2. Development of a charge-controlled memristor model

In order to detect edges, several methods are reported in the study. In 1986, John Canny proposed a computational method for image edge detection. He introduced the notion of nonmaximum suppression, which means that given the pre-smoothing filters, edge points are defined as points where the gradient magnitude assumes a local maximum in the gradient direction [2]. Although the method was developed in the early years of computer vision, it is still

Another method is based on anisotropic diffusion, which is a technique aiming at reducing the image noise without removing significant parts of the image such as edges, lines, or other details that are important for the interpretation of the image [3]. This method has evolved to nonlinear anisotropic diffusion, which consists in considering the original image as an initial state of a parabolic (diffusion-like) process and extracting filtered versions from its temporal

As a direct result, nonlinear resistive grids have been used to explicitly implement edge detection based on nonlinear anisotropic diffusion [5]. The nonlinear resistive grid and the elements of this processor are presented in Figure 1(a); the voltage sources represent each pixel of the image to be processed and the node voltages represent each pixel of the processed image. It is important to note that each branch in the grid is composed of a nonlinear resistive

Because of the temporal evolution of the procedure, memristive grids naturally fit the features needed for achieving edge detection [6, 7]. A memristive grid has the same structure of its resistive counterpart, but the nonlinear resistors have been substituted by memristors, as

Figure 1. Structure and components of the (a) resistive grid and (b) memristive grid.

in the state of art.

92 Advances in Memristor Neural Networks – Modeling and Applications

evolution [4].

element called fuse.

depicted in Figure 1(b).

Professor Leon O. Chua predicted in 1971 the existence of the fourth basic circuit element [8]. He called it memristor and defined it as a passive device with two terminals, which branch constitutive function relates the magnetic flux linkage and the electric charge. In 2008, the R. Stanley Williams group at Hewlett-Packard Laboratories presented a device whose behavior exhibits the memristance phenomenon [9].

Novel memristor applications became the main thrust in the search for better and more reliable models of the device that can predict the behavior of the electronic system application. With the goal of developing a memristor model that can achieve edge detection with the memristive grid, several features are pursued:


The modeling methodology can be described as follows: first, the nonlinear drift mechanism is expressed as a function of charge instead of time; then, a symbolic solution x qð Þ to the nonlinear equation is found, and finally, x qð Þ is used to generate the memristance expression.

The nonlinear drift mechanism that governs the functioning of the HP memristor [9] is given hereafter as the ordinary differential equation (ODE) which is expressed in terms of the charge derivative:

$$\frac{d\mathbf{x}(q)}{dq} = \eta \kappa f\_w(\mathbf{x}(q)) \tag{1}$$

where <sup>κ</sup> <sup>¼</sup> <sup>μ</sup>vRon <sup>Δ</sup><sup>2</sup> , μ<sup>v</sup> is the mobility of the charges in the doped region, Δ is the total length of the device, η describes the displacement direction of x qð Þ (η ¼ �1 or þ1), and Ron is the ON-sate resistance. Besides, f <sup>w</sup>ð Þ x qð Þ is the window function. We have selected the window given by [11]

$$f\_w = 1 - (2x(q) - 1)^{2k} \tag{2}$$

where k controls the level of linearity, as k increases, the linearity increases in the range 0 ! 1.

It is possible to find an analytical solution to Eq. (1) for k ¼ 1; however, for k > 1, the solution can only be assessed by resorting to numeric analysis methods [11]. In this chapter, we resort to the homotopy perturbation method (HPM) reported in [12, 13] to obtain a symbolic solution x qð Þ that contains the parameters of the memristor. In this method, different solutions are obtained for the choice made on the Joglekar exponent k and the order of the homotopy. Besides, it must be pointed out that a pair of solutions do indeed exist in every case because η takes values of þ1 and �1 depending on the direction of the charge displacement.

As an example of the solution, the equation obtained for order-1, k ¼ 3 and η ¼ �1 is given as follows:

$$\begin{split} \mathbf{x}\_{k1,\ell3,\eta^{-}} &= \left(X\_0^4 + X\_0^3 + X\_0^2 + X\_0\right)e^{-4\kappa\eta} - \left(3X\_0^4 + 2X\_0^3 + X\_0^2\right)e^{-8\kappa\eta} \\ &+ \left(3X\_0^4 + X\_0^3\right)e^{-12\kappa\eta} - X\_0^4 e^{-16\kappa\eta} \end{split} \tag{3}$$

where X<sup>0</sup> corresponds to the initial value of the state variable (when the charge is zero). It can be noted that the model only converges for positive values of q, and the function tends to 0 when q ! ∞.

The solution for η ¼ þ1 and positive values of q are given by

$$\begin{aligned} \mathbf{x}\_{\text{R1,O3},\eta^{+}} &= 1 + \left( -\mathbf{X}\_{0}^{4} + 5\mathbf{X}\_{0}^{3} - 10\mathbf{X}\_{0}^{2} + 10\mathbf{X}\_{0} - 4\right) \varepsilon^{-4\kappa\eta} + \left( 3\mathbf{X}\_{0}^{4} - 14\mathbf{X}\_{0}^{3} + 25\mathbf{X}\_{0}^{2} - 20\mathbf{X}\_{0} + 6\right) \varepsilon^{-8\kappa\eta} \\ &+ \left( -3\mathbf{X}\_{0}^{4} + 13\mathbf{X}\_{0}^{3} - 21\mathbf{X}\_{0}^{2} + 15\mathbf{X}\_{0} - 4\right) \varepsilon^{-12\kappa\eta} + \left( \mathbf{X}\_{0}^{4} - 4\mathbf{X}\_{0}^{3} + 6\mathbf{X}\_{0}^{2} - 4\mathbf{X}\_{0} + 1\right) \varepsilon^{-16\kappa\eta} \end{aligned} \tag{4}$$

In order to establish a comparison, the numerical solution to Eq. (1) is obtained with the Backward Euler method. Figure 2 shows the plots of the solution x qð Þ obtained with the numeric method and with HPM for homotopy orders 1–3 with k ¼ 1, 2 for both directions. Table 1 shows the values of the parameters used in these evaluations.

#### 2.1. Memristance expressions

Once the solution x qð Þ is obtained, it is substituted in the coupled resistor equivalent:

$$M(q) = R\_{on} \mathfrak{x}(q) + R\_{\mathfrak{q}\sharp} (1 - \mathfrak{x}(q)) \tag{5}$$

Mk1,O1,η� ¼

Table 1. Parameters for the plots of x qð Þ.

with Rd ¼ Roff � Ron.

order 3 (cyan).

