Hopf bifurcation in a general n-neuron ring network with n time delays

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1 Iranian Journal of Numerical Analysis and Opimizaion Vol 4, No. 2, (214), pp 15-3 Hopf bifurcaion in a general n-neuron ring newor wih n ime delays E. Javidmanesh and M. Khorshidi Absrac In his paper, we consider a general ring newor consising of n neurons and n ime delays. By analyzing he associaed characerisic equaion, a classificaion according o n is presened. I is invesigaed ha Hopf bifurcaion occurs when he sum of he n delays passes hrough a criical value. In fac, a family of periodic soluions bifurcae from he origin, while he zero soluion loses is asympoically sabiliy. To illusrae our heoreical resuls, numerical simulaion is given. Keywords: Ring newor; Sabiliy; Periodic soluion; Hopf bifurcaion; Time delay. 1 Inroducion Since Hopfield consruced a simplified neural newor (NN) model [7, 15], he dynamic behaviors (such as sabiliy, periodic oscillaory, limi cycles, bifurcaion and chaos) of coninuous-ime neural newors have received much aenion due o heir applicaions in opimizaion, signal processing, image processing, solving nonlinear algebraic equaions, paern recogniion, associaive memories and so on (see, [3, 1, 11, 18] and references herein). I is well nown ha ime delays exis in he signal ransmission, hus Marcus and Weservel proposed an NN model wih delays, based on he Hopfield NN model [12]. The ime delays are regarded as parameers; consequenly, periodic soluions ofen appear as soluions of delay differenial equaions (DDEs) hrough Hopf bifurcaion. For example, exisence of periodic soluions, in a special ype of DDEs, has been discussed in [9]. In [8, 16, 17], he auhors used Hopf bifurcaion heory o sudy some inds of neural newors. Corresponding auhor Received 1 February 214; revised 2 May 214; acceped 22 June 214 E. Javidmanesh Deparmen of Applied Mahemaics, Faculy of Mahemaical Sciences, Ferdowsi Universiy of Mashhad, Mashhad, Iran M. Khorshidi Deparmen of Applied Mahemaics, Faculy of Mahemaical Sciences, Ferdowsi Universiy of Mashhad, Mashhad, Iran. Mohsen.horshidi89@gmail.com 15

2 16 E. Javidmanesh and M. Khorshidi A ring newor is a newor opology in which each node connecs o exacly wo oher nodes, forming a single coninuous pahway for signals hrough each node a ring. Daa ravel from node o node, wih each node along he way handling every pace. Ring newors have been found in a variey of neural srucures such as cerebellum [5], and even in chemisry and elecrical engineering. In he field of neural newors, rings are sudied o gain insigh ino he mechanisms underlying he behavior of recurren newors. In [16], Wang and Han invesigaed he coninuous-ime bidirecional ring newor model. Many resuls have been repored in he lieraure on he dynamics of ring neural newors (see [2, 18]). The local and global sabiliy of ring newors wih delays have been discussed in [2, 18]. In [1], sabiliy analysis of a delayed ring newor model has been discussed. In his paper, we sudy Hopf bifurcaion for a ind of DDE sysem. In fac, we consider a general ring newor wih n neurons and n ime delays, which is described by he following DDE sysem: ẋ 1 () = r 1 x 1 () + g 1 (x 1 ()) + f 1 (x n ( τ n )), ẋ 2 () = r 2 x 2 () + g 2 (x 2 ()) + f 2 (x 1 ( τ 1 )), ẋ 3 () = r 3 x 3 () + g 3 (x 3 ()) + f 3 (x 2 ( τ 2 )), (1). ẋ n () = r n x n () + g n (x n ()) + f n (x n 2 ( τ n 2 )), ẋ n () = r n x n () + g n (x n ()) + f n (x n ( τ n )), where r i (i = 1, 2,..., n) denoes he sabiliy of inernal neuron processes, and x i () (i = 1, 2,..., n) represens he sae of he ih neuron a ime. f i and g i (i = 1, 2,..., n) are he acivaion funcion and nonlinear feedbac funcion, respecively. Also, τ i (i = 1, 2,...n) describes he synapic ransmission delay. Figure 1 shows sysem (1) schemaically. Figure 1: A general n-neuron ring wih n delays

