Neuromorphic Oscillators and Electronic Implementation of One Novel MFNN Model

Exploring the equivalent mathematical and physical representation of biological neurons is one of the most essential tasks for building the complex neural systems and networks. In this paper, we focus on constructing a novel memristive FitzHugh-Nagumo Neuron (MFNN) model to simulate the nonlinear oscillation behaviors and neuromorphic dynamics of some biological neurons. Then, the relationship between ranges of circuit parameter values and pinched hysteresis loop are presented. Moreover, its behaviors and neuromorphic nonlinear dynamics are analyzed and several essential curves are depicted, such as phase portraits, time-domain waveforms, Lyapunov exponent spectrums, Poincare maps, the bifurcation diagrams, and so on. Furthermore, the most important characteristics are the generation of memristive oscillators and bursting firing phenomenon. There is no doubt that the good agreement among theoretical analysis, simulation and experimental results verifies the practicability and flexibility of the configured MFNN model.


I. INTRODUCTION
Neuron model, as the basic unit of the neural network, its electronic properties are closely associated with information processing and propagation. In order to imitate the complex and rich nonlinear behaviors for one real biological neuron, studying its equivalent physical and mathematical models could be considered as the necessary step to explore the intelligent neuron network, even the next generation of neuromorphic devices. In the past few decades, relevant topics have become popular and published plenty of literature [1], [2], [3], [4]. On one hand, in the field of neural networks, the Hodgkin-Huxley (HH) model as one of the cornerstones of modern neuroscience was first published in 1952, which has provided useful research tools for explaining the mechanism of the nervous system and information communication. Subsequently, as a simplified 2D version of HH model, The associate editor coordinating the review of this manuscript and approving it for publication was Teerachot Siriburanon .
FitzHugh-Nagumo Neuron (FNN) model and its variants have been proposed and commonly used to simulate a detailed manner activation and deactivation dynamics of a spiking neuron, which range from Josephson junction to chemical reactions to cardiac and nerve cells [5], [6], [7], [8], [9], [10], [11], [12], [13]. Moreover, in 2017, A. Saha and U. Feudel studied two identical FNN oscillators which are coupled with one or two different delays and their transient chaotic dynamics were sandwiched between a bubbling transition and a blowout bifurcation [5]. In 2018, Z. M. Wu and his coworkers investigated bipartite networks of nonlocally coupled FNN oscillators, reported the existence of chimera states in bipartite networks [6]. In the same year, C. N. Takembo and his fellows obtained the analytical solutions describing localized nonlinear excitations in an improved diffusive FitzHugh-Nagumo cardiac tissue model [7]. Furthermore, integer-and fractional-order FNN models were successfully realized based on the low-voltage and low-power integrable CMOS circuits [8] or multiplier less digital platform [9]. Also, nonlinear behaviors, neuromorphic dynamics [10], [11], [12] and complex synchronization [13] were reported and references therein. On the other hand, in the field of nonlinear circuit theory, memristor and memristive devices were the hypothesis in 1971 by L. O. Chua and fabricated in 2008 by a group of researchers at the Hewlett-Packard laboratory [14], [26], which have attracted the research community and sparked enormous interests of scientists and engineers across many research areas, both theoretical and applied. In 2011, the novel characteristics on 'crossing-type pinched hysteretic loops' for the ideal memristors, memcapacitors and meminductors were published and proved [31], [32]. Subsequently, the autonomous implicit models yielding self-crossing trajectories (hysteresis with odd symmetry), typical for memory elements have also been presented [33]. Up to now, these theories revealed the features of the ideal memory elements. Besides, in 2012, a rigorous and comprehensive theoretic foundation for the memristive HH axon circuit was presented and demonstrated [15]. Correspondingly, the applications in the fields of memristor crossbar-based neuromorphic computing systems [16], synaptic circuits [17], [18], neural networks [19] and memory computing [20] have been expanded, just to name a few.
Nevertheless, plenty of research have been published on both the FNN models and their oscillators, there are still several shortcomings worthy to further research: i) Most existing works focus on the traditional circuit models but few involve mathematical representations and practical equivalent electric circuits based on memristive device(s). Therefore, the related research is still in its infancy; ii) Due to only few solidstate implemented memristors being commercially available, the emerging memristor emulators are considered as one of the most promising candidates to meet the requirements for mimicking the neuron cells. On this basis, some ideal memristor emulators were proposed [21] and the digitally implemented electronic circuits were designed based on field programmable gate arrays (FPGA) [22]. However, the roles of the generic and extended memristor models were ignored. Also, the 'crossing-type pinched hysteretic loops' and 'selfcrossing trajectories' for the ideal memory elements were analyzed in detail. But there is still an issue whether they could be available for the generalized and extended memory elements. Therefore, these previous conclusions are incomplete. In this paper, we devote more energy to supplementing the theories and seeking the way to configure the other types of memristor, which are more suitable for constructing the classical FFN model and exhibiting its complex phenomena; iii) Although plenty of the independent/universal emulators have been published. Most of them were designed with many electronic components and integrated chips [23], [24], [25], [26], [27], which could cause more and more uncertainty because of the inherent characteristics with the hardware devices. Therefore, simpler schematics, fewer components, and more targeted equivalent models are still research hotpots; iv) The FNN model with chaotic oscillators existed [22], which was a digitally implemented electronic circuit. However, a few analog implementations have not been found yet; v) When one memristor is utilized to mimic a FNN model, the dynamics of the oscillators and oscillation behaviors are still worthy to be further explored. Based on the above description, one equivalent MFNN circuit with commercially available analog components is introduced and analyzed as well as the bursting firings, neuromorphic oscillators, and experimental verification.
With these aims in mind, the paper is organized as follows. In Section II, the novel extended voltage controlled memristor is introduced. Then, the generation conditions and features for the nonlinear oscillators are analyzed. In Section III, the MFFN model is proposed. Also, the dynamical behaviors, bursting firings and bifurcation mechanisms are analyzed in detail, respectively. In Section IV, the simulation and verification are demonstrated. Finally, the paper is summarized in Section V.

