Theorems and Application of Local Activity of CNN with Five State Variables and One Port

Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems.


Introduction
Coupled nonlinear dynamical systems have been widely studied in recent years. However, the dynamical properties of these systems are difficult to deal with. Although the research on emergence and complexity has gained much attention during the past decades, the determination, prediction, and control of the complex patterns generated from highdimensional coupled nonlinear systems are still far from perfect. Nature abounds with complex patterns and structures emerging from homogeneous media, and the local activity is the origin of these complexities [1,2]. The cellular neural network (CNN), firstly introduced by Chua and Yang [3] as an implementable alternative to fully connected Hopfield neural network, has been widely studied for image processing, robotic, biological versions, and higher brain functions, and so on [3]. Many of the coupled nonlinear systems can be modeled and studied via the CNN paradigm [4]. The local activity proposed by Chua asserts that a wide spectrum of complex behaviors may exist if the cell parameters of the corresponding CNN are chosen in or nearby the edge of chaos [2,4]. There have been quite a few new methods developed for complex systems [5][6][7][8], and local activity has attracted the attention of many researchers. Now, local activity has been successfully applied to the research of complex patterns generated from several CNNs in physical, biological, and chemical domains, such as Fitzhugh-Nagumo equation [9], Brusselator equation [10], Gierer-Meinhart equation [11], Oregonator equation [12], Hodgkin-Huxley equation [13], Van Der Pol equation [14], the biochemical model [15], coupled excitable cell model [16], tumor growth and immune model [17], Lorenz model [18], advanced image processing [19], Rossler equation [20], images analysis [21,22], data prediction [23], neutron transport equation [24], vision safety [25], retinomorphic model [26], and theory research [27][28][29][30], and so forth.

Computational and Mathematical Methods in Medicine
Although Chua presents the main theorem of local activity at a cell equilibrium point [1,2], it is actually difficult to "test" directly the complex patterns of the high-dimensional coupled nonlinear systems, since the theorem contains no recipe for finding whether a variable actually exists or not. It is necessary to develop some mathematical criteria according to the numbers of the variables and ports; that is the topic addressed in this paper.
The remaining of this paper is organized as follows. The local activity of CNN is introduced in Section 2. A set of theorems for testing the local activity of reaction-diffusion CNN with five state variables and one port are set up in Section 3. As an application of the theorems, a coupled reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is introduced, aiming at describing HBV mutation in the therapeutic process. The bifurcation diagrams of this CNN are developed and some numerical simulations are presented in Section 4. Concluding remarks are given in Section 5.

Local Activity Theory of CNN
The CNN architecture is composed of a two-dimensional M × N array of cells. Each cell is denoted by C(i, j), where i = 1, 2, . . . , M, j = 1, 2, . . . , N. The dynamics of each cell is given by the equation: where x i j , y i j , u i j are the state, output, and input variables of the cell, respectively. a k,l , b k,l , z i j are the elements of the A-template, the B-template, and threshold, respectively. r is the radius of influence sphere. The output y i j is the piecewise linear function given by Clearly, CNN with different template elements may have different functions. A vast majority of active homogeneous media that are known to exhibit complexity are modeled by a reactiondiffusion partial differential equation (PDE): where X = (x 1 , x 2 , . . . x n ) is state variables, (x, y, z) is spatial coordinates, f i (x 1 , x 2 , . . . , x n ) is a coupled nonlinear vector function called the kinetic term, and D 1 , D 2 , . . . , D n are constants called diffusion coefficients. Replacing the Laplace in above formulation by its discrete version yields where where D = diag(D 1 , D 2 , . . . D n ),Ẋ α,β,γ denotes the state variable located at a point in three-dimensional space with spatial coordinates. Chua refers to the process of transforming a PDE into a reaction-diffusion CNN [2]. From Chua and his collaborators' point, PDEs are merely mathematical abstractions of nature, and the concept of a continuum is in fact an idealization of reality. Even the collection of all electrons in a solid does not form a continuum, because much volume separating the electrons from the nucleus represents a vast empty space [2]. Reaction-diffusion CNNs have been used to model some phenomena with important practical backgrounds, which were described by PDEs.
Generally speaking, in a reaction-diffusion CNN, every cell has n state variables, but only m (m ≤ n) state variables couple directly to their nearest neighbors via "reaction-diffusion". Consequently, each cell has the following state equa- where The cell equilibrium point The Jacobian matrix at the equilibrium point Q i has the following form: Computational and Mathematical Methods in Medicine 3 where A kl (Q i ) are called cell parameters and The local state equations at the cell equilibrium point Q i are defined viaV is called the admittance matrix at the cell equilibrium point Q i .

