IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137

Abstract—Memristors can be employed to mimic biological neural synapses or to describe electromagnetic induction effects. To exhibit the threshold effect of electromagnetic induction, this paper presents a threshold flux-controlled memristor and examines its frequency-dependent pinched hysteresis loops. Using an electromagnetic induction current generated by the threshold memristor to replace the external current in two-dimensional (2D) Hindmarsh-Rose (HR) neuron model, a three-dimensional (3D) memristive HR (mHR) neuron model with global hidden oscillations is established and the corresponding numerical simulations are performed. It is found that due to no equilibrium point, the obtained mHR neuron model always operates in hidden bursting firing patterns, including coexisting hidden bursting firing patterns with bi-stability also. In addition, the model exhibits complex dynamics of the actual neuron electrical activities, which acts like the 3D HR neuron model, indicating its feasibility. In particular, by constructing the fold and Hopf bifurcation sets of fast-scale subsystem, the bifurcation mechanisms of hidden bursting firings are expounded. Finally, circuit experiments on hardware breadboards are deployed and the captured results well match with the numerical results, validating the physical mechanism of biological neuron and the reliability of electronic neuron.

Index Terms—Bursting firing, bifurcation, chaotic dynamics, electromagnetic induction, memristor emulator, neuron model

I. INTRODUCTION

EMRISTOR, theoretically postulated by Chua, is an elementary circuit element defined by a constitutive

relation between magnetic flux φ and electric charge q [1], which can be effectively interpreted using Maxwell’s equations for a quasi-static expansion of the electromagnetic field quantities based on the electromagnetic field theory [2]. Like as the three classical circuit elements, memristor is a link between

M

the first-order electric and magnetic fields in the quasi-static

Manuscript received March 28, 2018; revised September 27, 2018; accepted ****, 2019. Date of publication ****, 2019; date of current version****, 2019. This work was supported by the National Natural Science Foundation of China under Grant Nos. 51777016, 61601062, and 61532020, and the Postgraduate Research & Practice Innovation Program of Jiangsu Province, China under Grant No. SJKY19_0191.

H. Bao and W. Liu are with the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (e-mail: charlesbao0319@gmail.com, wenboliu@nuaa.edu.cn).

A. Hu and B. Bao are with the School of Information Science and Engineering, Changzhou University, Changzhou 213164, China (e-mail: aihuanghu2016@163.com, mervinbao@126.com).

expansion [1], whose nonlinear resistance or conductance can be memorized by controlling the flow of the electric charge or the magnetic flux [2]. In fact, the magnetic flux and electric charge are two fundamental physical attributes to characterize a circuit element, i.e., φ and q are internal attributes related to the device material and its operating mechanism, respectively [3].

In recent years, numerous hardware implementations of neuromorphic circuits were developed through employing memristors to mimic the dynamical behaviors of biological neural synapses [4]–[9]. Supported by metastable memory state transition properties, short-term synaptic dynamics observed in biological synapses was emulated by solid-state TiO2 memristors [4]. Using memristors integrated into a dense, transistor-free crossbar circuit, an artificial neural network was experimentally demonstrated [5], which is an important step towards hardware implementation of complex neuromorphic networks. Regarded as synaptic emulators for neuromorphic computing, memristors with diffusive dynamics could enable direct exhibitions of both short- and long-term plasticity of biological synapses, standing for a development in real realization of neuromorphic functionalities [6]. Furthermore, Serb et al. [7] illustrated an unsupervised learning in a probabilistic neural network by utilizing solid-state memristors as multi-state synapses; Kumar et al. [8] performed neural- network-type implementations of analogue computing by utilizing grids of memristor devices with the synapse-like dynamics and adaptable conductivity; and Sangwan et al. [9] reported the experimental implementation of a multiterminal hybrid memristor and transistor, which could enable complex neuromorphic learning and the study of the physics of defect kinetics in two-dimensional (2D) materials. In short, memristors with basic neural functions are increasingly becoming the essential components of neuromorphic circuits.

The membrane potential in the neuron electrical activities can induce an electromagnetic induction current that behaves like a memristor synapse [10], [11]. When the external electro- magnetic radiations are imposed on neurons or the neuron electrical activities are considered with the electromagnetic induction, the memristors can be introduced into the classical neuron models for characterizing these induction effects [12], [13]. Under this strategy, some memristor-based neuron models were thereby proposed and complex dynamical behaviors of pattern selection, mode transition, and synchronous transition were disclosed [14]–[18]. Such memristive neuron models are valuable for better

Hidden Bursting Firings and Bifurcation Mechanisms in Memristive Neuron Model with

Threshold Electromagnetic InductionHan Bao, Aihuang Hu, Wenbo Liu, and Bocheng Bao

1

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137understanding chaotic dynamics of the actual electrical activities in biological neurons [12].

Extended from Hodgkin-Huxley neuron model [19], some simplified neuron models were presented to characterize the main dynamics of the neuron electrical activities [20]–[25], among them, the 2D and three-dimensional (3D) Hindmarsh- Rose (HR) models are effective and available for dynamical analysis in the neuron electrical activities due to simple algebraic structures [20], [21]. The 2D HR model is an especially simple neuron model. However, this simplified neuron model cannot mimic the rich nonlinear phenomena observed in biological neurons. There is evidence that neurons operating in a regime called ‘the edge of chaos’ may be central to complexity, learning efficiency, adaptability and analogue computation in brains [9], [26]. Since the electromagnetic induction effect is relevant to physical electrophysiological environment, a neuron model accompanying electromagnetic induction can be considered to generate complex dynamical patterns in biological neurons. Generally, the electromagnetic induction effect is bounded above and below [27], which is emulated by a threshold flux-controlled memristor considered in this paper. Introducing memristors into existing cellular neural networks and nonlinear chaotic circuits, numerous memristor-based neural networks [28]–[32] and chaotic circuits [33]–[35] were developed in the past few years. Similar to the above construction strategies, memristive neuron models can be obtained through introducing memristor synapses into some existing neuron models [10], [11], [14]–[18], [36]. With this context, a new memristive HR (mHR) neuron model with threshold electromagnetic induction is presented in this paper, which can exhibit complex dynamical behaviors of the neuron electrical activities better. Fortunately, chaotic and periodic bursting firings with global hidden oscillation can be emerged in the mHR neuron model, which has been rarely reported in the HR neuron model [36].