II. IMPLEMENTATION AND MECHANISM ANALYSIS OF THE NOVEL MEMRISTOR
In 2019, an oscillating circuit with a full-wave memristive rectifier was presented [34], [35], which includes a memristive circuit with 5 elements (4 diodes + 1 inductor). Up to now, its complex oscillations and two-parameter bifurcation behaviors have been found. As an active memristor, it has been used to build a simple third-order memristive diode bridge coupled with Sallen-Key lowpass filter [34]. However, the fields of the full-wave bridge are too limited to utilize in plenty of electronic circuit-engineering. Thus, one problem should be attention, that is, if any other oscillation circuits could be built and exhibit the similar nonlinear behaviors for some circuit systems? In this section, a novel memristive emulator is presented and analyzed, which could answer the above question.
The schematic diagram of the proposed emulator in Figure 1, which is composed of five electronic components: two parallel diodes (D 1 and D 2 ), one resistor (R 0 ), one capacitor (C 0 ), and one inductor (L 0 ). Two terminals (A and B) can be thought of as the interface to external circuits. When a periodic voltage (v M ) across is applied, a current flow (i M ) can be obtained; The potential difference across R 0 is denoted The voltage across (v C0 ) and the current flowing (i L0 ) are the variables of C 0 and L 0 ; Denoting v D1 (or v D2 ) and i D1 (or i D2 ) represent the voltage across and the current flowing the diodes, respectively.

A. THE MODEL OF SECOND-ORDER MEMRISTOR
From Figure1, the proposed memristor is easy to implement by off-the shelf components. Then, the modeling process is described in this subsection. Based on the constitutive relationship of the diode, which is the key factor for the oscillation mechanism. It can be characterized as follows  where I S and V T stand for the reverse saturation current and thermal voltage; n represents the emission coefficient.
The i Dk (k = 1, 2) represents the current flow of the diode. According to Kirchhoff's Current Law, the derivation process is illustrated as following Therefore, the following model can be given that According to the definition of the extended memristor in Reference [14], both the current flow (i M ) and the memristance (R M ) can be characterized as follows From Eq. (4), its mathematical model could exhibit the wealthy dynamics, including the neuromorphic oscillators, bursting phenomenon, etc. These characteristics express that the proposed memristor is suitable to configure the electronic neural unit, such as the FNN model.

