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Two-neuron-based non-autonomous memristive Hopfield neural network:

Numerical analyses and hardware experiments

Article  in  AEU - International Journal of Electronics and Communications · September 2018

DOI: 10.1016/j.aeue.2018.09.017

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Int. J. Electron. Commun. (AEÜ) 96 (2018) 66–74

Contents lists available at ScienceDirect

International Journal of Electronics andCommunications (AEÜ)

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Regular paper

Two-neuron-based non-autonomous memristive Hopfield neuralnetwork: Numerical analyses and hardware experiments

https://doi.org/10.1016/j.aeue.2018.09.0171434-8411/� 2018 Elsevier GmbH. All rights reserved.

⇑ Corresponding author.E-mail address: mervinbao@126.com (B. Bao).

Quan Xu, Zhe Song, Han Bao, Mo Chen, Bocheng Bao ⇑School of Information Science and Engineering, Changzhou University, Changzhou 213164, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 August 2018Accepted 14 September 2018

Keywords:memristive Hopfield neural network(mHNN)Non-autonomousNumerical simulationBreadboard experiment

This paper explores a two-neuron-based non-autonomous memristive Hopfield neural network (mHNN)through numerical analyses and hardware experiments. It is interested that the locus and stability of theAC equilibrium point for the mHNN change with the time evolution. Dynamical behaviors associatedwith the self-coupling strength of the memristive synapse are numerically investigated by bifurcationdiagrams, Lyapunov exponents and phase portraits. Particularly, bursting behaviors are revealed whenthe order gap exists between the natural frequency and external stimulus frequency. The interestingphenomena are illustrated through phase portraits, transmitted phase portraits, and time-domain wave-forms of two cases. Moreover, breadboard experimental investigations are carried out, which effectivelyverify the numerical simulations.

� 2018 Elsevier GmbH. All rights reserved.

1. Introduction

As a nonvolatile resistor with frequency-dependent pinchedhysteresis loops, memristor can remember the total electric chargepassed through it as time going [1]. The unique ability makesmemristor behaving like electronic synapse in neural networks[2–5]. Until now, numerous memristive neural networks have beenreported by introducing memristors into some classical neural net-works or replacing resistive synapse weights with memristivesynapse weights in these neural networks, within which richdynamical behaviors have been revealed by numerical and circuitsimulations [6–12]. Only the dynamical behaviors in small worldmemristive neural networks have been validated by hardwareexperimental measurements in recent years [9]. Unlike thosememristive neural networks employing memristors as mutual-connections between neurons, a new two-neuron-based non-autonomous memristor-based Hopfield neural network (mHNN)employing a memristor as a self-connection synapse is presented.Since memristors are used as the synapses of Hopfield neuralnetwork (HNN) [6–12], these memristor-based Hopfield neuralnetworks can be written as mHNN shortly for convenient.

The HNN is a kind of classical recurrent artificial neural net-work, which can be used in information processing and possibleengineering applications [13–19]. Thus, the dynamical behaviorsincluding chaos [7,9], hyperchaos [8], multi-stability [10] and

hidden attractors [11,20] have inspired great interest of research.Besides, studies of biological systems show that bursting behaviorplays crucial role in biological activities [21,22]. In the past fewyears, bursting behavior has been found in various physical sys-tems with two time scales including coupled chaotic systems[23,24], non-smooth Chua’s system [25], periodically forced sys-tem [26,27], electronic circuits [28,29], and so on. Thus, burstingbehavior may occur when the order gap exists between the naturalfrequency and external stimulus frequency in our proposed non-autonomous mHHN. Moreover, bursting behavior appears whenthe oscillation switches between quiescent state (QS) and the spik-ing state (SP), repetitively. Therefore, at least two bifurcationroutes should be involved to lead the oscillation switching fromQS to SP and vice versa [28]. Several bursting behaviors such asthe fold/fold and fold/Hopf busters have been revealed [26]. Foldbifurcation (FB) is a local bifurcation in which two fixed points ofa dynamical system collide and annihilated each other leadingthe oscillation to switching from QS to SP [26]. Hopf bifurcation(HB) is a critical point where a system’s stability switches fromunstable to stable state with the emergence of pure imaginaryeigenvalues, and a periodic solution arises, which means an equi-librium point of the system loses its stability.