The rest of the paper is organized as follows. In Sec. II, a simple threshold flux-controlled memristor is proposed and its frequency-dependent pinched hysteresis loops are examined numerically and experimentally. In Sec. III, a 3D mHR neuron model is established to numerically reveal hidden bursting firings. In Sec. IV, the bifurcation mechanisms of hidden bursting firings are comprehensively expounded by constructing the fold and Hopf bifurcation sets of fast-scale subsystem. In addition, hardware breadboards with commercially available components are developed and circuit experiments are performed to validate the hidden bursting firings reported in Sec. V. The last section concludes the paper.

II.THRESHOLD FLUX-CONTROLLED MEMRISTOR To exhibit the threshold effect of electromagnetic induction,

a threshold flux-controlled memristor is first considered. Hyperbolic tangent function is a monotonic continuous differentiable function, which is often used as a neuron activation function bounded above and below in Hopfield neural network [27]. In this context, a simple threshold

(a)

(b)Fig. 1. Circuit implementation for the threshold flux-controlled memristor emulator. (a) Operational amplifier and analog multiplier-based circuit schematic. (b) Equivalent circuit of the hyperbolic tangent function unit.

flux-controlled memristor is presented, which is described as

(1)Where φ is the inner magnetic flux, τ is the time constant, v and i are the input voltage and output current of the memristor, respectively. The memductance function W(φ) is expressed as

(2)The model (1) agrees with the definition of flux-controlled memristor [1], [3]. The memductance W(φ) in (2) is smoothly variable and indicates that the flux-controlled memristor is nonlinear and threshold. However, the memductance for the piecewise-linear memristors designed in [30], [33] is two- valued and non-smooth. In contrast, the memristance for the real nonlinear memristor given in [34], [37] is a linear function of electric charge.

Based on operational amplifier and analog multiplier [35], an implementation circuit of the threshold flux-controlled memristor characterized by (1) is designed, as depicted in Fig. 1(a), which contains an integrator U0 with the time constant τ = RC, a threshold function unit T0 labeled by –tanh with solid box, an analog multiplier M and a resistor R0. An equivalent circuit of the threshold function unit T0 is shown in Fig. 1(b), which is composed of a pair of differential transistors T1 and T2, a proportional constant current source I0, and two operational amplifiers Ui and Uo for the circuit unit gain control [38]–[40]. Note that the constant current source I0 is implemented by a differential transistor pair with three resistors and one voltage source of value about 1.1 mA. As for more details on the implementation of I0, please refer to [40]. Particularly, the presented threshold flux-controlled memristor emulator is simple and inductor-free, which make the circuit hardware gadget and suitable for IC design.

Operational amplifier AD711KN and analog multiplier AD633JNZ with ±15 V DC operation voltages, differential transistor pair MPS2222, resistor, and ceramic capacitor are selected. When the circuit parameters are determined as R =

–+U0

W(vφ)

M

v

W(vφ)vv

vφ

R0R C

i

i–tanh

T0

RF

T1–+

R

Ui

viT2

R

R R–+Uo

voRC RC

E

I0

R

2

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.290513710 kΩ, RF = 520 Ω, RC = 1 kΩ, and I0 = 1.1 mA, the input and

(a1) (a2)

(b1) (b2)Fig. 2. Frequency-dependent pinched hysteresis loops of the threshold flux- controlled memristor emulator, where (a1) and (b1) are numerical simulations and (a2) and (b2) are breadboard experiments. (a) A = 4 (or A = 4 V for experiments) and different frequencies, (b) F = 0.5 (or f = 500 Hz for experiments) and different amplitudes.

output relation of this circuit unit in Fig. 1(b) is described as

(3)where vi and vo are the input and output voltages, respectively.

According to the circuit schematic in Fig. 1(a), the threshold flux-controlled memristor emulator can be modeled as

(4)in which vφ is the output voltage of the integrator U0, and g is the gain of the multiplier M. The physical memductance function W(vφ) is then expressed as

(5)Introducing the threshold flux-controlled memristor emulator into a dynamical system and scaling the circuit parameters in a dimensionless form, one can get

(6)Thus, the circuit parameters of the threshold flux-controlled memristor emulator in Fig. 1(a) can be calculated as R = 10 kΩ, C = 100 nF, R0 = 1 kΩ, and g = 0.1.

Consider that a sinusoidal voltage source v = A sin(2πFt) is applied at the input port of the threshold memristor emulator shown in Fig. 1(a). When the amplitude A = 4 is fixed with different values of the frequency F as well as F = 0.5 is fixed with different values of A, the frequency-dependent pinched hysteresis loops of the threshold flux-controlled memristor are numerically simulated and drawn in Figs. 2(a1) and 2(b1), respectively. In breadboard experiments, a sinusoidal voltage source v = A sin(2πft) is generated by a function generator and its oscillating frequency is determined by f = F/RC [40]. Corresponding to the numerical simulations, the frequency- dependent pinched hysteresis loops with different values of the physical frequency and amplitude are experimentally

captured by a digital oscilloscope, as plotted in Figs. 2(a2) and 2(b2), respectively. These consistent results between numerical

(a1) (a2)

(b1) (b2)Fig. 3. The memductance curves against time for the threshold flux-controlled memristor emulator, where (a1) and (b1) are MATLAB simulations and (a2) and (b2) are PSIM simulations. (a) A = 4 (or A = 4 V for PSIM simulations) and different frequencies, (b) F = 0.5 (or f = 500 Hz for PSIM simulations) and different amplitudes.

simulations and breadboard experiments demonstrate three fingerprints of the frequency-dependent pinched hysteresis loop. To better visualize the experimental results, the memristor emulator output currents sensed by the current probe are magnified ten times by enwinding the test wire around the current inductive probe with ten turns.

