Circuit Design of Two Template Cellular Neural Networks for the Observation of Oscillation
Phenomena
Yasuteru Hosokawa1 , Yoko Uwate2 , and Yoshifumi Nishio2
1 Shikoku University
Furukawa, Ohjin-Cho, Tokushima 771-1192 Japan
Phone:+81-88-665-1300
E-mail: hosokawa@keiei.shikoku-u.ac.jp

2 Tokushima University
2-1 Minami-Josanjima-Cho, Tokushima 770-8506 Japan
Phone:+81-88-656-7470
E-mail: {uwate,nishio}@ee.tokushima-u.ac.jp

Abstractc(1,1)

Some interesting phenomena concerned with oscillations canbe observed in Two Template Cellular Neural Networks(TTCNN) propsed by Fujii et al [13]. However, these phenomena were confirmed on computer simulations from amathematical model only. A circuit experimentation is animportant factor for investigations of circuit systems. In thisstudy, a circuit of TTCNN have been designed. An oscillation, which is the same as the mathematical model, could beconfirmed by SPICE Simulations of the designed circuit.
1. Introduction

c(1,N)c(i-1,j-1)c(i-1,j)c(i-1,j+1)c(i,j-1)c(i,j)

c(i,j+1)c(i+1,j-1)c(i+1,j)c(i+1,j+1)c(M,1)

Cellular Neural Networks (CNN) proposed by Chua andYang [1][2] have many application posibilities. For instance,
image processing, pattern recognition, associative memoryand so on have been considered. The main features are an uniform structure consisting of local connections, continuoustime dynamics and parallel processing. Especially, the uniform structure is an import feature for the implementation as
an integrated circuit. On CNNs, all cells and signal wires
are located uniformly. Therefore, the implementation as anintegrated circuit is easy [3]-[5]. However, almost all modified CNNs [6]-[12], which have many advantages comparedto the conventional CNN, have a disadvantage in the point ofview from a circuit implementation. These CNNs need many
complicated signal wire layouts, many amplitudes, many operation signal and/or so on.
Two template CNN proposed by Fujii et al [13] is designed
by paying attention to a circuit implementation. Two different cloning template sets are located as a checkered pattern.
Thus, the structure is almost the same as the conventional
CNN. The difference of a conventional CNN is only operationsignals for values of a second cloning template set. Althoughsome interesting phenomena concerned with oscillations canbe observed, these were confirmed on computer simulations
from a mathematical model only.
In this study, a circuit of two template cellular neural net-c(M,N)

Figure 1: Structure of two template cellular neural networks.
Different cloning template sets are applied to magenta cells(Cell α) and blue cells (Cell β).

works is designed. An oscillation, which is the same as themathematical model, can be confirmed by SPICE Simulationsof the designed circuit.
2. Archtecture of Two Template CNN
The system model of Two Template CNN as shown inFig. 1 has a two-dimensional M by N array structure. Eachcell in the array is denoted as c(i, j), where (i, j) is the position of the cell, 1 ≤ i ≤ M and 1 ≤ j ≤ N . The couplingradius is assumed to be one. Cells having one cloning template are called as cell α and the other cells are called as cell
β. These two types of the cells are placed as a checkeredpattern.
A difference between the conventional CNN and the proposed system is only the number of signal lines for cloning
templates. Therefore, the proposed system is suitable for IC
implementation as the same as the conventional CNN.

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The state equations of the cells are given as follows:
1: For cell α.
Xdxij= − xij + Iα +Aα (i, j; k, l)ykl
dtc(k,l)X+
B α (i, j; k, l)ukl(1)

c(k,l)

2: For cell β.
Xdxij= − xij + Iβ +Aβ (i, j; k, l)ykl
dtc(k,l)X+B β (i, j; k, l)ukl(2)Figure 2: Minimum architecture of TTCNN for the oscillation.