B. PINCHED HYSTERESIS LOOPS
The fingerprints and frequencies are tested by the software Multisim 14.0 Version, which is industry standard SPICE simulation and circuit design software for analog, digital, and power electronics in education and research. Considering the following parameters are: I S = 6.8913nA, V T = 25mV, n = 1.8268, a periodic excitation sinusoidal input signal u M = U m sin(2πft), where U m = 1V is the amplitude, f is the repeating frequency. The components are shown in Table 1. And, the typical pinched hysteretic loops appear in the v M -i M co-ordinates. in Figure 2.
The time-domain waveforms, fingerprints and trajectory in the v-i plane are drawn in Figure2. Hereby, the CH1 and CH2 are the abbreviations for the channel one and channel two of the oscilloscope.
From Table 1, let us conclude that both components (R 0 , C 0 , L 0 ) and amplitude of the voltage (U m ) could impact

FIGURE 2. Nonlinear circuit based on the proposed memristor with
the existence of the pinched hysteresis loops and the frequency ranges of the proposed memristor: i) When U m increases, we could obtain a narrower frequency range, and vice versa; ii) When C 0 , L 0 , and U m are fixed and R 0 increases, we could obtain a wider frequency range, and vice versa; iii) When C 0 , R 0 , and U m are fixed and L 0 increases, a wider frequency range can be observed, and vice versa. Notably, all the testing results obtained in Table 1 come from the simulation via Multisim 14.0 instead of a circuit lab for reference only. Hereby, the fingerprint of the generalized memristive emulator and the phase portraits of the memristive Sallen-Key lowpass filter in [35] are captured in Figure 3. The following parameters are selected as U m =1V, L=50mF, and four 1N4148 diodes.
Observed between Figure 2(b) and 3(a), both fingerprints could emerge. Also, the same simulation results could be given in Figure 3(b) and (c). Compared with the works in [35], the proposed emulator has more flexible nonlinear behaviors.

C. THE OSCILLATOR WITH RC NETWORK
In this subsection, our task is to present the periodic oscillation phenomenon and nonlinear behaviors via a RC network Therefore, one RC oscillator circuit is designed, as shown in Figure 4. The mathematical model of Figure 2 is derived as following By substituting (1) and (4) into (5), setting v M = x, v C0 = y, and i L0 = z are the state variables.
Also, throughout the paper we fix the part of parameters as a = 1/(R 1 C 1 ), β=1/L 0 , d=2I S /C 1 , k 1 =2I S /C 0 , k 2 =2I S R 0 , u s (t) = U m cos(2πft), system (5) can be rewritten as follows Obviously, two equilibrium points (E ± ) in the system (6) at zero could be got, which are computed as E ± (∓aU m , 0, 0). The Jacobian matrix J evaluated as following The characteristic equation of J is By the Routh-Hurwitz criterion and the dissipation, the following conditions are satisfied [29,30]: The characteristic values = {λ 1 , λ 2 , λ 3 } of the Eq. (8) have one negative real root (λ 3 =kab) and one pair of complex conjugate roots with no real part (λ 1,2 = ±i). To ensure that the equilibrium points are two hyperbolic saddle points (or simple, saddle points).
From Figure 5, the oscillators, bursting and periodic oscillation phenomenon can occur quickly.

III. MEMRISTIVE FITZHUGH-NAGUMO NEURON MODEL
As a priority in improving the accuracy for the proposed MFFN, its circuit schematic diagram is constructed in Figure 6, which can be implemented by one capacitor, two resistors, one inductor, and the proposed memristor.
From Figure 6, the I EX represents the current flow of the extracellular medium. The memristor R M is linked across A and B. The voltage source (u s ) represents the electro-chemical gradients driving the flow of ions. Denoting the current flows (i s , i L1 , i C1 and i M ) stand for the physical variables of R 1 , L 1 , C 1 and R M .

