1. Introduction
The memristor is the fourth basic circuit element after resistors, capacitors, and inductors. Due to its unique nonlinear memory characteristics and the potential for nano-scale integration, it has sparked a widespread research craze in nonlinear circuits, chaotic systems, brain-inspired computing, and artificial intelligence. The theory of the memristor was first predicted by Professor Chua [1] in 1971. It was not until 2008 when Hewlett-Packard Laboratories achieved a physical prototype in TiO2 material [2] that its dynamic resistance-switching behavior showed similarities to the biomimicry of biological synapses, providing a revolutionary carrier for the next generation of information processing technologies.
The nonlinear characteristic of memristors gives them unique advantages in chaotic circuit design. For example, a memristor chaotic circuit [3] is designed by applying a flux-controlled memristor to replace the nonlinear resistor of the Chua circuit. Bao et al [4] proposed a memristor chaotic circuit based on an active flux-controlled memristor with a smooth, continuous, cubic nonlinearity. Wu and Wang [5] introduced a new memristor for building a simple novel memristor chaotic circuit. Ma et al [6] constructed a memristor chaotic circuit by using a memristor, a memcapacitor, and an inductor. Jin et al [7] developed a locally active memristor from a voltage-controlled memristor for building a simple memristor chaotic circuit. Guo et al [8] built an autonomous memristor chaotic circuit based on a nonlinear current-controlled memristor. These memristor chaotic circuits are obtained by using a memristor and other electrical elements, and they can generate rich nonlinear dynamics. In fact, there are also other chaotic circuits based on memristors. Such as multistability memristive chaotic circuits [9, 10], memristor chaotic circuits with hidden attractors [11–13], memristor chaotic circuits with coexisting attractors [14–16]. In addition, memristor chaotic systems are obtained by adding one or two memristors to a chaotic system. For instance, multi-stability memristive hyperchaotic systems [17, 18], memristive chaotic systems with hidden attractors [19, 20], memristor chaotic systems with coexisting attractors [21–23]. Moreover, memristive chaotic systems are implemented by using the analog circuit hardware [24–26] and digital circuit hardware [27–29], and they are also used in pseudo-random sequence generators [30–32] and image encryption algorithms [32–35].
Recently, memristors are widely used in neuroscience. For instance, magnetic flux-controlled memristor neural circuits [36, 37], charge-controlled neural circuit [38], dual memristors neural circuits [39, 40], neural circuits with memristive membrane [41–43], neural circuits with memristor ion channel [44–46]. These neural circuits are built from memristor-coupled nonlinear circuits. In addition, some memristor neurons are obtained by introducing one or two memristors into a neuron model presented by differential equations, for instance, HR memristive neuron [47], FHN memristive neuron [48], and memristor-coupled neuron systems [49, 50]. Furthermore, memristor neuron models are also implemented by using the analog circuit hardware [51, 52] and digital circuit hardware [53], and they are also used in pseudo-random sequence generators [54] and image encryption algorithms [55]. Moreover, there are many memristor neural networks [56–58]. For more research on memristors in neurons and networks, readers can refer to the reviews [59, 60]. In addition, reservoir computing based on the nonlinear dynamical properties of memristors in the temporal information processing and physical computing are researched in [61, 62].
Discrete systems have lower computational complexity and more diverse dynamic behaviors compared to continuous systems. Chaos is particularly typical in discrete maps. Analyzing the dynamics of discrete chaotic systems not only reveals the universal path of nonlinear systems moving from order to disorder, but also provides a rigorous mathematical framework for understanding the unpredictability of complex systems. Additionally, constructing neuron models in discrete time can significantly improve computational efficiency while retaining the key discharge characteristics of biological neurons. This discretization processing breaks through the step size limitations and hardware implementation bottlenecks of traditional continuous differential equations, making it possible to simulate large-scale neural networks. The discrete memristor is not a new type of device that has different physical properties from continuous-time memristors. Instead, it refers to a method or model for implementing the function and behavior of a memristor in a discrete-time system. The core idea is to use discrete-time iterative operations to simulate the continuous and history-dependent resistance state changes of a memristor. A discrete memristor has significant advantages in enhancing system performance, increasing dynamic complexity, improving computational efficiency, and achieving neuro-like functions. Therefore, discrete memristor chaotic systems and discrete memristor neurons are widely researched in recent years. For example, 2-dimensional discrete memristor chaotic system [63], 3-dimensional discrete memristor chaotic system [64], high-dimensional discrete memristor chaotic system [65], hyperchaotic discrete memristive systems [66–68]. In addition, discrete memristive chaotic systems are implemented based on digital circuit hardware [69–71], and they are used in image encryption algorithms [72–74]. Furthermore, there are many discrete memristor neurons proposed, such as the 2-dimensional memristive Chialvo neuron map [75], the diffusively delay-coupled memristive Chialvo neuron map [76], and the electrochemically coupled memristive Rulkov neuron map [77]. Moreover, there are many other memristive neuron maps [78–80] and memristive map neuron networks [81–83].
Metabolic energy maintains the vitality of biological neurons, while physical energy, named neuron energy, clarifies the dynamical mechanism for mode transition in electrical activities, and synchronization stability between neurons, formation of defects, and heterogeneity in the neural networks. Most of the neurons can simulate the electrical activities in many equivalent neural circuits, which are composed of capacitors, inductors, memristors, resistors, and nonlinear resistors. The field energy in the neural circuit is saved in the capacitive, inductive, and memristive elements. By applying a scale transformation on the physical variables in the nonlinear equations and field energy for the neural circuit, a dimensionless neuron model and Hamilton energy will be obtained for further nonlinear analysis. The energy function of the memristive map is defined in [84, 85]. Therefore, this review primarily introduces the discrete memristive models developed from the memristor-coupled nonlinear circuits and the verification of the chaotic maps expressed based on the mathematical iterative equations by applying the memristor-coupled nonlinear circuit. As is well known, a nonlinear circuit can be built by using a nonlinear electrical element and other electrical components. Memristor is a nonlinear electrical element, and it reveals the interrelationship between magnetic flux (φ) and charge (q). Figure 1 plotted the relationship diagram of the voltage (V), current (i), magnetic flux (φ), and charge (q).
Figure 1. The relationship among resistor (R), inductor (L), capacitor (C), and memristor (M). |
From figure 1, the variables (V, i, φ, q) have six mathematical equations, and they are described by applying electrical elements (R, L, C, M), Faraday’s Law of electromagnetic induction (dφ = Vdt), and the relationship between electric charge and current (dq = idt). A memristor is a nonlinear electronic component with the physical dimension of resistance. Memristors are classified into magnetic flux-controlled memristors (MFCM, the resistance value depends on the magnetic flux) and charge-controlled memristors (CCM, the resistance value depends on the charge).
The relationship between the current and voltage in MFCM is expressed by
$\begin{align}{i_M} = M\left(\varphi \right){v_M};\frac{{{\text{d}}\varphi }}{{{\text{d}}t}} = f\left(\varphi ,{v_M}\right),\end{align}$
where M(φ) denotes the memductance, and it can be described by applying a function with the magnetic flux. (iM, vM) are the output current and input voltage across the memristor, respectively. According to [59], the field energy of the MFCM can be calculated by using the field energy of an inductor with a suitable inductance LM, and then the field energy function of an MFCM can be obtained by $\begin{align}{W_M} = \frac{1}{2}{L_M}i_M^2 = \frac{1}{2}\varphi {i_M}.\end{align}$
Similarly, the volt–ampere characteristic of a CCM is estimated by
$\begin{align}{v_W} = W\left(q\right){i_W};\frac{{{\text{d}}q}}{{{\text{d}}t}} = g\left(q,{i_W}\right),\end{align}$
where W(q) means the mean-resistance. (iW, vW) denotes the input current and output voltage in the CCM, respectively. According to [59], the field energy of the CCM is estimated by using the field energy of an equivalent capacitor. Therefore, the field energy function of a CCM can be described as follows $\begin{align}{H_W} = \frac{1}{2}{C_W}v_W^2 = \frac{1}{2}q{v_W}.\end{align}$
2. Modeling of discrete chaotic systems based on memristor circuits
The discrete memristive chaotic system is a kind of chaotic system based on discrete memristors, and its research mainly focuses on how to utilize the nonlinear characteristics of memristors to enhance the chaotic characteristics of the system. This type of system can achieve more complex chaotic behaviors by introducing the discrete memristor model, such as expanding chaotic regions, increasing complexity, generating hyperchaotic sequences, and so on. The construction of discrete memristor chaotic systems is widely researched. However, it is very interesting research on the memristive chaotic maps built by the memristor-coupled nonlinear circuit.
