**4. RD-MCNN structure and static pattern formation**

The proposed RD-MCNN structure has a planar grid arrangement and it is composed of *m* � *n* identical cells where *m* is the number of rows, *n* is the number of columns, and *Ci*,*<sup>j</sup>* represents the cell at *i th* row and the *j th* column. All the cells are resistively coupled to the respective nearest neighbors in vertical and horizontal directions only, and the CNN structure assumes periodic boundary conditions; that is, the first and the last cells in each row, and respectively, in each column, are resistively coupled to each other as well. To study the emergence of pattern formation dynamics in the proposed RD-MCNN, among several structures investigated, here we present typical simulation results of a two-dimensional 51 � 51 structure. We define the same initial conditions (*T*<sup>0</sup> ¼ 300*K* and *Vc*<sup>0</sup> ¼ 0*V*) for all the cells unless otherwise stated, except for the cell, namely *C*26,26, which is in the center of the network. This exception for the definition of initial conditions is adopted to initiate the emergence of the transient behavior in the numerical simulations.

First of all, we investigate the effect of changing the DC operating point on the static pattern formation. To this end, we set the design parameters *Rb* to 2*k*Ω and *Rc* to 0*:*1*k*Ω, while we tune *Vb* to vary the location of the DC operating point of each cell. We present the emergent static patterns in **Figure 6**, where *Vb* ¼ 1*:*2*V* in **Figure 6(a)**, *Vb* ¼ 1*:*3*V* in **Figure 6(b)**, *Vb* ¼ 1*:*4*V* in **Figure 6(c)**, and *Vb* ¼ 1*:*5*V* in **Figure 6(d)**. Here, we plot the memristor voltage, or equivalently, the capacitor voltage, of each cell and assign a color code to its amplitude value, as depicted on the right side of each plot. It is possible to observe from **Figure 6** that a shift in the location of the operating point due to an increase in *Vb* may lead to static patterns featuring a reduced spread in capacitor voltage amplitude through the array,

*Pattern Formation in a RD-MCNN with Locally Active Memristors DOI: http://dx.doi.org/10.5772/intechopen.100463*

**Figure 6.**

*RD-MCNN output values represented by the cell memristor voltages obtained in a simulation of a twodimensional 51* � *51 RD-MCNN structure for Rb* ¼ 2*k*Ω*, Rc* ¼ 0*:*1*k*Ω*, while Vb* ¼ 1*:*2*V in (a), Vb* ¼ 1*:*3*V in (b), Vb* ¼ 1*:*4*V in (c), and Vb* ¼ 1*:*5*V in (d). All the cells have the same initial conditions except for the center cell C*26,26*. A shift in the memristor DC operating point may lead to static patterns with a reduced range in capacitor voltage amplitude, as mostly pronounced in (d).*

reflected by a gradual decrement in the variety of colors in the emergent patterns, as especially observed in **Figure 6(d)**.

Later on, we examine the effect of the coupling strength on the static pattern formation by varying the value of the coupling resistance *Rc*. Similarly, we set the design parameters *Vb* to 1*:*2*V* and *Rb* to 2*k*Ω while we tune the value of *Rc* to adjust the coupling strength. The results are illustrated in **Figure 7**, where *Rc* ¼ 0*:*25*k*Ω in **Figure 7(a)**, *Rc* ¼ 0*:*5*k*Ω in **Figure 7(b)**, *Rc* ¼ 1*k*Ω in **Figure 7(c)**, and *Rc* ¼ 2*k*Ω in **Figure 7(d)**. It can be seen that as *Rc* increases, first a deformation starts to occur in the outer parts of the static pattern of **Figure 7(a)**–**(c)**, while clearly, a new static pattern emerges, finally in **Figure 7(d)**. The final pattern in **Figure 7(d)** also shows that neighboring cells exhibit a sharper spread in the capacitor voltage amplitude, which is reflected by the clear color contrast observable in this plot, especially as compared to that of **Figure 7(a)**.

