Inhomogeneous lattice dynamical systems and the boundary effect

2.1 Constant cellular neural networks with nearest neighborhood

for $i\in \mathbb{Z}$. Denote the parameters that relate to the odd and even positions by ${\mathbb{T}}_o=[a_{-1;o},a_{0;o},a_{1;o}]$ and ${\mathbb{T}}_e=[a_{-1;e},a_{0;e},a_{1;e}]$, respectively. We call $\mathbb{T}=[{\mathbb{T}}_o,{\mathbb{T}}_e]$ the feedback template of (1), and $\mathbf{z}=[z_o,z_e]$ is the threshold. It is seen that the templates in (1) are periodic; the prescribed model is a generalization of the classical cellular neural network and is called the constant-type multiple cellular neural network.

where $c=o,e$, ${\mathbf{x}}_c\in {\mathbb{R}}^n$, $F({\mathbf{x}}_c)\in {\mathbb{R}}^n$ is a diagonal mapping (herein $F({\mathbf{x}}_o)={(f(x_1),\dots ,f(x_{2n-1}))}^t$ and $F({\mathbf{x}}_e)={(f(x_2),\dots ,f(x_{2n}))}^t$), and ${\beta }_c={(z_c,\dots ,z_c)}^t\in {\mathbb{R}}^n$. The sufficient conditions for the complete stability of (1) are given as follows. The extension of Theorem 2.1 can be seen in Theorem 2.5.

Theorem 2.1 A constant CNN is completely stable if, for$c=o,e$, one of the following conditions is satisfied.

S1 ${\mathbb{T}}_c$is symmetric.

The complete stability of (1) demonstrates that the investigation of the equilibrium solutions is essential. To make the discussion more clear, we focus on the mosaic solutions, i.e., $|x_i|>1$ for all i, and study the complexity of the output space $\mathbf{Y}=\{(y_i):y_i=f(x_i),i\in \mathbb{Z}\}$ of the mosaic solutions. We investigate the complexity of the output space in two aspects:

To achieve our target, we introduce the ordering matrix and transition matrix first. The ordering matrix is defined as

where ${\mathrm{\Sigma }}_n(X)$ consists of patterns of length n in X. Yielding $T_o$ and $T_e$, we derive the formula of ${\mathrm{\Gamma }}_n(\mathbf{Y})$ and $h(\mathbf{Y})$. For the general cases of constant CNNs, Theorem 2.2 is generalized by Lemma 2.11 and Theorem 2.13.

where${\rho }_o$and${\rho }_e$are the spectral radii of$T_o$and$T_e$, respectively.

Herein $D_{-}$ and $D_{+}$ relate to states ‘−’ (i.e., $y_0=y_{n+1}=-1$) and ‘+’ (i.e., $y_0=y_{n+1}=1$), respectively. Before presenting the formula of ${\mathrm{\Gamma }}_n^B(\mathbf{Y})$ and $h_B(\mathbf{Y})$ under the boundary condition $B=P,N,D_{-},D_{+}$, we introduce two operations of matrices.

With the introduction of the boundary matrices and the Kronecker and Hadamard products, we obtain Theorem 2.4 which reveals the formulae of exact number of patterns and topological entropy under the influence of three kinds of boundary conditions. The extension of Theorem 2.4 for general constant CNNs is demonstrated by Lemma 2.12 and Theorem 2.13.

Theorem 2.4 Suppose$n=2k+r$for some$k\ge 3$, $r=0,1$. $T_o$and$T_e$are the transition matrices of${\mathbf{Y}}_o$and${\mathbf{Y}}_e$, respectively. Then$h_B(\mathbf{Y})=h(\mathbf{Y})$, $B=P,N,D_{-},D_{+}$, if$T_o$and$T_e$are primitive matrices. Furthermore, the exact number of patterns of length n with boundary condition$B=P,N,D_{-},D_{+}$are as follows:

Herein$E_2=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$.

Herein$B=D_{-},D_{+}$relate to the conditions that the patterns on the boundary are ‘−’ and ‘+’, respectively.

