On the templates corresponding to cycle-symmetric connectivity in cellular neural networks

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c

World Scientific Publishing Company

ON THE TEMPLATES CORRESPONDING TO

CYCLE-SYMMETRIC CONNECTIVITY

IN CELLULAR NEURAL NETWORKS

CHIH-WEN SHIH∗ and CHIH-WEN WENG†

Department of Applied Mathematics, National Chiao Tung University,

Hsinchu, Taiwan, R.O.C. ∗_{[email protected]}

Received July 12, 2001; Revised October 26, 2001

In the architecture of cellular neural networks (CNN), connections among cells are built on linear coupling laws. These laws are characterized by the so-called templates which express the local interaction weights among cells. Recently, the complete stability for CNN has been extended from symmetric connections to cycle-symmetric connections. In this presentation, we investigate a class of two-dimensional space-invariant templates. We find necessary and sufficient conditions for the class of templates to have cycle-symmetric connections. Complete stability for CNN with several interesting templates is thus concluded.

Keywords: Neural network; complete stability; cycle-symmetric matrix.

1. Introduction

This investigation aims to explore the structures of templates in cellular neural networks (CNN), which yield cycle-symmetric connectivity among cells, and hence complete stability for the networks. We shall illustrate our results in the CNN proposed by Chua and Yang [1988]. Assume that the model is for-mulated on a two-dimensional n1× n2 array Tn := {(i, j) ∈ Z2_{|1 ≤ i ≤ n}

1, 1 ≤ j ≤ n2}. The circuit equation of a cell is given by

dxi,j dt =−xi,j+ X (k, `)∈Nr(i, j) A(i, j; k, `)fk,`(xk,`) + bi,j, (i, j)∈ Tn, (1)

where Nr(i, j) represents the r-neighborhood of the

cell at (i, j). That is,

Nr(i, j) ={(k, `)|max(|k − i|, |` − j|) ≤ r} . The feedback operator is represented by real num-bers A(i, j; k, `), (i, j) ∈ Tn, (k, `) ∈ Nr(i, j), and these real numbers constitute the template for CNN. This template describes the connection weights among cells. If (k, `) ∈ Nr(i, j), for some (i, j)∈ Tn and (k, `) not in Tn, then xk,` in (1) is determined by the imposed boundary condition, cf . [Thiran, 1993; Shih, 2000]. bi,j represent the terms from the control operator and threshold. fk,l is called output function. The standard output func-tion is given by fk,`(ξ) = f (ξ) := 1/2(|ξ+1|+|ξ−1|), (k, `)∈ Nr(i, j), (i, j)∈ Tn. This piecewise-linear function f results in lack of smoothness for the ∗_{Author for correspondence.}

Work partially supported by the National Science Council of Taiwan, R.O.C., the National Center for Theoretical Sciences, R.O.C., and the Lee and MTI Center for Networking Research.

†_{Work partially supported by The National Science Council of Taiwan, R.O.C.}

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Int. J. Bifurcation Chaos 2002.12:2957-2966. Downloaded from www.worldscientific.com

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vector field in (1). For details, please see [Chua & Yang, 1988; Chua, 1998; Lin & Shih, 1999].

Since there are finitely many cells, the indices {(i, j)} can be rearranged into a one-dimensional setting. Such an arrangement can be expressed by a bijection between the corresponding index sets, namely, from Tn onto {i ∈ Z1|1 ≤ i ≤ n1 × n2}. With this rearrangement, the coordinates {xi,j}, (i, j)∈ Tn become {xi}, 1 ≤ i ≤ n, n := n1× n2. Equation (1) is then recast into the following form.

dx

dt =F(x) := −x + Ay + b , (2) where y := (f1(x1), f2(x2), . . . , fn(xn)), b is a con-stant vector. In respecting (1), the n× n matrix A in (2) is generated from A(i, j; k, `). We call A the matrix of connection weights or connection matrix. Notably, CNN with coupling (templates) of any di-mension can be put into the form (2), as long as there are finitely many cells. The shortcoming for the expression (2) is that the templates features, that is, the local coupling relation among cells can no longer be read from the equation.