In addition, we focus on the effect of the initial conditions on pattern formation, as depicted in **Figure 8**, where we fix the values of all of the design parameters such that *Vb* ¼ 1*:*2*V*, *Rb* ¼ 2*k*Ω, and *Rc* ¼ 0*:*1*k*Ω. The cells located in the center of each side of the edges (i.e., *C*1,26,*C*26,1,*C*51,26,*C*26,51), or midpoint cells for short, have the same initial condition as the center cell *C*26,26 in **Figure 8(a)**, while only the corner cells (*C*1,1, *C*1,51, *C*51,1, *C*51,51) have the same initial condition as the center cell *C*26,26 in **Figure 8(c)**. On the other hand, only the midpoint cells share the same initial conditions in **Figure 8(b)**, while the center cell *C*26,26 features the same initial condition as the rest of the network. Similarly, only the corner cells share the same

#### **Figure 7.**

*RD-MCNN output values represented by the cell memristor voltages obtained in a simulation of a twodimensional 51* � *51 RD-MCNN structure for Rb* ¼ 2*k*Ω*, Vb* ¼ 1*:*2*V, while Rc* ¼ 0*:*25*k*Ω *in (a), Rc* ¼ 0*:*5*k*Ω *in (b), Rc* ¼ 1*k*Ω *in (c), and Rc* ¼ 2*k*Ω *in (d). All the cells have the same initial condition except for the center cell C*26,26*. As Rc increases, first a deformation occurs in the outer parts of the pattern observed in (a), as shown in (b) and (c), while a clearly new pattern emerges in (d). With reference to the pattern in (d), the capacitor voltages in neighboring cells exhibit a sharper change in the amplitude, which is practically reflected by the color contrast observable, especially as compared to that of (a).*

initial conditions in **Figure 8(d)**, while the center cell *C*26,26 features the same initial condition as the rest of the network cells. The difference between the patterns presented in **Figure 8(a)** and **(b)**, and equivalently in **Figure 8(c)** and **(d)**, shows the effect of a change in the number of cells sharing the same initial conditions. Likewise, the difference between the patterns presented in **Figure 8(a)** and **(c)**, and equivalently in **Figure 8(b)** and **(d)**, shows the effect of a change in the location (indeed, a rotation of half-side length) of cells sharing the same initial conditions, which can result in clearly different patterns. Since there exists a very high number of spatial permutations for the location and the number of cells with the same initial conditions, we conjecture that there may appear a large class of clearly distinguishable patterns, such as those presented in **Figure 8**, which results in a significant memory capacity of the network.

Lastly, we investigate the effect of the size of the network on the patterns generated through. For this purpose, we set the design parameters *Vb* to 1*:*2*V*, *Rb* to 2*k*Ω, and *Rc* to 0*:*2*k*Ω, while we define the same initial conditions for the corner cells as it is the case for the center cell. Then, we simulate structures of different size while preserving their square geometry, and depict the results in **Figure 9**, where *m* ¼ *n* ¼ 31 in **Figure 9(a)**, *m* ¼ *n* ¼ 41 in **Figure 9(b)**, *m* ¼ *n* ¼ 51 in **Figure 9(c)**, and *m* ¼ *n* ¼ 61 in **Figure 9(d)**. Once more, it can be concluded from **Figure 9** that the patterns generated across networks of different sizes and square geometry are clearly distinguishable one from the other.

*Pattern Formation in a RD-MCNN with Locally Active Memristors DOI: http://dx.doi.org/10.5772/intechopen.100463*

#### **Figure 8.**

*RD-MCNN output values represented by the cell memristor voltages obtained in a simulation of a twodimensional 51* � *51 RD-MCNN structure for Rb* ¼ 2*k*Ω*, Rc* ¼ 0*:*1*k*Ω *and Vb* ¼ 1*:*2*V. The cells in the middle of each side of the edges, or briefly the midpoint cells (C*1,26,*C*26,1,*C*51,26,*C*26,51*), have the same initial conditions as the center cell C*26,26 *in (a), while the corner cells (C*1,1*, C*1,51*, C*51,1*, C*51,51*) have the same initial conditions as the center cell C*26,26 *in (c). On the other hand, only the midpoint cells share the same initial conditions in (b), while the center cell C*26,26 *features the same initial condition as the rest of the network cells. Similarly, only corner cells share the same initial conditions in (d), while the center cell C*26,26 *features the same initial conditions as the rest of the network cells. A change in the location as well as in the number of cells sharing the same initial conditions can result in clearly different patterns, conferring the network a significant memory capacity.*
