VO2 Carbon Nanotube Composite Memristor-Based Cellular Neural Network Pattern Formation

electronics

Article VO2 Carbon Nanotube Composite Memristor-Based Cellular Neural Network Pattern Formation

Yiran Shen and Guangyi Wang *

Institute of Modern Circuits and Intelligent Information, Hangzhou Dianzi University, Hangzhou 310018, China; [email protected] * Correspondence: [email protected]

Abstract: A cellular neural network (CNN) based on a VO2 carbon nanotube memristor is proposed in this paper. The device is modeled by SPICE at first, and then the cell dynamic characteristics based on the device are analyzed. It is pointed out that only when the cell is at the sharp edge of chaos can the cell be successfully awakened after the CNN is formed. In this paper, we give the example of a 5 × 5 CNN, set specific initial conditions and observe the formed pattern. Because the generated patterns are affected by the initial conditions, the cell power supply can be pre-programmed to obtain specific patterns, which can be applied to the future information processing system based on complex space–time patterns, especially in the field of computer vision.

Keywords: VO2 carbon nanotube composite memristor; cellular neural network (CNN); von Neu- mann structure; local activity; edge of chaos

1. Introduction

Citation: Shen, Y.; Wang, G. VO2 The traditional processor uses the von Neumann structure, in which the storage unit Carbon Nanotube Composite and the processing unit are separated and connected by bus. In recent years, with the Memristor-Based Cellular Neural development of semiconductor technology, the speed of processing units has been greatly Network Pattern Formation. improved. However, due to the limitation of bus bandwidth, the operation speed of the Electronics 2021, 10, 1198. https:// whole processor is limited, which is called the “von Neumann bottleneck problem” [1,2]. In doi.org/10.3390/electronics10101198 order to solve this problem, inspired by the biological way of processing information, many pieces of literature have proposed various non-von Neumann processor solutions [3–5]. Academic Editor: Kris Campbell Typical bio-inspired computing relies on nonlinear networks that contain the same cells; each cell has a relatively simple structure and interacts with the surrounding cells. This Received: 10 April 2021 structure, called a CNN, a cellular neural network, has been applied in ﬁelds such as Accepted: 11 May 2021 Published: 18 May 2021 computer vision [6–8]. The concept of a CNN can be traced back to two articles by Chua in 1988 [9,10], which

Publisher’s Note: MDPI stays neutral respectively give the theory and application of CNNs. Recently, Itoh [11] summarized with regard to jurisdictional claims in some characteristics of CNNs in a long paper. Weiher et al. [12] discussed the pattern published maps and institutional affil- formation of CNNs based on the NbO2 memristor. iations. Recently, brain-like computing has become a hot topic in research [13–15], and the key is to find devices that can produce spike signals like neurons and that have very low power consumption. The VO2 carbon nanotube (CNT) composite device is a Mott memristor recently proposed [16]. It can generate periodic peak pulses with a pulse width of less than 20 ns, it is driven by a DC current or voltage, and it does not need additional Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. capacitance. It uses metal–carbon nanotubes as heaters, and, compared with the pure Mott This article is an open access article VO2 proposed earlier, adding CNTs can greatly reduce the transient duration and pulse distributed under the terms and energy and increase the frequency of a peak pulse by three orders of magnitude. conditions of the Creative Commons The VO2 nano crossbar device does not need the process of electric forming and has Attribution (CC BY) license (https:// low device size dispersion. For devices with a critical size between 50 and 600 nm, the creativecommons.org/licenses/by/ change coefficient of the switch threshold voltage is less than 13%, the switch durability 4.0/). is more than 26.6 million cycles, and the IV characteristics of the device are not changed

Electronics 2021, 10, 1198. https://doi.org/10.3390/electronics10101198 https://www.mdpi.com/journal/electronics Electronics 2021, 10, x FOR PEER REVIEW 2 of 14

