Journal of Signal Processing, Vol.18, No.4, pp.169-172, July 2014

SELECTED PAPER AT NCSP'14

Polygonal Structure Models of Ring Inhibitory Neural Network
Yuki Shoji, Yoko Uwate and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Phone: +81-88-656-7470, FAX: +81-88-656-7471
E-mail: {shoji, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract
The ring inhibitory neural network is a typical neural systemused to generate oscillation with a simple structure. Neurons in the ring inhibitory neural network are connected inone direction. In this study, we propose polygonal structure
models of the ring inhibitory neural network that are composed of multiple ring inhibitory neural networks. We confirm that the polygonal structure models based on triquetrous
and pentagonal models composed of three and five neurons
respectively can oscillate. In our simulations, the polygonalstructure models show various synchronization and oscillation phenomena based on their network topology.
1. Introduction
The synchronization phenomena of coupled oscillators
have been researched widely in the fields of communication
systems, robot engineering, biological systems, social networks, and so forth, mainly using the van der Pol oscillator and the Kuramoto model. On the other hand, the brain
network and other biological neural networks can also cause
synchronization. The oscillation and synchronization phenomena of neural networks arise from neurons. However,coupled oscillator models based on neurons have not 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 that canreplicate some functions of the human brain. In particular,a neural network with oscillation is applied to pattern recognition and walking rhythm outbreak circuits [1]-[4]. A ringinhibitory neural network is a typical oscillation model witha simple structure and it is composed of neurons connected
in one direction. The weights between the neurons are defined as negative because the neurons composing the system
are inhibitory [5].
In this study, we focus on large-scale networks based on
the ring inhibitory neural network as the neural oscillator.
We propose polygonal structure models of the ring inhibitoryneural network and investigate their synchronization and oscillation phenomena.

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2. Oscillation Systems
2.1 Ring inhibitory neural network
Neural networks with oscillation are classified into two
types: networks with self-connection and without selfconnection such as ring inhibitory neural networks. The ringinhibitory neural network is composed by connecting neurons in one direction. Figure 1 shows a general model of thering inhibitory neural network. The dynamical system of thismodel is described by Eqs. (1) and (2). Moreover, the output
function is defined as the arctangent.
duiτ= −ui + Wji f (uj )(1)
dtf (u) = tan−1 u(2)Here u is the internal value of each neuron and τ is the time
constant. Neurons in the ring inhibitory neural network areconnected unidirectionally with the negative weight Wji . The
oscillation phase of this system is the same as the number ofneurons m. Oscillation is caused if the weight Wji is set
to a negative value. In addition, this model can oscillate if
the number of neurons is odd. We explain this condition byfocusing on the equilibrium point. ue which has to satisfy
Eq. (3). The state of N1 is substituted sequentially in accordance with Eq. (4), when the number of neurons is m.
− ui + Wji f (uj ) = 0(3)u1 = Wm1 f (Wmm−1 f (Wm−1m−2 · · · W21 f (u1 ) · · ·)) (4)Equation (4) is a monotonically decreasing function if thenumber of neurons m is odd. When the function f is monotonically decreasing, the system has only one equilibriumpoint.

Figure 1: Ring inhibitory neural network
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Figs. 5(a) and 5(b) include triquetrous and pentagonal structures. In this study, we classify these models into pentagonalmodels, because they generate five-phase synchronization.
In addition, polygonal structure models can be classified
into two kinds, symmetric and asymmetric models, by thenumbers of inputs and outputs. The polygonal models areclassified as symmetric if the numbers of inputs and outputsof each neuron are equal as in Fig. 3(a). On the other hand,an asymmetric model does not satisfy this condition such as
det|A − λE| = 0(5)the model in Fig. 3(b).
Next, we focus on the stability and equilibrium point of
The equilibrium point is unstable when the ring inhibitory
thetwo polygonal models based on the triquetrous model.
neural network shows a limit cycle. This point is stable if theItisimportant that we analyze the equilibrium point to unreal parts of all eigenvalues are negative. Therefore, the ring
derstandthe characteristic behavior of the system. Frominhibitory neural network form = 3 shows a limit cycle if theEqs.
(2)
and(6), the equilibrium point ue of each system isweight is set to Wji < −2.0.
the origin. Each model can oscillate if each neuron generates a limit cycle around the origin. Therefore, the equilibrium point must be unstable. The eigenvalue λ can be foundfrom Eq. (5) if the Jacobian matrix of the right side of Eq. (6)is the matrix A. The oscillation condition of the asymmetric polygonal model for m = 7 is Wji < −1.1. Similarly,
thesymmetric polygonal model for m = 6 is unstable when
Figure 2: Dynamics of triquetrous ring inhibitory neural netW
< −1.0.
jiwork (m = 3, W = −3.0, τ = 1.0)
Figure 2 shows the dynamics of the ring inhibitory neural network comprising three neurons. This model generates
three-phase waveform. From Eqs. (1) and (2), the equilibrium
point ue of the ring inhibitory neural network is the origin. Inaddition, the stability of the equilibrium point depends on theweight Wji . We define the Jacobian matrix of the right sideof Eq. (1) as matrix A. An eigenvalue λ of matrix A at theequilibrium point is calculated as follows.

