⟩18
3.4 Strong coupling with a periodic external force
When the intensity of coupling with a periodic external force κ is large, the following results in
Figure 3.5 are obtained. The resuls are different from the results when κ is small. The procedure
in Chapter 3.2 applied for this case (Figure3.6). However, Figure 3.6 cannot explain Figure 3.5.
In the case of the width of frequency range of the band-pass filter β = 0.01, when the center
frequency of the band-pass filter ωf is around 0.6, |a1| is the largest, but the actual oscillation
frequency of the observation phase does not equal to ω1 (Figure 3.5(a) and Figure 3.6(a)). In
the case of the width of frequency range of the band-pass filter β = 0.22, |a1| is always the most
large, but the actual oscillation frequency of the observation phase does not constantly equals to
ω1 (Figure 3.5(b) and Figure 3.6(b)). These difference occurs because the analytical calculation
described in Chapter 3.2 is not accurate enuough when the intensity of coupling with periodic
external force κ is large. There is a need to consider the higher order terms of κ.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
(a)
κ=0.1 β=0.01
frequency
frequency 0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
(b)
κ=0.1 β=0.22fr
equency
frequencyFigure 3.5: Each line represents the actual oscillation frequency of the observation phase which is calcu-
lated using a time-series data of an oscillator whose natural frequency is ω1. horizontal axis: ωf, vertical
axis:⟨Φωf
⟩
0
0.001
0.002
0.003
0.004
0.005
0 0.2 0.4 0.6 0.8 1
(a)
κ=0.2
β=0.01
|a~ 1|, |
a~ 2|, |
a~ 3|
ωf
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.2 0.4 0.6 0.8 1
(b)
κ=0.2
β=0.22
|a~ 1|, |
a~ 2|, |
a~ 3|
ωfFigure 3.6: ω1 = 0.1,Ω = 0.7, κ = 0.1. red line: the coefficient of the ω1 frequency component |a1|,blue line: the coefficient of the |2ω1 − Ω| frequency component |a2|, purple line: the coefficient of the Ω
frequency component |a3|
19
3.5 Relationship between phase lag and the frequency a band-
pass filter passes
The BiQuad filter is a linear operation. In order to determine Ck1, Ck2, Ck3 in Equation(3.9)
which indicate the phase lag, I fit each frequency component.
The observed data
F1(t) = cos(0.1t+ π), (3.16)
F2(t) = sin(0.7t), (3.17)
F3(t) = sin(0.5t), (3.18)
are filtered as
F1(t) = a1 cos(0.1t+ C1), (3.19)
F2(t) = a2 sin(0.7t+ C2), (3.20)
F3(t) = a3 sin(0.5t+ C3). (3.21)
The coefficients a1, a2, a3 are assumed to be greater or equal 0, the phase lag parameter C1, C2, C3
is
C1 =
π2
(ωf < 0.1
)π(ωf = 0.1
)3π2
(ωf > 0.1
) , (3.22)
C2 =
−π
2
(ωf < 0.7
)0(ωf = 0.7
)π2
(ωf > 0.7
) , (3.23)
C3 =
−π
2
(ωf < 0.5
)0(ωf = 0.5
)π2
(ωf > 0.5
) . (3.24)
From these results, when the observed data are given as F (t) = sin (ωt+ C), the filtered data
are given as F (t) = sin(ωt+ C + C
)and C is as
C =
−π
2
(ωf < ω
)0(ωf = ω
)π2
(ωf > ω
) . (3.25)
The phase lag depends on magnitude correlation between the center frequency ωf and does not
depend on the width of frequency range of the band-pass filter.
20
Chapter 4
Multiple Oscillators
In this chapter, I check that synchronization can be detected using observation phase when the
system consists of multiple oscillators.
First, I consider a system of 100 oscillators, whose natural frequencies are uniformly distributed
between 0 and 1. In the absence of a periodic external force, i.e. the intensity of coupling with
periodic external force κ equals to 0, if the width of frequency range of the band-pass filter β is
large, the graph of the actual oscillation frequency of the observation phase has specific structure
(Figure 4.1(a)). When the intensity of coupling with periodic external force κ equals to 0.1 and
the frequency of the periodic external force Ω equals to 0.7, if the width of frequency range of
the band-pass filter β is small, plateau does not exist and synchronization cannnot be detected.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
(a)
κ=0.0
frequency
frequency 0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
(b)
κ=0.1
frequency
frequency
Figure 4.1: purple: β = 1.0 × 10−5, blue: β = 0.01, red: β = 0.22. horizontal axis: ωf, vertical axis:⟨Φωf
⟩
Next, I set the width of frequency range of the band-pass filter β equals to 0.22. The natural
frequency of the oscillators distribute as follows.
