**3. The single cell**

To achieve a compact single cell design, a basic solution is to establish oscillatory dynamics by implementing a minimal second-order system accompanied by a simple bias circuitry. Since the memristor itself implements a first-order system with inductive dynamics, an efficient solution to increase the order of the system would be to include an additional capacitor into the cell design. Additionally, we prefer to realize the proper biasing *via* a DC voltage source and a bias resistor. Consequently, the proposed single cell is depicted in **Figure 3(a)** where *Vb* is the DC voltage source, *Rb* is the bias resistor, *Rs* is the series resistor, and *C* is the parallel capacitor, while the open circle denotes the coupling node.

## **3.1 Stability analysis of the single cell**

Pattern formation through a RD-MCNN prerequisites stable single-cell dynamics in the isolated case. Therefore, in this section, we perform a parametric stability analysis for the uncoupled single cell, which is also the first step of parameter extraction of the circuit element values. A straightforward procedure of stability analysis would require performing linearity analysis on the state equations of the second-order single cell, followed by the derivation of the eigenvalues, to express the stability conditions [37]. In this work, we present a circuit theoretical approach where we employ the AC equivalent circuit of the single cell, which is depicted in **Figure 3(b)**, and derive the stability conditions for the uncoupled case. Adopting this approach, it is possible to express in an efficient way, the stability conditions of the isolated cell in terms of linear circuit element values and of AC element values of the memristor in a parametric form, which enables a direct evaluation of the results. Finally, although the results to be derived should hold for any equilibrium point on the DC I-V curve, without loss of generality, we assume that the memristor is biased in the NDR region, where *R*<sup>1</sup> <0 and the device is locally active, which is essential for the emergence of complexity.

To derive the stability conditions of the single-cell circuit in **Figure 3(a)**, we firstly obtain the AC equivalent of it, simply by replacing the memristor with its small-signal equivalent (previously introduced in **Figure 1(b)**) and by assuming the DC voltage source as a short-circuit element, resulting in the circuit given in **Figure 3(b)**. Second, we calculate the impedance function *Z s*ð Þ, which is illustrated in **Figure 3(b)**, and given in (3). Here, we would like to note that the poles of *Z s*ð Þ directly correspond to the eigenvalues of the state equations of the single cell.

#### **Figure 3.**

*(a) The single cell of the RD-MCNN structure. Vb is the DC voltage source, Rb is the bias resistor, Rs is the series resistor, C is the parallel capacitor. The open circle denotes the coupling node. (b) AC equivalent of the circuit in (a). The small-signal transfer function Z s*ð Þ *is the impedance seen through the coupling node.*

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

$$Z(s) = \frac{sL(R\_1 + R\_2)R\_b + R\_1R\_2R\_b}{s^2LC(R\_1 + R\_2)R\_b + s[L(R\_1 + R\_2 + R\_b) + R\_1R\_2R\_bC] + (R\_1 + R\_b)R\_2} = \frac{N(s)}{D(s)}\tag{3}$$

The stability of *Z s*ð Þ can be determined by examining the location of the poles of *Z s*ð Þ, or equivalently, the roots of *D s*ð Þ via Routh-Hurwitz criteria. At this point, it is timely to recall from **Figure 2** that *L*>0, *R*<sup>2</sup> > 0, and ð Þ *R*<sup>1</sup> þ *R*<sup>2</sup> >0, which reveals the fact that the coefficient of the *s* <sup>2</sup> term of *D s*ð Þ is readily positive. Consequently, to guarantee the (asymptotic) stability of *Z s*ð Þ, remaining the coefficients of the *D s*ð Þ polynomial have to be nonnegative, as dictated by Routh-Hurwitz criteria. Therefore, we have ð Þ *R*<sup>1</sup> þ *Rb* >0 and *L R*ð <sup>1</sup> þ *R*<sup>2</sup> þ *Rb*Þ þ *R*1*R*2*RbC*> 0 that respectively imply that the bias resistor *Rb* has to be larger than the magnitude of the inverse of the slope of the DC I-V curve (i.e., *Rb* >j j *R*<sup>1</sup> ), while *C* has to be smaller than a critical *Cmax* value, which is given in Eq. (4). Here, we would like to note that *D s*ð Þ would simplify in the absence of the parallel capacitor *C*, while the first stability condition ð Þ *R*<sup>1</sup> þ *Rb* >0 would remain the same, as shown in [38], but Eq. (4) would be unnecessary.

$$C < \frac{L(R\_1 + R\_2 + R\_b)}{-R\_1 R\_2 R\_b} = C\_{\text{max}}\tag{4}$$

In **Figure 4(a)**, *Cmax* is shown (in blue) as a function of the equilibrium point of the memristor current *IQ*�*NDR* across the entire NDR region, for two different values of *Rb*. It can be seen from **Figure 4(a)** that the smaller value of *Rb* results in a larger value for *Cmax*, which would be beneficial during a hardware realization, while we remind that *Rb* should be kept larger than j j *R*<sup>1</sup> for a stable operation.

#### **3.2 Local activity analysis of the single cell**

The small-signal behavior of an uncoupled cell, which is biased at a locally passive operating point, can be represented by a positive real complexity function (or similarly, by a positive real transfer function for the 1-port coupling case). Subsequently, the AC equivalent circuit of the same cell at the given operating point would be strictly composed of passive linear elements, which inherently results in stable dynamics. Therefore, a diffusive coupling (i.e., the coupling established *via* a resistor) obtained between these identical cells would similarly result in an

#### **Figure 4.**

*(a) Cmax vs. IQ*�*NDR (blue curves) and Rc*� *max vs. IQ*�*NDR (orange curves). (b) Cmax \* Rc*� *max product. It can be seen from both graphs that smaller values of Rb results in larger values for Cmax, Rc*� *max , and Cmax \* Rc*� *max values, while the last product term is more influenced from Rb value.*