Figure 2. An MFCM-coupled nonlinear circuit without an inductor. NR is a nonlinear resistor; C denotes a capacitor, and M(φ) means an MFCM. |
In figure 2, the equation of the circuit state variable is described as follows
$\begin{equation}\left\{ \begin{aligned} C\frac{{{\text{d}}V}}{{{\text{d}}t}} &= \frac{r}{\rho }\left(V - \frac{{{V^2}}}{{{V_0}}}\right) - \varphi V,\\ \frac{{{\text{d}}\varphi }}{{{\text{d}}t}} &= \alpha \varphi + \beta V.\\ \end{aligned} \right.\end{equation}$
The dimensionless equation in equation (5 ) is described by
$\begin{equation}\left\{ \begin{aligned} \frac{{{\text{d}}x}}{{{\text{d}}t}} &= - cwxr + \left(x - {x^2}\right),\\ \frac{{{\text{d}}w}}{{{\text{d}}t}} &= \beta x + aw.\\ \end{aligned} \right.\end{equation}$
The parameters and variables in equation (6 ) can be calculated as follows
$\begin{equation}\left\{ \begin{aligned} x &= \frac{V}{{{V_0}}},w = \frac{\varphi }{{\rho C{V_0}}},\tau = \frac{t}{{\rho C}},\\ c &= {\rho ^2}C{V_0},a = \alpha \rho C.\\ \end{aligned} \right.\end{equation}$
The field energy function of the electrical elements in figure 2 is estimated by
$\begin{equation}\left\{ \begin{aligned}W&=\frac{1}{2}C{V^2} + \frac{1}{2}\varphi {i_M},\\ H&=\frac{W}{{CV_0^2}}=\frac{1}{2}{x^2}+\frac{1}{2}c{w^2}x.\\\end{aligned}\right.\end{equation}$
Furthermore, a memristor chaotic map is built based on equation (6 ), and the result is described as follows
$\begin{equation}\left\{ \begin{aligned}{y_{n + 1}} &= \lambda \left({y_n} - y_n^2\right) - \gamma {y_n}{z_n},\\ {z_{n + 1}} &= \delta {y_n} + \mu {z_n}.\\\end{aligned}\right.\end{equation}$
Similarly, parameters and variables in equation (9 ) are calculated by
$\begin{equation}\left\{\begin{aligned}{y_n}&=\frac{{r\Delta\tau}}{{1+r\Delta\tau}}{x_n},{z_n}=\frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{w_n},\\ \lambda &=1+r\Delta\tau,\delta=a\Delta\tau+1,\gamma=c\Delta\tau,\mu=b\Delta\tau.\\\end{aligned}\right.\end{equation}$
In addition, the energy function of the memristive chaotic map is calculated
$\begin{align}{H_n} = \frac{1}{2}{y_n} + \frac{1}{2}\gamma z_n^2{y_n}.\end{align}$
Figure 3. A CCM-coupled nonlinear circuit without a capacitor. NR is a nonlinear resistor; L denotes an inductor, and M(q) means a CCM. |
The equations of a CCM-coupled nonlinear circuit without a capacitor in figure 3 are described as follows
$\begin{equation}\left\{\begin{aligned}L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}}&=r\rho \left({i_L} - \frac{{\rho i_L^2}}{{{V_0}}}\right) - \left(\alpha + 3\beta {q^2}\right){i_L} - E,\\ \frac{{{\text{d}}q}}{{{\text{d}}t}} &= aq + b{i_L}.\\\end{aligned}\right.\end{equation}$
The dimensionless equations for the memristor-coupled nonlinear circuit are expressed by variables as follows
$\begin{equation}\left\{ \begin{aligned}\frac{{{\text{d}}y}}{{{\text{d}}t}}&=r\left(y - {y^2}\right) - d - \left(\alpha ^{\prime} + \beta ^\prime{u^2}\right)y,\\ \frac{{{\text{d}}u}}{{{\text{d}}t}} &= by + a^{\prime} u.\\\end{aligned}\right.\end{equation}$
The parameters and variables in equation (13 ) can be calculated as follows
$\begin{equation}\left\{ \begin{aligned}y &=\frac{{\rho {i_L}}}{{{V_0}}},u=\frac{{{\rho ^2}q}}{{L{V_0}}},\tau =\frac{{\rho t}}{L},\\ \alpha ^{\prime} &= \frac{\alpha }{\rho },\beta ^{\prime} = \frac{{3\beta {L^2}V_0^2}}{{{\rho ^5}}},a^{\prime} = \frac{{aL}}{\rho },d = \frac{E}{{{V_0}}}.\\ \end{aligned}\right.\end{equation}$
The field energy for the memristor-coupled nonlinear circuit in figure 3 and its dimensionless energy function are calculated by
$\begin{align}\left\{ \begin{gathered} {W_2} = \frac{1}{2}Li_L^2 + \frac{1}{2}{C_M}V_M^2 = \frac{1}{2}Li_L^2 + \frac{1}{2}q{V_M},\\ {H_2} = \frac{{{W_2}}}{{V_0^2\left(L/{\rho ^2}\right)}} = \frac{1}{2}{y^2} + \frac{1}{2}\left(\alpha ^{\prime} u + \beta ^\prime{u^3}\right)y.\\ \end{gathered} \right.\end{align}$
Furthermore, the parameters and variables in equation (13 ) are defined as follows
$\begin{align}\left\{ {\begin{array}{*{20}{l}} {{w_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{y_n},{v_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{u_n},\chi = 1 + r\Delta \tau ,\xi = {\alpha ^{\prime} }\Delta \tau ,} \\ {\varepsilon = \frac{{{\beta ^{\prime} }{{\left(1 + r\Delta \tau \right)}^2}}}{{{r^2}\Delta \tau }},\kappa = a\Delta \tau + 1,\eta = b\Delta \tau ,{d^{\prime} } = d\Delta \tau .} \end{array}} \right.\end{align}$
A two-dimensional CCM map is described by
$\begin{equation}\left\{ \begin{aligned}{w_{n + 1}} &= \chi \left({w_n} - w_n^2\right) - \left(\zeta + \varepsilon v_n^2\right){w_n} - d^{\prime} ,\\ {v_{n + 1}}&=\kappa{w_n}+\eta {v_n}.\\\end{aligned}\right.\end{equation}$
The energy function for the two-dimensional discrete memristor system in equation (17 ) can be obtained as follows
$\begin{align}{H^{\prime}_n} = \frac{1}{2}{w_n} + \frac{1}{2}\left(\xi + \varepsilon v_n^2\right){w_n}{v_n}.\end{align}$
Figure 4. A dual memristor-coupled nonlinear circuit without an inductor. NR is a nonlinear resistor, C denotes a capacitor, M(φ) means an MFCM, and M(q) is a CCM. |
The circuit equations of the dual memristor-coupled nonlinear circuit in figure 4 are obtained as follows
$\begin{equation}\left\{\begin{aligned}C\frac{{{\text{d}}V}}{{{\text{d}}t}}&=\frac{r}{\rho }\left(V - \frac{{{V^2}}}{{{V_0}}}\right) - \sin qV - \varphi V,\\ \frac{{{\text{d}}\varphi }}{{{\text{d}}t}} &= bV + a\varphi ,\\ \frac{{{\text{d}}q}}{{{\text{d}}t}} &= \sin qV + dq.\\\end{aligned}\right.\end{equation}$
To obtain the dimensionless equations of equation (19 ), the parameters and variables are redefined in equation (20 ).
$\begin{align}\left\{ \begin{gathered} x = \frac{V}{{{V_0}}},w = \frac{\varphi }{{\rho C{V_0}}},u = \frac{q}{{C{V_0}}},\tau = \frac{t}{{\rho C}},\\ a = {\rho ^2}C{V_0},k = C{V_0},c = a\rho C,d^{\prime} = dC\rho ,\alpha = \rho .\\ \end{gathered} \right.\end{align}$
As a result, equation (19 ) can be reexpressed by
$\begin{equation}\left\{\begin{aligned}\frac{{{\text{d}}x}}{{{\text{d}}t}}&=- awxr + \left(x - {x^2}\right) - \alpha \sin \left(ku\right)x,\\ \frac{{{\text{d}}w}}{{{\text{d}}t}} &= bx + cw,\\ \frac{{{\text{d}}u}}{{{\text{d}}t}} &= \alpha \sin \left(ku\right)x + d^{\prime} u.\\\end{aligned}\right.\end{equation}$
The field energy in figure 4 and its dimensionless energy are calculated as follows
$\begin{align}\left\{ \begin{gathered} W = \frac{1}{2}C{V^2} + \frac{1}{2}\varphi {i_{M1}} + \frac{1}{2}qV,\\ H = \frac{W}{{CV_0^2}} = \frac{1}{2}{x^2} + \frac{1}{2}a{w^2}x + \frac{1}{2}ux.\\ \end{gathered} \right.\end{align}$
To build a dual memristive chaotic map, the parameters and variables in equation (21 ) are reupdated by
$\begin{equation}\left\{\begin{aligned}{y_n}&=\frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{x_n},{z_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{w_n},\\{v_n} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{u_n},\lambda = 1 + r\Delta \tau ,\\ b &= \frac{{1 + r\Delta \tau }}{r},\beta = \alpha \Delta \tau ,\delta = k\frac{{1 + r\Delta \tau }}{{r\Delta \tau }},\\a &= c\Delta \tau + 1;d = d^{\prime}\Delta \tau + 1,c = b\Delta \tau .\end{aligned}\right.\end{equation}$
A dual memristor discrete system can be expressed as follows
$\begin{equation}\left\{\begin{aligned}{y_{n + 1}}&=\lambda \left({y_n} - y_n^2\right) - b{y_n}{z_n} - \beta \sin \left(\delta {v_n}\right){y_n},\\ {z_{n + 1}} &= a{y_n} + c{z_n};\\ {v_{n + 1}} &= d{v_n} + \beta \sin \left(\delta {v_n}\right){y_n}.\\\end{aligned}\right.\end{equation}$
The energy function of the discrete memristor system in equation (24 ) is described by
$\begin{align}{H_n} = \frac{1}{2}{y_n} + \frac{1}{2}bz_n^2{y_n} + \frac{1}{2}{v_n}{y_n}.\end{align}$
Figure 5. A controllable memristor circuit without an inductor. NR is a nonlinear resistor, C denotes a capacitor, M(φ) means an MFCM, and is denotes an external current. |
The circuit equation presented in figure 5 can be obtained as follows
$\begin{align}\left\{ {\begin{array}{*{20}{l}} {C\frac{{{\text{d}}V}}{{{\text{d}}t}} = {i_s} + \frac{r}{\rho }\left( {V - \frac{{{V^2}}}{{{V_0}}}} \right) - \varphi V,} \\ {\frac{{{\text{d}}\varphi }}{{{\text{d}}t}} = \alpha \varphi + \beta V.} \end{array}} \right.\end{align}$
The variables in equation (26 ) are updated by
$\begin{align}x = \frac{V}{{{V_0}}},w = \frac{\varphi }{{\rho C{V_0}}},\tau = \frac{t}{{\rho C}}.\end{align}$
Therefore, the equations in equation (26 ) are described as follows 28 ) are defined by
$\begin{equation}\left\{ \begin{aligned}\frac{{{\text{d}}x}}{{{\text{d}}\tau }} &= {u_s} - cwx + r\left(x - {x^2}\right),\\ \kern-24pt \frac{{{\text{d}}w}}{{{\text{d}}\tau }} &= \beta x + aw,\end{aligned}\right.\end{equation}$
where parameters in equation ( $\begin{align}c = {\rho ^2}C{V_0},a = \alpha \rho C,{u_s} = \frac{{\rho {i_s}}}{{{V_0}}}.\end{align}$
For the memristive circuit in figure 5, its physical field energy is calculated as follows
$\begin{align}W = \frac{1}{2}C{V^2} + \frac{1}{2}{L_M}i_M^2 = \frac{1}{2}C{V^2} + \frac{1}{2}\varphi {i_M}.\end{align}$
The energy corresponding to equation (30 ) can be updated by
$\begin{align}H = \frac{W}{{CV_0^2}} = \frac{1}{2}{x^2} + \frac{1}{2}c{w^2}x.\end{align}$
The variables in equation (28 ) are further defined by
$\begin{align}{y_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{x_n},{z_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{w_n}.\end{align}$
And then a memristive map is given as follows 33 ) can be calculated by
$\begin{align}\left\{ {\begin{array}{*{20}{l}} {{y_{n + 1}} = {i_s} + \lambda \left( {{y_n} - y_n^2} \right) - \gamma {y_n}{z_n},} \\ {{z_{n + 1}} = \delta {y_n} + \mu {z_n},} \end{array}{\text{ }}} \right.\end{align}$
where the parameters in equation ( $\begin{align}\mu = b\Delta \tau ,\lambda = 1 + r\Delta \tau ,\delta = a\Delta \tau + 1,\gamma = c\Delta \tau ,{i_s} = \frac{{r{{\left(\Delta \tau \right)}^2}}}{{1 + r\Delta \tau }}\!.\end{align}$
Moreover, an energy function of the chaotic map is defined as follows
$\begin{align}{H_n} = \frac{1}{2}{y_n} + \frac{1}{2}\gamma z_n^2{y_n}.\end{align}$
In summary, modeling discrete chaotic systems based on memristor circuits serves as an important bridge connecting cutting-edge electronic devices with nonlinear dynamics theory. The core idea lies in utilizing the unique memory characteristics and nonlinear resistance-switching behavior of memristors to construct discrete chaotic systems that are physically feasible and functionally rich. By ingeniously combining memristors with simple linear circuit elements (such as capacitors and inductors), more complex second-order or higher-order discrete systems can be constructed. For instance, in a circuit composed of electronic components like memristors and capacitors, a two-dimensional nonlinear mapping can be established. This circuit achieves an accurate transformation from continuous physical dynamics to discrete mathematical iterations.