The layout of the paper is as follows. In Section 2, a simplifiedmemristor emulator is presented and pinched hysteresis loopsare numerically and experimentally validated. In Section 3, theconnection topology and mathematical model of a two-neuron-based non-autonomous mHNN are described, as well as the ACequilibrium points and the changing of their stability with the time

Q. Xu et al. / Int. J. Electron. Commun. (AEÜ) 96 (2018) 66–74 67

evolution are conducted. In Section 4, dynamical behaviors associ-ated to the self-coupling strength are numerically revealed bybifurcation diagrams, Lyapunov exponents, and phase portraits.In Section 5, busting behaviors are explored by phase portraits,transformed phase portraits, and time-domain waveforms. In Sec-tion 6, hardware experimental measurements are performed tofurther verify the numerically simulated dynamical behaviors inthe non-autonomous mHNN. Finally, the conclusions of this workare drawn.

2. Simplified memristor emulator

In order to explore the action mechanisms of memristive self-connection synapse weight in the non-autonomous mHNN conve-niently, a simplified memristor emulator deriving from the Ref.[30] is proposed, as shown in Fig. 1. The simplified memristor emu-lator consists of only six discrete components including an integra-tor U1 connected two resistors R and a capacitor C, an analoguemultiplier M1 and a resistor Ra. It is important to stress that theresistor R in parallel to the integrating capacitor C is employed toavoid DC voltage integral drift [30]. Compare to the memristoremulator proposed in [30], the simplified memristor emulatorhas the merits of simpler circuit realization and mathematicalmodel.

The mathematical model of the simplified memristor emulatorcan be obtained by employing Kirchhoff’s circuit laws and the con-stitutive relations of the discrete components are expressed as

i ¼ W v0ð Þv ¼ gRav0v

dv0=dt ¼ � 1RC v0 þ vð Þ ð1Þ

where v and i represent the voltage and current at the input port ofthe simplified memristor emulator, respectively. v0 is the innerstate variable and g is the gain of the multiplier M1. For the sakeof leading the memristor emulator into a HNN system conveniently,the mathematical model can be scaled into a dimensional form by

s ¼ t=RC; a ¼ gR=Ra ð2Þ

Fig. 1. Circuit realization of the simplified memristor emulator.

i, m

A

v, V

f = 500 Hz

f = 10 kHz

f = 1 kHz

(a)

Fig. 2. Numerically simulated pinched hysteresis loops of the simplified memristor emdifferent amplitudes.

Obviously, the memductance function W(v0) has a linear formand can be expressed as

W v0ð Þ ¼ av0 ð3ÞBy selecting g = 1, R = 10 kX, Ra = 10 kX and C = 100 nF as typi-

cal circuit parameters, the memductance function is calculated asW(v0) = v0.

With the typical circuit parameters, the frequency-dependentpinched hysteresis loops of the simplified memristor emulatoris numerically simulated by (1). Herein, a sinusoidal voltagesource v = Vmsin(2pft) is employed, where Vm and f are the ampli-tude and frequency of the stimulus, respectively. When Vm = 3 Vis fixed and f is respectively selected as 500 Hz, 1 kHz and10 kHz, while when f = 1 kHz is kept and Vm is respectivelyselected as 2 V, 3 V and 4 V, the voltage-current relations arenumerical simulated and plotted in Fig. 2(a) and (b), respectively.Fig. 2(a) shows that the hysteresis loop pinched at the origin andshrinks as the frequency increasing continuously. Whereas Fig. 2(b) manifests that the pinched hysteresis loop is regardless of theamplitude. The simulated results explain that the simplifiedmemristor emulator can meet the three fingerprints of memristor[31].

To verify the frequency-dependent pinched hysteresis loops, ahardware circuit on breadboard is made. All the circuit parame-ters are the same as those in the numerical simulations. Tek-tronix AFG 3102C is taken as a sinusoidal voltage source andthe phase portraits in the v–i plane are captured by a TektronixTDS 3034C digital oscilloscope in XY mode with 600 mV/div in Xdirection and 2.5 mA/div in Y direction. It should be noted thatfor better observing in hardware experimental measurements,all the output currents sensed by current probe are magnifiedfifty times by enwinding the measured wire around the currentinductive probe with fifty turns. The experiment results shownin Fig. 3 are consistent well with those revealed by numericalsimulations.

