**1. Introduction**

An important feature of complex systems is the emergence of spatiotemporal patterns, which can be observed in numerous physical systems consisting of homogeneous media [1]. Among many examples, the emergence phenomenon can occur as a result of oscillatory kinetics in a chemical reaction [2], self-organization of biological organisms [3], mechanical vibration on liquid surfaces [4], or mineral precipitation on geologic surfaces [5]. In particular, network dynamics of pattern formation is considered to be the key attribute of information processing and memory storage in biological neural networks [6] and, hence, is the main concern in neuroscience. Following the consequences of the seminal paper of Turing [7] where he introduced the chemical basis of morphogenesis, the mechanism behind pattern formation dynamics has been extensively studied in various scientific branches. On one hand, different mathematical models have been proposed to elucidate the analytical principles of pattern formation dynamics [8]. On the other hand, there

has been an ambiguity on the conceptual definition of the complexity phenomenon, which was previously described as symmetry breaking, instability of the homogenous, exchange of energy, or self-organization. In [9], Chua has proposed the theory of local activity, as the origin of complexity, where he quantitatively defined the mathematical principle behind the emergence of complex patterns in a homogenous medium. In this way, the theory of local activity assembled various definitions under the same framework and enabled the quantitative investigation of pattern formation dynamics, especially through electrical circuits. Since then, several works dealing with pattern formation dynamics on electrical hardware have referred to the theory of local activity to perform robust and quantitative analysis (e.g., see [10]). Mathematically, a well-known method to realize pattern formation dynamics is to implement reaction–diffusion partial differential equations (RD-PDEs) [11]. Various physical systems adopting reaction–diffusion equations have been shown to create well-known spatiotemporal phenomena such as traveling waves or clustering patterns [12]. Therefore, circuit implementations of RD-PDEs would be a reasonable approach to capture pattern formation dynamics on electrical hardware. Cellular nonlinear networks (CNNs), which can be described as homogeneous structures composed of evenly spaced and locally coupled identical cells, are prominent hardware solutions to implement the RD-PDEs. In particular, the reaction dynamics of an RD-PDE can be implemented by the identical cells of a CNN, while the diffusive dynamics of the same RD-PDE can be successfully discretized using the central difference approach and then implemented *via* resistive coupling between the identical neighboring cells, resulting in the so-called RD-CNN structure [13]. Motivated by this approach, emergent phenomena, such as Turing patterns or auto waves, have already been demonstrated across RD-CNNs [14].

The high-speed data transfer capability of modern mobile communication systems has led to the introduction of 5G and, in the future, 6G networks successively in a short period, which has enabled a new era in technology such as Industry 4.0 and Internet-of-Things (IoT) [15]. In parallel with the requirements of the new technological applications and the performance criteria of the corresponding hardware realizations, the design of bio-inspired neuromorphic systems utilizing the inmemory-computing principle, which stands as an alternative option to the design of conventional von-Neumann computing architectures where memory and processor units are separated from each other, has recently gained a lot of attention [16]. Similarly, there is a huge research effort in academia and industry for developing efficient fabrication techniques to implement the new in-memory computing circuit elements such as resistive switching memories, which can apparently be described as memristors [17]. The memristor, a two-terminal circuit element that was theoretically hypothesized 50 years ago, can be briefly considered as a nonlinear resistor with inherent memory dynamics [18]. Due to recent developments in physical implementations of different types of memory devices, the modeling and analysis of memristors and memristive systems have also gained attention, resulting in comprehensive circuit and system theoretical investigations [19]. Accordingly, the availability of manufactured nanoscale memristors emerges as a key enabler for the implementation of memory and processing units realized in the same place of compact hardware, and thus, provides the opportunity to design novel bio-inspired systems with in-memory-computing capabilities. In this way, it can be possible to overcome the end of Moore's law by engineering the information processing architectures, rather than downscaling the semiconductor device dimensions, which practically has come to an end.

Consistent with the above-mentioned text, locally active memristors have already been utilized in the design and application of spiking neural cells

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

demonstrating promising results [20], while prominent artificial neural network designs employing memristor crossbar arrays have been presented in several works [21]. Besides, the theory of memristor cellular nonlinear networks (MCNNs) has been comprehensively investigated in [22–25] and the formation of Turing patterns with reaction–diffusion MCNNs (RD-MCNNs) has been introduced in [26]. In addition, pattern formation utilizing locally active NbO memristors in an MCNN has been presented in [27]. In a previous work [28], we have presented the modelbased analytical investigation of dynamic pattern formation in a RD-MCNN, exploiting a manufactured locally active memristor, while the proposed single cell structure, as well as the subsequent results, is different from those presented here. Similarly, in [29], we have presented preliminary results of the mathematical investigation of static pattern formation in a RD-MCNN structure. The goal of this work is to extend the content of the previously presented works and to introduce an analytical design procedure for the implementation of static pattern formation across a compact RD-MCNN structure while taking into account the theory of local activity for the mathematical treatment. To enable the locally active dynamics, we employ a nanoscale locally active generic memristor model in the design of the basic network cell. The proposed RD-MCNN has a planar grid form and is composed of locally coupled identical cells. The compact unit cell consists of a locally active memristor in parallel with a linear capacitor, besides the bias circuitry, namely a DC voltage source and its series resistor. We first introduce the mathematical definition of the locally active generic memristor employed and, then, study the main characteristics of its AC equivalent circuit. Later on, we perform a quick stability analysis and determine stability criteria for the single cell. Consequently, we apply the theory of local activity in order to extract the parameter space with locally active, edge-of-chaos, and sharp-edge-of-chaos domains, performing all the necessary calculations parametrically. The corresponding parameter space domains are illustrated in terms of intrinsic network characteristics such as the cell DC operating point, the cell capacitance, and the coupling resistance. Essentially, we adopt a circuit theoretical approach, regarding the stability analysis of the single cell and the destabilization process after the coupling is established, which promotes an efficient investigation of the criteria to be derived. Finally, we carry out numerical simulations where we demonstrate the emergence of pattern formation across the proposed RD-MCNN structure for various values of the design parameters.