3 Hopf bifurcaion in a general n-neuron ring newor wih n ime delays 17 The properies of periodic soluions are also imporan in many applicaions. In fac, various local periodic soluions can arise from he differen equilibrium poins of ring newors by applying Hopf bifurcaion echnique, herefore he sudy of Hopf bifurcaion is very imporan. In [16], he simplified hree-neuron bidirecional ring newor has been invesigaed. Fourneuron and five-neuron newors wih muliple delays have been sudied in [8, 17]. In his paper, we discuss Hopf bifurcaion on sysem (1) generally, no for a paricular n. To he bes of our nowledge, i has no been done before. We would lie o poin ou ha in [1], alhough sabiliy analysis of he sysem has been presened, bu he auhors didn discuss Hopf bifurcaion on he sysem in all cases. Firs, we ae he sum of he delays τ 1 + τ τ n as a parameer τ. By classifying based on n, we sudy he associaed characerisic equaion. To calculae he criical value τ for Hopf bifurcaion, we generalize and modify he mehods proposed in [8, 9]. Then we will show ha he zero soluion loses is sabiliy and Hopf bifurcaion occurs when τ passes hrough he criical value τ. We would lie o poin ou ha i is he firs ime o deal wih Hopf bifurcaion analysis of sysem (1). This paper is organized in five secions. In Sec.2, we give he necessary preliminaries. In Sec.3, we will sudy Hopf bifurcaion on sysem (1). To illusrae he resuls, some numerical simulaions are presened in Sec.4. Finally, in Sec.5, some main conclusions are saed. 2 Preliminaries 2.1 Delay Differenial Equaion There is always a ime delay in many naural phenomena, because a finie ime is required o sense informaion and hen reac o i. DDEs are differenial equaions in which he derivaives of some unnown funcions a presen ime are dependen on he values of he funcions a previous imes. A general delay differenial equaion for x() R n aes he form ẋ() = f(, x(), x ), (2) where x (θ) = x( + θ) and τ θ. Observe ha x (θ) wih τ θ represens a porion of he soluion rajecory in a recen pas. In his equaion f is a funcional operaor from R R n C 1 (R, R n ) o R n. Similar o ODEs, many properies of linear DDEs can be characerized and analyzed using he characerisic equaion. The linearizaion of sysem (2) a he equilibrium poin x is ẋ() = A x() + A 1 x( τ 1 ) + + A m x( τ m ), (3)

4 18 E. Javidmanesh and M. Khorshidi where A j = D j+1 f(x,..., x ), (j =, 1,..., m) and D j f is he Jacobian of f corresponding o is jh componen. Subsiude x() = e λ v, v R n ino (3), we have [λi A m A j e λτ j ]e λ v =. j=1 Therefore he characerisic equaion associaed wih (2) is For furher deails, see [4]. de(λi A m A j e λτ j ) =. (4) j=1 2.2 Hopf bifurcaion In his secion, we sudy bifurcaions ha occur in C 1 sysem ẋ = f(x, µ), (5) depending on a parameer µ R, a nonhyperbolic equilibrium poins. In he following, we will give definiion of a srucurally sable vecor field or dynamical sysem. Definiion 2.1. Le E be an open subse of R n. A vecor field f C 1 (E) is said o be srucurally sable if here is an ε > such ha for all g C 1 (E) wih f g < ε, f and g are opologically equivalen on E; i.e., here is a homeomorphism H : E E which maps rajecories of ẋ = f(x) ono rajecories of ẋ = g(x), and preserves heir orienaion by ime. In his case, we also say ha he dynamical sysem ẋ = f(x) is srucurally sable. If a vecor field f C 1 (E) is no srucurally sable, hen f is said o be srucurally unsable. The qualiaive behavior of he soluion se of sysem (5) depending on a parameer µ R, changes as he vecor field f passes hrough a poin in he bifurcaion se or as he parameer µ varies hrough a bifurcaion value µ. A value µ of he parameer µ in equaion (5) for which he C 1 vecor field f(x, µ ) is no srucurally sable is called a bifurcaion value. For more informaion see [13]. Theorem 2.2. Suppose ha sysem (5), has an equilibrium (x, µ ) a which he following properies are saisfied: a : D x f µ (x ) has a simple pair of pure imaginary eigenvalues and oher eigenvalues have negaive real pars. Therefore, here exis a smooh