Electronics 2021, 10, 1198 2 of 13 change coefficient of the switch threshold voltage is less than 13%, the switch durability is more than 26.6 million cycles, and the IV characteristics of the device are not changed significantly. The VO2 device technology in the process of non-electric formation acceler- signiﬁcantly.ates the development The VO2 device of an technologyactive memristor in the process neuron of circuit. non-electric It ca formationn simulate accelerates the most theknown development neuron dynamics of an active and memristorclear the way neuron for the circuit. realization It can simulateof a large-scale the most integrated known neuron dynamics and clear the way for the realization of a large-scale integrated circuit circuit (IC). In addition, the VO2 memristor is superior to its NbO2 counterpart in both (IC). In addition, the VO2 memristor is superior to its NbO2 counterpart in both switch switch speed and switching energy. The simulated Mott transition in the VO2 is 100 times speed and switching energy. The simulated Mott transition in the VO2 is 100 times faster faster than in the NbO2 and consumes only about one-sixth (16%) of the energy [17]. than in the NbO2 and consumes only about one-sixth (16%) of the energy [17]. In this paper, a cellular neural network based on a VO2 CNT is proposed. As a cell In this paper, a cellular neural network based on a VO2 CNT is proposed. As a cell itself, a VO2 CNT is set to be stable and static, that is, in “sleep”. If the memristor is on the itself, a VO CNT is set to be stable and static, that is, in “sleep”. If the memristor is “edge of chaos”,2 two or more cells are connected by the RC coupling, which will make it on the “edge of chaos”, two or more cells are connected by the RC coupling, which will in the state of “wake-up”, namely, in the dynamic oscillation mode. The second part of make it in the state of “wake-up”, namely, in the dynamic oscillation mode. The second this paper is the modeling of VO2 CNT composite devices, providing the SPICE model. part of this paper is the modeling of VO CNT composite devices, providing the SPICE The third part is the cell circuit, which gives2 the analysis process of the decoupled circuit. model. The third part is the cell circuit, which gives the analysis process of the decoupled circuit.The response The response of the memristor of the memristor is expanded is expanded near the near operating the operating point and point the and small-signal the small- signalequivalent equivalent circuit circuitis given. is Based given. on Based this, on the this, influence the inﬂuence of coupling of coupling R and CR deviceand C param-device parameterseters on the oninput the impedance input impedance of one port of one is analyzed. port is analyzed. The fourth The part fourth is the part CNN is the simula- CNN simulation,tion, which describes which describes the pattern the patternformation formation characteristics characteristics of a CNN of composed a CNN composed of memris- of memristor-basedtor-based cells. cells. Compared with the traditional CMOS realization of CNN, the memristor counterpart can largely save power and chipchip area,area, andand provideprovide ultra-highultra-high processingprocessing speeds.speeds.

2. VO 2 CarbonCarbon Nanotube Nanotube Compos Compositeite Device Device Modeling

The structure of of the the VO VO22 carboncarbon nanotube nanotube composite composite device device is isshown shown in inFigure Figure 1.1 It. Itcontains contains a transverse a transverse active active region, region, which which is defined is defined by a byVO a2 VOthin2 metalthin metal strip with strip a with thick- a thicknessness of about of about 5 nm 5(as nm shown (as shown in the in red the region red region of the of figure), the figure), and its and two its ends two endsare con- are connectednected with with Pd electrodes Pd electrodes (as shown (as shown in the in blue the blueregion region of the of figure). the figure). This is This a planar is a planar Mott Mottmetal-insulator metal-insulator transition transition device. device. The aligned The aligned carbon carbon nanotubes nanotubes (as shown (as shown in the inblack the blackline) were line) first were grown first grown on quartz on quartz substrate, substrate, and then and thentransferred transferred to the to surface the surface of a VO of a2 VOthin2 metalthin metal strip stripbefore before the whole the whole device device was formed. was formed.

Figure 1. A VO 2 carboncarbon nanotube composite device structure. The redred partpart isis thethe VOVO22 and the black lineblack over line it over represents it represents the carbon the carbon nanotube. nanotube.

As a Mott metal-insulator transition device, it satisﬁessatisfies thethe followingfollowing equationequation [[16]:16]:

√ βφvdβ− vm/d−φ 2 m( ) i = ATekT kT iATe=m 2  m i v +v2 /R T−T (1) dT = m m m CNT − amb (1) dt ivC+−th v2 RC Tth R Tth dT =−mm m CNT amb dtC CR The ﬁrst formula of Equation (1) is the currentth emission th th equation of the Schottky diode, and the second formula is Newton’s cooling law, where, im and νm are the current and voltageThe through first formula the memristor of Equation NDR (1) device, is the cu respectively;rrent emissionT is equation the absolute of the temperature Schottky di- of theode, device; and theT ambsecondis the formula ambient is temperature;Newton's coolingA and law,β are where, scaling ݅m constants;and ݒm ared theis thecurrent effective and devicevoltage length;throughk theis Boltzmann memristor constant;NDR device,φ is respectively; the energy barrier;T is the absoluteCth is the temperature heat capacity of (effectivethe device; thermal Tamb is the mass); ambientRth is temperature; the thermal A resistance; and ߚ areR scalingCNT is theconstants; resistance d is valuethe effective of the CNT (carbon nanotube), taking 600 kΩ. The second expression of Equation (1) represents the electrical and thermal coupling between RCNT and VO2. The parameters of the device are shown in Table1.

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device length; k is Boltzmann constant; φ is the energy barrier; ܥth is the heat capacity

(effective thermal mass); ܴth is the thermal resistance; ܴCNT is the resistance value of the

Electronics 2021, 10, 1198 CNT (carbon nanotube), taking 600 kΩ. The second expression of Equation (1) represents 3 of 13 the electrical and thermal coupling between ܴCNT and VO2. The parameters of the device are shown in Table 1. Table 1. Device parameters. The data is cited directly from the literature [16]. Table 1. Device parameters. The data is cited directly from the literature [16].

