Permanent oscillations in a 3-node recurrent neural network model

Neurocomputing 74 (2010) 274–283

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Permanent oscillations in a 3-node recurrent neural network model Chunhua Feng a, Christian O’Reilly b, Re´jean Plamondon b, a b

Department of Mathematics, Guangxi Normal University, Guilin, Guangxi 541004, PR China ´ lectrique, E ´ cole Polytechnique de Montre´al, Montre´al, Que´bec, Canada H3C 3A7 Laboratoire Scribens, De´partement de Ge´nie E

a r t i c l e in f o

a b s t r a c t

Article history: Received 13 January 2009 Received in revised form 29 January 2010 Accepted 11 March 2010 Communicated by V. Jirsa Available online 2 April 2010

In this paper we discuss the existence of oscillations in a specific recurrent neural network: the 3-node network with two weight parameters and one time delay. Simple and practical criteria for fixing the range of the parameters in this network model are derived. Computer simulations are used to show typical patterns and to point out that these criteria are only sufficient conditions for such oscillations to happen. & 2010 Elsevier B.V. All rights reserved.

Keywords: Recurrent neural networks Time delays 3-node networks

1. Introduction Dynamical recurrent neural networks have been used in many applications such as signal processing [19,28,35,38,39,41,47]; speech and language processing [11,16,44]; optimal control and estimation [6,8,12,14,18,22,29], and neuroscience [7,38]. In this latter field, for example, studies dealing with the motor control of human locomotion use dynamic recurrent neural networks to mimic the coordination of the lower limb [10]. It was also shown that a dynamical recurrent neural network can be trained to predict the position of the hand from neural recording of primates executing a reaching task [37]. The extensive use of these networks mainly relies on their powerful learning capabilities and their dynamical behavior. Doya and Yoshizawa [13], Yang and Dillon [46], and Pearlmutter [30] have shown that certain configurations are even capable of learning time varying periodic signals. About 10 years ago, Ruiz et al. [36], and Townley et al. [40] have studied a class of recurrent networks which are able to learn and replicate autonomously time varying signals: 8 0 x ðtÞ ¼ x1 ðtÞ þtanhðx2 ðtÞÞ > > > 10 > > x ðtÞ ¼ x2 ðtÞ þtanhðx3 ðtÞÞ > > > 2 > <........................ x0n1 ðtÞ ¼ xn1 ðtÞ þ uðtÞ > > > > 0 > > x > n ðtÞ ¼ xn ðtÞ þw1 tanhðx1 ðtÞÞ þ    þ wn1 tanhðxn1 ðtÞÞ > > : yðtÞ ¼ tanhðx ðtÞÞ n

 Corresponding author.

E-mail address: [email protected] (R. Plamondon). 0925-2312/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2010.03.002

ð1Þ

where the x0i ðtÞ are the first time derivatives of the input xi(t), the tanhðxi ðtÞÞ are the activation functions, and the wi are the weights or the parameter of the system. This network operates as follows: First, it receives a teaching signal u(t) from the external environment. During that period, the network weights are self-adapted and y(t) tracks u(t). When the adaptation is completed, the system switches autonomously from the learning stage into a unity feedback configuration stage so that y(t) replace u(t) and the periodic pattern is sustained. Such configuration was used by Ruiz et al. to study in detail the specific 3-node recurrent network described as 8 0 x ðtÞ ¼ x1 ðtÞ þ tanhðx2 ðtÞÞ > < 1 x02 ðtÞ ¼ x2 ðtÞ þ tanhðx3 ðtÞÞ ð2Þ > : x0 ðtÞ ¼ x ðtÞ þ w tanhðx ðtÞÞ þ w tanhðx ðtÞÞ 3 1 1 2 2 3 Ruiz et al. [36] have shown that the recurrent network model (2) can possess an attractive limit cycle and they specified the variation ranges of the parameters. Townley et al. [40], from a computational point of view, have used bifurcation techniques and monotone dynamical systems theory to show that recurrent networks can learn and then replicate a permanent oscillation. Recently, Gao and Zhang [18] have displayed an effective technique to determine the number and the distribution of its equilibrium points and further studied the qualitative properties of these equilibria. These previous studies did not deal with time delays, although systems with delays are omnipresent in biological and physical systems and these time lags cannot be neglected in many applications.1 When time delays are taken into account, some authors, on the one hand, have studied the convergence and the stability of various 1 For a comprehensive discussion of time delay phenomena and their impact on systems’ behavior, see [25].

C. Feng et al. / Neurocomputing 74 (2010) 274–283

recurrent networks [1,9,21,26]. For example, Li and Liao [23] have defined a set of sufficient conditions that guarantees the global exponential stability of some recurrent systems. On the other hand, it is well known that time delays may result in oscillatory behaviors or network instability [4,22,24,47–49]. Be´lair et al. [4] have investigated the stability for a single delay neural network model and indicated that time delay may induce oscillations in the system. However, the authors did not specify any conditions for such oscillations to happen. In a recent paper [15], we have shown that oscillatory behavior requires some relationship between the time delay and the w1 and w2 parameters. The present paper shows that time delay induces oscillations when the parameters of a 3-node system take on some limited positive and negative values. We derive three sets of sufficient conditions to guarantee the existence of oscillations for such a recurrent neural network.

2. Permanent oscillation in time delay recurrent neural networks model Up to now, at least five classes of activation function have been proposed for neural networks [45] with the function tanh(x) being just a special case. In the following, to perform a more general analysis, we consider a 3-node closed loop network system with a general activation function f(x). This system, schematized in Fig. 1, is represented by the following equation: 8 0 x ðtÞ ¼ x1 ðtÞ þf ðx2 ðtt2 ÞÞ > < 1 x02 ðtÞ ¼ x2 ðtÞ þf ðx3 ðtt3 ÞÞ ð3Þ > : x0 ðtÞ ¼ x ðtÞ þw f ðx ðtt ÞÞ þw f ðx ðtt ÞÞ 3 1 1 1 2 2 2 3 where xðtÞ ¼ ðx1 ,x2 :x3 ÞT A R3 , w1 ,w2 A R, and ti 40 are constants.