ωk =k
100(0 ≤ k ≤ 99) (4.1)
ωk =k
100(0 ≤ k ≤ 199) (4.2)
21
ωk =k
25(0 ≤ k ≤ 49) (4.3)
ωk =2k
15(0 ≤ k ≤ 14) (4.4)
In the absence of a periodic external force, if the distribution of the natural frequency is Equation
(4.1), (4.2) and (4.3), the graph of the actual oscillation frequency of the observation phase has
specific structure (Figure 4.2(a)). When the interval of the natural frequency ωk+1−ωk is large,
the specific structure in the graph of the actual oscillation frequency of the observation phase
does not appear. By conparing the result in Figure 4.2(a) using Equation (4.1) and one using
Equation (4.2), when the center frequency of the filter ωf is small, the component of the large
natural frequency does not influence the actual oscillation frequency of the observation phase.
The actual oscillation frequency of the observation phase is up to the max natural frequency of
the oscillator.
When the intensity of coupling with periodic external force κ equals to 0.1 and the frequency of
the periodic external force Ω equals to 0.7, synchronization can be detected in all the destribution
described above.
0
0.5
1
1.5
2
0 0.5 1 1.5 2
(a)
κ=0.0 β=0.22
frequency
frequency 0
0.5
1
1.5
2
0 0.5 1 1.5 2
(b)
κ=0.1 β=0.22
frequency
frequency
Figure 4.2: red: ωk =k
100(0 ≤ k ≤ 99), purple: ωk =
k
100(0 ≤ k ≤ 199), green: ωk =
k
25(0 ≤ k ≤ 49),
blue: ωk =2k
15(0 ≤ k ≤ 14). horizontal axis: ωf, vertical axis:
⟨Φωf
⟩
From these results, there is a need to set the appropriate width of the frequency range of
the band-pass filter in order to detect synchronization and it depends on the distribution of the
natural frequency of the oscillation.
22
Chapter 5
Summary and Future Work
In this study, I aim to verify whether synchronization can be detected using a band-pass filter
even when only the average dynamics of constituent oscillators can be observed and the filter
characteristics. I generated time-series data using simple mathematical models describing oscil-
lators with interaction and periodic external forcing. By filtering the observed data, observation
phase is calculated.
First, I considered a system consisting of an oscillator under periodic external force. I checked
that regardless of the frequency a band-pass filter passes, the actual oscillation frequency of the
observation phase equals to the natural frequency of the oscillator in the absence of periodic ex-
ternal force. In the presence of external force, the actual oscillation frequency of the observation
phase equals to the frequency of periodic external force regardless of the center frequency of the
filter ωf when the oscillator well synchronizes with periodic external force. However, when the
oscillator does not perfectly synchronize with the periodic external force, the actual oscillation
frequency of the observation phase depends on the frequency a band-pass filter passes. The
value of the plateaus are ω1, |2ω1−Ω|,Ω, where ω1 is the natural frequency of the oscillator and
Ω is the frequency of a periodic external force. In order to explain these results, I perform an
analytical calculation. By examining magnitude relationship of the coefficient of each frequency
component, the results are explained qualitatively. Even if the intensity of coupling with a
periodic external force κ is large, the results can be explained qualitatively by considering the
higher order terms of κ.
Next, I consider the system consisting of multiple oscillators. Synchronization can be detected
when the width of frequency range of the band-pass filter is appropriate. The appropriate width
of frequency range of the band-pass filter depends on the distribution of the natural frequency
of the oscillators.
Future studies targets the coupled oscillator systems.
As a future perspective, the knowledge of this study is applied to actual data. In order to
do this, some proposal of the method is necessary. First, the method for finding an appropriate
center frequency of the band-pass filter ωf is needed. Second is how to find the appropriate
width of frequency range of the band-pass filter. And more, the parameter of the mathematical
model which represents the dynamics of the system can be inferred in good accuracy.
23
Acknowledgements
I express my sincere thanks to Prof. K. Kudo for the chief examiner. She gave me advice at the
presentation opportunity. I am grateful to Prof. N. Aubert Kato for sub-chief examiner. He has
read this paper and gives us useful comments. I also thank Prof. H. Kori from the University
of Tokyo for his elaborated guidance and invaluable discussions. And thanks to the laboratory
members.
24
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