Open access peer-reviewed chapter

Pattern Formation in a RD-MCNN with Locally Active Memristors

By Ahmet Samil Demirkol, Alon Ascoli, Ioannis Messaris and Ronald Tetzlaff

Submitted: February 18th 2021Reviewed: September 15th 2021Published: October 26th 2021

DOI: 10.5772/intechopen.100463

Downloaded: 56

Abstract

This chapter presents the mathematical investigation of the emergence of static patterns in a Reaction–Diffusion Memristor Cellular Nonlinear Network (RD-MCNN) structure via the application of the theory of local activity. The proposed RD-MCNN has a planar grid structure, which consists of identical memristive cells, and the couplings are established in a purely resistive fashion. The single cell has a compact design being composed of a locally active memristor in parallel with a capacitor, besides the bias circuitry, namely a DC voltage source and its series resistor. We first introduce the mathematical model of the locally active memristor and then study the main characteristics of its AC equivalent circuit. Later on, we perform a stability analysis to obtain the stability criteria for the single cell. Consequently, we apply the theory of local activity to extract the parameter space associated with locally active, edge-of-chaos, and sharp-edge-of-chaos domains, performing all the necessary calculations parametrically. The corresponding parameter space domains are represented in terms of intrinsic cell characteristics such as the DC operating point, the capacitance, and the coupling resistance. Finally, we simulate the proposed RD-MCNN structure where we demonstrate the emergence of pattern formation for various values of the design parameters.

Keywords

  • pattern formation
  • memristor
  • reaction–diffusion
  • cellular nonlinear networks
  • destabilization
  • local activity
  • complexity

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 viaresistive 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 in-memory-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 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, 23, 24, 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 model-based 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.

Advertisement

2. The locally active generic memristor

Nanoscale memristors with locally active (i.e., S-shaped) DC current–voltage (I–V) characteristics have been utilized in the design of oscillatory neuron cells [30, 31]. Furthermore, it was shown in [32, 33, 34] that memristor models in a generic form are capable of representing the dynamics of these nanoscale devices accurately. Similarly, in this work, we adopt a simplified generic memristor model to implement locally active dynamics, which helps to reduce the complexity and the simulation time of large-scale networks employing such devices. In addition, the adoption of the generic form enables simplified calculations related to the derivation of the AC model of the device, further promoting the clarification of the results.

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), Rsis the series resistance, Tis the state variable representing the temperature, and GT=g0expg1/Tis the temperature-dependent memductance function. We would like to note that Eqs. (1) and (2) define a sole memristor core with current imc=imand voltage vmc=vm1+RsGT1. The parameter values of the given model employed during the numerical simulations can be found in Table 1.

g0/Sg1/KRs/ΩCT/JK1gT/WK1T0/K
5103170020010146.67107300

Table 1.

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

im=vmGT1+RsGT=vmcGTE1
CTdTdt=vmcimgTTT0E2

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).

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 resistorRs, is depicted in the inset figure. (b) AC equivalent circuit of the entire memristor whereR1=Rs+R1i.

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 Rsto obtain the AC equivalent circuit of the overall memristor. Essentially, we set Rs=0Ωin Eqs. (1) and (2) so that the updated equation set represents the core 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, R1irepresents the inverse of the slope of the DC I-V curve for Rs=0Ωand naturally gets negative values in the NDR region. Furthermore, the quantity R1i+R2corresponds to the instantaneous resistance, VT0/IT0, of the core memristor, which always gets positive values at a given temperature T0, since the I-V curve lies either in the first or in the third quadrant. The same explanations are still valid also for Rs0Ωif R1iis replaced by R1=R1i+Rs. Lastly, Lrepresents the dynamics of the core device, while Land R2have positive values at all equilibrium points. The graphical representation of the small-signal element values of Figure 1(b) versus the DC current IQcan 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 ofFigure 1(b)are depicted as a function of the DC equilibrium currentIQ, while the parameter values are adopted fromTable 1during the numerical simulations. (a)R1vs. DC current where positive values ofR1are depicted in blue color and negative values ofR1are depicted in orange color. The orange part ofR1curve one-to-one corresponds to the NDR region of the DC I-V curve. (b)R2vs.IQwhereR2always gets positive values. (c)R1+R2vs.IQ.R1+R2corresponds to the instantaneous resistanceVT0/IT0of the memristor and always gets positive values. (d)Lvs.IQwhereLrepresents the dynamics of the memristor and takes positive values for all current values.