Equation (2) can also be interpreted as a ma-trix form of (1). Indeed, let each of x, y and b be an n1 × n2 matrix and let A be a linear op-erator on the space of n1 × n2 matrices. That A acts on y yields an n1 × n2 matrix Ay, which is defined by (Ay)ij = P(k,`)∈TnA(i, j; k, `)yk,` = P (k,`)∈TnA(i, j; k, `)fk,`(xk,`), where A(i, j; k, `) = ( A(i, j; k, `) if (k, `)∈ Nr(i, j), 0 otherwise. (3) Indeed, A can also be regarded as an n× n ma-trix with each of its column and row indexed by elements of Tn. For notational convenience, if u = (i, j), v = (k, `), we write A(u; v) for the (u, v)-entry of A.

By complete stability, we mean that every solu-tion of the system tends to an equilibrium as time goes to infinity. The complete stability for (1) with standard output function was first studied in [Chua & Yang, 1988] when the model was proposed. [Lin & Shih, 1999] has overcome the nondifferentiability of the Lyapunov function and provided a rigorous proof for complete stability of CNN in the generic case, that is, the isolated equilibria case. The proof relies on the construction of a Lyapunov function and the use of LaSalle’s invariance principle. The

basic assumption in these investigations is the sym-metry condition of the circuit parameters in (1):

A(i, j; k, `) = A(k, `; i, j),

for all (k, `), (i, j)∈ Tn. (4) With this assumption, if (1) is imposed with certain discrete-type boundary conditions, A is always symmetric, as (1) is reformulated into the form (2). Recently, the complete stability for (2) (hence (1)) has been extended to the so-called cycle-symmetric connections, cf. [Shih, 2001]. Cycle-symmetric matrices can be described as follows. Let A = [A(u; v)] be an n× n matrix with either A(u; v) = 0 or A(u; v)A(v; u) 6= 0 for u, v ∈ Tn. There corresponds an undirected graph whose ver-tex v is joined to the verver-tex u by the edge uv if and only if A(u; v) 6= 0 and A(v; u) 6= 0. With the abuse of notation, we denote this graph also by Tn. Let u1, . . . , un be n distinct vertices. Then the sequence u1u2· · · u`u1is a cycle (of length `) if any two consecutive vertices have an edge. Sometimes we treat the cycle as the edge set {u1u2, u2u3, . . . , u`−1u`, u`u1}. An n × n matrix A is cycle-symmetric if A satisfies the following two conditions (H1), (H2).

(H1) A(u; v)A(v; u) > 0, if A(u; v)6= 0 , (5) (H2) Q uv∈C A(u; v) = Q uv∈C A(v; u),

for any cycle C , (6) where Π denotes the product. A is called sign-symmetric if A satisfies (H1). It is straightforward to verify that symmetric A satisfies (H1) and (H2). Convergence of dynamics (complete stability herein) for several neural networks has been ex-tended to connection matrices satisfying (H1) and (H2), see [Fiedler & Gedeon, 1998; Gedeon, 1999]. An interesting linear algebra theorem states that a square matrix A is similar to a symmetric matrix by a positive diagonal matrix (that is, there exists D = diag(d1, . . . , dn), di > 0 such that DAD−1 is symmetric) if and only if A satisfies both (H1) and (H2). This theorem has simplified the verifica-tions of convergence in the above-mentioned stud-ies, cf. [Shih & Weng, 2000].

Notably, in addition to extending the complete stability to cycle-symmetric connection matrices, Shih [2001] has also addressed the complete sta-bility for (2) with other sigmoidal output func-tions, and the case of nonisolated equilibria. As for

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the characterization and classification of equilibria and their output patterns for CNN, the readers are referred to [Juang & Lin, 1997, 2000; Shih, 1998; Ban et al., 2001].

In this investigation, we shall explore the struc-tures of space-invariant templates (see Sec. 2) which yield cycle-symmetric connection among cells. Restated, we plan to study the question: what kind of A(i, j; k, `) in (1) would yield a cycle-symmetric A in (2)?