2.2 Stability of constant cellular neural networks

The rest of this section extends the results in Section 2.1. To make the paper compact, we introduce the general setting for multi-dimensional inhomogeneous LDS and then concentrate on the one-dimensional case. The elucidation of multi-dimensional systems will be investigated in another paper.

where $\mathbf{i}\in {\mathbb{Z}}^D$, and ${\mathcal{N}}_{\mathbf{i}}$, which is a finite subset of ${\mathbb{Z}}^D$, indicates the neighborhood for neuron $x_{\mathbf{i}}$. The piecewise linear function $f(x)=\frac{1}{2}(|x+1|-|x-1|)$ is called the output function; $\mathbf{z}=[z_{\mathbf{i}}]$ refers to the threshold, and the feedback template$\mathbb{T}={[{\mathbb{T}}_{\mathbf{i}}]}_{\mathbf{i}\in {\mathbb{Z}}^D}$ stores the weight of local interaction between neurons, where ${\mathbb{T}}_{\mathbf{i}}={[a_{\mathbf{k};\mathbf{i}}]}_{\mathbf{k}\in {\mathcal{N}}_{\mathbf{i}}}$.

An inhomogeneous CNN-based LDS is called a constant CNN if the neighborhood , the template , and z are periodic up to shifts. More precisely, there exists $\ell \in \mathbb{N}$ such that ${\mathcal{N}}^{\prime }=\{{\mathcal{N}}_{\mathbf{i}}^{\prime }:\mathbf{i}\in {\mathbb{Z}}^D\}$, ${\mathbb{T}}^{\prime }={[{\mathbb{T}}_{\mathbf{i}}^{\prime }]}_{\mathbf{i}\in {\mathbb{Z}}^D}$, and $\mathbf{z}={[z_{\mathbf{i}}]}_{\mathbf{i}\in {\mathbb{Z}}^D}$ satisfy ${\mathcal{N}}_{\mathbf{i}+\mathbf{j}\ell }^{\prime }={\mathcal{N}}_{\mathbf{i}}^{\prime }$, ${\mathbb{T}}_{\mathbf{i}+\mathbf{j}\ell }^{\prime }={\mathbb{T}}_{\mathbf{i}}^{\prime }$, and $z_{\mathbf{i}+\mathbf{j}\ell }=z_{\mathbf{i}}$ for $\mathbf{i},\mathbf{j}\in {\mathbb{Z}}^D$, where

${\mathbf{K}}_{\mathbf{i}}^{\prime }={\mathbf{K}}_{\mathbf{i}}-\mathbf{i}=\{\mathbf{j}-\mathbf{i}:\mathbf{j}\in {\mathbf{K}}_{\mathbf{i}}\},\mathbf{K}=\mathcal{N},\mathbb{T}.$

where $1\le \overline{i}\le \ell$ and . Without loss of generality, we assume ${\mathcal{N}}_i=\{-d,\dots ,0,\dots ,d\}$ for some $d\in \mathbb{N}$, $1\le i\le \ell$. In this case, the feedback template of (7) is $\mathbb{T}={[{\mathbb{T}}_j]}_{1\le j\le \ell }$, where ${\mathbb{T}}_j=[a_{-d;j},\dots ,a_{0;j},\dots ,a_{d;j}]$. A stationary solution $x={(x_i)}_{i\in \mathbb{Z}}$ is called a mosaic solution if $|x_i|>1$ for all $i\in \mathbb{Z}$, and $y={(y_i)}_{i\in \mathbb{Z}}={(f(x_i))}_{i\in \mathbb{Z}}$ is called a mosaic pattern. A system of ordinary differential equations is said to be completely stable if every trajectory tends to an equilibrium point. Theorem 2.5 infers that a constant CNN is a completely stable system. (We remark that Theorem 2.5 is an extension of Theorem 2.1.)

(It is easily seen that $j_i=j+\ell i$. We reindex the coordinates of neurons to clarify the upcoming investigation.) To prove Theorem 2.5, we consider two kinds of feedback templates separately. For the case that the feedback template of a classical CNN is symmetrical, Forti and Tesi demonstrated that it is completely stable.

