Memory Analysis for Memristors and Memristive Recurrent Neural Networks

I. Introduction

THE memristor was first defined by Chua [1] and can be described by the following mathematical model [2]

where $u(t)$, $y(t)$ are input and output of memristive systems, respectively. $x(t)$ is the state variable, $f(x(t),u(t),t)$ is a $n$-dimensional vector function and $g(x(t), u(t), t)$ is the generalized system response. Williams and his colleagues transform the concept of memristors into the physical devices [3], whose structure diagram is shown in Fig. 1. The memristor is composed of a two-layer ${\rm{TiO}}_2$ thin film, two platinum contacts, a doped region $R_{\rm on}$, and an undoped region $R_{\rm off}$. D, w are the thickness of the film and the width of the doped region, respectively. Later, Chua points out that all two-terminal non-volatile memory devices based on resistance switching are memristors, regardless of the device material and physical operating mechanisms [4].

Figure  1.  The schematic diagram of the HP memristor. (a) The diagram of the HP memristor model. (b) The circuit symbol of the memristor, showing the positive and negative polarities

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The memristor has various applications for its nano-scale size and memory property. For example, it is used to implement chaotic circuits [5], [6], memristor oscillators [7], and neural synapses [8]. Snider et al. adopt memristors in neuromorphic applications to simulate learning, adaptive and spontaneous behaviors and to implement synaptic weights in artificial neural networks [9], [10]. Pershin and Di Ventra give an experimental demonstration for associative memory with memristive neural networks [11]. Then the memristor is employed as a nonvolatile memory storage device [12], [13]. Furthermore, it has also been used to simulate the human brain’s hierarchical temporal memory, short-term, long-term memory [14], [15] and memristive recurrent neural networks. Meanwhile, memristors have also been harnessed for image processing, adaptive filters, digital logic, neuromophic engineering, digital and quantum computation [16], [17], etc.

The dynamic properties of memristors are the foundation for its applications; thus memristors with different materials and configurations are made for dynamic analysis experiments [18], [19]. Williams et al. [3] present the mathematical model for memristors and show its fingerprint characteristic with a pinched hysteresis current i-voltage v loop. Based on Williams’ mathematical model of memristors, Wang [20] derives the formula of the internal state $x(t)$ and obtains the analytical expression of the current $i(t)$ and the voltage $v(t)$. Considering the doped materials’ nonlinear drift, Biolek et al. [21] introduce window functions and give the SPICE model of memristors. Using Bernoulli dynamics, Drakakis et al. [22] derive the analytic description, $I_{\rm mres} = f(V_{\rm mres})$ which defines the relation between the current $I_{\rm mres}$ and the voltage $V_{\rm mres}$ under the assumption of nonlinear dopant drift. Wang et al. [23] and Zhang et al. [24] propose a piecewise linear (PWL) memristance model for studying dynamic properties of memristors. For single memristors, Biolek et al. [25], [26]. demonstrate a methodology to obtain the analytical solution of a memristor’s voltage/current response under the current/ voltage excitation. For the properties of multiple memristors, Bao et al. [27] give the voltage-current relationship of parallel memristors. Kim et al. [28], [29] analyze the composite behavior of multiple connected memristors under the assumption that all memristors should reach a stable state. Then they construct a memristor emulator which could be connected in serial, parallel, or hybrid, simplifying the study of multiple memristors.

