2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Polygonal Structure Model of Ring Inhibitory Neural Network
Yuki Shoji, Yoko Uwate and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima,Tokushima, Japan
Phone/FAX:+81-88-656-7470
E-mail: {shoji, uwate, nishio}@ee.tokushima-u.ac.jp

with connecting neuron to one direction. Figure 1 shows thenetwork model of the ring inhibitory neural network. The dyThe ring inhibitory neural network is a typical model to gen- namical system of this model is described as Eqs. (1) and (2).
erate oscillation with a simple structure. Neurons of the ring Moreover, the output function is defined to arctangent asinhibitory neural network are connected in one direction in- Eq. (2).
hibitory. In this study, we propose the ring inhibitory neudui
ral network with polygonal structure. This model is comτ= −ui + Wji f (uj )(1)dt
posed by some ring inhibitory neural networks as convenf (u) = tan−1 u(2)tional model. We confirm that polygonal models which basedon the conventional triquetrous / pentagonal model composed
Where τ is the time constant. Neurons of the ring inby three and five neurons can oscillate. In these simulahibitoryneural network are connected unidirectionally withtions, polygonal structure models show various oscillations
theweight
Wji of negative value. The oscillation phase of
by changing symmetry and circulation conditions.
this model is same as the number of neuron m. Oscillation iscaused if the weight Wji is set to negative value. In addition,
1. Introduction
this model can oscillate if the number of neuron is odd. WeThe neural network is calculation models which can repli- explain about this term to focus on equilibrium point. Thecate some functions of human brain. This is applied to data equilibrium point u0 has to fulfill Eq. (3). The state of N1 isprocessing, data mining, and so on. In addition, the oscil- substituted sequentially as follows, when the number of neulation model of neural network is applied to pattern recog- ron is m.
Abstract

nition and walking rhythm outbreak circuit[1]-[4]. The ringinhibitory neural network is a one of typical oscillation modelwith a simple structure and it is composed by connected neurons in one direction. Weights between each neuron are defined as nagative because neurons composing the model areinhibitory[5].
In this study, we propose the polygonal ring inhibitory neural network. The structure of polygonal model is constructedby combined conventional models under some condition. Futuremore, we explain about oscillation and synchronization
phenomena of polygonal structure models.

− ui + Wji f (uj ) = 0(3)u1 = Wm1 f (Wmm−1 f (Wm−1m−2 · · · W21 f (u1 ) · · ·)) (4)Equation (4) is monotone decreasing function if the number of neuron m is odd. In case the function f is monotonic
decrease, systems have only one equilibrium point.
Figure 2 shows dynamics of the ring inhibitory neural network which is comprised by three neurons. This model oscillates three-phase waveform. From Eqs. (1) and (2), the equilibrium point u0 of the ring inhibitory neural network is theWm1

2. Oscillation System
2.1 Ring Inhibitory Neural Network

N1NmW21N2W32

Neural networks with limit cycle are classified in twomodel. One is the network model which has self-connection,and the another one does not have self-connection like ringinhibitory neural networks. We apply the ring inhibitory neural network which is comprised of connecting neurons in one
direction. The ring inhibitory neural network is composed-5-N3W43

N4N6N5W65

W54

Figure 1: Ring inhibitory neural network.

origin. In addition, stability of the equilibrium point depends
on the weight Wji . We define jacobian matrix of the rightside of Eq. (1) as matrix A. The eigenvalue λ of matrix A atthe equilibrium point is calculated from the following Eq. (5).
det|A − λE| = 0(5)
The equilibrium point is unstable when the ring inhibitoryneural network shows limit cycle. This point is stable if realparts of all eigenvalues are negative. Therefore, the modelcauses hop bifurcation, and it shows limit cycle if the weightis set to Wji < −2.0.
1.9