Advertisement

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 viaa DC voltage source and a bias resistor. Consequently, the proposed single cell is depicted in Figure 3(a) where Vbis the DC voltage source, Rbis the bias resistor, Rsis the series resistor, and Cis the parallel capacitor, while the open circle denotes the coupling node.

Figure 3.

(a) The single cell of the RD-MCNN structure.Vbis the DC voltage source,Rbis the bias resistor,Rsis the series resistor,Cis the parallel capacitor. The open circle denotes the coupling node. (b) AC equivalent of the circuit in (a). The small-signal transfer functionZsis the impedance seen through 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 R1<0and 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 Zs,which is illustrated in Figure 3(b), and given in (3). Here, we would like to note that the poles of Zsdirectly correspond to the eigenvalues of the state equations of the single cell.

Zs=sLR1+R2Rb+R1R2Rbs2LCR1+R2Rb+sLR1+R2+Rb+R1R2RbC+R1+RbR2=NsDsE3

The stability of Zscan be determined by examining the location of the poles of Zs, or equivalently, the roots of Dsvia Routh-Hurwitz criteria. At this point, it is timely to recall from Figure 2 that L>0, R2>0,and R1+R2>0, which reveals the fact that the coefficient of the s2term of Dsis readily positive. Consequently, to guarantee the (asymptotic) stability of Zs, remaining the coefficients of the Dspolynomial have to be nonnegative, as dictated by Routh-Hurwitz criteria. Therefore, we have R1+Rb>0and LR1+R2+Rb+R1R2RbC>0that respectively imply that the bias resistor Rbhas to be larger than the magnitude of the inverse of the slope of the DC I-V curve (i.e., Rb>R1), while Chas to be smaller than a critical Cmaxvalue, which is given in Eq. (4). Here, we would like to note that Dswould simplify in the absence of the parallel capacitor C, while the first stability condition R1+Rb>0would remain the same, as shown in [38], but Eq. (4) would be unnecessary.

C<LR1+R2+RbR1R2Rb=CmaxE4

In Figure 4(a), Cmaxis shown (in blue) as a function of the equilibrium point of the memristor current IQNDRacross the entire NDR region, for two different values of Rb. It can be seen from Figure 4(a) that the smaller value of Rbresults in a larger value for Cmax, which would be beneficial during a hardware realization, while we remind that Rbshould be kept larger than R1for a stable operation.

Figure 4.

(a)Cmaxvs.IQNDR(blue curves) andRcmaxvs.IQNDR(orange curves). (b)Cmax*Rcmaxproduct. It can be seen from both graphs that smaller values ofRbresults in larger values forCmax,Rcmax, andCmax*Rcmaxvalues, while the last product term is more influenced fromRbvalue.

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 viaa resistor) obtained between these identical cells would similarly result in an eventually stable structure since the combination of passive circuit elements through a resistive coupling cannot give rise to instability. Thus, to be able to observe the emergence of complexity in a diffusively coupled homogenous network, it is vital to accommodate locally active cells, which cannot be represented by positive real complexity functions in the small-signal regime. This fact at the same time implies that local activity can be simply judged as a violation of local passivity [9]. Furthermore, a locally active cell is defined to be on the edge-of-chaos, if it is biased at an asymptotically stable operating point. Most importantly, an uncoupled cell, which is poised on the edge-of-chaos and therefore considered as a “dead” or “silent” cell due to its stability, can be potentially destabilized from its quite state viaresistive coupling, which leads to the generation of complex patterns across the homogenous medium. Correspondingly, the edge-of-chaos domain involves a subset called the sharp-edge-of-chaos domain, which defines the set of parameters that destabilize the cell after coupling is introduced. Strictly speaking, a one-port cell is said to be locally active if and only if its small-signal transfer function (e.g., Zsgiven by Eq. (3) in our case) satisfies any of the conditions below [9]:

