Edge of Chaos in Integro-Differential Model of Nerve Conduction

Abstract

In this paper, we consider an integro-differential model of nerve conduction which presents the propagation of impulses in the nerve’s membranes. First, we approximate the original problem via cellular nonlinear networks (CNNs). The dynamics of the CNN model is investigated by means of local activity theory. The edge of chaos domain of the parameter set is determined in the low-dimensional case. Computer simulations show the bifurcation diagram of the model and the dynamic behavior in the edge of chaos region. Moreover, stabilizing control is applied in order to stabilize the chaotic behavior of the model under consideration to the solutions related to the original behavior of the system.

1. Introduction

Computational properties in neuroscience concerning dynamical systems depend on the bifurcation behavior of the system. For some values of the parameters neurons can be excitable, whereas for other values they can fire spikes periodically. In the theory of dynamical systems [1], these two types of behavior can be related to a stable equilibrium and a limit cycle attractor. The transition from one to another dynamical behavior depends on a change in the parameters.

The FitzHugh–Nagumo equation [2] is one of the most famous models describing the mathematical explanation of the impulse behavior in nerve membranes. This equation or the Nagumo active pulse transmission equation can be unified in the frames of one- or two-dimensional reaction–diffusion cellular nonlinear networks (RD-CNNs) consisting of dynamical units which are in fact a degenerate case of Chua’s oscillator [3,4]. Cellular nonlinear networks [5] represent a new class of information processing systems with very important potential applications in information processing (see Figure 1). CNNs take some of their properties from neurobiology and integrated circuits. A CNN is defined by linear locally coupled dynamical systems which we call cells (the squares in Figure 1). The CNN dynamics consists of the dynamic behavior of the system units called cells, nonlinear dynamics of the synaptic laws between the cells in the nearest neighborhood, and of boundary and initial conditions [5,6]. The coupling between cells is given in Figure 1 (in pink is given the coupling between a cell and its neighbor cells). The dynamical behavior of a CNN is presented by nonlinear ordinary differential or difference equations (ODEs) which we call the state equations of cells. Therefore, the dynamics of a CNN is given by a system of connected ordinary differential equations. A CNN is able to exhibit very complex dynamics, for example, chaotic behavior, pattern formation, nonlinear oscillation, and wave propagation [7,8]. For some symmetric templates [6], the equations governing CNN dynamics are very similar to a spatial discretization of a nonlinear partial differential equation, encountered, for instance, in reaction–diffusion systems that can model pattern formation mechanisms in chemical reactions or biological growth. The compact form of RD-CNN allows very complex systems to be modeled by them [9,10].

CNNs [5,6] can approximate some complex nonlinear dynamical systems, in which behavior like bifurcations and chaos occur. RD-CNNs [11] may change their dynamic behavior from stable to unstable if the cell self-feedback coefficients [5] change to a critical value. In this case, we say that it is a loss of stability and the birth of limit cycles.

The FitzHugh–Nagumo equation is a simplification of the classical Hodgkin–Huxley equations [12] for nerve conduction. In this paper, we shall prove that taking the appropriate circuit parameters gives the possibility for a CNN to become a more general and versatile model of nerve conduction. The generalization of the FitzHugh–Nagumo equation for the propagation of the voltage pulse through the nerve axon [13] is given below:

$$u_t-D{\nabla }^{2}u=\sigma u(u-\alpha )(1-u)-\beta {\int }_0^{t}g\left(u(s,x)\right)ds,$$

(1)

where $0<x<1$, $0<\alpha <\frac{1}{2}$—sodium potential, $\sigma >0$—membrane capacity, $\beta >0$—potassium equilibrium potential, $D>0$ denotes the diffusion coefficient, and nonlinear function g depends on u. The equation under consideration (1) is actually a nonlinear parabolic integro-differential equation, in which $u(x,t)$ stands for a nerve axon of the membrane. We have that $u=0$ describes the resting state of the nerve. The classical Nagumo equation is $u_t=u(u-\alpha )(1-u)+u_{xx}$. When $u=0$ we obtain the resting state, and for $u=1$ the excited state of the nerve. For this equation, it has been proved [14,15] that both $u=0$ and $u=1$ are stable, whereas $u=\alpha$ is unstable. Therefore, there is a threshold phenomenon. The model without an integral part thus predicts that a finite localized stimulus can be sufficient to trigger a wave front such that the nerve is in its rest state before its passage and its excited state afterward. Thus, the information is passed along the nerve. For a model of nerve conduction to be realistic there must be a mechanism for returning to the rest state, so that the nerve may again be excited by a stimulus. This is the role of the integral part.

