Let us consider equations (1) and (2). When *u*(*x*)and *v*(*x*) are represented by voltages on the RD hardware, the gradient (diffusion terms in the RD model) is represented by linear resistors. We discretize first equation of (1) in the following way:
$$\frac{d{u}_{s}}{dt}=\frac{{g}_{u}({u}_{s-1}-{u}_{s})+{g}_{u}({u}_{s+1}-{u}_{s})}{\mathrm{\Delta}{x}^{2}}+{f}_{u}(.),$$(21)

where *s* is the spatial index, *Δx* is the discrete step in space, terms *g*_{u}(*u*_{s–1} – *u*_{s})and *g*_{u}(*u*_{s}+1-* u*_{s})represent respectively current flowing into the *s*th node from (*s* – l)th and (*s +* l)th nodes via two resistors whose conductance is represented by *g*_{u}. The spatial Laplacian V^{2} in equation (1) can be approximated as
$$\begin{array}{l}{\mathrm{\nabla}}^{2}u(x)=\frac{{u}_{s-1}+{u}_{s+1}-2{u}_{s}}{\mathrm{\nabla}{x}^{2}}.\end{array}$$

Here, we introduce the memristor model [7], in which the resistors are replaced with memristors. The resulting point dynamics is given by
$$\begin{array}{l}\frac{d{u}_{s}}{dt}=\frac{{g}_{u}({w}_{s}^{L})({u}_{s-1}-{u}_{s})+{g}_{u}({w}_{s}^{R})({u}_{s+1}-{u}_{s})}{\mathrm{\Delta}{x}^{2}}+{f}_{u}\left(.\right);\\ \frac{dvj}{dt}={f}_{v}\left(.\right),\end{array}$$(22)

where *g*_{u}(.)denotes the monotonically increasing function defined by:
$$\begin{array}{l}{g}_{u}\left({w}_{s}^{L,R}\right)={g}_{\mathit{m}\mathit{i}\mathit{n}}+\left({g}_{max}-{g}_{\mathit{m}\mathit{i}\mathit{n}}\right)\frac{1}{1+{e}^{-\beta {w}_{s}^{L,R}}},\end{array}$$(23)

where *β* denotes the gain, *g*_{min} and *g*_{max} denote the minimum and maximum coupling strengths, respectively, ${w}_{S}^{L.R}$ denote the variables for determining the coupling strength ( *L -* leftward, *R -* rightward). Finally, we introduce the following memristive dynamics for ${w}_{S}^{L,R}$:
$$\begin{array}{l}\tau \frac{d{w}_{s}^{L,R}}{dt}={g}_{u}({w}_{s}^{L,R}).{\eta}_{1}.({u}_{s-1}-{u}_{s}),\end{array}$$(24)

where the right-hand side represents the current of the memristors [7], *n*_{1}denotes the polarity coefficient -*n*_{1} ${\eta}_{1}=+1:{w}_{s}^{L},{\eta}_{1}=-1:{w}_{s}^{R}.$

In this section we shall identify the values of the cell parameters for which the interconnected RD system may exhibit complexity. The necessary condition for a nonconservative system to exhibit complexity is to have its cell locally active [3]. The theory which will be presented below offers a constructive analytical method for uncovering local activity. The precisely defined parameter domain in which the model can exhibit complex behavior is called the edge of chaos. The RD system operating near the edge of chaos, where the cells are not locally active, is also linearly asymptotically stable. In particular, constructive and explicit mathematical inequalities can be obtained for identifying the region in the CNN parameter space where complexity phenomena may emerge, as well as for localizing in further into a relatively small parameter domain called edge of chaos where the potential for emergency is maximized. By restricting the cell parameter space to the local activity domain, a major reduction in the computing time required by the parameter search algorithms is achieved [3].