The memristive recurrent neural networks (MRNNs) are presented by replacing linear resistors with memristors in classical recurrent neural networks circuits. There are some compound results about the dynamical characteristics of the MRNNs [30]–[33]. Furthermore, we found that the MRNNs are a family of neural networks [34]. The MRNNs can be region stable and convergent to a sub neural network in the family of neural networks. Such a convergent result is dependent on the initial values of memristive synapses and network states. Hence, it is important to locate the initial states of memristive synapses and analyze the memory property of memristors. Although memory analysis has been discussed in the existing literature, determining how to locate the state of a memristor is scarcely discussed. With this motivation, we investigate the memory property of a memristor based on the relation between its voltage $v(t)$ and current $i(t)$ and give the method to locate the initial states of the memristive synapses. Our analysis comprehensively includes memristors under the assumptions of both linear and nonlinear dopant drift. We also extend the methods to obtain the initial states of two memristors connected in series and parallel, whose initial states can be obtained simultaneously with only a few measurements and one integration. SPICE simulations have been conducted for each presented method. The simulation results convincingly confirm the viability of our approaches. The rest of this paper is organized as follows: in Section II, we analyze the memory property of the memristor with linear and nonlinear dopant drift under a current and a voltage source, respectively. Further discussion on the memory property of two series- or parallel-connected memristors, as well as the algorithm to locate the initial states of memristive synapses, are provided in Section III. Finally, Section IV concludes the paper.

II. Memory Analysis for One Memristor

In this section, we discuss a method to compute the initial value of a single memristor under voltage and current sources by using the memristor models with linear and nonlinear dopant drift.

A Linear Dopant Drift With Current Excitation

In this section, we consider the memory of single memristive synapses based on Williams’s memristor model [3] as follows:

where $i(t)$, $\mu_V$ are the current through the device and the average ionic mobility, respectively; $v(t)$ is the applied voltage source. Let $x(t) = {w(t)}/{D}$ be the state variable of the memristor, as in (1); then, (2) can be rewritten as

with $x(t)\in [0,1]$. Let $M(t) = R_{\rm on}x(t)+R_{\rm off}(1-x(t))$. A pinched hysteresis loop figure, the fingerprint characteristic of the memristor, can be obtained by applying a sinusoidal current source $i(t) = i_0\sin(\omega t)$ to the memristor. An HSPICE simulation is conducted and the result is shown in Fig. 2. The simulation parameters are set as following: $i_0 =$ 200 μA, $\omega = 2{\text{π}} \;{\rm{rad/s}}$, $x(t_0) = 0.1$, $t_0 = 0 \;{\rm{s}}$, $R_{\rm on} = 100 \Omega$, $r = R_{\rm off}/R_{\rm on} =$ 160, $D = 10^{-6}\;{\rm{cm}}$, $\mu_V = 10^{-10} \;{\rm{cm}}^2/{\rm{sV}}$.

Figure  2.  The dynamical characteristics of the linear HP memristor. (a) The linear memristor’s fingerprint characteristic: pinched hysteresis loop figure. (b) The change of the state $x(t)$, applied current source $i(t)$ and the corresponding voltage $v(t)$ of the memristor

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Remark 1: From (2) and (3), the memristance is variable in the interval $[R_{\rm on}, R_{\rm off}]$. The memristance will be changed when the voltage or the current source is applied to the device. The initial value of the memristor is the memristance $M(t_0) = R_{\rm on}x(t_0)+R_{\rm off}(1-x(t_0))$ before the voltage or the current source is applied. The initial value can be memorized by the memristor and it affects memristance variation. This point, however, has not been discussed in the literature. The simulation parameters are chosen by using those in [3]. We compute the initial state of the memristor with the voltmeter-ammeter method and consider two cases, i.e., current excitation and voltage excitation, as shown in Fig. 3. The developed methods are for linear and nonlinear dopant drift.

Figure  3.  Circuits for measuring the initial state of a memristor, under the excitation of (a) a current source and (b) a voltage source. $V$ and $I$ are voltmeter and ammeter, respectively

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Let $\xi = \mu_V\dfrac{R_{\rm on}}{D^2}$, then we get

and

We apply a current source at time $t = t_0$ and let $q(t) = \int i(t)dt$, $q(t_0) = 0$. Then we integrate both sides of (5) for $\forall t>t_0$, thus we have

and

Substituting (7) into (3),

and then

where $i(t)$ is the known current source, $q(t) = \int i(t)dt$, and $v(t)$ can be measured with a voltmeter. $R_{\rm off}$, $R_{\rm on}$, $\xi$ are known parameters. Therefore we get the formula for the initial state of a memristor with linear dopant drift.