This presentation is organized as follows. A key lemma of this work is given in Sec. 2. The main result of this investigation, on two-dimensional tem-plates corresponding to cycle-symmetric connec-tivity, is summarized in Theorem 3.2 in Sec. 3. We further extend our results to CNN with one-dimensional and three-one-dimensional templates and other two-dimensional templates in Sec. 4. We compare our results with earlier studies on the templates and connection matrices which yield

complete stability or almost complete stability for CNN in Sec. 5.

2. Preliminaries

In this presentation, we plan to investigate the tem-plates which have uniform local behaviors, that is, the connection among cells depends only on their relative positions. Restated, A(i, j; k, `) = m(k− i; ` − j). Such a kind of template is called space-invariant template. A 3× 3 space-invariant template is characterized by a 3× 3 matrix

M =      m(−1; 1) m(0; 1) m(1; 1) m(−1; 0) m(0; 0) m(1; 0) m(−1; −1) m(0; −1) m(1; −1)      . (7) Accordingly, A(i, j; k, `) = ( m(k− i; ` − j) if |k − i| ≤ 1 and |` − j| ≤ 1 , 0 otherwise . (8)

Notably, the index of the entry of M is not stan-dard. We use this index to make the representa-tion of A in (8) more realizable. Observe that this is the case r = 1 in Eq. (1). Note that if m(i; j) = m(−i; −j), then A is a symmet-ric matrix, that is, A(u; v) = A(v; u), where u = (i, j), v = (k, `). Recall that

A(u; v)6= 0 iff uv is an edge in the graph Tn. (9) In this case, m(v− u) 6= 0 by (8), and we say that the edge uv has type “v− u”, where the subtrac-tion is the usual vector subtracsubtrac-tion. Type “u− v” is called the opposite type of “v− u”.

We shall determine all the templates M in (7) that ensure A to be cycle-symmetric in the next section. The complete stability for CNN with these templates is thus concluded. We need the follow-ing technical lemma to prove our main result. This lemma states that a cycle C of Tnof length at least 5 can be decomposed into two cycles by a path of Tn, cf . Figs. 1 and 2. We omit the proof since it is not particularly illuminating.

Lemma 2.1. Let C : u1u2· · · usu1 be a cycle of length s≥ 5 in Tn. Then there is a path P : v1· · · vτ

3 u 4 u 1 u 2 u

Fig. 1. The cycle u1u2u3u4u1 cannot be decomposed into two cycles.

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3 u 4 u 1 u 2 u 5 u 6 u

Fig. 2. The cycle u1u2u3u4u5u6u1 is decomposed into two cycles by the edge u2u5.

in Tn such that the following (a) and (b) hold: (a) v1 = up and vτ = uq for some p, q (1 ≤ p ≤

q− 1 ≤ s − 1).

(b) If τ = 2, then edge v1v2 6= edge u1us. If τ > 2, then v2, . . . , vτ−1 belong to the bounded region inside of C.

In fact, the path P in Lemma 2.1 can be cho-sen such that each of its edge has the same or the opposite type of some edge in the cycle C.

3. The Main Result

Throughout this section, we assume that A is defined by (8), and

m(i; j) = 0 if and only if m(−i; −j) = 0

(i, j =−1, 0, 1) . (10) 1

u

2

u

₃ 1

u

2

u

3

u

4

u

1

u

2

u

3

u

4

u

1

u

2 3

u

Fig. 3. Only four basic cycles C need to be checked in (6), if A is generated from a 3× 3 space-invariant template.

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The assumption (10) on M ensures the condition A(i, j; k, `) = 0 if and only if A(k, `; i, j) = 0. The following lemma is immediate from (5), (8) and (10).

Lemma 3.1. A is sign-symmetric if and only if m(i; j)m(−i; −j) ≥ 0, i, j = −1, 0, 1, and (10) holds.