Lemma 2. A reaction-diffusion CNN cell is called locally active at the equilibrium point Q i if and only if, its admittance matrix
at Q i satisfies at least one of the following four conditions [4].
where ω 0 is any real number.
(3) Y Q (s) has a simple pole s = iω p on the imaginary axis, where its associated residue matrix: is either a complex number or a negative real number.
(4) Y Q (s) has a multiple pole on the imaginary axis.
Definition 3. The cell equilibrium point Q i is called stable if and only if, all the real parts of eigenvalue λ i of Jacobian matrix at the equilibrium point Q i are negative [2].

Definition 4.
A "reaction-diffusion" CNN with n state variables and m ports is said to be operating on the "edge of chaos" with respect to an equilibrium point Q i if and only if, Q i is both locally active and stable when I a = 0 [4].
Using the above lemma and definitions, the bifurcation of CNN with respect to an equilibrium point can be divided into three parts: the edge of chaos domains (the locally active and stable domains), the locally active and unstable domains, and the locally passive domains. Numerical simulations indicated that many complex dynamical behaviors, such as oscillatory patterns, chaotic patterns, or divergent patterns, may emerge if the selected cell parameters are located in or nearby the edge of chaos domains.

Analytical Criteria for Local Activity of CNN with Five State Variables and One Port
For the reaction-diffusion CNN with five state variables and one port, its local state equations have the forṁ where 11 ], A ab = a 12 a 13 a 14 a 15 , The corresponding CNN cell admittance matrix Y Q (s) is given by [1].
where T, T 1 , K, K 1 , L, L 1 , Δ, Δ 1 are the parameters of a i j 's.

Theorem 5.
A necessary and sufficient condition for Y Q (s) to satisfy condition (1) in Lemma 2 is that ∃s, such that g(s) = 0 (Re[s] > 0), and any one of the following conditions holds.

Computational and Mathematical Methods in Medicine
Proof. Obviously proved. Denote Theorem 6. Let the following parameters be defined as in Theorem 5, then Y H Q (iw) < 0 for some w = w 0 ∈ R if any one of the following conditions holds.
(2) If a 11 = 0, then Let Computational and Mathematical Methods in Medicine So, if any one of conditions (1) (2) in Lemma 2. This completes the proof.
Theorem 7. For j = 1, or 2, let Then Y Q (s) satisfies condition (3) of Lemma 2, if any one of the following conditions holds.
, and any one of the following conditions holds.
and any one of the following conditions holds for j = 1, or 2.
If Y Q (s) has a simple pole s = iω on the imaginary axis, where its associated residue is either a complex number or a negative real number, then k 1 / = 0, so f (s) / = 0, which implies that iω is not a zero point of f (s) = 0, iω is not a removed pole of Y Q (s).
(II) If Y Q (s) has a simple pole s = 0 and two conjugate poles ±iω (ω / = 0) on the imaginary axis, and another pole is a / = 0.
Therefore, when any one of conditions (I)-(IV) holds, Y Q (s) satisfies condition (3) in Lemma 2. This completes the proof.

Theorem 8. Y Q (s) has a multiple pole on the imaginary axis if any one of the following conditions holds.
(IV) T = L = 0, K > 0, Δ = (K/2) 2 , and any one of the following conditions holds.
When any one of Theorems 5-8 holds, Y Q (s) satisfies Lemma 2, which implies that the reaction-diffusion CNN with five state variables and one port at the equilibrium point is active.
These theorems can be implemented by a computer program for calculating the bifurcation diagram of the general corresponding CNN to determine emergence of complex dynamic patterns of the corresponding CNN.

