**2.1 The model definition**

We introduce the memristor model equations, namely the I–V relationship in Eq. (1), and the state equation in Eq. (2), where *im* (*vm*) is the memristor current (voltage), *Rs* is the series resistance, *T* is the state variable representing the temperature, and *G T*ð Þ¼ *<sup>g</sup>*<sup>0</sup> <sup>∙</sup> *exp* �*g*1*=<sup>T</sup>* is the temperature-dependent memductance function. We would like to note that Eqs. (1) and (2) define a sole memristor core with current *imc* <sup>¼</sup> *im* and voltage *vmc* <sup>¼</sup> *vm* <sup>∙</sup>ð Þ <sup>1</sup> <sup>þ</sup> *Rs* <sup>∙</sup> *G T*ð Þ �<sup>1</sup> . The parameter values of the given model employed during the numerical simulations can be found in **Table 1**.

$$i\_m = \upsilon\_m \bullet \frac{G(T)}{1 + R\_s \bullet G(T)} = \upsilon\_{mc} \bullet G(T) \tag{1}$$

$$C\_T \frac{dT}{dt} = v\_{mc} \bullet i\_m - \mathbf{g}\_T \bullet (T - T\_0) \tag{2}$$

The S-shaped DC I-V curve under the current sweep and the schematic of the complete device, which can be depicted as the series combination of the core memristor and *Rs*, are introduced in **Figure 1(a)**. At this point, we would like to point out that the negative differential resistance (NDR) region, which hosts the peculiar dynamics of the locally active memristor, is highlighted with the orange color in **Figure 1(a)**.

### **2.2 The AC equivalent circuit**

Since the small-signal equivalent of the memristor plays an important role during the forthcoming circuit theoretical stability analysis, it is crucial to obtain the AC equivalent circuit of the memristor device. For this purpose, we first derive the small-signal equivalent of the core memristor and then combine it in series with *Rs* to obtain the AC equivalent circuit of the overall memristor. Essentially, we set *Rs* ¼ 0Ω in Eqs. (1) and (2) so that the updated equation set represents the core


**Table 1.**

*Parameter values for Eqs. (1) and (2).*

#### **Figure 1.**

*(a) DC I-V curve of the locally active memristor, with NDR region highlighted. The schematic of the memristor, as the series combination between the core device and the series resistor Rs, is depicted in the inset figure. (b) AC equivalent circuit of the entire memristor where R*<sup>1</sup> ¼ *Rs* þ *R*1*i.*

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

memristor only. Then, we linearize Eqs. (1) and (2) by deriving their first-order Taylor series expansions. As the next step, we apply Laplace transform to the linearized equations, and after a suitable rearrangement of the Laplace equations, we express the impedance function of the core memristor in Foster's first form [35] RL circuit configuration. The final configuration of the AC equivalent circuit of the entire memristor is depicted in **Figure 1(b)**.

Regarding the AC equivalent circuit, *R*1*<sup>i</sup>* represents the inverse of the slope of the DC I-V curve for *Rs* ¼ 0Ω and naturally gets negative values in the NDR region. Furthermore, the quantity ð Þ *R*1*<sup>i</sup>* þ *R*<sup>2</sup> corresponds to the instantaneous resistance, *V T*ð Þ<sup>0</sup> *=I T*ð Þ<sup>0</sup> , of the core memristor, which always gets positive values at a given temperature *T*0, since the I-V curve lies either in the first or in the third quadrant. The same explanations are still valid also for *Rs* 6¼ 0Ω if *R*1*<sup>i</sup>* is replaced by *R*<sup>1</sup> ¼ *R*1*<sup>i</sup>* þ *Rs*. Lastly, *L* represents the dynamics of the core device, while *L* and *R*<sup>2</sup> have positive values at all equilibrium points. The graphical representation of the small-signal element values of **Figure 1(b)** versus the DC current *IQ* can be found in **Figure 2**, where we use parameter values as given in **Table 1** during the numerical simulations. Finally, a similar procedure regarding the derivation of the AC equivalent circuit of a generic memristor including a detailed investigation and graphical illustration can be found in [36] as well.

**Figure 2.**

*Small signal elements values of Figure 1(b) are depicted as a function of the DC equilibrium current IQ , while the parameter values are adopted from Table 1 during the numerical simulations. (a)* ∣*R*1∣ *vs. DC current where positive values of R*<sup>1</sup> *are depicted in blue color and negative values of R*<sup>1</sup> *are depicted in orange color. The orange part of* ∣*R*1∣ *curve one-to-one corresponds to the NDR region of the DC I-V curve. (b) R*<sup>2</sup> *vs. IQ where R*<sup>2</sup> *always gets positive values. (c) R*ð Þ <sup>1</sup> þ *R*<sup>2</sup> *vs. IQ . R*ð Þ <sup>1</sup> þ *R*<sup>2</sup> *corresponds to the instantaneous resistance V T*ð Þ<sup>0</sup> *=I T*ð Þ<sup>0</sup> *of the memristor and always gets positive values. (d) L vs:IQ where L represents the dynamics of the memristor and takes positive values for all current values.*