3. Modeling of discrete neurons based on memristor circuits
The discrete memristor neuron model can be established by combining the nonlinear characteristics of memristors with the dynamic characteristics of discrete-time systems. The discrete memristor neuron model provides new perspectives and methods for the modeling, analysis, and application of neural networks, and it has extensive research value and engineering potential. Therefore, the modeling of the memristive neuron map is becoming a research hotspot. In [89], a CCM neuron map is developed from a CCM-coupled nonlinear circuit. Figure 6 plots a CCM-coupled nonlinear circuit.
Figure 6. CCM-coupled nonlinear circuit. NR is a nonlinear resistor, C denotes a capacitor, L means an inductor, R is a linear resistor, E denotes a voltage source, and M(q) is a CCM. |
The output equations in figure 6 are described by
$\begin{equation}\left\{\begin{aligned}C\frac{{{\text{d}}{V_C}}}{{{\text{d}}t}}&=-{i_L}+\frac{r}{\rho }\left({V_C} - \frac{{V_C^2}}{{{V_0}}}\right) - {\frac{{{V_C}}}{{{k_M}\left(\alpha + 3\beta {q^2}\right)}}_M},\\ L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}} &= {V_C} + E - {i_L}R,\\ \frac{{{\text{d}}q}}{{{\text{d}}t}} &= {\frac{{{V_C}}}{{{k_M}\left(\alpha + 3\beta {q^2}\right)}}_M} + \xi q.\\\end{aligned}\right.\end{equation}$
The field energy in figure 6 is calculated by
$\begin{align}W = \frac{1}{2}CV_C^2 + \frac{1}{2}L{i_L}^2 + \frac{1}{2}{C_M}V_M^2 = \frac{1}{2}CV_C^2 + \frac{1}{2}L{i_L}^2 + \frac{1}{2}q{V_M}.\end{align}$
Furthermore, the variables and parameters in equations (36 ) and (37 ) can be reupdated as follows
$\begin{align}\left\{ \begin{gathered} x = \frac{{{V_C}}}{{{V_0}}},y = \frac{{\rho {i_L}}}{{{V_0}}},z = \frac{q}{{C{V_0}}},\tau = \frac{t}{{\rho C}},a = \frac{{{k_M}\alpha }}{\rho },\\ b = \frac{{3{k_M}{C^2}V_0^2\beta }}{\rho },c = \frac{{\rho {C^2}}}{L},d = \frac{E}{{{V_0}}},e = \frac{R}{\rho },f = \xi \rho C.\\ \end{gathered} \right.\end{align}$
As a result, a CCM neuron is expressed by
$\begin{equation}\left\{ \begin{aligned}\frac{{{\text{d}}x}}{{{\text{d}}\tau }} &= - y + rx\left(1 - x\right) - \frac{x}{{a + b{z^2}}},\\ \frac{{{\text{d}}y}}{{{\text{d}}\tau }} &= c\left(x - ey + d\right),\\ \frac{{{\text{d}}z}}{{{\text{d}}\tau }} &= fz + \frac{x}{{a + b{z^2}}}.\\ \end{aligned} \right.\end{equation}$
The dimensionless energy function of a CCM neuron can be presented as follows
$\begin{align}H = \frac{W}{{CV_0^2}} = \frac{1}{2}{x^2} + \frac{1}{{2c}}{y^2} + \frac{1}{2}xz.\end{align}$
To obtain a CCM neuron map, the discrete variables of the neuron presented in equation (39 ) can be expressed as
$\begin{equation}\left\{\begin{aligned}{x_n}&=\frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{x^{^{\prime}}_n},{y_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{y^{^{\prime}}_n},{z_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{z^{^{\prime}}_n},\\\eta &= 1 + r\Delta \tau ,\delta ^{\prime} = \Delta \tau ,a^{\prime} = \frac{a}{{\Delta \tau }},\\ b^{\prime} &= \frac{{{{\left(1 + r\Delta \tau \right)}^2}}}{{{r^2}\Delta {\tau ^3}}}b,c^{\prime} = c\Delta \tau ,d^{\prime} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}d,\\e^{\prime} &= \frac{1}{{c\Delta \tau }} - e,f^{\prime} = f\Delta \tau + 1.\\ \end{aligned}\right.\end{equation}$
Furthermore, a CCM map neuron is described as follows
$\begin{equation}\left\{ \begin{aligned}{x_{n + 1}} &= \eta {x_n}\left(1 - {x_n}\right) - \delta ^\prime{y_n} - \frac{{{x_n}}}{{a^{\prime} + b^{^{\prime}}z_n^2}},\\ {y_{n + 1}} &= c^{\prime}\left({x_n} + d^{\prime} + e^\prime{y_n}\right),\\ {z_{n + 1}} &= \frac{{{x_n}}}{{a^{\prime} + b^{^{\prime}}z_n^2}} + f^\prime{z_n}.\\ \end{aligned} \right.\end{equation}$
In addition, the energy function of the CCM map neuron presented in equation (42 ) can be expressed by
$\begin{align}{H_n} = \frac{{{W_n}}}{{CV_0^2}} = {H^{\prime}_n} + {H^{^{\prime\prime}}_n} + {H^{^{\prime\prime}\!^{\prime}}_n} = \frac{1}{2}x_n^2 + \frac{1}{{2c^{\prime}}}y_n^2 + \frac{1}{2}{x_n}{z_n}.\end{align}$
Figure 7. A MFCM-coupled nonlinear circuit. NR is a nonlinear resistor, C denotes a capacitor, L means an inductor, R is a linear resistor, E denotes a voltage source, and M(φ) is an MFCM. |
The equations of an MFCM-coupled nonlinear circuit are described by
$\begin{equation}\left\{\begin{aligned}C\frac{{{\text{d}}{V_C}}}{{{\text{d}}t}}&={i_L} + \frac{r}{\rho }\left({V_C} - \frac{{V_C^2}}{{{V_0}}}\right) - \left({\alpha _1} + 3{\beta _1}{\phi ^2}\right){V_C},\\ L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}} &= E - V - {i_L}R,\\ \frac{{{\text{d}}\phi }}{{{\text{d}}t}}&=A\phi + BV.\\ \end{aligned}\right.\end{equation}$
The variables and parameters in equation (44 ) are redefined in equation (45 ).
$\begin{align}\left\{ \begin{gathered} x = \frac{V}{{{V_0}}},y = \frac{{{i_L}\rho }}{{{V_0}}},\tau = \frac{t}{{\rho C}},\varphi ^{\prime} = \frac{\phi }{{\rho C{V_0}}},\alpha ^{\prime} = {\alpha _1}\rho ,\\ \beta ^{\prime} = {\beta _1}{\rho ^3}{C^2}{V_0}^2,A^{\prime} = A\rho C,a = \frac{E}{{{V_0}}},b = \frac{{{\rho ^2}C}}{L},c = \frac{R}{\rho }.\\ \end{gathered} \right.{\kern 1pt} \end{align}$
A dimensionless neuron is updated as follows
$\begin{equation}\left\{ \begin{aligned} \frac{{{\text{d}}x}}{{{\text{d}}\tau }} &= r\left(x - {x^2}\right) + y - x\left(\alpha ^{\prime} + 3\beta ^\prime{{\varphi ^{\prime}}^2}\right),\\ \frac{{{\text{d}}y}}{{{\text{d}}\tau }} &= b\left( - cy + a - x\right),\\ \frac{{{\text{d}}\varphi ^{\prime}}}{{{\text{d}}\tau }} &= Bx + A^{\prime}\varphi ^{\prime}.\\ \end{aligned} \right.{\kern 1pt}\end{equation}$
Field energy of an MFCM neuron and its dimensionless energy function are calculated as follows
$\begin{align}\left\{ \begin{gathered} W = \frac{1}{2}C{V^2} + \frac{1}{2}Li_L^2 + \frac{1}{2}{L_M}i_M^2 = \frac{1}{2}C{V^2} + \frac{1}{2}Li_L^2 + \frac{1}{2}\phi {i_M},\\ H = \frac{{{W_1}}}{{CV_{_0}^2}} = \frac{1}{2}{x^2} + \frac{1}{{2b}}{y^2} + \frac{1}{2}\left(\alpha ^{\prime}\varphi ^{\prime} + 3\beta ^\prime{{\varphi ^{\prime}}^3}\right)x.\\ \end{gathered} \right.\end{align}$
The discrete variables and parameters in equation (46 ) are redefined in equation (48 ) for obtaining a map neuron.