Suppose A is sign-symmetric. We shall prove that the following conditions (i)–(iv) are equivalent to A being cycle-symmetric. Conditions (i) m(0; 1)m(1; 0)m(−1; −1) = m(1; 1)m(−1; 0)m(0; −1); (ii) m(1; 0)m(0; −1)m(−1; 1) = m(1; −1)m(0; 1)m(−1; 0); (iii) m(0;−1)2_m(_{−1; 1)m(1; 1)} = m(−1; −1)m(1; −1)m(0; 1)2; (iv) m(−1; 0)2m(1; −1)m(1; 1) = m(−1; −1)m(−1; 1)m(1; 0)2.

Notably, conditions (i)–(iv) say that to de-termine whether A is cycle-symmetric, one only needs to examine (6) for the four basic cycles shown in Fig. 3. In addition, in (i), m(0; 1)m(1; 0) m(−1; −1) = m(1; 1)m(−1; 0)m(0; −1) is ex-actly the condition that M is cycle-symmetric, and in (ii), m(1; 0)m(0;−1)m(−1; 1) = m(1; −1) m(0; 1)m(−1; 0) is the condition that

Mt    0 0 1 0 1 0 1 0 0   

is cycle-symmetric, where Mt is the transpose of M. See (6) and (7).

Theorem 3.2. Assume that m(i; j)m(−i; −j) ≥ 0, i, j =−1, 0, 1, and (10) holds. Then A defined by (8) is cycle-symmetric if and only if Condi-tions (i)–(iv) hold.

Proof. (⇒) Suppose A is cycle-symmetric. To prove (i), set u1 = (1, 1), u2 = (1, 2), u3 = (2, 2), and apply (8) to (6) with the cycle u1u2u3u1. Note that both sides of (i) are zero if u1u2u3u1is not a cy-cle. Similarly, setting u1 = (1, 2), u2= (2, 2), u3= (2, 1), and considering the cycle u1u2u3u1 prove (ii). To prove (iii), set u1 = (2, 3), u2 = (2, 2), u3 = (2, 1), u4 = (1, 2), and apply (8) to (6) with

the cycle u1u2u3u4u1. Similarly, (iv) can be proved by setting u1 = (3, 2), u2 = (2, 2), u3 = (1, 2), u4 = (2, 1), and considering the cycle u1u2u3u4u1. (⇐) Suppose (i)–(iv) hold. We shall prove (6) for any cycle C: u1· · · usu1 of Tn. Note that the area of enclosed region of C is V = k/2 for some posi-tive integer k. We prove by induction on k. Suppose k = 1. Observe that this is equivalent to s = 3, and u1u2u3u1is a triangle. There are at most eight pos-sible triangles of area 1/2 starting with u1. For each triangle, we find Conditions (i)–(ii) are enough to obtain (6) for the cycle C. To prove the case k = 2, we prove a more general case that s = 4. There are essentially three types of cycles of length 4: a square of area 1, a square of area 2, and a triangle of area 1 with its base of length 2. For squares, (6) is a trivial equality. For a triangle, Conditions (iii) and (iv) are enough to obtain (6) for the cycle C. Now suppose k > 2 and s≥ 5. Let P = v1· · · vτ be a path satisfying (a) and (b) in Lemma 2.1. Define the following three paths P1, P2, P3.

P1 : u1u2· · · up P2 : upup+1· · · uq P3 : uquq+1· · · usu1,

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see Fig. 4. Let P−1 be the reversed path of P. Note that P1∪ P ∪ P3 and P−1∪ P2 both are cycles with smaller enclosed areas. By induction,

Y ujuj+1∈P1∪P ∪P3 A(uj; uj+1) = Y ujuj+1∈P1∪P ∪P3 A(uj+1; uj) , (12) and Y ujuj+1∈P−1∪P2 A(uj; uj+1) = Y ujuj+1∈P−1∪P2 A(uj+1; uj) . (13)

Multiplying the same sides of (12) and (13) together, we have Y ujuj+1∈P1∪P2∪P3∪P ∪P−1 A(uj; uj+1) = Y ujuj+1∈P1∪P2∪P3∪P ∪P−1 A(uj+1; uj) . (14)

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1 u s u q u 1 P P 3 P 2 P P u 1 P : P2: 3 P : P:

Fig. 4. A cycle u1u2· · · usu1 decomposed into two cycles.