Analysis and Simulations of Reaction-Diffusion CNN of HBV Mutation-Selection Model
Life systems consist of locally coupled homogeneous media. Mostly, dynamics of life systems are suitable to be described via locally connected reaction-diffusion CNNs. It may be expected that reaction-diffusion CNN will become a promising candidate for modeling life phenomena. In Chapter 11 "Timing the emergence of resistance" (Page 110) of the book "Virus dynamic: mathematical principles of immunology and virology" (Oxford university press), dy n dt = bevx + b n xv n − ay n , dv n dt = k n y n − uv n , where the five variables-x, y, v, y n , v n represent the numbers of uninfected cells, infected cells infected by normal virus, normal virus, infected cells infected by mutated virus, and mutant viruses, respectively. λ is the rate of reproduction of uninfected cells. Uninfected cells die at rate dx and become infected at rate bxv by normal virus and infected at rate b n xv n by mutated virus. Infected cells infected by normal and mutated virus are removed at rate ay and ay n , respectively. Normal virus is produced at rate ky and removed at rate uv, mutated virus is produced at rate ky n and removed at rate uv n . e is the rate constant describing the probability of mutation of virus (usual 10 −5 -10 −3 ), a, b, b n , d, e, k, k n , u, λ are positive constants. The model was briefly analyzed in Nowak's book. The reaction-diffusion CNN of HBV mutation selection of model has the form: Let equation (38) be zeros (D 1 = 0) and solve it, we can get the two equilibrium points: where x 0 = au/((1 − e)bk) and Q 1 , Q 2 stand for the patient's complete recovery and HBV persistent infection, respectively. Consequently, the Jacobian matrix at the equilibrium point In Figures 1 and 2, the domains are coded as follows: edge of chaos (locally active and stable) domain (shown red), locally active and unstable domain (shown green) and locally passive domain (shown blue). From Figure 1(a), we can see that the bifurcation at equilibrium point Q 1 does not exist at the edge of chaos domain.
In the following discussions, we select some parameters in No. 8-14 and u = 5, k = 12.5. The simulation results are shown in Figures 3, 4, 5, 6, and 7. During the simulation, we reached a new conclusion.
From Table 1 and Figures 3-7, we can conclude that (I) when k is smaller (less than 5), (1) these parameters are located in the green domain (the local and unstable domains); (2) regardless of the value of u, the dynamic pattern of equation (37)   No. 1 variable in equation (37), that is, the number of uninfected cells increases as compared to the initial number. All these indicate that the potency is perfect except for some virus mutation. The potency is ideal.
(II) When k is larger (greater than 5) but less than a threshold value (according to initial values and parameters, for example 12.5 in Figure 6), we can conclude the following.
(1) These parameters are located in the red domain (edge of chaos). (III) When k < 40 and greater than a threshold value (according to initial values and parameters), (1) these parameters are located in the red domain (edge of chaos); (2) regardless of the u value, the dynamic pattern of equation (37)   mutant virus and of the uninfected cells decrease, and the number of normal virus increases. These imply that although the drug can prevent the mutation of the HBV effectively, it may destroy uninfected cells and the liver. The potency is not ideal.

Conclusions
The local activity of CNN has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells. Based on the local activity principle, the analytic criteria for the local activity in reaction-diffusion CNN with five state variables and one port are set up. The analytical criteria include four theorems, which provide the inequalities involving the parameters of the CNN. The inequalities can be used for calculating the bifurcation diagram to determine emergence of complex dynamic patterns of the reaction-diffusion CNN. As an application example, a reaction-diffusion CNN of HBV mutation-selection model is analyzed and simulated, and the bifurcation diagrams are calculated. Numerical simulations show this CNN model may explain certain complex mutant conditions during the therapy. We conclude that the local activity theory provides a practical tool for the study of the complex dynamics of certain coupled nonlinear systems.