The generalization of the FitzHugh–Nagumo equation comes from one very important application in cardiology. The action of the cardiac Purkinje fiber model [12] of the heart was described by the Hodgkin–Huxley equation. The experiments provided for this model and corresponding computer simulations explain why a heart with a normal heart rate can stop beating suddenly. The simulations of Hodgkin–Huxley model provided in [16] show that oscillatory patterns as well as chaotic patterns may appear.

Till now, Equation (1) has not been studied via local activity theory. In [17], the FitzHugh–Nagumo equation is studied via a coupled neural network, but the edge of chaos domain is not determined theoretically. The novelty of this paper is first of all in approximating the integro-differential Equation (1) into an RD-CNN grid, and secondly, in applying local activity theory for determination of the edge of chaos parameter domain. The experiments [16] show that the cell parameters located near the edge of chaos domain lead the heart to stop beating. One example of the classical FitzHugh–Nagumo model can be found in [6]. In comparison to Equation (1), Poincare–Andronov–Hopf bifurcation is proved for this example. Here, we apply local activity theory, which gives us a better understanding of nerve conduction.

In this paper, we shall study the dynamics of (1) by applying the local activity theory. We construct a CNN architecture for integro-differential Equation (1) in Section 2. For the integro-differential Equation (1), there are two possible cases: (1) a one-dimensional CNN model, and (2) a two-dimensional CNN model. We study the second one. We investigate the dynamics of the CNN model using the theory of local activity. The main principles of this theory are presented in Section 3. In Section 4 we provide the main theorems for determination of the edge of chaos region in the parameter space of a cell in which complex behavior occurs. In Section 5 we apply the constructive algorithm for studying the chaos in the two-dimensional CNN model and we present some computer simulations. Stabilization of the model is addressed in Section 6 by constructing the feedback control.

2. Cellular Nonlinear Network Model of Nerve Conduction

Integro-differential Equation (1) can be presented by an RD-CNN with a degenerate Chua oscillator in cells [5]. For this, we map $u(x,t)$ into a CNN in such a way that the state voltage of a CNN cell $u_k$ [6] at a grid point k is obtained by $u(kh,t)$, $h=\Delta x$. Then, the second spatial partial derivative can be written as

$$u_{xx}\approx \frac{1}{h^2}[u(x+h,t)-u(x,t)-(u(x,t)-u(x-h,t))]=\frac{1}{h^2}[u_{k+1}-2u_k+u_{k-1}].$$

(2)

Therefore, in a one-dimensional case, we obtain the following CNN model:

(1) CNN cell dynamics:

$$\frac{d{u}_j}{dt}-I_j^s=\sigma u_j(u_j-\alpha )(1-u_j)-\beta {\int }_0^t{u}_{j}ds,1\le j\le N.$$

(3)

(2) CNN synaptic law:

$$I_j^s=\frac{1}{h^2}(u_{j-1}-2u_j+u_{j+1}).$$

(4)

We choose the grid size of our CNN model to be $h=1$. We denote the nonlinearity by $n\left(u_j\right)=u_j(u_j-\alpha )(1-u_j)$. Substituting (4) into (3) we obtain

$$\frac{d{u}_j}{dt}-(u_{j-1}-2u_j+u_{j+1})=n\left(u_j\right)-\beta {\int }_0^t{u}_{j}ds,1\le j\le N.$$

(5)

which is the one-dimensional CNN model of (1).