8 ><

>:

Rdð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>2</sup> <sup>e</sup><sup>4</sup>κ<sup>q</sup> � ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup><sup>8</sup>κ<sup>q</sup> � � <sup>þ</sup> Ron <sup>q</sup> <sup>≤</sup> <sup>0</sup>

Charge-Controlled Memristor Grid for Edge Detection http://dx.doi.org/10.5772/intechopen.78610 95

Figure 2. Plots of x qð Þ for k ¼ 1, 2. Numerical solution (red) and HPM solutions for order 1 (blue), order 2 (violet), and

<sup>μ</sup><sup>v</sup> <sup>m</sup><sup>2</sup> ð Þ <sup>=</sup>Vs <sup>Δ</sup> <sup>ð</sup>nm<sup>Þ</sup> <sup>κ</sup> ð Þ <sup>m</sup>=As <sup>X</sup><sup>0</sup> <sup>1</sup> � <sup>10</sup>�<sup>14</sup> <sup>10</sup> <sup>10</sup>, <sup>000</sup> <sup>0</sup>:<sup>5</sup>

RdX<sup>0</sup> <sup>X</sup>0e�8κ<sup>q</sup> � ð Þ <sup>X</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup> <sup>e</sup>�4κ<sup>q</sup> � � <sup>þ</sup> Roff <sup>q</sup> <sup>&</sup>gt; <sup>0</sup>

(6)

The expressions for the memristance for order-1 with k ¼ 1 � 5 are given hereafter.

Expressions for η ¼ �1

Figure 2. Plots of x qð Þ for k ¼ 1, 2. Numerical solution (red) and HPM solutions for order 1 (blue), order 2 (violet), and order 3 (cyan).


Table 1. Parameters for the plots of x qð Þ.

$$M\_{k1,O1,\eta^{-}} = \begin{cases} R\_d (X\_0 - 1) \left[ (X\_0 - 2)e^{4\kappa q} - (X\_0 - 1)e^{8\kappa q} \right] + R\_{on} & q \le 0 \\\\ R\_d X\_0 \left[ X\_0 e^{-8\kappa q} - (X\_0 + 1)e^{-4\kappa q} \right] + R\_{\eta\theta} & q > 0 \end{cases} \tag{6}$$

with Rd ¼ Roff � Ron.

<sup>f</sup> <sup>w</sup> <sup>¼</sup> <sup>1</sup> � ð Þ <sup>2</sup>x qð Þ� <sup>1</sup> <sup>2</sup><sup>k</sup> (2)

where k controls the level of linearity, as k increases, the linearity increases in the range 0 ! 1. It is possible to find an analytical solution to Eq. (1) for k ¼ 1; however, for k > 1, the solution can only be assessed by resorting to numeric analysis methods [11]. In this chapter, we resort to the homotopy perturbation method (HPM) reported in [12, 13] to obtain a symbolic solution x qð Þ that contains the parameters of the memristor. In this method, different solutions are obtained for the choice made on the Joglekar exponent k and the order of the homotopy. Besides, it must be pointed out that a pair of solutions do indeed exist in every case because η

As an example of the solution, the equation obtained for order-1, k ¼ 3 and η ¼ �1 is given as

0e �16κq

where X<sup>0</sup> corresponds to the initial value of the state variable (when the charge is zero). It can be noted that the model only converges for positive values of q, and the function tends to 0

In order to establish a comparison, the numerical solution to Eq. (1) is obtained with the Backward Euler method. Figure 2 shows the plots of the solution x qð Þ obtained with the numeric method and with HPM for homotopy orders 1–3 with k ¼ 1, 2 for both directions.

Once the solution x qð Þ is obtained, it is substituted in the coupled resistor equivalent:

The expressions for the memristance for order-1 with k ¼ 1 � 5 are given hereafter.

�4κ<sup>q</sup> � <sup>3</sup>X<sup>4</sup>

�4κ<sup>q</sup> <sup>þ</sup> <sup>3</sup>X<sup>4</sup>

�12κ<sup>q</sup> <sup>þ</sup> <sup>X</sup><sup>4</sup>

<sup>0</sup> <sup>þ</sup> <sup>2</sup>X<sup>3</sup>

<sup>0</sup> � <sup>14</sup>X<sup>3</sup>

M qð Þ¼ Ronx qð Þþ Roffð Þ 1 � x qð Þ (5)

<sup>0</sup> <sup>þ</sup> <sup>6</sup>X<sup>2</sup> <sup>0</sup> � <sup>4</sup>X<sup>0</sup> <sup>þ</sup> <sup>1</sup> <sup>e</sup>

<sup>0</sup> � <sup>4</sup>X<sup>3</sup>

<sup>0</sup> <sup>þ</sup> <sup>25</sup>X<sup>2</sup> <sup>0</sup> � <sup>20</sup>X<sup>0</sup> <sup>þ</sup> <sup>6</sup> <sup>e</sup>

e

<sup>0</sup> <sup>þ</sup> <sup>X</sup><sup>2</sup> 0 �8κq

(3)

�8κq

(4)

�16κq

<sup>0</sup> þ X<sup>0</sup>

�12κ<sup>q</sup> � <sup>X</sup><sup>4</sup>

takes values of þ1 and �1 depending on the direction of the charge displacement.

<sup>0</sup> <sup>þ</sup> <sup>X</sup><sup>2</sup>

e

xk1,O3,η� <sup>¼</sup> <sup>X</sup><sup>4</sup>

94 Advances in Memristor Neural Networks – Modeling and Applications

<sup>0</sup> <sup>þ</sup> <sup>5</sup>X<sup>3</sup>

<sup>0</sup> <sup>þ</sup> <sup>13</sup>X<sup>3</sup>

<sup>þ</sup> <sup>3</sup>X<sup>4</sup>

The solution for η ¼ þ1 and positive values of q are given by

<sup>0</sup> � <sup>10</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>10</sup>X<sup>0</sup> � <sup>4</sup> <sup>e</sup>

Table 1 shows the values of the parameters used in these evaluations.

<sup>0</sup> � <sup>21</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>15</sup>X<sup>0</sup> � <sup>4</sup> <sup>e</sup>

<sup>0</sup> <sup>þ</sup> <sup>X</sup><sup>3</sup>

<sup>0</sup> <sup>þ</sup> <sup>X</sup><sup>3</sup> 0 e

follows:

when q ! ∞.

xk1,O3,η<sup>þ</sup> <sup>¼</sup> <sup>1</sup> þ �X<sup>4</sup>

þ �3X<sup>4</sup>

2.1. Memristance expressions

Expressions for η ¼ �1

Mk2,O1,η� ¼ Rdð Þ X<sup>0</sup> � 1 <sup>2</sup>X<sup>3</sup> <sup>0</sup> <sup>þ</sup> <sup>3</sup>X<sup>0</sup> � <sup>8</sup> � �<sup>e</sup> κ<sup>q</sup> � <sup>3</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup> κq �2ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> <sup>e</sup><sup>24</sup>κ<sup>q</sup> � <sup>2</sup> ð Þ Xo � <sup>1</sup> <sup>3</sup> e κq þ Ron q ≤ 0 RdX<sup>0</sup> X3 e �32κ<sup>q</sup> � <sup>2</sup>X<sup>2</sup> e �24κ<sup>q</sup> <sup>þ</sup> <sup>3</sup>X0<sup>e</sup> �16κq � 1 <sup>2</sup>X<sup>3</sup> � <sup>6</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>9</sup>X<sup>0</sup> <sup>þ</sup> <sup>3</sup> � �<sup>e</sup> �8κq þ Roff q > 0 >>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>: (7) Mk3,O1,η� ¼ Rdð Þ X<sup>0</sup> � 1 X<sup>5</sup> � <sup>20</sup>X<sup>4</sup> <sup>þ</sup> <sup>20</sup>X<sup>3</sup> þ15X<sup>0</sup> � 46 B@ CAe κq �5ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup><sup>24</sup>κ<sup>q</sup> � <sup>20</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> e κq � 20 ð Þ Xo � <sup>1</sup> <sup>3</sup> e κ<sup>q</sup> � <sup>4</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup> e κq � 16 ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup> e κq þ Ron q ≤ 0 RdX<sup>0</sup> <sup>X</sup><sup>5</sup> e �72κ<sup>q</sup> � <sup>4</sup>X<sup>4</sup> e �60κ<sup>q</sup> <sup>þ</sup> <sup>X</sup><sup>3</sup> e �48κq � 20 <sup>X</sup><sup>2</sup> e �36κ<sup>q</sup> <sup>þ</sup> <sup>5</sup>X0<sup>e</sup> �24κq � 1 <sup>16</sup>X<sup>5</sup> � <sup>60</sup>X<sup>4</sup> <sup>þ</sup> <sup>100</sup>X<sup>3</sup> � <sup>100</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>75</sup>X<sup>0</sup> <sup>þ</sup> <sup>15</sup> � �<sup>e</sup> �12κq þ Roff q > 0 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: (8)