Deleting the common terms, it follows that Y ujuj+1∈P ∪P−1 A(uj; uj+1) = Y ujuj+1∈P ∪P−1 A(uj+1; uj) . (15)

which is nonzero ensured by (9), we have (6). Notably, A is sign-symmetric by Lemma 3.1. This completes the proof.

Corollary 3.3. In addition to the assumptions of Theorem 3.2, suppose m(0; 1)m(−1; 0) 6= 0 holds.

Then A is cycle-symmetric if and only if Conditions (i) and (ii) hold.

Proof. A is sign-symmetric by Lemma 3.1. Mul-tiply the same sides of the equalities in Conditions (i) and (ii), we obtain

m(1; 0)2m(−1; −1)m(−1; 1)m(0; 1)m(0; −1) = m(−1; 0)2m(1; 1)m(1;−1)m(0; 1)m(0; −1) .

(16) Observe that m(0; 1)m(0; −1) 6= 0, by the as-sumptions. Dividing both sides of (16) by m(0; 1) m(0; −1), we have (iv). Multiply the opposite sides of equalities in Conditions (i) and (ii), we obtain

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m(0; 1)2m(−1; −1)m(1; −1)m(−1; 0)m(1; 0) = m(0;−1)2m(−1; 1)m(1; 1)m(−1; 0)m(1; 0) .

(17) Observe m(−1; 0)m(1; 0) 6= 0. Dividing both sides of (17) by m(0;−1)m(0; 1), we obtain (iii).

Combining Theorem 3.2 with the theorem in [Shih, 2001], we have the following conclusion. Corollary 3.4. The complete stability of CNN holds for any sign-symmetric templates which sat-isfy Conditions (i)–(iv). In particular, the com-plete stability of CNN holds for the following sign-symmetric square-crossed and diagonal-crossed templates: M =    0 m(0; 1) 0 m(−1; 0) m(0; 0) m(1; 0) 0 m(0; −1) 0   , M =    m(−1; 1) 0 m(1; 1) 0 m(0; 0) 0 m(−1; −1) 0 m(1; −1)    ,

where m(i; j)m(−i; −j) ≥ 0, and m(−i; −j) = 0 whenever m(i; j) = 0, i, j =−1, 0, 1.

Remarks

(a) For the case of symmetric template, that is, m(i; j) = m(−i; −j), −1 ≤ i, j ≤ 1, (10) and (i)–(iv) hold obviously. Thus, cycle-symmetric connectivity indeed generalizes symmetric connectivity. (b) M =    1 0 1 2 m(0; 0) 1 2 0 2   

satisfies Conditions (i)–(iii), but does not satisfy Condition (iv). This shows (iv) is necessary in Theorem 3.2. Similarly, (iii) is necessary.

4. Other Templates Corresponding to

Cycle-Symmetric Connection

In this section, we would like to explore more tem-plates corresponding to cycle-symmetric connection in CNN. We shall consider 1× 3 and 1 × 5 one-dimensional templates, 5× 5 square-crossed and diagonal-crossed two-dimensional templates. Our results can even be extended to CNN with three-dimensional templates.

The evolution equation of CNN with one-dimensional templates is given by

dxi dt =−xi+ X k∈Nr(i) A(i; k)fk(xk) + bi, i∈ Tn, (18) where Tn := {i ∈ Z1|1 ≤ i ≤ n}. As r = 1, we consider the space-invariant template denoted by M1×3 = [m−1 m0 m1]. Restated, A(i; k) = mk−i, for i ∈ Tn and k ∈ N1(i). Accordingly, for each i ∈ Tn, we define A(i; k) = A(i; k) if k ∈ N1(i), and zero if k ∈ Tn \ N1(i). It is obvious that A is a tridiagonal n× n matrix. Assume that m₋₁m1> 0, then the graph induced by A can only have cycles of length two and (H2) in (6) is trivially satisfied.

For the case r = 2, we consider the following space-invariant template

M1×5 = [m−2 m−1 m0 m1 m2] .

