Configurations of steady states for Hopfield-type neural networks E. Kaslik a,b,∗ St. Balint a a Faculty
of Mathematics and Computer Science, West University of Timi¸soara Bd. V. Parvan nr. 4, 300223, Timi¸soara, Romania b L.A.G.A,
UMR 7539, Institut Galil´ee, Universit´e Paris 13 99 Avenue J.B. Cl´ement, 93430, Villetaneuse, France
Abstract The dependence of the steady states on the external input vector I for the continuoustime and discrete-time Hopfield-type neural networks of n neurons is discussed. Conditions for the existence of one or several paths of steady states are derived. It is shown that, in some conditions, for an external input I there may exist at least 2n exponentially stable steady states (called configuration of steady states), and their regions of attraction are estimated. This means that there exist 2n paths of exponentially stable steady states defined on a certain set of input values. Conditions assuring the transfer of a configuration of exponentially stable steady states to another configuration of exponentially stable steady states by successive changes of the external input are obtained. These results may be important for the design and maneuvering of Hopfield-type neural networks used to analyze associative memories. Key words: Hopfield neural network, associative memory, steady states, control PACS: 62M45, 92B20, 93C15, 93C55, 37B25
1
Introduction
In this paper, we consider the continuous-time Hopfield-type neural network defined by the following system of nonlinear differential equations: x˙i = −ai xi +
n X
Tij gj (xj ) + Ii
i = 1, n
(1)
j=1
∗ Corresponding author. Email addresses:
[email protected] (E. Kaslik),
[email protected] (St. Balint).
Preprint submitted to Elsevier Science
16 February 2006
In [1] a semi-discretization technique has been presented for (1), leading to discrete-time neural networks which faithfully preserve the characteristics of (1), i.e. the steady states and their stability properties. Despite this fact, in this paper, we will consider a more general class of discrete time Hopfield-type neural networks, defined by the following discrete system: xip+1 = xip − ai xip +
n X
Tij gj (xjp ) + Ii
∀i = 1, n, p ∈ N
(2)
j=1
In equations (1)-(2) we have that ai > 0, Ii denote the external inputs, T = (Tij )n×n is the interconnection matrix, gi : R → R (i = 1, n) represent the neuron input-output activations. For simplicity, we will suppose that the activation functions gi are of class C 1 on R and gi (0) = 0, for i = 1, n. The systems (1)-(2) can be written in the matrix forms: x˙ = −Ax + T g(x) + I
(3)
xp+1 = xp − Axp + T g(xp ) + I
(4)
where x = (x1 , x2 , ..., xn )T ∈ Rn , A = diag(a1 , ..., an ) ∈ Mn×n , I = (I1 , ..., In )T ∈ Rn and g : Rn → Rn is given by g(x) = (g1 (x1 ), g2 (x2 ), ..., gn (xn ))T . Since neural networks like (1) have been first considered in [2,3], they have received much attention because of their applicability in problems of optimization, signal processing, image processing, solving nonlinear algebraic equation, pattern recognition, associative memories and so on. The qualitative analysis of neural dynamics plays an important role in the design of practical neural networks. To solve problems of optimization, neural control and signal processing, neural networks have to be designed in such way that, for a given external input, they exhibit only one globally asymptotically stable steady state. This matter has been treated in [4–9,1] and the references therein. On the other hand, if neural networks are used to analyze associative memories, the existence of several locally asymptotically stable steady states is required, as they store information and constitute distributed and parallel neural memory networks. In this case, the purpose of the qualitative analysis is the study of the locally exponentially stable steady states (existence, number, regions of attraction) so as to ensure the recall capability of the models. Conditions for the local asymptotic stability of the steady states (and estimations of their regions of attraction) for Hopfield-type neural networks have been derived din [10–15] using single Lyapunov functions and in [16,17] using vector Lyapunov functions. 2
The aim of this paper is to show, that in some conditions, for some values of the external input I there exists a configuration of at least 2n asymptotically stable steady states for (1)-(2). Therefore, there exist 2n paths of steady states defined on a set of values of the external input. Estimations of the regions of attraction of these steady states are given. Finally, we address the problem of controllability of these configurations of steady states.
