Synchronization in Dynamical Oscillatory Networks
with Non-Uniform Coupling Distributions
Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— In this study, synchronization observed in dynamical oscillatory networks with non-uniform coupling distributions
is investigated. The coupling states (on/off) of all connections
are stochastically determined at every certain time with thecoupling probability. We focus on the heavy tail type of coupling
distribution for the network. By using computer simulations, we
confirm that the dynamical network with the heavy tail type of
coupling distribution can hardly achieve global synchronization.

I. I NTRODUCTION
The synchronization phenomena observed from coupledoscillators are suitable models to analyze the natural phenomena [1],[2]. Therefore, many researchers have proposed
different coupled oscillatory networks and have discovered
many interesting synchronization phenomena [3],[4].
Recently, synchronization in dynamical networks with timevarying topology has been extensively investigated [5],[6]instead of static networks (i.e., network connections are fixed
constants). This is because the real-world complex networks
change their topologies with time. In these studies, novel
synchronization phenomena have been observed and theoretical approaches (such as Lyapunov function [5] and basinstability [6]) are used to explain the obtained synchronization
phenomena. However, a node in a complex network is expressed by a mathematical model in most studies of synchronization of dynamical networks. Although, it is very important
to use mathematical model for the dynamical networks inorder to understand the synchronization states by approachingtheoretical methods, we also need to consider physical models
for future engineering applications.
Therefore, we have investigated synchronization in coupledelectrical circuits systems with stochastic coupling [7]. In
Ref. [7], we have confirmed that the novel synchronization
states can be observed in polygonal oscillatory networks withon-off coupling by changing the coupling probability.
In this study, we focus on the brain networks as one of
dynamical complex networks. Because, we would like topropose modeling of synchronization in brain by using coupledelectrical oscillatory circuits, in order to make clear the mechanism of functional operation in brain. Structural and functional
brain networks are explored using graph theory and the brain
network structures have been made clear [8]. Furthermore,Song et al. have reported that the synaptic connectivity in local
cortical circuits has heavy tail distribution [9]. We also apply

978-1-4673-6853-7/17/$31.00 ©2017 IEEE

846

this heavy tail characteristics of coupling distribution for the
proposed system in this study. First, a schematic diagram of abrain network in Ref. [8] is used as a simple network modelto understand detailed synchronization phenomena. Then, we
extend the proposed network to a real brain network of themacaque visual cortex. By using computer simulations, we
confirm that the dynamical network with heavy tail type of
coupling distribution can hardly achieve global synchronization.
II. P ROPOSED S YSTEM
A. Network Model
A network model composed of 13 nodes and 22 edges isshown in Fig. 1. There are two important hubs in this network,“Connector hub” and “Provincial hub”. The both hubs are
high-degree nodes. “Connector hub” shows a diverse connectivity by connecting two sub-networks. “Provincial hub”
primarily connects nodes in the same sub-network.
The coupling state (on/off) of adjacent nodes is determinedstochastically. Each edge has a coupling probability (𝑝). The
network topology is updated at every certain time (𝜏 =100). In
this study, the node is expressed by van der Pol oscillator
as shown in Fig. 2(a). The oscillators are coupled by aresistor (see Fig. 2(b)). Namely, two coupled oscillators tend
to synchronize with in-phase state.
7
5

8

4

6
9

3

10

13

1
2

Connector hub

12

11

Provincial hub

Fig. 1. Network model (node: 13, edge: 22, a schematic diagram of a brain
network [8]).

Next, we develop the expression for the circuit equationsof the network model. The 𝑣𝑘 − 𝑖𝑅𝑘 characteristics of thenonlinear resistor are approximated by the following thirdorder polynomial equation,
𝑖𝑘 = −𝑔1 𝑣𝑘 + 𝑔3 𝑣𝑘 3 (𝑔1 , 𝑔3 > 0),(𝑘 = 1, 2, ...13).