Corresponding to the results depicted in Fig. 2, the aforementioned sinusoidal voltage sources with different frequencies and amplitudes are applied at the input port of the threshold memristor emulator depicted in Fig. 1(a). Realizing the model (1) with MATLAB, the curves of the memductance function W(φ) in (2) against time are displayed in Figs. 3(a1) and 3(b1), respectively. As the memductance is very difficult to be measured in breadboard experiments, PSIM circuit simulation software is employed to draw the curves of the physical memductance. Similarly, using the implementation circuit of the memristor emulator shown in Fig. 1(a), the curves of the physical memductance function W(vφ) in (5) against time by PSIM circuit simulations are displayed in Figs. 3(a2) and 3(b2), respectively. PSIM circuit simulations are consistent well with MATLAB numerical simulations. The curves shown in Fig. 3 imply that the memductance of the flux-controlled memristor emulator is bounded in the region of 0 mS ≤ W(vφ) < 1 mS, i.e., the presented memristor emulator is passive.

III. HIDDEN MEMRISTIVE NEURON MODEL

A. Description of the mHR Neuron ModelThe 2D HR neuron model was simplified from the classical

Hodgkin-Huxley model [19] by Hindmarsh and Rose [20],

v

i (×1

0−3) F = 10

F = 0.5

F = 2

v, 800 mV/div

i ,1

mA

/div f = 10 kHz

f = 500 Hz

f = 2 kHz

v

i (×1

0−3)

v, 800 mV/div

i ,1

mA

/div

A = 3

A = 2

A = 4

A = 3 V

A = 2 V

A = 4 V

F = 10

F = 0.5 F = 2

t, ms0 1 2 43

-0.2

0.8

f = 10 kHz

f = 500 Hz f = 2 kHz

0.60.40.2

0

τ

W(v

φ), m

S

W(φ

)

1

t, ms0 1 2 43-0.2

W(v

φ), m

S

0.80.60.40.2

0

τ

W(φ

)

A = 3 A = 2

A = 4

A = 3 V A = 2 V

A = 4 V1

3

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137which is mathematically described as

(7)where x is the membrane potential and y is spiking variable, a,

Fig. 4. Hidden dynamical behaviors with the specified parameters a = 1, b = 3, c = 1, and d = 5, the upper is the bifurcation diagram of the maxima X of the membrane potential x and the lower is the first two Lyapunov exponents.

b, c and d are positive constants and I is the external current. Additionally, a = 1 and c = 1 are determined [41]–[44].

To better emulate complex dynamical behaviors of the neuron electrical activities, a 3D HR neuron model was proposed by introducing an extra third equation to improve the two-dimensional HR neuron model (7) [21], which can be written as

(8)where z is the bursting variable and x1 is the resting potential, as well as r and s are positive constants but r is very small.

Motivated by the aforementioned design scheme of the three- dimensional HR neuron model, a novel 3D memristive HR neuron model is presented by referring to the previously published literatures [10], [11], [14]–[18]. To construct this mHR neuron model, the threshold flux-controlled memristor in (1) is considered to emulate the electromagnetic induction effect on the membrane potential. When the current induced by the memristor is used to substitute the external current I in the two-dimensional HR neuron model (7), the mHR neuron model can be achieved and expressed as

(9)where φ is the flux variable representing the time integral of the membrane potential x. The term mtanh(φ)x stands for the electromagnetic induction effect on the membrane potential and m indicates the electromagnetic induction strength. It should be mentioned that a nonlinear threshold memductance

function is utilized and the external current I is removed in the presented mHR neuron model (9), but a linear memductance function is considered and the external current I is still retained in the memristive HR neuron model reported in [36].

With the mathematical model (9), it is obvious that there is no symmetry property in the 3D mHR neuron model. Setting the

Fig. 5. Asymmetric chaotic bursting firing with m = 1. (a) Time sequences of three state variables and (b) phase orbits in the y – x and φ – x planes.

Fig. 6. Asymmetric periodic bursting firing with m = 1.4. (a) Time sequences of three state variables and (b) phase orbits in the y – x and φ – x planes.

left side of (9) as zero, it can be easily found that there is no any equilibrium point existing in such the mHR neuron model, neither stable nor unstable as c = 1 [41]–[45]. Due to no equilibrium point available, the mHR neuron model is a special nonlinear dynamical system owning the specific hidden attractors, i.e., the hidden oscillating patterns [35], [46]–[49].

X

m

Lyap

unov

Exp

onen

ts

L2

L1

x

Time

y, φ

y

φ(a)

x

y φ

x

(b)

Time

xy,

φ

y

φ(a)

x x

φy (b)

4

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137B. Hidden Dynamical Behaviors and Bursting Firings

The parameters in (9) are specified as follows: a = 1, b = 3, c = 1, and d = 5. When the electromagnetic induction strength m is increased in the region [0.4, 1.6], the bifurcation diagram of the maxima X of the membrane potential x and first two Lyapunov exponent spectra for the initial conditions (0, 0, 0) are plotted in Fig. 4, from which hidden dynamical behaviors with period, chaos, period doubling cascades, reverse period

(a)

(b)Fig. 7. Two-parameter dynamical behaviors in the m – b parameter space with a = 1, c = 1, and d = 5. (a) Bifurcation diagram depicted by the periodicities of the membrane potential x. (b) Dynamical map described by the largest Lyapunov exponent.

doubling cascades, periodic window, and boundary crisis are found. The results illustrate that due to the introduction of the threshold flux-controlled memristor, the mHR neuron model (9) exhibits complex hidden dynamics for better demonstrating the bursting firing activities of the membrane potential.

Without lose of generality, m = 1 and 1.4 are set as two representative examples. The mHR neuron model can exhibit asymmetric chaotic and periodic bursting firings, whose time sequences of three state variables and phase orbits in the y – x and φ – x planes are shown in Figs. 5 and 6, respectively. The bursting firings are originally used to describe a type of the neuron electrical activities, which appears when the oscillation alternates between the quiescent state and repetitive spiking state [50]. Analogous to the classical 3D HR neuron model [43], [44], the state variables x and y are two fast variables,

whereas the state variable φ is a slow variable. Therefore, the model (9) involves two time scales and a fast-slow bifurcation analysis method can be utilized to study the bifurcation mechanisms of the asymmetric chaotic and periodic bursting firings.