275

If we choose f(xi) such that it is bounded and that f(0) ¼0, then 0 is an equilibrium point of the system (3). If x(t) is a solution of the system (3) then, considering that the f(xi) are bounded, there exist Mi 40 (i¼1,2,3) such that x0i ðtÞ o 0 whenever xi ðtÞ Z Mi and x0i ðtÞ 4 0 whenever xi ðtÞ r Mi (i¼1,2,3). Let M¼ max{M1, M2, M3}, then jxi ðtÞj r M. Hence, x(t) is globally bounded. The linearization of this system around x¼ 0 is given by 8 0 z ðtÞ ¼ z1 ðtÞ þ az2 ðtt2 Þ > < 1 z02 ðtÞ ¼ z2 ðtÞ þ az3 ðtt3 Þ ð4Þ > : z0 ðtÞ ¼ z ðtÞ þ w az ðtt Þ þ w az ðtt Þ 3 1 1 1 2 2 2 3 where a ¼ df ðxÞ=dxjx ¼ 0 40. According to [18], the system (3) without delay has one, two or three equilibrium points which depend on the selected values of w1 and w2. Note that time delays cannot change the number or the location of the equilibrium points. This means that whether the time delayed system (3) has one or more than one equilibrium points depend on the selected parameters. Let (xn1i, xn2i, xn3i) (i¼1,2,3) be equilibria of (3). Since  ,x ,x ,Þ a ð0,0,0Þ, then we (0,0,0) is one equilibrium point, if ðx1i 2i 3i can make a change of variables y1 ¼x1  xn1i, y2 ¼ x2 xn2i, y3 ¼x3  xn3i, that linearizes the system at the equilibrium point and makes this transformed system equivalent to (4). Thus, although in the following we only discuss the oscillation of (0,0,0), our results are general for any equilibrium point (xn1, xn2, xn3). Lemma 1. The bounded solutions of the systems (3) and (4) have the same oscillatory behaviors. Proof. Note that the system (4) may have unbounded solutions. Here, we only discuss the oscillatory behavior of the bounded solutions of system (3) and (4). Since f(xi) are activation functions such that f(0) ¼0, expanding f(xi) in a Taylor series about xi ¼0 00 gives f ðxi ðtti ÞÞ ¼ axi ðtti Þ þ ðx2i ðtti Þ=2Þf ðxi Þ, where xi lies between 0 and xi. As xi approaches 0, the function x2i will become infinitesimally small. This means that the behavior of the systems (3) and (4) will be almost the same. If the solutions of system (3) are oscillatory, then the solutions of system (4) will also be oscillatory. & Thus, in the following we will only discuss the behavior of system (4). According to the method of Baldi and Atiya [3], a system with n delays can be reduced to a system with only one delay by a transformation of variables. So, we assume that t1 ¼ t2 ¼ t3 ¼ t 40. Then, system (4) becomes 8 0 z ðtÞ ¼ z1 ðtÞ þ az2 ðttÞ > < 1 z02 ðtÞ ¼ z2 ðtÞ þ az3 ðttÞ ð5Þ > : z0 ðtÞ ¼ z ðtÞ þ w az ðttÞ þ w az ðttÞ 3

3

1

1

2

2

We can rewrite Eq. (5) as z0 ðtÞ ¼ EzðtÞ þ AzðttÞ

ð6Þ T

where E is the 3  3 identity matrix, z¼(z1, z2, z3) and 0 1 0 a 0 B 0 0 aC A¼@ A aw1 aw2 0 The characteristic equation of (6) is detðlE þ EAelt Þ ¼ 0

ð7Þ

Eq. (7) can be expanded into a transcendental form ðl þ 1Þ3 a3 w1 e3lt a2 w2 ðl þ 1Þe2lt ¼ 0 3

Fig. 1. Schematic representation of the 3-node closed loop system according to (3).

2

ð8Þ

It can be shown that when a w1 +a w2 ¼1, l ¼ 0 is a root of Eq. (8). So, a sufficient condition for the existence of a nontrivial solution of Eq. (8) is a3 w1 þ a2 w2 a 1.

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C. Feng et al. / Neurocomputing 74 (2010) 274–283

From now on, we will adopt the following norm for the vector and matrix of the system (5): JzJ ¼ max1 r i r 3 jzi ðtÞj, P JAJ ¼ max1 r i r 3 3j ¼ 1 jaij j. The measure2mðAÞ of the matrix A is then defined by

mðAÞ ¼ limþ y-0

JEþ yAJ1

y

ð9Þ

which, for the chosen norms, reduces to mðAÞ ¼ max1 r i r 3 P faii þ j a i jaij jg and mðEÞ ¼ 1. Theorem 1. Assuming the following set of conditions for the weights of system (6): (i) w1 a0, (w1+w2)a41 or (w1+w2)ao 1, and a3|w1|+a2|w2|41; (ii) JAJte ¼ aðjw1 j þjw2 jÞte 4et .