ji

2.2 Polygonal structure models
In this section, we explain the polygonal structure models.
The conventional model of the ring inhibitory neural network
can generate only an m-phase waveform if the number of neurons is m. We propose a ring inhibitory neural network withpolygonal structures. The polygonal models are composedby connecting multiple conventional models under a certain
condition. The neurons in this system stimulate each othercomplicated and show various oscillation phenomena. We
can confirm the synchronization of neurons by investigatinga polygonal structure. Moreover, the oscillation and synchronization are affected by the network topology. The dynamicalsystem of this network is described by Eq. (6). The weights
Wji between coupled neurons are all negative, whereas theweight Wji between disconnected neurons is defined as zero.
m∑dui= −ui +Wji f (uj )
(6)
τdt
j=1

Moreover, we define some connection condition. The connections of neurons are set to one direction, which does
not disturb circulation. This connection method follows abase model such as a conventional triquetrous or pentagonal model. In other words, the output of the neurons circulates through three neurons, for example, N1 , N2 , andN4 , as shown in Fig. 3(a). The network models of polygonal structure based on triquetrous and pentagonal models areshown in Figs. 4 and 5, respectively. The connection methods of the symmetric and asymmetric model with m = 10 in

170(a) Symmetric model(b) Asymmetric model

Figure 3: Polygonal models for m = 63. Simulation Results
In our simulation, we investigate the dynamics of polygonal structure models such as those Figs. 4 and 5 by the RungeKutta method. The weight Wji is set to Wji = −3.0. The
time constant τ is set to τ = 2.0.
3.1 Polygonal models based on m = 3
In this section, we give the simulation results for the polygonal structure models based on the conventional model composed of three neurons.
The polygonal model for m = 6 in Fig. 4(a) is classified asa symmetric model. The oscillation of this polygonal modelis shown in Fig. 6(a). This model for m = 6 has a symmetric structure. It generates a three-phase waveform, and eachwave pattern has the same amplitude. In contrast to the asymmetric model, the neurons at symmetric positions cause complete synchronization such as states u1 and u5 . This complete
synchronization is by solving the circuit equation.
By contrast, the neurons in the asymmetric model for m =6 in Fig. 4(b) cause different oscillation phenomena. The os-

Journal of Signal Processing, Vol. 18, No. 4, July 2014

(a) Symmetric model
for m = 6(c) Asymmetric model for m = 7(b) Asymmetric modelfor m = 6(d) Asymmetric model for
m=9Figure 4: Polygonal models based on m = 3 (triquetrousmodel)

for m = 7 in Fig. 4(c). The asymmetric model for m = 7
generates three wave patterns, and each pattern has a different
amplitude. Neurons with the same numbers of inputs andoutputs cause complete synchronization such as u1 , u5 andu6 . This becomes clear by solving the circuit equation.
The polygonal model for m = 9 in Fig. 4(d) is also classified as an asymmetric model. Figure 6(e) shows the oscillation of this model. This model generates nine wave pattern,and each wave pattern has a different amplitude. This model
does not cause complete synchronization between each neuron, in contrast to the asymmetric model for m = 7.