Mk4,O1,η� ¼

Rdð Þ X<sup>0</sup> � 1

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

RdX<sup>0</sup>

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

 

� 70

� <sup>112</sup>

� 17

 <sup>X</sup><sup>7</sup> e

þ X5 e

þ <sup>X</sup><sup>3</sup> e

<sup>þ</sup>7X0e�32κ<sup>q</sup>

 B@

� 1  ð Þ Xo � <sup>1</sup> <sup>3</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>7</sup>

�128κ<sup>q</sup> � <sup>32</sup>

 B@ X<sup>7</sup>

�420X<sup>4</sup>

� <sup>560</sup>X<sup>6</sup>

�7ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup><sup>32</sup>κ<sup>q</sup> � <sup>14</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup>

e

e κ<sup>q</sup> � <sup>32</sup>

e κq

> e �80κq

e �48κq

� <sup>1120</sup>X<sup>6</sup>

� <sup>1470</sup>X<sup>2</sup>

<sup>þ</sup> <sup>2352</sup>X<sup>5</sup>

� <sup>2940</sup>X<sup>4</sup>

þ 735X<sup>0</sup> þ 105

 CAe �16κq

 <sup>X</sup><sup>6</sup> e �112κq

�96κ<sup>q</sup> � <sup>28</sup>X<sup>4</sup>

�64κ<sup>q</sup> � <sup>14</sup>X<sup>2</sup>

X<sup>7</sup>

<sup>þ</sup>2450X<sup>3</sup>

<sup>þ</sup> <sup>210</sup>X<sup>3</sup>

 <sup>þ</sup> <sup>672</sup>X<sup>5</sup> 

κ<sup>q</sup> � <sup>28</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>6</sup>

þ 105X<sup>0</sup> � 352

e<sup>48</sup>κ<sup>q</sup>

e κq

> e κq

 CAe κq

Charge-Controlled Memristor Grid for Edge Detection http://dx.doi.org/10.5772/intechopen.78610

þ Ron q ≤ 0

þ Roff q > 0

(9)

Mk4,O1,η� ¼ Rdð Þ X<sup>0</sup> � 1 X<sup>7</sup> � <sup>560</sup>X<sup>6</sup> <sup>þ</sup> <sup>672</sup>X<sup>5</sup> �420X<sup>4</sup> <sup>þ</sup> <sup>210</sup>X<sup>3</sup> þ 105X<sup>0</sup> � 352 B@ CAe κq �7ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup><sup>32</sup>κ<sup>q</sup> � <sup>14</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> e<sup>48</sup>κ<sup>q</sup> � 70 ð Þ Xo � <sup>1</sup> <sup>3</sup> e κ<sup>q</sup> � <sup>28</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup> e κq � <sup>112</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup> e κ<sup>q</sup> � <sup>32</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>6</sup> e κq � 17 ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>7</sup> e κq þ Ron q ≤ 0 RdX<sup>0</sup> <sup>X</sup><sup>7</sup> e �128κ<sup>q</sup> � <sup>32</sup> <sup>X</sup><sup>6</sup> e �112κq þ X5 e �96κ<sup>q</sup> � <sup>28</sup>X<sup>4</sup> e �80κq þ <sup>X</sup><sup>3</sup> e �64κ<sup>q</sup> � <sup>14</sup>X<sup>2</sup> e �48κq <sup>þ</sup>7X0e�32κ<sup>q</sup> � 1 X<sup>7</sup> � <sup>1120</sup>X<sup>6</sup> <sup>þ</sup> <sup>2352</sup>X<sup>5</sup> � <sup>2940</sup>X<sup>4</sup> <sup>þ</sup>2450X<sup>3</sup> � <sup>1470</sup>X<sup>2</sup> þ 735X<sup>0</sup> þ 105 B@ CAe �16κq þ Roff q > 0 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: (9)

Mk2,O1,η� ¼

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

Mk3,O1,η� ¼

Rdð Þ X<sup>0</sup> � 1

Advances in Memristor Neural Networks – Modeling and Applications

>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>:

Rdð Þ X<sup>0</sup> � 1

 <sup>X</sup><sup>5</sup> e

� 20 <sup>X</sup><sup>2</sup> e

� 1 <sup>16</sup>X<sup>5</sup>

RdX<sup>0</sup>

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

 X3 e

� 1 <sup>2</sup>X<sup>3</sup>

 

� 20

� 16

 B@

RdX<sup>0</sup>

 <sup>2</sup>X<sup>3</sup>

<sup>0</sup> <sup>þ</sup> <sup>3</sup>X<sup>0</sup> � <sup>8</sup> � �<sup>e</sup>

e

<sup>e</sup><sup>24</sup>κ<sup>q</sup> � <sup>2</sup> 

�24κ<sup>q</sup> <sup>þ</sup> <sup>3</sup>X0<sup>e</sup>

 <sup>þ</sup> <sup>20</sup>X<sup>3</sup> 

�2ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup>

�32κ<sup>q</sup> � <sup>2</sup>X<sup>2</sup>

 � <sup>6</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>9</sup>X<sup>0</sup> <sup>þ</sup> <sup>3</sup> � �<sup>e</sup>

� <sup>20</sup>X<sup>4</sup>

þ15X<sup>0</sup> � 46

�5ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup><sup>24</sup>κ<sup>q</sup> � <sup>20</sup>

ð Þ Xo � <sup>1</sup> <sup>3</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup>

�36κ<sup>q</sup> <sup>þ</sup> <sup>5</sup>X0<sup>e</sup>

� <sup>60</sup>X<sup>4</sup>

�72κ<sup>q</sup> � <sup>4</sup>X<sup>4</sup>

e

e κq

�24κq

<sup>þ</sup> <sup>100</sup>X<sup>3</sup>

 � <sup>100</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>75</sup>X<sup>0</sup> <sup>þ</sup> <sup>15</sup> � �<sup>e</sup>

e �60κ<sup>q</sup> <sup>þ</sup>

X<sup>5</sup>

κ<sup>q</sup> � <sup>3</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>

ð Þ Xo � <sup>1</sup> <sup>3</sup>

�16κq

�8κq

 CAe κq

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup>

κ<sup>q</sup> � <sup>4</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup>

 <sup>X</sup><sup>3</sup> e �48κq

e κq

> e κq

e κq

κq

þ Ron q ≤ 0

(7)

