Inhomogeneous lattice dynamical systems and the boundary effect

This study considers the dynamics of cellular neural network-based inhomogeneous lattice dynamical systems (CNN-based ILDS). The influence of three kinds of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, is elucidated. We reveal that the complete stability of CNN-based ILDS and, under some prescriptions, the topological entropies of CNN-based ILDS with/without the boundary condition are identical.

MSC:37B10.

In the past few decades, the standard cellular neural networks (CNNs) introduced by Chua and Yang [1] have been one of the most investigated paradigms for neural information processing [2]. In a wide range of applications, the CNNs are required to be completely stable, i.e., each trajectory should converge toward some stationary state. In the study of stationary solution, the investigation of mosaic solutions is most essential in CNNs due to the learning algorithm and training processing. More abundant output patterns make the learning algorithm more efficient. Mathematically, the study of the mosaic solutions is reasonable due to the following two facts: (1) complete stability of a wide range of parameters, and (2) the output function of CNNs is a piecewise linear function with constant value for |x|\ge 1; namely,

The outputs y=(f(x_i)), called patterns, are essential for understanding CNN systems. Traditionally, the template for CNNs is homogeneous (also known as isotropic), i.e., the template is space-invariant. However, there are more and more CNNs using inhomogeneous templates to describe some of the problems that arise from the biological and ecological contexts [3–8], skeletonization [9], image processing [10, 11], artificial locomotion control [12], and delayed-type CNN [13–16]. Some new and interesting phenomena of pattern formation and spatial chaos were also found in inhomogeneous multi-layer neural networks. In this paper, the entropy with/without the boundary effect for stable patterns of inhomogeneous CNN is investigated. Entropy is a quantity used for measuring the complexity of the output patterns and it plays an important role in learning algorithm. Surprisingly, such a topic reveals the deep connection with symbolic dynamical systems (SDS). In 1-d CNN, it has been proved that the space of the mosaic solutions (defined later) forms a 1-d subshift of finite type (SFT, [17]). Recently, it has also been proved that the mosaic solutions of a multi-layer CNN (MCNN) form a sofic space[18–20], which is a factor of SFT. The mosaic solutions of inhomogeneous CNN, indeed, produce new shift spaces in SDS. To clarify the investigation of inhomogeneous CNNs, we concentrate our discussion on two classes, and the methodology can be applied in a general case. More specifically, two types of inhomogeneous CNN, constant and arithmetic CNN, are presented herein. It is proved that the space of the mosaic solutions forms a new class in SDS (Theorem 2.10 and Theorem 3.5), called a multiple shift space, which was initiated from the study of the arithmetic regression property in the number theory of mathematics [21–24]. The complexity (topological entropy) can be computed due to the equivalence of the mosaic solutions and multiple shift spaces (Theorem 2.13 and Theorem 3.7). The positivity of entropy unveils the spatial chaos for given systems and pattern formation for zero entropy. Such topics, e.g., pattern formation or synchrony phenomena on LDS, have been investigated by many mathematicians and physicists [25–30].

Besides the entropy formula being established, the boundary effect for constant CNNs and arithmetic CNNs are also considered. Three types of boundary conditions, periodic, Dirichlet, and Neumann, are proposed to a given constant CNN and arithmetic CNN. Sufficient conditions are found for the preservation of entropy under the boundary constraint (Theorem 2.13 and Theorem 3.7), i.e., h_P=h_D=h_N=h. This extends the results in the classical CNNs (cf.[31, 32]). The preservation of entropy under the boundary constraint is unavoidable [33]; since the number of nodes in a lattice is infinite, one usually uses the finite approximation method to exploit the statistical properties of the whole lattice.

Some related topics are also addressed herein. It is known that the mosaic solution of single/multi-layer template-invariant CNNs is constrained by the so-called separation property, namely, not all but some of the patterns that satisfy this property will appear as the mosaic solution for a given CNN [34]. However, more combinations of mosaic patterns will help the learning and training process to be more efficient. It is believed that the template-variant or the multi-layer CNN will achieve this goal. In mathematical language, it means that h(\mathbb{T}) will be ϵ-dense in [0,log2] when parameter runs all of the parameter space, where h(\mathbb{T}) denotes the entropy function according to the parameter . It is proved that constant CNNs possess the ϵ-dense property (Theorem 2.14), and it seems that arithmetic CNNs also satisfy the ϵ-dense property by numerical computation (Conjecture 3.8). We believe that further interesting applications of the results presented (or of the generalizations) can be obtained.