Parameter VO2 CNT Device Parameter VO CNT Device Tamb 2962 K β −−40.50.5 T amb 3.3×⋅⋅ 10 eV296 m K V − −0.5 × 4−−92· 0.5 · A β 3.31.710×⋅ 10eV A Km V A 1.7 × 10−9 A · K−2 d × −6 d 5105 × 10 m−6 m 81− Rth ×⋅8 −1 Rth 2.52.5 10× 10 KK W· W −−17−17 1 −1 Cth Cth 5105×⋅× 10 JKJ · K RS RS 55005500 Ω Ω VS VS 10 V10 V RCNTRCNT 600,000600,000 Ω Ω φ φ 0.580.58 eV eV −5 −1 k 8.62 × −−10 eV · K k 8.62×⋅ 1051 eV K

AccordingAccording to Equation to Equation (1), the (1),SPICE the SPICEmodel model of the of device the device can canbe beestablished, established, as as shown shown inin Figure Figure 2 2 1

B1 B2 C1 RCNT I=Ivo2() 5e-17 600000 I=dTdt() 2 .FUNC dTdt()=-((Tamb-v(T))/Rth+(Ivo2()*(v(1)-v(2))+(V(1)-V(2))**2/RCNT))/Cth .FUNC Ivo2()=A*v(T)**2*exp((beta*sqrt((v(1)-v(2))/d)-phi)/kB/v(T)) .PARAM Tamb=296 .PARAM A=1.7e-9 .PARAM d=5e-6 .PARAM Rth=2.5e8 .PARAM Cth=5e-17 .PARAM beta=3.3e-4 .PARAM kB=8.62e-5 .PARAM phi=0.58 .PARAM RCNT=600000 .IC v(T)=Tamb Figure 2. SPICEFigure model 2. SPICE of the model device. of the device. Electronics 2021, 10, x FOR PEER REVIEW 4 of 14 The quasi-staticThe quasi-static volt-ampere volt-ampere characteristics characteristics of the device of the devicecan be canobtained be obtained by using by using the the SPICESPICE model, model, as shown as shown in Figure in Figure 3. 3.

FigureFigure 3. Quasi-static 3. Quasi-static volt-ampere volt-ampere characteristics characteristics of the of theVO VO2 carbon2 carbon nanotube nanotube composite composite device. device.

TheThe simulation simulation measurement measurement method method of of the the quasi-static quasi-static volt-ampere volt-ampere characteristics characteristics of of thethe device device is is to to use use a a sinusoidal sinusoidal voltage voltage source source with a frequency frequency of 1 Hz and amplitude of of 10 V to excite excite the device through a small series resistor, andand thenthen drawdraw thethe V–IV–I relationshiprelation- ship curve by measuring the current flowing through the device. When you look at Figure 3, you can clearly see a hysteresis region, which is a typical feature of the local active memristor.

3. Cell Circuit

Figure 4 is the designed circuit of a memristor cell, in which RS is the bias resistor,

M is the VO2 memristor and RCNT is the carbon nanotube resistance. The parallel resistor and capacitor combination is the circuit coupled with adjacent cells, and it is also a part of the cell.

Figure 4. Cell circuit. The part shown in the dotted box is the VO2 carbon nanotube composite

device, Rs is the bias resistor and R and C are the coupling resistor and capacitor, respectively.

According to Figure 4, the circuit equation can be obtained by using the Kirch- hoff current law, the Kirchhoff voltage law and Equation (1), as follows:

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Figure 3. Quasi-static volt-ampere characteristics of the VO2 carbon nanotube composite device. The simulation measurement method of the quasi-static volt-ampere characteristics Electronics 2021, 10, 1198 of the device is to use a sinusoidal voltage source with a frequency of 1 Hz and amplitude4 of 13 of 10 V to excite the device through a small series resistor, and then draw the V–I relationship curve by measuring the current flowing through the device. When you look at Figure 3, you can clearly see a hysteresis region, which is a typical feature of the local active curvememristor. by measuring the current ﬂowing through the device. When you look at Figure3 , you can clearly see a hysteresis region, which is a typical feature of the local active memristor. 3. Cell Circuit 3. Cell Circuit Figure 4 is the designed circuit of a memristor cell, in which RS is the bias resistor, Figure4 is the designed circuit of a memristor cell, in which RS is the bias resistor, M M is the VO2 memristor and RCNT is the carbon nanotube resistance. The parallel resistor is the VO2 memristor and RCNT is the carbon nanotube resistance. The parallel resistor andand capacitorcapacitor combinationcombination isis thethe circuitcircuit coupledcoupled with with adjacent adjacent cells, cells, and and it it is is also also a apart partof of thethe cell.cell.