To show hidden dynamical behaviors in the mHR neuron model intuitively, two-parameter bifurcation diagram depicted by the periodicities of the membrane potential x and two- parameter dynamical map described by the largest Lyapunov exponent in the m – b plane are plotted [51], as shown in Figs. 7(a) and 7(b), respectively, where the initial conditions (0, 0, 0)

Fig. 8. Coexisting phenomenon of hidden chaotic and periodic bursting firings with bi-stability. (a) Attracting basins in the x(0) – y(0) plane with φ(0) = 0 and the x(0) – φ(0) plane with y(0) = 0. (b) Coexisting phase orbits in the φ – x plane and coexisting time sequences of x.

are assigned. In the Fig. 7(a), the red regions labeled by CH stand for chaotic behaviors and the other color regions labeled by P1 ~ P12 represent periodic behaviors with different periodicities. Whereas in the Fig. 7(b), the color regions with positive largest Lyapunov exponent denote chaotic behaviors and the other color regions indicate periodic behaviors. The dynamical behaviors revealed by the two-parameter bifurcation diagram in Fig. 7(a) and two-parameter dynamical map in Fig. 7(b) are congruent, which perfectly manifest how hidden dynamical behaviors evolve in the m – b plane.

Since the 2D HR neuron model can not exhibit chaotic dynamics [41], it needs to be mentioned that any hidden chaotic bursting firing in the mHR neuron model is induced by the electromagnetic induction effect of the membrane potential, implying the existence of complex dynamics in neuron with the threshold electromagnetic induction.

C.Attracting Basin and Coexisting Hidden AttractorsWhen the system parameters are set as a = 1, b = 3.2, c = 1,

d = 5, and m = 1.4, the interesting coexisting behaviors of chaotic attractor and periodic limit cycle are emerged from the mHR neuron model under different initial conditions. Two attracting basins in the x(0) – y(0) and x(0) – φ(0) planes, coexisting phase orbits in the φ – x plane, and coexisting time

b

m

b

m

y(0)

φ(0)

x(0) x(0)(a)

x x

φ Time(b)

5

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137sequences of the membrane potential x are displayed in Fig. 8. In Fig. 8(a), the moving trajectories started from the red regions oscillate in chaotic patterns and those from the blue regions are in periodic patterns, demonstrating the bi-stability phenomenon of coexisting hidden attractors. In Fig. 8(b), the red and blue orbits are initiated from the initial conditions (0, 0, 0) and (0, –5, 0), respectively, illustrating the coexistence of chaotic and periodic bursting firings in the mHR neuron model also. Correspondingly, the Lyapunov exponents for the chaotic bursting firing are 0.0279, 0 and −3.8660, respectively. In contrast, that corresponding to the periodic bursting firing are 0, −0.1381 and −3.7741, respectively.

Fig. 9. Equilibrium points of the fast-scale subsystem illustrated by the real roots of (10).

IV. BIFURCATION MECHANISMS OF BURSTING FIRINGS

From the time sequences of three state variables in Figs. 5 and 6, it can be observed that the state variables x and y are two fast variables, whereas the state variable φ is a slow variable. Therefore, the presented mHR neuron model can be divided into two subsystems of the fast-scale subsystem described by the first two equations of (9) and the slow-scale subsystem expressed by the third equation of (9) [50].

A. Fast-slow Bifurcation AnalysisThe equilibrium points for the fast-scale subsystem are

given by , in which is achieved by numerically solving the real root of the following cubic polynomial equation

(10)According to Cardano’s Formula, the solutions of (10) can

be described as

(11)

where , ,

and . For Δ > 0, only a real root exists in (10) and is calculated from the first equation of (11). For Δ = 0, two real roots are in (10) and they are given by the first two equations of (11). Whereas for Δ < 0, there are three real roots

in (10) and they can be expressed by the three equations of (11).

Consider a = 1, b = 3, c = 1, and d = 5 are the specified parameters and take m = 1 and 1.4 as two examples. With respect to the slow variable φ, the real roots in (10) can be illustrated in Fig. 9. It can be concluded that with the increase of the slow variable φ, the sign of Cardan discriminant Δ changes from negative to positive via zero, leading to the fact that the number of the equilibrium points EFS of the fast-scale subsystem has a transition from three to two and to one.

To analyze the stability of the equilibrium points of the fast-scale subsystem, the Jacobian matrix at EFS is deduced as

(12)

Fig. 10. The fold and Hopf bifurcation sets of the fast-scale subsystem with the m and φ evolutions.

The characteristic polynomial is thereby formulated as

(13)

where

.Since the coefficients μ and η of (13) are related to the slow

variable φ nonlinearly, the stability of the fast-scale subsystem is associated with the slow-scale subsystem. Therefore, based on (13), bifurcation behaviors of two fast variables caused by the slow variable φ can be investigated.

At Δ = 0, a small perturbation of the slow variable φ can cause the tangent point of the equilibrium points to disappear or to split into two crossing points, which results in that a fold bifurcation occurs. With Δ = 0, the fold bifurcation set of the fast-scale subsystem can be derived. Of course, another simple method can be used due to the fact that the fold bifurcation involves a zero eigenvalue. In this way, the characteristic polynomial (13) can be rewritten as

(14)i.e., there yields η = 0. Thus, with the condition in (10), the fold bifurcation set can be described as

(15)For the specified parameters a = 1, b = 3, c = 1, and d = 5, the fold bifurcation set can be simplified as

φ

φ = 0.1417

x

φ = 0.1009

m = 1.0

m = 1.4

φm

FB Set

HB Set

F1

F2 H2

*

* ▪

6

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137

(16)The Hopf bifurcation of the fast-scale subsystem is

associated with the existence of a pair of pure imaginary eigenvalues. Then, (13) can be rewritten as

(17)i.e., one gets μ = 0. Thus, with the condition in (10), the Hopf bifurcation set can be achieved as

(18)For the specified parameters a = 1, b = 3, c = 1, and d = 5, the Hopf bifurcation set in the considered region of m (0.4 ≤ m ≤ 1.6) can be determined as

(19)When the electromagnetic induction strength m and slow

variable φ vary in the region [0.4, 1.6] and [–2, 8], respectively,

Fig. 11. Bifurcation mechanisms of chaotic and periodic bursting firings at (a) m = 1 and (b) m = 1.4, where F1 and F2 are the fold bifurcation points, H2

is the Hopf bifurcation point, and EP1 and EP2 are the equilibrium points of the fast-scale subsystem with the φ variation.

the fold bifurcation set of (16) and Hopf bifurcation set of (19) in the φ – m plane are numerically plotted in Fig. 10, which demonstrate that the possible bifurcation route of the fast-scale subsystem takes place as the strength m and slow variable φ evolve gradually.