We organize the material in this paper as follows. Section 2 introduces the concepts of general inhomogeneous CNN-based LDS and constant-type multiple CNNs. Stability, partition of the parameter space and the equivalence of mosaic solutions with a multiple shift space are discussed therein. This together with the exact number of mosaic solutions under the boundary constraint (Lemma 2.12) is used to derive the entropy formula and entropy preservation property. Parallel discussions for arithmetic-type multiple CNNs can be found in Section 3. Some one- and two-dimensional examples are addressed in Section 4, and we leave the discussion in Section 5.

In this section, we investigate a specified type of inhomogeneous LDS named constant-type multiple cellular neural network (constant CNN). To clarify the elucidation, Section 2.1 concentrates on the constant CNNs with nearest neighborhood. The general cases of constant CNNs and deeper architecture are investigated in the rest of this section.

2.1 Constant cellular neural networks with nearest neighborhood

First we consider the LDS realized as

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.

A system of ordinary differential equations is called completely stable if each of its solution x approaches an equilibrium state. Let {\mathbf{x}}_o, {\mathbf{x}}_e denote the collection of cells in odd and even coordinates, respectively. Express (1) as

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, forc=o,e, one of the following conditions is satisfied.

S1 {\mathbb{T}}_cis symmetric.

S2 {\mathbf{A}}_c(i,i)>1andK_c^{-1}(i,j)\ge 0for all i, j, where

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:

• {\mathrm{\Gamma }}_n(\mathbf{Y}): The exact number of patterns of length n.

• h(\mathbf{Y}): The topological entropy of the output space.

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

herein the pattern ‘−’ stands for the state y_i=-1 and ‘+’ stands for y_i=1. Let

For c=o,e, define the transition matrix T_c of {\mathbf{Y}}_c by

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.

Theorem 2.2 Supposen=2k+rfor somek\ge 3, r=0,1. T_oandT_eare the transition matrices of{\mathbf{Y}}_oand{\mathbf{Y}}_e, respectively. Then

where\parallel A\parallel ={\mathrm{\Sigma }}_{i,j}|A(i,j)|for any nonnegative matrixA=(A(i,j))\in {\mathbb{R}}^{m\times n}. Moreover, the topological entropy of Y is

where{\rho }_oand{\rho }_eare the spectral radii ofT_oandT_e, respectively.

In the meantime, it is natural to elucidate the influence of boundary conditions on the exact number of patterns of length n and topological entropy. Three types of boundary conditions, periodic, Neumann, and Dirichlet boundary conditions, are considered. To reflect the influence of the boundary conditions, we introduce three boundary matrices. Let

The periodic boundary matrix R^P is a 16\times 16 matrix defined by

The Neumann boundary condition infers zero flux on both sides of the space. The left and right Neumann boundary matrices are then defined by

respectively. Furthermore, the Dirichlet boundary condition indicates that both sides of the space are constant states and the corresponding boundary matrices are

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.

Definition 2.3

1. 1.

   Suppose that A\in {\mathcal{M}}_{p\times q}(\mathbb{R}) is a p\times q matrix and B\in {\mathcal{M}}_{r\times s}(\mathbb{R}) is an r\times s matrix. The Kronecker product A\otimes B\in {\mathcal{M}}_{pr\times qs} is defined by

   A\otimes B={(A(i,j)B)}_{1\le i\le p,1\le j\le q}.
2. 2.

   Suppose that A,B\in {\mathcal{M}}_{p\times q}(\mathbb{R}) are p\times q matrices. The Hadamard product A\circ B\in {\mathcal{M}}_{p\times q}(\mathbb{R}) is defined by

   A\circ B={(A(i,j)B(i,j))}_{1\le i\le p,1\le j\le q}.