Next we verify this method with an HSPICE simulation. Predetermining $x(t_0)$ to be 0.28, we run the circuit in Fig. 3(a) for 2.73 s, while other parameters are the same with those in Fig. 2. The simulation process is presented in Fig. 4. After 2.73 s, the current $i(2.73)$ and voltage $v(2.73)$ across the memristor are –198.4229 μA and –1.16128 V, respectively. Since the current source is sinusoidal, we get $q(2.73) = 3.582\times$$10^{-5}{\rm{C}}$ by integrating $i(t)$ from $t = 0$ to t = 2.73 s. Therefore we can apply the mathematical method and get $x(t_0) = 0.28$ which matches the value we predetermined. The result verifies that our method of applying current excitation to determine the initial state of a linear memristor is feasible.

Figure  4.  Simulation process of a linear memristor under the current source for 2.73 s

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B Linear Dopant Drift With Voltage Excitation

In this section, we consider the voltage excitation and drive the formula for $x(t_0)$. In (3), let $\beta = D^2/\mu_V$, and then

We apply a voltage source at time $t = t_0$ and let $\varphi(t) = \int v(t)dt$, $r = R_{\rm off}/R_{\rm on},$ and $\varphi(t_0) = 0$. Then we integrate both sides of (10) for $\forall t>t_0$,

where

It is easy to find that the constant $c$ of the integration is dependent on initial value of $x(t)$ at $t = t_0$. Therefore the memory effect of memristors is attributed to the integration constant $c$. Next we deduce the analytic expression of $i(t)$ and $v(t)$. Since $x(t)\in [0, 1]$, from (11), we get

Differentiating (13) with respect to time $t$, then we obtain

in which constant $c$ is not removed. Based on (3), we have

From (15), we can obtain $c$ by solving (15) with $i(t)\neq 0$ as

Then by (12), the initial state $x(t_0)$ can be obtained

where $v(t)$ is the applied voltage source, $i(t)$ can be measured with an ammeter, $r$, $R_{\rm on}$, $\, \beta$ are known and $\varphi(t)$ can be calculated by $\varphi(t) = \displaystyle\int v(t)dt$. So $x(t_0)$ can be deduced from (17) if it is unknown.

The result can be easily examined with a simulation. We simulate the circuit Fig. 3(b) on HSPICE. The initial state of the linear memristor is predetermined as 0.37. The applied voltage is a simple sinusoidal voltage source $v(t) = v_0\sin(\omega t)$, where $v_0 = 1 {\rm{V}}$, $\omega = 2{\text{π}} \;{\rm{rad/s}}$. The other simulation parameters are set as following: $t_0 = 0 \;{\rm{s}}$, $R_{\rm on} = 100 \Omega$, $r = 160$, $D = 10^{-6}\;{\rm{cm}}$, $\mu_V = 10^{-10} \;{\rm{cm}}^2/{\rm{sV}}$. We run the simulation for 4.21 s. At the end of the simulation, we get $v(4.21) =$ 968.5832 mV and $i(4.21) = 120.7629 \;{{\text{μ}}\rm{A}}$. Because the applied voltage source is sinusoidal, $\varphi(t)$ can be calculated from the integration from $t = 0$ to $t = 4.21 \;{\rm{s}}$: $\varphi(4.21) = 0.1196\;{\rm{Wb}}$. Thus we can apply the mathematical method in (17) and obtain

Figure  5.  Simulation process of a linear memristor under the voltage source for 4.21 s.

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which is the same with what we predetermined for $x(t_0)$. From the result, the viability of our method has been examined by the simulation.

C Nonlinear Dopant Drift With Current Excitation

In this section, we will show the methods to determine the initial state $x(t_0)$ of the memristor under the assumption of nonlinear dopant drift. For the nonlinear memristor, the descriptive model should be adjusted from (3) to

where $f(x(t))$ is a window function such that $f(0) = f(1) = 0$ ensures no drift at boundaries. The window function in model (18) is

in which $p$ is a positive integer. $f(x)$ is shown in Fig. 6 for $p = 1, 2, 5$, respectively. As $p$ increases, the curve get flatter in the middle and becomes steeper at the boundaries. If not specified, all nonlinear memristors are configured as $p = 1$ in the rest of this paper.