R t 0    t e x ðsÞ ds ¼ xðt Þxðt eÞ ¼ xðt Þ 4 0, but on the other hand, 0 x ðtÞ ¼ xðtÞ þ aðjw1 jþ jw2 jÞxðttÞ o 0 ðt A ½t e,t Þ meaning that R t 0 t e x ðsÞ ds o 0, which leads to a contradiction. Consider the following integrals with the assumption that t is sufficiently small, i.e. t o e: Z t t0 þ t

½xðsÞ þaðjw1 j þ jw2 jÞxðstÞ ds Z t

¼

xðsÞ ds þ

t0 þ t

Z t  e ¼

xðsÞ dsþ

t0 þ t

Z

t  e

xðsÞ ds þ

t0 þ t

aðjw1 jþ jw2 jÞxðsÞ dsþ

Z t  t

Proof. Note that if a 4 0 and aðw1 þw2 Þ o 1 or a(w1+w2) 41, then aðjw1 jþ jw2 jÞ 4 1 and JAJ ¼ aðjw1 j þjw2 jÞ. Let us consider the system (6) and suppose that there exists a solution z(t)¼(z1(t), z2(t), z3(t))T which is bounded and nonoscillatory on R. It will then follow that there exists a t0 40 such that no component of z(t) has a zero for t 4t0 þ t. Therefore, jz1 ðtÞj þ jz2 ðtÞj þ jz3 ðtÞj 4 0 ðt 4 t0 þ tÞ. It is clear that for equation dx=dt ¼ x, as x 4 0 then djxj=dt ¼ dx=dt ¼ x ¼ jxj, and if x o 0 then djxj=dt ¼ dðxÞ= dt ¼ x ¼ jxj, thus djxj=dt ¼ jxj always holds. Therefore, for t Zt0 þ t we have 8 djz1 ðtÞj > > > > dt r jz1 ðtÞj þ ajz2 ðttÞj > > > < djz ðtÞj 2 r jz2 ðtÞj þ ajz3 ðttÞj ð10Þ > dt > > > > djz3 ðtÞj > > r jz3 ðtÞj þ ajw1 Jz1 ðttÞj þajw2 Jz2 ðttÞj : dt Setting yðtÞ ¼ jz1 ðtÞj þ jz2 ðtÞj þjz3 ðtÞj, for t Zt0 þ t, one can obtain ð11Þ

Now, for t Z t0 þ t, considering the scalar delay differential equation: ð12Þ

with uðsÞ ¼ yðsÞ, s A ½t0 ,t0 þ t, the Theorem of Differential Inequality [20] gives yðtÞ r uðtÞ

Z t  t

t0

Then all bounded solutions of system (6) are oscillatory on R.

duðtÞ ¼ mðEÞuðtÞ þ JAJuðttÞ ¼ uðtÞ þ aðjw1 j þ jw2 jÞuðttÞ dt

aðjw1 jþ jw2 jÞxðsÞ ds

t0

þ

dyðtÞ r mðEÞyðtÞ þJAJyðttÞ ¼ yðtÞ þ aðjw1 jþ jw2 jÞyðttÞ dt

Z t  t

ð13Þ

In fact, if tn is the first point in the interval ½t0 þ t,1Þ such that yðt  Þ 4 uðt  Þ, and if xðtÞ ¼ yðtÞuðtÞ, we then have dxðtÞ=dt ¼ dyðtÞ=dtduðtÞ=dt ¼ ðyðtÞuðtÞÞ þ aðjw1 jþ jw2 jÞðyðttÞuðttÞÞ ¼ xðtÞ þ aðjw1 j þjw2 jÞxðttÞ. Since xðt  Þ ¼ yðt  Þuðt  Þ 40, there exists, because of the continuity of xðtÞ, a small e 4 0 such that, in the interval D ¼ ðt  e,t  þ eÞ, we have xðtÞ 40 ðt A DÞ and xðt eÞ ¼ 0. Also, in the interval ½t0 ,t  eÞ we have xðtÞ ¼ yðtÞuðtÞ r0. If e o t, then t  e A ½t0 ,t  þ eÞ. In this case, if we integrate both sides of the above representation from t  e to tn, we obtain

Z t t  t

Z t  e t0 þ t

xðsÞ ds

aðjw1 j þjw2 jÞxðsÞ ds

aðjw1 jþ jw2 jÞxðsÞ ds t  e Z t  e ½xðsÞ þ aðjw1 jþ jw2 jÞxðsÞ ds ¼ þ

t0 þ t

Z t  t

þ Z

t  e t0 þ t

½xðsÞ þaðjw1 j þ jw2 jÞxðsÞ ds aðjw1 jþ jw2 jÞxðsÞ dsþ

þ t0

Z t t  t

xðsÞ ds

The value of each of the four integrals on the right hand side can be analyzed to show that their sum is less than or equal to 0. For the first one, remembering that aðjw1 j þ jw2 jÞ 4 1, the inequality ½xðsÞ þaðjw1 j þ jw2 jÞxðsÞ o 0 holds in the interval ½t0 þ t,t  e R t  e and therefore t0 þ t ½xðsÞ þaðjw1 j þjw2 jÞxðsÞ dso 0 also holds. For the second term, since e and t are both sufficiently R t  t small, t e ½xðsÞ þ aðjw1 jþ jw2 jÞxðsÞ ds-0. Also, in the interval ½t0 ,t0 þ t, we have that xðtÞ ¼ 0 and then the third integral is null R t þt (i.e. t00 aðjw1 j þjw2 jÞxðsÞ ds ¼ 0). Finally, for the fourth integral, in the interval D ¼ ðt  e,t  þ eÞ, the inequality xðtÞ 4 0 holds for R t ðt A DÞ which implies that t e xðsÞ ds o0. Therefore, the sum of these four integrals is less than 0 which leads again to a R t contradictory conclusion, namely that t0 þ t ½xðsÞ þ aðjw1 j þ jw2 jÞ xðstÞ dsr 0. Moreover, it can be shown that all bounded solutions of system (6) and (12) have sufficiently large zeros on ½t0 þ t,1Þ. Suppose that this is not the case, then the characteristic equation associated with (12), given by

l ¼ mðEÞ þJAJelt

ð14Þ

  will have a real nonpositive root, say l (If l 4 0, then the system (6) will have an unbounded solution which contradicts the boundedness of solution of system (6).). Thus, we have from (14):    jl j ¼ jmðEÞ þJAJejl jt j Z JAJejl jt 1