þ Roff q > 0

þ Ron q ≤ 0

þ Roff q > 0

(8)

�12κq

Mk5,O1,η� ¼ Rdð Þ X<sup>0</sup> � 1 X<sup>9</sup> � <sup>6048</sup>X<sup>8</sup> <sup>þ</sup> <sup>9792</sup>X<sup>7</sup> � <sup>9408</sup>X<sup>6</sup> <sup>þ</sup>6048X<sup>5</sup> � <sup>2520</sup>X<sup>4</sup> <sup>þ</sup> <sup>840</sup>X<sup>3</sup> þ 315X<sup>0</sup> � 1126 @ Ae κq �9ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup><sup>40</sup>κ<sup>q</sup> � <sup>24</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> e<sup>60</sup>κ<sup>q</sup> �56ð Þ Xo � <sup>1</sup> <sup>3</sup> <sup>e</sup><sup>80</sup>κ<sup>q</sup> � <sup>504</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup> e κq � <sup>672</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup> e κ<sup>q</sup> � <sup>128</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>6</sup> e κq � <sup>576</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>7</sup> e κ<sup>q</sup> � <sup>32</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>8</sup> e κq � <sup>256</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>9</sup> e κq þ Ron q ≤ 0 RdX<sup>0</sup> <sup>X</sup><sup>9</sup> e �200κ<sup>q</sup> � <sup>32</sup>X<sup>8</sup> e �180κq þ X7 e �160κ<sup>q</sup> � <sup>128</sup>X<sup>6</sup> e �140κq þ X5 e �120κ<sup>q</sup> � <sup>504</sup> X4 e �100κq <sup>þ</sup>56X<sup>3</sup> e�80κ<sup>q</sup> � <sup>24</sup>X<sup>2</sup> e�60κ<sup>q</sup> <sup>þ</sup>9X0e�40κ<sup>q</sup> � 1 X<sup>9</sup> � <sup>10080</sup>X<sup>8</sup> <sup>þ</sup> <sup>25920</sup>X<sup>7</sup> � <sup>40320</sup>X<sup>6</sup> <sup>þ</sup>42336X<sup>5</sup> � <sup>31752</sup>X<sup>4</sup> <sup>þ</sup> <sup>17640</sup>X<sup>3</sup> � <sup>7560</sup>X<sup>2</sup> þ 2835X<sup>0</sup> þ 315 @ Ae �20κq þ Roff q > 0 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: (10)

Mk2,O1,η<sup>þ</sup> ¼

Mk3,O1,η<sup>þ</sup> ¼

RdX<sup>0</sup>

>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>:

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

RdX<sup>0</sup>

 X3 e

Rdð Þ X<sup>0</sup> � 1

 <sup>X</sup><sup>5</sup> e

� 20 <sup>X</sup><sup>2</sup> e

� 1

Rdð Þ X<sup>0</sup> � 1

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

� 1 <sup>2</sup>X<sup>3</sup>

> <sup>2</sup>X<sup>3</sup>

κ<sup>q</sup> � <sup>2</sup>X<sup>2</sup>

 � <sup>6</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>9</sup>X<sup>0</sup> <sup>þ</sup> <sup>3</sup> � �<sup>e</sup>

�2ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup>

κ<sup>q</sup> <sup>þ</sup> <sup>5</sup>X0<sup>e</sup>

<sup>165</sup> � <sup>60</sup>X<sup>4</sup>

 <sup>16</sup>X<sup>5</sup>

� 20

� 16

κq

<sup>þ</sup> <sup>100</sup>X<sup>3</sup>

 <sup>þ</sup> <sup>20</sup>X<sup>3</sup> <sup>0</sup> <sup>þ</sup> <sup>15</sup>X<sup>0</sup> � <sup>46</sup> � �<sup>e</sup>

� <sup>20</sup>X<sup>4</sup>

�5ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>�24κ<sup>q</sup> � <sup>20</sup>

ð Þ Xo � <sup>1</sup> <sup>3</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup>

e

e �72κq

 � <sup>100</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>75</sup>X<sup>0</sup> <sup>þ</sup> <sup>15</sup> � �<sup>e</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup>

�48κ<sup>q</sup> � <sup>4</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup>

e �36κq

> e �60κq

e κ<sup>q</sup> <sup>þ</sup> <sup>X</sup><sup>3</sup> e κq

κ<sup>q</sup> � <sup>4</sup>X<sup>4</sup>

e

<sup>0</sup> <sup>þ</sup> <sup>3</sup>X<sup>0</sup> � <sup>8</sup> � �<sup>e</sup>

κ<sup>q</sup> <sup>þ</sup> <sup>3</sup>X0<sup>e</sup>

<sup>e</sup>�24κ<sup>q</sup> � <sup>2</sup> 

κq

þ Roff q ≤ 0

Charge-Controlled Memristor Grid for Edge Detection http://dx.doi.org/10.5772/intechopen.78610

> > κq

�12κq

þ Ron q > 0

þ Roff q ≤ 0

þ Ron q > 0

(13)

(12)

�16κq

κq

�8κ<sup>q</sup> � <sup>3</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>

ð Þ Xo � <sup>1</sup> <sup>3</sup>

e �32κq

Expressions for η ¼ þ1

$$M\_{\rm k1,O1,\eta^{+}} = \begin{cases} R\_d X\_0 \left[ X\_0 e^{8\kappa q} - (X\_0 + 1) e^{4\kappa q} \right] + R\_{\rm eff} & q \le 0 \\\\ R\_d (X\_0 - 1) \left[ (X\_0 - 2) e^{-4\kappa q} - (X\_0 - 1) e^{-8\kappa q} \right] + R\_{\rm on} & q > 0 \end{cases} \tag{11}$$