A. THE MATHEMATICAL REPRESENTATION OF MFNN MODEL
The mathematical model can be shown as follows.
By substituting (1) and (4) into (10), setting v C1 = x, L 1 i L1 = y, v C0 = z, and L 0 i L0 = w are the state variables. Also, the other parameters are fixed as following: ρi M = i, a=1/(R 1 C 1 ), b = R 2 /L 1 d=2I S /C 1 , e=1/(L 0 C 1 ), f=1/C 0 , k 1 = 2I S /C 0 , k 2 =2I S R 0 . After affine transformation of system (10) is shown as follows Obviously, two equilibrium points of system (11) are E + (−aU m , 0, 0, 0) and E − (aU m , 0, 0, 0), respectively. The character-istic equation is list as follows det(λI − J ) The four eigenvalues are resolved as in (13), shown at the bottom of the page. The Eq. (12) is composed of the product of two quadratic polynomials. According to the eigenvalues in Eq. (13), Routh-Hurwitz criterion and the dissipation, the following conditions are satisfied as: When the above relationships are satisfied, we could obtain one pair of negative real roots (λ 3,4 ) and one pair of complex conjugate roots with a positive real part (λ 1,2 ). Also, the following conclusion could be got: i) Both the equilibrium points are hyperbolic saddle points (or simply, saddle points); ii) According to the nature of the excitation voltage source (u S ), the location and the number of equilibrium points can be determined. In other words, when a DC source (such as E or −E) is applied, only one equilibrium point can be occurred (i.e., E + or E − ); While, when a bipolar periodic voltage response is applied, two equilibrium points (i.e., E + and E − ) could occur quickly; iii) Impact of the equilibrium points on both the location and the number of strange attractors. Intuitively, when a=−0.8, b =0.0667, d=k=3.6 × 10 −4 , u S (t)=10sin(2πft), f=400Hz, the plots of the equilibria, strange attractors, and time-domain waveforms of the MFNN model are illustrated in Figure 7.
From Figure 7, two equilibrium points could be obtained, that is, E + (-10,0,0,0) and E − (10,0,0,0). Also, the doubleattractor and the periodic bursting oscillation phenomenon are exhibited quickly. Likewise, if the input signal is a DC source (U + =10V or U − =-10V), the equilibria, strange attractors and time-domain waveforms are depicted in Figure 8. As Figure 8 presented, the single equilibrium point is obtained. Precisely, when U + =10V is applied, the single attractor can be clearly observed, which locate at E + (−10, 0,0,0); Likewise, when U − = −10V is applied, the single attractor could occur and locate at E − (10,0,0,0). Different types of strange attractors occur, and different bursting phenomena could be observed. Specifically, the following key conclusions are discovery, which can highlight the unique nonlinear dynamics of the configured MFNN model: i) Both the AC and DC excitation response could induce the nonlinear oscillation for the MFNN model; ii) The locations of the equilibrium point are determined by the amplitude and frequency of the input excitation. Compared with the other neural model, the designed MFNN model is more suitable to build the neural networks or complex systems with the characteristics of bursting excited, association, long and short-term memory, etc. Meanwhile, the other nonlinear properties are analyzed as the following subsection.

B. DYNAMICS BEHAVIORS
As we all know, the Lyapunov exponent (LE) is the first important factor to directly demonstrate the nonlinear behaviors. Now, LEs are depicted by Wolf's method in Figure 9. Also, the Hausdroff (Lyapunov) dimension is calculated as When an excitation voltage u S (t)=10sin(0.02t) is applied, one poincare map can be drawn, which manifests the orbits of the oscillators by a series of isolated points (See Figure 9). Both Figure 7 and Figure 10, there exist two timescales of symmetric chaotic double-oscillator for two variables u c1 and u c0 , which could cause the division of two subsystems, named a fast-spiking subsystem and a slow modulation subsystem. Furthermore, from Figure 7 (c), When the conditions Eq. (14) are satisfied, the proposed MFNN model (11) manifests a repetitive spiking state via alternately moving trajectory between two equilibrium regions, which is also moving around only one slow modulation (unstable focus subsystem) region. Similar bursting behaviors have appeared in the biological neuron, such as repetitive spiking -quiescence -spiking, back and forth. Therefore, the presented phenomenon in both Figure 7 and Figure 10 could demonstrate the existence of biological characteristics of the periodic bursting firing activities in the MFNN model, which also reflects the feasibility to design the neuron network models.

C. BIFURCATION MECHANISMS
Next, as the other essential factor to clearly manifest the neuromorphic dynamics, i.e., bifurcation, for the MFNN model should be discussed in this subsection.
From the previous analysis, we know that the existence of chaotic oscillation is the key point to analyze the nonlinear dynamics. Supposing the other parameter values, such as the amplitude and frequency of the external forcing current, are  fixed, the bifurcation diagrams of a and b are depicted in Figure 11.
Observed from Figure 11 (a) that the region of parameter a can be divided into three subregions, as following: i) For the first part a [−1, -0.8283], the double-oscillator, bursting firing behaviors exist, but two clear fast-spiking regions and an unclear slow modulation region (see Figure 12(a) and (b)); ii) For the second part a (−0.8283, −0.5354], the double-oscillator, bursting firing behaviors exist, also two clear fast-spiking regions and a slow modulation region (see Figure 12(c) and (d)); iii) For the last part a (−0.5354, 0), both memristive oscillator and bursting firing behaviors disappear (see Figure 12(e) and (f)). The phase portraits in u C1 -u C0 and the time-domain waveforms of u C1 (t) are depicted in Figure 12.
Similarly, from Figure 11 (b), the region of parameter b could also be divided into three parts, as following: i) For the first part b [0,0.7576]U (4.495,5], the double-oscillator, bursting firing, two clear fast-spiking regions and an unclear slow modulation region are obtained, which is similar to Figure 12 (2.374, 4.495] the similar dynamics behaviors exist with the phenomenon in Figure 12 (c) and (d)), including the double-oscillator, bursting firing, fast-spiking regions and modulation region.