3. Two-neuron-based non-autonomous mHNN

3.1. Model description

Hopfield neural network can be described by a set of nonlinearordinary differential equations corresponded to n-neurons [9]. Theconnection topology for a two-neuron-based non-autonomousmHNN is considered in our work, as shown in Fig. 4. Herein, anexternal stimulus input I(s) having the form of sinusoidal functionI(s) = Imsin(2pFs) is employed, where Im and F are the stimulus-associated parameters of the amplitude and frequency,respectively.

i, m

A

v, V

Vm = 4 V

Vm = 2 V

Vm = 3 V

(b)

ulator in the v–i plane: (a) Vm = 3 V with different frequencies; (b) f = 1 kHz with

f = 500 Hz

f = 10 kHz

f = 1 kHz

Vm = 4 V

Vm = 2 V

Vm = 3 V

(b)(a)

Fig. 3. Experimentally captured pinched hysteresis loops of the simplified memristor emulator in the v–i plane: (a) Vm = 3 V with different frequencies; (b) f = 1 kHz withdifferent amplitudes.

–2.4

2.8

4Neuron1

Neuron2

–kW

I( )

Fig. 4. Connection topology of the two-neuron-based non-autonomous mHNN.

68 Q. Xu et al. / Int. J. Electron. Commun. (AEÜ) 96 (2018) 66–74

Denote x1 and x2 as the state variables of two neurons. Themathematical model of the proposed two-neuron-based non-autonomous mHNN is described as

dx1=ds ¼ �x1 � kWtanh x1ð Þ þ 2:8tanh x2ð Þ þ I sð Þdx2=ds ¼ �x2 � 2:4tanh x1ð Þ þ 4tanh x2ð Þdx3=ds ¼ �x3 � tanh x1ð Þ

ð4Þ

where k is a positive constant standing the self-coupling strength ofthe simplified memristor and W = x3 stands for the memristive self-connection synapse weight of neuron 1. Thus, the proposed smallworld mHNN modeled by (4) is a three-dimensional non-autonomous memristive dynamical system.

3.2. Equilibrium point and stability analysis

The equilibrium point of the mHNN model (4) can be expressed

as S ¼ x�1; x

�2; x

�3

� �, in which

x�1 ¼ arctanh � 5

12x�2 þ 5

3tanh x�2

� �h i

x�3 ¼ 5

12x�2 � 5

3tanh x�2

� � ð5Þ

where x�2 is numerically solved by

arctanh 512x

�2� 5

3tanh x�2

� �h iþk 5

12x�2� 5

3tanh x�2

� �h i2þ 14

5 tanh x�2

� �þ I sð Þ¼0

ð6Þ

Herein, (6) contains I(s), which leads to that x�1, x

�2 and

x�3changes with the time evolution. Thus, S ¼ x

�1; x

�2; x

�3

� �can be

termed as AC equilibrium point [32].

By linearizing (4) around the AC equilibrium point, the Jacobianmatrix of the two-neuron-based non-autonomous mHNN isobtained as

J ¼

�1� kx�3sech

2 x�1

� �2:8sech2 x

�2

� ��ktanh x

�1

� �

�2:4sech2 x�1

� ��1þ 4sech2 x

�2

� �0

�sech2 x�1

� �0 �1

266664

377775

ð7Þ

The characteristic equation evaluated at S ¼ x�1; x

�2; x

�3

� �can be

derived as

k3 þ a1k2 þ a2kþ a3 ¼ 0 ð8Þ

where

a1 ¼ 3� 4sech2 x�2

� �þ kx

�3sech

2 x�1

� �

a2 ¼ 3þ k 2x�3 � tanh x

�1

� �h isech2 x

�1

� �� 8sech2 x

�2

� �

þ 6:72� 4kx�3

� �sech2 x

�1

� �sech2 x

�2

� �

a3 ¼ 1þ k x�3 � tanh x

�1

� �h isech2 x

�1

� �� 4sech2 x

�2

� �

þ 4ktanh x�1

� �þ 6:72� 4kx

�3

h isech2 x

�1

� �sech2 x

�2

� �

ð9Þ

The external stimulus input I(s) changes in the range of [–Im, Im]

with the time evolution, so one can get the values of x�2 and explore

the stability of AC equilibrium point for the specified range of I(s)numerically. k = 0.2 and k = 0.45 with Im = 2 are taken as two casesto explore the locus and stability of the AC equilibrium point, asshown in Fig. 5.