$\begin{equation}\left\{\begin{aligned}{z_n}&=\frac{{r\Delta\tau}}{{1 + r\Delta \tau }}{x_n},{w_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{y_n},{\varphi _n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{\varphi ^{^{\prime}}_n},\\ \alpha &= \alpha ^{\prime}\Delta \tau ,\lambda = 1 + r\Delta \tau , \beta = \frac{{\beta ^{\prime}{{\left(1 + r\Delta \tau \right)}^2}}}{{\Delta \tau {r^2}}},\\k &= \Delta \tau ,d = \frac{{r\Delta {\tau ^2}}}{{1 + r\Delta \tau }}a,f = c\Delta \tau - \frac{1}{b},\\ \eta &= A^{\prime}\Delta \tau + 1,\mu = B\Delta \tau .\\ \end{aligned}\right.\end{equation}$
As a result, an MFCM map neuron can be described as follows
$\begin{equation}\left\{\begin{aligned}{z_{n + 1}}&=\lambda \left({z_n} - z_n^2\right) - {z_n}\left(\alpha + 3\beta {\varphi _n}^2\right) + k{w_n},\\ {w_{n + 1}} &= b\left(d - k{z_n} - f{w_n}\right),\\ {\varphi _{n + 1}} &= \eta {\varphi _n} + \mu {z_n}.\\ \end{aligned}\right.\end{equation}$
In addition, the energy function of the discrete memristive neuron map is calculated by
$\begin{align}{H_n} = \frac{1}{2}{z_n}^2 + \frac{1}{{2b}}{w_n}^2 + \frac{1}{2}\left(\alpha {\varphi _n} + 3\beta {\varphi _n}^3\right){z_n}.\end{align}$
Figure 8. A dual memristor-coupled nonlinear circuit. NR is a nonlinear resistor, C denotes a capacitor, L means an inductor, R is a linear resistor, M(φ) denotes an MFCM, and M(q) is a CCM. |
The equivalent dynamical equations in figure 8 are presented as
$\begin{equation}\left\{ \begin{aligned} C\frac{{{\text{d}}V}}{{{\text{d}}t}} &= - {i_L} + \frac{r}{r}\left( {V - \frac{{{V^2}}}{{{V_0}}}} \right) - \varphi V,\\ L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}} &= V - q{i_M} - R{i_L},\\ \frac{{{\text{d}}q}}{{{\text{d}}t}} &= aq + b{i_M},\\ \frac{{{\text{d}}\varphi }}{{{\text{d}}t}} &= a\varphi + b{V_\varphi }.\\ \end{aligned} \right.\end{equation}$
Furthermore, the variables in equation (52 ) are replaced by
$\begin{align}x = \frac{V}{{{V_0}}},y = \frac{{\rho {i_L}}}{{{V_0}}},z = \frac{q}{{C{V_0}}},w = \frac{\varphi }{{\rho C{V_0}}},\tau = \frac{t}{{\rho C}}.\end{align}$
As a result, the dynamical equations in equation (51 ) are expressed as follows 53 ) are obtained by
$\begin{equation}\left\{ \begin{aligned} \frac{{{\text{d}}x}}{{{\text{d}}t}} &= - y + r\left( {x - {x^2}} \right) - cwx,\\ \frac{{{\text{d}}y}}{{{\text{d}}t}} &= d\left( {x - ezy - gy} \right)\!,\\ \frac{{{\text{d}}z}}{{{\text{d}}t}} &= a^{\prime} z + by,\\ \frac{{{\text{d}}w}}{{{\text{d}}t}} &= a^{\prime} w + bx,\\ \end{aligned} \right.\end{equation}$
where parameters in equation ( $\begin{align}c = {\rho ^2}C{V_0},d = \frac{{{\rho ^2}C}}{L},e = \frac{{C{V_0}}}{\rho },g = \frac{R}{\rho },\alpha ^{\prime} = \rho \alpha C,a^{\prime} = a\rho C.\end{align}$
The corresponding field energy of the dual memristor-coupled nonlinear circuit is calculated as follows
$\begin{align}W &= {W_C} + {W_L} + {W_M} + {W_\varphi }\nonumber\\ & = \frac{1}{2}C{V^2} + \frac{1}{2}Li_L^2 + \frac{1}{2}{q^2}{i_L} + \frac{1}{2}{\varphi ^2}V.\end{align}$
Furthermore, the dimensionless energy in equation (55 ) is obtained by
$\begin{align}H & = \frac{W}{{CV_0^2}} = {H_C} + {H_L} + {H_M} + {H_\varphi } \nonumber\\ & = \frac{1}{2}{x^2} + \frac{1}{{2d}}{y^2} + \frac{1}{2}e{z^2}y + \frac{1}{2}c{w^2}x.\end{align}$
The variables in equation (53 ) are redefined as follows
$\begin{align}{x^{\prime}_n} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{x_n},{y^{\prime}_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{y_n},{z^{\prime}_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{z_n},\nonumber\\ {w^{\prime}_n} & = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{w_n}.\end{align}$
As a result, a dual memristor map neuron can be presented as follows
$\begin{equation}\left\{ \begin{aligned} {x^{^{\prime}}_{n + 1}} &= - {\lambda _1}{y^{^{\prime}}_n} + {r_1}\left( {{x^{^{\prime}}_n} - {x^{^{\prime}}_n}^2} \right) - {c_1}{w^{^{\prime}}_n}{x^{^{\prime}}_n},\\ {y^{^{\prime}}_{n + 1}} &= {d_1}\left( {{x^{^{\prime}}_n} - {e_1}{z^{^{\prime}}_n}{y^{^{\prime}}_n} - {g_1}{y^{^{\prime}}_n}} \right)\!,\\ {z^{^{\prime}}_{n + 1}} &= {\alpha _1}{z^{^{\prime}}_n} + {\beta _1}{y^{^{\prime}}_n},\\ {w^{^{\prime}}_{n + 1}} &= {a_1}{w^{^{\prime}}_n} + {b_1}{x^{^{\prime}}_n}.\\ \end{aligned} \right.\end{equation}$
The parameters in equation (58 ) can be obtained as follows
$\begin{equation}\begin{aligned}\left\{ {\begin{array}{*{20}{l}}{{\lambda_1}= \Delta \tau ,{r_1} = 1 + r\Delta \tau ,{c_1} = c\frac{{1 + r\Delta \tau }}{r},{d_1} = d\Delta \tau ,}\\{{e_1} = e\frac{{1 + r\Delta \tau }}{{r\Delta \tau }}, {g_1} = g - \frac{1}{{d\Delta \tau }},{\alpha _1} = 1 + \alpha ^{\prime}\Delta \tau ,} \\{{\beta_1}=\beta \Delta \tau ,{a_1} = 1 + a^{\prime}\Delta \tau \!,{b_1} = b\Delta \tau.}\end{array}}\right.\end{aligned}\end{equation}$
Similarly, the energy function of the dual memristor map neuron in equation (58 ) can be calculated by
$\begin{align}{H_n} = \frac{1}{2}x^{^{\prime} 2}_n + \frac{1}{{2{d_1}}}y^{^{\prime} 2}_n + \frac{1}{2}{e_1}z^{^{\prime} 2}_n y^{^{\prime}_n} + \frac{1}{2}{c_1}w^{^{\prime} 2}_n x^{^{\prime}_n}.\end{align}$
Figure 9. A memristive circuit without a capacitor. NR is a nonlinear resistor, L1 and L2 are two inductors, R is a linear resistor, and M(q) is a CCM. |
Based on Kirchhoff’s laws, the circuit equations in figure 9 are given as follows
$\begin{equation}\left\{ \begin{aligned} {L_1}\frac{{{\text{d}}{i_{L1}}}}{{{\text{d}}t}} &= r\rho \left({i_{L1}} - \frac{{\rho i_{L1}^2}}{{{V_0}}}\right) - q\left({i_{L1}} + {i_{L2}}\right)\!,\\ {L_2}\frac{{{\text{d}}{i_{L2}}}}{{{\text{d}}t}} &= - q\left({i_{L1}} + {i_{L2}}\right) - R{i_{L2}},\\ \frac{{{\text{d}}q}}{{{\text{d}}t}} &= \alpha q + \beta \left({i_{L1}} + {i_{L2}}\right)\!.\\ \end{aligned} \right.\end{equation}$
The variables and parameters in equation (61 ) are defined by
$\begin{align}\left\{ \begin{gathered} x = \frac{{\rho {i_{L1}}}}{{{V_0}}},y = \frac{{\rho {i_{L2}}}}{{{V_0}}},z = \frac{{{\rho ^2}q}}{{{L_1}{V_0}}},\tau = \frac{{\rho t}}{{{L_1}}},\\ c = \frac{{{L_1}}}{{{L_2}}},a = \frac{{{L_1}{V_0}}}{{{\rho ^3}}},b = \frac{R}{\rho },\alpha ^{\prime} = \frac{{{\alpha _1}{L_1}}}{\rho }.\\ \end{gathered} \right.\end{align}$
According to equations (61 ) and (62 ), an oscillator is described by
$\begin{equation}\left\{ \begin{aligned} \frac{{{\text{d}}x}}{{{\text{d}}\tau }} &= r\left(x - {x^2}\right) - az\left(x + y\right)\!,\\ \frac{{{\text{d}}x}}{{{\text{d}}\tau }} &= c\left[ - az\left(x + y\right) - by\right]\!,\\ \frac{{{\text{d}}z}}{{{\text{d}}\tau }} &= \alpha ^{\prime} z + \beta \left(x + y\right)\!.\\ \end{aligned} \right.\end{equation}$
The physical field energy for figure 9 and its dimensionless energy function H can be calculated by
$\begin{equation}\left\{ \begin{aligned} W &= {W_{L1}} + {W_{L2}} + {W_M} = \frac{1}{2}{L_1}i_1^{^2} + \frac{1}{2}{L_2}i_2^{^2} + \frac{1}{2}q{V_M},\\ H &= \frac{{{W_{L1}} + {W_{L2}} + {W_M}}}{{V_0^2{L_1}/{\rho ^2}}} = {H_{L1}} + {H_{L2}} + {H_M}\\ &= \frac{1}{2}{x^2} + \frac{1}{{2c}}{y^2} + \frac{1}{2}a{z^2}\left(x + y\right).\\ \end{aligned} \right.\end{equation}$
Furthermore, a group of new discrete variables and parameters in equation (63 ) is defined as follows
$\begin{equation}\left\{ \begin{aligned} {x_n} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{u_n},{y_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{v_n},{z_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{w_n},\\ \delta &= \frac{{a\left(1 + r\Delta \tau \right)}}{r},\lambda = 1 + r\Delta \tau ,\gamma = \frac{{bc\Delta \tau - 1}}{c},\\ \kappa &= 1 + \alpha ^{\prime}\Delta \tau ,\eta = \beta \Delta \tau . \end{aligned} \right.\end{equation}$
A memristive map of the memristive oscillator in equation (63 ) can be described by
$\begin{equation}\left\{ \begin{aligned} {u_{n + 1}} &= \lambda \left({u_n} - u_n^2\right) - \delta {w_n}\left({u_n} + {v_n}\right)\!,\\ {v_{n + 1}} &= c\left[ - \delta {w_n}\left({u_n} + {v_n}\right) - \gamma {v_n}\right]\!,\\ {w_{n + 1}} &= \kappa {w_n} + \eta \left({u_n} + {v_n}\right)\!.\\ \end{aligned} \right.\end{equation}$
Furthermore, a similar energy function is calculated by
$\begin{align}{H_n} = \frac{1}{2}u_n^2 + \frac{1}{{2c}}v_n^2 + \frac{1}{2}\delta w_n^2\left({u_n} + {v_n}\right)\!.\end{align}$
The current for the nonlinear resistor and memristor in figure 10 is estimated as follows