2
Paths of steady states
A steady state x = (x1 , x2 , ..., xn )T of (1)-(2) corresponding to the external input I = (I1 , I2 , ..., In )T is a solution of the equation: −Ax + T g(x) + I = 0
(5)
For a given external input vector I ∈ Rn the system (5) may have one solution, several solutions or it may happen that it has no solutions. On the other hand, for any state x ∈ Rn there exists a unique external input I(x) ∈ Rn such that x is a steady state of (1)-(2) corresponding to the input I(x). Clearly, the external input function I : Rn → Rn is of class C 1 or Rn and is defined by I(x) = Ax − T g(x)
(6)
The set I defined by I = {I ∈ Rn /∃x ∈ Rn : I = Ax − T g(x)}
(7)
is the collection of those inputs I for which the system (5) has at least one solution. If I = Rn then for any input I ∈ Rn the system (5) has at least one solution. If I is strictly included in Rn then there exist input vectors I for which system (5) has no solution. In the followings, before defining the concept of path of steady states, a few preliminary results will be presented concerning the existence of steady states of (1)-(2) in rectangles included in Rn . Proposition 1 If ∆ is a rectangle in Rn , i.e. for i = 1, n there exist αi , βi ∈ R, αi < βi such that ∆ = (α1 , β1 ) × (α2 , β2 ) × ... × (αn , βn ) and det(A − T Dg(x)) 6= 0 for any x ∈ ∆ then the function I|∆ (the restriction of the external input function to ∆) is injective. 3
PROOF. Let be x0 , x00 ∈ ∆, x0 6= x00 . For every i = 1, n there exists ci ∈ (x0i , x00i ) such that gi (x0i ) − gi (x00i ) = gi0 (ci )(x0i − x00i ). Therefore, we have I(x0 ) − I(x00 ) = A(x0 − x00 ) − T (g(x0 ) − g(x00 )) = (A − T Dg(c))(x0 − x00 )(8) where c = (c1 , c2 , ..., cn )T ∈ (x01 , x001 ) × (x02 , x002 ) × ... × (x0n , x00n ) ⊂ ∆. Hence, the matrix A − T Dg(c) is non-singular, which provides that I(x0 ) 6= I(x00 ). Thus, I|∆ is injective. 2 Corollary 2 Let be a rectangle ∆ ⊂ Rn such that det(A − T Dg(x)) 6= 0 for any x ∈ ∆. Then for any I ∈ I(∆) the system (1)-(2) has a unique steady state in ∆. In [9], it has been shown that in certain conditions (similar to those from the following Corollary) for a given input vector I ∈ Rn the neural network defined by (1)-(2) has a unique steady state in Rn . Corollary 3 Let be a rectangle ∆ ⊂ Rn . If gi0 (s) > 0 for any s ∈ R and i = 1, n and: Tii −
ai 0 gi (xi )
+
X
|Tji | < 0
∀i = 1, n, ∀x ∈ ∆
(9)
i6=j
then for any I ∈ I(∆) the system (1)-(2) has a unique steady state in ∆. If Ii > 0 for any i = 1, n then the coordinates of the steady state are positive. Next, we define the concept of path of steady states and we give some sufficient conditions for the existence of one ore several paths of steady states for (1)-(2). Definition 4 We call path of steady states for (1)-(2) a continuous function ϕ : U ⊂ I → Rn such that −Aϕ(I) + T g(ϕ(I)) + I = 0
∀I ∈ U
(10)
i.e. ϕ(I) is a steady state of (1)-(2) for any input I ∈ U . If ϕ is of class C 1 on U then we say that ϕ is a C 1 -path. Let be an external input I ? ∈ I and x? ∈ Rn a steady state corresponding to this input, i.e. −Ax? + T g(x? ) + I ? = 0. The following theorem holds for the existence of a C 1 -path of steady states of (1)-(2) which contains x? : Theorem 5 If the matrix A − T Dg(x? ) is non-singular, then there exists a unique maximal domain V ? ⊂ Rn containing x? , a unique maximal domain U ? ⊂ Rn containing I ? and a uniquely defined function ϕ : U ? → V ? of class C 1 having the following properties: 4