Observed from Fig. 5(a), it can be found that the locus of theonly AC equilibrium point changes with the time evolution.According to the eigenvalues calculated by (8), the stabilities ofthe AC equilibrium points can be explicated and divided into threetypes for k = 0.2 including stable node-focus (SNF), unstable saddlefocus (USF), and stable node (SN). It is worth nothing that there aretwo Hopf bifurcation points (HBPs) having a real eigenvalue and apair of pure imaginary eigenvalues in (8). The locus and stability ofthe AC equilibrium points with the time evolution are listed inTable 1. The first and second columns are the rang of I(s) and cor-responding stability of the AC equilibrium point, as well as thethird and fourth columns show several AC equilibrium points

USF

SNF

HBP

HBP

SN2x

USF

SNFHBP

2x

I ( ) I ( )

FBP

HBPFBPSN

S1

S2 S3

(a) (b)

Fig. 5. Numerically simulated AC equilibrium points and stability analysis: (a) k = 0.2; (b) k = 0.45.

Table 1AC equilibrium points, the corresponding eigenvalues, and their stability for k = 0.2.

Range of I(s) Stability I(s) and AC equilibrium point Eigenvalues

[�2, �0.9952] SNF I(s) = �1.5585: (1.2189, 1.7520, �0.8393) �0.7036 ± j0.4041, �1.0896�0.9952 HBPs I(s) = �0.9952: (1.1878, 0.9293, �0.8299) ±j0.4289, �1.08150.7131 I(s) = 0.7131: (�1.0531, �0.8191, 0.7830) ±j0.3269, �0.8807(�0.9952, 0.7131) USF I = 0: (0, 0, 0) 1 ± j1.6492, –1(0.7131, 0.7191) SNF I(s) = 0.7178: (�1.1388, �0.8880, 0.8140) �0.1020 ± j0.1692, �0.8697(0.7191, 0.7781) SN I(s) = 0.7520: (�1.4128, �1.1862, 0.8881) �0.3368, �0.6603, �0.7920(0.7781, 2) SNF I(s) = 1.1360: (�1.3086, �1.6320, 0.8639) �0.7964 ± j0.4146, �0.8834

Q. Xu et al. / Int. J. Electron. Commun. (AEÜ) 96 (2018) 66–74 69

and eigenvalues for a single value of I(s) in the correspondingrange, respectively.

Whereas for k = 0.45, the evolutional law of AC equilibriumpoint is different from that of k = 0.2 in a small range of I(s). WhenI(s) is in the range of (0.5370, 0.5641), there are three AC equilib-rium points marked as S1, S2 and S3 for each I(s). As a whole, thenumber of the AC equilibrium points is changed with I(s) in therange of [–2, 2], leading to the appearance of fold bifurcation points(FBPs) in Fig. 5(b), where the up-right is an enlarged view of therectangle zone. FB is restricted by a3 = 0, a1 > 0, and a1a2 � a3 > 0,making it jumps between different AC equilibrium points. Sum-mary of the locus and stability of AC equilibrium points with thetime evolution for k = 0.45 is listed in Table 2.

In particular, the AC equilibrium point and its stability changewith the time evolution, leading to that an unstable manifold ofthe unstable AC equilibrium point may be attracted into a self-excited attractor in the proposed non-autonomous mHNN [32].The existence of HBP and FBP may result in the occurrence ofbursting behavior in the proposed mHNN when two time scalesare involved [26].

Table 2AC equilibrium points, the corresponding eigenvalues, and their stability for k = 0.45.

Range of I(s) Stability I(s) and AC eq

[�2, �1.1851] SNF I(s) = �1.5520�1.1851 HBPs I(s) = �1.18510.5634 I(s) = 0.5634:(�1.1851, 0.5370) USF I(s) = �0.36800.5370 FBPs I(s) = 0.5370:0.5641 I(s) = 0.5641:(0.5370, 0.5634) USF I(s) = 0.5425:

USF S2(�1.2317, �SNF S3(�1.3814, �

(0.5634, 0.5639] SNF I(s) = 0.5638:USF S2 (�0.9702, �SNF S3 (�1.4245, �

(0.5639, 0.5641) SNF I(s) = 0.5640:SN S2 (�0.9552, �SNF S3 (�1.4248, �

(0.5641, 2) SNF I(s) = 0.57: (�

4. Dynamics associated with k

With the model (4), numerical explorations for the two-neuron-based non-autonomous mHNN are performed by MATLAB ODE23algorithm with the time-step Ds = 0.01 and initial conditions (0,0, 0.1). The dynamical behaviors associated with k are performedby bifurcation diagrams and corresponding Lyapunov exponents,where the Wolf’s method [33] is employed to calculated the Lya-punov exponents.

