Cellular Nonlinear Networks (CNN) are analogue dynamic processor arrays, which are made of cells. Let us consider a two–dimensional grid with 3 x 3 neighborhood system shown on Figure 1.

Fig 1

3 x 3 neighborhood CNN

One of the key features of CNN is that the individual cells are nonlinear dynamical systems, but the coupling between them is linear. Roughly speaking, one could say that these arrays are nonlinear but have a linear spatial structure, which makes the use of techniques for their investigation common in engineering or physics attractive.

We will give general definition of a CNN which follows the original one [2]:

For the analysis of this CNN model (6) we shall apply the describing function method. In this case the state variable u_kl(t) are functions depending on three arguments - two discrete space arguments k, l, and one continuous time t. We use the following double Fourier transform F(s, z₁, z₂)of functions ƒ_kl (t),continuous in time and discrete in space: $$F(s,z_1,z_2)={\displaystyle \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{z}_1^{-k}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{z}_2^{-l}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_{kl}(t)exp(-st)dt.}$$(7)

Applying the above transform to our CNN model (6) we obtain: $$\begin{array}{ll}\mathrm{s}\mathrm{U}& =-(b+1)U+U^{2}V+D_1[z_{1}U+z_1^{-1}U+z_{2}U+z_2^{-1}U-4U],\\ \mathrm{s}\mathrm{V}& =bU-U^{2}V+D_2[z1V+z_1^{-1}V+z2V+z_2^{-1}V-4V],\end{array}$$(8)

where s = iω₀, z₁ = exp(iΩ₁), z₂ = exp(iΩ₂), ω₀, Ω₁, Ω₂ are temporal and two spatial frequencies, respectively and $i=\sqrt{-1}$ is the identity.

According to the describing function method [6], the transfer function H(s, z₁, z₂) can be obtained from the above equation as: $$H(s,z_1,z_2)={\displaystyle \frac{D_2(z1+z_{12}^{-1}+z+z_2^{-1}-4)-s}{s+1-D_1(z+z_{12}^{-1}+z+z_2^{-1}-4)}=\frac{U}{V}.}$$(9)

Since we are looking for eventual periodic solutions, we shall suppose that the state has the following form: $$u_{kl}(t)=U_{m_0}sin({\omega }_{0}t+k{\mathrm{\Omega }}_1+l{\mathrm{\Omega }}_2),$$(10)

where the temporal frequency is ${\displaystyle {\omega }_0=\frac{2\pi }{T_0},T_0>0}$ is the minimal period. If we take the periodic boundary conditions for our CNN model (6), making the array circular, we get: $${\mathrm{\Omega }}_1+{\mathrm{\Omega }}_2=\frac{2K\pi }{n},0\le K\le n-1,n=M.M$$(11)

According to the describing function method [6], we shall approximate the output υkl(t) by the fundamental component of its Fourier expansion: $$v_{kl}(t)\approx V_{m_0}sin({\omega }_{0}t+k{\mathrm{\Omega }}_1+l{\mathrm{\Omega }}_2),$$(12)

with $$V_{m_0}=\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}f(U_{m_0}sin\psi )d\psi ,$$(13)

where ƒ is the piecewise-linear sigmoid output function $f(x)={\displaystyle \frac{1}{2}(|x+1|-|x-1|).}$ So we obtain: V_mo = 2U_mo. We shall express now the transfer function Η from (9) in terms of ω₀, Ω ₁, Ω₂: $$H_{{\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2}({\displaystyle {\omega }_0)=\frac{D_2(2cos{\mathrm{\Omega }}_1+2cos{\mathrm{\Omega }}_2-4)-i{\omega }_0}{i{\omega }_0+1-D_1(2cos{\mathrm{\Omega }}_1+2cos{\mathrm{\Omega }}_2-4)}.}$$(14)

According to (9) and (14) the following constraints on its real and imaginary parts of the transfer function hold: $$\mathfrak{R}(H)=\frac{D_{2}B-D_2{D}_1{B}^2-{\omega }_0^2}{(1-D_{1}B{)}^2+{\omega }_0^2}=\frac{U_{m_0}}{V_{m_0}};$$(15) $$\mathfrak{J}(H)=\frac{(D_{1}B-1-D_{2}B){\omega }_0}{(1-D_{1}B{)}^2+{\omega }_0^2}=0,$$(16)

where Β = 2cosΩ₁ + 2cosΩ₂ - 4.

