2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Synchronizations and Oscillations of Negative Coupled Neural Oscillators
Yuki Shoji† , Yoko Uwate† and Yoshifumi Nishio††Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {shoji, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Neurons transmit signals through the chemical and electrical synapses such as resistance coupling thatact as a neurotransmitter. This electrical synapse functions
such as the resistor coupling. By contrast, the coupling vander Pol oscillators generate complex synchronization phenomena. In this study, we focus on the negative coupledneural oscillator such as a ring and polygonal topology. We
give the simulation results and investigate comparison withthe coupled neural oscillators to van der Pol oscillators.

1. Introduction
The synchronization phenomena of coupled oscillators
have been researched widely in the fields of communication systems, robot engineering, biological systems, socialnetworks, and so forth, mainly using the van der Pol oscillator and the Kuramoto model. On the other hand, thebrain network and other biological neural networks canalso cause synchronization. The oscillation and synchronization phenomena of neural networks arise from neurons.
However, coupled oscillator models based on neurons havenot been investigated suﬃciently. It is important to investigate oscillator models based on neurons in order to reveal
various complex synchronization phenomena in the brain.
Artificial neural networks are calculation models thatcan replicate some functions of the human brain. In particular, a neural network with oscillation is applied to patternrecognition and walking rhythm outbreak circuits [1]-[2].
Meanwhile, neurons transmit signals through the synapses
that act as a neurotransmitter each other. Synaptic connection methods are classified into two types: a chemical synapse and an electrical synapse, the so-called ”gapjunction”. The chemical synapse denotes the general combination of a mathematic neuron model. By contrast, theelectrical synapse functions such as a resistor coupling andpromotes synchronization between the neurons.
Here the van der Pol circuit is one of the oscillation electric circuit. The coupled oscillator composed by some vander Pol circuits shows complex synchronization and oscillation phenomenon. In this paper, we propose negative
coupled neural oscillators as a ring and polygonal topology. We discuss various synchronizations and comparewith these phenomena of the coupled neural oscillators tothe van der Pol oscillators [3][4].

2. Oscillation Circuit
2.1. Neural oscillator
First of all, we explain the general neural oscillator composed by two components. The general neuronal oscillator
is shown in Fig. 1. This model is composed by the excitatory neuron N1 and the inhibitory one N2 . The weight W21
is defined as the positive weight W21 = W positive and W12 isdefined as the negative weight W12 = Wnegative . Moreover,the neuron N1 has a self-connection with the weight W11 of
the positive weight W11 = W positive . The dynamical systemof this general model is described by Eq. (1). Here u is theinternal value of each neuron and τ is the time constant.
The output function is defined as the arctangent.
du1

τ
= −u1 + W12 f (u2 ) + W11 f (u1 )dt du2τ
= −u2 + W21 f (u1 )(1)
dt

f (u) = tan−1 uFigure 2 shows oscillation of the general model with theweight W positive = 2.0 and Wnegative = −2.0. The neurons inthis general model generates oscillation each.
The negative coupled neural oscillators are composed bysome general oscillators coupled each other. In the coupled neural oscillator, each excitatory neuron Nodd is connected with the resister such as the electrical synapse. Theequation of the electrical synapse is described by Eq. (2).
W21W11N1N2W12Figure 1: The general neural oscillator.
u1 u27.8
5.2
2.60.0-2.6-5.2-7.8tFigure 2: Oscillation of the general neural oscillator(W positive = 2.0, Wnegative = −2.0, τ = 2.0).

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Here the parameter Wgap denotes the weight of the electrical synapse such as a resistance value. V pre is membranepotential of a previous neuron.
r

Igap = −Wgap (U − U pre )Wgap

N1W21W12N5W56

N3 W33Wgap
W65W34

N6il3

Cv3

r
L

Lr

ir2

2nd oscillator
il2Cv2L

Figure 4: The negative coupled van der Pol oscillator as aring topology.

3. Simulation Results

3.1. Ring topology

W43

3.1.1. Neural oscillator

N4

Figure 3: The negative coupled neural oscillator as a ringtopology.

2.2. van der Pol oscillator
In this section, we introduce the van der Pol oscillator.
As a characteristic, the van der Pol oscillator is similar tothe neural oscillator. Figure 4 shows the negative coupledvan der Pol oscillator as a ring topology. The resistancer is defined as a negative value. We assume that the vk −irK characteristics of the nonlinear resistor is given by thefollowing equation
irK = −g1 vk + g3 v3k

3rd oscillator
ir3v1

In our simulation, we investigate dynamics of the models by the Runge-Kutta method of 4-order with the time
distance h = 0.05.