2 FigureFigure 4.4.Cell Cell circuit.circuit. The The part part shown shown in in the the dotted dotted box box is is the the VO VO2 carbon carbon nanotube nanotube composite composite device, device, R is the bias resistor and R and C are the coupling resistor and capacitor, respectively. Rs is the biass resistor and R and C are the coupling resistor and capacitor, respectively.

AccordingAccording toto FigureFigure4 ,4, the the circuit circuit equation equation can can be obtainedbe obtained by usingby using the Kirchhoffthe Kirch- currenthoff current law, thelaw, Kirchhoff the Kirchhoff voltage voltage law and law Equation and Equation (1), as (1), follows: as follows:

2 dT = vmim+vm/RCNT − T−Tamb dt Cth Cth Rth dvC = − vC + C dt R iA√ β v /d−φ (2) 2 m 0 = −im + AT e kT 0 = −VS + RSim + (1 + RS/RCNT)vm − RSiA

vA = vC + vm (3) In order to make it easy to write a MATLAB simulation program, Equations (2) and (3) can be further arranged into a matrix form, as follows:

. Ex = f(x, i ) A (4) vA = Cx

where T T x= (x1 x2 x3 x4) = (T vC vm im) (5)  1 0 0 0   0 C 0 0  E =   (6)  0 0 0 0  0 0 0 0

 2  vmim+vm/RCNT − T−Tamb Cth Cth Rth  − vC +   R i√A  f(x, iA) =  β v /d−φ  (7)  2 m   −im + AT e kT  −VS + RSim + (1 + RS/RCNT)vm − RSiA Electronics 2021, 10, 1198 5 of 13

C= [0 1 1 0] (8) The necessary conditions for the local activity of resistive-coupled reaction–diffusion CNNs (RD-CNNs) are given in [18]. In order to apply the theory to the designed network, the input impedance Z of port A must be calculated. Therefore, it is necessary to linearize Equation (4) near the operating point without coupling, that is to say, the small-signal equivalent circuit analysis condition is satisﬁed under the condition of a zero-port current. The memristor equation is expanded near the operating point.

T = T0 + δT vm = VM + δvm (9) im = IM + δim

Here, T0, VM and IM are the state, voltage and current values of the DC operating point, respectively. We can expand the expression of the memristor current near the operating point:

im = IM + δim 0 0 0 (10) = a00(Q) + a11(Q)δT + a12(Q)δvm + h.o.t

where √ − ( β VM/d φ ) 0 2 kT0 IM = a00(Q) = im = AT0 e √ √ β VM/d−φ 0 ∂i β VM/d−φ ( ) = m | = ( − ) kT0 (11) a11 Q ∂T Q 2T0 √k Ae β VM/d−φ 0 ∂im AβT0 1 kT a (Q) = |Q = √ √ e 0 12 ∂vm 2k d VM

h.o.t. is the higher-order term relative to δT and δvm. If |δT| 1 and |δvm| 1, we can get the following approximate linear relationship:

0 0 δim = a11(Q)δT + a12(Q)δvm (12)

Next, we will expand the memristor equation of state with a Taylor series near the operating point VM, T0)Q :

dT 0 0 dt |Q = f (T, vm) = f (T0 + δT, VM + δvm) = f (T0, VM) + b11(Q)δT + b12(Q)δvm + h.o.t (13)

where √ √ β VM/d−φ 0 ∂ f (T,vm) 1 VM β VM/d−φ kT b (Q) = |Q = − + (2T0 − )Ae 0 11 ∂T Cth Rth Cth √ k √ (14) √ VM/d−φ VM/d−φ 0 ∂ f (T,vm) 1 AβT0 kT kT 2VM b (Q) = |Q = ( √ VMe 0 + AT0e 0 + ) 12 ∂vm Cth 2 d RCNT

Notice that f (T0, VM) = 0, because (T0, VM) is a point on the DC V–I curve. Ignoring the high-order small term, let us linearize the memristor state Equation (13) near the operating point: d(δT) = b0 (Q)δT + b0 (Q)δv (15) dt 11 12 m Taking Laplace transform for Equations (12) and (15), we obtain

ˆ 0 ˆ 0 im(s) = a11(Q)T(s) + a12(Q)vˆm(s) ˆ 0 ˆ 0 (16) sT(s) = b11(Q)T(s) + b12(Q)vˆm(s) Electronics 2021, 10, x FOR PEER REVIEW 7 of 14