B. Bifurcation Mechanism of Bursting FiringsAs for m = 1 and 1.4, two examples are used for elaborating

the bifurcation mechanisms of the chaotic and periodic bursting firings in the 3D mHR neuron model.

Referring to Fig. 10, one can see that only one fold bifurcation occurs in the fast-scale subsystem with the φ evolution during a cycle when m = 1. The fold bifurcation

point marked as F1 locates at φ = 0.1417. The phase orbit of the mHR neuron model (9) overlapped with the equilibrium points EP1 of the fast-scale subsystem are depicted in Fig. 11(a). However, when m = 1.4, one fold bifurcation along with one Hopf bifurcation appear simultaneously in the fast-scale subsystem with the φ evolution during a cycle. The fold bifurcation point marked as F2 and Hopf bifurcation point marked as H2 locate at φ = 0.1009 and 1.0613, respectively. Similarly, the phase orbit of the mHR neuron model (9) overlapped with the equilibrium points EP2 of the fast-scale subsystem are drawn in Fig. 11(b).

For the asymmetric chaotic bursting firing in Fig. 11(a), the fold bifurcation occurs at φ = 0.1417. When φ < 0.1417, three equilibrium points appear, in which the equilibrium point colored in blue is an unstable focus, those colored in dark cyan and brown yellow are an unstable saddle and a stable node, respectively. That makes the mHR neuron model (9) goes into a repetitive spiking state via the fold bifurcation and its moving trajectory rises to the upper. And when φ > 0.1417, only an unstable focus colored in blue exists in the fast-scale subsystem

Fig. 12. Operational amplifiers and analog multipliers-based implementation circuit of the mHR neuron model.

and the moving trajectory oscillates around the unstable focus. With the increase of φ, the attractive force of the unstable focus becomes weak, resulting in that the moving trajectory turns back and initiates the next cycle at the fold bifurcation point F1. Hence, in the asymmetric chaotic bursting firing, there is no quiescent state due to the absence of the Hopf bifurcation.

As for the asymmetric periodic bursting firing in Fig. 11(b), the fold bifurcation occurs at φ = 0.1009. When φ < 0.1009, there are three equilibrium points including an unstable focus colored in green, an unstable saddle colored in pink, and a stable node colored in purple. Due to the occurrence of the fold bifurcation, the mHR neuron model (9) goes into a repetitive spiking state and its moving trajectory jumps to the upper. While when φ > 0.1009, the fast-scale subsystem only has an unstable focus colored in green and then the unstable focus changes into a stable node via the Hopf bifurcation point H2 at φ = 1.0613, resulting in the appearance of a quiescent state. The quiescent state ends once the trajectory moves at the fold bifurcation point F2 and a new cycle of the periodic bursting firing emerges again. Therefore, the asymmetric periodic bursting firing starts from the repetitive spiking state to quiescent state and back via the fold and Hopf bifurcations.

The biological neurons have similar bursting behaviors, such as repetitive spiking, quiescence, spiking again, and so on [50]. The results given in Fig. 11 well demonstrate the

φ

x

φ = 0.1417*F1EP1(a)

φ

x

φ = 0.1009*F2

EP2▪

H2

φ = 1.0613

(b)

R1

R2

–+U1

C–+U2

vx

10 kΩ

–vx

10 kΩ

M1

M2

R3

vx

vy

–+U3

C–+U4

vy

10 kΩ

–vy

10 kΩ

R6

VcW(vφ)

R4

R5

M3

7

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137biological properties of bursting firing activities in neuron, reflecting the feasibility of the presented mHR neuron model.

V. HARDWARE BREADBOARD AND CIRCUIT EXPERIMENTS

A nonlinear dynamical system of ordinary differential equations in a polynomial form can be physically implemented by an analog electronic circuit using commercially available discrete components of resistors, capacitors, operational amplifiers, and analog multipliers [27], [35]. An electronic neuron constructed by circuit components can emulate the biological properties of a neuron extremely well [25], [52], [53]. In addition, the circuit implementation of a neuron can effectively accelerate IC design [54], which are salutary to neuron-based engineering applications.

A. Physical Circuit Designs and Parameter SelectionsBased on the circuit implementation of the presented

threshold flux-controlled memristor, the circuit schematic of the new 3D mHR neuron model described by (9) can be designed, as depicted in Fig. 12. Obviously, the implementation circuit equations of the mHR neuron model in Fig. 12 are established as

Fig. 13. Experimentally captured asymmetric chaotic bursting firing with R0 = 1 kΩ. (a) Time sequences of three state variables and (b) phase orbits in the vy

– vx and vφ – vx planes.

Fig. 14. Experimentally captured asymmetric periodic bursting firing with R0

= 0.71 kΩ. (a) Time sequences of three state variables and (b) phase orbits in the vy – vx and vφ – vx planes.

(20)where vx, vy, and vφ are three circuit variables, Vc is an external bias voltage, and g1, g2, and g3 are three control gains of the multipliers M1, M2, and M3, respectively.

Comparing (20) with (9), the linear element parameters in Fig. 12 can be determined by

(21)Suppose that the integral time constant τ = RC = 10 kΩ × 10 nF = 0.1 ms, i.e., R = 10 kΩ and C = 10 nF, and the multiplier gains g = g1 = 0.1 and g2 = g3 = 1. For the aforementioned model parameters, the circuit parameters for the implementation circuit of the mHR neuron model are obtained by calculating (21) as R2 = R5 = R6 = 10 kΩ, R0 = 1 kΩ (adjustable), R1 = 3.33 kΩ, R3 = 1 kΩ, R4 = 2 kΩ, and V c = 1 V.

B. Hardware Breadboard and Circuit ExperimentsUsing the circuit schematics shown in Figs. 1 and 12, a

hardware breadboard with commercially available components is welded. The operational amplifiers AD711JN and analog multipliers AD633JN are selected and supplied by ±15 V voltage modules. The DC voltage V c is provided by an off-the-peg DC power supply and the experimental results are captured by a digital phosphor oscilloscope.