5 Hopf bifurcaion in a general n-neuron ring newor wih n ime delays 19 b : curve of equilibria (x(µ), µ) wih x(µ ) = x. The eigenvalues λ(µ), λ(µ) of D x f µ (x(µ)) which are imaginary a µ = µ vary smoohly wih µ. Furhermore, if, d dµ (Reλ(µ)) µ=µ = d, hen Hopf bifurcaion will occur. Proof. see [6]. When he parameer µ passes hrough he criical value µ, according heorem 2.2, Hopf bifurcaion occurs. In fac, a family of periodic soluions appear or disappear. If he equilibrium poin is asympoically sable for µ < µ, when µ passes hrough µ, a family periodic soluions bifurcae from he equilibrium poin. In his case, we say ha supercriical Hopf bifurcaion occurs. Bu, if a branch of periodic soluions exis for µ < µ and while µ passes hrough µ, hese periodic soluions disappear, we say ha subcriical Hopf bifurcaion happens. For more deails, see [6]. 3 Main resuls To esablish he main resuls for sysem (1), i is necessary o mae he following assumpion f i, g i C 1, f i () = g i () =, for i = 1, 2,..., n. (6) I is easily seen ha he origin (,,...,) is an equilibrium poin of (1). Under he hypohesis (6), he linearizaion of (1) around he origin gives ẋ 1 () = 1 x 1 () + f 1()x n ( τ n ), ẋ 2 () = 2 x 2 () + f 2()x 1 ( τ 1 ),. ẋ n () = n x n () + f n()x n 2 ( τ n 2 ), ẋ n () = n x n () + f n()x n ( τ n ), where i = r i g i (), i = 1, 2,..., n. The characerisic equaion of (2) is where λ n + a 1 λ n a n λ + a n + be λτ =, (8) (7)

6 2 E. Javidmanesh and M. Khorshidi Denoe a 1 = n i, a 2 = i=1 1 i<j n i j,, a n = i j l... m, 1 i < j < l <... < m n }{{} n-1 n n n a n = i, b = f i (), τ = τ i. (9) i=1 hen equaion (8) becomes i=1 i=1 p(λ) = λ n + a 1 λ n a n λ + a n, p(λ) + be λτ =. (1) To sudy Hopf bifurcaion, i is necessary o discuss he exisence of pure imaginary roos of (1). Leing λ = iω, and subsiuing his ino (1), we have A + ib + b(cosωτ isinωτ) =, (11) where A = Re{p(iω)}, B = Im{p(iω)}. (12) Separaing he real and imaginary pars of (11), we ge and We rewrie he equaions (13) and (14), as follows: and A + bcosωτ =, (13) B bsinωτ =. (14) A = bcosωτ, (15) B = bsinωτ. (16) Squaring boh sides of (15) and (16), and adding hem up gives A 2 + B 2 = b 2. (17) Now, according o (12), i is easy o use compuer o calculae he roos of (17). Then we can ge he ime delay τ, by subsiuing ω in (15). To find he soluions for equaion (17), we consider he following cases: Case (a) : n = 4 ( N), Case (b) : n = ( N),

7 Hopf bifurcaion in a general n-neuron ring newor wih n ime delays 21 Case (c) : n = ( N {}), Case (d) : n = ( N {}), In case (a), from (12) and he definiion of p(λ), we can ge { A = ω n a 2 ω n 2 + a 4 ω n a n, B = a 1 ω n + a 3 ω n 3 a 5 ω n a n ω. (18) Using (17) and (18), we obain (ω n a 2 ω n 2 + a 4 ω n a n ) 2 + ( a 1 ω n + a 3 ω n 3 a 5 ω n a n ω) 2 = b 2. (19) In case (b), he definiion of p(λ) and (12) lead o { A = a 1 ω n a 3 ω n 3 + a 5 ω n a n, B = ω n a 2 ω n 2 + a 4 ω n 4 (2)... + a n ω. Subsiuing (2) in (17) gives (a 1 ω n a 3 ω n 3 +a 5 ω n 5...+a n ) 2 +(ω n a 2 ω n 2 +a 4 ω n 4...+a n ω) 2 = b 2. (21) In case (c), A and B can be calculaed as follows: { A = ω n + a 2 ω n 2 a 4 ω n a n, B = a 1 ω n a 3 ω n 3 + a 5 ω n a n ω. (22) From he equaions (17) and (22), we have ( ω n + a 2 ω n 2 a 4 ω n a n ) 2 + (a 1 ω n a 3 ω n 3 + a 5 ω n a n ω) 2 = b 2. (23) In case (d), from he definiion of p(λ) and (12), we can compue { A = a 1 ω n + a 3 ω n 3 a 5 ω n a n, B = ω n + a 2 ω n 2 a 4 ω n 4 (24) a n ω. By using he equaions (17) and (24), we can ge ( a 1 ω n +a 3 ω n 3 a 5 ω n a n ) 2 +( ω n +a 2 ω n 2 a 4 ω n a n ω) 2 = b 2. (25) In all he above cases, afer simplificaion, we can easily see ha he equaions (19), (21), (23) and (25) lead o ω 2n + e 1 ω 2n 2 + e 2 ω 2n e n ω 2 + e n =, (26) where he coefficiens e i (i = 1, 2,..., n), can be calculaed as follows:

