**5. Conclusion**

In this chapter, we have presented the mathematical investigation of static pattern formation across a resistively coupled RD-MCNN structure *via* the application of the theory of local activity. The considered networks have identical cells in a compact form, each of which is composed of a DC bias voltage source, a bias resistor, a locally active memristor, and a parallel capacitor. The memristor was represented through a generic model, which helped to reduce the numerical complexity and to shorten the simulation time of large-scale networks. A useful AC equivalent circuit could be derived for the locally active device, which facilitated further calculations related to the small signal model of the network cell. We have adopted a systematic circuit theoretical approach that we applied for the stability analysis of the isolated cell and for the extraction of its parameter values for its operation on the edge-of-chaos and sharp-edge-of-chaos domains. In this way, we have performed a simple, fast, and robust analysis, which at the same time allowed an efficient interpretation of the results. All the calculations were performed in parametric form, which allowed a deep investigation of the results, making it possible to extract the related sets of parameters in terms of the cell characteristics, namely the DC operating point and the capacitor value, as well as the value of the

**Figure 9.**

*Simulation results of various m* � *n RD-MCNN structures for Rb* ¼ 2*k*Ω*, Rc* ¼ 0*:*2*k*Ω *and Vb* ¼ 1*:*2*V, where m* ¼ *n* ¼ 31 *in (a), m* ¼ *n* ¼ 41 *in (b), m* ¼ *n* ¼ 51 *in (c), and m* ¼ *n* ¼ 61 *in (d). The corner cells have the same initial conditions as the center cell for all the structures. Although the four structures share a square geometry, they differ in size, which leads to clearly distinguishable patterns.*

coupling resistance. The proposed RD-MCNN was shown to generate diverse patterns while we have extensively investigated the effect of design parameters, initial conditions, and the size of the network on the patterns generated therein. Furthermore, the mathematical and circuit theoretical approach employed during the investigation of pattern formation dynamics in the RD-MCNN under focus in this chapter can be easily adopted to RD-MCNNs with different cell designs.

Finally, the content of this work can be extended to the investigation of the role of the network geometry and variety in cells' initial conditions on pattern formation. Furthermore, the impact of memristor non-idealities such as variability in memristor behavior from cycle to cycle, as well as from device to device, and endurance degradation and short- and long-term reliability issues shall be further examined to achieve a robust design. Such non-idealities, essentially, can affect the NDR characteristics of the memristors and narrow the width of the NDR region, or the devices can even get stuck in one of the high-resistance or low-resistance locally passive regimes where they act as dead cells without any dynamics. Preliminary simulation results show that endowing memristors with narrower NDR width negatively affect the patterns' color contrasts, while the presence of the high- or lowresistance locally passive devices results in the formation of color clusters within the patterns. Interestingly, similar color clusters emerge also when some of the array memristors' initial conditions are randomly selected, which suggests a possible strategy to compensate for the disturbing action of locally passive cells on the formation of predefined patterns through a dynamic conditioning of their initial conditions. Similarly, the low color contrast quality in the patterns, which occur due *Pattern Formation in a RD-MCNN with Locally Active Memristors DOI: http://dx.doi.org/10.5772/intechopen.100463*

to the existence of memristors featuring a narrower NDR region, can be improved by reprogramming dynamically the bias voltage sources in these "defect" cells. In conclusion, proper control strategies including cells' dynamic biasing and initial condition conditioning, as well as reconfiguration of the network array, may be considered as a solution to overcome the harmful effects which memristor nonidealities may induce on the emergence of predefined patterns in the proposed RD-MCNNs. Future studies will also be devoted to envision an application, where the capability of our cellular medium to generate a variety of steady-state static patterns may be useful to our modern society.