0.0

-1.9

t

Figure 2: Dynamics of triquetrous ring inhibitory neural network (m = 3, Wji = −3.0, τ = 1.0).
2.2 Polygonal Structure ModelIn this section, we explain about the polygonal structure
model. In this study, we propose to extend the polygonal structure from the ring inhibitory neural network. The
polygonal models are comprised by connecting some ringinhibitory neural networks under certain condition. We canconfirm synchronization of neurons and characteristic oscillation by applying polygonal structure. The dynamical systemof this model is described in Eq. (6). Weights Wji between
coupled neurons are unified negative value each. Whereas,
the weight Wji between disconnected neurons is defined aszero.
m∑duiτ
= −ui +Wji f (uj )
(6)
dt
j=1

from Eq. (5) if jacobian matrix of the right side of Eq. (6)is matrix A. The oscillation conditon of m = 7 asymmetry
polygonal model is Wji < −1.1. In the same way, the oscillation condition of m = 6 symmetry polygonal model isWji < −1.0.
Moreover, Fig. 4(a) shows the non-oscillation networkmodel which removed some combination from m = 9 polygonal model. This model can not oscillate in spite of having a
unstable condition. Such phenomenon is confirmed in many
non-oscillation models. Now, we define the loop number L.
The loop number L means the number of neurons of the paththrough the neurons most in a single stroke. For example, theloop number of the m = 7 polygonal model is L = 3 by binding of solid lines like a Fig. 3(a). While the m = 9 polygonalmodel has L = 9 due to pass nine neurons when trace solidlines like a Fig. 3(b). Moreover, the loop number is related
to the oscillation condition. The polygonal model can not oscillate except some models if the loop number L is even for
example the non-oscillation model in Fig. 4(d). While models
which have odd loop numbers like Figs. 4(a), 4(b), and 4(c)can oscillate.
Futuremore, the polygonal model can be split into three
categories, asymmetry models of partial, the one of entire circulation, and symmetry models. The network is determinedas the symmetry model if the number of input and output of
all neurons is the same. The models of Figs. 4(d) and 5(b) areclassified into the symmetry model. The asymmetry modelunder the neuronal number m > L is defined asymmetry ofpartial circulation. The models of Figs. 4(a), 5(a) and 5(c)are classified into the asymmetry model of partial circulation.
On the other hand, the asymmetry model of entire circulationis the case of m = L. The model of Fig. 4(a) is classified
into asymmetry model of entire circulation. The oscillationof polygonal model varies among these models. From this,the loop number L is similar as the number of neuron in theconventional models.

In addition, we define some condition of connections.
Connections of neuron are set to one direction which does
not disturb circulation. This connection method obeys thebase model like a triquetrous and pentagonal conventional
network. In other words, the output of neuron circulate
through three neurons, for example N1 , N2 and N4 in
Fig. 3(a). Network models of polygonal structure based on(a) m = 7 model (L = 3).
(b) m = 9 model (L = 9).
triquetrous / pentagonal ring neural networks are shown on
Figure 3: Polygonal model and circulation route.
Figs. 4 and 5 each.
Next, we focus on stability and the equilibrium point of
the polygonal models based on triquetrous network. It is
important that we analyze the equilibrium point to understand the characteristic of the behavior of the system. From 3. Simulation Result
Eqs. (2) and (6), the equilibrium point u0 of the polygonal
In this study, we confirm oscillation by the Runge-Kutta
model is the origin. This model can oscillate if each neuron causes limit cycle around the origin. Therefore, the equi- method. The weight Wji is set to Wji = −3.0. The time
librium point must be unstable. The eigenvalue λ is found constant τ is set to τ = 2.0.

-6-and each wave pattern has the different amplitude. In addition
to this, neurons which have the same number of input andoutput synchronize like u1 , u5 and u6 . This becomes clear by
calculating the circuit equation.
From the m = 6 polygonal model of Fig. 4(b), this modelis classified into a symmetry model. The loop number L ofthis model is L = 6. The waveform which is oscillated by(a) m = 7 asymmetry.
this model is shown on Fig. 6(a). This model oscillate three(b) m = 6 symmetry.
phase waveform, and each wave pattern has the same amN1
plitude only in this case. As a different from the asymmetry
model, only this model can oscillate in spite of having an evenN3
N2loop number. Neurons at the symmetry position synchronize
like u1 and u5 . This becomes clear by calculating a circuit
N4
N5
N6equation.
From the m = 9 polygonal model of Fig. 4(c), this model
N8N7
is classified into an asymmetry model of entire circulation.
N9Figure 6(c) shows the oscillation of this model. The asym(c) m = 9 asymmetry.
metry model of entire circulation oscillates nine-phase wave(d) Non-oscillation model.
form, and each wave pattern has the different amplitude. As
Figure 4: Polygonal model based on m = 3 (triquetrous different features with the m = 7 and m = 6 models, this osmodel).
cillates L-phase wave without some models if the loop number is L only in this case.
N1
The model of Fig. 4(d) can not oscillate, because thismodel is asymmetric and the loop number is L = 6. FromN5N2
this result, we consider that the asymmetric model of even L
can not oscillate.
N4N3(a) m = 6 asymmetry.