Figure  6.  Illustration for window function $f(x) = 1-(2x-1)^{2p}$ with $p = 1, 2, 5$

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Taking $p = 1$ in (19) and combing with (18), we get

The fingerprint characteristic of the memristor with nonlinear dopant drift, a bow-tie shape $i-v$ figure, can be generated by applying a sinusoidal current source $i(t) = i_0\sin(\omega t)$ to the memristor. Based on (20), an HSPICE simulation is performed and the result is shown in Fig 7. The simulation parameters are set as following: $i_0 = 800 \;{{\text{μ}}\rm{A}}$, $\omega = 2{\text{π}} \;{\rm{rad/s}}$, $r = 160$, $D = 10^{-6}\;{\rm{cm}}$, $\mu_V = 10^{-10} \;{\rm{cm}}^2/{\rm{sV}}$.

Figure  7.  The dynamical characteristics of the nonlinear HP memristor (a) The nonlinear memristor’s fingerprint characteristic: bow-tie shape figure. (b) The change of the state $x(t)$, applied current source $i(t)$ and corresponding voltage $v(t)$ of a nonlinear memristor

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Let $\xi = \mu_V {R_{\rm on}}/{D^2}$ and simplify (20) as

then

For the initial time $t_0$, $q(t_0) = 0$, we integrate both sides of (22) for $\forall t>t_0$

Let

and solve (23) for $x(t)$, we have

From (24), we can find the determinant relation between the constant $c$ and $x(t_0)$. In other words, $c$ includes the history information of $x(t)$. Now substitute (25) into (20), and we get

from which $c$ can be calculated by

Therefore the initial state $x(t_0)$ of the nonlinear memristor is obtained by solving (24),

where $i(t)$ is the current source, $q(t)$ can be calculated by $q(t) = \int i(t)dt$, $v(t)$ can be measured by the voltmeter, $R_{\rm off}$, $R_{\rm on}$, $\xi$ are known. Thus with only a voltmeter and related calculation, the initial state of a memristor, under the assumption of nonlinear dopant drift, can be obtained.

The result of the HSPICE simulation agrees with the method. We preset $x(t_0) = 0.53$, and then run the circuit in Fig. 3(a) for 3.67 s. The parameters of the circuit are kept the same with those in Fig. 7. The simulation result is shown in Fig. 8. At the end of the simulation, the current $i(3.67)$ and voltage $v(3.67)$ of the memristor are –701.0453 μA and –75.32032 mV, respectively. Because the current source is sinusoidal, we can calculate $q(3.67) = 1.8866\times10^{-4}{\rm{C}}$ by integrating $i(t)$ from $t = 0$ to $t = 3.67\;{\rm{s}}$. Therefore

Figure  8.  Simulation process of a nonlinear memristor under the current source for 3.67 s

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The simulation result matches the value we predetermined for $x(t_0)$. That means the method is applicable for the calculation of initial states of memristors under the nonlinear dopant drift assumption.

D Nonlinear Dopant Drift With Voltage Excitation

For nonlinear memristors, the initial state $x(t_0)$ can also be acquired through voltage excitation. From (20), we have

where $\, \beta = D^2/\mu_V$. Substitute $i(t)$ with (29) into (20), and we get

We apply the voltage source at $t = t_0$ and let $\varphi(t) = \int v(t)dt$, $r = R_{\rm off}/R_{\rm on},$ and $\varphi(t_0) = 0$. Then we integrate both sides of (30) for $\forall t>t_0$

Let

then (31) can be simplified to

The relation between $c$ and $x(t_0)$ in (32) claims that the initial state of a nonlinear memristor can be calculated from the integration constant $c$. That is to say, the memory effect of nonlinear memristors can be represented by an integration constant. Next we deduce how to calculate $c$. Change (33) to

where $v(t)$ is the applied voltage source, $i(t)$ can be measured with an ammeter, $\, \beta$ and $r$ are known, $\varphi(t)$ can be integrated from the applied voltage $v(t)$, and $\varphi(t) = \int v(t)dt$. The state $x(t)$ at time $t$ can also be calculated from (20)

Combining (34) and (35), $c$ is obtained by

Then from (32), $x(t_0)$ can be acquired by solving an equation

Since (37) is $r$-order, where $r$ is undetermined, the analytical solution of $x(t_0)$ is not easy to get. So a numerical solution is recommended.