i. ϕ(I ? ) = x? ; ii. −Aϕ(I) + T g(ϕ(I)) + I = 0 for any I ∈ U ? ; iii. the matrix A − T Dg(x) is non-singular on V ? . PROOF. Direct consequence of the implicit function theorem and the continuous dependence of det(A − T Dg(x)) on x. 2 Let be the set C defined by C = {x ∈ Rn / det(A − T Dg(x)) = 0}
(11)
and the set G = Rn \ C. The set G is an open set and for any x ∈ G we have that det(A − T Dg(x)) 6= 0. Let be {Gα }α the set of the open connected components of G, i.e. for any α S the set Gα 6= ∅ is open and connected, Gα = G and Gα0 ∩ Gα00 = ∅ if α0 6= α00 . α
Theorem 6 If Gα is a rectangle in Rn then there exists a unique function ϕα : Hα → Gα of class C 1 having the following properties: i. Hα = I(Gα ) where I(x) = Ax − T g(x) for any x ∈ Rn ; ii. Aϕα (I) − T g(ϕα (I)) = I for any I ∈ Hα ; iii. the matrix Dϕα (I) is non-singular on Hα . PROOF. According to Proposition 1, I|Gα is injective. Consider Hα = I(Gα ) and remark that I|Gα is a bijection of class C 1 . Now we can consider ϕα = (I|Gα )−1 which satisfies ii. and iii. 2 It is clear that for every α, the function ϕα given by Theorem 6 is C 1 -path of steady states of (1)-(2). Remark 7 If the set Gα is not a rectangle in Rn , consider ∆α the largest rectangle included in Gα . For the rectangle ∆α the statements of Theorem 6 are fulfilled, i.e. there exists a unique C 1 -path of steady states ϕα : I(∆α ) → ∆α of (1)-(2). S
In this way, it has been shown that Rn can be decomposed as Rn = ( Gα ) ∪ C α
and sufficient conditions have been found assuring that for an input vector I in a certain set Hα ⊂ Rn there exists a unique steady state in the largest rectangle ∆α included in Gα . This is the case in general, when several paths of steady states exist. This kind of a result can be important in the design of Hopfield-type neural networks used to analyze associative memories. 5
3
Controllability
Definition 8 A change at a certain moment of the external input from I 0 to I 00 is called maneuver and it is denoted by I 0 7→ I 00 . We say that the maneuver I 0 7→ I 00 made at t = t0 is successful on the path ϕα : Hα → ∆α of (1) if I 0 , I 00 ∈ Hα and if the solution of the initial value problem x˙ = Ax + T g(x) + I 00
x(t0 ) = ϕα (I 0 )
(12)
tends to ϕα (I 00 ) as t → ∞. We say that the maneuver I 0 7→ I 00 made at k = k0 is successful on the path ϕα : Hα → ∆α of (2) if I 0 , I 00 ∈ Hα and if the solution of the initial value problem xp+1 = Bxp + T g(xp ) + I 00
xk0 = ϕα (I 0 )
(13)
tends to ϕα (I 00 ) as p → ∞. The system (1)-(2) is said to be controllable along a path of steady states if any two steady states belonging to the path can be transferred one in the other by a finite number of successive maneuvers. If there exists a unique path of globally exponentially stable steady states ϕ : Rn → Rn of (1)-(2) then any maneuver I 0 7→ I 00 is successful on the path ϕ, therefore, the system (1)-(2) is controllable along the path ϕ. If the steady states ϕα (I) of the path ϕα are only locally exponentially stable, then it may happen that some maneuvers are not successful along this path. In such cases, it is appropriate to use the following result [18–20]: Theorem 9 For two steady states ϕα (I ? ) and ϕα (I ?? ) belonging to the path ϕα : Hα → ∆α of locally exponentially stable steady states of (1)-(2), there exists a finite number of values of the external inputs I 1 , I 2 , ..., I p ∈ I(∆α ) such that all the maneuvers I ? 7→ I 1 7→ I 2 7→ ... 7→ I p 7→ I ??