$\begin{equation}\left\{ \begin{aligned} {i_{NR}} &= - \frac{r}{\rho }\left({V_1} - \frac{{{V_1}^2}}{{{V_0}}}\right)\!\!,\\ {V_M} &= M\left(q\right){i_M} = \frac{{\csc \left(q\right)}}{{{\lambda _1}}} \cdot {i_M} = {V_1}.\\ \end{aligned} \right.\end{equation}$
Figure 10. A memristive neural circuit with a thermosensitive membrane. NR is a nonlinear resistor, C1 and C2 denote two capacitors, L means an inductor, Rs and R are two linear resistors, RT denotes a thermistor, Vs is a voltage source and M is a CCM. |
The field energy in figure 10 is described by
$\begin{align}W = \frac{1}{2}{C_1}{V_1}^2 + \frac{1}{2}{C_2}{V_2}^2 + \frac{1}{2}L{i_L}^2 + \frac{1}{2}q{V_M}.\end{align}$
The correlation between physical variables in figure 10 is obtained as follows
$\begin{equation}\left\{ \begin{aligned} {C_2}\frac{{{\text{d}}{V_2}}}{{{\text{d}}t}} &= {i_s} - \frac{{{V_2} - {V_1}}}{{{R_T}}},\\ {C_1}\frac{{{\text{d}}{V_1}}}{{{\text{d}}t}} &= \frac{{{V_2} - {V_1}}}{{{R_T}}} - {i_L} - {i_{NR}} - {i_M},\\ L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}} &= {V_1} - {i_L}R,\\ \frac{{{\text{d}}q}}{{{\text{d}}t}} &= {\lambda _2}q + {i_M}.\\ \end{aligned} \right.\end{equation}$
The new variables and parameters are defined as follows
$\begin{equation}\left\{ \begin{aligned} x &= \frac{{{V_2}}}{{{V_0}}},y = \frac{{{V_1}}}{{{V_0}}},z = \frac{{\rho {i_L}}}{{{V_0}}},\tau = \frac{t}{{\rho {C_2}}},{u_s} = \frac{{\rho {V_s}}}{{{R_s}{V_0}}},\\a &= \frac{{\rho R{C_2}}}{L}, b = \frac{\rho }{{{R_s}}} + \frac{\rho }{{{R_T}}}, c = \frac{\rho }{{{R_T}}},d = \frac{{{\rho ^2}{C_2}}}{L},\\ \alpha &= \frac{{{C_2}}}{{{C_1}}},u = \frac{q}{{{C_2}{V_0}}},f = \rho {\lambda _1},k = {C_2}{V_0}, g = {\lambda _2}\rho {C_2}.\\ \end{aligned} \right.\end{equation}$
Consequently, the dynamic equation of a memristive neuron can be described by
$\begin{equation}\left\{ \begin{aligned} \frac{{{\text{d}}x}}{{{\text{d}}\tau }} &= {u_s} - bx + cy,\\ \frac{{{\text{d}}y}}{{{\text{d}}\tau }} &= \alpha \left[ {c\left( {x - y} \right) - z + r\left( {y - {y^2}} \right) - f\sin \left(ku\right)y} \right],\\ \frac{{{\text{d}}z}}{{{\text{d}}\tau }} &= dy - az,\\ \frac{{{\text{d}}u}}{{{\text{d}}\tau }} &= gu + f\sin \left(ku\right)y.\\ \end{aligned} \right.\end{equation}$
The dimensionless Hamilton energy function H corresponding to the field energy W in equation (69 ) is presented as follows
$\begin{align}H = \frac{W}{{{C_2}V_0^2}} = \frac{1}{2}{x^2} + \frac{1}{{2\alpha }}{y^2} + \frac{1}{{2d}}{z^2} + \frac{1}{2}uy.\end{align}$
The new parameters and discrete variables are defined by
$\begin{equation}\left\{ \begin{aligned} {x_n}^{\prime} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{x_n},{y_n}^{\prime} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{y_n},{z_n}^{\prime} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{z_n},\\ {u_n}^{\prime} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{u_n}, {u_s}^{\prime} = \frac{{r\Delta {\tau ^2}}}{{1 + r\Delta \tau }}{u_s},{b_1} = b\Delta \tau - 1,\\{c_1} &= c\Delta \tau ,{\alpha _1} = \alpha \Delta \tau , {d_1} = 1 + \frac{1}{{\alpha r\Delta \tau }}, {g_1} = a\Delta \tau - 1,\\{l_1} &= k\frac{{1 + r\Delta \tau }}{{r\Delta \tau }},{l_2} = d\Delta \tau ,{l_3} = g\Delta \tau + 1, {l_4} = f\Delta \tau .\\ \end{aligned} \right.\end{equation}$
A map neuron of equation (72 ) can be described as follows
$\begin{equation}\left\{ \begin{aligned} {x^{^{\prime}}_{n + 1}} &= {u^{^{\prime}}_s} - {b_1}{x^{^{\prime}}_n} + {c_1}{y^{^{\prime}}_n},\\ {y^{^{\prime}}_{n + 1}} &= {\alpha _1}\left[c\left({x_n}^{\prime} - {y_n}^{\prime} \right) - {z^{^{\prime}}_n} + r\left({d_1}{y^{^{\prime}}_n} - {y^{^{\prime}}_n}^2\right)\right. \\ &\left.- f\sin \left({l_1}{u^{^{\prime}}_n}\right){y^{^{\prime}}_n}\right],\\ {z^{^{\prime}}_{n + 1}} &= {l_2}{y^{^{\prime}}_n} - {g_1}{z^{^{\prime}}_n},\\ {u^{^{\prime}}_{n + 1}} &= {l_3}{u^{^{\prime}}_n} + {l_4}\sin \left({l_1}{u^{^{\prime}}_n}\right){y^{^{\prime}}_n}.\\ \end{aligned} \right.\end{equation}$
A similar energy function for a map neuron can be presented by
$\begin{align}{H_n} = \frac{1}{2}x^{^{\prime} 2}_n + \frac{1}{{2{\alpha _1}}}y^{^{\prime} 2}_n + \frac{1}{{2{l_2}}}z^{^{\prime} 2}_n + \frac{1}{2}{u^{^{\prime}}_n}{y^{^{\prime}}_n}.\end{align}$
Figure 11. Schematic diagram for a neural circuit with a hybrid ion channel. NR is a nonlinear resistor, C denotes a capacitor, L means an inductor, R is a linear resistor, and M(q) is a CCM. |
The relationship of the variables in figure 11 is described as follows
$\begin{equation}\left\{ \begin{aligned} C\frac{{\text{d}V}}{{\text{d}t}} &= \frac{r}{\rho }\left(V - \frac{{{V^2}}}{{{V_0}}}\right) - {i_L},\\ L\frac{{\text{d}{i_L}}}{{\text{d}t}} &= V - \left(\alpha + 3\beta {q^2}\right){i_L} - {i_L}R,\\ \frac{{\text{d}q}}{{\text{d}t}} &= k{i_L} + \varepsilon q.\\ \end{aligned} \right.\end{equation}$
The field energy in figure 11 is estimated by
$\begin{align}W = {W_1} + {W_2} + {W_3} = \frac{1}{2}C{V^2}\; + \frac{1}{2}Li_L^2 + \frac{1}{2}q{V_M}.\end{align}$
To minimize the influence of physical dimensions, the dimensionless variables and parameters are adopted as follows
$\begin{align}\left\{ \begin{gathered} x = \frac{V}{{{V_0}}},y = \frac{{\rho {i_L}}}{{{V_0}}},\tau = \frac{t}{{\rho C}},c = \frac{{{\rho ^2}C}}{L},u = \frac{q}{{C{V_0}}},\\ \alpha = \frac{R}{\rho },f = \varepsilon \rho C,a = \frac{\alpha }{\rho },b = \frac{{3\beta {C^2}V_0^2}}{\rho }.\\ \end{gathered} \right.\end{align}$
Consequently, the memristive neural circuit defined in equation (77 ) gives rise to an oscillator whose dynamics are captured by the following dimensionless model in equation (80 ),
$\begin{equation}\left\{ \begin{aligned} \frac{{\text{d}x}}{{{\text{d}}\tau }} &= r\left(x - {x^2}\right) - y,\\ \frac{{\text{d}y}}{{\text{d}\tau }} &= c\left[x - \alpha y - \left(a + b{u^2}\right)y\right],\\ \frac{{\text{d}u}}{{\text{d}\tau }} &= ky + fu.\\ \end{aligned} \right.\end{equation}$
Thus, the field energy in equation (78 ) is expressed in dimensionless form as the Hamilton energy function in equation (81 ),
$\begin{align}\left\{ \begin{gathered} {H_C} = \frac{1}{2}{x^2};{H_L} = \frac{1}{{2c}}{y^2};{H_M} = \frac{1}{2}\left(au + b{u^3}\right)y,\\ H = \frac{W}{{C{V_0}^2}} = \frac{1}{2}{x^2} + \frac{1}{{2c}}{y^2} + \frac{1}{2}\left(au + b{u^3}\right)y.\\ \end{gathered} \right.\end{align}$
The parameters and variables in equation (80 ) are updated as follows
$\begin{equation}\left\{ \begin{aligned} {Z_n} &= \frac{{r\vartriangle \tau }}{{1 + r\vartriangle \tau }}{x_n},\;{W_n} = \frac{{r\vartriangle \tau }}{{1 + r\vartriangle \tau }}{y_n},\;{V_n} = \frac{{r\vartriangle \tau }}{{1 + r\vartriangle \tau }}{u_n},\\ \lambda &= 1 + r\vartriangle \tau ,\gamma = \vartriangle \tau ; d = \alpha \vartriangle \tau - \frac{1}{c},\\ a^{\prime} &= a\vartriangle \tau ,\;b^{\prime} = b\frac{{{{\left(1 + r\vartriangle \tau \right)}^2}}}{{{r^2}\vartriangle \tau }}, \beta = f\vartriangle \tau + 1,\;\delta = k\vartriangle \tau .\\ \end{aligned} \right.\end{equation}$
And then, a memristive map is given as
$\begin{equation}\left\{ \begin{aligned} {Z_{n + 1}} &= \lambda \left({Z_n} - {Z_n}^2\right) - \gamma {W_n},\\ {W_{n + 1}} &= c\left[\gamma {Z_n} - d{W_n} - \left(a^{\prime} + b^{\prime} V_n^2\right){W_n}\right],\\ {V_{n + 1}} &= \delta {W_n} + \beta {V_n}.\\ \end{aligned} \right.\end{equation}$
An energy function of the discrete form in equation (83 ) can be obtained as follows
$\begin{align}\left\{ \begin{aligned} & {H_{Cn}} = \frac{1}{2}Z_n^2,{H_{Ln}} = \frac{1}{{2c}}W_n^2,{H_{Mn}} = \frac{1}{2}\left(a^\prime{V_n} + b^\prime{V_n}^3\right){W_n},\\ & {H_n} = \frac{1}{2}Z_n^2 + \frac{1}{{2c}}W_n^2 + \frac{1}{2}\left(a^\prime{V_n} + b^\prime{V_n}^3\right){W_n}{\text{.}} \end{aligned} \right.\end{align}$
Li et al [95] considered a scenario in which the intracellular and extracellular media exhibit distinct electromagnetic field gradients and energy levels on either side of the cell membrane. To emulate the resulting variability in capacitive properties and the influence of membrane material characteristics, they incorporated two capacitors connected via a piezoelectric component. Additionally, two memristors were introduced into the neural circuit illustrated in figure 12 to account for specific field-induced effects.