In order to get the numerical solution of $x(t_0)$, an algorithm is presented as follows. First, construct a function $z(x)$ according to (37)

For $x(t_0) \in (0,1)$, iterate $x(t_0)$ from 0 to 1, with a small increment $\delta$ (e.g., $\delta = 0.0001$) in each iteration. Notice that the precision of the numerical solution is dependent on $\delta$: the smaller $\delta$, the better the accuracy. During the iteration, $x(t_0)$ is regarded as the independent variable $x$ to calculate $z(x)$ in (38). Since the derivative of $z(x)$

$z(x)$ is monotonically increasing with the increment of $x$ in every step. Continue the iteration until obtaining a $\hat{x}$ such that $z(\hat{x})>0$, then compare the absolute value of $z(\hat{x})$ and $z(\hat{x}-\delta)$ to select the smaller one. As a result, the corresponding $x$ to this smaller $z(x)$ is the numerical solution of $x(t_0)$ we are looking for.

An HSPICE simulation is conducted to verify this approach. The simulation circuit is the same with Fig. 3(b). Predetermining the initial state of the nonlinear memristor as 0.41, we apply a simple sinusoidal voltage source $v(t) = v_0\sin(\omega t)$, where $v_0 = 1.2\;{\rm{V}}$, $\omega = 2{\text{π}} \;{\rm{rad/s}}$. The other simulation parameters are set as following: $t_0 = 0 \;{\rm{s}}$, $R_{\rm on} = 100\; \Omega$, $r = 160$, $D = 10^{-6}\;{\rm{cm}}$, $\mu_V = 10^{-10} \;{\rm{cm}}^2/{\rm{sV}}$. The simulation lasted for 3.73 s and we get $v(3.73) = -1.190538 \;{\rm{V}}$ and $i(3.73) = -243.4537 \;{{\text{μ}}\rm{A}}$. Since the applied voltage source is sinusoidal, $\varphi(t)$ can be calculated from the integration from $t = 0$ to $t = 3.73\;{\rm{s}}$: $\varphi(3.73) = 0.2149\;{\rm{Wb}}$. Thus we can apply our mathematical method in (36)–(38) and obtain

Figure  9.  Simulation process of a nonlinear memristor under the voltage source for 3.73 s.

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The calculation result of $x(t_0)$ is the same with what we predetermined. The feasibility of our method has been examined from the simulation.

Remark 2: From the analysis above, the integration constant $c$ includes the information of the initial value $x(t_0)$. The memory is attributed to the integration constant $c$, which means that the initial value $x(t_0)$ can be computed by the integration constant $c$. For different models of the memristors, the formulas of the integration constant $c$ are different. The accuracy of the initial value computation is dependent on the model of the device.

E Memory Analysis for MRNNs

The model of the MRNNs is obtained by replacing linear resistors with memristors and can be described by the following differential systems

where $u_i(t)$, $i = 1, 2, \ldots, n$, are the states of the network, $R_i, M_{ij}(t)$, $i, j = 1, 2, \ldots, n$, are linear resistances and memristances, respectively; $f_j(s)$ and $I_i$, $i, j = 1, 2, \ldots, n$, $s\in {\mathbb R}$ are activation functions and bias currents, respectively.

According to the property of memristor, MRNNs are a cluster of neural networks. When the power is off, MRNNs can store their historic state. In order to analyze their memory property, i.e., computing initial values of every memristors, we use Algorithm 1 for the memory analysis for MRNNs.