(1)

ir
n

i

v

C

NR

For every set of parameter values of the coupling probability,
we simulate the network for 100 different distribution patterns.

LR

A. Global Synchronization(b) Coupling method.

(a) van der Pol oscillator.

Fig. 2.

Circuit model.

The normalized circuit equations governing the circuit are
expressed as[𝑘th oscillator]
⎧
(
)
∑
1
𝑑𝑥𝑘


= 𝜀 1 − 𝑥𝑘 2 𝑥 𝑘 − 𝑦𝑘 − 𝛾(𝑦𝑘 − 𝑦𝑛 )


3
⎨ 𝑑𝜏
𝑛∈𝑆𝑘
(2)
𝑑𝑦 𝑘


 𝑑𝜏 = 𝑥𝑘⎩(𝑘 = 1, 2, ...13).
In these equations, 𝛾 is the coupling strength, 𝜀 denotes thenonlinearity of the oscillators and 𝑦𝑛 denotes the current ofoscillators connected with 𝑘th oscillator. For the computer
simulations, we calculate Eq. (2) using the fourth-order RungeKutta method with the step size ℎ = 0.005. The parameter ofthis circuit model are fixed as 𝜀 = 0.1.
B. Coupling Strength Distribution
Here, we consider three different types of coupling distribution for the network model as follows.
1) Uniform distribution2) Gaussian distribution3) Heavy tail distribution
In the case of Uniform distribution, all edges have the same
coupling strength (𝛾 = 0.02). Gaussian and Heavy taildistributions for the simulations are shown in Fig. 3. The total
coupling strength of three different coupling distributions is
set to the same value.
10

10

Coupling distribution
Uniform
Gaussian
Heavy tailGlobal synchronization ratio [%]100.00100.0099.10

Figure 4 shows the coupling probability dependency of
global synchronization. We can see that three coupling strength
distribution have different characteristics for global synchronization. When the coupling probability is small, the networkwith Gaussian distribution has larger global synchronization
ratio than the others. The network with Uniform distribution
achieve to 100 % global synchronization, when the coupling
probability is set to 𝑝 = 0.4. After that, the network with
Gaussian distribution reaches to 100 % synchronization. In
the case of the network with Heavy tail distribution, it canhardly become 100 % global synchronization by increasing
the coupling probability.
We can say that the network with Heavy tail coupling
distribution synchronizes to avoid global synchronization. The
difference of global synchronization between the network with
Heavy tail coupling distribution and other networks becomeslarge when the coupling probability is set to 𝑝=0.3 to 0.6.
Namely, the coupling distribution has a huge effect on the
dynamical networks which are not classified in the both of
random networks (𝑝: small) and static networks (𝑝=1.0).

4
2

120

6

Global synchronization ratio [%]60

TABLE IG LOBAL S YNCHRONIZATION R ATIO OF S TATIC N ETWORK

8
Number of edges

Number of edges

8First, we explain the simulation results of static networkmodel (𝑝=1.0) for applying the three coupling distributions.
Table I summarizes global synchronization ratio of three
coupling distributions. We observe that the networks with
Uniform and Gaussian distributions achieve 100 % global synchronization. While, the network with Heavy tail distributions
fails to achieve 100 % global synchronization.

4
2

0.005

0.01

0.015 0.02 0.025
Coupling strength

0.03

0.035

(a) Gaussian.
Fig. 3.

000.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Coupling strength(b) Heavy tail.
Coupling strength distribution.

100

80

60

Uniform
Gaussian
Heavy tail

40

20

0

0

0.2

0.4

0.6

0.8

1

Coupling probability: p

III. S YNCHRONIZATION R ESULTS
For the computer simulations, we simulate the network for
105 time steps and we fix a certain time interval (𝜏 =100,000)
for checking final synchronization state of the network. In
order to analyze synchronization state, we define the synchronization as the following equation.
∣𝑦𝑘 − 𝑦𝑛 ∣ < 0.01

(𝑘 ∈ 𝑆𝑛 ).