IV. SIMULATION VERIFICATION
With the circuit schematics in Figure1 and Figure 6, the proposed MFNN model can be implemented by several offthe-shelf components, the following experiments are tested. The circuit parameters are chosen as: f=400Hz, R 0 =1k ,  C 0 = C 1 =10nF, L 0 = L 1 =50mH, R 1 =-3k , and R 2 =5k . When the applied sinusoidal voltage source is u(t) = U m sin(2πft), the amplitude is U m =1V, the phase portraits and time-domain waveforms are captured as Figure 13.
While the voltage signal is U + = 1V (or U − = −1V), the curves are depicted as Figure 14.
All the experimental measurements are consistent with the theoretical analysis of Figure 7, Figure 8, and Figure 12. Meanwhile, the results of the different types of voltage response could lead to the different dynamic behaviors, including the position and the equilibrium points, neuromorphic oscillators, two fast-spiking regions and the only slow modulation region, and so on. According to the accuracy of the circuit parameters and the performances of the inductor, there little differences could occur.

V. CONCLUSION
In order to explore one effective method to imitate the busting firings, spiking oscillation, and construct the electrical circuit of one FFN neuron model with the memristor. Firstly, a simplest second-order extended memristor is introduced. It consists of only five electronic components. Then, based on the constitutive relationship of the diode, the expression of memristance R M is derived. Secondly, the characteristics of a designed memristor are presented in detail, such as the rules of its parameter values, the relationships between the pinched hysteresis loop and frequency bandwidth. Thirdly, the nonlinear RC network is configured to manifest the feature of memristive oscillators and conditions. Finally, the parameter conditions, phase portraits, time-domain, Lyapunov exponent spectrums, Poincare map and the bifurcation diagrams are depicted in detail. Based on these curves, the bursting firings, spiking, and bifurcation behaviors are illustrated. Specifically, for the MFNN model, we demonstrated the neuromorphic oscillation phenomenon can be induced by the periodic AC response. Also, the nonlinear behaviors consistently exist when this excitation disappears. Furthermore, the theoretical analysis is verified by the experimental results, which provide valid experimental data to support the practicability and flexibility of the MFNN model in biological neural network models.
As one of the important classical neural models, the proposed MFNN might be considered as the key point to build the synapses, some complex neural networks, and so on. Moreover, in the view of applications of the MFNN model, such as coupling synchronization, bipartite networks, magnetic flow effect, Low-voltage low-power integrable CMOS circuit implementation, and so on, the following research would become the focus of further research. Also, based on the previous research of our group, the other memristive systems could be introduced and depth analysis in order to explore their applications on the novel biological neuron, long-/short memory model, brain-like neural network, etc. He currently works as a Minister with Aerospace Times Feihong Technology Company Ltd., engaged in the research and development for many key scientific technologies and preresearch projects, and his work was supported by many funds. His current research interests include UAV overall design, research and development project management, and research and development technology management. He has authored over 20 papers in these fields.
YUHAN QIAN received the B.S. degree in automation engineering from Jilin University, Jilin, China, in 2017, and the M.S. degree in control engineering from the University of Science and Technology Beijing, Beijing, China, in 2020.
In 2020, he joined Aerospace Times Feihong Technology Company Ltd., as a Designer. His research interests include object detection, object tracking, object localization, embedded systems, and image processing. He has authored over 20 papers in these areas.
XINSHI LIU received the master's degree in control engineering from the University of Science and Technology Beijing, Beijing, China, in 2020.
He currently works as a Designer with Aerospace Times Feihong Technology Company Ltd., engaged in the research and development for many key scientific technologies and preresearch projects, and his work was supported by a number of funds. His current research interests include intelligent perception for unmanned systems, multi-agent cooperative control, and intelligent sensing and detection. He has authored over ten papers in these fields.
SHUAISHUAI WANG received the bachelor's degree in aircraft design and engineering from the Beijing University of Aeronautics and Astronautics, in 2014, and the master's degree in aerospace science and technology from the National University of Defense Technology, in 2017. He is currently an Engineer with Aerospace Times Feihong Technology Company Ltd. His main research interests include dynamics analysis, guidance law design, and advanced control methods of unmanned aerial vehicles. He has authored over ten papers in these fields.