Algorithm 1 Memory analysis for MRNNS

For the MRNNs with $n^2$ memristive synapses $M_{ij}$, $x_{ij}(t_0)$, $i, j = 1, 2, \ldots, n$ represent the initial states of $n^2$ memristive synapses, and $u_i(t)$ $i = 1, 2, \ldots, n$ are voltage values, and the states of MRNNs.

1. Derive analytical expressions of $u_i(t)$ $i = 1, 2, \ldots, n$ with (39);

2. Compute $f(u_{j}(t))-u_i(t)$, for $i, j = 1, 2, \ldots, n$ for some time $t$;

3. Use the above developed voltmeter-ammeter method to obtain $x_{ij}(t_0)$, $i, j = 1, 2, \ldots, n$.

Remark 3: From (39), coefficients of MRNN are variable in the interval $\left[{1}/{R_{\rm off}}, {1}/{R_{\rm on}}\right]$. If the MRNN can be convergent to one sub-network, the convergent result is dependent on the initial value $u_i(t_0)$ of the network state and the initial value $M_{ji}(t_0)$ of memristive synapses. It is necessary for us to locate the initial state of memristive synapses, i.e., analyzing the memory property of the memristor. It is difficult to obtain an accurate value of the voltage between two terminals of memristors. In future works, we will design suitable observers to obtain the voltage value of memristors. Memristors are nano-scale nonlinear resistors with stationary $R_{\rm on}$ and $R_{\rm off}$. In practical applications, we need to connect two or more memristors in series or parallel to obtain different memristors with different memristances. Therefore, it is necessary to analyze the memory of two or more memristors in series or parallel.

III. Memory Properties of Two Memristors Interconnection

In this section, we will discuss memory properties of two series and parallel memristors. As discussed in Section II, one measurement value ($i(t)$ or $v(t)$) and one integration value ($q(t)$ or $\varphi(t)$) is needed to determine the initial state of the memristor. However, we do not need to conduct two measurements and two integrations for two memristors when they are connected in series or parallel, because the memristors connected in series share the same current, and the ones in parallel share the same voltage. Our approach is valid for $n$ ($n>1$) memristors connected in series and parallel: for the series connection case, $n$ measurements for $n$ individual voltages and an integration for the common charge are required; for the parallel connection case, we need $n$ measurements for $n$ individual currents and an integration for the shared magnetic flux. For the purpose of simplicity and without loss of generality, we only discuss two series and parallel memristors.

A Two Memristors in Series

Firstly we discuss the memory property of two memristors $M_1$ and $M_2$ in series and discuss the method in finding initial states of $M_1$ and $M_2$. Denote $x_1(t_0)$, $x_2(t_0)$ as initial states of $M_1$ and $M_2$. Then one can attach a given current source $i(t)$ to them and measure the corresponding voltage $v_1(t)$, $v_2(t)$ of $M_1$, $M_2$. When two memristors are connected in series, we should consider the memristors’ polarities as shown in Fig. 10. In the situation of Fig. 10(a), $M_1$ and $M_2$ connected in series share the same polarity with the current source and voltmeters. We can directly apply (28) in Section II to each memristor. In Fig. 10(b), while $M_1$ shares the same polarity with the excitation, $M_2$ is opposite to the current source. Hence calculation methods for the initial state of $M_2$ should be adjusted. A negative sign should be added to $i(t)$, $v(t)$ and $q(t)$ to offset the polarity difference of $M_2$. Since the existence of opposite polarity is more universal for series connected memristors, in this subsection, we only discuss the situation in Fig. 10(b).

Figure  10.  Two memristors in series with (a) the same polarity and (b) the opposite polarity

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If the discussed memristors are under the assumption of linear dopant drift, we calculate the integration constant $c_{1}$ and the initial state $x_1(t_0)$ of $M_1$. For $M_2$, the memristor opposite to the one connected,

where $\bar{c}$ and $\bar{x}(t_0)$ are the integration constant and the initial state of a memristor connected in opposite polarity, respectively. So $c_{2}$ and $x_2(t_0)$ of $M_2$ can be obtained following (40).