We shall apply the theory of local activity in order to study dynamic behaviour of RD-CNN model with memristors coupling in the following form:
$$\begin{array}{}\frac{d{u}_{kl}}{dt}& ={f}_{1}({u}_{kl},{v}_{kl})+{D}_{1}({w}^{L,R})({u}_{k-1l}+{u}_{k+1l}+{u}_{kl-1}+{u}_{kl+1}-4{u}_{kl});\\ \frac{d{v}_{kl}}{dt}& ={f}_{2}({u}_{kl},{v}_{kl})+{D}_{2}({w}^{L,R})({v}_{k-1l}+{v}_{k+1l}+{v}_{kl-1}+{v}_{kl+1}-4{v}_{kl}),\end{array}$$(25)

where *D*_{p}(*w*^{L,R})denotes the monotonically increasing function defined as
$$\begin{array}{l}{D}_{p}({w}^{L,R})={D}_{min}({D}_{max}-{D}_{min}).\frac{1}{1-{e}^{-\beta {w}_{p}^{L,R}}},p=1,2,\end{array}$$(26)

*f*_{1}(*u*_{kl},v_{kl}) = *a*-(*b* + 1)*u*_{kl}+*v*_{kl}u^{2}_{kl} , f_{2}(*u*_{kl},v_{kl}) = *bu*_{kl}-*v*_{kl}u^{2}_{kl}. Then the memristive dynamics is defined as in (24).

We develop the following constructive algorithm for determining the edge of chaos domain.

1. Map the RD-BCNN with memristor coupling into its discrete-space version:
$$\begin{array}{l}\frac{d{u}_{kl}}{dt}={f}_{1}({u}_{kl},{v}_{kl})+{D}_{1}({w}_{kl}^{L,R})({u}_{k-1l}+{u}_{k+1l}+{u}_{kl-1}+{u}_{kl+1}-4{u}_{kl})\phantom{\rule{thickmathspace}{0ex}};\\ \frac{d{v}_{kl}}{dt}={f}_{2}({u}_{kl},{v}_{kl})+{D}_{2}({w}_{kl}^{L,R})({v}_{k-1l}+{v}_{k+1l}+{v}_{kl-1}+{v}_{kl+1}-4{v}_{kl}),\end{array}$$(27)

2. Find the equilibrium points of RD-BCNN model (27). According to the theory of dynamical systems equilibrium points *u*, v** are these for which
$$\begin{array}{l}{f}_{1}({u}^{\ast},{v}^{\ast})=0;\\ {f}_{2}({u}^{\ast},{v}^{\ast})=0.\end{array}$$(28)

In general, system (28) may have one, two or three real roots and these roots are functions of the cell parameters *a, b*. In the case the Brusselator cell we have a unique equilibrium point ${E}_{1}=({u}_{1}^{\ast},{v}_{1}^{\ast})=(a,{\displaystyle \frac{b}{a}).}$

3. Calculate the cell coefficients of the Jacobian matrix of (28) about the system equilibrium point *E*_{1}, namely,
$$\begin{array}{l}{a}_{11}({E}_{1})=b-1;\\ {a}_{12}({E}_{1})={a}^{2};\\ {a}_{21}({E}_{1})=-b;\\ {a}_{22}({E}_{1})=-{a}^{2}.\end{array}$$

4. Calculate the trace *Tr*(*E*_{1})and the determinant Δ*(E*_{1})of the Jacobian matrix for the unique equilibrium point.

#### Definition 4.1

*Stable and locally active region SLAR*(*E*_{1})* at the equilibrium point E*_{1} for the RD-BCNN with memristor coupling model is such that
$$\begin{array}{l}{a}_{22}>0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}4{a}_{11}{a}_{22}<({a}_{12}+{a}_{21}{)}^{2}\end{array}$$

*and*
$$\begin{array}{l}Tr<0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\Delta}>0.\end{array}$$

5. Edge of chaos

In the literature, the so-called edge of chaos (EC) means a region in the parameter space of a dynamical system where complex phenomena and information processing can emerge. We shall try to define more precisely this phenomena till now known only via empirical examples.