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 Supposen=2k+rfor somek\ge 3, r=0,1. T_oandT_eare the transition matrices of{\mathbf{Y}}_oand{\mathbf{Y}}_e, respectively. Thenh_B(\mathbf{Y})=h(\mathbf{Y}), B=P,N,D_{-},D_{+}, ifT_oandT_eare primitive matrices. Furthermore, the exact number of patterns of length n with boundary conditionB=P,N,D_{-},D_{+}are as follows:

• The periodic boundary condition:

  {\mathrm{\Gamma }}_n^P(\mathbf{Y})=\{\begin{array}{ll}\parallel (T_o^{k-1}\otimes T_o)\circ R^P\parallel \cdot \parallel T_e^{k-2}\parallel ,& r=0;\\ \parallel (T_o^{k-1}\otimes T_e^{k-1})\circ R^P\parallel ,& r=1.\end{array}

  (3)

• The Neumann boundary condition:

  {\mathrm{\Gamma }}_n^N(\mathbf{Y})=\{\begin{array}{ll}\parallel L^N\circ (T_o^k\otimes T_e^{k-1})\circ R^N\parallel ,& r=0;\\ \parallel ((L^N\circ (T_o^k\otimes T_e^{k-1}))\otimes T_e^{k-1})\circ (E_2\otimes R^N)\parallel ,& r=1.\end{array}

  (4)

HereinE_2=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right).

• The Dirichlet boundary condition:

  {\mathrm{\Gamma }}_n^B(\mathbf{Y})=\{\begin{array}{ll}\parallel (L^B{T}_e^{k-1})\otimes (T_o^{k-1}{R}^B)\parallel ,& r=0;\\ \parallel T_o^{k-1}\parallel \cdot \parallel L^B{T}_e^k{R}^B\parallel ,& r=1.\end{array}

  (5)

HereinB=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.

A D-dimensional inhomogeneous CNN-based LDS is realized as

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

It is seen that the constant CNNs generalize the concept of the classical CNNs that were introduced in [1, 35]. More precisely, a classical CNN is a constant CNN with \ell =1. The essential description of a one-dimensional constant CNN is presented in the following form:

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.)

Theorem 2.5 Suppose that(\mathbb{T},\mathbf{z})is the template of (7) and the system is written as

Then a constant CNN is completely stable if, for1\le j\le \ell, one of the following conditions is satisfied.

1. (1)

   {\mathbb{T}}_j is symmetric.

2. (2)

   K_j is nonsingular and K_j^{-1}\ge 0, where K_j is defined in (10).

Let \mathrm{\Lambda }=\{1,\dots ,\ell \} be a finite index set. The one-dimensional lattice ℤ can be decomposed into ℓ non-overlapping subspaces

Equation (7) can then be restated as

(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.

For the case that the feedback template is not symmetrical, suppose that a CNN with n-neurons is described as follows:

where x\in {\mathbb{R}}^n, A is an n\times n constant matrix with diagonal elements satisfying

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])

Let K be an n\times n matrix satisfying

for1\le i,j\le n. A classical CNN with asymmetric feedback template is completely stable if K is nonsingular andK^{-1}\ge 0, herein a matrixA\ge 0means thatA(i,j)\ge 0for all i, j.

It comes immediately from Theorem 2.7 that if the feedback template A=[a_{-1},a_0,a_1] of a CNN is asymmetric, then the system is completely stable provided there exists a positive constant r such that

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. □

For a fixed template, the collection of mosaic patterns \mathbf{Y}=\{y={(y_i)}_{i\in \mathbb{Z}}:y_i=f(x_i),|x_i|>1\} is called the output space of (7). Since the neighborhood {\mathcal{N}}_i is finite for each i, the output space is determined by the so-called admissible local patterns. Suppose that y is a mosaic pattern, for each j\in \mathrm{\Lambda } and i\in \mathbb{Z}, the necessary and sufficient condition for y_{j_i}=1 is

and the necessary and sufficient condition for y_{j_i}=-1 is

Set

The set of admissible local patterns ℬ of a constant CNN is then

Similar to the discussion in [17], the output space Y can be represented as

(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])

There is a positive integer K(n) and a unique set of open subregions {\{P_k\}}_{k=1}^K satisfying

1. (i)

   {\mathcal{P}}^n={\bigcup }_{k=1}^K{\overline{P}}_k,

2. (ii)

   P_i\cap P_j=\mathrm{\varnothing } if i\ne j,

3. (iii)

   (\mathbb{T},z) and ({\mathbb{T}}^{\prime },z^{\prime })\in P_k for some k if and only if \mathcal{B}(\mathbb{T},z)=\mathcal{B}({\mathbb{T}}^{\prime },z^{\prime }).