Theorem 4.1. CNN with space-invariant template M1×3 or M1×5 is completely stable if M1×3 satis-fies m₋₁m1 > 0, and M1×5 satisfies m−imi > 0, i = 1, 2 and m2m2₋₁ = m−2m21.

Proof. The case with template M1×3 has already been illustrated. For the template M1×5, consider a cycle C in Tn. When moving along the cycle C in a fixed direction, there are four types of edges: “1”, “2”, “−1”, “−2”. They represent the coordi-nate difference between two nodes connected by an edge. More precisely, if u1u2 is an edge of type “1” (resp. “2”), then u2 is in the east of u1with coordi-nate distance 1 (resp. 2). Similarly, the edge u1u2 of types “−1” or “−2” means that u2 is in the west of u1 with coordinate distance 1 or 2, respectively. Suppose the edge of type “i” appears nitimes in C. Then necessarily,

n1+ 2n2− n−1− 2n−2 = 0 , (19) since C is a cycle. To verify (6), we need to show

mn1 1 mn22m n₋₁ −1 mn−2−2 = mn1−1m n₋₂ 2 mn−11mn−22 . (20) It is clear that (20) holds from (19) if we as-sume that m2m2₋₁ = m−2m21. This completes the proof.

The result in Theorem 4.1 can be easily generalized to larger two-dimensional templates. Consider the following square-crossed 5× 5 template

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M+₅_×5 =          0 0 m(0; 2) 0 0 0 0 m(0; 1) 0 0 m(−2; 0) m(−1; 0) m(0; 0) m(1; 0) m(2; 0) 0 0 m(0; −1) 0 0 0 0 m(0; −2) 0 0          . (21)

Theorem 4.2. CNN with space-invariant template M+₅_×5 is completely stable if m(−i; 0)m(i; 0) > 0 and m(0;−i)m(0; i) > 0, i = 1, 2, and m(2; 0)m(−1; 0)2 = m(−2; 0)m(1; 0)2, m(0; 2)× m(0;−1)2_{= m(0;}_{−2)m(0; 1)}2_.

The idea of the proof is rather straightforward. Pick any node u of a cycle C in the graph induced by the matrix A defined from M+₅_×5. Since M+₅_×5 is square-crossed, u can go east, west, north and south directions. Starting from u, in order to come back to itself, the total length that u travels to the east should be the same as that traveled by u to the west, while the total length that u travels to the south should coincide with the length that u travels to the north. The verification of the theo-rem then follows from similar argument as in the proof of Theorem 4.1.

It is not difficult to see that the above re-sult also holds for the diagonal-crossed templates.

Furthermore, we can extend the above theo-rems to CNN with three-dimensional templates. Notice that there are six neighboring cells con-nected to each cell at (i1, i2, i3) ∈ Z3, as the orthogonal template M⊥₃_×3×3 in Fig. 5 is consid-ered. The conditions in Theorem 4.2 can eas-ily be extended to a three-dimensional version to conclude the completely stability for CNN. Restated, if m(1; 0; 0)m(−1; 0; 0) > 0, m(0; 1; 0) m(0; −1; 0) > 0, m(0; 0; 1)m(0; 0; −1) > 0, then M⊥₃_×3×3 is a template which yields complete stability for CNN. One can also consider a two-dimensional template with six connecting neigh-bors for each cell. For example, M in (7) with m(−1; 1) = m(1; −1) = 0. In addition to m(i; j)m(−i; −j) > 0, the Condition (i) in Sec. 3 still needs to be satisfied for complete stabil-ity of CNN. This comparison indicates that three-dimensional template M⊥₃_×3×3 requires weaker circuit parameter condition in concluding the complete stability of CNN. m(-1;0;0) m(0;0;1) m(0;0;0) m(0;-1;0) m(1;0;0) m(0;0;-1) m(0;1;0)

i

j

k

Fig. 5. An orthogonal three-dimensional template M⊥3×3×3.