An HSPICE simulation is conducted to examine this method for linear memristors in series. We predetermine the initial states $x_1(t_0)$, $x_2(t_0)$ to be 0.15 and 0.75, respectively. The simulation circuit is Fig. 10(b), where the current source $i(t) = i_0\sin(\omega t)$, $i_0 = 200 \;{{\text{μ}}\rm{A}}$, and $\omega = 2{\text{π}} \;{\rm{rad/s}}$. The other simulation parameters are the same as the ones in Section II: $t_0 \!=\! 0\;{\rm{s}}$, $R_{\rm on} \!=\! 100\;\Omega$, $r \!=\! 160$, $D \!=\!10^{-6}\;{\rm{cm}}$, $\mu_V\! =\! 10^{-10} \;{\rm{cm}}^2/{\rm{sV}}$. We then run the circuit for 3.13 s. $i(3.13) = 145.7937 \;{{\text{μ}}\rm{A}}$, $v_1(3.13)$ and $v_2(3.13)$ are measured as 1.752215 V and 826.8759 mV, respectively. Since the current source is sinusoidal, we get $q(3.13) = 1.0041\times10^{-5}{C}$ by integrating $i(t)$ from $t = 0$ to $t = 3.13 \;{\rm{s}}$. With the presence of $i(3.13)$, $v_k(3.13)$ and $q(3.13)$, $k = 1,2$, the initial states of $M_1$, $M_2$ are obtained according to (40),

The results are coincident with the values we preset for $x_1(t_0)$, $x_2(t_0)$, representing the validity of the methods (40).

For series connected memristors under the assumption of nonlinear dopant drift, the integration constant $c_{1}$ and the initial state $x_1(t_0)$ of $M_1$ can be calculated from (28). As for the opposite connected $M_2$, (28) should be adjusted to

Hence $c_{2}$ and $x_2(t_0)$ of $M_2$ can be obtained from (41).

This approach for series connected memristors with the nonlinear dopant drift can also be verified with an HSPICE simulation. We preset the initial states $x_1(t_0)$, $x_2(t_0)$ to be 0.21 and 0.47, respectively. The simulation circuit and parameters are identical to the ones above for linear memristors, except the magnitude of the current source is $i_0 = 800 \;{{\text{μ}}\rm{A}}$. Run the circuit for 4.11 s. At the end of the simulation, $i(4.11) = 509.9392 \;{{\text{μ}}\rm{A}}$, $v_1(4.11)$ and $v_2(4.11)$ are measured as 4.420901 V and 6.407382 V, respectively. Since the current source is sinusoidal, we can get $q(4.11) = 2.9219\times10^{-5}{\rm{C}}$ by integrating $i(t)$ from $t = 0$ to $t = 4.11\;{\rm{s}}$. Now we have $i(4.11)$, $v_k(4.11)$ and $q(4.11)$, $k = 1,2$, the initial states of $M_1$, $M_2$ can be calculated according to (28) and (41)

The predetermined values for $x_1(t_0)$, $x_2(t_0)$ are obtained from the simulation, showing the feasibility of methods (28) and (41).

B Two Memristors in Parallel

In this subsection, we study the property of two memristors in parallel and give the formulas to calculate the initial states of $M_1$, $M_2$. A common voltage source $v(t)$ is applied to them and the corresponding currents $i_1(t)$, $i_2(t)$ of $M_1$, $M_2$ can be measured. When two memristors are connected in parallel, we should consider the memristors’ polarities as shown in Fig. 11. In Fig. 11(a), $M_1$ and $M_2$ connected in parallel share the same polarity with the voltage source and the ammeters. Equations (17) or (36), and (37) in Section II can be applied to each memristor. In Fig. 11(b), however, $M_2$ is opposite to the voltage excitation, contrary to the regular connection of $M_1$. Therefore the calculation methods for the initial state of $M_2$ should be adjusted accordingly. A negative sign is added to $v(t)$, $i(t)$ and $\varphi(t)$ to compensate for the polarity difference of $M_2$. We only discuss the situation in Fig. 11(b) in this subsection, because the existence of opposite polarity is more general for parallel connected memristors.