#### Definition 4.2

*RD-BCNN with memristor coupling is said to be operating on the edge of chaos EC iff there is at least one equilibrium point which is both locally active and stable*.

In the case of Brusselator cell we prove the following theorem:

#### Theorem 4.3

*RD-BCNN with memristor coupling* (27) *is operating in the EC regime iff the following conditions for the parameters are satisfied:* –4(*b*– 1)*a*^{2} < (*a*^{2} – *b*)^{2} and *b* < *a*^{2} + 1. *In this parameter set the unique equilibrium point Ε*_{1}* is both locally active and stable*.

*Proof*. The Jacobian matrix *A* of the system (28) is:
$$A=\left[\begin{array}{ll}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right],$$(29)

where ${a}_{11}={\displaystyle \frac{\mathrm{\partial}{f}_{1}}{\mathrm{\partial}{u}_{kl}}=b-1,{a}_{12}=\frac{\mathrm{\partial}{f}_{1}}{\mathrm{\partial}{v}_{kl}}={a}^{2},{a}_{21}=\frac{\mathrm{\partial}{f}_{2}}{\mathrm{\partial}{u}_{kl}}=-b,{a}_{22}=\frac{\mathrm{\partial}{f}_{2}}{\mathrm{\partial}{v}_{kl}}=-{a}^{2}.}$

Since we found the unique equilibrium point of the RD-BCNN to be ${E}_{1}=(a,{\displaystyle \frac{b}{a}),}$ then
$$\begin{array}{l}Tr({E}_{1})=-{a}^{2}+b-1,\end{array}$$(30)
$$\begin{array}{l}\mathrm{\Delta}({E}_{1})={a}^{2}.\end{array}$$(31)

Now we shall calculate *SLAR*(*E*_{1})according to Definition 4.1: *SLAR*(*E*_{1}): *-4*(*b*-1)*a*^{2} < (*a*^{2}*-b*^{2})^{2}and *b* < *a*^{2}+ 1. Therefore, according to Definition 4.2, the parameter set in which the RD-BCNN with memristor coupling is operating in edge of chaos regime is: –4(*b –* 1*)a*^{2} < (*a*^{2} – *b*^{2})^{2}and *b* < *a*^{2}+ 1. Theorem is proved.

After simulations, we obtain the following edge of chaos region for the RD-BCNN with memristor coupling:

The local activity domain and the edge of chaos domain are shown on Fig. 2 for the following parameter set: *a* ∊ [–5, 5] and *b* ∊ [–0.4, 0.4]. Most of the cell parameters points, which were found capable of self-organization are located nearby the bifurcation boundary separating the stable region from the unstable region. This simulation provides very useful information concerning the potential of the cell to exhibit complexity. All simulations obtained in the paper use MATCNN software [8].

Fig 2 Edge of chaos region in RD-BCNN with memristor coupling

Now, we shall present a short discussion of the dynamic nonhomogeneous patterns which have emerged from the RD-BCNN with memristor coupling for several cell parameter points chosen arbitrarily within, or nearby, the edge of chaos domain. After extensive computer simulations we obtain the following results:

The dynamic patterns presented in Figure 3 are associated with cell parameter points located within the EC region, which corresponds to the locally active and stable domain. Dynamic behaviour in this case will correspond to cells oscillating independently of each other with uncorrelated phases. In a visual representation of the type shown in Figs. 3 such a nonsynergetic behavior would be seen as a change in the shape of the patterns during the temporal evolution. At a glance, one can see that this is clearly in all of these cases we have obtained an emergent, complex behavior.

Fig 3 Dynamic patterns in RD-BCNN with memristor coupling

During the simulation procedure the initial state of all Brusselator cells is set to be inactive state. After stimulating the center node, the excitable waves propagate outwards, resulting in the aggregation of patterns. Figure 3 shows the surface wave patterns of RD-BCNN with memristor coupling. Although the center was stimulated, wave propagates asymmetrically due to memristive effects with same polarity over the medium. The velocity of waves propagation was differed depending on the direction of the wave propagation.

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