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5. Discussions and Conclusions

There are several previous works addressing the ma-trices of connection weights which yield complete stability for CNN. Gilli [1994] has studied a class of connection matrices. A matrix in this class can be transformed to a symmetric matrix by multiplying a positive diagonal matrix from its left. Restated, let A be an n× n matrix. If there exists a positive diagonal matrix Λ such that ΛA is symmetric, then the CNN (2) with connection matrix A was proved to be completely stable. However, in addition to that some arguments therein remain to be rigor-ously justified, proving that limt→∞x(t) = 0 does˙ not imply complete stability. Nevertheless, there is a linear algebra theorem which relates this class of n× n matrices to our cycle-symmetric matrices.

Gedeon [1999] studied the neural networks of Lotka–Volterra type and Grossberg’s models. In deriving the convergence (complete stability herein) for the systems, he proposed the same class of ma-trices of connection weights. Let us describe the setting. A square matrix A = [aij] is called combi-natorially symmetric if aij 6= 0 implies aji 6= 0. We quote Theorem 3.1 in [Gedeon, 1999] as follows.

Let A be a real, combinatorially symmetric, irreducible matrix. There is a positive diagonal ma-trix Λ such that ΛA is a symmetric mama-trix if and only if

(a) there is a spanning tree T of the graph G(A) such that for every edge eij ∈ T , we have aijaji > 0, and

(b) if CT is a chord cycle in the graph G(A) with respect to tree T , then ΠCTaij = ΠCTaji. Though (a) is a little weaker than our condition (H1), (a) and (b) together are equivalent to (H1) with (H2) in (5), (6). Furthermore, the asser-tion that there exists a positive diagonal matrix Λ such that ΛA is a symmetric matrix is equiv-alent to that there exists a positive diagonal ma-trix D such that DAD−1 is symmetric. Indeed, if DAD−1 is symmetric with D being positive di-agonal, then DDAD−1D is symmetric, that is, D2A is symmetric and Λ = D2 is positive diago-nal. On the other hand, if there is a positive di-agonal matrix Λ such that ΛA is symmetric, then (√Λ)−1ΛA(√Λ)−1 = √ΛA(√Λ)−1 is symmetric and D =√Λ is positive diagonal. With these struc-tures recognized, it is no surprise that some of the templates discussed in [Gilli, 1994] coincide with ours. However, our Conditions (i)–(iv) in Sec. 3

have provided more complete descriptions on the 3× 3 templates which yield this class of connection matrices, hence complete stability for CNN.

As continuously differentiable increasing out-put functions are considered, Chua and Wu [1992] derived a class of templates for almost completely stable CNN, by applying the theory in cooperative systems [Hirsch, 1985]. The result is stated as fol-lows. Let J be a diagonal matrix with entries either 1 or−1. Then CNN with n×n connection matrix A is stable almost everywhere if the equilibria are iso-lated and J AJ = J AJ−1is non-negative. With the same kind of output functions or standard output function (piecewise-linear), CNN with some of the templates discussed therein, such as sign-symmetric square-crossed or diagonal-crossed templates, are in fact completely stable, instead of merely almost sta-ble, according to our Corollary 3.4. In [Wu & Chua, 1997], assuming continuously differentiable increas-ing output functions again, CNN with connection matrix A is proved to be completely stable if there exist diagonal matrices D = diag(d1, . . . , dn) and T = diag(t1, . . . , tn) with diti > 0 such that DAT is symmetric. It is not difficult to verify that such n× n matrix A also satisfies our (H1) and (H2) and vice versa. Though these two descriptions are equivalent, the conditions (H1) and (H2) are more concrete and explicit. Examining (H1) and (H2) is more straightforward than finding these diagonals D and T such that DAT is symmetric. Moreover, all the results in this presentation hold for CNN with standard output function as well as continu-ously differentiable increasing output functions, as indicated in [Shih, 2001].

As a conclusion, the present investigation has illustrated that generalizing symmetric connectiv-ity to cycle-symmetric connectivconnectiv-ity among cells is not merely a process of changing variables to the equation or an application of a linear algebra the-orem. It indeed provides important information on the structures of space-invariant templates which yield complete stability for CNN.

Acknowledgment

The authors would like to thank the referee for call-ing their attention to the works by Gilli and Chua and Wu.

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