ð15Þ

Using the formula ex Zex ðx Z 0Þ, then 1Z

  JAJejl jt JAJet  t  eð1 þ jl jÞt Z JAJet  t  e  ¼  1 þjl j ð1 þ jl jÞ  t

ð16Þ

2

Considering a linear and delayed differential system of the form dxðtÞ=dt ¼ BxðtÞþ AxðttÞ, where t 40 is a constant and A and B denote n  n matrices with, respectively, real constant elements aij and bij (i,j¼1,2,y,n), the P norms of vectors and matrices can be defined as follow: JxðtÞJ ¼ ni¼ 1 jxi ðtÞj; P P JAJ ¼ maxj ni¼ 1 jaij j; JBJ ¼ maxj ni¼ 1 jbij j. In such a case, the measure mðBÞ of the matrix B is defined by mðBÞ ¼ limy-0 þ ðJE þ yBJ1Þ=y where E is the identity matrix. For the chosen norms, this may reduce to mðBÞ ¼ max1 r j r n ½bjj þ P i ¼ 1,j a j jbij j and mðEÞ ¼ 1. The measure of the matrix A can be defined similarly. For more details on the measure of matrix, the reader can consult [43].

The latter inequality means that JAJte ¼ aðjw1 j þ jw2 jÞte ret and this contradicts the condition (ii) of Theorem 1. Now, since u(t) has sufficiently large zeros, then z(t) also has sufficiently large zeros. But we also need to show that these zero points of the bounded solution z(t) are discrete (i.e. that each of these solutions are unstable). As discussed above, only the instability of the trivial

C. Feng et al. / Neurocomputing 74 (2010) 274–283

277

20 15 10

w2

5 0 −5 −10 −15 −20 −20

−15

−10

−5

0 w1

5

10

15

20

20 Fig. 3. Simulink model of the system (6).

15

Letting t-þ 1, e-0, one can get 8 > < s1 r as2 s2 r as3 > : s r ajw js þ ajw js

10

w2

5 0

3

1

2

−10

s2 r a2 ðjw1 js1 þ jw2 js2 Þ

−15

ð20Þ

2

The system (20) is equivalent to ( s1 r as2

−5

−20 −20

1

ð21Þ

Since a 4 0 and a3 jw1 j þ a2 jw2 j 4 1, then the system (21) has nontrivial positive solutions. Thus, there exist a si 40 such that −15

−10

−5

0 w1

5

10

15

20

lim supjzi ðtÞj ¼ si 4 0

t- þ 1

ð22Þ

Fig. 2. Theoretical sustainability of oscillations. The different symbols are used to indicate that a parameter configuration should lead to oscillations according to one of the three theorems (3 : Theorem 1 , .: Theorem 2,  : Theorem 3). (a) t ¼ 2, (b) t ¼ 6.

This means that the null solution of system (20) is unstable. So, z(t) has sufficiently large and discrete zeros. In other words, there exists a sequence ftk ginf k ¼ 1 ðtk otk þ 1 Þ such that z(tk)¼ 0 (k¼1,2,y). This means that z(t) is oscillatory on R, which contradicts the assumptions of a nonoscillatory behavior of z(t). Thus, there exists an oscillatory solution for the system (6). &

solution of system (6) needs to be investigated. Assume that

A supplementary set of conditions can be derived to assure the oscillatory behavior of the system (6). In [36], the relation ð2w2 w1 Þ o8 is derived as a criteria for the stability of the trivial solution of a system without delay. The converse relation, namely ð2w2 w1 Þ 4 8, can be used as a criteria for obtaining unstable solutions for a system without delay. Since an unstable solution in a system without delay cannot be stabilized by the addition of a delay, the solutions related to the last criteria are also unstable in (6). The complete oscillatory conditions based on this assertion are given in the following theorem.

lim supjzi ðtÞj ¼ si

t- þ 1

ði ¼ 1,2,3Þ

ð17Þ

According to the definition of the superior limit, for a sufficiently small constant e 4 0, there exists a t1 4 0 such that, for any large t ðt 4t1 þ 2tÞ, jzi ðtÞj r ð1 þ eÞsi (i¼1,2,3). So as t 4t1 þ 2t, we have 8 0 z ðtÞ þz1 ðtÞ r að1 þ eÞs2 > < 1 z02 ðtÞ þz2 ðtÞ r að1 þ eÞs3 > : z0 ðtÞ þz ðtÞ r ajw jð1 þ eÞs þ ajw jð1 þ eÞs 3 1 1 2 2 3

ð18Þ Theorem 2. Assume that, for the system (6), the following relationships hold:

and we obtain 8 t > < jz1 ðtÞj rað1þ eÞð1e Þs2 jz2 ðtÞj rað1þ eÞð1et Þs3 > : jz ðtÞj rajw jð1 þ eÞð1et Þs þ ajw jð1 þ eÞð1et Þs 3 1 1 2 2

(i) w1 a 0 and ðw1 þw2 Þa o1,ð2w2 w1 Þa 48; (ii) JAJte ¼ aðjw1 j þjw2 jÞte4 et . ð19Þ Then the bounded solutions of this system are oscillatory on R.

C. Feng et al. / Neurocomputing 74 (2010) 274–283

20

20

15

15

10

10

5

5 w2

w2

278

0

0

−5

−5

−10

−10

−15

−15

−20 −20 −15 −10

−5

0

5

10

15

20

−20 −20 −15 −10

−5

w1

0

5

10

15

20

w1

Fig. 4. Stability of the system (6) simulated with Simulink (3 : sustained oscillation, .: nonoscillatory saturated output,  : decaying output). (a) t ¼ 2, (b) t ¼ 6.