Figure  11.  Two memristors in parallel with (a) the same polarity and (b) the opposite polarity

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First we discuss the memristors in parallel under the linear dopant drift assumption; the integration constant $c_{1}$ and the initial state $x_1(t_0)$ of $M_1$ can be calculated from (17). While $c_{2}$ and $x_2(t_0)$ of the opposite connected memristor $M_2$ can be obtained

A parallel memristors circuit simulation is conducted to examine this method. We preset the initial states $x_1(t_0)$, $x_2(t_0)$ to be 0.33 and 0.67, respectively. The simulation circuit is in Fig. 11(b), where the applied voltage is a simple sinusoidal voltage source $v(t) = v_0\sin(\omega t)$, $v_0 = 1 \;{\rm{V}}$, $\omega = 2{\text{π}} \;{\rm{rad/s}}$. The other simulation parameters are the same with those in series connection. Run the simulation for 5.21 s. Then $v(5.21) =$ 968.5831 mV, $i_1(5.21)$ and $i_2(5.21)$ are measured as 109.9511 μA and 118.6726 μA, respectively. We can also get $\varphi(5.21) = 0.1196\;{\rm{Wb}}$ by integrating $v(t)$ from $t = 0$ to $t = 5.21$s since the voltage source is sinusoidal. With the existence of $v(5.21)$, $i_k(5.21)$ and $\varphi(5.21)$, $k = 1,2$, the initial states of $M_1$, $M_2$ are obtained according to (17) and (42),

The correctness of our approach is examined from the consistency of the result and preset values.

Then we should consider the situation when two memristors under the assumption of nonlinear dopant drift are connected in parallel. The integration constant $c_{1}$ of $M_1$ can be calculated from (36), and the initial state $x_1(t_0)$ can be obtained from solving (37). The numerical algorithm to determine the solution of (37) has been described in Section II-D. As for the opposite connected $M_2$, (36) should be adjusted to

to get the integration constant $c_{2}$ of $M_2$. And $x_2(t_0)$, the initial state of $M_2$, can also be obtained from solving the equation

We simulate the nonlinear memristors in parallel to test this algorithm. Predetermining the initial states $x_1(t_0)$, $x_2(t_0)$ to be 0.45 and 0.54, respectively. The circuit and parameters are the same with the previous ones, except the magnitude of the voltage source $v_0 = 1.2 \;{\rm{V}}$. The simulation is lasted for 5.77 s. At the end of the simulation, $v(5.77) = -1.190538 \;{\rm{V}}$, $i_1(5.77)$ and $i_2(5.77)$ are measured as –230.0400 μA and –115.2139 μA, respectively. Since the voltage source is sinusoidal, we can calculate $\varphi(5.77) = 0.1670\;{\rm{Wb}}$ by integrating $v(t)$ from $t = 0$ to $t = 5.77 \;{\rm{s}}$. Now we get $v(5.77)$, $i_k(5.77)$ and $\varphi(5.77)$, $k = 1,2$, the initial states of $M_1$, $M_2$ can be obtained following the calculation of (36), (43) and the solution of (37), (44),

The results are in accordance with the predetermined values.

Remark 4: For two memristors connected in series or parallel, the total initial memristance can be computed if the initial values $x_1(t_0)$, $x_2(t_0)$ are obtained, respectively. We focus on the memory analysis for two memristors in series or parallel, i.e., the total initial memristance computation. This is difference from the property analysis of two series or paprallel memristors in the existing papers.

IV. Concluding Remarks

In this paper, we discuss the memory property of memristors by deriving the formula for the initial value formula and the voltmeter-ammeter method. Then we analyze two series and parallel memristors' memory. According to the developed memory analysis method, we give the algorithm for locating the initial values of all memristive synapses of the MRNN (39). Our analysis shows that the integration constant $c$ in the expression plays an important role in the memory of the electronic device. The accuracy may be improved for the computation of the initial values if the state observer can be designed for the MRNN. This will be our future work.