(14)
are successful on the path ϕα . Theorem 9 states that the system (1)-(2) is controllable along the path ϕα of locally exponentially stable steady states. In fact, the transfer from a steady 6
state ϕα (I ? ) to a steady state ϕα (I ?? ) is made through the regions of attraction of the states ϕα (I 1 ), ϕα (I 2 ), ..., ϕα (I n ), ϕα (I ?? ). Corollary 10 If ϕα :
T α∈Γ
T α∈Γ
Hα 6= ∅ for a certain set Γ of indexes α and the paths
Hα → ∆α (α ∈ Γ) are locally exponentially stable, then for two
configurations of steady states {ϕα (I ? )}α∈Γ and {ϕα (I ?? )}α∈Γ where I ? , I ?? ∈ T Hα there exists a finite number of external input vectors I 1 , I 2 , ..., I p ∈ α∈Γ T α∈Γ
Hα such that the maneuvers I ? 7→ I 1 7→ I 2 7→ ... 7→ I p 7→ I ??
(15)
transfer the configuration {ϕα (I ? )}α∈Γ into the configuration {ϕα (I ?? )}α∈Γ . Example 11 Let be the following Hopfield-type neural network [12]: x˙1 = −x1 + 17 ln 4 tanh x2 + I1 15
(16)
x˙ = −x + 17 ln 4 tanh x + I 2 2 1 2 15
It is easy to see that for (I1 , I2 ) = (0, 0) the system (16) has three steady states: (0, 0)T which is unstable and (ln 4, ln 4)T and (− ln 4, − ln 4)T which are locally exponentially stable. The external input function is I : R2 → R2 defined by I(x1 , x2 ) = (x1 −
17 ln 4 17 ln 4 tanh x2 , x2 − tanh x1 )T 15 15
(17)
The Jacobi matrix of the system is
−1
17 ln 4 15(cosh x2 )2
17 ln 4 15(cosh x1 )2
−1
(18)
which is non-singular if and only if cosh x1 cosh x2 6= set G has two open connected components:
17 ln 4 . 15
It follows that the
G− = {x = (x1 , x2 )T ∈ R2 : cosh x1 cosh x2
17 ln 4 } 15
(20)
All the steady states belonging to the set G− are unstable, as the eigenvalues 4 of the Jacobi matrix in a point x = (x1 , x2 )T ∈ G− are ± 15 cosh17xln1 cosh − 1, x2 one of them being positive, and the other negative. 7
In the other connected component G+ there exist at least two paths of steady states, one of them containing (ln 4, ln 4)T and the other containing (− ln 4, − ln 4)T . All the steady states belonging to G+ are locally exponentially stable. We denote by ϕ the path included in G+ which contains the steady state (ln 4, ln 4)T . Let us analyze some characteristics of the steady states belonging to the path ϕ which correspond to external inputs of the form I = (I1 , I2 )T with I1 = I2 . One can prove that the steady states which correspond to a given input (I1 , I1 )T are of the form (x1 , x1 )T . It is obvious that for any steady state (x1 , x1 )T from the first bisector it corresponds an input (I1 , I1 )T where I1 (x1 ) = x1 − 1715ln 4 tanh x1 .
4
3
2 4 1
3 2 1
Fig. 1. Estimates of the regions of attraction of xi , i = 1, 4