Proof. Note that when ð2w2 w1 Þa 4 8, then ðjw1 jþ jw2 jÞa 41, and JAJ is still equivalent to aðjw1 j þjw2 jÞ. From the condition (ii), we know that system (6) has sufficiently large zeros. We will show that these zeros are discrete using a method similar to the proof of Theorem 1. Considering that for large t ðt 4 t1 þ 2tÞ the inequality jzi ðtÞj o ð1 þ eÞsi holds, then, in this case, we can consider the following system: 8 0 z ðtÞ þz1 ðtÞ ¼ að1 þ eÞz2 ðtÞ > < 1 z02 ðtÞ þz2 ðtÞ ¼ að1 þ eÞz3 ðtÞ ð23Þ > : z0 ðtÞ þz ðtÞ ¼ að1 þ eÞw z ðtÞ þ að1 þ eÞw z ðtÞ 3 1 1 2 2 3 The oscillation of system (23) implies the oscillation of (6). The characteristic equation associated with (23) is given by ðl þ 1Þ3 að1þ eÞw2 ðl þ 1Það1 þ eÞw1 ¼ 0

ð24Þ

Taking the limit e-0 and expanding the cubic polynomial (24), 3 2 we get the equivalent form l þ3l þ ð3aw2 Þl þ 1aðw1 þ w2 Þ ¼ 0, on which the Routh–Hurwitz criterion [17] can be directly applied to determine the stability conditions associated to the system (23) without having to resolve explicitly for the roots of (24). This analysis shows that if aðw1 þ w2 Þ o1 and að2w2 w1 Þ 48, the polynomial (24) will have one negative zero and two zeros with positive real parts, implying that the trivial solution of system (23) is unstable. Thus, there exists at least a si 4 0 and this means that the zeros of system (6) are discrete. Therefore, the system (6) generates permanent oscillations. & Theorem 3. Assume the following set of conditions for the system (6): (i)(a) w1 a0 and jlM ðAÞ=lm ðAÞj 4 1, if A has only real eigenvalues; (i)(b) w1 a0 and ð1 þ lÞelt 4 1, if A has complex eigenvalues; (ii) JAJte ¼ aðjw1 j þjw2 jÞte 4et . where lm ðAÞ, lM ðAÞ denote, respectively, the minimum and maximum eigenvalues of matrix A. Then the bounded solutions of system (6) are oscillatory on R. Proof. As explained in the proof of Theorem 1, we know from condition (ii) that system (6) has sufficiently large zeros. We want to show that the trivial solution is unstable. In other words, there is at least one eigenvalue of A which has a positive real part. Since A is a 3  3 matrix, there are two possible situations regarding the configuration of its eigenvalues: (1) all three of them are real; (2) one eigenvalue is real and the other two are complex conjugates. In case (1), because jlM ðAÞ=lm ðAÞj 4 1, there is at least a positive

real eigenvalue of A. In case (2), from Be´lair’s Lemma 2.1 [5], if l is a root of Eq. (7), then there is an eigenvalue d of the matrix A for which d ¼ ð1 þ lÞelt . Since ð1 þ lÞelt 4 1, l must have a positive real part. In both cases, at least one of the eigenvalues of A has a positive real part. Therefore, the trivial solution is unstable and the system (6) is oscillatory on R. & 3. Simulation results To highlight the behavior of system (6) under various sets of conditions, simulations have been performed on Matlab using f ðxÞ ¼ 0:5  ðjx þ1jjx1jÞ as activation function.3 Therefore, a ¼1 in the following. Notice that, in theses these simulations, the time unit can be arbitrarily chosen as long as t and the time axis are expressed in the same unit. In this paper we used the millisecond everywhere. Simulation results are separated in two subsections. The first one discusses how oscillation sustainability is affected by the variation of the neural network parameters (i.e. t, w1 and w2) while the second one shows some examples of time varying signals obtained for a few combinations of the network parameters. 3.1. Oscillation sustainability Three different simulation conditions have been considered to analyze oscillation sustainability. For each of these simulations, the system has been evaluated for every couple (w1, w2) with wi ¼  20, 18, y, 18, 20 and with a time delay of 2 and 6 ms. In the first simulation, the direct application of the inequalities stated in Theorems 1–3 are tested (see Fig. 2). This shows what parameter combinations are analytically known to result in oscillations. However, it should be noticed that the boundedness of solutions of the system (6) is not tested by these inequalities. Therefore, the solutions may not be oscillating if they are not bounded. A second simulation has been performed using a Simulink model (see Fig. 3) to show what parameter combination generates oscillations when numerically tested. For this purpose, two criteria were considered to confirm that the system was oscillating. The first criterion was fulfilled if the ratio of the root 3 Using an iteration method [18,36] to discuss the situation of equilibria is invalid when the activation function takes 0:5  ðjx þ 1jjx1jÞ. This is due to the linearity of the activation function.

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mean square (RMS) value of the last 100t ms of the output signal over the RMS value of its first 100t ms was greater than 0.01. The second criterion was respected if at least four zero crossings were detected in the last 100t ms of the simulation. The simulation was 2000t ms long. When both criteria were met, the parameter combination was considered to generate sustained oscillations. If the first criterion was fulfilled but not the second, the combination resulted in a nonoscillatory saturated output. Finally, if the first condition was not met, the combination generated a decaying output. The results of these simulations are shown in Fig. 4. Notice that a lot of solutions which were indicated in Fig. 2 as oscillatory, if bounded, gives nonoscillatory saturated output. These are due to the unbounded solutions saturated by the bounded activation function. The two preceding simulations show under which conditions the system (6) is stable for different combinations of parameters. However, these simulations are based on an important assumption, namely that the weight parameter can be learned by the recurrent neural networks. A third set of simulations has been conducted to consider the added complications related to weight learning. This simulation has two steps. First, the specified weights are used to derive a teaching signal. Second, this signal is fed to the network which is expected to adapt its weights from initial null values to the weight values specified at the first step. At the end of the simulation, oscillation sustainability is assessed as described in the previous simulation. Results obtained with this scheme are reported in Fig. 5. As can be seen, many more solutions are decaying than what is shown in Fig. 4. This is due to learning difficulties.