Wgap

N2

W55

C(2)

The coupling with the general resister promotes synchronization. By contrast, we consider the negative coupled oscillators connected by the negative resistances. In the coupled neural oscillator, the excitatory neurons Nodd are connected each other to compose a topology. Figure 3 shows
the negative coupling neural oscillator as a ring topology.
In our simulations, the parameters of neural oscillators aredefined as W positive = 3.0, Wnegative = −3.0, Wgap = −0.18,
τ = 2.0.
W11

ir1 il1(g1 , g3 > 0), (k = 1, 2, 3).

The negative coupled neural oscillator as a ring topology
in Fig. 3 is composed by connecting the excitatory neurons N1 , N3 , and N5 . Figure 5(a) shows oscillations of
the negative-coupled neural oscillator as a ring topology.
The amplitude of each oscillation is larger and frequency islower than one of the general neural oscillator. This neuraloscillator generates oscillation with three-phase synchronization in the whole excitatory neurons.

(a) Oscillation (the excitatory neurons N1 , N3 and N5 ).

(3)

The normalized circuit equations governing the circuit inFig. 4 are expressed as dx1k
= −yk + (1 − xk2 )x − γ(2xk − xk+1 − xk−1 )dτ3dylk= xk(k0 = k3 , k4 = k1 ).
 dτ(4)wheret=√LCτ, vk =
 = g1√√

g1
3g3 , ilk =LC,

γ=r

√
√

g13g3

√

LC yl k,(b) Phase planes between the negative
coupled neurons.

Figure 5: The behavior of the negative-coupled neural oscillator as a ring topology (Wgap = −0.18, W positive = 2.0,
Wnegative = −2.0, τ = 2.0).

3.1.2. van der Pol oscillator

CL.

Here γ denotes the nonlinearity of the oscillation and 
is the coupling strength. The resistor r is defined as thenegative value. In our simulations, the parameters of vander pol oscillators are defined as  = 0.1, γ = −0.1.

The negative coupled van der Pol oscillator as a ring
topology in Fig. 4 is composed by connecting van der Pol
oscillator through the negative resisters. Figure 6 shows
the oscillation of the this coupled oscillator. The oscillators generates oscillation with three-phase synchronizationin the whole system. This synchronization phenomenon is- 828 -common to the neural and the van der Pol oscillator. Thissynchronization phenomenon is similar to one of the coupled van der Pol oscillator in the references [3]. From theabove, the negative coupled oscillators as a ring topology
show three-phase synchronization in any models.

W77 N7Wgap

Wgap
W78

N1 W11Wgap
Wgap

W87

W12

N2N5W55
W56

W21

N2N3 W33

Wgap
W65

W34

N6W43

N4

Figure 7: The negative coupled neural oscillator as 3 − 3polygonal topology.

Figure 6: The oscillation of the van der Pol oscillator as aring topology ( = 0.1, γ = −0.1).
(a) Oscillation (the excitatory neurons N1 , N3 , N5 and N7 ).

3.2. Polygonal topology
In this section, we focus on the negative coupled oscillator as a polygonal topology. We propose the symmetric3 − 3 and the asymmetric 3 − 5 polygonal topology.
3.2.1. 3 − 3 neural oscillator
Figure 7 shows the negative coupled neural oscillator as
a 3 − 3 polygonal topology. The excitatory neurons N1 , N3 ,N5 , and N7 are connected each other through the negative
coupling without a connection between the neurons N1 toN5 . The connection between the neurons N3 to N7 is sharedwith two ring topology.
Figure 8 shows the behavior of the neural oscillator as
3 − 3 polygonal topology. The oscillation phenomenon on
this condition are shown in Figs. 8(a) and phase planes areshown in 8(b). The neurons neighbored each other without
the neurons N3 and N7 synchronize with anti-phase state.
By contrast, the excitatory neurons N3 and N7 cause synchronization phenomenon with in-phase. These phenomena can be confirmed by phase planes in 8(b).

(b) Phase planes between the negative coupled neurons.

Figure 8: The behavior of the neural oscillator as a 3 −3 polygonal topology (Wgap = −0.18, W positive = 2.0,
Wnegative = −2.0, τ = 2.0).
4rd oscillator

1st oscillatorr

ir1 il1
r

ir3

il3

Cv3

ir4

il4

Cv4

r
L

Lr

r

ir2

il2

Cv2

3rd oscillator

Cv1

LL

2nd oscillator

Figure 9: The van der Pol oscillator as a 3 − 3 polygonal
topology.

