Inhomogeneous lattice dynamical systems and the boundary effect
© Ban and Chang; licensee Springer. 2013
Received: 18 April 2013
Accepted: 24 October 2013
Published: 21 November 2013
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.
The outputs , 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 , image processing [10, 11], artificial locomotion control , 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, ). 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., . This extends the results in the classical CNNs (cf.[31, 32]). The preservation of entropy under the boundary constraint is unavoidable ; 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 . 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 will be ϵ-dense in when parameter runs all of the parameter space, where 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.
2 Constant cellular neural networks
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
for . Denote the parameters that relate to the odd and even positions by and , respectively. We call the feedback template of (1), and 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 , , is a diagonal mapping (herein and ), and . 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, one of the following conditions is satisfied.
S1 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., for all i, and study the complexity of the output space of the mosaic solutions. We investigate the complexity of the output space in two aspects:
: The exact number of patterns of length n.
: The topological entropy of the output space.
where consists of patterns of length n in X. Yielding and , we derive the formula of and . For the general cases of constant CNNs, Theorem 2.2 is generalized by Lemma 2.11 and Theorem 2.13.
whereandare the spectral radii ofand, respectively.
Herein and relate to states ‘−’ (i.e., ) and ‘+’ (i.e., ), respectively. Before presenting the formula of and under the boundary condition , we introduce two operations of matrices.
- 1.Suppose that is a matrix and is an matrix. The Kronecker product is defined by
- 2.Suppose that are matrices. The Hadamard product is defined by
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 Supposefor some, . andare the transition matrices ofand, respectively. Then, , ifandare primitive matrices. Furthermore, the exact number of patterns of length n with boundary conditionare as follows:
The periodic boundary condition:(3)
The Neumann boundary condition:(4)
The Dirichlet boundary condition:(5)
Hereinrelate 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 , and , which is a finite subset of , indicates the neighborhood for neuron . The piecewise linear function is called the output function; refers to the threshold, and the feedback template stores the weight of local interaction between neurons, where .
where and . Without loss of generality, we assume for some , . In this case, the feedback template of (7) is , where . A stationary solution is called a mosaic solution if for all , and 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.)
is nonsingular and , where is defined in (10).
(It is easily seen that . 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 ()
A classical CNN with symmetric feedback template is completely stable.
is a diagonal mapping from to , and is a constant vector. Takahashi and Chua proposed a criterion to determine whether a CNN is completely stable.
Theorem 2.7 ()
for. A classical CNN with asymmetric feedback template is completely stable if K is nonsingular and, herein a matrixmeans thatfor all i, j.
Proof of Theorem 2.5 Suppose ; 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 . 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, .)
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 is fixed, there are finitely many equivalent classes of templates and z so that the basic sets of admissible local patterns are constrained. Let be the parameter space of the classical CNNs, where . Theorem 2.8 indicates that the can be partitioned into a finite number of subregions such that each subregion has the same mosaic patterns.
Theorem 2.8 ()
and for some k if and only if .
Hereis the closure of P in.
Theorem 2.9 (Separation property)
and for some k if and only if .
Proof Similar to the proof of Theorem 2.5, a constant CNN is reduced to a classical CNN whenever , hence Theorem 2.9 is performed in this case. When , the basic set of admissible local patterns of (7) is the ordered union of the basic set of admissible local patterns . More specifically, is isomorphic to the direct product , where is the parameter space of (7) j , the subsystem of (7) restricting to the cells . Since, for , each parameter space is partitioned into a finite number of equivalent subregions by Theorem 2.8, is then the union of a unique set of open subregions which satisfies conditions (i) to (iii). This derives the desired result. □
which is invariant under σ. The set 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 . Observe that Y is topologically conjugated to the direct product of the output spaces of the classical CNNs, that is, , where is determined by . This derives Theorem 2.10, which indicates that the output space of a constant CNN is a multiple subshift for some parameters.
ifandfor, where Ω is a SFT that comes from the output space of the classical CNN with respect to template.
2.3 Boundary effect on constant cellular neural networks
(7) n -N: constant CNNs with Neumann boundary condition on ;
(7) n -P: constant CNNs with periodic boundary condition on ;
(7) n -D: constant CNNs with Dirichlet boundary condition on .
These boundary conditions are discrete analogues of the ones in PDEs; to be specific, a pattern satisfies: (i) the Neumann boundary condition if and ; (ii) the periodic boundary condition if ; (iii) the Dirichlet boundary condition if and are prescribed.
Since , the total number of patterns of finite length in a constant CNN relates to the number of patterns in the subspaces. For each , there is a transition matrix that is implemented for the investigation of the subspace a (cf. 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, anddenotes the number of patterns of length q in X.
- (i)Periodic boundary matrix . More precisely,
Dirichlet boundary matrices , , , and stands for the left/right Dirichlet boundary condition that is given by ‘−’ and ‘+’, respectively.
- (iii)Neumann boundary matrices , . More precisely,
Here ⊗ is the Kronecker product, E is a matrix with entries being 1’s, I is the identity matrix, and is a column vector with entries being 1’s. Suppose that M is a matrix. Define by letting all the even/odd columns be zero vectors. Furthermore, indicates the matrix obtained from M by setting each of the lower-/upper-half rows as a zero vector.
Herein 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.