If, for a given value of Ω1, 0 ≤ Ω1 < 2π, we can find a solution (ω₀, Ω₂, U_mo, V_mo ) to these two equations, the describing function method predicts the existence of a periodic solution with an amplitude approximately U_mo and a minimal period $T_0={\displaystyle \frac{2\pi }{{\omega }_0}}$ [6]. Therefore from (11), (15) and (16) we get the following system: $$\begin{array}{l}{\mathrm{\Omega }}_1+{\mathrm{\Omega }}_2=\frac{2K\pi }{n};\\ D_{1}B-1-D_{2}B=0;\\ \frac{D_{2}B-D_2{D}_1{B}^2-{\omega }_0^2}{{(1-D_{1}B)}^2+{\omega }_0^2}=\frac{1}{2}.\end{array}$$(17)

From the second equation of (17) we obtain $B={\displaystyle \frac{1}{D_1-D_2}.}$ Then combining with the first equation of (17) we get: $$\begin{array}{}{\mathrm{\Omega }}_1=\frac{K\pi }{n}+arccos[\frac{4(D_1-D_2)+1}{4cos\frac{K\pi }{n}(D_1-D_2)}],\\ {\mathrm{\Omega }}_2=\frac{K\pi }{n}-arccos[\frac{4(D_1-D_2)+1}{4cos\frac{K\pi }{n}(D_1-D_2)}],0\le K\le n-1.\end{array}$$

Solving the third equation of (17) with respect to ω₀ we obtain: $${\omega }_0=\sqrt{\frac{B(D_2+4D_1)-B^2(D_2{D}_1+2D_1^2)-2}{3}}.$$

Therefore the following theorem has been proved:

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 sth node from (s – l)th and (s + l)th nodes via two resistors whose conductance is represented by g_u. The spatial Laplacian V² in equation (1) can be approximated as $$\begin{array}{l}{\mathrm{\nabla }}^{2}u(x)=\frac{u_{s-1}+u_{s+1}-2u_s}{\mathrm{\nabla }{x}^2}.\end{array}$$

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], n₁denotes the polarity coefficient -n₁ ${\eta }_1=+1:w_s^L,{\eta }_1=-1:w_s^R.$

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: $$\begin{array}{}\frac{d{u}_{kl}}{dt}& =f_1(u_{kl},v_{kl})+D_1(w^{L,R})(u_{k-1l}+u_{k+1l}+u_{kl-1}+u_{kl+1}-4u_{kl});\\ \frac{d{v}_{kl}}{dt}& =f_2(u_{kl},v_{kl})+D_2(w^{L,R})(v_{k-1l}+v_{k+1l}+v_{kl-1}+v_{kl+1}-4v_{kl}),\end{array}$$(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)

f₁(u_kl,v_kl) = a-(b + 1)u_kl+v_klu²_kl , f₂(u_kl,v_kl) = bu_kl-v_klu²_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: $$\begin{array}{l}\frac{d{u}_{kl}}{dt}=f_1(u_{kl},v_{kl})+D_1(w_{kl}^{L,R})(u_{k-1l}+u_{k+1l}+u_{kl-1}+u_{kl+1}-4u_{kl});\\ \frac{d{v}_{kl}}{dt}=f_2(u_{kl},v_{kl})+D_2(w_{kl}^{L,R})(v_{k-1l}+v_{k+1l}+v_{kl-1}+v_{kl+1}-4v_{kl}),\end{array}$$(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 E₁, 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(E₁)and the determinant Δ(E₁)of the Jacobian matrix for the unique equilibrium point.

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)a² < (a² – b)² and b < a² + 1. In this parameter set the unique equilibrium point Ε₁ 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(E₁)according to Definition 4.1: SLAR(E₁): -4(b-1)a² < (a²-b²)²and b < a²+ 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)a² < (a² – b²)²and b < a²+ 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.

In this paper, we studied RD CNN model with Brusselator cell. We started by studying the dynamics of RD-BCNN, applying describing function method and we compare the obtained results with the classical ones for RD Brusselator equation [5]. Then we introduced memristor dynamics in our RD-BCNN and determined the edge of chaos domain of the parameter set in which the model exhibits complex dynamical behavior. We also provided extensive computer simulations of the dynamic patterns which can be obtained in the edge of chaos domain.

Among the demonstrated behaviour (Fig. 3), the nonuniform spatial pattern generation is applied to investigation on detecting global motion of excitable waves, because the memristors [7] accumulated the directions of waves on the media, which resulted in detecting majority of wave directions at every spatial point. This application is not limited to the analysis of RD systems, but the idea can be transferred to analyzing much more complex systems like brain networks, social networks and so on.

Recent laboratory observations suggesting that chaotic regimes may in fact represent the ground state of central nervous system point to the intriguing possibility of exploiting and controlling chaos for future scientific and engineering applications [3].

This first author is supported by Fellowship at Technical University of Dresden.