When the self-coupling strength k increases from 0.05 to 0.45,the bifurcation diagrams of the system variable x1 are plotted inFig. 6(a) and the first Lyapunov exponent is shown in Fig. 6(b),respectively. Fig. 6 manifests that complex dynamical behaviorsare emerged in the two-neuron-based non-autonomous mHNN,including period, chaos, period-doubling bifurcation, and periodicwindow.

For several values of k, the phase portraits in the x1 � x2 planeare numerically simulated by MATLAB software, as shown inFig. 7. The orbits are initialized by the initial conditions (0, 0,0.1), which are the same as those used in bifurcation diagrams in

uilibrium points Eigenvalues

: (1.3565, 1.5608, �0.8756) �0.5495 ± j0.4053, �1.1620: (1.2422, 0.9777, �0.8461) ±j0.4668, �1.543(�0.9057, �0.7054, 0.7191) ±j0.2763, �0.6337: (0.2799, 0.2231, �0.2728) 0.9937 ± j1.5904, –1.0665(�1.3224, �1.0582, 0.8674) 0, –0.7807 ± j0.3167(�0.9438, �0.7345, 0.7370) 0, �0.1547, �0.5642S1(�0.7301, �0.5721, 0.6231) 0.2584 ± j0.6973, –0.75710.9681, 0.8431) 0.1158, –0.7314 ± j0.31401.1326, 0.8813) –0.1121, –0.8066 ± j0.3140S1 (�0.9217, �0.7176, 0.7267) –0.6108, –0.0292 ± j0.20730.7547, 0.7488) 0.1281, –0.4520 ± j0.10471.2129, 0.8905) –0.2433, –0.8228 ± j0.3154S1 (�0.9343, �0.7272, 0.7326) �0.5875, –0.0548 ± j0.12600.7432, 0.7421) 0.0796, �0.3016, –0.52111.2136, 0.8906) �0.2444, –0.8229 ± j0.31541.4301, �1.2284, 0.8917) �0.2694, �0.8247 ± j0.3165

x 1

k

Firs

t Lya

puno

v Ex

pone

nt

k

zero line

(a) (b)

Fig. 6. MATLAB numerical simulations of the dynamical behaviors with k increasing for the initial conditions (0, 0, 0.1): (a) Bifurcation diagrams of the system variable x1; (b)first Lyapunov exponent.

x1

x 2

x1

x 2

(a) (b)

x1

x 2

x1

x 2

(c) (d)

Fig. 7. Numerically simulated phase portraits in the x1 � x2 plane for different k: (a) Period-2 limit cycle at k = 0.08; (b) period-4 limit cycle at k = 0.12; (c) chaotic attractor atk = 0.25; (d) period-5 limit cycle at k = 0.29.

70 Q. Xu et al. / Int. J. Electron. Commun. (AEÜ) 96 (2018) 66–74

Fig. 6(a). These MATLAB numerical simulations just verify the strik-ing dynamics of limit cycles, chaotic attractors, and periodic win-dow emerging from the two-neuron-based non-autonomousmHNN.

5. Bursting behavior

With consideration of an order gap existing between the excit-ing frequency of external stimulus and the natural frequency ofautonomous system, the proposed two-neuron-based non-autonomous mHNN can be called as generalized autonomous sys-tem (GAS) [26]. It leads to the occurrence of bursting oscillationswith the alternates between QS and SP. Under the concept ofGAS, the external stimulus I(s) = Imsin(2pFs) in (4) can be treatedas a slow-varying parameter. Due to the boundedness of sinusoidalfunction, the slow-varying parameter I(s) changes in the range of[�Im, Im]. Herein, Im = 2 which is the same as that in Section 3.