According to the first formula of (16) and Equation (17), the admittance function of the memristor can be obtained as follows:

isˆ () aQbQ′′() () YsQ(, ) m =+11 12 a′ ( Q ) − ′ 12 Electronics 2021, 10, 1198 vsˆm () sbQ11 ( ) 6 of 13 1 (18) =+ aQ′ ( ) (())−bQ′ 12 1 + 11 s ′′ ′′ aQbQaQbQ11() 12 () 11 () 12 () The Laplace transforms of δT, δim and δvm are Tˆ (s), iˆm(s) and vˆm(s). According to the second formula of (16), we obtain: Change Equation (18) into the following form: 0 b (Q)vˆm(s) Tˆ (s) = 12 (17) − 110 ( ) YsQ(, )=+s b11 Q + (19) sLRs sp R According to the ﬁrst formula of (16) and Equation (17), the admittance function of the memristor can be obtained as follows: YsQ(, ) is the admittance of the memristor. The small signal equivalent circuit of the ˆ ( ) a0 (Q)b0 (Q) memristor is shown in Figure 5. L i,m Rs and11 R 12 are defined0 as follows: Y(s, Q) , s ( )s = 0p + a12(Q) vˆm s s−b11(Q) = 1 + 0 ( ) (18) 1(−b0 (Q)) a12 Q s L 1= + 11 a0 (Qs )b0 (Q) a0 (Q)b0 (Q) 11 12 ′′11 12 aQbQ11() 12 () (())−bQ′ Change Equation (18) into the followingR = form:11 s ′′ (20) aQbQ11() 12 () 1 1 Y(s, Q) = 1 + (19) R = p sL′s + Rs Rp aQ12 () Y(s, Q) is the admittance of the memristor. The small signal equivalent circuit of the memristorConsidering is shown the in DC Figure bias5 of. Lthes, R cells and circuit,Rp are according deﬁned asto follows:Equation (2), and that the R = and C parameter values do not affect the DC operating point of the system, let iA 0 , 1 fx(,i )== fx (,0)0, respectively. Then,Ls =we 0use MATLAB0 to solve the DC operating point A a11(Q)b12(Q) (− 0 ( )) （ = b11 Q Q, TV0 ,MQ ,) (3.049042473697802eR +s 03,0.006060366246039)= 0 0 , and substitute the operat-(20) a11(Q)b12(Q) 1 ing point value into Equation (20), weRp can= evaluate0 Ls , Rs and Rp respectively. a12(Q)

Figure 5. Small signal equivalent circuit of the memristor. Figure 5. Small signal equivalent circuit of the memristor. Considering the DC bias of the cell circuit, according to Equation (2), and that the The equivalent impedance of port A in the frequency domain is obtained. R and C parameter values do not affect the DC operating point of the system, let iA = 0, f(x, iA) = f(x, 0) = 0 , respectively. Then, we use MATLAB to solve the DC operating point Q, T0, VM, )Q = (3.049042473697802e + 03,11 0.006060366246039), and substitute the Zj()=()ω Zs =+ operating point value into Equation (20),sj= weω can evaluate Ls, Rs and Rp respectively. (21) YsQ(, ) 11+ The equivalent impedance of port A in the frequency domain is obtained. RsCs= jω

In [18], Professor Chua derived the local 1active condition1 of resistive coupled RD- Z(jω) = Z(s)| = + (21) CNNs based on the reaction–diffusions= jequation.ω Y(s, QIt )is said1 + that1 the cell is in the local active R sC s=jω region if the input impedance of the one port cell satisfies at least one of the following conditions:In [18], Professor Chua derived the local active condition of resistive coupled RD-CNNs based(a) on There the reaction–diffusion is a pole in the right equation. half plane; It is said that the cell is in the local active region if the input(b) There impedance are higher-order of the one port poles cell on satisfies the imaginary at least oneaxis; of the following conditions: (a) There is a pole in the right half plane; (b) There are higher-order poles on the imaginary axis; (c) There is a pole s = jωp of order one on the imaginary axis and lim (s − jωp)Z(s) is s→jωp a negative real number or a complex number with non-zero imaginary part; (d) There is at least one angular frequency value ω, such that the real part of the impedance Z(jω) is less than zero. Electronics 2021, 10, 1198 7 of 13

Through calculation, the system impedance Z(s, Q) has the following form:

A(s − z )(s − z ) Z(s, Q) = 1 2 (22) (s − p1)(s − p2)