The resistance R0 is used for adjusting the electromagnetic induction strength m. First, R0 is fixed as 1 kΩ, i.e., m = 1. Corresponding to Fig. 5, the time sequences and phase orbits of hidden asymmetric chaotic bursting firing are experimentally captured and shown in Fig. 13. Next, R0 is turned as 0.71 kΩ, i.e., m = 1.4. Corresponding to Fig. 6, the time sequences and phase orbits of hidden asymmetric periodic bursting firing are experimentally captured, as shown in Fig. 14. The experimental results imply that hidden asymmetric bursting firing behaviors also can be captured from the circuit experiments of the mHR neuron model as well, which well validate the numerical simulations.

t, 4 ms/div

v xv y

,vφ

vφ

vy

0.5

V/d

iv2

V/d

iv

(a)

(b)vy, 1 V/div

v x, 6

25 m

V/d

iv

vφ, 1 V/div

v x, 6

25 m

V/d

iv

t, 4 ms/div

v xv y

,vφ

vφ

vy

0.5

V/d

iv2

V/d

iv

(a)

(b)vy, 1 V/div

v x, 0

.5 V

/div

vφ, 0.8 V/div

v x, 0

.5 V

/div

8

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137

VI. CONCLUSION

In this paper, we presented a simple threshold flux-controlled memristor and designed the memristor emulator for exhibiting the threshold effect of electromagnetic induction in biological neurons. Then, its frequency-dependent pinched hysteresis loops were further examined. By replacing the external current with the electromagnetic induction current generated from the designed memristor in two-dimensional HR neuron model, we synthesized a 3D hidden mHR neuron model. Based on numerical simulations of the mathematical models and circuit experiments deployed on hardware breadboards, hidden chaotic and periodic bursting firings of the mHR neuron model were inspected in detail and confirmed commendably. In addition, bifurcation mechanisms of hidden bursting firings are comprehensively expounded by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem with the fast-slow bifurcation analysis. Consequently, the presented 3D hidden mHR neuron model can effectively exhibit complex dynamical behaviors of the electrical activities in biological neurons with threshold electromagnetic induction.

Due to the technical difficulties in fabricating nanometer- size devices, the real memristors with a discrete form are still commercially unavailable nowadays [55]. Thus, the memristor modeling and its emulator designing are very important for investigating complex dynamics of the neuron electrical activities and developing engineering applications of the electrical neural networks [32], [55]–[57] and chaos-based pseudorandom number generator [58]. In near future, we will aim to employ a real memristor to build the neuromorphic circuit of the mHR neuron model.

REFERENCES

[1] L. O. Chua, “Memristor—The missing circuit element,” IEEE Trans. Circuit Theory, vol. CT–18, no. 5, pp. 507–519, 1971.

[2] K. Eshraghian, O. Kavehei, K. R. Cho, J. M. Chappell, A. Iqbal, S. F. Al-Sarawi, and D. Abbott, “Memristive device fundamentals and modeling: Applications to circuits and systems simulation,” Proc. IEEE, vol. 100, no. 6, pp. 1991–2007, 2012.

[3] F. Z. Wang, “A triangular periodic table of elementary circuit elements,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 60, no. 3, pp. 616–623, 2013.

[4] R. Berdan, E. Vasilaki, A. Khiat, G. Indiveri, A. Serb, and T. Prodromakis, “Emulating short-term synaptic dynamics with memristive devices,” Sci. Rep., vol. 6, p. 18639, 2016.

[5] Z. Wang, S. Joshi, S. E. Savel’Ev, H. Jiang, M. Rivu, P. Lin, M. Hu, N. Ge, J. P. Strachan, Z. Li, Q. Wu, M. Barnell, G. L. Li, H. L. Xin, R. S. Williams, Q. Xia, and J. J. Yang, “Memristors with diffusive dynamics as synaptic emulators for neuromorphic computing,” Nat. Mater., vol. 16, no. 1, pp. 101–108, 2017.

[6] M. Prezioso, F. Merrikhbavat, B. D. Hoskins, G. C. Adam, K. K. Likharev, and D. B. Strukov, “Training and operation of an integrated neuromorphic network based on metal-oxide memristor,” Nature, vol. 521, no. 7550, pp. 61–64, 2015.

[7] A. Serb, J. Bill, A. Khiat, R. Berdan, R. Legenstein, and T. Prodromakis, “Unsupervised learning in probabilistic neural networks with multi-state metal-oxide memristive synapses,” Nat. Commun., vol. 7, p. 12611, 2016.

[8] S. Kumar, J. P. Strachan, and R. S. Williams, “Chaotic dynamics in nanoscale NbO2 Mott memristor for analogue computing,” Nature, vol. 548, no. 7667, pp. 318–321, 2017.

[9] V. K. Sangwan, H. S. Lee, H. Bergeron, I. Balla, M. E. Beck, K. S. Chen, and M. C. Hersam, “Multi-terminal memtransistors from

polycrystalline monolayer molybdenum disulfide,” Nature, vol. 554, no. 25747, pp. 500–504, 2018.

[10] M. Lv, C. N. Wang, G. D. Ren, and J. Ma, “Model of electrical activity in a neuron under magnetic flow effect,” Nonlinear Dyn., vol. 85, no. 3, pp. 1479–1490, 2016.

[11] F. Wu, C. Wang, X. Ying, and J. Ma, “Model of electrical activity in cardiac tissue under electromagnetic induction,” Sci. Rep., vol. 6, no. 28, p. 41598, 2016.

[12] J. Ma and J. Tang, “A review for dynamics in neuron and neuronal network,” Nonlinear Dyn., vol. 89, no. 3, pp. 1569–1578, 2017.

[13] Y. Wang, J. Ma, Y. Xu, F. Wu, and P. Zhou, “The electrical activity of neurons subject to electromagnetic induction and Gaussian white noise,” Int. J. Bifurcation Chaos, vol. 27, no. 2, p. 1750030, 2017.

[14] Y. Xu, Y. Jia, M. Y. Ge, L. L. Lu, L. J. Yang, and X. Zhan, “Effects of ion channel blocks on electrical activity of stochastic Hodgkin-Huxley neural network under electromagnetic induction,” Neurocomputing, vol. 283, pp. 196–204, 2018.