Mk2,O1,η<sup>þ</sup> ¼ RdX<sup>0</sup> X3 e κ<sup>q</sup> � <sup>2</sup>X<sup>2</sup> e κ<sup>q</sup> <sup>þ</sup> <sup>3</sup>X0<sup>e</sup> κq � 1 <sup>2</sup>X<sup>3</sup> � <sup>6</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>9</sup>X<sup>0</sup> <sup>þ</sup> <sup>3</sup> � �<sup>e</sup> κq þ Roff q ≤ 0 Rdð Þ X<sup>0</sup> � 1 <sup>2</sup>X<sup>3</sup> <sup>0</sup> <sup>þ</sup> <sup>3</sup>X<sup>0</sup> � <sup>8</sup> � �<sup>e</sup> �8κ<sup>q</sup> � <sup>3</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup> �16κq �2ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> <sup>e</sup>�24κ<sup>q</sup> � <sup>2</sup> ð Þ Xo � <sup>1</sup> <sup>3</sup> e �32κq þ Ron q > 0 >>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>: (12) Mk3,O1,η<sup>þ</sup> ¼ RdX<sup>0</sup> <sup>X</sup><sup>5</sup> e κ<sup>q</sup> � <sup>4</sup>X<sup>4</sup> e κ<sup>q</sup> <sup>þ</sup> <sup>X</sup><sup>3</sup> e κq � 20 <sup>X</sup><sup>2</sup> e κ<sup>q</sup> <sup>þ</sup> <sup>5</sup>X0<sup>e</sup> κq � 1 <sup>165</sup> � <sup>60</sup>X<sup>4</sup> <sup>þ</sup> <sup>100</sup>X<sup>3</sup> � <sup>100</sup>X<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>75</sup>X<sup>0</sup> <sup>þ</sup> <sup>15</sup> � �<sup>e</sup> κq þ Roff q ≤ 0 Rdð Þ X<sup>0</sup> � 1 <sup>16</sup>X<sup>5</sup> � <sup>20</sup>X<sup>4</sup> <sup>þ</sup> <sup>20</sup>X<sup>3</sup> <sup>0</sup> <sup>þ</sup> <sup>15</sup>X<sup>0</sup> � <sup>46</sup> � �<sup>e</sup> �12κq �5ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>�24κ<sup>q</sup> � <sup>20</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> e �36κq � 20 ð Þ Xo � <sup>1</sup> <sup>3</sup> e �48κ<sup>q</sup> � <sup>4</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup> e �60κq � 16 ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup> e �72κq þ Ron q > 0 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: (13)

Mk5,O1,η� ¼

Rdð Þ X<sup>0</sup> � 1

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

RdX<sup>0</sup>

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

Expressions for η ¼ þ1

 

 @

Advances in Memristor Neural Networks – Modeling and Applications

�56ð Þ Xo � <sup>1</sup> <sup>3</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>7</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>9</sup>

�200κ<sup>q</sup> � <sup>32</sup>X<sup>8</sup>

�160κ<sup>q</sup> � <sup>128</sup>X<sup>6</sup>

�120κ<sup>q</sup> � <sup>504</sup> X4 e �100κq

e�80κ<sup>q</sup> � <sup>24</sup>X<sup>2</sup>

X<sup>9</sup>

<sup>þ</sup>42336X<sup>5</sup>

� <sup>672</sup>

� <sup>576</sup>

� <sup>256</sup>

 <sup>X</sup><sup>9</sup> e

þ X7 e

þ X5 e

<sup>þ</sup>56X<sup>3</sup>

� 1 

Mk1,O1,η<sup>þ</sup> ¼

<sup>þ</sup>9X0e�40κ<sup>q</sup>

 @

> ><

> >:

X<sup>9</sup>

<sup>þ</sup>6048X<sup>5</sup>

�9ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup><sup>40</sup>κ<sup>q</sup> � <sup>24</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup>

e

e

e κq

e �180κq

> e �140κq

e�60κ<sup>q</sup>

� <sup>10080</sup>X<sup>8</sup>

� <sup>31752</sup>X<sup>4</sup>

<sup>þ</sup> <sup>25920</sup>X<sup>7</sup>

<sup>þ</sup> <sup>17640</sup>X<sup>3</sup>

 � <sup>40320</sup>X<sup>6</sup> 

RdX<sup>0</sup> <sup>X</sup>0e<sup>8</sup>κ<sup>q</sup> � ð Þ <sup>X</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup> <sup>e</sup><sup>4</sup>κ<sup>q</sup> � � <sup>þ</sup> Roff <sup>q</sup> <sup>≤</sup> <sup>0</sup>

Rdð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>2</sup> <sup>e</sup>�4κ<sup>q</sup> � ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>�8κ<sup>q</sup> � � <sup>þ</sup> Ron <sup>q</sup> <sup>&</sup>gt; <sup>0</sup>

� <sup>7560</sup>X<sup>2</sup>

þ 2835X<sup>0</sup> þ 315

 Ae �20κq

� <sup>6048</sup>X<sup>8</sup>

� <sup>2520</sup>X<sup>4</sup>

<sup>e</sup><sup>80</sup>κ<sup>q</sup> � <sup>504</sup>

<sup>þ</sup> <sup>9792</sup>X<sup>7</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup>

κ<sup>q</sup> � <sup>128</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>6</sup>

κ<sup>q</sup> � <sup>32</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>8</sup>

<sup>þ</sup> <sup>840</sup>X<sup>3</sup>

e<sup>60</sup>κ<sup>q</sup>

e κq

> e κq

e κq

 � <sup>9408</sup>X<sup>6</sup> 

þ 315X<sup>0</sup> � 1126

 Ae κq

þ Ron q ≤ 0

þ Roff q > 0

(10)

(11)

Mk4,O1,η<sup>þ</sup> ¼ RdX<sup>0</sup> <sup>X</sup><sup>7</sup> e κ<sup>q</sup> � <sup>32</sup> <sup>X</sup><sup>6</sup> e κ<sup>q</sup> <sup>þ</sup> X5 e κq �28X<sup>4</sup> e<sup>80</sup>κ<sup>q</sup> <sup>þ</sup> <sup>X</sup><sup>3</sup> e κ<sup>q</sup> � <sup>14</sup>X<sup>2</sup> e κ<sup>q</sup> <sup>þ</sup> <sup>7</sup>X0<sup>e</sup> κq � 1 X<sup>7</sup> � <sup>1120</sup>X<sup>6</sup> <sup>þ</sup> <sup>2352</sup>X<sup>5</sup> � <sup>2940</sup>X<sup>4</sup> <sup>þ</sup>2450X<sup>3</sup> � <sup>1470</sup>X<sup>2</sup> þ 735X<sup>0</sup> þ 105 @ Ae κq þ Roff q ≤ 0 Rdð Þ X<sup>0</sup> � 1 X<sup>7</sup> � <sup>560</sup>X<sup>6</sup> <sup>þ</sup> <sup>672</sup>X<sup>5</sup> � <sup>420</sup>X<sup>4</sup> <sup>þ</sup>210X<sup>3</sup> þ 105X<sup>0</sup> � 352 @ Ae �16κq �7ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>�32κ<sup>q</sup> � <sup>14</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> e�48κ<sup>q</sup> � 70 ð Þ Xo � <sup>1</sup> <sup>3</sup> e �64κ<sup>q</sup> � <sup>28</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup> e �80κq � <sup>112</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup> e �96κ<sup>q</sup> � <sup>32</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>6</sup> e �112κq � 17 ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>7</sup> e �128κq þ Ron q > 0 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: (14) Mk5,O1,η<sup>þ</sup> ¼ RdX<sup>0</sup> <sup>X</sup><sup>9</sup> e κ<sup>q</sup> � <sup>32</sup>X<sup>8</sup> e κ<sup>q</sup> <sup>þ</sup> X7 e κq �128X<sup>6</sup> e<sup>140</sup>κ<sup>q</sup> <sup>þ</sup> X5 e κ<sup>q</sup> � <sup>504</sup> X4 e κq <sup>þ</sup>56X<sup>3</sup> e<sup>80</sup>κ<sup>q</sup> � <sup>24</sup>X<sup>2</sup> e<sup>60</sup>κ<sup>q</sup> <sup>þ</sup> <sup>9</sup>X0e<sup>40</sup>κ<sup>q</sup> � 1 X<sup>9</sup> � <sup>10080</sup>X<sup>8</sup> <sup>þ</sup> <sup>25920</sup>X<sup>7</sup> � <sup>40320</sup>X<sup>6</sup> <sup>þ</sup> <sup>42336</sup>X<sup>5</sup> �31752X<sup>4</sup> <sup>þ</sup> <sup>17640</sup>X<sup>3</sup> � <sup>7560</sup>X<sup>2</sup> þ 2835X<sup>0</sup> þ 315 @ Ae κq þ Roff q ≤ 0 Rdð Þ X<sup>0</sup> � 1 X<sup>9</sup> � <sup>6048</sup>X<sup>8</sup> <sup>þ</sup> <sup>9792</sup>X<sup>7</sup> � <sup>9408</sup>X<sup>6</sup> <sup>þ</sup> <sup>6048</sup>X<sup>5</sup> �2520X<sup>4</sup> <sup>þ</sup> <sup>840</sup>X<sup>3</sup> þ 315X<sup>0</sup> � 1126 @ Ae �20κq �9ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>�40κ<sup>q</sup> � <sup>24</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> e�60κ<sup>q</sup> �56ð Þ Xo � <sup>1</sup> <sup>3</sup> <sup>e</sup>�80κ<sup>q</sup> � <sup>504</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup> e �100κq � <sup>672</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup> e �120κ<sup>q</sup> � <sup>128</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>6</sup> e �140κq � <sup>576</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>7</sup> e �160κ<sup>q</sup> � <sup>32</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>8</sup> e �180κq � <sup>256</sup> ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>9</sup> e �200κq þ Ron q > 0 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: (15)