(a) Oscillation of symmetric model for m = 6(b) Oscillation of asymmetric model for m = 6 (N1 , N4 and N6 )(a) Symmetricm = 10models

for(c) Oscillation of asymmetric model for m = 6 (N2 , N3 and N5 )(b) Asymmetric models for (c) Asymmetric models
m = 10for m = 6

Figure 5: Polygonal models based on m = 5 (pentagonalmodel)(d) Oscillation of asymmetric model for m = 7

cillation phenomena of the asymmetric model for m = 6
are shown in Figs. 6(b) and 6(c). This asymmetric modelfor m = 6 generates two kinds of three-phase waveform.
These three-phase waveforms are generated by neurons withthe same numbers of inputs and outputs. Specifically, neurons(e) Oscillation of asymmetric model for m = 9
N1 , N4 and N6 , which have one input and output generate athree-phase waveform. Moreover, the three-phase waveform Figure 6: Oscillation of the polygonal models based on theof neurons N2 , N3 and N5 is the same as the oscillation of triquetrous network for m = 3 (W = −3.0, τ = 2.0)
jithe symmetric model for m = 6.
Figure 6(d) shows the oscillation of the asymmetric model

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3.2 Polygonal models based on m = 5In this section, we give the simulation results for the polygonal models based on m = 5 (pentagonal model). The conventional model for m = 5 generates a five-phase waveform,
and the neurons of this model have the same amplitude.
The model in Fig. 5(a) is the symmetric model for m = 10.
(a) Oscillation of symmetric model for m = 10
This model is symmetric, because the numbers of the inputs and outputs of all the neurons are the same. The oscillation of this model is shown in Fig. 7(a). The model form = 10 generates a five-phase waveform, and each wave
pattern has the same amplitude, similarly to the symmetricmodel in Fig. 4(a). In addition, the neurons located at symmetrical position generate the same wave such as neurons N1(b) Oscillation of asymmetric model for m = 10 (N1 , N2 , N3 ,and N8 . These models have the same characteristic as theN4 and N5 )polygonal models based on model for m = 3.
Figure 5(b) shows the asymmetric model for m = 10. The
oscillation of this model is shown in Figs. 7(b) and 7(c). Twokinds of five-phase wave are generated simultaneously. The
input and output values of neurons N1 to N5 are both three
and those of neurons N6 to N10 are both two. Moreover, thefive-phase waveform of N6 to N10 is the same as the oscilla(c) Oscillation of asymmetric model for m = 10 (N6 , N7 , N8 ,tion of the symmetric model for m = 10. These characterisN9 and N10 )tics are also the same as those of the polygonal models basedon the network model for m = 3.
The network model in Fig. 5(c) is a combination of the triquetrous and pentagonal models. This model is asymmetricand generates the same oscillation phenomenon as the asymmetric model for m = 7 in Fig. 4(c). Figure 7(d) shows theoscillation of this model. This model generates five-pattern(d) Oscillation of asymmetric model for m = 6
oscillations and causes phase-synchronization between neurons. Moreover, neurons with the same numbers of inputs andoutputs synchronize completely such as states u1 and u6 . The Figure 7: Oscillation of the polygonal models based on theamplitude of neuron N4 is larger than the other neurons. A pentagonal network for m = 5 (Wji = −3.0, τ = 2.0)
similar phenomenon was confirmed for the triquetrous asymmetric model for m = 7. The neurons in an asymmetric6, No. 4, pp. 941-948, 1995.
model with more inputs oscillate with a larger amplitude.
[2] S. Still and G. L. Masson: Traveling waves in a ring of4. Conclusion
three inhibitory coupled model neurons, Neurocomputing, Vols. 26-27, pp. 533-539, 1999.
In this study, we focused on the large-scale networks basedon the neural oscillator and proposed polygonal structure[3] G. B. Ermentrout and C. C. Chow: Modeling neural osmodels of the ring inhibitory neural network. These modcillations, physiology and behavior, Vols. 4-5, No. 77, pp.
els cause synchronization which cannot be confirmed using
629-633, 2002.
coupled van der Pol oscillators. In addition, the polygonalstructure models can be split into three categories and gener- [4] G. Endo, J. Morimoto, J. Nakanishi and G. Cheng: An
ate different waveforms in each model. Multi phase oscillaempirical exploration of a neural oscillator for biped lotion was confirmed in some polygonal models. In future, wecomotion control, Robotics and Automation, Proceedwill investigate general rules for the synchronization of eachings of ICRA, Vol. 3, pp. 3036-3042, 2004.
neuron and the oscillation conditions in detail.
[5] Y. Nakamura, Y. Nakano and H. Kawakami: N-phase oscillation of a neural oscillator connected with twisted ring
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Journal of Signal Processing, Vol. 18, No. 4, July 2014