Figure 3: Streamlined minimum architecture of TTCNN
for the oscillation.

c(k,l)

A{αβ} (i, j; k, l), B {αβ} (i, j; k, l) and I{αβ} are called as thefeedback coefficient, the control coefficient and the bias current, respectively. The variables u, x and y are the input, stateand output variables of the cell, respectively. Aα , B α , Aβand B β are 3 × 3 matrices, which can be described for example:
Aα =Aα (i, j; i − 1, j − 1)
 Aα (i, j; i, j − 1)
Aα (i, j; i + 1, j − 1)

Aα (i, j; i − 1, j)
Aα (i, j; i, j)
Aα (i, j; i + 1, j)

Aα (i, j; i − 1, j + 1)
Aα (i, j; i, j + 1)  .
Aα (i, j; i + 1, j + 1)

One of the parameter sets obtained oscillation phenomena
is p = 2, q = 4, and r = 1. In this case, the architecture
can be converted into Fig. 3. In Fig. 2, each cell has four/twolines connecting another cell, because the boundary condition is set as periodic. Additionally, cloning template values of these four/two lines are same. Therefore, by changing
four/two times cloning template value of one line and eliminating the other lines, Fig. 2 is streamlined into Fig. 3. Thisarchitecture is designed as an electric circuits.
4. Circuit Design(3)

The output equation of the cell is given as follows:yij = 0.5(|xij + 1| − |xij − 1|).

(4)

3. Settings for Oscillation Phenomena in TTCNNIn this section, settings for Oscillation Phenomena inTTCNN are described. In order to simplify the system parameters, cloning templates are set as symmetry as follows.
−pAα =  q−ppAβ = −qp

−pq  , B α = 0, Iα = 0,−p−qp−r −q  , B β = 0, Iβ = 0.
−qpqrq(5)Figure 4 shows a circuit schematic of Cell α1 as an example schematic of Cells. The cell obtains input singals
from output of each cell including itself. All input signals
are multiplied by cloning template values and these values
are summed. These are realized by adder-substractor partof Fig. 4. Basically, the cloning template values are set as
values of Ra1−1 , Ra1−2 , Ra1−3 , and Ra1−4 . Specifically,
R1R1R1R1p = Ra1−3, q = Ra1−2= Ra1−4, and r = Ra1−1. In case ofa negative value, an inverter is also added. The inverter consists of OPamp and two resistors. For instance, one inverter
consists of OPamp U6 , resisters R10 , and R11 . The voltage as
multiplied and summed signals is converted into a current forthe state value. The state value is shown as voltage C1 in theRC part. The limiter part as shown in Fig. 4 converts the statevalue to output value. Namely, This part means the output
function and value 1 in this system is defined as a threasholdvoltage of the diode in this part.
A difference from other cells is only adder-substracter part.
Since this part includes cloning template values, registancevalues and the number and the position of the inveter are difference points. The other parts are complimentally same.

Boundary conditions are set as periodic. The input values donot concern to the result because B α = B β = 0.
By results of many computer simulations, the minimum
number of an observation of oscillatory phenomena is four.
Namely, the simplest architecture for an oscillation consists 5. SPICE Simulationof four cells as shown in Fig. 2. In the conventional CNN,
Figure 5 shows one of SPICE simulation result of Fig. 3
symmetry cloning templates could not oscillate. This point isincluding Fig. 4. Each state values of cells are shown. Folone of the important feature of TTCNN.

- 542 -Figure 4: Circuit schematic of Cell α1 of the designed circuit.