In the two-dimensional case, the solution $u(x,y,t)$ of (1) is a continuous function of the time t and the space variables $x,y$. Function $u(x,y,t)$ can be represented by a set of functions $u_{k,l}$ which are defined as $u_{k,l}=u(k{h}_x,l{h}_y,t)$, where $h_x$ and $h_y$ denote the space intervals in the x and y coordinates. Then, the Laplacian of $u(x,y,t)$ with respect to x and y can be replaced by

$$u_{xx}+u_{yy}\approx \frac{1}{4}[u_{i,j-1}\left(t\right)+u_{i,j+1}\left(t\right)+u_{i-1,j}\left(t\right)+u_{i+1,j}\left(t\right)]-u_{i,j}.$$

Therefore, we obtain the two-dimensional discretized Laplacian template $A_2=\left(\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ 1\hfill & -4\hfill & 1\hfill \\ 0\hfill & 1\hfill & 0\hfill \end{array}\right)$. The CNN model of the two-dimensional integro-differential Equation (1) is

$$\frac{d{u}_{k,l}}{dt}-A_2\ast u_{k,l}=n\left(u_{k,l}\right)-\beta {\int }_0^{t}g\left(u_{k,l}\right)\left(s\right)ds,1\le k,l\le N,$$

(6)

where ∗ is the convolution operator [6].

In this paper, we study the dynamics of the two-dimensional CNN model (6) of nerve conduction by applying the local activity theory.

3. Local Activity Principle

We shall explain the basics of local activity theory by studying the following reaction–diffusion system (RD) written in the vector form:

$$\begin{array}{ccc}\hfill {\dot{U}}_a& =& f_a(U_a,U_b)+D{\nabla }^2{U}_a\hfill \\ \hfill {\dot{U}}_b& =& f_b(U_a,U_b),\hfill \end{array}$$

(7)

where $U_a\in {\mathbf{R}}^m$, $U_b\in {\mathbf{R}}^{(n-m)}$ are vectors, the diffusion coefficients are denoted by a $D-m\times m$ diagonal matrix, and $f_a\in {\mathbf{R}}^m$, $f_b\in {\mathbf{R}}^{(n-m)}$ are the kinetics vectors. It is known that dynamical systems have some tunable control parameters $\mu ={[{\mu }_1,{\mu }_2,\dots ,{\mu }_p]}^T$ which are associated with the changing behavior of the system. We include them in system (7) and obtain the following system of kinetic equations:

$$\begin{array}{ccc}\hfill {\dot{U}}_a& =& f_a(U_a\left(r\right),U_b\left(r\right);\mu )\hfill \\ \hfill {\dot{U}}_b& =& f_b(U_a\left(r\right),U_b\left(r\right)),\hfill \end{array}$$

(8)

which are stated for each grid point $r=(i,j,k)$.

When $D>0$ the diffusion term can play a stabilizing role in the above reaction–diffusion equations. Then, the origin of any complex behavior of (8) can be understood from the cell kinetic equations. We shall prove in a rigorous way that the kinetic equations are not locally active for the parameters $\mu \in {\mathbf{R}}^n$. In this case, RD-CNN cannot exhibit any complex behavior independently of the choice of diffusion coefficient $D>0$.

Furthermore, we can find explicit mathematical criteria [15,18,19] for testing any cell kinetic equation for local activity. In this way, we shall determine the active parameter domain A of the parameter space $\mu \in {\mathbf{R}}^n$, in which kinetic equations are locally active. The rest $P={\mathbf{R}}^n\backslash A$ is the passive domain of the parameters, in which we obtain homogeneous solutions of the reaction–diffusion equations. It is very important to point out that the local activity is defined only with respect to m port variables $U_a$ and it does not involve the diffusion coefficients in the reaction–diffusion equations. For this purpose, we define an interaction term $I_a=D{\nabla }^2{U}_a$, which will act as the input vector in the diffusion-driven ports:

$$\begin{array}{ccc}\hfill {\dot{U}}_a\left(r\right)& =& f_a(U_a\left(r\right),U_b\left(r\right);\mu )+I_a\hfill \\ \hfill {\dot{U}}_b\left(r\right)& =& f_b(U_a\left(r\right),U_b\left(r\right);\mu ),\hfill \end{array}$$

(9)

These equations are called force kinetic equations [15,19].