When F = 0.0005 is selected, an order gap exists between theexternal stimulus frequency and natural frequency. For k = 0.2 and0.45, bursting behaviors can be explored through phase portraits,transformed phase portrait [26], and time-domain waveform, asshown in Fig. 8. The first LEs for the two cases are LE1 = �0.5751and LE1 = �0.5261, which manifests that the non-autonomousmHNN runs in periodic oscillations. At the first roughly glace, thephase portrait in the x1 � x3 plane and time-domain waveform ofthe variable x2 for the two cases are very similar, as shown inFig. 8(a) and (b). Thus, the transformed phase portrait is employedto explicate the difference between their bifurcation mechanisms.For k = 0.2, only HB occurs, leading to that the mHNN keeps therepetitive spiking oscillation and quiescent oscillation. The trajecto-ries are dense in the dot-line rectangle due to that the spiking oscil-lation is always around the AC equilibrium points, as illustrated inFig. 8(c1). Whereas for k = 0.45, the existence of HB and FB leadsto that the trajectories not only are around the AC equilibriumpoints, but also transmit and deviate the AC equilibrium points.

x1

x 3

HBP

HBP

x1

x 3

HBP

HBP

FBPs

(a1) (a2)

x 2

(×104) (×104)x 2

I

x 2

HBP

HBP

I

x 2 HBPHBP

FBPs

HBP

FBPs

(b1) (b2)

(c1) (c2)

Fig. 8. Numerically simulated phase portraits, transformed phase portraits, and time-domain waveform for different k: (a) phase portrait in the x1 � x3 plane; (b) time-domain waveform of variable x2; (c) transformed phase portraits in the I � x2 plane. (a1), (b1) and (c1) for k = 0.2, as well as (a2), (b2) and (c2) for k = 0.45.

Q. Xu et al. / Int. J. Electron. Commun. (AEÜ) 96 (2018) 66–74 71

The trajectories are dense in the up and down dot-line rectangle, asshown in Fig. 8(c2), where an enlarged view of the crowded part forthe one HBP and two FBPs are illustrated. The asymmetric Hopf/

RF

VEE

T1

RT

–R

Ui

vinT2

T3 T4

RT

R

R

R

–Uo

RW

voutRC RC

VCC

I0

R

(a)

Fig. 9. Circuit implementation: (a) Inverting hyperbolic tangent function uni

Hopf bursting and fold/Hopf bursting mechanisms are explored inthe proposed non-autonomous mHNN, which are different fromthose symmetric ones revealing in [25,26,29].

(b)

t circuit; (b) scheme of the two-neuron-based non-autonomous mHNN.

72 Q. Xu et al. / Int. J. Electron. Commun. (AEÜ) 96 (2018) 66–74

6. Circuit implementation and hardware experiments

The hyperbolic tangent function circuit unit is realized by adual-transistor pair of T1 and T2, a module of current source I0,and two operational amplifier circuits for controlling gains [9,34].Four bipolar transistors MPS2222, two operational amplifiersTL082CP with ±15 V DC voltage sources, and eleven resistors areemployed in Fig. 9(a). The circuit parameters are selected asR = 10 kX, RF = 520X, RC = 1 kX, RT = 2 kX and RW = 9.8 kX. With

v1

1 V/Div

v 21

V/D

iv

(a)

v1

1 V/Div

v 21

V/D

iv

(c)

Fig. 10. Experimentally captured phase portraits for different R1: (a) Period-2 limit cycR1 = 40 kX; (d) period-5 limit cycle at R1 = 34.5 kX.

v1

500 mV/Div

v 325

0 m

V/D

iv

(a)

Fig. 11. Experimentally captured periodic bursting oscillation for k = 0.2, f = 0.5 Hz, anvariable x2.

the specified circuit parameters, the output of the hyperbolic tan-gent function circuit can be deduced as

mout ¼ �tanh minð Þ ð10Þwhere vin and vout denote the input voltage and output voltage ofthe hyperbolic tangent function circuit unit in Fig. 9(a).