Figure 12. A neural circuit with a piezoelectric membrane. C1 and C2 denote two capacitors, L1 and L2 are two inductors, R1 and R2 are two linear resistors, PC denotes a piezoelectric ceramic, E1 and E2 are two voltage sources, and M(φ) denotes two MFCM. |
The correlation of physical variables in figure 12 is described as follows
$\begin{equation}\left\{ \begin{aligned} {C_1}\frac{{\text{d}{V_{C1}}}}{{\text{d}t}} &= {i_{M1}} - {i_{L1}} - {i_s},\quad\\ {C_2}\frac{{\text{d}{V_{C2}}}}{{\text{d}t}} &= {i_{M2}} - {i_{L2}} + {i_s},\\ {L_1}\frac{{\text{d}{i_{L1}}}}{{\text{d}t}} &= {V_{C1}} + {E_1} - {i_{L1}}{R_1},\quad\\ {L_2}\frac{{\text{d}{i_{L2}}}}{{\text{d}t}} &= {V_{C2}} + {E_2} - {i_{L2}}{R_2},\\ \frac{{\text{d}{\phi _1}}}{{\text{d}t}} &= {a_1}{\phi _1} + b{V_{C1}},\quad \\ \frac{{\text{d}{\phi _2}}}{{\text{d}t}} &= {\eta _1}{\phi _2} + \mu {V_{C2}}.\\ \end{aligned} \right.\end{equation}$
The current in equation (85 ) is respectively approached by
$\begin{equation}\left\{ \begin{aligned} {i_s} &= \left({V_{C1}} - {V_{C2}}\right)/{R_S},\\ {i_{M1}} &= {M_1}\left({\phi _1}\right){V_{C1}} = \left({\alpha _1} + 3{\beta _1}{\phi _1}^2\right){V_{C1}},\\ {i_{M2}} &= {M_2}\left({\phi _2}\right){V_{C2}} = {\phi _2}{V_{C2}}.\\ \end{aligned} \right.\end{equation}$
The dimensionless variables and parameters are defined in equation (87 ).
$\begin{equation}\left\{\begin{aligned}x^{\prime}&=\frac{{{V_{C1}}}}{{{E_1}}},y^{\prime} = \frac{{{V_{C2}}}}{{{E_1}}},z^{\prime} = \frac{{{i_{L1}}{R_1}}}{{{E_1}}},w^{\prime} = \frac{{{i_{L2}}{R_1}}}{{{E_1}}},\\ {\varphi _1} &= \frac{{{\phi _1}}}{{{R_1}{C_1}{E_1}}}\!\!, {\varphi _2} = \frac{{{\phi _2}}}{{{R_1}{C_1}{E_1}}}\!,\\ \tau &= \frac{t}{{{R_1}{C_1}}},\alpha ^{\prime} = {\alpha _1}{R_1},\beta ^{\prime} = {\beta _1}{C_1}^2{E_1}^2{R_1}^3,\\ \alpha {^{^{\prime\prime}}} &= {R_1}^2{E_1}{C_1}, {\delta _1} = \frac{{{R_1}}}{{{R_S}}},{\delta _2} = \frac{{{C_1}}}{{{C_2}}},\\ {\delta _3} &= \frac{{{R_1}^2{C_1}}}{{{L_1}}},{\delta _4} = \frac{{{R_1}^2{C_1}}}{{{L_2}}},{\delta _5} = \frac{{{E_2}}}{{{E_1}}},\\ {\delta _6} &= \frac{{{R_2}}}{{{R_1}}},a = {a_1}{R_1}{C_1}, \eta = {\eta _1}{R_1}{C_1}.\\ \end{aligned} \right.\end{equation}$
Therefore, a neuron with piezoelectric membrane can be described by
$\begin{equation}\left\{\begin{aligned}\frac{{{\text{d}}x^{\prime}}}{{{\text{d}}\tau }} &= \left(\alpha ^{\prime} + 3\beta ^\prime{\varphi _1}^2\right)x^{\prime} - z^{\prime} - {\delta _1}\left(x^{\prime} - y^{\prime}\right),\\ \frac{{{\text{d}}y^{\prime}}}{{{\text{d}}\tau }} &= {\delta _2}\left[\alpha ^{^{\prime\prime}}{y^{\prime}}{\varphi _2} - w^{\prime} + {\delta _1}\left(x^{\prime} - y^{\prime}\right)\right],\\ \frac{{{\text{d}}z^{\prime}}}{{{\text{d}}\tau }} &= {\delta _3}\left(x^{\prime} + 1 - z^{\prime}\right),\\ \frac{{{\text{d}}w^{\prime}}}{{{\text{d}}\tau }} &= {\delta _4}\left(y^{\prime} + {\delta _5} - {\delta _6}w^{\prime}\right),\\ \frac{{{\text{d}}{\varphi _1}}}{{{\text{d}}\tau }} &= a{\varphi _1} + bx^{\prime},\\ \frac{{{\text{d}}{\varphi _2}}}{{{\text{d}}\tau}} &= \eta {\varphi _2} + \mu y^{\prime}.\\ \end{aligned}\right.\end{equation}$
The physical energy in figure 12 and its equivalent dimensionless energy form are presented as follows
$\begin{equation}\left\{\begin{aligned}W&=\frac{1}{2}{C_1}{V_{C1}}^2 + \frac{1}{2}{C_2}{V_{C2}}^2 + \frac{1}{2}{L_1}i_{L1}^2 + \frac{1}{2}{L_2}i_{L2}^2 \\&+ \frac{1}{2}{L_{M1}}i_{M1}^2 + \frac{1}{2}{L_{M2}}i_{M2}^2\\ &= \frac{1}{2}{C_1}{V_{C1}}^2 + \frac{1}{2}{C_2}{V_{C2}}^2 + \frac{1}{2}{L_1}i_{L1}^2 + \frac{1}{2}{L_2}i_{L2}^2 \\&+ \frac{1}{2}{\phi _1}{i_{M1}} + \frac{1}{2}{\phi _2}{i_{M2}},\\ H &= \frac{W}{{{C_1}E_1^2}} = \frac{1}{2}{{x^{\prime}}^2} + \frac{1}{{2{\delta _2}}}{{y^{\prime}}^2} + \frac{1}{{2{\delta _3}}}{{z^{\prime}}^2} \\ &+ \frac{1}{{2{\delta _4}}}{{w^{\prime}}^2} + \frac{1}{2}\left(\alpha ^\prime{\varphi _1} + 3\beta ^{\prime}\varphi _1^3\right)x^{\prime} + \frac{1}{2}\alpha ^{^{\prime\prime}}\varphi _2^2y^{\prime}.\\ \end{aligned}\right.\end{equation}$
The parameters and variables in equation (88 ) are given as follows
$\begin{equation}\left\{ \begin{aligned}{x_n} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{x^{^{\prime}}_n},{y_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{y^{^{\prime}}_n},\\{z_n} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{z^{^{\prime}}_n}, {w_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{w^{^{\prime}}_n}, \\ {u_n} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{\varphi _1}_n,{v_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{\varphi _2}_n,\\{\alpha _1} &= \frac{{1 + r\Delta \tau }}{r}\alpha ^{^{\prime\prime}}, \beta = \frac{{\beta ^{\prime}{{(1 + r\Delta \tau )}^2}}}{{\Delta \tau {r^2}}}, \\ \alpha &= \alpha ^{\prime}\Delta \tau ,{\lambda _1} = \Delta \tau ,{\lambda _2} = {\delta _1}\Delta \tau ,\\{\lambda _3} &= 1 - \frac{1}{{{\lambda _2}}}, {\lambda _4} = \frac{1}{{{\delta _2}{\lambda _2}}} - 1,{\lambda _5} = {\delta _3}\Delta \tau , \\ {\lambda _6} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }},{\lambda _7} = 1 - \frac{1}{{{\lambda _5}}},{\lambda _8} = {\delta _4}\Delta \tau ,\\{\lambda _9} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{\delta _5}, {\lambda _{10}} = {\delta _6} - \frac{1}{{{\lambda _8}}}, \\ {a_2} &= 1 + a\Delta \tau ,{b_1} = b\Delta \tau ,{\eta _2} = 1 + \eta \Delta \tau ,\\{\mu _1} &= \mu \Delta \tau , {{u^{^{\prime\prime}}}_{{\text{ext}}}} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{u^{^{\prime}}_{{\text{ext}}}}. \\ \end{aligned}\right.\end{equation}$
And then a map neuron can be described as follows
$\begin{equation}\left\{ \begin{aligned}{x_{n + 1}} &= \left(\alpha + 3\beta {u_n}^2\right){x_n} - {\lambda _1}{z_n} - {\lambda _2}\left({\lambda _3}{x_n} - {y_n}\right),\\ {y_{n + 1}} &= {\delta _2}\left[{\alpha _1}{y_n}{v_n} - {\lambda _1}{w_n} + {\lambda _2}\left({x_n} - {\lambda _4}{y_n}\right)\right],\\ {z_{n + 1}} &= {\lambda _5}\left({x_n} + {\lambda _6} - {\lambda _7}{z_n}\right),\\ {w_{n + 1}} &= {\lambda _8}\left({y_n} + {\lambda _9} - {\lambda _{10}}{w_n}\right),\\ {u_{n + 1}} &= {a_2}{u_n} + {b_1}{x_n},\\ {v_{n + 1}} &= {\eta _2}{v_n} + {\mu _1}{y_n}.\\ \end{aligned}\right.{\kern 1pt}\end{equation}$
Similarly, a energy function can be estimated as
$\begin{align}{H_n} & = \frac{1}{2}{x_n}^2 + \frac{1}{{2{\delta _2}}}{y_n}^2 + \frac{1}{{2{\lambda _5}}}{z_n}^2 \nonumber\\ &\quad + \frac{1}{{2{\lambda _8}}}{w_n}^2 + \frac{1}{2}\left(\alpha {u_n} + 3\beta {u_n}^3\right){x_n} + \frac{1}{2}{\alpha _1}v_n^2{y_n}.\end{align}$
Figure 13. A memristor-coupled neural circuit. NR is a nonlinear resistor, C denotes a capacitor, L means an inductor, R is a linear resistors and M(φ) is a MFCM. |
The relations of the physical variables are approached as follows
$\begin{equation}\left\{\begin{aligned}C\frac{{{\text{d}}V}}{{{\text{d}}t}} &= \frac{r}{\rho }\left(V - \frac{{{V^2}}}{{{V_0}}}\right) - {i_L},\\ L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}} &= V - {i_L}R - {V_M},\\ \frac{{{\text{d}}\varphi }}{{{\text{d}}t}} &= A\varphi + B{V_M}.\\ \end{aligned}\right.\end{equation}$
The parameters and variables in equation (93 ) are updated by
$\begin{align}\left\{ \begin{gathered} x = \frac{V}{{{V_0}}},y = \frac{{\rho {i_L}}}{{{V_0}}},\tau = \frac{t}{{C\rho }},z = \frac{\varphi }{{{V_0}C\rho }},\xi = \frac{1}{{\mu L{V_0}}},\\ \alpha = \frac{{C{\rho ^2}}}{L},\beta = \frac{{C\rho R}}{L},A^{\prime} = AC\rho ,B^{\prime} = \frac{B}{{\mu C{\rho ^2}{V_0}}}.\\ \end{gathered} \right.\end{align}$
Therefore, the circuit equations in equation (93 ) can be replaced as follows
$\begin{align}\left\{ \begin{gathered} \frac{{{\text{d}}x}}{{{\text{d}}\tau }} = r\left(x - {x^2}\right) - y,\\ \frac{{{\text{d}}y}}{{{\text{d}}\tau }} = \alpha x - \beta y - \xi \frac{y}{z},\\ \frac{{{\text{d}}z}}{{{\text{d}}\tau }} = A^{\prime} z + B^{\prime}\frac{y}{z}.\\ \end{gathered} \right.\end{align}$
From figure 13, the field energy and its dimensionless form are given by
$\begin{equation}\left\{\begin{aligned}W&=\frac{1}{2}C{V^2}\; + \frac{1}{2}Li_L^2 + \frac{1}{2}\varphi {i_M} \\&= CV_0^2\left(\frac{1}{2}{x^2} + \frac{1}{{2\alpha }}{y^2} + \frac{1}{2}yz\right),\\ H &= \frac{W}{{CV_0^2}} = \frac{1}{2}{x^2} + \frac{1}{{2\alpha }}{y^2} + \frac{1}{2}yz.\\ \end{aligned}\right.\end{equation}$
The variables (x, y, z) and parameters are linear transformed as follows
$\begin{equation}\left\{\begin{aligned}{u_n}&=\frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{x_n},{v_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{y_n},\\{w_n} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{z_n}, \lambda = 1 + r\Delta \tau,\\ a &= 1 - \beta \Delta \tau ,b = \frac{{\xi r{{\left(\Delta \tau \right)}^2}}}{{1 + r\Delta \tau }},c = 1 + A^{\prime}\Delta \tau,\\d &= \Delta \tau , e = \alpha \Delta \tau ,f = \frac{{B^{\prime} r{{\left(\Delta \tau \right)}^2}}}{{1 + r\Delta \tau }}.\\ \end{aligned}\right.\end{equation}$
Therefore, a new map neuron can be presented by
$\begin{equation}\left\{ \begin{aligned}{u_{n + 1}} &= \lambda \left({u_n} - {u_n}^2\right) - d{v_n},\\ {v_{n + 1}} &= e{u_n} + a{v_n} - b{v_n}/{w_n},\\ {w_{n + 1}} &= c{w_n} + f{v_n}/{w_n}.\\ \end{aligned}\right.\end{equation}$
The energy function for map neuron in equation (98 ) can be approached as follows
$\begin{equation}\left\{ \begin{aligned}H_n^C &= \frac{1}{2}{u_n}^2,H_n^L = \frac{1}{{2e}}{v_n}^2,H_n^M = \frac{1}{2}{v_n}{w_n},\\ {H_n} &= H_n^C + H_n^L + H_n^M = \frac{1}{2}{u_n}^2 + \frac{1}{{2e}}{v_n}^2 + \frac{1}{2}{v_n}{w_n}.\\ \end{aligned}\right.\end{equation}$