Theorem 2.6 ([36])

A classical CNN with symmetric feedback template is completely stable.

$F(x)={(f(x_1),f(x_2),\dots ,f(x_n))}^t\in {\mathbb{R}}^n$ is a diagonal mapping from ${\mathbb{R}}^n$ to ${\mathbb{R}}^n$, and $\beta ={({\beta }_1,{\beta }_2,\dots ,{\beta }_n)}^t$ is a constant vector. Takahashi and Chua proposed a criterion to determine whether a CNN is completely stable.

Theorem 2.7 ([37])

for$1\le i,j\le n$. A classical CNN with asymmetric feedback template is completely stable if K is nonsingular and$K^{-1}\ge 0$, herein a matrix$A\ge 0$means that$A(i,j)\ge 0$for all i, j.

Proof of Theorem 2.5 Suppose $|\mathrm{\Lambda }|=1$; in this case, a constant CNN is deduced to be a classical CNN. Theorem 2.6 infers that a constant CNN is completely stable if the feedback template is symmetrical. Whenever is asymmetric, the system is still completely stable if the matrix K defined in (10) is nonsingular and $K^{-1}\ge 0$. It is indicated via (8) that a constant CNN can be decomposed into ℓ independent CNN subsystems, the complete stability of a constant CNN comes from the complete stability of every subsystem. □

(Recall that in the above equation, $j_i=j+\ell i$.)

One of the important research issues in the circuit theory is the learning problem. That is to say, mathematically, for what and how many phenomena the constant CNNs are capable of exhibiting. Theorem 2.9 infers that once $|\mathcal{N}|=max\{|{\mathcal{N}}_j|:j\in \mathrm{\Lambda }\}$ is fixed, there are finitely many equivalent classes of templates and z so that the basic sets of admissible local patterns $\mathcal{B}(\mathbb{T},\mathbf{z})$ are constrained. Let ${\mathcal{P}}^n=\{(\mathbb{T},z):\mathbb{T}=[a_{-d},\dots ,a_d]\in {\mathbb{R}}^{2d+1},z\in \mathbb{R}\}$ be the parameter space of the classical CNNs, where $n=2d+2$. Theorem 2.8 indicates that the ${\mathcal{P}}^n$ can be partitioned into a finite number of subregions such that each subregion has the same mosaic patterns.

Theorem 2.8 ([34])

Here$\overline{P}$is the closure of P in$P^n$.

Let $\mathcal{P}=\{(\mathbb{T},z):\mathbb{T}=[{\mathbb{T}}_1,\dots ,{\mathbb{T}}_{\ell }],z=[z_1,\dots ,z_{\ell }]\}$ be the parameter space of (7). The following theorem demonstrates that is also partitioned into a finite number of equivalent subregions.

Theorem 2.9 (Separation property)

Proof Similar to the proof of Theorem 2.5, a constant CNN is reduced to a classical CNN whenever $|\mathrm{\Lambda }|=1$, hence Theorem 2.9 is performed in this case. When $|\mathrm{\Lambda }|\ge 2$, the basic set of admissible local patterns $\mathcal{B}(\mathbb{T},z)$ of (7) is the ordered union of the basic set of admissible local patterns $\mathcal{B}({\mathbb{T}}_j,z_j)$. More specifically, is isomorphic to the direct product ${\prod }_{j\in \mathrm{\Lambda }}{\mathcal{P}}_j$, where ${\mathcal{P}}_j$ is the parameter space of (7) _j , the subsystem of (7) restricting to the cells ${\{x_{C\ell +j}\}}_{C\in \mathbb{Z}}$. Since, for $j=1,\dots ,\ell$, each parameter space ${\mathcal{P}}_j$ is partitioned into a finite number of equivalent subregions by Theorem 2.8, is then the union of a unique set of open subregions ${\{P_k\}}_{k=1}^K$ which satisfies conditions (i) to (iii). This derives the desired result. □