3. Characterization of the model

Ap is the amplitude of the sinusoidal stimuli.

k ¼ 1, 5 and ω ¼ 1; 2; 5; 10.

frequency ω ! ∞ can be expressed as

Table 2. Parameter values used in the characterization.

lim<sup>ω</sup>!<sup>∞</sup> Mki,Oj

where ki and Oi are the selected k and homotopy order, respectively.

Figure 3. Frequency behavior of the pinched hysteresis loops for (a) k ¼ 1 and (b) k ¼ 5.

values for k ¼ 1 � 5.

The developed model is tested in order to verify that it fulfills the main fingerprints of the device [10]. The nominal values of the HP memristor [9] are used, as shown in Table 2, where

On one side, the v tð Þ-i tð Þ characteristic of a memristor must be a pinched hysteresis loop (PHL). Besides, the area of the PHL must decrease with the frequency. In the limit as the frequency tends to infinity, the memristor behaves as a linear resistor. In Figure 3, the PHLs are shown for

Figure 4 shows the area as a function of the frequency. It can be verified that the lobe area decreases monotonically with the frequency from a critical value ωc. Table 3 shows these

On the other side, as the frequency tends to infinity, the value of the memristance becomes constant and the device acts as a linear resistor [10]. The limit of the memristance when the

<sup>μ</sup><sup>v</sup> <sup>m</sup><sup>2</sup> ð Þ <sup>=</sup>Vs <sup>Δ</sup> <sup>ð</sup>nm<sup>Þ</sup> <sup>κ</sup> ð Þ <sup>m</sup>=As Ron <sup>ð</sup>Ω<sup>Þ</sup> Roff <sup>ð</sup>Ω<sup>Þ</sup> <sup>X</sup><sup>0</sup> Ap <sup>ð</sup>μA<sup>Þ</sup> <sup>η</sup> � <sup>10</sup>�<sup>14</sup> <sup>10</sup> <sup>10</sup>, <sup>000</sup> <sup>100</sup> <sup>16</sup> � <sup>10</sup><sup>3</sup> <sup>0</sup>:5 40 <sup>þ</sup><sup>1</sup>

<sup>¼</sup> <sup>X</sup>0Ron <sup>þ</sup> ð Þ <sup>1</sup> � <sup>X</sup><sup>0</sup> Roff <sup>¼</sup> Rinit (16)

Charge-Controlled Memristor Grid for Edge Detection http://dx.doi.org/10.5772/intechopen.78610 # 3. Characterization of the model

Mk4,O1,η<sup>þ</sup> ¼

Mk5,O1,η<sup>þ</sup> ¼

RdX<sup>0</sup>

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

 <sup>X</sup><sup>9</sup> e

�128X<sup>6</sup>

<sup>þ</sup>56X<sup>3</sup>

� 1 

Rdð Þ X<sup>0</sup> � 1

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

RdX<sup>0</sup>

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

 <sup>X</sup><sup>7</sup> e κ<sup>q</sup> � <sup>32</sup> <sup>X</sup><sup>6</sup> e κ<sup>q</sup> <sup>þ</sup>

Advances in Memristor Neural Networks – Modeling and Applications

�28X<sup>4</sup>

� 1 

Rdð Þ X<sup>0</sup> � 1

e<sup>80</sup>κ<sup>q</sup> <sup>þ</sup> <sup>X</sup><sup>3</sup> e

 @

 

� 70

� <sup>112</sup>

� 17

 X5 e

κ<sup>q</sup> � <sup>32</sup>X<sup>8</sup> e κ<sup>q</sup> <sup>þ</sup> X7 e κq

e<sup>140</sup>κ<sup>q</sup> <sup>þ</sup>

e<sup>80</sup>κ<sup>q</sup> � <sup>24</sup>X<sup>2</sup>

X<sup>9</sup>

 @

�56ð Þ Xo � <sup>1</sup> <sup>3</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>7</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>9</sup>

� <sup>672</sup>

� <sup>576</sup>

� <sup>256</sup>

 @

 

�31752X<sup>4</sup>

X<sup>9</sup>

�2520X<sup>4</sup>

X<sup>7</sup>

 @

<sup>þ</sup>2450X<sup>3</sup>

X<sup>7</sup>

<sup>þ</sup>210X<sup>3</sup>

ð Þ Xo � <sup>1</sup> <sup>3</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>5</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>7</sup>

� <sup>1120</sup>X<sup>6</sup>

� <sup>1470</sup>X<sup>2</sup>

� <sup>560</sup>X<sup>6</sup>

�7ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>�32κ<sup>q</sup> � <sup>14</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup>

e

e �96κ<sup>q</sup> � <sup>32</sup>

e �128κq

κ<sup>q</sup> � <sup>504</sup> X4 e κq

<sup>þ</sup> <sup>25920</sup>X<sup>7</sup>

� <sup>7560</sup>X<sup>2</sup>

<sup>þ</sup> <sup>9792</sup>X<sup>7</sup>

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup>

�120κ<sup>q</sup> � <sup>128</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>6</sup>

�160κ<sup>q</sup> � <sup>32</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>8</sup>

þ 315X<sup>0</sup> � 1126

e�60κ<sup>q</sup>

e �100κq

> e �140κq

e �180κq

� <sup>40320</sup>X<sup>6</sup>

� <sup>9408</sup>X<sup>6</sup>

þ 2835X<sup>0</sup> þ 315

e<sup>60</sup>κ<sup>q</sup> <sup>þ</sup> <sup>9</sup>X0e<sup>40</sup>κ<sup>q</sup>