Now, let us define the equilibrium states. By setting the state change to zero we obtain

$$\begin{array}{ccc}\hfill 0& =& f_a(U_a\left(r\right),U_b\left(r\right);\mu )+I_a\hfill \\ \hfill 0& =& f_b(U_a\left(r\right),U_b\left(r\right);\mu ),\hfill \end{array}$$

(10)

After solving these equations for each fixed parameter $\mu$ we usually obtain multiple equilibrium points E for each input $I_a$. Let us linearize the kinetic equations around each equilibrium point, so we obtain

$$\begin{array}{ccc}\hfill \frac{d{u}_a\left(t\right)}{dt}& =& a_{11}\left(E\right)u_a\left(t\right)+a_{12}\left(E\right)u_b\left(t\right)+i_a\hfill \\ \hfill \frac{d{u}_b\left(t\right)}{dt}& =& a_{21}\left(E\right)u_a\left(t\right)+a_{22}\left(E\right)u_b\left(t\right),\hfill \end{array}$$

(11)

where $a_{11}\left(E\right)=\frac{\partial f_a}{\partial u_a}{|}_E$, $a_{12}\left(E\right)=\frac{\partial f_a}{\partial u_b}{|}_E$, $a_{21}\left(E\right)=\frac{\partial f_b}{\partial u_a}{|}_E$, $a_{22}\left(E\right)=\frac{\partial f_b}{\partial u_b}{|}_E$.

Let us now define the local activity at a cell equilibrium point E. We take a continuous input function of time $i_a\left(t\right)$, for $t\ge 0$ and the initial conditions $u_a\left(0\right)=0$, $u_b\left(0\right)=0$. Then, the solution of the linearized kinetic equations at the equilibrium point of the cell E can be considered as an infinitesimal state which we denote by $u_a\left(t\right)$ and $u_b\left(t\right)$ for $t\ge 0$. Consider the local power flow $p\left(t\right)=u_a\left(t\right)·i_a\left(t\right)$ to be the rate of change of energy at time t at the equilibrium point of the cell E. Actually, $p\left(t\right)$ represents the scalar product between the two vectors $u_a\left(t\right)$ and $i_a\left(t\right)$. Let $w\left(t\right)$ be the total energy.

Definition 1

([15,18,19]). We say that the cell is locally active at an equilibrium point of the cell E if, and only if, there exists continuous input time function $i_a\left(t\right)\in {\mathbf{R}}^m$, $t\ge 0$, and for the finite time τ, $0<\tau <\infty$ one can find a network energy flowing out of the cell at $t=\tau$, and the cell has zero energy at $t=0$. Then, the following inequality is satisfied:

$$w\left(t\right)={\int }_0^{\tau }{u}_a\left(t\right)·i_a\left(t\right)dt<0,$$

(12)

where $u_a\left(t\right)$ denotes a solution of (11) at E having initial states $u_a\left(0\right)=0$, $u_b\left(0\right)=0$.

Now, we shall adjust Definition 1 to the RD equations. We say that the RD are locally active if, and only if, their cells are locally active at the equilibrium point. If they are not, we say that they are locally passive [18,19].

Naturally, one very important question arises: How to understand the local activity principle from Definition 1? First of all, the assumption of zero energy at $t=0$ is very important, because in case the cell has some stored energy at $t=0$ it can be discharged outside the circuit even when it is locally passive. This actually is the cell’s ability to act as a source of small-signal energy which leads to a larger energy signal.

The inequality $w\left(t\right)={\int }_0^{\tau }{u}_a\left(t\right)·i_a\left(t\right)dt<0$ defines mathematically the local activity at an equilibrium point E. However, this condition is difficult to test because it is not clear if the function $i_a\left(t\right)$ exists. For this purpose, we need to develop a constructive procedure by introducing some mathematical criteria in a complex domain $\mathbf{C}$ via the Laplace transform. After applying the Laplace transform to (11) we obtain

$$\begin{array}{c}\hfill \left|\begin{array}{c}s{\overline{u}}_a\left(s\right)=a_{11}\left(E\right){\overline{u}}_a\left(s\right)+a_{12}\left(E\right){\overline{u}}_b\left(s\right)+{\overline{i}}_a\left(s\right)\hfill \\ s{\overline{u}}_b\left(s\right)=a_{21}\left(E\right){\overline{u}}_a\left(s\right)+a_{22}\left(E\right){\overline{u}}_b\left(s\right),\hfill \end{array}\right.\end{array}$$