which is invariant under σ. The set $X_{\mathrm{\Omega }}$ is called a multiple subshift if Ω is a subshift. Equation (8) together with the proof of Theorem 2.9 asserts that the output space Y of a constant CNN is decomposed into subspaces ${\mathbf{Y}}_1,\dots ,{\mathbf{Y}}_{\ell }$. Observe that Y is topologically conjugated to the direct product of the output spaces ${\mathbf{Y}}_j$ of the classical CNNs, that is, $\mathbf{Y}\cong {\prod }_{j\in \mathrm{\Lambda }}{\mathbf{Y}}_j$, where ${\mathbf{Y}}_j$ is determined by ${\mathcal{B}}_j=({\mathcal{B}}_j(+),{\mathcal{B}}_j(-))$. This derives Theorem 2.10, which indicates that the output space of a constant CNN is a multiple subshift for some parameters.

if${\mathbb{T}}_i={\mathbb{T}}_j$and$z_i=z_j$for$1\le i,j\le \ell$, where Ω is a SFT that comes from the output space of the classical CNN with respect to template$({\mathbb{T}}_1,z_1)$.

2.3 Boundary effect on constant cellular neural networks

These boundary conditions are discrete analogues of the ones in PDEs; to be specific, a pattern $y_1\cdots y_n$ satisfies: (i) the Neumann boundary condition if $y_{n+1}=y_n$ and $y_0=y_1$; (ii) the periodic boundary condition if $y_{n+1}=y_1$; (iii) the Dirichlet boundary condition if $y_0$ and $y_{n+1}$ are prescribed.

Since $\mathbf{Y}\cong {\prod }_{j\in \mathrm{\Lambda }}{\mathbf{Y}}_j$, the total number of patterns of finite length in a constant CNN relates to the number of patterns in the subspaces. For each $j\in \mathrm{\Lambda }$, there is a transition matrix $T_j$ that is implemented for the investigation of the subspace ${\mathbf{Y}}_j$^a (cf.[17] and Section 4). Lemma 2.11 elucidates the exact number of mosaic patterns of length n of a constant CNN without the influence of the boundary condition. The verification is straightforward and is omitted.

where$j\in \mathrm{\Lambda }$, and${\mathrm{\Gamma }}_q(X)$denotes the number of patterns of length q in X.

Here ⊗ is the Kronecker product, E is a $2^{2d}\times 2^{2d}$ matrix with entries being 1’s, I is the $2^{2d}\times 2^{2d}$ identity matrix, and ${\mathbf{1}}_k$ is a $k\times 1$ column vector with entries being 1’s. Suppose that M is a $2k\times 2k$ matrix. Define $M^{(o)}/M^{(e)}$ by letting all the even/odd columns be zero vectors. Furthermore, $M^{(u)}/M^{(l)}$ indicates the matrix obtained from M by setting each of the lower-/upper-half rows as a zero vector.

Herein $\parallel M\parallel$ refers to the 1-norm of the matrix M. Lemma 2.12 demonstrates the explicit formulae of the number of patterns of length n with boundary conditions.

otherwise.

Here$E_2$is a$2\times 2$matrix with entries being 1’s, and ∘ means the Hadamard product.

Proof We address the proof of ${\mathrm{\Gamma }}_n^P(\mathbf{Y})$, where the other cases can be verified in an analogous method.

and completes the proof. □

The existence of $h(\mathbf{Y})$ comes immediately from the submultiplicativity of ${\{{\mathrm{\Gamma }}_n(\mathbf{Y})\}}_{n\in \mathbb{N}}$, which can be verified by applying Lemma 2.11. Theorem 2.13 declares the formula of the topological entropies of the constant CNNs, and the relation between the topological entropies of the constant CNNs and the classical CNNs.

Theorem 2.13$h(\mathbf{Y})=\frac{1}{\ell }{\sum }_{j\in \mathrm{\Lambda }}h({\mathbf{Y}}_j)$. Moreover, $h_B(\mathbf{Y})=h(\mathbf{Y})$for$B\in \{P,N,D_{-},D_{+}\}$provided${\mathbf{Y}}_j$is mixing for all$j\in \mathrm{\Lambda }$.