[15] G. D. Ren, Y. Xu, and C. N. Wang, “Synchronization behavior of coupled neuron circuits composed of memristors,” Nonlinear Dyn., vol. 88, no. 2, pp. 893–901, 2017.

[16] M. Lv and J. Ma, “Multiple modes of electrical activities in a new neuron model under electromagnetic radiation,” Neurocomputing, vol. 205, pp. 375–381, 2016.

[17] L. Lu, Y. Jia, W. Liu, and L. Yang, “Mixed stimulus-induced mode selection in neural activity driven by high and low frequency current under electromagnetic radiation,” Complexity, vol. 2017, p. 7628537, 2017.

[18] M. Y. Ge, Y. Jia, Y. Xu, and L. J. Yang, “Mode transition in electrical activities of neuron driven by high and low frequency stimulus in the presence of electromagnetic induction and radiation,” Nonlinear Dyn., vol. 91, no. 1, pp. 515–523, 2018.

[19] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” J. Physiol., vol. 117, no. 4, pp. 500–544, 1952.

[20] J. L. Hindmarsh and R. M. Rose, “A model of the nerve impulse using two first-order differential equations,” Nature, vol. 296, no. 5853, pp. 162–164, 1982.

[21] J. L. Hindmarsh and R. M. Rose, “A model of neuronal bursting using three coupled first order differential equations,” Proc. Roy. Soc. London B Biol. Sci., vol. 221, no. 1222, pp. 87–102, 1984.

[22] Y. Yi and X. F. Liao, “Filippov Hindmarsh-Rose neuronal model with threshold policy control,” IEEE Trans. Neural Netw. Learn., vol. 30, no. 1, pp. 306–311, 2019.

[23] C. R. Laing and C. C. Chow, “A spiking neuron model for binocular rivalry,” J. Comput. Neurosci., vol. 12, no. 1, pp. 39–53, 2002.

[24] K. Tsumoto, H. Kitajima, T. Yoshinaga, K. Aihara, and H. Kawakami, “Bifurcations in Morris–Lecar neuron model,” Neurocomputing, vol. 69, no. 4–6, pp. 293–316, 2006.

[25] R. Behdad, S. Binczak, A. S. Dmitrichev, V. I. Nekorkin, and J. M. Bilbault, “Artificial electrical Morris–Lecar neuron,” IEEE Trans. Neural Netw. Learn., vol. 26, no. 9, pp. 1875–1884, 2015.

[26] L. O. Chua, V. Sbitnev, and H. Kim, “Neurons are poised near the edge of chaos,” Int. J. Bifurcation Chaos, vol. 22, no. 4, p. 1250098, 2012.

[27] B. C. Bao, H. Qian, J. Wang, Q. Xu, M. Chen, H. G. Wu, and Y. J. Yu, “Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network,” Nonlinear Dyn., vol. 90, no. 4, pp. 2359–2369, 2017.

[28] S. B. Ding, Z. S. Wang, and H. G. Zhang, “Dissipativity Analysis for stochastic memristive neural networks with time-varying delays: A discrete-time case,” IEEE Trans. Neural Netw. Learn., vol. 29, no. 3, pp. 618–630, 2018.

[29] Y. Sheng, F. L. Lewis, and Z. G. Zeng. “Exponential stabilization of fuzzy memristive neural networks with hybrid unbounded time-varying delays,” IEEE Trans. Neural Netw. Learn., vol. 30, no. 3, pp. 739–750, 2019.

[30] A. Buscarino, C. Corradino, L. Fortuna, M. Frasca, and L. O. Chua, “Turing patterns in memristive cellular nonlinear networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol.63, no. 8, pp. 1222–1230, 2016.

[31] E. Bilotta, P. Pantano, and S. Vena, “Speeding up cellular neural network processing ability by embodying memristors,” IEEE Trans. Neural Netw. Learn., vol. 28, no. 5, pp. 1228–1232, 2017.

[32] X. F. Hu, G. Feng, S. K. Duan, and L. Liu, “A memristive multilayer cellular neural network with applications to image processing,” IEEE Trans. Neural Netw. Learn., vol. 28, no. 8, pp. 1889–1901, 2017.

9

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137[33] A. Buscarino, L. Fortuna, M. Frasca, L. V. Gambuzza, and G. Sciuto,

“Memristive chaotic circuits based on cellular nonlinear networks,” Int. J. Bifurcation Chaos, vol. 22, no. 3, p. 1250070, 2012.

[34] L. V. Gambuzza, L. Fortuna, M. Frasca, and E. Gale, “Experimental evidence of chaos from memristors,” Int. J. Bifurcation Chaos, vol. 25, no. 8, p. 1550101, 2015.

[35] B. C. Bao, H. Bao, N. Wang, M. Chen, and Q. Xu, “Hidden extreme multistability in memristive hyperchaotic system,” Chaos, Solitons Fractals, vol. 94, pp. 102–111, 2017.

[36] B. C. Bao, A. H. Hu, H. Bao, Q. Xu, M. Chen, and H. G. Wu, “Three-dimensional memristive Hindmarsh-Rose neuron model with hidden coexisting asymmetric behaviors,” Complexity, vol. 2018, p. 3872573, 2018.

[37] H. Kim, M. P. Sah, C. Yang, S. Cho, and L. O. Chua, “Memristor emulator for memristor circuit applications,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 59, no. 10, pp. 2422–2431, 2012.

[38] X. Y. Hu, C. X. Liu, L. Liu, J. K. Ni, and S. L. Li, “An electronic implementation for Morris Lecar neuron model,” Nonlinear Dyn., vol. 84, no. 4, pp. 2317–2332, 2016.

[39] S. K. Duan and X. F. Liao, “An electronic implementation for Liao’s chaotic delayed neuron model with non-monotonous activation function,” Phys. Lett. A, vol. 369, no. 1, pp. 37–43, 2007.

[40] B. C. Bao, H. Qian, Q. Xu, M. Chen, J.Wang, and Y. J. Yu, “Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network,” Front. Comput. Neurosci., vol. 11, no. 81, pp. 1–14, 2017.

[41] E. Kaslik, “Analysis of two- and three-dimensional fractional-order Hindmarsh-Rose type neuronal models,” Fract. Calc. Appl. Anal., vol. 20, no. 3, pp. 623–645, 2017.