  1. Zshas a pole with positive real part, that is, Res>0.

  2. Zshas multiple poles on the imaginary() axis.

  3. Zshas a simple pole s=jωpon the imaginaryaxis and the residue associated with this pole, specifically rjωp=limsjωpZssjωp, is either a negative real number or a complex number.

  4. ReZ<0for some ω.

To confirm the locally active dynamics of the complexity function Zsof Eq. (3), which is associated with the uncoupled cell of Figure 3(a), a direct approach would be to check whether these four rigorously defined criteria are satisfied or not, an approach we had applied in a previous work [28]. However, in this work, we apply a quick inspection method of local activity criteria, and rather check if Zsclearly violates local passivity. In consistent with this approach, it can be seen that Zspossesses a right half plane (RHP) zero (i.e., z=R1R2/LR1+R2>0, for R1<0), a feature that cannot be realized with any locally passive transfer function. Thus, without a need for a further examination, we can infer that once it is biased in the NDR region, the uncoupled cell operates in the locally active regime. Furthermore, since the uncoupled cell is designed to be asymptotically stable (viatuning Rband Caccordingly), we can directly conclude that it is both locally active and on the edge-of-chaos across the entire NDR region. Finally, we would like to note that the RHP zero is a strong indicator of a possible destabilization scenario after the coupling is established.

3.3 Destabilization analysis after the introduction of coupling

For the emergence of pattern formation, it is essential that the identical cells lose stability after the resistive coupling is established between them. In the previous section, we have studied the stability of the uncoupled cell, while in this section, we investigate the destabilization process after a resistive array is added to couple the elements of the network. Since a direct stability analysis that requires exploring the complete network itself would be complicated and time-consuming, it would be beneficial to apply a simplified stability analysis. Considering highly accurate illustrative examples, the stability analysis of two-cell [37] and three-cell [29] arrays can be found in the literature. In this work, we present a quick inspection method, which is conceptually introduced in [9] and has been implemented through an example in [28], where the idea is to terminate the coupling node of the single cell with a coupling resistor, resulting in the circuit depicted in Figure 5(a) for the case study presented here.

Figure 5.

(a) A simplified scenario for the resistively coupled single cell whereRcstands for the coupling resistor. (b) AC equivalent circuit of the cell in (a). The poles ofZcsshall be investigated for analyzing the stability of the circuit in (a) at a given equilibrium point.

To investigate the possible destabilization scenarios of the resistively terminated cell in Figure 5(a), we follow the same circuit theoretical approach as in the previous subsections and examine the corresponding AC equivalent circuit, which is shown in Figure 5(b). Here, the poles of the impedance function Zcscoincide with the eigenvalues of the second-order system equivalent of the circuit in Figure 5(a) at a given equilibrium point and should be examined to determine the destabilization conditions. At this point, it should be noted that the AC equivalent circuits depicted in Figures 3(b) and 5(b) are qualitatively equivalent to each other, while Rcappears additionally in parallel with Rbin Figure 5(b). Therefore, the expression of Zcs, readily given in Eq. (5), is identical to the expression of Zs, where the term Rbis replaced with the parallel equivalent term Rbc, that is, Rbc=RbRc/Rb+Rc.