(13)

where $s=\kappa +i\omega$ is a complex number in the domain $\mathbf{C}$. We solve the last equation of (13) and we obtain

$${\overline{u}}_b\left(s\right)={(s\mathbf{1}-a_{22}\left(E\right))}^{-1}{a}_{21}\left(E\right){\overline{u}}_a\left(s\right).$$

After substituting it into the first equation of (13) we obtain the input function in the following form:

$${\overline{i}}_a\left(s\right)=X_E\left(s\right){\overline{u}}_a,$$

where $X_E\left(s\right)=[(s\mathbf{1}-a_{11}\left(E\right))-a_{12}\left(E\right){(s\mathbf{1}-a_{22}\left(E\right))}^{-1}{a}_{21}\left(E\right)]$ denotes the complexity matrix at the equilibrium point, and $\mathbf{1}$ represents the identity matrix. The complexity matrix gives us the necessary computational procedure independent of whether an input function exists or not. In the simple case $m=1,n=2$, the complexity matrix can be obtained as the following scalar rational function:

$$X_E\left(s\right)=\frac{s^2-(a_{11}\left(E\right)+a_{22}\left(E\right))s+a_{11}\left(E\right)a_{22}\left(E\right)-a_{12}\left(E\right)a_{21}\left(E\right)}{s-a_{22}\left(E\right)}$$

If we consider the case $m=1,n>2$, then the numerator and the denominator are polynomials which depend on s. Therefore, we can present them as products with poles:

$$X_E\left(s\right)=\frac{\kappa (s-z_1)(s-z_2)\dots (s-z_{\alpha })}{(s-p_1)(s-p_2)\dots (s-p_{\beta })},$$

where $s=z_1,z_2,\dots ,z_{\alpha }$ are $\alpha$ zeros of $X_E\left(s\right)$ and $s=p_1,p_2,\dots ,p_{\beta }$ are $\beta$ poles of $X_E\left(s\right)$, being complex numbers.

In the case when $m>1$, $X_E\left(s\right)$ is an $m\times m$ matrix with elements being rational functions of s:

$$X_E\left(s\right)=\left[\begin{array}{cccc}{X}_{11}\left(s\right)\hfill & X_{12}\left(s\right)\hfill & \dots \hfill & X_{1m}\left(s\right)\hfill \\ X_{21}\left(s\right)\hfill & X_{22}\left(s\right)\hfill & \dots \hfill & X_{2m}\left(s\right)\hfill \\ \dots \hfill & \dots \hfill & \dots \hfill & \dots \hfill \\ X_{m1}\left(s\right)\hfill & X_{m2}\left(s\right)\hfill & \dots \hfill & X_{mm}\left(s\right)\hfill \end{array}\right].$$

Now, we shall introduce the main theorem for local passivity [15,18,19]:

Theorem 1.

The cell is locally passive at equilibrium point E if and only if one of the following conditions is true:

(1) 

Consider the plane $Re\left[s\right]>0$. Then, the complexity matrix $X_E\left(s\right)$ should have a pole in it;

(2) 

Consider the imaginary axis. Then, the complexity matrix $X_E\left(s\right)$ should have a multiple pole in it;

(3) 

Consider the imaginary axis. Then, the complexity matrix $X_E\left(s\right)$ should have a single pole $s=i{\omega }_p$ in it. Moreover, $K_E\left(i{\omega }_p\right)=li{m}_{s\to i{\omega }_p}(s-i{\omega }_p)X_E\left(s\right)$ could be either a real number, or a complex number;

(4) 

$Re\left[X_E\left(i\omega \right)\right]<0,\omega \in (-\infty ,\infty ).$

Theorem 1 gives the necessary and sufficient conditions for local passivity of the cell. In other words, the uncoupled cell is locally passive at a cell equilibrium point if and only if the complexity function $X\left(E\right)$ is a positive real function. From Figure 2 it can be seen that the left-half s–plane is mapped into the closed right-half $X\left(E\right)$-plane. If the cells are locally passive the RD cannot have any complex dynamical behavior. Below, we give the definition of complexity given in [15,18,19]:

Definition 2.