Figure 14. Memristor-coupled neural circuit. NR is a nonlinear resistor, C denotes a capacitor, L means an inductor, E denotes a voltage source, R is a linear resistors and M(q) is a CCM. |
In figure 14, the circuit equations are described as follows
$\begin{equation}\left\{\begin{aligned}C\frac{{{\text{d}}V}}{{{\text{d}}t}}&={i_s} - {i_L} + \frac{r}{\rho }\left(V - \frac{{{V^2}}}{{{V_0}}}\right) - \sin \left(q\right){V_M},\\ L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}} &= V - R{i_L} + E,\\ \frac{{{\text{d}}q}}{{{\text{d}}t}}&=\beta q +{i_M}.\\\end{aligned}\right.\end{equation}$
The dimensionless variables and parameters in equation (100 ) are given in equation (101 ).
$\begin{equation}\left\{\begin{aligned}x&=\frac{V}{{{V_0}}},y=\frac{{\rho{i_L}}}{{{V_0}}},z = \frac{q}{{C{V_0}}},\tau = \frac{t}{{\rho C}}, \\a &= \frac{E}{{{V_0}}},b = \frac{R}{\rho }, c = \frac{{{\rho ^2}C}}{L}, \xi = \frac{\rho }{{{R_s}}},\\{u_s} &= \frac{\rho }{{{V_0}}}{i_s},\alpha = \rho ,k = C{V_0},{k_q} = \beta \rho C.\\ \end{aligned}\right.\end{equation}$
And then an oscillator model is presented by
$\begin{equation}\left\{\begin{aligned}\frac{{{\text{d}}x}}{{{\text{d}}\tau }} &= - yr\left(x - {x^2}\right) + {u_s} - \alpha \sin \left(kz\right)x,\\ \frac{{{\text{d}}y}}{{{\text{d}}\tau }} &= c\left( {x - by + a} \right)\!,\\ \frac{{{\text{d}}z}}{{{\text{d}}\tau }} &= \alpha \sin \left(kz\right)x + {k_q}z.\\ \end{aligned}\right.\end{equation}$
The physical energy in the neural circuit in figure 14 and its dimensionless form is approached by
$\begin{align}\left\{ \begin{gathered} {E_{CLM}} = \frac{1}{2}C{V^2} + \frac{1}{2}L{i^2} + \frac{1}{2}qV,\\ H = \frac{{{E_{CLM}}}}{{CV_0^2}} = \frac{1}{2}{x^2} + \frac{1}{{2c}}{y^2} + \frac{1}{2}zx.\\ \end{gathered} \right.\end{align}$
The variables from the oscillator model in equation (102 ) are linear transformed in equation (104 ).
$\begin{equation}\left\{\begin{aligned}{u_n}&=\frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{x_n},{v_n} = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{y_n}, \\ {w_n} &= \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}{z_n}, \gamma = \frac{{r{{\left( {\Delta \tau } \right)}^2}}}{{1 + r\Delta \tau }}{u_{s}},\\ \delta &= \Delta \tau ,\lambda = 1 + r\Delta \tau ,\sigma = \alpha \Delta \tau ,\\ \kappa &= \frac{{1 + r\Delta \tau }}{{r\Delta \tau }}k, d = c\Delta \tau , s = \frac{{r\Delta \tau }}{{1 + r\Delta \tau }}a,\\ e &= - \frac{1}{{c\Delta \tau }} + b,f = {k_q}\Delta \tau + 1.\\ \end{aligned} \right.\end{equation}$
And then a neuron map is obtained as follows
$\begin{equation}\left\{\begin{aligned}{u_{n + 1}}&=\lambda \left({u_n} - {u_n}^2\right) - \delta {v_n} - \sigma \sin \left(\kappa {w_n}\right){u_n} + \gamma ,\\ {v_{n + 1}} &= d\left( {{u_n} + s - e{v_n}} \right),\\ {w_{n + 1}} &= \sigma \sin \left(\kappa {w_n}\right){u_n} + f{w_n}.\\ \end{aligned}\right.\end{equation}$
Consequently, a similar discrete energy function for equation (105 ) can be approached by
$\begin{align}{H_n} = \frac{1}{2}u_n^2 + \frac{1}{{2d}}v_n^2 + \frac{1}{2}{u_n}{v_n}.\end{align}$
In summary, modeling discrete neurons based on memristor circuits is a highly promising technical path in the field of neuromorphic computing. The core lies in leveraging the inherent physical properties of memristors to naturally simulate the key behaviors of biological neurons. The core principle of this work is to discretize the dynamic process of neurons. The continuous processes, such as membrane potential accumulation, threshold firing, and refractory period of biological neurons, are transformed into iterative operations in discrete time steps.
4. Memristor-coupled circuit design of chaotic map
Design of the chaotic maps is interesting research, and many chaotic maps have been proposed by using different methods of mathematics and control. However, the physical circuit expression of the chaotic map systems described by mathematical models is very important research. At first, a two-dimensional chaotic map system [98] is described by
$\begin{equation}\left\{\begin{aligned}{x_{n + 1}}&={\mu _1}{x_n} - {\mu _2}x_n^2 - c{x_n}{y_n},\\ {y_{n + 1}} &= b{y_n} + a{x_n}.\\ \end{aligned}\right.\end{equation}$
At first, the map equations presented in equation (107 ) are equivalent to the form of the differential equations by applying the linear transformation, and the variables in equation (107 ) are defined as follows
$\begin{align}{w_n} = \frac{{1 + \Delta \tau }}{{\Delta \tau }}{x_n},{u_n} = \frac{{1 + \Delta \tau }}{{\Delta \tau }}{y_n}.\end{align}$
As a result, equation (107 ) is updated by
$\begin{align}\left\{\begin{aligned}{w_{n + 1}}\frac{{\Delta \tau }}{{1 + \Delta \tau }} &= \mu \left({w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }} - {\left({w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }}\right)^2}\right)\\ &- c{w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }}{u_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }},\\ {u_{n + 1}}\frac{{\Delta \tau }}{{1 + \Delta \tau }} &= b{u_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }} + a{w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }}.\\ \end{aligned}\right.\end{align}$
Furthermore, equation (109 ) is rewritten as follows
$\begin{align}\left\{ \begin{aligned} & {w_{n + 1}}\frac{{\Delta \tau }}{{1 + \Delta \tau }} - {w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }} = \mu \left({w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }} \right.\\ & \left.- {\left({w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }}\right)^2}\right) - c{w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }}{u_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }} - {w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }},\\ & {u_{n + 1}}\frac{{\Delta \tau }}{{1 + \Delta \tau }} - {u_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }} = b{u_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }}\\ & + a{w_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }} - {u_n}\frac{{\Delta \tau }}{{1 + \Delta \tau }}. \end{aligned} \right.\end{align}$
According to dx/dτ = (xn+1–xn)/Δτ, equation (110 ) is updated as follows
$\begin{align}\left\{ \begin{array}{*{20}{l}} {\frac{{{\text{d}}w}}{{{\text{d}}\tau }} = \frac{1}{{\Delta \tau }}({\mu _1}w - {\mu _2}{w^2}\frac{{\Delta \tau }}{{1 + \Delta \tau }}) - cwu\frac{{\Delta \tau }}{{1 + \Delta \tau }} - w),} \\[8pt] {\frac{{{\text{d}}u}}{{{\text{d}}\tau }} = \frac{1}{{\Delta \tau }}(bu + aw - u).} \end{array}\right.\end{align}$
In fact, equation (111 ) can be simplified by
$\begin{align}\left\{ {\begin{array}{*{20}{l}} {\frac{{{\text{d}}w}}{{{\text{d}}\tau }} = {\mu _1^{\prime} }w - {\mu _2^{\prime} }{w^2} - {c^{\prime} }wu,} \\[8pt] {\frac{{{\text{d}}u}}{{{\text{d}}\tau }} = {b^{\prime} }u + {a^{\prime} }w,} \end{array}} \right.\end{align}$
where $\begin{align}{\mu _1^{\prime} }&= \frac{{{\mu _1} - 1}}{{\Delta \tau }},{\mu _2^{\prime} }= \frac{{{\mu _2}}}{{1 + \Delta \tau }},c^{\prime} = \frac{c}{{1 + \Delta \tau }},\nonumber\\ b^{\prime} & = \frac{{b - 1}}{{\Delta \tau }},a^{\prime} = \frac{a}{{\Delta \tau }}.\end{align}$
The dimension of the dimensionless variables is defined as follows 112 ) can be updated by
$\begin{align}w = \frac{v}{{{v_0}}},u = \frac{\varphi }{{\rho C{v_0}}},\tau = \frac{t}{{\rho C}},\end{align}$
where v0 is the reference voltage, C means the capacitance of a capacitor, ρ denotes the resistance of a resistor, and then equation ( $\begin{equation}\left\{ \begin{array}{*{20}{l}} C\frac{{{\text{d}}v}}{{{\text{d}}t}} = \frac{1}{\rho }\left({\mu _1^{\prime} }v - {\mu _2^{\prime} }\frac{{{v^2}}}{{{v_0}}}\right) - \frac{{c^{\prime}}}{{C{\rho ^2}{v_0}}}v\varphi ,\\ \frac{{{\text{d}}\varphi }}{{{\text{d}}t}} = \frac{{b^{\prime}}}{{C\rho v_0^2}}\varphi + \frac{{a^{\prime}}}{{v_0^2}}v.\end{array} \right.\end{equation}$
Furthermore, equation (115 ) is simplified as follows
$\begin{align}\left\{ \begin{array}{*{20}{l}} {C\frac{{{\text{d}}v}}{{{\text{d}}t}} = \frac{1}{\rho }\left({\mu _1^{\prime} }v - {\mu _2^{\prime} }\frac{{{v^2}}}{{{v_0}}}\right) - \alpha v\varphi ,} \\ {\frac{{{\text{d}}\varphi }}{{{\text{d}}t}} = \beta \varphi + \gamma v,} \end{array}\right.\end{align}$
where $\begin{align}\alpha = \frac{{c^{\prime}}}{{C{\rho ^2}{v_0}}},\beta = \frac{{b^{\prime}}}{{C\rho v_0^2}},\gamma = \frac{{a^{\prime}}}{{v_0^2}}.\end{align}$
In fact, the higher-order terms in equation (116 ) can be defined as a nonlinear resistor as follows 116 ) can be defined as an MFCM in the following, 116 ) and Kirchhoff’s law, the current relationship between electronic components in a circuit is described as follows
$\begin{align}{i_{NR}} = - \frac{1}{\rho }\left({\mu _1^{\prime} }v - {\mu _2^{\prime} }\frac{{{v^2}}}{{{v_0}}}\right),\end{align}$
where ρ is the resistance of the nonlinear resistor, $\left({\mu _1^{\prime}},\, {\mu _2^{\prime}}\right)$ denotes dimensionless parameters of the nonlinear resistor, v0 means cut-off voltage and v is the voltage across the nonlinear resistor. The nonlinear terms and second equation in the equation ( $\begin{equation}\left\{\begin{aligned}{i_M}&=\alpha v\varphi ,\\ \frac{{{\text{d}}\varphi }}{{{\text{d}}t}} &= \beta \varphi + \gamma v,\\ \end{aligned}\right.\end{equation}$
where φ is magnetic flux, v is the voltage across the memristor, and (α, β, γ) denotes the parameters related to the memristor material. Based on equation ( $\begin{align}{i_M} = - {i_{NR}} - {i_C}.\end{align}$
Therefore, the circuit corresponding to the discrete system presented in equation (107 ) is designed in figure 15.