Here\overline{P}is the closure of P inP^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)

There is a positive integer K and a unique set of open subregions {\{P_k\}}_{k=1}^K satisfying

1. (i)

   \mathcal{P}={\bigcup }_{k=1}^K{\overline{P}}_k,

2. (ii)

   P_i\cap P_j=\mathrm{\varnothing } if i\ne j,

3. (iii)

   (\mathbb{T},z) and ({\mathbb{T}}^{\prime },z^{\prime })\in P_k for some k if and only if \mathcal{B}(\mathbb{T},z)=\mathcal{B}({\mathbb{T}}^{\prime },z^{\prime }).

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. □

Let q\ge 2 be an integer, and let Ω be a subset of the symbolic space {\mathrm{\Sigma }}_m={\{0,\dots ,m-1\}}^{\mathbb{Z}} which is invariant under the shift map \sigma :{\mathrm{\Sigma }}_m\to {\mathrm{\Sigma }}_m defined by \sigma {(x)}_i=x_{i+1}. Denote

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.

Theorem 2.10 Given a set of templates(\mathbb{T},\mathbf{z}), where\mathbb{T}=[{\mathbb{T}}_1,\dots ,{\mathbb{T}}_{\ell }]and\mathbf{z}=[z_1,\dots ,z_{\ell }]. Let Y be the solution space of the constant CNN with respect to(\mathbb{T},\mathbf{z}). Then

if{\mathbb{T}}_i={\mathbb{T}}_jandz_i=z_jfor1\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

This subsection elucidates the influence of the boundary condition on the exact number of mosaic patterns of finite length and on the growth rate as the length increases. The investigation starts with formulating the number of patterns. Denote by {\mathbb{Z}}_{n\times 1}=\{k\in \mathbb{Z}:1\le k\le n\} the coordinates of the neurons. In this case, the boundary sites are \mathbb{B}=\{0,n+1\}. For the constant CNNs on {\mathbb{Z}}_{n\times 1}, the following three types of boundary conditions are considered:

1. (i)

   (7) _n -N: constant CNNs with Neumann boundary condition on {\mathbb{Z}}_{n\times 1};

2. (ii)

   (7) _n -P: constant CNNs with periodic boundary condition on {\mathbb{Z}}_{n\times 1};

3. (iii)

   (7) _n -D: constant CNNs with Dirichlet boundary condition on {\mathbb{Z}}_{n\times 1}.

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.

Lemma 2.11 Forn\in \mathbb{N}, writen=k\ell +rfor somek\in \mathbb{Z}andr=0,\dots ,\ell -1. Then

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

Let {\mathrm{\Sigma }}_n^B(\mathbf{Y}) denote the collection of output patterns of length n with boundary condition B, where B=P,N, and D stands for the periodic, Neumann, and Dirichlet boundary conditions, respectively. To find the exact number {\mathrm{\Gamma }}_n^B(\mathbf{Y})=|{\mathrm{\Sigma }}_n^B(\mathbf{Y})|, we introduce the following boundary matrices.

1. (i)

   Periodic boundary matrix R^P={\mathbf{1}}_{2^{2d}}^t\otimes (\left(\begin{array}{c}{E}^{(o)}\\ E^{(e)}\end{array}\right)\otimes {\mathbf{1}}_{2^{2d-1}}). More precisely,

   R^P=\left(\begin{array}{ccc}{E}^{(o)}& \cdots & E^{(o)}\\ ⋮& ⋮\\ E^{(o)}& \cdots & E^{(o)}\\ E^{(e)}& \cdots & E^{(e)}\\ ⋮& ⋮\\ E^{(e)}& \cdots & E^{(e)}\end{array}\right)\in {\mathcal{M}}_{2^{4d}\times 2^{4d}}(\mathbb{R}).
2. (ii)

   Dirichlet boundary matrices L^{D_{-}}=I^{(u)}, L^{D_{+}}=I^{(l)}, R^{D_{-}}=I^{(o)}, and R^{D_{+}}=I^{(e)} stands for the left/right Dirichlet boundary condition that is given by ‘−’ and ‘+’, respectively.