In the Figure 1 we have represented four steady states which belong to the first bisector: x1 = (1, 1)T for which I11 = −0.196566, x2 = (ln 4, ln 4) which corresponds to I12 = 0, x3 = (3, 3) for which I13 = 1.43664 and x4 = (4, 4) which corresponds to I14 = 2.42992. All these steady states belong to G+ , therefore, they are locally exponentially stable. In Figure 1, the estimates of the regions of attraction of each steady state xi , i = 1, 4 are also presented. These estimates have been found using the method proposed in [21]. One can see that x1 is in the estimate of the region of attraction of x4 , therefore, the maneuver I : (I11 , I11 )T 7→ (I14 , I14 )T is successful and transfers the neural network from the steady state x1 to the steady state x4 directly. On the other hand, x4 does not belong to the estimate of the region of attraction of x1 , therefore, we are not sure that the direct maneuver I : (I14 , I14 )T 7→ (I11 , I11 )T is successful. On the other hand, we observe that x4 ∈ Da (x3 ), x3 ∈ Da (x2 ) and x2 ∈ Da (x1 ) (where Da denotes the region of attraction), hence the neural network can be transferred from x4 to x1 by the following successive maneuvers: I : (I14 , I14 )T 7→ (I13 , I13 )T 7→ (I12 , I12 )T 7→ (I11 , I11 )T 8
(21)
4
Configurations of 2n steady states
In this section, we will consider the following hypothesis for the activation functions: (H1 ) The activation functions gi , i = 1, n are bounded: |gi (s)| ≤ 1
for any s ∈ R, i = 1, n
(22)
(H2 ) There exists α ∈ (0, 1) such that the functions gi , i = 1, n satisfy: gi (s) ≥ α if s ≥ 1 and gi (s) ≤ −α if s ≤ −1
(23)
(H3 ) The derivatives of the activation functions gi , i = 1, n satisfy: |gi0 (s)| < P n
ai
j=1
∀|s| ≥ 1
(24)
|Tji |
These hypothesis are not too restraining. Indeed, if the activation functions are bounded, but not by 1, one can consider the new activation functions gi and replace the matrix T by the matrix (Tij sup |gj (s)|)n×n . These sup |gi (s)| s∈R
s∈R
new activation functions satisfy hypothesis (H1 ). More, the activation functions gi are usually chosen to satisfy the following conditions: gi (s) → 1 as s → ∞, gi (s) → −1 as s → −∞ and gi0 (s) → 0 as s → ±∞. Hence, for α ∈ (0, 1) there exists M > 0 such that for any i = 1, n one has: • gi (s) ≥ α if s ≥ M , gi (s) ≤ −α if s ≤ −M ai • |gi0 (s)| < P for |s| ≥ M . n |Tji |
j=1
If M ≤ 1, then hypothesis (H2 ) and (H3 ) hold for the activation functions gi . If M > 1, consider the suitable rescaling in the system (3): y = (3) becomes: y˙ = −Ay +
1 1 T g(M y) + I M M
1 x. M
System
(25)
which describes a neural network having the activation function g˜(y) = M1 g(M y) and the external input I˜ = M1 I. The functions g˜i satisfy hypothesis (H2 ) and (H3 ): 9
• g˜i (s) ≥ α ˜ if s ≥ 1 and g˜i (s) ≤ −˜ α if s ≤ −1 where α ˜= ai 0 • |˜ gi (s)| < P for |s| ≥ 1 n
α M
∈ (0, 1)
|Tji |
j=1
The following theorem provides a bound for the set of steady states of (1)-(2) corresponding to an input I: Theorem 12 If hypothesis (H1 ) is fulfilled, then for any input vector I ∈ Rn the following statements hold: i. There exists at least one steady state of (1)-(2) (corresponding to I) in the rectangle ∆I = [−M1 , M1 ] × [−M2 , M2 ] × ... × [−Mn , Mn ] of Rn , where Mi =
n X 1 (|Ii | + |Tij |) for any ai j=1
i = 1, n
(26)
ii. Every steady state of (1)-(2), corresponding to I, belongs to the rectangle ∆I defined above. iii. If in addition det(A − T Dg(x)) 6= 0 for any x ∈ ∆I then the system (1)-(2) has a unique steady state, corresponding to I, and it belongs to ∆I .