lowing settings are applied. R1 = R3 = R6 = R8 = R9 =R11 = R12 = Rα1−1 = 10.0 [kΩ]. R1 = Ra1−1 = 40[kΩ], R2 = R5 = 100 [kΩ], R3 = R6 = R7 = R8 =
R9 = R10 = R11 = R12 = R13 = R14 = R15 = 10[kΩ],R4 = 1[kΩ], and Ra1−2 = Ra1−3 = Ra1−4 = 5[kΩ]. Thesimulator is LTspice [14]. Voltages of capasitors VCCellα1 =0.5, VCCellα2 = −0.5, VCCellβ1 = 0.5, and VCCellβ2 = 0.5
are set as initial values. An oscillation can be obsereved. Theamplitudes of Cells α and β are about 4[V] and 2[V]. Thebias of Cells α and β are about ±3[V] and 0[V]. In this system, value 1 is about 0.5[V], bacause it is defined as a diodethreashold voltage. Therefore, the amplitudes of Cells α andβ are about 8 and 4. The bias of Cells α and β are about ±6
and 0. This result is almost the same as a corresponding resultof mathematical model as shown in Fig. 6.
6. Conclusion[2] L. O. Chua and L. Yang, ”Cellular neural networks: Applications,” IEEE Trans. Circuits and Systems, vol. 35,
no. 10, pp. 1273–1290, 1988.
[3] L. Yang, L. O. Chua and K. R. Krieg, ”VLSI implementation of cellular neural networks,” Proc. Int. Symp. Circuits and Systems, vol. 3, pp. 2425–2427, 1990.
[4] J. E. Varrientos, E. Sanchez-Sinencio and J. RamirezAngulo, ”A current-mode cellular neural network implementation,” IEEE Trans. Circuits and Systems II, vol. 40,
no. 3, pp. 147–155, 1993.
[5] G. C. Cardarilli, R. Lojacono, M. Salerno and F. Sargeni, ”VLSI implementation of a Cellular Neural Network with programmable control operator,” Proc. Midwest Symp. Circuits and Systems, vol. 2, pp. 1089–1092,
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[6] T. Roska and L. O. Chua, ”Cellular neural networks withIn this study, a circuit of TTCNN which consists of four
nonlinear and delay-type template elements,” Proc. Int.
cells have been designed. By SPICE simulation, an oscillaWorkshop on Cellular Neural Networks and Their Applition which is the same as the mathematical model could becations, pp. 12–25, 1990.
confirmed. Thus, the circuit design is reasonable.
As future works, a circuit implementation and increasing [7] T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, ”Fuzzy
cellular neural networks: theory,” Proc. Int. Workshop
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Acknowledgment[8] W. Porod, C. S. Lent, G. Tóth, H. Luo, A. Csurgay, Y.F. Huang and R.-W. Liu, ”Quantum-dot cellular nonlinThis work was partly supported by JSPS Grant-in-Aid forear networks: computing with locally-connected quanScientific Research 16K06357.
tum dot arrays,” Proc. Int. Symp. Circuits and Systems,
vol. 1, pp. 745–748, 1997.
References[9] K. Tsuruta, Z. Yang, Y. Nishio, A. Ushida, ”Small-world
cellular neural networks for image processing applica[1] L. O. Chua and L. Yang, ”Cellular neural networks: Thetions,” Proc. European Conf. on Circuit Theory and Deory,” IEEE Trans. Circuits and Systems, vol. 35, no. 10,
pp. 1257–1272, 1988.
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Figure 6: Computer simulation result of the mathematicalmodel of TTCNN. The number of cell is four. p = 2,q = 4, and r = 1. Time step is 0.01. Initial values are
x1 = x2 = x3 = 1 and x4 = −1. These variables arecorresponding to internal state values of each cell.

8642[14] Linear Technology, ”LTspice,”0

http://www.linear.com/designtools/software/#LTspice
Accessed Jan. 2018.

-2-4-6-800.005

0.01

0.015

0.02

Figure 5: SPICE Simulation result of the designed circuit.
Vertical axes shows voltages [V] of four capasitors. First line:
Cell α1, Second line: Cell β1, Third line: Cell β2, and Fourthline: Cell α2. Horizontal axes shows time [s].

[10] A. Bouzerdoum and R. B. Pinter, ”Shunting inhibitory
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pp. 215-221, 1993.
[11] P. Arena, S. Baglio, L. Fortuna and G. Manganaro,”Complexity in a two-layer CNN,” Proc. Int. Workshop Cellular Neural Networks and their Applications,pp. 127–132, 1996.
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[13] J. Fujii, Y. Hosokawa, and Y. Nishio, ”Wave Phenomena
in Cellular Neural Networks Using Two Kinds of Template Sets,” Proc. of International Symposium on Nonlinear Theory and its Applications, pp. 23-26, 2007.

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