Zcs=sLR1+R2Rbc+R1R2Rbcs2LCR1+R2Rbc+sLR1+R2+Rbc+R1R2RbcC+R1+RbcR2=NcsDcsE5

To claim instability of Zcs, at least one coefficient of the Dcspolynomial has to be negative, as dictated by Routh-Hurwitz criteria. Here, it is straightforward to see that the coefficient of the s2term of Dcsis readily positive. Similarly, it is possible to show after some algebraic rearrangement that the coefficient of the sterm remains positive as long as the stability precondition given by Eq. (4), with Rbreplaced by Rbc, is satisfied. Thus, for instability, the constant term of Dcs, namely R1+RbcR2, has to be negative. Starting with the inequality R1+RbcR2<0and replacing Rbcwith RbRc/Rb+Rcdirectly give us the destabilization condition introduced by Eq. (6).

Rc<R1RbR1+Rb=Rcmax=R1RbE6

In Figure 4(a), we plot (in orange) Rcmaxas a function of the equilibrium point of the memristor current IQNDRacross the entire NDR region, for two different values of Rb. Similar to the characteristics of Cmax, the smaller value of Rbresults in a larger value for Rcmax, which would relieve the design constraints for an hardware realization. On the other hand, Eq. (6) reveals the fact that considering the memristor model-dependent quantities, Rcmaxdepends only on the slope of the I-V curve (i.e., R11), while Cmaxdepends on other quantities (see Eq. (4)) as well. Besides, as a function of IQNDR, the characteristics of Cmaxincrease (decrements), while the characteristics of Rcmaxdecrease (increments). Moreover, it is interesting to mention that the condition derived in Eq. (6) is qualitatively equivalent to the results presented in [29, 37], indicating the fact that the simplification considered in this work is accurate and provides a quick insight into the destabilization scenario of the coupled cell in Figure 5(a).

Considering a typical time constant based design approach, we characterize the term RcmaxCmax, introducing the exact and an approximate product in Eq. (7), and plot these quantities in Figure 4(b) as a function of IQNDR, for two different values of Rb. Here, we would like to note that the approximate product term L/R1+Rbis obtained under the assumption R1+R2+RbR2.

RcmaxCmax=R1RbR1+RbLR1+R2+RbR1R2Rb=LR1+R2+RbR2R1+RbLR1+RbE7

It can be seen from Figure 4(b) that the approximated time constant term given in Eq. (7) follows the original expression very closely, especially for the smaller value of Rb. Moreover, the approximated value is independent of R2, but rather dependent on the dynamic quantity L, or in other words, the switching speed of the memristor device. As a final and crucial remark, referring to the instability region of parameters, that is, the sharp-edge-of-chaos domain, it is possible to rigorously define the necessary circuit design variables by taking into account the parametric equations given by Eqs. (4) and (6).

Advertisement

4. RD-MCNN structure and static pattern formation

The proposed RD-MCNN structure has a planar grid arrangement and it is composed of m×nidentical cells where mis the number of rows, nis the number of columns, and Ci,jrepresents the cell at ithrow and the jthcolumn. 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 51×51structure. We define the same initial conditions (T0=300Kand Vc0=0V) for all the cells unless otherwise stated, except for the cell, namely C26,26, 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 Rbto 2kΩand Rcto 0.1kΩ, while we tune Vbto vary the location of the DC operating point of each cell. We present the emergent static patterns in Figure 6, where Vb=1.2Vin Figure 6(a), Vb=1.3Vin Figure 6(b), Vb=1.4Vin Figure 6(c), and Vb=1.5Vin 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 Vbmay lead to static patterns featuring a reduced spread in capacitor voltage amplitude through the array, reflected by a gradual decrement in the variety of colors in the emergent patterns, as especially observed in Figure 6(d).

Figure 6.

RD-MCNN output values represented by the cell memristor voltages obtained in a simulation of a two-dimensional 51 × 51 RD-MCNN structure forRb=2kΩ,Rc=0.1kΩ, whileVb=1.2Vin (a),Vb=1.3Vin (b),Vb=1.4Vin (c), andVb=1.5Vin (d). All the cells have the same initial conditions except for the center cellC26,26. 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).