� <sup>10080</sup>X<sup>8</sup>

<sup>þ</sup> <sup>17640</sup>X<sup>3</sup>

� <sup>6048</sup>X<sup>8</sup>

�9ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>e</sup>�40κ<sup>q</sup> � <sup>24</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup>

e

e

e �200κq

<sup>þ</sup> <sup>840</sup>X<sup>3</sup>

<sup>e</sup>�80κ<sup>q</sup> � <sup>504</sup>

 X5 e κq

<sup>þ</sup> <sup>2352</sup>X<sup>5</sup>

<sup>þ</sup> <sup>672</sup>X<sup>5</sup>

�64κ<sup>q</sup> � <sup>28</sup>ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>4</sup>

þ 105X<sup>0</sup> � 352

κ<sup>q</sup> <sup>þ</sup> <sup>7</sup>X0<sup>e</sup>

 � <sup>2940</sup>X<sup>4</sup> 

> � <sup>420</sup>X<sup>4</sup>

> > e�48κ<sup>q</sup>

e �80κq

> e �112κq

 <sup>þ</sup> <sup>42336</sup>X<sup>5</sup> 

 <sup>þ</sup> <sup>6048</sup>X<sup>5</sup>   Ae κq

 Ae �20κq

ð Þ <sup>X</sup><sup>0</sup> � <sup>1</sup> <sup>6</sup>

þ 735X<sup>0</sup> þ 105

κq

þ Roff q ≤ 0

þ Ron q > 0

þ Roff q ≤ 0

þ Ron q > 0

(14)

(15)

 Ae κq

 Ae �16κq

κ<sup>q</sup> � <sup>14</sup>X<sup>2</sup> e The developed model is tested in order to verify that it fulfills the main fingerprints of the device [10]. The nominal values of the HP memristor [9] are used, as shown in Table 2, where Ap is the amplitude of the sinusoidal stimuli.

On one side, the v tð Þ-i tð Þ characteristic of a memristor must be a pinched hysteresis loop (PHL). Besides, the area of the PHL must decrease with the frequency. In the limit as the frequency tends to infinity, the memristor behaves as a linear resistor. In Figure 3, the PHLs are shown for k ¼ 1, 5 and ω ¼ 1; 2; 5; 10.

Figure 4 shows the area as a function of the frequency. It can be verified that the lobe area decreases monotonically with the frequency from a critical value ωc. Table 3 shows these values for k ¼ 1 � 5.

On the other side, as the frequency tends to infinity, the value of the memristance becomes constant and the device acts as a linear resistor [10]. The limit of the memristance when the frequency ω ! ∞ can be expressed as

$$\lim\_{\omega \to \infty} \left( M\_{k\_{\circ}, O\_{\uparrow}} \right) = X\_0 R\_{\text{or}} + (1 - X\_0) R\_{\text{off}} = R\_{\text{init}} \tag{16}$$

where ki and Oi are the selected k and homotopy order, respectively.


Table 2. Parameter values used in the characterization.

Figure 3. Frequency behavior of the pinched hysteresis loops for (a) k ¼ 1 and (b) k ¼ 5.

Figure 4. PHL lobe area as a function of the frequency (units of area in <sup>μ</sup>m2) for (a) <sup>k</sup> <sup>¼</sup> 1 and (b) <sup>k</sup> <sup>¼</sup> 5.


Table 3. Critical frequencies for different values of k.

#### 3.1. Comparison with other models

Several models are already reported in the study, which have been developed for different applications. A first scheme is reported in [14] in the form of a macro-model implemented in the SPICE circuit simulator. The second model is reported in [15], which is a mathematical model implemented in MATLAB. Figure 5 shows the v tð Þ� i tð Þ characteristics of these models and our charge-controlled model. For the sake of comparison, the model Mk1,O<sup>3</sup> is used.

3.2. Memristance-charge characteristic

Figure 6. Memristance-charge characteristics.

4. Memristive grid for edge detection

sharper transition.

Figure 6 shows the M � q curves of the model for both cases of η. It can be seen that for η ¼ �1, the memristance tends to Roff in the positive range of the charge and tends to Ron in the negative range of q. On the contrary, when η ¼ þ1, the memristance tends to Ron in the positive range of the charge and to Roff in the negative range. Besides, the curves with higher k show a

Charge-Controlled Memristor Grid for Edge Detection http://dx.doi.org/10.5772/intechopen.78610 103

Figure 1(b) shows the memristive grid used for edge detection. In fact, each fuse of the grid consists of two memristors in an anti-series connection, that is, the series connection of two memristors connected back to back, as shown in Figure 7(a). The combined M � q characteristic of the memristive fuse has the shape depicted in Figure 7(b). Ideally, the ON-state memristance is zero and the slope from the ON-state to the OFF-state around Qt is infinite. In

Figure 5. Comparison of the v tð Þ� i tð Þ curves.

Figure 6. Memristance-charge characteristics.

3.1. Comparison with other models

Figure 5. Comparison of the v tð Þ� i tð Þ curves.

Table 3. Critical frequencies for different values of k.

102 Advances in Memristor Neural Networks – Modeling and Applications

Several models are already reported in the study, which have been developed for different applications. A first scheme is reported in [14] in the form of a macro-model implemented in the SPICE circuit simulator. The second model is reported in [15], which is a mathematical model implemented in MATLAB. Figure 5 shows the v tð Þ� i tð Þ characteristics of these models and our charge-controlled model. For the sake of comparison, the model Mk1,O<sup>3</sup> is used.

Figure 4. PHL lobe area as a function of the frequency (units of area in <sup>μ</sup>m2) for (a) <sup>k</sup> <sup>¼</sup> 1 and (b) <sup>k</sup> <sup>¼</sup> 5.

k 12345 ω<sup>c</sup> 0:947 1:252 1:656 2:013 2:828

#### 3.2. Memristance-charge characteristic

Figure 6 shows the M � q curves of the model for both cases of η. It can be seen that for η ¼ �1, the memristance tends to Roff in the positive range of the charge and tends to Ron in the negative range of q. On the contrary, when η ¼ þ1, the memristance tends to Ron in the positive range of the charge and to Roff in the negative range. Besides, the curves with higher k show a sharper transition.

#### 4. Memristive grid for edge detection

Figure 1(b) shows the memristive grid used for edge detection. In fact, each fuse of the grid consists of two memristors in an anti-series connection, that is, the series connection of two memristors connected back to back, as shown in Figure 7(a). The combined M � q characteristic of the memristive fuse has the shape depicted in Figure 7(b). Ideally, the ON-state memristance is zero and the slope from the ON-state to the OFF-state around Qt is infinite. In practice, Mon has a very low value, and Moff takes a very high value. The value of Qt defines the degree of smoothing: more smoothing is related to larger Mq, which implies longer settling times in the edge detection. Besides, the memristance threshold Mth, which is related to Qt, is selected to define which pixel is identified as an edge of the original image.

Figure 7(c) shows the M � q characteristic of the fuse to be used in the memristive grid. The parameters of the model are recast in Table 4.