(3)

847

Fig. 4.

Global synchronization.

B. Cluster Synchronization
Here, we focus on cluster synchronization related withconnector and provincial hubs in the network. We check thesynchronization of the edges with the both hubs. If all edges

60
40
20

0.2

0.3 0.4 0.5 0.6 0.7
Coupling probability: p

0.8

0.9

Edges with connector hub
Edges with provincial hub
Other edges

100

80

60

40

6-13

7-8

6-7

1-6

3-6

9-13

12-13

5-6

1-2

7-13

8-10

11-12

2-3

9-11

4-5

9-10

3-4

1-4

1-5

8-9

20

Edge

80

(c) Heavy tail.

60

Fig. 6.

40

Synchronization of all edges (𝑝=0.3).

20
0

25

0.1

0.2

0.3 0.4 0.5 0.6 0.7
Coupling probability: p

0.8

0.9
20
15
10
5
0

80

Coupling strength60(a) Standard.

40
25

25

20

20

20
0

0.1

0.2

0.3 0.4 0.5 0.6 0.7
Coupling probability: p

0.8

0.9

(c) Heavy tail.

15
10Strongest
coupling

0Strongest
coupling

0

0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20

Cluster synchronization.

10
5

5

Fig. 5.

15

Coupling strength

Coupling strength

848(c) Pattern 2.
25

20

20
Number of edges

25

15
10

Strongest
coupling

10Strongest
coupling

5
0

0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20

0

15

0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2

5

Coupling strength

Coupling strength(d) Pattern 3.
Fig. 7.

(e) Pattern 4.
Heavy tail distributions.

120

Global synchronization ratio [%]D. Heavy Tail Distribution
In this section, we investigate the characteristics of Heavy
tail coupling distribution. We consider four patterns of Heavy
tail coupling distribution as shown in Fig. 7. The position of
Heavy tail is changed from 𝛾=0.07 to 0.19 (step size: 0.04).
The simulation result of global synchronization is shown in
Fig. 8. From this figure, we can see that global synchronization
ratio decreases by increasing the distance of the heavy tail(from pattern 1 to pattern 4). Namely, we consider that the
coupling distribution of Heavy tail has important role for
global synchronization in dynamical networks.

Number of edges

C. Synchronization Ratio of Edge
In order to investigate the effect of connector and provincial hubs, we calculate average synchronization ratio for alledges. Figure 6 shows the simulation results of three couplingstrength distributions when the coupling probability is fixedwith 𝑝=0.3. From this figure, we confirm that the edges
connected with the provincial hub are easy to synchronize.
While, the edges connected with the connector hub are difficult
to synchronize. Because the synchronization ratio of edges ofconnector hub is smaller than the other edges.

(b) Pattern 1.

100

80

60

Standard
Pattern 1
Pattern 2
Pattern 3
Pattern 4

40

20

0
0.1

0.2

0.3

0.4

0.5

0.6

0.7

Coupling probability: p

Fig. 8.

Global synchronization.

0.8

0.9

6-7

7-8

Average synchronization ratio [%]

120

6-13

3-6

1-6

5-6

1-2

9-13

2-3

12-13

4-5

8-10

7-13

1-4

9-11

11-12

8-9

1-5

3-4

9-10

9-12

10-11

6-7

3-6

1-6

6-13

7-8

5-6

9-13

8-10

12-13

11-12

0

Edge

9-12

Average synchronization ratio [%]Average synchronization ratio [%]20(b) Gaussian.

0

Global sync.
Connector hub sync.
Provincial hub sync.

100

40

(a) Uniform.

(b) Gaussian.

120

60

Edge

10-11

100

10-11

8-9

9-12

9-11

4-5

9-10

3-4

7-13

2-3

1-5

1-4

0

Global sync.
Connector hub sync.
Provincial hub sync.