3.2.2. 3 − 3 van der Pol oscillator
Figure 9 shows the negative coupled van der Pol oscillator as a 3 − 3 polygonal topology. The 1st and 3rd oscillator
is also shared with two ring topology such as neural oscillator as a 3 − 3 polygonal topology.
The behavior of the van der Pol oscillator as 3 − 3 polygonal topology is shows in Fig. 10. This coupled oscillator also cause synchronization with the neighbor oscillators. The 2nd oscillator synchronized only the 4th oscillator with in-phase state, and synchronizations in other couplings show anti-phase state. This synchronization phenomenon is similar to one of the coupled van der Pol oscillator such as the references [4]. The negative coupledoscillators as a 3 − 3 polygonal topology also show samesynchronization regardless of each model.

Figure 10: The oscillation of the van der Pol oscillator as a3 − 3 polygonal topology ( = 0.1, γ = −0.1).

3.2.3. 3 − 5 neural oscillator
Figure 11 shows the negative coupled oscillator as a 3−5
polygonal topology. We give the simulation results of theneural and van der Pol oscillator as this topology.
Figure 12 shows the behavior of the neural oscillator asa 3 − 3 polygonal topology. As characteristics of the asymmetrical negative coupled neural oscillator as a polygonal- 829 -topology, the shared neurons between two rings only synchronize with in-phase state. By contrast, other connections synchronize with anti-phase state.
In the case of a polygonal topology, coupled oscillator
can synchronized with in-phase and anti-phase if both ring
topologies are composed by odd neural oscillators. In othercases, each neural oscillator can not synchronize and increase the lineality.
N1, N2(1st oscillator)

in-phase in negative coupled neural oscillator as a polygonal topology synchronization, nevertheless the power consumption is minimum if it is 80 degrees. From the above,synchronizations of this coupled neural oscillator are notdepended on the power consumption.

N7, N8(4th oscillator)
N9, N10(5th oscillator)(a) Oscillation (the 1st, 2nd and 3rd oscillators).

N5, N6(3rd oscillator)

N3, N4(2nd oscillator)

N11, N12(6th oscillator)

Figure 11: The negative coupled oscillator as a 3 − 5 polygonal topology.

(b) Oscillation (the 1st, 2nd, 4th, 5th and 6th oscillators).

Figure 13: The behavior of the van der Pol oscillator as a
3 − 5 polygonal topology ( = 0.1, γ = −0.1).

4. Conclusion(a) Oscillation (the excitatory neurons N1 , N3 and N5 ).

(b) Phase planes between negativecoupled neurons.

Figure 12: The behavior of the neural oscillator as a
3 − 5 polygonal topology (Wgap = −0.18, W positive = 2.0,
Wnegative = −2.0, τ = 2.0).

In this study, we have focused on oscillations and synchronizations of negative coupled neural oscillators. Some
negative coupled neural oscillators as a polygonal topology cause synchronization that can not be confirmed in thecoupled van der Pol oscillators. On the other hand, neuraloscillators as a ring topology shows similar synchronizations. In future, we will investigate general rules for thesynchronization of each neuron and the oscillation conditions in detail.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aidfor Challenging Exploratory Research 26540127.

3.2.4. 3 − 5 van der Pol oscillator
Figure 13 shows the behavior of the van der Pol oscillator as a 3 − 5 polygonal topology. In the contrast to theprevious neural oscillator, the van der Pol oscillator in thistopology can not cause synchronization with in-phase andanti-phase. Each oscillation has various phase diﬀerences.
These synchronized states close to the one of the coupled
polygonal oscillatory networks in the references [4]. In
this reference, it is introduced that the phase diﬀerences
are caused in order to decrease the power consumption of
the coupling resistors. These negative coupled neural oscillators are able to say the same thing.
In Ref [5], the power consumption in the whole system isintroduces as a cause of the phase diﬀerences. Specifically,
the phase diﬀerence at the shared connection and others
are varied in order to make this power consumption minimum. However, the shared neurons are synchronized withReferences[1] D. Wang: Emergent synchrony in locally coupled neuraloscillators, IEEE Transactions on Neural Networks, Vol.6,No.4, pp. 941-948, 1995.
[2] G. Endo, J. Morimoto, J. Nakanishi and G. Cheng: An empirical exploration of a neural oscillator for biped locomotion
control, Robotics and Automation, Proceedings of ICRA,
Vol.3, pp. 3036-3042, 2004.
[3] Y. Uwate, Y. Nishio, R. Stoop: Synchronization in three coupled van der Pol oscillators with diﬀerent coupling strength.
2010 International Workshop on Nonlinear Circuits, Communication and Signal Processing, pp.109-112, 2010.
[4] Y. Uwate, Y. Nishio: Synchronization in several types of coupled polygonal oscillatory networks. IEEE Transactions onCircuits and Systems, Vol.59, No. 5, pp.1042-1050, 2012.

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