With the mathematical model (4), the two-neuron-based mem-ristive HNN can be physically implemented by discrete circuitcomponents, as shown in Fig. 9(b). The memristor W represents

v1

1 V/Divv 2

1 V

/Div

(b)

v1

1 V/Div

v 21

V/D

iv

(d)

le at R1 = 125 kX; (b) period-4 limit cycle at R1 = 83.3 kX; (c) chaotic attractor at

t1 ms/Div

v 21

V/D

iv

(b)

d A = 2 V: (a) Phase portraits in the x1 � x3 plane; (b) time-domain waveform for

Q. Xu et al. / Int. J. Electron. Commun. (AEÜ) 96 (2018) 66–74 73

the simplified memristor emulator in Fig. 1 and the two circuitmodules Ta and Tb marked by –tanh with solid box are the invert-ing hyperbolic tangent function circuit units drawn in Fig. 9(a).

The circuit in Fig. 9(b) has three dynamic elements of thecapacitors, corresponding to three state variables of v1, v2, and v0,respectively. Therefore, the circuit state equations for the two-neuron-based memristive HNN can be established as

RCdv1dt ¼ �v1 � gR

R1v0tanh v1ð Þ þ R

R2tanh v2ð Þ � R

R3v s tð Þ

RCdv2dt ¼ �v2 � R

R4tanh v1ð Þ þ R

R5tanh v2ð Þ

RCdv0dt ¼ �v0 � tanh v1ð Þ

ð11Þ

Supposing that the integrating time constant RC = 1 ms, theresistance and the capacitance can be selected as R = 10 kX andC = 100 nF, respectively. The resistances are calculated asR2 = R/2.8 = 3.571 kX, R3 = R = 10 kX, R4 = R/2.4 = 4.167 kX, andR5 = R/4 = 2.5 kX. For different k, the resistances R1 can be calcu-lated by R1 = Raa = R/k.

The sinusoidal voltage stimulus is employed as vs(t) = A sin(2pft), where A = Im and f = F/RC. Tektronix AFG 3102C is taken asa sinusoidal voltage stimulus. It should be addressed that one aux-iliary voltage follower circuit realized by an operation amplifierAD711KN is hired to isolate the applied sinusoidal voltage sourcein the hardware circuit.

With the circuit schematic in Fig. 9, an analogue electronic cir-cuit is practically set up by some commercially available compo-nents on breadboards. By tuning the resistances R1, theexperimentally captured attractors in the v1–v2 plane for differentk are shown in Fig. 10. The experimental results are captured by aTektronix TDS 3034C digital oscilloscope in XY mode with 1 V/divin X direction and 1 V/div in Y direction. Whereas for theexperimental verification of the bursting behavior, the outputof Tektronix AFG 3102C is set to A = 2 V andf = 0.0005/0.001 Hz = 0.5 Hz. The Tektronix TDS 3034C digital oscil-loscope works in XY mode with 500 mV/ div in X direction and250 mV/div in Y direction for better observing in busting behavior.

It is emphasized that the desired initial capacitor voltages (0 V,0 V, 0.1 V) are difficult to assign in the hardware circuit, which arerandomly sensed through turning on the hardware circuit powersupplies again [30,35]. Ignoring the minor deviations caused bythe calculation error, parasitic circuit parameters and active devicenon-idealities, the experimental results shown in Fig. 10 are wellconsistent well with the numerical simulations in Fig. 7. Moreover,Fig. 11(a) and (b) coincides well with the numerical simulations inFig. 8(a1) and (b1), respectively. The experimentally capturedresults further validate the existing dynamical behaviors in thetwo-neuron-based non-autonomous mHNN.

7. Conclusions

A two-neuron-based non-autonomous mHNN is presented,upon which the dynamical behaviors depending on the self-coupling strength of memristive self-connection synapse andstimulus frequency are revealed by numerical simulations andexperimental measurements. Due to that the locus and stabilityof the AC equilibrium point change with the time evolution, com-plex dynamical behaviors including period, chaos, period-doublingbifurcation, and periodic window are explored. Specifically, theorder gap between the natural frequency and external stimulusleads to the occurrence of bursting behavior. The presenting two-neuron-based non-autonomous mHNN has simple connectiontopology and feasible hardware realization, which can be takenas a paradigm for demonstrating the dynamical behavior inHopfield neural network experimentally.

Acknowledgement

This research issue was supported by the grants from theNational Natural Science Foundations of China under Grant Nos.61801054, 51777016, 61601062, 61705021, 11602035, and51607013, and the Natural Science Foundations of Jiangsu Pro-vince, China under Grant No. BK20160282.

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