(a) Uniform.

Synchronization ratio [%]

20

80

0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20

0.1

40

Edges with connector hub
Edges with provincial hub
Other edges

100

Number of edges

0

60

120

0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20

80

80

Number of edges

100

120

Edges with connector hub
Edges with provincial hub
Other edges

100

Number of edges

Global sync.
Connector hub sync.
Provincial hub sync.

Synchronization ratio [%]Synchronization ratio [%]

120

120

1-2

of the connector hub are synchronized, we call the stateconnector hub synchronization. Similarly, if all edges of theprovincial hub are synchronized, we call the state provincial
hub synchronization.
Figure 5 shows the simulation results of global, connector
hub and provincial hub synchronization when three couplingstrength distributions are applied to the network. In the casesof Uniform and Gaussian coupling distributions, the difference between connector and provincial hubs synchronization
becomes 0 by increasing the coupling probability. While, inthe case of Heavy tail coupling distribution, there are somedifferences between connector and provincial hubs synchronization even if the coupling probability becomes large.

Finally, the proposed network is applied to real networkmodel in brain. The modified brain network of the macaquevisual cortex [10] is shown in Fig. 9. This brain network is
composed of 30 nodes and 152 edges. We investigate global
synchronization of the network when three different coupling
distributions are applied. In the case of Uniform distribution,
all edges have the same coupling strength (𝛾 = 0.002).
Gaussian and Heavy tail distributions for the simulations areshown in Fig. 10.

we confirm that the effect of Heavy tail coupling distribution
can be prominently visible for the dynamical network.
It is known that serious symptom such as epilepsia can becaused by global synchronization in brain. We consider that
Heavy tail coupling distribution has important role to avoid
serious symptom in normal brain.
120

Global synchronization ratio [%]IV. A PPLYING R EAL N ETWORK IN B RAIN

100

Uniform
Gaussian
Heavy tail

80

60

40

20

0

0

0.2

0.4

0.6

0.8

1

Coupling probability: p

Fig. 11.

Brain network of macaque visual cortex [10] (node: 30, edge: 152).

60

50

50
Number of edges

60

40
30
20

40
30
20
10

10

Coupling strength

6

02
0.

01
0.

01

8

0.

00

Coupling strength(a) Gaussian.
Fig. 10.

0.

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035

2

0

0

0.
0
0. 01
00
2
0.
00
4

Number of edges

Fig. 9.

(b) Heavy tail.

V. C ONCLUSION
We have investigated synchronization state in the dynamical
oscillatory networks with non-uniform coupling distributions.
We considered three different types of coupling strength;
Uniform, Gaussian and Heavy tail distributions. It was confirmed that the coupling distribution of Heavy tail has different
characteristics with other distributions. Namely, the dynamical
network with Heavy tail coupling distribution tends to avoid
global synchronization.
For the future work, we would like to consider the influenceof additional noises, frequency and parameter errors in order
to explain the mechanism of real brain network. Applying theoretical analysis to synchronization of the proposed networkis also our future work.
R EFERENCES

Coupling strength distribution.

Table II summarizes global synchronization ratio of thestatic networks. We observe that the network with Uniform
distribution achieves 100 % global synchronization. While,the network with Gaussian and Heavy tail distributions failto achieve 100 % global synchronization.
TABLE IIG LOBAL S YNCHRONIZATION R ATIO OF S TATIC N ETWORK
Coupling distribution
Uniform
Gaussian
Heavy tailGlobal synchronization of brain network.

Global synchronization ratio [%]100.0099.62
91.11

Figure 11 shows the simulation results of global synchronization ratio. By increasing the coupling probability, global
synchronization ratio of Uniform and Gaussian coupling distributions increases. While, in the case of Heavy tail coupling
distribution, global synchronization ratio is smaller than othercoupling distributions with whole range of the coupling probability. This results have similar characteristics with the above
simple dynamical network. By applying the real brain network,

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