N1N1N5(a) Oscillation of m = 7.

N2N3N4(b) m = 10 symmetry.

u2,u3,u7
u1,u5,u6

tN2

N5N4

u4

7.8
5.2
2.6
0.0
-2.6
-5.2
-7.8

7.8
5.2
2.6
0.0
-2.6
-5.2
-7.8

N3(c) m = 10 asymmetry.

Figure 5: Polygonal model based on m = 5 (pentagonalmodel).

t(b) Oscillation of m = 6.

7.8
5.2
2.60.0
-2.6
-5.2
-7.8

3.1 Polygonal Model based on m = 3u2,u6 u3,u4 u1,u5

In this section, we explain about simulation results ofthe polygonal structure model based on conventional modelt
which was composed by three neurons.
(c) Oscillation of m = 9.
From the m = 7 polygonal model of Fig. 4(a), this modelis classified into an asymmetry model of partial circulation. Figure 6: Oscillation of the polygonal model based on m = 3This model has the loop number L = 3 and oscillates the triquetrous network (Wji = −3.0, τ = 2.0).
waveform like Fig. 6(a). The m = 7 polygonal asymmetry
model of partial circulation oscillates three-phase waveform,-7-

3.2 Polygonal Model based on m = 5

u1=u6

In this section, we explain about simulation results of thepolygonal model based on m = 5 (pentagonal model). Them = 5 conventional model oscillates five-phase wave, andneurons of this model have unified amplitude.
The network model in Fig. 5(a) shows the combination of
the triquetrous and pentagonal models. This model fulfills(a) m=6 asymmetry.
condition of asymmetry model of partial circulation. Figu3=u10
u4=u6
u2=u9 u1=u8 u5=u3
ure 7(a) shows an oscillation of this model. The m = 10
7.8
polygonal model oscillates five-phase wave, and wave pat5.2
2.6
terns have the different amplitude each. Neurons which have0.0
the same number of input and output synchronize as u1 and-2.6-5.2
u6 . This becomes clear by calculating a circuit equation. The-7.8
feature of this model follows an asymmetry model of partialtcirculation.
(b) m=10 symmetry.
The network model in Fig. 5(b) is the m = 10 symmetric model. This model fulfills condition of the symmetry
model, because values of input and output of all neurons are
the same. The oscillation of this model shows in Fig. 7(b).
The m = 10 model oscillate five-phase wave, and wave patterns have the same amplitude each like the symmetry modelof Fig. 4(b). In addition, neurons located on symmetry os(c) m=10 asymmetry (N1 , N2 , N3 , N4 and N5 ).
cillate same wave like N1 and N8 . These models have samecharacteristic as the polygonal models based on m = 3 network model.
Figure 5(c) shows the network model of m = 10 asymmetry. The oscillation of m = 10 asymmetric model is shown inFigs. 7(c) and 7(d). Two kinds of five-phase wave are oscil(d) m=10 asymmetry (N6 , N7 , N8 , N9 and N10 ).
lated simultaneously. The input and output value of N1 to N5
are three and the one of N6 to N10 neurons are two. While theloop number L is L 6= 5, the waves oscillated by this model Figure 7: Oscillation of the polygonal model based on m=5
show five-phase.
pentagonal network (Wji =-3.0, τ =2.0).
4. Conclusion
Locomotion Control, Robotics and Automation, 2004.
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