where A is the real coefficient. According to Chua’s theory [18], the system described by conditions (a)–(c) is unstable, while the system is a stable local active system when condition (d) is satisfied along with the poles being in the left half plane, that is, the system is located in the edge of chaos. In addition, according to the theory of signals and systems, Y(s, Q) is the transfer function of the system. When the poles of Y(s, Q), corresponding to the zeroes, is located in the right half plane of the complex plane, the system will be unstable, and the system is said to be on the sharp edge of chaos. Specifically, because the numerator of the input impedance is a quadratic polynomial, its zeroes correspond to two cases: the system has two positive zeroes (two zeroes whose real part is greater than zero; they are conjugate zeroes, i.e., z1 6= z2, Re(z1) > 0 and Re(z2) > 0) or one positive real zero (zero is a positive real number as multiple roots, (z1 = z2) > 0), corresponding to dynamic pattern and static pattern, respectively. In Figure6, yellow indicates that the system is in the local passive area and the input impedance does not meet Condition (d); the other areas are in the local active area and the input impedance meets Condition (d). Blue indicates that the system is at the sharp edge of chaos. At this time, the impedance function of the system has two positive zeroes (two zeroes whose real parts are greater than zero; they are conjugate zeroes) or one positive real zero (zero is a positive real number as multiple roots), which correspond to the dark blue and light blue regions, respectively. Other regions in the figure, that is, when green corresponds to other cases of zeroes, indicate that the system is at the edge of chaos. In Figure6, the edge region of chaos, that is, the light green Region II, contains coupling parameters that may not destabilize the system because the resulting local input impedance of the cell does not have the zero point of the positive real part. On the other hand, the union of the dark blue Region III and the light blue Region IV represents the sharp edge of chaos. In the dark blue Region III, the local input impedance of the cell allows a pair of complex conjugate zeroes with positive real parts, which makes the system unstable and leads to the formation of a dynamic pattern in a steady state. In the light blue Region IV, a zero point of the local input impedance of a single cell is positive and real, which triggers the instability of the system and gradually leads to a static mode. It has been proved by literature [12] that when the cell is in the sleeping mode, the "cell" equation has only a steady-state homogeneous solution; only when the cell is in the sharp edge of chaos can it be successfully “awakened” when it is connected to the CNN; that is to say, the system equation has a non-homogeneous solution, and the light blue region is in so-called “static wake-up”. When a cell is connected to a network, it has a non-homogeneous static stable solution, which is different from when the cell is isolated; the dark blue region is called a dynamic wake-up, which causes oscillation when the cell is connected to the network, so it has a dynamic oscillation solution, which is different from the static solution when the cell is isolated. Because the capacitance value of static wake-up is too large, it is not practical for an integrated system. Only a dynamic wake-up is considered. Electronics 2021, 10, x FOR PEER REVIEW 9 of 14 Electronics 2021, 10, 1198 8 of 13 Electronics 2021, 10, x FOR PEER REVIEW 9 of 14

Figure 6. Input impedance characteristic diagram depending on R and C parameters. I—passive FigureFigureregion; 6.6. II—edge Input impedance of chaos region;characteristic characteristic III—sharp diagram diagram edge dependingof depending chaos region on on R R (dynamic);andand CC parameters.parameters. IV—sharp I—passive I—passive edge of region;region;chaos II—edge regionII—edge (static). of of chaos chaos region; region; III—sharp III—sharp edge edge of of chaos chaos region region (dynamic); (dynamic); IV—sharp IV—sharp edge edge of chaosof chaos region (static). region (static). 4. CNN and Simulation 4.4. CNNCNNConnect andand SimulationSimulation two cells, and the common resistance and capacitance can be equivalent to twoConnectConnect identical twotwo series cells,cells, connections andand thethe commoncommon [12] byresistance resistancethe circuit andand principle, capacitancecapacitance so as cancanto distribute bebe equivalentequivalent them toto to twotwoeach identicalidentical cell, as shown seriesseries connectionsconnectionsin Figure 7. [[12]12] byby thethe circuitcircuit principle,principle, soso asas toto distributedistribute themthem toto eacheach cell,cell, asas shownshown inin FigureFigure7 .7.  = =RR2 Re  =2R (23) RR =2 (23) Ce =CCC/2 /2 (23) CC = /2

(a) (a)

(b) (b) FigureFigure 7. 7.The The common common resistor resistor and and capacitor capacitor are are equivalent equivalent to two to two parts parts in the in series. the series. (a) is ( thea) is original the configurationFigureoriginal 7. Theconfiguration whilecommon (b) is resistorwhile its equivalent (b and) is itscapacitor circuit.equivalent are equivalentcircuit. to two parts in the series. (a) is the original configuration while (b) is its equivalent circuit.

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Ω  =Ω = Select R = 1 M , C = 720 p, corresponding to =ΩR 2M  =, C 360 p ; the two cells are SelectSelectR R= = 11 MMΩΩ,,C C= = 720720 p,p, correspondingcorresponding to ReR= 22MMΩ, Ce, C= 360360p p;; the the two two cells cells are are in sleeping mode before coupling, as shown in Figure 8. In Figure 8b, the voltage decreases inin sleeping sleeping mode mode before before coupling, coupling, as as shown shown in in Figure Figure8 .8. In In Figure Figure8b, 8b, the the voltage voltage decreases decreases from 10 V to 0 V, and the cell does not oscillate. fromfrom 10 10 V V to to 0 0 V, V, and and the the cell cell does does not not oscillate. oscillate.