3. (iii)

   Neumann boundary matrices L^N={\mathbf{1}}_{2^{2d}}^t\otimes (\left(\begin{array}{c}{E}^{(u)}\\ E^{(l)}\end{array}\right)\otimes {\mathbf{1}}_{2^{2d-1}}), R^N={\mathbf{1}}_{2^{2d-1}}^t\otimes ((E^{(o)}{E}^{(e)})\otimes {\mathbf{1}}_{2^{2d}}). More precisely,

   L^N=\left(\begin{array}{ccc}{E}^{(u)}& \cdots & E^{(u)}\\ ⋮& ⋮\\ E^{(u)}& \cdots & E^{(u)}\\ E^{(l)}& \cdots & E^{(l)}\\ ⋮& ⋮\\ E^{(l)}& \cdots & E^{(l)}\end{array}\right),R^N=\left(\begin{array}{ccccc}{E}^{(o)}& E^{(e)}& \cdots & E^{(o)}& E^{(e)}\\ ⋮& ⋮& ⋮& ⋮\\ E^{(o)}& E^{(e)}& \cdots & E^{(o)}& E^{(e)}\end{array}\right).

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.

Recall that a set function \chi :2^{\mathbb{R}}\to {\{0,1\}}^{\mathbb{R}} is defined by \chi (E)(x):={\chi }_E(x)=1 if and only if x\in E for E being a nonempty subset of ℝ. For n\in \mathbb{N}, define

It is seen that k(n) is a nonnegative integer. To clarify the formulae of the exact number of patterns of length n of constant CNNs with boundary conditions, we introduce some notations first. Suppose n=k\ell +r, where 0\le r\le \ell -1. For j\in \mathrm{\Lambda }, set

and

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.

Lemma 2.12 Letn=k\ell +r, where0\le r\le \ell -1. Supposek\ge 2d+1, then the exact number{\mathrm{\Gamma }}_n^B(\mathbf{Y})with boundary conditionB\in \{P,N,D_{-},D_{+}\}are as follows:

1. (i)

   The periodic boundary condition:

   {\mathrm{\Gamma }}_n^P(\mathbf{Y})=\{\begin{array}{ll}\parallel (T_1^{K_1(n)}\otimes T_1)\circ R^P\parallel {\prod }_{j>1}{m}_j(n),& r=0;\\ \parallel (T_1^{K_1(n)}\otimes T_{r+1}^{K_{r+1}(n)+1})\circ R^P\parallel {\prod }_{j\ne 1,r+1}{m}_j(n),& \mathit{\text{otherwise}}.\end{array}

   (17)
2. (ii)

   The Dirichlet boundary condition:

   (18)

whereB=D_{-}/D_{+}means the pattern on the boundary is ‘-/+’.

1. (iii)

   The Neumann boundary condition:

   {\mathrm{\Gamma }}_n^N(\mathbf{Y})=\{\begin{array}{ll}\parallel L^N\circ (T_1^{K_1(n)+1}\otimes T_{\ell }^{K_{\ell }(n)+1})\circ R^N\parallel {\prod }_{j\ne 1,\ell }{m}_j(n),& r=0;\\ \parallel [(L^N\circ (T_1^{K_1(n)+1}\otimes T_{\ell }^{K_{\ell }(n)+1}))\otimes T_2^{K_2(n)+1}]\\ \circ (E_2\otimes R^N)\parallel {\prod }_{j=3}^{\ell -1}{m}_j(n),& r=1;\\ \parallel [T_{\ell -1}^{K_{\ell -1}(n)+1}\otimes (L^N\circ (T_1^{K_1(n)+1}\otimes T_{\ell }^{K_{\ell }(n)+1}))]\\ \circ (E_2\otimes R^N)\parallel {\prod }_{j=2}^{\ell -2}{m}_j(n),& r=\ell -1\end{array}

   (19)

and

otherwise.

HereE_2is a2\times 2matrix 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.