3.2. Example of output signals With the same routines used in the third set of simulations of Section 3.1, a few examples of output signals have been generated. The first of these examples (see Fig. 6) depicts the

oscillations for different delays and for weights fixed to w1 ¼  8 and w2 ¼ 3. Notice that in this case, w1 þ w2 o 1 and 2w2 w1 4 8 and therefore condition (i) of Theorem 2 is satisfied. When the time delay changes from 2 to 6 ms, the system start oscillating. As can be shown, an increasing delay results in a slower oscillation frequency. It should be noted that when the time delay is equal to 6 ms, the condition (ii) of Theorem 2 does not hold, but the system is still oscillating. This implies that our theorem defines sufficient conditions for determining oscillations in such a time delay system. Fig. 7 shows the oscillations for the same delay but for different values of the parameters w1 and w2. When the time delay is fixed to 2 ms, and when w1 ¼4 and w2 ¼ 18, the system is stable. However, when we fix w1 ¼ 4 and slightly change the value of the parameter w2 (e.g. w2 ¼  24), the recurrent neural network loses its stability and generates oscillations. For w1 ¼4, w2 ¼  24, nondecaying oscillations are obtained. Note that since w1 þw2 ¼ 20 o 1 and 2w2 w1 ¼ 52 o8, in this case, and according to Lemma 3.1 of [36], the trivial solution would be stable in a system without delay. However, the conditions of Theorem 1 are satisfied for the values w1 ¼4, w2 ¼  24. Therefore, the oscillations in our system were induced by the time delay. Fig. 8 displays oscillations for different negative values of the parameter w1, for t ¼ 2 ms and w2 ¼  1.2. In these conditions, the system generates a permanent oscillation when w1 ¼ 8. Notice that, in this case, the trivial solution is stable for a system without delay because w1 þ w2 ¼ 9:2o 1 and 2w2 w1 ¼ 5:6 o 8. Nevertheless, oscillations appear in our delayed system. This implies that delayed recurrent neural network have a different behavior than its nondelayed counterpart. In fact, two types of oscillations should be distinguished in a delayed neural network. The first type, which could be labeled ‘‘delay-unrelated oscillation’’, happens regardless of the presence of a delay. In the second case, there is ‘‘delay induced oscillation’’. This type of oscillation is possible whenever (1), the trivial solution is stable when t ¼ 0, and (2), there exists a value of t a0

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for which the system generates permanent oscillations. According to Be´lair’s Corollary 2.8 [5], a necessary and sufficient condition for the occurrence of delay induced oscillations in our system is that Reðlmax Þ o1 o jlmax j, where lmax denotes the maximum eigenvalues of matrix A. In this view, when selecting w1 ¼4, w2 ¼  24 in Fig. 7 and calculating the eigenvalues of matrix A, it results that jlmax j ¼ 9:5942 and Reðlmax Þ ¼ 0:1272. Thus, Reðlmax Þ o 1 ojlmax j. Similarly, for w1 ¼  8, w2 ¼  1.2 in Fig. 8, jlmax j ¼ 1:8005 and Reðlmax Þ ¼ 0:8946 is obtained. Therefore Reðlmax Þ o 1 ojlmax j holds again. Thus, according to Be´lair’s result, the delay in the system should induce permanent oscillations, which is indeed the case as shown in Figs. 7 and 8.

4. Conclusions We have proposed three theorems to ensure the existence of oscillations in a 3-node time delay recurrent neural network model with general activation function. Our simulations implie that the above theorems are only sufficient conditions for inducing oscillations in such a time delay system. The condition (ii) of the above theorems guarantees that the system has sufficiently large zero points. If the system has already sufficiently

large zero points under condition (i), then condition (ii) can be neglected. Fig. 6a–c may reflect this case. Our simulations also imply that the construction of the recurrent neural networks is very complex. A little change of its parameter values can induce a large change in the behavior of such a dynamical system. Time delay induced oscillations depends on the values of the parameters. However, it is still unclear under which conditions, time delays will surely induce oscillations, especially, for n-node recurrent networks. This will be the topic of a follow up study. It should also be noted that, although the current study addresses only the case where a common time delay is applied to the neural network (i.e. t1 ¼ t2 ¼ t3 ), this is not a limitation. Indeed, as discussed by Baldi and Atiya in [3], a system with different time delays (i.e. t1 a t2 a t3 ) may be represented in an equivalent form using a common delay. Also, it should be noticed that the oscillation behavior is related to the value of t since it affects the oscillation frequency as it is shown in Fig. 6. From a more general perspective, this work is a part of our research effort to understand how human movements are controlled. So far, we have been successful in describing the global behavior of neuromuscular systems [31–34] and the origin of the typical velocity profiles that are observed in rapid upper limb movements [2,27,42]. The next step is to understand how such a system can be controlled

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by higher level networks. For the production of oscillations, recurrent neural networks seem to be potential candidates and the present study on a 3-node network support that hypothesis.

Acknowledgments This research was supported by Grant RGPIN-915 from the NSERC to Re´jean Plamondon. This project has been carried out while the first author was working as a visiting researcher at Laboratoire Scribens, E´cole Polytechnique de Montre´al.