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 Vbto 1.2Vand Rbto 2kΩwhile we tune the value of Rcto adjust the coupling strength. The results are illustrated in Figure 7, where Rc=0.25kΩin Figure 7(a), Rc=0.5kΩin Figure 7(b), Rc=1kΩin Figure 7(c), and Rc=2kΩin Figure 7(d). It can be seen that as Rcincreases, 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).

Figure 7.

RD-MCNN output values represented by the cell memristor voltages obtained in a simulation of a two-dimensional 51 × 51 RD-MCNN structure forRb=2kΩ,Vb=1.2V, whileRc=0.25kΩin (a),Rc=0.5kΩin (b),Rc=1kΩin (c), andRc=2kΩin (d). All the cells have the same initial condition except for the center cellC26,26. AsRcincreases, 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).

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=1.2V, Rb=2kΩ, and Rc=0.1kΩ. The cells located in the center of each side of the edges (i.e., C1,26,C26,1,C51,26,C26,51), or midpoint cells for short, have the same initial condition as the center cell C26,26in Figure 8(a), while only the corner cells (C1,1, C1,51, C51,1, C51,51) have the same initial condition as the center cell C26,26in Figure 8(c). On the other hand, only the midpoint cells share the same initial conditions in Figure 8(b), while the center cell C26,26features the same initial condition as the rest of the network. Similarly, only the corner cells share the same initial conditions in Figure 8(d), while the center cell C26,26features 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.

Figure 8.

RD-MCNN output values represented by the cell memristor voltages obtained in a simulation of a two-dimensional 51 × 51 RD-MCNN structure forRb=2kΩ,Rc=0.1kΩandVb=1.2V. The cells in the middle of each side of the edges, or briefly the midpoint cells (C1,26,C26,1,C51,26,C26,51), have the same initial conditions as the center cellC26,26in (a), while the corner cells (C1,1,C1,51,C51,1,C51,51) have the same initial conditions as the center cellC26,26in (c). On the other hand, only the midpoint cells share the same initial conditions in (b), while the center cellC26,26features 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 cellC26,26features 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.

Lastly, we investigate the effect of the size of the network on the patterns generated through. For this purpose, we set the design parameters Vbto 1.2V, Rbto 2kΩ, and Rcto 0.2kΩ, 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=n=31in Figure 9(a), m=n=41in Figure 9(b), m=n=51in Figure 9(c), and m=n=61in 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.

Figure 9.

Simulation results of variousm×nRD-MCNN structures forRb=2kΩ,Rc=0.2kΩandVb=1.2V, wherem=n=31in (a),m=n=41in (b),m=n=51in (c), andm=n=61in (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.

Advertisement

5. Conclusion

In this chapter, we have presented the mathematical investigation of static pattern formation across a resistively coupled RD-MCNN structure viathe 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 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 low-resistance 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 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 non-idealities 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.

Advertisement

Acknowledgments

We kindly acknowledge the support for the Book Processing Charge by the Open Access Publication Fund of Saxon State and University Library Dresden (SLUB). This work was also supported by the German Research Foundation (DFG) under Project No. 411647366.

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Ahmet Samil Demirkol, Alon Ascoli, Ioannis Messaris and Ronald Tetzlaff (October 26th 2021). Pattern Formation in a RD-MCNN with Locally Active Memristors, Memristor - An Emerging Device for Post-Moore’s Computing and Applications, Yao-Feng Chang, IntechOpen, DOI: 10.5772/intechopen.100463. Available from:

chapter statistics

56total chapter downloads

1Crossref citations

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

Related Content

This Book

Next chapter

Development of Compute-in-Memory Memristive Crossbar Architecture with Composite Memory Cells

By Mehri Teimoory, Amirali Amirsoleimani, Arash Ahmadi and Majid Ahmadi

Related Book

First chapter

Boundary Perturbation Theory for Scattering in Layered Rough Structures

By Pasquale Imperatore, Antonio Iodice and Daniele Riccio

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us