Suppose that r=0. It is seen from Lemma 2.11 that

At the same time, k\ge 2d+1 indicates that K_j(n)>0 for all j. A straightforward examination demonstrates that

and {\mathrm{\Gamma }}_k({\mathbf{Y}}_j)=m_j(n) for j=2,\dots ,\ell. Therefore, we have

If r>0, then y_{n+1}=y_1 and

where {\mathrm{\Gamma }}_{k+1}^P({\mathbf{Y}}_1,{\mathbf{Y}}_{r+1}) refers to the number of patterns

with y_{r+1;k+1}=y_{1;1}, and y_{1;1}\cdots y_{1;k+1} and y_{r+1;1}\cdots y_{r+1;k+1} are patterns of length k+1 in {\mathbf{Y}}_1 and {\mathbf{Y}}_{r+1}, respectively. It is verified that

This derives

and completes the proof. □

Next, to study the influence of boundary conditions on the exact number of patterns of finite length, we consider the effect on the growth rate of the number of patterns; more specifically, the topological entropy of the output space Y. The topological entropy h(X) of a space X is defined by

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

Proof For n\in \mathbb{N}, there exists a unique n_{\ell }\in \mathbb{Z} such that

Lemma 2.11 infers that

Applying the squeeze theorem, we have

This completes the first part of the proof.

To evaluate the boundary effect on the topological entropy of Y, we demonstrate that h_P(\mathbf{Y})=h(\mathbf{Y}). The other cases can be done analogously. Let τ denote the smallest integer such that T_j^{\tau }>0 for j\in \mathrm{\Lambda }, restated, T_j^{\tau }(p,q)>0 for 1\le p,q\le 2^{2d}. According to the definition of R^P,

Suppose n=k\ell +r. Lemma 2.12 implements

if r=0, and

otherwise. On the other hand, it is easily checked that

The above observation derives that

and thus we have h_B(\mathbf{Y})=h(\mathbf{Y}). □

The following theorem comes immediately from Theorem 2.13, the proof is omitted.

Theorem 2.14 The set of topological entropies of the constant CNNs is dense in the closed interval[0,log2]. More precisely, givenϵ>0and\lambda \in [0,log2], there exists a constant CNN such that.

This section elucidates another kind of inhomogeneous CNN-based LDS named arithmetic-type multiple cellular neural network (arithmetic CNN). It is seen that the templates of a constant CNN are periodic; in other words, the number of distinct templates is finite. This section investigates inhomogeneous CNNs whose number of distinct templates is infinite. First we consider a one-dimensional LDS with nearest neighborhood to interpret the idea of our methodology, then the derived results are generalized to general cases in the rest of this section.

3.1 Arithmetic cellular neural networks with nearest neighborhood

To clarify the study of an inhomogeneous LDS with nearest neighborhood, we consider the following system,

where i\ge 0 and q\ge 1 is odd. The feedback template \mathbb{T}=[{\mathbb{T}}_1,{\mathbb{T}}_3,{\mathbb{T}}_5,\dots ] consists of infinitely many subtemplates {\mathbb{T}}_q=[a_{0;q},a_{1;q}] for q=2k+1, k\ge 0, and the threshold \mathbf{z}=[z_1,z_3,z_5,\dots ] is an infinite vector. An inhomogeneous CNN realized as (22) is called the arithmetic CNN.

Similar to the discussion in the previous subsection, we demonstrate that arithmetic CNNs are completely stable. Let {\mathbf{x}}_q=(x_q,x_{2q},\dots ,x_{2^{n_q-1}q}) denote the collection of cells related to initial coordinate q. Express (22) as

where {\mathbf{A}}_q and {\beta }_q are similar to those defined in the previous subsection. A sufficient condition for the complete stability of (22) is presented as Theorem 3.1, which is a special case of Theorem 3.4.

Theorem 3.1 An arithmetic CNN is completely stable if{\mathbf{A}}_q(i,i)>1andK_q^{-1}(i,j)\ge 0for all q, i, j, where

Following the complete stability of an arithmetic CNN is the spatial complexity of the output space and the influence of boundary conditions. Note that the output space \mathbf{Y}=\{(y_i):y_i=f(x_i),i\in \mathbb{N}\} is different from the one in the previous subsection. The ordering matrix is then defined as

After redefining the ordering matrix, we obtain a sequence of transition matrices T_q corresponding to {\mathbf{Y}}_q=\{(y_{q{2}^{i-1}}):(y_i)\in \mathbf{Y}\} for q=2k+1, k\ge 0. The following theorem exhibits the computation of {\mathrm{\Gamma }}_n(\mathbf{Y}) and h(\mathbf{Y}). Furthermore, Theorem 3.2 is generalized to Theorem 3.7 for general arithmetic CNNs.