[42] M. Storace, D. Linaro, and L. E. De, “The Hindmarsh-Rose neuron model: Bifurcation analysis and piecewise-linear approximations,” Chaos, vol. 18, no. 3, p. 033128, 2008.

[43] S. R. D. Djeundam, R. Yamapi, T. C. Kofane, and M. A. Azizalaoui, “Deterministic and stochastic bifurcations in the Hindmarsh-Rose neuronal model,” Chaos, vol. 23, no. 3, p. 033125, 2013.

[44] G. Innocenti and R. Genesio, “On the dynamics of chaotic spiking- bursting transition in the Hindmarsh-Rose neuron,” Chaos, vol. 19, no. 2, p. 023124, 2009.

[45] S. Lakshmanan, C. P. Lim, S. Nahavandi, M. Prakash, and P. Balasubramaniam, “Dynamical analysis of the Hindmarsh-Rose neuron with time delays,” IEEE Trans. Neural Netw. Learn., vol. 28, no. 8, pp. 1953–1958, 2017.

[46] S. Jafari, J. C. Sprott, and S. M. R. H. Golpayegani, “Elementary quadric chaotic flows with no equilibria,” Phys. Lett. A, vol. 377, pp. 699–702, 2013.

[47] V. T. Pham, C. Volos, S. Jafari, Z. C. Wei, and X. Wang, “Constructing a novel no-equilibrium chaotic system,” Int. J. Bifurcation Chaos, vol. 24, no. 5, p. 1450073, 2014.

[48] F. R. Tahir, S. Jafari, V. T. Pham, C. Volos, and X. Wang, “A novel no-equilibrium chaotic system with multiwing butterfly attractors,” Int. J. Bifurcation Chaos, vol. 25, no. 4, p. 1550056, 2015.

[49] A. P. Kuznetsov, S. P. Kuznetsov, E. Mosekilde, and N. V. Stankevich, “Co-existing hidden attractors in a radio-physical oscillator system,” J. Phys. A, Math. Theor., vol. 48, no. 12, p. 125101, 2015.

[50] E. M. Izhikevich, “Neural excitability, spiking and bursting,” Int. J. Bifurcation Chaos, vol. 10, no. 6, pp. 1171–1266, 2000.

[51] P. C. Rech, “Period-adding and spiral organization of the periodicity in a Hopfield neural network,” Int. J. Mach. Learn. Cybern., vol. 6, no. 1, pp. 1–6, 2015.

[52] R. D. Pinto, P. Varona, A. R. Volkovskii, A. Szücs, H. D. Abarbanel, and M. I. Rabinovich, “Synchronous behavior of two coupled electronic neurons,” Phys. Rev. E, vol. 62, no. 2, pp. 2644–2656, 2000.

[53] L. E. De and M. Hasler, “Oscillations and oscillatory behavior in small neural circuits,” Biol. Cybern., vol. 95, no. 6, pp. 537–554, 2006.

[54] N. Dahasert, I. Öztürk, and R. Kiliç, “Experimental realizations of the HR neuron model with programmable hardware and synchronization applications,” Nonlinear Dyn., vol. 70, no. 4, pp. 2343–2358, 2012.

[55] V. Ntinas, I. Vourkas, A. Abusleme, G. C. Sirakoulis, and A. Rubio, “Experimental study of artificial neural networks using a digital memristor simulator,” IEEE Trans. Neural Netw. Learn., vol. 29, no. 10, pp. 5098–5110, 2018.

[56] Y. J. Zheng, S. X. Li, R. Yan, H. J. Tang, and K. C. Tan, “Sparse temporal encoding of visual features for robust object recognition by

spiking neurons,” IEEE Trans. Neural Netw. Learn., vol. 27, no. 11, pp. 2327–2336, 2016.

[57] P. M. Sheridan, C. Du, and W. D. Lu, “Feature extraction using memristor networks,” IEEE Trans. Neural Netw. Learn., vol. 27, no. 11, pp. 2327–2336, 2016.

[58] C. Q. Li, D. D. Lin, J. H. Lü, F. Hao, “Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography,” IEEE MultiMedia, vol. 25, no. 4, pp. 46–56, 2018.

Han Bao received the B.S. degree from Finance and Economics University of Jiangxi, China, in 2015 and the M.S. degree from Changzhou University, China in 2018. He is currently pursuing the Ph.D. degree in nonlinear system analysis and computer measurement technology at Nanjing University of Aeronautics and Astronautics, China. His research interest includes memristive neuromorphic

circuit, computer science, and artificial intelligence.

Aihuang Hu received the B.S. degree in computer science and technology from Wenzheng College of Soochow University, Suzhou, China, in 2016. She is currently pursuing the M.S. degree in computer application technique at Changzhou University, China. Her research interest includes memristor-based neuron models, memristive chaotic circuits, and circuit

implementation.

Wenbo Liu received the B.S. and M.S. degrees in electronic engineering from the Harbin Shipbuilding Engineering Institute, Harbin, China, in 1990 and 1993, respectively. She obtained the Ph.D. degree from the Department of Measurement and Testing, Nanjing University of Aeronautics and Astronautics, Nanjing, China in 2001.

Since 1993, she has been working at the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China, where she was promoted to full professor in 2006. Her research interests include control and measurement technology, signal processing and information processing methods, and nonlinear system analysis and design.

10

IEEE Transactions on Neural Networks and Learning Systemshttps://doi.org/10.1109/TNNLS.2019.2905137

Bocheng Bao received the B.S. and M.S. degrees in electronic engineering from the University of Electronics Science and Technology of China, Chengdu, China, in 1986 and 1989, respectively. He obtained the Ph.D. degree from the Department of Electronic Engineering, Nanjing University of Science and Technology, Nanjing, China, in 2010.

He has more than 20 years’ experience in industry and worked in several enterprises as Senior Engineer and General Manager. From June 2008 to January 2011, he was a Professor in the School of Electrical and Information Engineering, Jiangsu University of Technology, Changzhou, China. Then he works in the School of Information Science and Engineering, Changzhou University, Changzhou, China as a full professor. From June 2013 to December 2013, he visited the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. His research interests include bifurcation and chaos, analysis and simulation in neuromorphic circuits, power electronic circuits, and nonlinear circuits and systems.

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