- (i)The periodic boundary condition:(17)
- (ii)The Dirichlet boundary condition:(18)
- (iii)The Neumann boundary condition:(19)
Hereis amatrix with entries being 1’s, and ∘ means the Hadamard product.
Proof We address the proof of , where the other cases can be verified in an analogous method.
and completes the proof. □
The existence of comes immediately from the submultiplicativity of , 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. Moreover, forprovidedis mixing for all.
This completes the first part of the proof.
and thus we have . □
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. More precisely, givenand, there exists a constant CNN such that.
3 Arithmetic cellular neural networks
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
where and is odd. The feedback template consists of infinitely many subtemplates for , , and the threshold is an infinite vector. An inhomogeneous CNN realized as (22) is called the arithmetic CNN.
where and 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.
After redefining the ordering matrix, we obtain a sequence of transition matrices corresponding to for , . The following theorem exhibits the computation of and . Furthermore, Theorem 3.2 is generalized to Theorem 3.7 for general arithmetic CNNs.
To simplify the formulae of , the following theorem presents the specific case. The general case is postponed to Lemma 3.6 and Theorem 3.7.
Theorem 3.3 Supposefor some k and Y is the output space of an arithmetic CNN. Then, , ifare primitive for all q. Furthermore, the exact number of patterns of length n with boundary conditionare as follows:
The periodic boundary condition:(24)
The Neumann boundary condition:(25)
The Dirichlet boundary condition:(26)
3.2 Stability of arithmetic cellular neural networks
In this case, the feedback template consists of infinitely many smaller templates , and the threshold is . Similar to the discussion in the previous section, Theorem 3.4 asserts that an arithmetic CNN is completely stable. The proof is omitted.
Then the system is completely stable iffor all, wherecomes fromdefined in (10).
Recall that the output space Y of a constant CNN can be decomposed into finitely many subspaces such that is a SFT for each j. In other words, the output space of a constant CNN extends the concept of SFTs. The output space of an arithmetic CNN is decomposed into countable subspaces; more precisely, , where is determined by the basic set of admissible local patterns . Theorem 3.5 demonstrates that the output space of an arithmetic CNN is a generalization of the so-called multiplicative shifts.
Then is called a multiplicative subshift. We define a semigroup action on by the following. For any and , the action is given by . It is seen that is invariant under the action. In other words, defines a multiplicative subshift.
A straightforward examination indicates that the output space Y of an arithmetic CNN is a multiplicative subshift if the neighborhood and the templates of (27) are invariant; restated, and for all . The proof is omitted.
where Ω is the SFT that comes from the output space of the classical CNN with respect to the template.
3.3 Boundary effects on arithmetic cellular neural networks
Recall that is the transition matrix of .
where I is the identity matrix. The formulae of are addressed as follows and the demonstration is omitted.
- (i)The periodic boundary condition:(37)
- (ii)The Dirichlet boundary condition:(38)
- (iii)The Neumann boundary condition:(39)
Theorem 3.7 formulates the topological entropy of the output space of an arithmetic CNN with/without boundary conditions.
whereis defined in (33). Furthermore, forprovidedis mixing for.
Proof The calculation of is presented; the effect of the boundary condition on the topological entropy can be elucidated via similar discussion, and as with the proof of Theorem 2.13, thus is omitted.
This completes the proof. □
The numerical experiment asserts that, similar to Theorem 2.14, the set of topological entropies of the arithmetic CNNs is dense in the closed interval . The theoretical proof of the following conjecture is not complete yet.
Conjecture 3.8 Givenand, there exists an arithmetic CNN such that.
4.1 One-dimensional cellular neural networks
where is the golden mean and is the maximal root of .
The topological entropy of constant CNNs with 2-components and templates being given by and
4.2 Two-dimensional constant cellular neural networks
A detailed and complete investigation is postponed to the upcoming manuscript.
The present paper studies two types of one-dimensional inhomogeneous CNN-based LDS, say, constant- and arithmetic-type multiple CNNs, which are a generalization of the classical CNNs. Sufficient conditions for the complete stability of constant and arithmetic CNNs are revealed. Since there is a wide range of parameters making the system completely stable, it is essential to investigate the complexity of mosaic patterns of the given system. A systematic methodology is proposed to interpret the exact number of mosaic patterns of inquired length and the topological entropy of the output space. Furthermore, the exact number of mosaic patterns and the topological entropy of the output space under the influence of three boundary conditions, say, the periodic, Neumann, and Dirichlet boundary conditions, are obtained. Remarkably, the boundary condition does not influence the topological entropy under some presumption.
The reveal that the set of topological entropies of the output spaces of constant CNNs is dense in the close interval indicates how rich phenomena constant CNNs could exhibit. Although there is a lack of rigorous proof for the density of the set of topological entropies of arithmetic CNNs, numerical experiments assert an affirmative result.
The methodology we propose in this investigation can be applied to multi-dimensional cases. A detailed discussion is on-going.
a Notably every subspace is a subshift of finite type (SFT) in symbolic dynamical systems, and thus can be studied via the graph theory and matrix theory. The reader is referred to  and  for more details.
b An arithmetic CNN is a classical CNN for the case that , this makes the requirement essential.
We thank the anonymous referees for their valuable comments that helped improve the quality and readability of the paper. Ban is partially supported by the National Science Council, ROC (Contract No. NSC 102-2628-M-259-001-MY3). Chang is grateful for the partial support of the National Science Council, ROC (Contract No. NSC 102-2115-M-035-004-).
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