We say that a spatially continuous or discrete medium which is made of identical cells interacting with each other in a small neighborhood and having identical interaction laws exhibit complex behavior if the homogeneous medium leads to the occurrence of non-homogeneous static or spatio-temporal patterns when homogeneous initial and boundary conditions are applied.

Therefore, RD can present complex dynamical behavior if, and only if, the corresponding continuous equations, or their discretized versions, have at least one non-homogeneous static or spatio-temporal solution having homogeneous initial and boundary conditions.

4. Edge of Chaos in RD-CNN Systems

As we stated in the previous section, all solutions of the RD-CNN systems will converge to a unique steady state when $t\to \infty$ if the cells are strictly locally passive [15,18,19]. Consequently, such equations cannot produce any type of complexity [15,18,19]. Below, we give the definition of the edge of chaos in RD-CNN systems:

Definition 3.

RD systems with $I_a=0$ work in the edge of chaos if and only if one or more of its cell equilibrium points are not only locally active but asymptotically stable as well. We call the edge of chaos parameter set Ω of all locally active parameters $\mu \in {\mathbf{R}}^m$ having the above property.

It is possible to show that an RD-CNN system oscillates in the edge of chaos regime when appropriate diffusion coefficients are chosen. But it is not always possible to find such diffusion coefficients in order to destabilize the homogeneous solution. In fact, we can prove that such a set of diffusion coefficients exists only for a subset of the edge of chaos parameter domain, which we shall call the sharp edge of chaos.

Below, we shall consider two cases of a sharp edge of chaos one-port variable and two-port variables and we shall state the following theorems:

Theorem 2.

We say that the sharp edge of chaos parameter domain is identical to the edge of chaos of the complexity function X(E) of any RD-CNN when we have only a one-port variable.

Theorem 3.

The sharp edge of chaos in the two-port variables case can be determined by one of the three conditions for the $jk$-th coefficients $X_{jk}\left(i\omega \right)$, $j,k=1,2$ of the $2\times 2$ complexity matrix $X\left(E\right)$ for $\omega ={\omega }_0$:

(1).

$Re\left[X_{11}\right]<0$;

(2).

$Re\left[X_{22}\right]<0$;

(3).

$4(Re\left[X_{11}\right]Re\left[X_{22}\right]+Im{X}_{12}Im\left[X_{21}\right]\times (Re\left[X_{11}\right]Re\left[X_{22}\right]2]Re\left[X_{21}\right]<{(Re\left[X_{12}\right]Im\left[X_{21}\right]-Re\left[X_{21}\right]Im\left[X_{12}\right])}^2$

It follows from the above statements that we have an edge of chaos regime but not a sharp edge of chaos regime if the cell is not destabilized by any locally passive coupling networks. We have that the sharp edge of chaos domain is a subset of the edge of chaos domain, which is a subset of the locally active domain.

Following the above theory of local activity, the below constructive algorithm for the determination of the edge of chaos is developed [20]:

(1).

Determine the equilibrium points of the RD-CNN model under consideration;

(2).

Find the cell coefficients of the Jacobian matrix of the linearized system for all equilibrium points of RD-CNN;

(3).

Calculate the trace and the determinant of the Jacobian matrix of the linearized system for all equilibrium points of RD-CNN;

(4).

Determine the stable and locally active region for all equilibrium points;

(5).

Find the edge of chaos region (following Definition 3) looking for at least one equilibrium point that is not only locally active but also stable.

We apply this algorithm in the next section.

5. Edge of Chaos in RD-CNN Model of Nerve Conduction

As we can see from the above two sections, the theory of local activity gives an exact answer to the following question: for which values of the cell the interconnected system can produce complexity. The answer is stated in Theorem 1; i.e., the necessary condition for an RD system to have complex behavior is to have its cell locally active. The theory that is presented above follows [15,18,19] and allows us to develop a very simple analytical method for the determination of local activity. In our particular case for the RD-CNN model, we are able to determine the domain of cell parameters in order to obtain the local activity of the cells which exhibit complexity. This precisely determined parameter domain is called the edge of chaos [15,18,19].