References [1] S. Arik, V. Tavsanoglu, Equilibrium analysis of delayed CNNs, IEEE Trans. Circuits Syst. I 47 (2000) 571–574. [2] C.G. Atkeson, J.M. Hollerbach, Kinematic features of unrestrained vertical arm movements, J. Neurosci. 5 (1985) 2318–2330. [3] P. Baldi, A.F. Atiya, How delays affect neural dynamics and learning, IEEE Trans. Neural Networks 5 (4) (1994) 612–621. [4] J. Be´lair, S.J. Campbell, V.D. Driessche, Frustration, stability and delay-induced oscillations in a neural network model, SIAM J. Appl. Math. 56 (1) (1996) 245–255. [5] J. Be´lair, Stability in a model of a delayed neural network, J. Dyn. Differential Equations 5 (4) (1993) 607–623. [6] T.G. Barbounis, J.B. Theocharis, Locally recurrent neural networks for long-term wind speed and power prediction, Neurocomputing 69 (4–6) (2006) 466–496. [7] E. Bieberich, Recurrent fraction neural networks: a strategy for the change of local and global information processing in the brain, Biosyst. 66 (3) (2002) 145–164. [8] J. Cao, X. Lin, Study of hourly and daily solar irradiation forecast using diagonal recurrent wavelet neural networks, Energy Conver. Manage. 49 (6) (2008) 1396–1406. [9] J. Cao, Global stability analysis in delayed cellular neural networks, Phys. Rev. E 59 (5) (1999) 5940–5944. [10] G. Cheron, F. Leurs, A. Bengoetxea, J.P. Draye, M. Destre´e, B. Dan, A dynamic recurrent neural network for multiple muscles electromyographic mapping to elevation angle of the lower limb in human locomotion, J. Neurosci. Methods 129 (2003) 95–104. [11] M. Delgado, M.C. Pegalajar, A multiobjective genetic algorithm for obtaining the optimal size of a recurrent neural network for grammatical inference, Pattern Recognition 38 (9) (2005) 1444–1456. [12] N. Djarfour, T. Aifa, K. Baddari, A. Mihoubi, J. Ferahtia, Application of feedback connection artificial neural network to seismic data filtering, C. R. Geoscience 340 (6) (2008) 335–344. [13] K. Doya, S. Yoshizawa, Adaptive neural oscillator using continuous time backpropagation learning, Neural Networks 2 (1989) 375–385. [14] A. Evsukoff, S. Gentil, Recurrent neuro-fuzzy system for fault detection and isolation in nuclear reactors, Adv. Eng. Inf. 19 (1) (2005) 55–66. [15] C. Feng, R. Plamondon, Existence of oscillations in recurrent neural networks with time delay models, DCDIS A Suppl. Adv. Neural Networks 14 (SI) (2007) 111–118. [16] T.D. Ganchev, D.K. Tasoulis, M.N. Vrahatis, N.D. Fakotakis, Generalized locally recurrent probabilistic neural networks with application to text-independent speaker verification, Neurocomputing 70 (2007) 1424–1438. [17] F.R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959. [18] B. Gao, W. Zhang, Equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function, IEEE Trans. Neural Networks 19 (5) (2008) 782–794. [19] S.L. Goh, D.P. Mandic, An augmented CRTRL for complex-valued recurrent neural networks, Neural Networks 20 (10) (2007) 1061–1066. [20] P. Hartman, Ordinary Differential Equations, Birkhauser, Boston, Basel, Stuttgart, 1982 26–27. [21] Q. Hong, J. Peng, Z.B. Xu, Nonlinear measures: a new approach to exponential stability analysis for Hopfield-type neural networks, IEEE Trans. Neural Networks 12 (2001) 360–370. [22] A. Karami-Mollaee, M.R. Karami-Mollaee, A new approach for instantaneous pole placement with recurrent neural networks and its application in control of nonlinear time-varying systems, Syst. Control Lett. 35 (5) (2006) 385–395. [23] C. Li, X. Liao, New algebraic conditions for global exponential stability of delayed recurrent neural networks, Neurocomputing 64 (2005) 319–333. [24] Y. Liu, Z. You, L. Cao, On the almost periodic solution of cellular neural networks with distributed delays, IEEE Trans. Neural Networks 18 (1) (2007) 295–300. [25] N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge University Press, Cambridge, 1989. [26] S. Mohamad, K. Gopasamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput. 135 (2003) 17–38. [27] H. Nagasaki, Asymmetric velocity and acceleration profiles of human arm movements, Exp. Brain Res. 74 (1989) 319–326.