R1 R2 R1 R2 5500 5500 5500 5500 a b a V1 1 b 1 V2 V1 1 1 V2

10 2 2 10 10 2 2 10

.tran 0 3u 0 .tran 0 3u 0

(a) (a)

(b) (b) Figure 8. The two cells are uncoupled and in sleeping mode; (a) is the configuration of two uncou- FigureFigure 8. 8.The The two two cells cells are are uncoupled uncoupled and and in in sleeping sleeping mode; mode; (a) ( isa) the is the conﬁguration configuration of two of two uncoupled uncoupled cells (b) shows there is no oscillation. cellspled ( bcells) shows (b) shows there isthere no oscillation.is no oscillation. After coupling, it is in wake-up mode, as shown in Figure 9. Figure 9b shows that the AfterAfter coupling,coupling, it it is is inin wake-upwake-up mode, mode, as as shown shown in in Figure Figure9 .9. Figure Figure9b 9b shows shows that that the the voltage drops from the initial 10 V to oscillate near 0 V. voltagevoltage drops drops from from the the initial initial 10 10 V V to to oscillate oscillate near near 0 0 V. V.

R3 R3

R1 2Meg R2 R1 2Meg R2 5500 C1 5500 5500 C1 5500 a b a b V1 1 360p 1 V2 V1 1 360p 1 V2

10 2 2 10 10 2 2 10

.tran 0 3u 0 .tran 0 3u 0

(a) (a) Figure 9. Cont.

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(b) (b) FigureFigure 9.9. TheThe cells are in wake-up mode mode after after coupling. coupling. (a ()a )is is configuration conﬁguration of of two two coupled coupled cells. cells. In InFigure(b (),b ),we we9. see The see there cells there isare isosci in oscillation llationwake-up in in modethe the circuit. circuit.after coupling. (a) is configuration of two coupled cells. In (b), we see there is oscillation in the circuit. ItIt cancan bebe seen seen fromfrom Figures Figures8 8and and9 that9 that when when the the cell cell is is in in the the sleeping sleeping mode, mode, both both It can be seen from Figures 8 and 9 that when the cell is in the sleeping mode, both cellscells do do notnot oscillate,oscillate, andand thethe differentialdifferential equation equation of of thethe systemsystem hashas onlyonly aa trivialtrivial solution,solution, cells do not oscillate, and the differential equation of the system has only a trivial solution, thatthat is, is, a a zero zero solution. solution. WhenWhen thethe cellcell isis inin wake-up wake-up mode,mode, thethe differentialdifferential equation equation of of the the that is, a zero solution. When the cell is in wake-up mode, the differential equation of the cell system has oscillatory solutions, and different cells get different oscillatory solutions. cellcell system system has has oscillat oscillatoryory solutions, solutions, and and different different cells cells get gedifferentt different oscillatory oscillatory solutions. solutions. In order to observe the characteristics of a CNN composed of cells, 5 × 5 cells are InIn order order to to observe observe the the characteristics characteristics of aof CNN a CNN composed composed of cells, of cells, 5 × 55 cells × 5 cellsare are arrayed, as shown in Figure 10, and the DC excitation of each cell rises to 10 V in 1 u seconds. arrayed,arrayed, as as shown shown in in Figure Figure 10, 10, and and the the DC DC excitation excitation of each of each cell cellrises rises to 10 to V 10in V1 uin 1 u The cells in the central position (3,3) are deliberately raised to 10 V in 0.9 u seconds. The seconds.seconds. The The cells cells in in the the central central position position (3,3) (3,3) are aredeliberately deliberately raised raised to 10 to V 10 in V0.9 in u 0.9sec- u sec- voltageonds.onds. The The response voltage voltage ofresponse response 25 cells of is of25 observed, 25cells cells is observed,is as observed, shown as inshown as Figure shown in 11Figure in. Figure 11. 11.

Figure 10. CNN composed of 5 × 5 cells.

ElectronicsElectronics 2021 2021, 10, 10, x, FORx FOR PEER PEER REVIEW REVIEW 1212 of of 14 14

Electronics 2021, 10, 1198 11 of 13 FigureFigure 10. 10. CNN CNN composed composed of of 5 5× ×5 5cells. cells.

FigureFigureFigure 11. 11. 11. Terminal TerminalTerminal voltage voltage voltage response response response of of ofeach each each memristor. memristor. memristor.