Let us rewrite the two-dimensional CNN model of nerve conduction (6) in the following vector form:

$$\begin{array}{c}\hfill \left|\begin{array}{c}\frac{dV}{dt}=D{A}_2\ast \frac{DU}{dt}\sigma n^{\prime }\left(U\right)-\beta g\left(U\right)=F_1+I\hfill \\ \frac{dU}{dt}=V,\hfill \end{array}\right.\end{array}$$

(14)

where $(U,V)$ are vectors obtained by lining up the state variables $u,v$ in a row [13], $n^{\prime }\left(u\right)=\frac{dn\left(u\right)}{dt}$, $I=D{A}_2\ast \frac{DU}{dt}$ is the interaction term, and according to the theory of local activity in order to obtain the edge of chaos domain we shall take $I=0$ (uncoupled cell), $F_1=\sigma n^{\prime }\left(U\right)-\beta g\left(U\right)$.

According to the above algorithm, we are looking for the equilibrium points of (14). We can obtain one, two, or three equilibrium points E, being functions of the cell parameters $\alpha ,\beta$, and $\sigma$. Let us now calculate the four cell coefficients $a_{11},a_{12},a_{21}$, and $a_{22}$ at each equilibrium point.

The trace $Tr\left(E_s\right)$ and the determinant $\Delta \left(E_s\right)$, $s=1,2$ of the Jacobian matrix $J=\left[\begin{array}{cc}{a}_{11}\hfill & a_{12}\hfill \\ a_{21}\hfill & a_{22}\hfill \end{array}\right]$ are calculated according to [18].

According to [15,18,19], a locally active cell at an equilibrium point $E_s$ is such that it satisfies at least one of the following four conditions:

(1).

$Tr\left(E_s\right)>0$ or $\Delta \left(E_s\right)<0$;

(2).

$Tr\left(E_s\right)>a_{22}$ or ($Tr\left(E_s\right)\le a_{22}$ and $a_{22}\Delta \left(E_s\right)$);

(3).

$Tr\left(E_s\right)=0$ and $\Delta \left(E_s\right)>0$ and $a_{22}\ne 0$;

(4).

$Tr\left(E_s\right)=0$ and $\Delta \left(E_s\right)=0$ and $a_{22}\ne 0$.

We see that if $a_{22}=0$, then conditions (1)–(4) lead to (2) which is always satisfied in the $Tr\left(E_s\right)\le 0$ and $\Delta \left(E_s\right)\ge 0$ quadrant of the $Tr-\Delta$ plane. It follows that the cell is locally passive only if $a_{22}\le 0$. After extensive calculations, we obtain the following bifurcation diagram in equilibrium point $E_1=(0,0)$ (see Figure 3).

Remark 1.

From the bifurcation diagram shown in Figure 3 we obtain the characterization of a specific cell parameter point giving information about the number of equilibrium points, stability, and local activity.

The homogeneous interaction between many identical cells is capable of describing many processes arising in nature. It concerns cells such as electronic circuits, biological cells, physical devices, dynamical systems, artificial life-like cells, and other abstract ones. The main feature of all the above cells is that they can exhibit collective complex behavior under certain conditions. Consequently, the dynamics of the system can be obtained not only by summing up the functions of its parts but can be more complicated. Life is very complex by itself [19].

The following proposition holds:

Proposition 1.

The two-dimensional CNN model of nerve conduction (14) works in the edge of chaos domain if and only if the conditions in the system parameters satisfy $g^{\prime }\left(0\right)>0$ and $1-\beta <\sigma (1-\alpha )$. In this parameter set, the equilibrium point $E_1=(0,0)$ is both locally active and asymptotically stable.

The simulations shown in Figure 4 show the chaotic behavior of our CNN model in the edge of chaos regime.

The simulation in Figure 4 shows that in the local activity region near the edge of chaos we have a spatio-temporal solution which oscillates in time. The peaks show that spatio-temporal patterns appear in our model.

Remark 2.

We can see that each cell of our model (14) should be defined by strictly monotone increasing functions—the state variables in the case of equilibrium states. This leads to stronger mathematical conditions [15,18,19] than in the case when we apply a homeomorphic function in ${\mathbf{R}}^n$, $n\ge 2$. By definition, we shall call the dynamical system a strictly monotone-increasing system if and only if the associated complexity matrix $X\left(E\right)$ has a positive-definite Hermitian matrix $X^H\left(E\right)$ for all $\omega \in (-\infty ,\infty )$, and for all $u_a\in {\mathbf{R}}^n$. This can be considered as the general concept of a strictly monotone-increasing function to the function space [15,18,19].