[28] L. Olien, J. Be´lair, Bifurcations, stability and monotonicity properties of a delayed neural network model, Physica D 102 (1997) 349–363. [29] A.G. Parlos, K. Kim, R.M. Bharadwaj, Sensorless detection of mechanical faults in electromechanical systems, Mechatronics 14 (4) (2004) 357–380. [30] B.A. Pearlmutter, Gradient calculations for dynamic recurrent neural networks, IEEE Trans. Neural Networks 6 (5) (1995) 1212–1228. [31] R. Plamondon, A kinematic theory of rapid human movements. Part I. Movement representation and generation, Biol. Cybern. 72 (1995) 295–307. [32] R. Plamondon, A kinematic theory of rapid human movements. Part II. Movement time and control, Biol. Cybern. 72 (1995) 309–320. [33] R. Plamondon, A kinematic theory of rapid human movements: Part III. Kinetic outcomes, Biol. Cybern. 78 (1998) 133–145. [34] R. Plamondon, C.H. Feng, A. Woch, A kinematic theory of rapid human movement. Part IV: a formal mathematical proof and new insights, Biol. Cybern. 89 (2003) 126–138. [35] D.V. Prokhorov, Training recurrent neurocontrollers for robustness with derivative-free Kalman filter, IEEE Trans. Neural Networks 17 (6) (2006) 1606–1616. [36] A. Ruiz, D.H. Owens, S. Townley, Existence, learning, and replication of periodic motion in recurrent neural networks, IEEE Trans. Neural Networks 9 (4) (1998) 651–661. [37] J. Sanchez, D. Erdogmus, M.A.L. Nicolelis, J. Wessberg, J.C. Principe, Interpreting spatial and temporal neural activity through a recurrent neural network brain–machine interface, IEEE Trans. Neural Syst. Rehabil. Eng. 2 (13) (2005) 213–219. [38] J.S. Shieh, C. Chou, S. Huang, M. Kao, Intracranial pressure model in intensive care unit using a simple recurrent neural network through time, Neurocomputing 57 (2004) 239–256. [39] J.B. Theocharis, A high-order recurrent neuro-fuzzy system with internal dynamics: application to the adaptive noise cancellation, Fuzzy Set Syst. 157 (4) (2006) 471–500. [40] S. Townley, A. Ilchmann, M.G. Weib, W. Mcclements, A.C. Ruiz, D.H. Owens, D. Pratzel-Wolters, Existence and learning of oscillations in recurrent neural networks, IEEE Trans. Neural Networks 11 (1) (2000) 205–214. [41] E.D. Ubeyli, Recurrent neural networks employing Lyapunov exponents for analysis of Doppler ultrasound signals, Expert Syst. Appl. 34 (4) (2008) 2538–2544. [42] Y. Uno, M. Kawato, R. Suzuki, Formation and control of optimal trajectory in human multijoint arm movement, Biol. Cybern. 61 (1989) 89–101. [43] M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, New Jersey, 1978. [44] G.D. Wu, C.T. Lin, A recurrent neural fuzzy network for word boundary detection in variable noise-level environments, IEEE Trans. Syst. Man Cybern. Part B Cybern. 31 (1) (2001) 84–97. [45] J. Xu, Y.Y. Cao, Y. Sun, J. Tang, Absolute exponential stability of recurrent neural networks with generalized activation function, IEEE Trans. Neural Networks 19 (6) (2008) 1075–1089. [46] H. Yang, T.S. Dillon, Exponential stability and oscillation of Hopfield graded response neural networks, IEEE Trans. Neural Networks 5 (5) (1994) 719–729. [47] H. Yousefi, M. Hirvonen, H. Handroos, A. Soleymani, Application of neural network in suppressing mechanical vibration of a permanent magnet linear motor, Control Eng. Prac. 16 (7) (2008) 787–797. [48] W. Yu, J. Cao, Stability and Hopf bifurcations on a two-neuron system with time delay in the frequency domain, Int. J. Bifur. Chaos 17 (4) (2007) 1355–1366. [49] W. Yu, J. Cao, Stability and Hopf bifurcations analysis on a four-neuron BAM neural network with time delays, Phys. Lett. A 351 (1) (2007) 64–78.

Chunhua Feng received his Ph.D. (2005) degree from the E´cole Polytechnique de Montre´al. He is currently an associate professor of Mathematics at the Guangxi Normal University of China. His present research interests include periodic solutions and almost periodic solutions of time delay systems, stability and oscillation of recurrent neural networks, and mathematical biology.

Christian O’Reilly received B.Ing. in Electrical Engineering (2007) and is currently pursuing a Ph.D. degree in Biomedical Engineering at the E´cole Polytechnique de Montre´al. He is a student member of the Institute of Electrical and Electronics Engineers. His principal research interests are related to the modeling of the human movements, the characterization of neuromuscular systems, and the development of computerized tool for motor dysfunction diagnoses.

C. Feng et al. / Neurocomputing 74 (2010) 274–283 Re´jean Plamondon received a B.Sc. degree in Physics, and M.Sc.A. and Ph.D. degrees in Electrical Engineering from Universite´ Laval, Que´bec, P.Q., Canada, in 1973, 1975 and 1978, respectively. In 1978, he joined the faculty of the E´cole Polytechnique, Universite´ de Montre´al, Montre´al, P.Q., Canada, where he is currently a Full Professor. He has been the Head of the Department of Electrical and Computer Engineering from 1996 to 1998 and the Chief Executive Officer of E´cole Polytechnique from 1998 to 2002. He is now the Head of Laboratoire Scribens at this institution. Over the last 30 years, Professor Plamondon has been involved in many pattern recognition projects, particularly in the field of on-line and off-line handwriting analysis and processing. He has proposed many original solutions, based on exhaustive studies of human movements generation and perception, to problems related to the design of automatic systems for signature verification and handwriting recognition, as well as interactive electronic penpads to help children learning handwriting and powerful methods for analyzing and interpreting neuromuscular signals. His main contribution has been the development of a kinematic theory of rapid human movements which can take into account, with the help of a unique basic equation

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called a delta-lognormal function, the major psychophysical phenomena reported in studies dealing with rapid movements. The theory has been found successful in describing the basic kinematic properties of velocity profiles as observed in finger, hand, arm, head and eye movements. Professor Plamondon has studied and analyzed these biosignals extensively in order to develop creative and powerful methods and systems in various domains of engineering. Full member of the Canadian Association of Physicists, the Ordre des Inge´nieurs du Que´bec, the Union national des e´crivains du Que´bec, Dr Plamondon is an also active member of several international societies. He is a Fellow of the Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS; 1989), of the International Association for Pattern Recognition (IAPR; 1994) and of the Institute of Electrical and Electronics Engineers (IEEE; 2000). From 1990 to 1997, he was the President of the Canadian Image Processing and Pattern Recognition Society and the Canadian representative on the board of Governors of IAPR. He has been the President of the International Graphonomics Society (IGS) from 1995 to 2007. He has been involved in the planning and organization of numerous international conferences and workshops and has worked with scientists from many countries. He is the author or co-author of more than 300 publications and owner of four patents. He has edited or co-edited four books and several Special Issues of scientific journals. He has also published a children book, a novel and three collections of poems.