ItIt canIt can can be be beseen seen seen from from from Figure Figure Figure 11 11 11 that that that after after after a ashort a short short transient transient transient response, response, response, each each each cell cell cell reaches reaches reaches steady-statesteady-statesteady-state oscillation. oscillation. oscillation. According According According to to toFigure Figure Figure 11, 11, 11 the ,the the network network network pattern pattern pattern diagram diagram diagram at at atdifferent different different timestimestimes can can can be be bemade, made, made, as as asshown shown shown in in inFigure Figure Figure 12, 12, 12 in ,in inwhich which which different different different colors colors colors represent represent represent different different different terminalterminalterminal voltages voltages voltages of of ofthe the the cellular cellular cellular memristor. memristor. memristor. Figure Figure Figure 12 12 12is is justis just just a areproduction a reproduction reproduction of of ofFigure Figure Figure 11 11 11 fromfromfrom another another another view view view of of ofthe the the spatial spatial spatial distribution distribution distribution of of ofthe the the sampled sampled sampled voltage voltage voltage at at ata aspecific a specific specific time, time, time, where,where,where, in in inthe the the vertical vertical vertical axis, axis, axis, “1” “1” “1” represents represents represents th the the ehighest highest highest voltage voltage voltage of of ofthe the the cell, cell, cell, and and and other other other values values values areareare just just just the the the normalized normalized normalized voltage voltage relating relating to toto that thatthat cell cellcell respectively. respectively.respectively. This This This is is called called the the pattern pattern of ofof thethe the network.network. network. From From From the the the figure, figure, figure, it canit itcan becan seenbe be seen thatseen that the that initialthe the initial initial conditions conditions conditions can affect can can theaffect affect network the the networknetworkpattern pattern formation,pattern formation, formation, so that so the so that powerthat the the sourcepower power voltagesource source ofvoltage voltage each cellof of each caneach becell cell programmed can can be be pro- pro- to grammedgrammedobtain a to specific to obtain obtain pattern, a aspecific specific so pattern, as pattern, to complete so so as as to to the complete complete information the the information information processing. processing. processing.

(a()a t) =t =3.386 3.386 us us

Figure 12. Cont.

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(b) t = 4.407 us

FigureFigure 12. 12. PatternPattern formation formation diagrams diagrams of ofthe the network. network. They They are are directly directly derived derived from from Figure Figure 11. 11. ((aa)) is is at at t t= = 3.386 3.386 us us and and (b (b) )is is at at t t= = 4.407 4.407 us. us.

5.5. Conclusions Conclusions

InIn this this paper, paper, a a CNN CNN based based on on a VO2 carboncarbon nanotube nanotube composite composite memristor memristor was was proposed.proposed. The The analysis analysis shows shows that that the the coupling coupling RCRC parametersparameters will will affect affect the the network network patternpattern formation. formation. According According to to the the local local active active and and edge edge of of chaos chaos theory theory proposed proposed by by ProfessorProfessor Chua, itit isis pointed pointed out out that that only only when when the the cell cell is in is the in sharp the sharp edge ofedge chaos of regionchaos regioncan the can cell the be cell successfully be successfully awakened awakened after formingafter forming the network, the network, that is,that the is, differential the differ- entialequation equation of the of system the system has ahas non-homogeneous a non-homogeneous solution. solution. If theIf the system system equation equation has has a anon-homogeneous non-homogeneous static static solution, solution, it it isis saidsaid thatthat thethe systemsystem is in static mode; if the the system system equationequation has has an an oscillatory oscillatory solution, solution, it it is is sa saidid that that the the system system is is in in dynamic dynamic mode. mode. When When thethe cell cell is is biased biased in in other other regions, regions, the the differenti differentialal equation equation of of the the system system has has only only a a trivial trivial solution,solution, that that is, is, a azero zero solution, solution, and and the the cell cell is in is the in the sleeping sleeping mode. mode. At the At same the same time, time, the formationthe formation of the of themode mode is also is also affected affected by by the the initial initial conditions. conditions. In In our our experiment, experiment, we we deliberately made the cells located in the middle of the 5 × 5 CNN network earlier than deliberately made the cells located in the middle of the 5 × 5 CNN network earlier than other cells, so that we could program the cell power supply voltage to get a specific mode other cells, so that we could program the cell power supply voltage to get a specific mode and complete the information processing. In summary, the architecture proposed in this and complete the information processing. In summary, the architecture proposed in this paper can be applied to future computing occasions based on complex space–time patterns, paper can be applied to future computing occasions based on complex space–time pat- especially in the field of computer vision. terns, especially in the field of computer vision. Author Contributions: Conceptualization, Y.S.; Formal analysis, Y.S.; Funding acquisition, G.W.; Author Contributions: Conceptualization, Y.S.; Formal analysis, Y.S.; Funding acquisition, G.W.; Methodology, Y.S.; Supervision, G.W. All authors have read and agreed to the published version of Methodology, Y.S.; Supervision, G.W. All authors have read and agreed to the published version of the manuscript. the manuscript. Funding: This project was supported by the National Natural Science Foundation of China (Grant Funding:No. 61771176). This project was supported by the National Natural Science Foundation of China (Grant No. 61771176). Conflicts of Interest: The authors declare no conflict of interest.

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