PROOF. The set of steady states of (1)-(2) corresponding to I are given by equation (5), which is equivalent to x = A−1 (I + T g(x))
(27)
Let be the function h : Rn → Rn defined by h(x) = A−1 (I + T g(x)). For any x ∈ Rn and i = 1, n, one has n n X X 1 1 |hi (x)| = | (Ii + Tij gj (xj ))| ≤ (|Ii | + |Tij |) = Mi ai ai j=1 j=1
(28)
Therefore, h(Rn ) ⊂ ∆I , which proves ii. More, one gets that h(∆I ) ⊂ ∆I , and as h is a continuous function, Brouwer’s fixed point theorem guarantees the existence of at least one steady state of (1)-(2) corresponding to I in ∆I , so the statement i holds. Statement iii follows directly from Corollary 2. 2
For every ε ∈ {±1}n we define the rectangle ∆ε = J(ε1 ) × J(ε2 ) × ... × J(εn ) where J(1) = (1, ∞) and J(−1) = (−∞, −1). For continuous-time Hopfield-type neural networks, the following theorem holds: 10
Theorem 13 (the continuous case) Suppose that hypothesis (H1 ) and (H2 ) hold. If the external input I ∈ Rn satisfies |Ii | < Tii α − ai −
X
|Tij | ∀i = 1, n
(29)
i6=j
then the following statements hold: i. In every rectangle ∆ε , ε ∈ {±1}n , there exists at least one steady state of (1) corresponding to the input I. ii. Every ∆ε , ε ∈ {±1}n , is invariant to the flow of system (1). iii. More, if hypothesis (H3 ) holds as well, then the steady state of (1) corresponding to the input I, which lies in the rectangle ∆ε , ε ∈ {±1}n is unique, it is exponentially stable and its region of attraction includes ∆ε . PROOF. Let be an input I satisfying (29) and ε ∈ {±1}n . i. Consider the function h : Rn → ∆I defined by h(x) = A−1 (I + T g(x)) and the rectangle ∆I given in the Theorem 12. For x ∈ ∆ε we have that εi xi ≥ 1 for any i = 1, n and therefore εi hi (x) = ≥
X εi (Tii gi (xi ) + Tij gj (xj ) + Ii ) ≥ ai j6=i X 1 (Tii α − |Tij | − |Ii )| > 1 ai j6=i
(30)
This means that hi (x) ∈ J(εi ) for any i = 1, n and therefore, hi (x) ∈ ∆ε . We have just proved that h(∆ε ) ⊂ ∆ε ∩ ∆I and Brouwer’s fixed point theorem guarantees the existence of at least one steady state of (1) corresponding to the input I in ∆ε ∩ ∆I . ii. Let be x0 ∈ ∆ε . Suppose that there exists t0 ≥ 0 and i ∈ 1, n such that xi (t0 ) = εi , where xi (t) = xi (t; x0 , I). Consider yi (t) = εi (−ai xi (t) + Pn j=1 Tij gj (xj (t)) + Ii ). Based on (29) and hypothesis (H1 ), we have that yi (t0 ) = εi (−ai εi + Tii gi (εi ) + ≥ −ai + Tii α −
X
X
Tij gj (xj (t0 )) + Ii ) ≥
j6=i
|Tij | − |Ii | > 0
(31)
j6=i
Therefore, there exists t1 > t0 such that yi (t) > 0 for any t ∈ [t0 , t1 ]. This implies that εi x˙i (t) = yi (t) > 0 for any t ∈ [t0 , t1 ] and therefore the function εi xi is strictly increasing on [t0 , t1 ]. Hence εi xi (t) > εi xi (t0 ) = ε2i = 1 for any 11
t ∈ (t0 , t1 ]. This means that xi (t) ∈ J(εi ) for any t ∈ (t0 , t1 ]. It follows that the solution x(t; x0 , I), x0 ∈ ∆ε will remain in ∆ε for any t ≥ 0. ai iii. Consider βi such that |gi0 (s)| ≤ βi < P n
for any |s| ≥ 1 and i = 1, n.
|Tji |
j=1
We will first show that the steady state of (1), corresponding to the input I, which lies in ∆ε is unique. Suppose the contrary, i.e. there exist two steady states x, y ∈ ∆ε , x 6= y of (1). For every i = 1, n, one has: ai |xi − yi | = |
n X
Tij (gj (xj ) − gj (yj ))| ≤
j=1
n X
|Tij |βj |xj − yj |
(32)
aj |xj − yj |
(33)