In other words, a homogeneous and non-conservative medium is not able to produce complexity unless the cells, or the coupling network, are locally active.

6. Stabilizing Feedback Control for RD-CNN Model

In this section, we shall design a linear feedback control for the RD- CNN model (6) in order to stabilize its chaotic behavior to the solution related to the original behavior of the initial system (1).

Let us extend the RD-CNN model (6) by adding to each cell the local linear feedback:

$$\frac{d{u}_{k,l}}{dt}-A_2\ast u_{k,l}=\sigma n\left(u_{k,l}\right)-\beta {\int }_0^{\tau }g\left(u_{k,l}\right)dt-\eta u_{k,l},$$

(15)

where $\eta$ is the feedback control coefficient, which should be equal for all cells. The problem is to prove that this simple, and very useful for implementation, feedback can stabilize the dynamics of RD-CNN model (6) for which we obtain complex dynamical behavior in the previous section. In the following theorem, we give sufficient conditions on the feedback coefficient values which provide the stability of RD-CNN model (6). The following theorem holds:

Theorem 4.

Suppose that all parameters of RD-CNN system (6) including feedback coefficient η (15) have positive values. Then, its linearized model is asymptotically stable for all $\eta >0$.

Proof. 

First of all, we define the quadratic Lyapunov function in the following way $L\left(z\right)=\frac{1}{2}{z}^{T}z$. Then, its derivative along the linearized control CNN is $\frac{dL\left(z\right)}{dt}=\frac{1}{2}{z}^T(J^T\left(\eta \right)+J\left(\eta \right))z=-z^{T}Q\left(\eta \right)z$. Therefore, $\frac{dL\left(z\right)}{dt}<0$ possesses positive definiteness of $Q\left(\eta \right)$. It can be shown that $Q\left(\eta \right)$ is positive definite if $\eta >0$. For this we calculate the eigenvalues of $J\left(\eta \right)$ with respect to the values of the feedback coefficient $\eta$. The stability of the linear system implies that the eigenvalues ${\lambda }_j^i$, $i=1,\dots ,4$ should satisfy the inequality $ma{x}_{i}Re{\lambda }_j^i<0$.  □

Extensive simulations of the stabilized two-dimensional RD-CNN model (6) are given in Figure 5.

Remark 3.

In the simulations of Figure 5, we obtain a gradient-like solution for our Equation (15). In our statement above, we assume that the duration of feedback control is very small in comparison to the change of states of the RD-CNN system. Such an assumption is applicable in most biological and chemical processes and other natural phenomena. It leads to a hybrid, discrete-continuous dynamic system, in which the discrete-time dynamics is represented by a difference equation in the form of jumps [21].

7. Discussion

In our investigations, we developed and applied the local activity theory to an RD-CNN system. It can be generalized for other dynamical systems describing spatially continuous or discrete media. Moreover, the developed simple algorithm can be applied to any system consisting of cells with interactions between them described by deterministic mathematical models [15,22]. The main difficulty of such a problem is to obtain necessary and sufficient conditions in order for the investigated system to have at infinity unique steady state solutions. It is clear from the above-presented theory that a homogeneous non-conservative system cannot produce complexity if the cells or the coupling system is not locally active.

It is known that a diffusion process is able to equalize differences, but actually, an originally dead or inactive cell can become alive or active when we have a coupling with other cells by diffusion [23]. Such a phenomenon seems artificial, but it is possible to prove it mathematically and rigorously. It is also confirmed with different applications in reality.

8. Conclusions

This paper deals with the integro-differential model of nerve conduction. We approximate the model in one- and two-dimensional cases via CNN discretization. The local activity principle is applied in order to determine the edge of chaos region of the model. Feedback control is constructed in order to stabilize the chaotic behavior of the model. Future investigations will a ddress applying the principle of local activity to other systems than reaction–diffusion. This is a very challenging problem in the determination of the edge of chaos.