Figure 15. An MFCM-coupled nonlinear circuit without an inductor. NR is a nonlinear resistor, C denotes a capacitor, and M is an MFCM. |
The result in figure 8 illustrates that a map system of a mathematical model is verified by using circuits designed with physical electronic components (a nonlinear resistor, an MFCM, and a capacitor). In fact, the dimension of the dimensionless variables in equation (112 ) can also be defined by 112 ) can be updated by
$\begin{align}w = \frac{{\rho {i_L}}}{{{v_0}}},u = \frac{{q{\rho ^2}}}{{L{v_0}}},\tau = \frac{{\rho t}}{L},\end{align}$
where v0 is the reference voltage, L means the inductance of the induction coil, ρ denotes the resistance of the resistor, and then equation ( $\begin{equation}\left\{\begin{aligned}L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}} &= \frac{{{\mu _1^{\prime} }}}{{v_0^2}}\rho {i_L} - \frac{{{\mu _2^{\prime} }}}{{v_0^2}}\frac{{{\rho ^2}i_L^2}}{{{v_0}}} - \frac{{{\rho ^3}c^{\prime}}}{{Lv_0^3}}q{i_L},\\ \frac{{{\text{d}}q}}{{{\text{d}}t}} &= \frac{{b^{\prime}\rho }}{L}q + a^\prime{i_L}.\\ \end{aligned}\right.\end{equation}$
Furthermore, equation (122 ) is simplified as follows
$\begin{align}\bigg\{ \begin{array}{*{20}{l}} {L\frac{{{\text{d}}{i_L}}}{{{\text{d}}t}} = \eta \rho {i_L} - \varsigma \frac{{{\rho ^2}i_L^2}}{{{v_0}}} - \kappa q{i_L},} \\ {\frac{{{\text{d}}q}}{{{\text{d}}t}} = \delta q + \lambda {i_L},} \end{array}\end{align}$
where $\begin{align}\eta = \frac{{{\mu _1^{\prime} }}}{{v_0^2}},\varsigma = \frac{{{\mu _2^{\prime} }}}{{v_0^2}},\kappa = \frac{{{\rho ^3}c^{\prime}}}{{Lv_0^3}},\delta = \frac{{b^{\prime}\rho }}{L},\lambda = a^{\prime}.\end{align}$
The higher-order terms in equation (123 ) are considered as a nonlinear resistor, and it is defined by 123 ) are defined as a CCM in the following. 123 ) and Kirchhoff’s law, the voltage relationship between electronic components in a circuit is described as follows
$\begin{align}{v_{NR}} = \eta \rho {i_L} - \varsigma \frac{{{\rho ^2}{i_L}^2}}{{{v_0}}},\end{align}$
where ρ is the resistance of the nonlinear resistor, (η, ζ) denotes dimensionless parameters of the nonlinear resistor, v0 means the cut-off voltage, and iL is the current across the nonlinear resistor. The nonlinear terms and the second equation in equation ( $\begin{equation}\left\{\begin{aligned}{v_M}&=\kappa q{i_L},\\ \frac{{{\text{d}}q}}{{{\text{d}}t}} &= \delta q + \lambda {i_L},\\ \end{aligned}\right.\end{equation}$
where q is charge, iL is the current across the memristor, and (κ, δ, λ) denotes the parameters related to the memristor material. Based on equation ( $\begin{align}{v_L} = {v_{NR}} - {v_M}.\end{align}$
Therefore, the circuit corresponding to the discrete system presented in equation (107 ) is designed in figure 16.
Figure 16. A CCM-coupled nonlinear circuit without a capacitor. NR is a nonlinear resistor, L means an inductor, and M is a CCM. |
A brief summary of the modeling steps of the memristive map based on the memristor-coupled circuit is described as follows:
| Step 1: Design a memristor-coupled circuit by connecting and paralleling the memristor with other circuit components. | |
| Step 2: Calculate the output state equation of the memristor-coupled circuit based on Kirchhoff’s laws. | |
| Step 3: Based on the physical dimensions of electronic components, the dimensional calculation of circuit equations is carried out to obtain dimensionless dynamic equations. | |
| Step 4: The dimensionless dynamic equation is transformed linearly to obtain the corresponding map model. |
Similarly, the physical verification steps of the chaotic map based on the memristor-coupled circuit are described as follows:
| Step 1: Based on the physical quantities and mathematical models of the circuit components, the map equations are dimensionally transformed to obtain the corresponding differential equation forms. | |
| Step 2: Simplify the form of the differential equation obtained in the first step. | |
| Step 3: The differential equation is transformed into a reasonable corresponding mathematical model of electronic components. | |
| Step 4: Based on Kirchhoff’s laws and the results obtained in the third step, a reasonable physical circuit for the electronic components can be designed. |
5. Open problems
(1) The map model obtained based on the memristor-coupled circuit does not exhibit hyperchaotic behavior. Establishing hyperchaotic memristor maps or neuron maps based on memristor coupling circuits is an open research topic.
Potential physical and mathematical causes: The generation of hyperchaos requires that the system’s phase space has sufficient dimensions to extend and fold the trajectory. Although the introduction of discrete memristors can increase the mathematical dimension of the mapping, this coupling method does not always effectively generate exponential divergence simultaneously in multiple directions. Some coupling methods may merely ‘project’ chaotic behavior onto new dimensions rather than generating independent chaotic oscillations in multiple dimensions. For instance, simple substitution or series connection of memristors may not be sufficient to create the complex phase space structure required for multiple positive Lyapunov exponents.
(2) The two-dimensional map system has been physically verified by the memristor-coupled circuit. The memristor-coupled circuits of three-dimensional or higher-dimensional map models have been confirmed to be an open research field.
Key implementation challenges: Converting three-dimensional or higher-dimensional discrete mappings into physical circuits means that nonlinear coupling relationships among multiple state variables need to be precisely implemented. This usually involves a large number of operational amplifiers, analog multipliers, memristor model circuits, and addition/subtraction circuits. As the dimension increases, the complexity of the circuit grows exponentially, leading to a significant increase in power consumption, volume, and cost. At the same time, it puts forward extremely high requirements for the layout and wiring of the circuit board to reduce the influence of parasitic parameters
(3) The memristor-coupled circuit design for chaotic maps, without addressing the circuit implementation of neuron maps. The design of the circuit implementation of neuron maps is an open research direction.
Deep-seated challenges and opportunities: memristors mainly play the role of a ‘coupling device’ that generates and enhances nonlinearity in chaotic mapping. However, to construct a neuron model, the memristor needs to simulate the complex electrochemical behavior of biological neurons, including the transmembrane migration of ions, the generation and propagation of action potentials, the refractory period, and the conversion of multiple discharge modes (such as peak discharge and cluster discharge). This requires that the memristor model not only possess nonlinearity but also have complex internal dynamics and multi-time-scale characteristics, which poses higher demands on device physics and materials science.
6. Conclusions
This review provides systematic theoretical and methodological support for solving the problem of converting between the two types of systems based on electronic components by establishing a bidirectional design framework between continuous and discrete nonlinear systems. Research shows that the dynamics of continuous circuits, such as memristor oscillators, can be precisely reduced to discrete mapping models through modeling, and the connection topology of circuit components directly determines the dynamic form of the discrete system. Conversely, the iterative equation of the target discrete system can also be physically implemented through a modular circuit structure, thereby supporting reverse design from the computational model to the hardware system. This framework not only profoundly reveals the intrinsic connection between continuous and discrete dynamics in nonlinear circuits but also provides an extensible methodological foundation for the design of dedicated nonlinear hardware systems for applications such as secure communication, true random number generation, and brain-like computing.
Data availability
No any data are used.
Conflict of interest
The authors declare no conflict of interest with this publication.
Authorship contribution statement
Feifei Yang: Writing-original draft, Methodology, Writing-final version. Xinlin Song: Supervision, Methodology. Ge Zhang: Methodology. Huiping Yin: Supervision, Funding acquisition. Jiangxing Chen: Supervision.