j=1
Therefore, n X
ai |xi − yi | ≤
i=1
n X n X
|Tij |βj |xj − yj |
0. Therefore, we have that V (t) ≤ e−kt V (0).
Hence V (t) → 0 exponentially as t → ∞. This means that x(t; x0 , I) → xI,ε as t → ∞. Thus xI,ε is exponentially stable and its region of attraction includes ∆ε . 2 For discrete-time Hopfield-type neural networks, the following theorem holds: Theorem 14 (the discrete case) Suppose that hypothesis (H1 ) and (H2 ) hold and that ai ∈ (0, 1), i = 1, n. If the external input I ∈ Rn satisfies |Ii | < Tii α − ai −
X
|Tij | ∀i = 1, n
(36)
i6=j
then the following statements hold: i. In every rectangle ∆ε , ε ∈ {±1}n , there exists at least one steady state of (2) corresponding to the input I. ii. Every ∆ε , ε ∈ {±1}n , is invariant to the map x 7→ f (x, I). iii. More, if hypothesis (H3 ) holds as well, then the steady state of (2) corresponding to the input I, which lies in the rectangle ∆ε , ε ∈ {±1}n is unique, it is exponentially stable and its region of attraction includes ∆ε . PROOF. Let be an input I satisfying (36) and ε ∈ {±1}n . i. similar to the proof of Theorem 13(i). ii. Let be x ∈ ∆ε . One has to prove that f (x, I) ∈ ∆ε . Using (36), hypothesis (H1 ) and (H2 ), for any i = 1, n it results that: εi fi (x, I) = εi (xi − ai xi + Tii gi (xi ) + ≥ 1 − ai + Tii α −
X
X
Tij gj (xj ) + Ii ) ≥
j6=i
|Tij | − |Ii | > 1
(37)
j6=i
Therefore, f (x, I) ∈ ∆ε . ai iii. Suppose that |gi0 (s)| ≤ βi < P n
for any |s| ≥ 1 and i = 1, n. The
|Tji |
j=1
uniqueness of the steady state of (2) corresponding to the input I, which lies 13
in the rectangle ∆ε is shown in the same way as in Theorem 13(iii). This unique steady state will be denoted by xI,ε . Let’s prove that xI,ε is exponentially stable and its region of attraction includes ∆ε . Consider the function V : Rn → R+ defined by V (x) =
n X
|xi − xI,ε i |
∀x ∈ Rn
(38)
i=1
On ∆ε , the function V satisfies:
V (f (x, I)) =
n X
(|fi (x, I) − xI,ε i | =
i=1
= ≤ ≤ =
n X i=1 n X i=1 n X i=1 n X
|(1 − ai )(xi − xI,ε i )+ (1 − ai )|xi − xI,ε i |+ (1 − ai )|xi − xI,ε i |+
n X
Tij (gj (xj ) − gj (xI,ε j ))| ≤
j=1 n n XX i=1 j=1 n X n X
|Tij ||gj (xj ) − gj (xI,ε j )| ≤ |Tij |βj |xj − xI,ε j | =
i=1 j=1
(1 − ai + βi
i=1
where k = max(1−ai +βi i=1,n
n X
|Tji |)|xj − xI,ε j | ≤ kV (x)
(39)
j=1 n P j=1
|Tji |) ∈ (0, 1). From ii. we have that V (f p (x, I)) ≤
k p V (x) for any p ∈ N. Hence V (f p (x, I)) → 0 as p → ∞. This means that f p (x, I) → xI,ε as p → ∞. Thus xI,ε is exponentially stable and its region of attraction includes ∆ε . 2 Remark 15 From Theorem 12, it follows that if there exists an input I satisfying |Ii | ≤ ai −
n X
|Tij |
∀i = 1, n
(40)
j=1
then there exists al least one steady state of (1)-(2) corresponding to I belonging to the rectangle [−1, 1]n , and there are no other steady states corresponding to I outside this rectangle. The existence of such an input implies that ai > |Tii | for any i = 1, n. On the other hand, Theorems 13-14 guarantee that if there exists α ∈ (0, 1) 14
and an input I satisfying |Ii | < Tii α − ai −
X
|Tij | ∀i = 1, n
(41)
i6=j
then there exist 2n steady states corresponding to I outside the rectangle [−1, 1]n (in every rectangle ∆ε ). The existence of such an input implies that ai < Tii α for any i = 1, n. It is easy to see that the above conditions oppose. We will denote by U the set defined by U = {I ∈ Rn /|Ii | < Tii α − ai −
X
|Tij |, ∀i = 1, n}
(42)
i6=j
The consequence of Theorems 13 and 14 can be summarized as follows: Theorem 16 Suppose that hypothesis (H1 ), (H2 ) and (H3 ) hold for the activation functions gi , i = 1, n and that Tii α − ai −
X
|Tij | > 0
∀i = 1, n
(43)
i6=j