The importance of a smart selection on the M � q characteristic resides in the fact that it allows us to achieve an appropriate smoothing preprocessing [16]. A figure of merit of great significance is the relation between the smoothing level L and the branch memristance in the grid, Mbranch. In fact, L is a space constant that serves to measure the smoothing as a number of pixels:

$$\lambda = \frac{M\_{\text{branch}}}{R\_{in}}; \quad \text{ } \varsigma = \cosh^{-1}\left(1 + \frac{\lambda}{2}\right); \quad L = \frac{1}{\varsigma}; \tag{17}$$

Figure 8 shows a single node of the grid (node Ni,j). Herein, the voltage source di,j is associated with the pixel i, j, which takes values between 0 ! 1 V. KCL analysis of the output node Ni,j

Parm. <sup>μ</sup><sup>v</sup> <sup>m</sup><sup>2</sup> ð Þ <sup>=</sup>Vs Ron <sup>ð</sup>Ω<sup>Þ</sup> <sup>Δ</sup> <sup>ð</sup>nm<sup>Þ</sup> Rinit <sup>ð</sup>Ω<sup>Þ</sup> <sup>X</sup><sup>0</sup> Roff <sup>ð</sup>Ω<sup>Þ</sup> Mon <sup>ð</sup>Ω<sup>Þ</sup> Moff <sup>ð</sup>Ω<sup>Þ</sup> Rin <sup>ð</sup>Ω<sup>Þ</sup> Value <sup>1</sup> � <sup>10</sup>�<sup>14</sup> 1 10 1:1 0:999 1100 2:2 1101 50

Iin þ Ii�1,j þ Iiþ1,j þ Ii,j�<sup>1</sup> þ Ii,jþ<sup>1</sup> ¼ 0 (18)

þ

ui,jþ<sup>1</sup> � ui,j Mi,jþ<sup>1</sup>

Charge-Controlled Memristor Grid for Edge Detection http://dx.doi.org/10.5772/intechopen.78610

¼ 0 (19)

105

ui,j�<sup>1</sup> � ui,j Mi,j�<sup>1</sup>

yields

This can be established as

Table 4. Nominal parameter values.

di,j � ui,j Rin

Figure 8. Current contributions at node Ni,j.

þ

ui�1,j � ui,j Mi�1,j

þ

where Mi�<sup>1</sup>,j, Miþ<sup>1</sup>,j, Mi,j�<sup>1</sup>, Mi,jþ<sup>1</sup> are the memristances incident to the node.

rule. In a last step, the memristance is updated in the charge-controlled model.

uiþ1,j � ui,j Miþ1,j

þ

For an m � n image, KCL analysis on the complete grid yields a system of m � n DAEs that is solved for the nodal voltages ui,j. Moreover, the associated charges of the memristors are calculated by the numerical integration of their currents by using the trapezoidal integration

Some additional considerations must be taken into account for processing images with a memristive grid, due to the fact that the memristive grid implements a nonlinear anisotropic method. Namely, the method needs a stop criterion to find a solution [17]. The images processed with the memristive grid are in gray scale, and a threshold to stop the process is selected.

#### 4.1. Solving the memristive grid

The equations emanating from the memristive grid form a set of differential algebraic equations (DAEs) that is solved with MATLAB. The number of pixels of the image determines the size of the grid and therefore the number of DAEs.

Figure 7. Memristive fuse: (a) anti-series connection, (b) schematic M-q characteristic, and (c) M-q characteristic of the fuse.


Table 4. Nominal parameter values.

practice, Mon has a very low value, and Moff takes a very high value. The value of Qt defines the degree of smoothing: more smoothing is related to larger Mq, which implies longer settling times in the edge detection. Besides, the memristance threshold Mth, which is related to Qt, is

Figure 7(c) shows the M � q characteristic of the fuse to be used in the memristive grid. The

The importance of a smart selection on the M � q characteristic resides in the fact that it allows us to achieve an appropriate smoothing preprocessing [16]. A figure of merit of great significance is the relation between the smoothing level L and the branch memristance in the grid, Mbranch. In fact, L is a space constant that serves to measure the smoothing as a number of pixels:

; <sup>ς</sup> <sup>¼</sup> cosh�<sup>1</sup> <sup>1</sup> <sup>þ</sup>

Some additional considerations must be taken into account for processing images with a memristive grid, due to the fact that the memristive grid implements a nonlinear anisotropic method. Namely, the method needs a stop criterion to find a solution [17]. The images processed with the memristive grid are in gray scale, and a threshold to stop the process is

The equations emanating from the memristive grid form a set of differential algebraic equations (DAEs) that is solved with MATLAB. The number of pixels of the image determines the

Figure 7. Memristive fuse: (a) anti-series connection, (b) schematic M-q characteristic, and (c) M-q characteristic of the fuse.

λ 2 

; L <sup>¼</sup> <sup>1</sup> ς

; (17)

selected to define which pixel is identified as an edge of the original image.

parameters of the model are recast in Table 4.

104 Advances in Memristor Neural Networks – Modeling and Applications

selected.

4.1. Solving the memristive grid

size of the grid and therefore the number of DAEs.

<sup>λ</sup> <sup>¼</sup> Mbranch Rin

Figure 8 shows a single node of the grid (node Ni,j). Herein, the voltage source di,j is associated with the pixel i, j, which takes values between 0 ! 1 V. KCL analysis of the output node Ni,j yields

$$I\_{in} + I\_{i-1,j} + I\_{i+1,j} + I\_{i,j-1} + I\_{i,j+1} = 0 \tag{18}$$

This can be established as

$$\frac{d\_{i,j} - u\_{i,j}}{R\_{\text{in}}} + \frac{u\_{i-1,j} - u\_{i,j}}{M\_{i-1,j}} + \frac{u\_{i+1,j} - u\_{i,j}}{M\_{i+1,j}} + \frac{u\_{i,j-1} - u\_{i,j}}{M\_{i,j-1}} + \frac{u\_{i,j+1} - u\_{i,j}}{M\_{i,j+1}} = 0 \tag{19}$$

where Mi�<sup>1</sup>,j, Miþ<sup>1</sup>,j, Mi,j�<sup>1</sup>, Mi,jþ<sup>1</sup> are the memristances incident to the node.

For an m � n image, KCL analysis on the complete grid yields a system of m � n DAEs that is solved for the nodal voltages ui,j. Moreover, the associated charges of the memristors are calculated by the numerical integration of their currents by using the trapezoidal integration rule. In a last step, the memristance is updated in the charge-controlled model.

Figure 8. Current contributions at node Ni,j.

As shown in Eq. (17), the level of smoothing depends on the rate Mbranch Rin , that is, the equivalent of each memristance arriving to the node Ni,j divided by the input resistance. The initial condition of the memristive grid is Mon Rin ¼ 0:044 which corresponds to L ¼ 4:78 pixels.

The dynamics of the grid comes from the time-dependent behavior of the memristance, which implies that the value of Mbranch increases with t causing in turn a low level of smoothing. In fact, after a long period of time, the output image gets closer to the original image. It clearly results that a stop criterion is needed.

This criterion is the smoothing time tsmooth, since it defines when the smoothing level of the output image is reached. At this point, the edges are determined by those nodes in the grid where the fuses have reached Mth. This threshold is referred to as a fraction of the maximum value of the memristance. A percentage of 2 of Moff has been used, allowing edges to be detected when the output image still retains a high level of smoothing. As a result, edge detection can be efficiently performed even for images with high levels of noise.
