## 2 Two Dimensional Generalized Discrete Memristor Coupling Model  
In 2014, [PERSON] [29] proposed three mathematical expressions of time-varying memristor (i.e., CMM), namely three different memristor models. At the same time, according to the input signal of the memristor, each model is divided into charge-controlled memristor and flux-controlled memristor. The mathematics expression of these are shown in Table 1, where \(M(\cdot)\) is the memristance function based on the charge-controlled memristor, while \(W(\cdot)\) is the memristance function based on the magnetic flux-controlled memristor. In the ideal model, \(M(\cdot)\) is only a function of charge \(q\), and the derivative of charge \(q\) is the current \(i\) through the device, while \(W(\cdot)\) is only a function of magnetic flux \(\phi\). The \begin{table}
\begin{tabular}{c c c} \hline \hline
**CMs** & **Charge-Controlled** & **Flux-Controlled** \\ \hline ideal & \(\left\{\begin{array}{l}v(t)=M(q)i(t)\\ dq/dt=i(t)\\ v(t)=M(x)i(t)\\ dx/dt=f(x,i)\\ \end{array}\right.\) & \(\left\{\begin{array}{l}i(t)=W(\phi)v(t)\\ d\phi/dt=v(t)\\ i(t)=W(x)v(t)\\ dx/dt=v(x,v)\\ \end{array}\right.\) \\ generalized & \(\left\{\begin{array}{l}v(t)=M(x,i)i(t)\\ dx/dt=i(x,i)\\ \end{array}\right.\) & \(\left\{\begin{array}{l}i(t)=W(\phi)v(t)\\ d\phi/dt=v(t)\\ i(t)=W(x)v(t)\\ dx/dt=v(x,v)\\ \end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mathematical models of three continuous-time memristors. derivative of magnetic flux \(\phi\) is the voltage \(v\) at both ends of the device. In the general model, \(M(\cdot)\) is only a function of the system state variable \(x\), and the derivative of the system variable \(x\) is a function of the state variable \(x\) and the current \(i\) flowing through the device, denoted by \(f(x,i)\), while \(W(\cdot)\) is also a function of the system state variable \(x\), and the derivative of the system variable \(x\) is a function of the state variable \(x\) and the voltage \(v\) both ends of the device, denoted by \(g(x,v)\). In the generalized model, \(M(\cdot)\) is the function of the system state variable \(x\) and the current \(i\) flowing through the device, and the derivative of the system variable \(x\) is a function of the state variable \(x\) and the current \(i\) flowing through the device, denoted by \(f(x,i)\), while \(W(\cdot)\) is the function of the system state variable \(x\) and the voltage \(v\) at both ends of the device, and the derivative of the system variable \(x\) is a function of the state variable \(x\) and the voltage \(v\) both ends of the device, denoted by \(g(x,v)\). In this paper, we take the ideal charge-controlled memristor model as the research object, so we can get a charge-controlled memristor characterized by voltage \(v(t)\) and current \(i(t)\), and the mathematics expression is as follows  
\[\left\{ \begin{array}{rl}&v(t)=M(q)i(t)\\ &dq(t)/dt=i(t)\end{array} \right. \tag{1}\]  
where \(v(t)\) and \(i(t)\) is the voltage and the input current of the memristor, and \(q(t)\) is the charge the memristor at time \(t\). The relationship between charge \(q(t)\) and input current \(i(t)\) is written as  
\[q(t)=q(t_{0})+\int_{t_{0}}^{t}i(\tau)d\tau, \tag{2}\]  
where \(q(t_{0})\) is the initial charge of memristor. After discretization by the backward- Euler method, the mathematical expression of the discretized model is as follows:  
\[\left\{ \begin{array}{rl}&v_{n}=M(q_{n})i_{n}\\ &q_{n+1}=q_{0}+h\sum\limits_{j=1}^{n}i_{j}\end{array} \right. \tag{3}\]  
where \(h\) represents the step size required for each iteration. Generally, suppose \(h=1\). In this paper, the memristance function \(M(q_{n})\) is based on \(TiO_{2}\) memristor model, and the mathematical expression of \(M(q_{n})\) for the function can be obtained as  
\[M(q_{n})=(R_{1}q_{n}+R_{2}q_{n}{}^{2}), \tag{4}\]  
where \(R_{1}=R_{OFF}\), \(R_{2}=R_{OFF}\mu_{V}R_{ON}/D^{2}\).