More, in the discrete case, suppose that ai ∈ (0, 1), i = 1, n. The following statements hold: i. U 6= ∅ and in each rectangle ∆ε , ε ∈ {±1}n there exists a unique path of exponentially stable steady states ϕε : U → ∆ε for (1)-(2). ii. For every I ∈ U and ε ∈ {±1}n , the region of attraction of ϕε (I) includes ∆ε . iii. The maneuver I 0 7→ I 00 , with I 0 , I 00 ∈ U is successful along every path of steady states ϕε : U → ∆ε , ε ∈ {±1}n . It transfers the configuration of steady states ϕε (I 0 ) to the configuration of steady states ϕε (I 00 ). iv. The system (1)-(2) is controllable along every path ϕε : U → ∆ε , ε ∈ {±1}n . Example 17 Consider the following neural network: x˙1 = −a1 x1 + b1 g(x1 ) + b2 g(x2 ) + I1
x˙2 = −a2 x2 + b2 g(x1 ) + b1 g(x2 ) + I2
where g : R → R, g(s) =
2 π
arctan( π2 s). 15
(44)
Let be α = g(1) ' 0.63. One can easily check that the activation function g satisfies the hypothesis (H1 ) and (H2 ). Let be β = g 0 (1) ' 0.28. We have that 0 < g 0 (s) ≤ β for any |s| ≥ 1. If β(|b1 | + |b2 |) < ai < αb1 − |b2 |
i = 1, 2
(45)
then the activation function g satisfies hypothesis (H3 ) and the condition (43) of Theorem 16 is also fulfilled. The set U is defined in this case by: U = {I ∈ R2 /|Ii | < αb1 − |b2 | − ai , i = 1, 2}
(46)
We will consider b1 = 1000, b2 = −0.5 and a1 = a2 = αb1 − |b2 | − 300. In this case, we have U = (−300, 300) × (−300, 300). Therefore, we have four paths of exponentially stable steady states of (44), which we denote by ϕε : U → ∆ε , ε ∈ {±1}2 . In Figure 2, the gray rectangles represent the four sets of steady states ϕε (U ). The four spirals in Figure 2 represent the steady states corresponding to the inputs Iu = (20u cos u, 20u sin u) with u ∈ [0, 4π]. 4
3
2
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
4
Fig. 2. The sets of steady states ϕε (U ) for (44)
Example 18 We consider a discrete Hopfield-type neural network with the non-monotone activation function g(x) = tanh(5x) tanh(10x2 − 1): x1
p+1
x2
p+1
= 0.5x1p + 20g(x1p ) − g(x2p ) + I1 =
0.5x2p
−
g(x1p )
+
20g(x2p )
+ I2
(47)
It has been shown [22] that in some cases, the absolute capacity of an associative neural network can be improved by using non-monotone activation 16
functions instead of the usual sigmoid ones. Consider α = f (1) ∈ (0, 1). It can be verified that the activation function g satisfies hypothesis (H1 ), (H2 ) and (H3 ) and that condition (43) of Theorem 16 is also fulfilled. The set U is in this case the rectangle U = {I ∈ R2 /|Ii | < 18.4982, i = 1, 2}
(48)
Therefore, we have four paths of exponentially stable steady states of (47), which we denote by ϕε : U → ∆ε , ε ∈ {±1}2 . In Figure 3, the gray rectangles represent the four sets of steady states ϕε (U ). In Figure 3, we have also represented the four steady states corresponding to the input I = (0, 0)T , namely: (38, 38)T , (−42, 42)T , (42, −42)T and (−38, −38)T and the four steady states corresponding to the input I = (10, 10)T , namely: (58, 58)T , (−22, 62)T , (62, −22)T and (−18, −18)T . The maneuver I : (0, 0)T 7→ (10, 10)T transfers the configuration of steady states {(38, 38)T , (−42, 42)T , (42, −42)T , (−38, −38)T } to the configuration of steady states {(58, 58)T , (−22, 62)T , (62, −22)T , (−18, −18)T }. 75
50
25
0
-25
-50
-75 -75
-50
-25
0
25
50
75
Fig. 3. The sets of steady states ϕε (U ) for (47) and the maneuver I : (0, 0)T 7→ (10, 10)T
5
Conclusions
For continuous and discrete-time Hopfield-type neural networks, conditions ensuring the existence of 2n paths of exponentially stable steady states defined on a certain set of external input values have been derived. More, it has been shown that the system is controllable along each of these paths of steady 17
states. Finding similar conditions for Cohen-Grossberg neural networks may constitute a direction for future research.
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