8 22 E. Javidmanesh and M. Khorshidi e 1 = a 2 1 2a 2, e 2 = a 2 2 2a 1 a 3,, e n = a 2 n 2a n a n 2, e n = a 2 n b 2. (27) Leing z = ω 2, hen equaion (26) becomes z n + e 1 z n + e 2 z n e n z + e n =. (28) Thus, he fac ha equaion (28) has posiive roos is a necessary condiion for he exisence of he pure imaginary roos of equaion (8). Le h(z) = z n + e 1 z n + e 2 z n e n z + e n. (29) In he following, we will give lemma o esablish he disribuion of posiive real roos of equaion (28). Lemma 3.1. If e n <, hen equaion (12) has a leas one posiive roo. Proof. Since h() = e n < and lim z h(z) = +. Hence, here exiss a z > such ha h(z ) =. Now, from (15) we ge { 1 (cos ω ( A τ (j) = 1 (2π cos ω ( A b ) + 2jπ) if B b b ) + 2jπ) if B b (j =, ±1, ±2,...; = 1, 2,..., n), (3) < where w = z, and wihou loss of generaliy, z posiive roos of (28). Therefore, we can define ( = 1,..., n) are he τ = τ () { ()} = min τ, ω = ω. (31) {1,2,...,n} So, wih he help of relaions (17) and (31), ω and τ are obained. Now, we show ha sysem (1) undergoes Hopf bifurcaion a he origin when τ = n i=1 τ i passes hrough τ. In all cases, he sabiliy and Hopf bifurcaion can be analyzed analogously. Le λ(τ) = α(τ) + iω(τ), (32) be he roo of equaion (8) near τ = τ (j) Then, he following lemma holds. saisfying α(τ (j) (j) ) =, ω(τ ) = ω. Lemma 3.2. Suppose h (z ), where h(z) is defined by (4) and z = ω2. Then ±iω is a pair of simple purely imaginary roos of equaion (3) when τ = τ (j). Moreover, dre ( λ(τ) ) dτ τ=τ (j). Proof. From (17) and (12), equaion (26) can be ransformed ino he following form

9 Hopf bifurcaion in a general n-neuron ring newor wih n ime delays 23 p(iω)p(iω) b 2 =. (33) From (29), we have h(ω 2 ) = p(iω)p(iω) b 2. (34) Differeniaing boh sides of equaion (34) wih respec o ω, we obain 2ωh (ω 2 ) = i[p (iω)p(iω) p(iω)p (iω)]. (35) If iω is no simple, hen ω mus saisfy ha implies d (j) [p(λ) + be λτ ] λ=iω =, dλ p (iω ) + b( τ (j) Togeher wih equaion (1), we have Thus, by (33), (35) and (36), we obain Im(τ (j) ) = Im { )e iτ (j) ω =. τ (j) = p (iω ) p(iω ). (36) (iω p ) } { = Im p(iω ) (iω p )p(iω ) } p(iω )p(iω ) = i p (iω )p(iω ) p (iω )p(iω ) 2p(iω )p(iω ) = ω h (ω 2 ) p(iω ) 2. Since τ (j) is real, i.e. Im(τ (j) ) =, we have (z h ) = (ω 2 h ) =. Therefore, we ge a conradicion o he condiion h (z ). This proves he firs conclusion in he lemma. Differeniaing boh sides of equaion (1) wih respec o τ, we obain which implies dλ(τ) dτ [ p (λ) bτe λτ ] dλ dτ bλe λτ =, bλ = p (λ)e λτ bτ = bλ [ p (λ)e λτ bτ ] p (λ)e λτ bτ 2 = λ [ p (λ)p(λ) b 2 τ ] p (λ)e λτ bτ 2. I follows ogeher wih (35) ha

10 24 E. Javidmanesh and M. Khorshidi d(reλ(τ)) Re {λ [ p (λ)p(λ) b 2 τ ]} τ=τ (j) dτ τ=τ = p (λ)e λτ bτ 2 (j) = iω [ + p (iω )p(iω ) p (iω )p(iω ) ] 2 p (iω )e iω τ (j) bτ (j) 2 = = ω 2 h (ω 2 ) p (iω )e iω τ (j) ω 2 h (z ) p (iω )e iω τ (j) bτ (j) 2. bτ (j) 2 Thus, dre(λ(τ)) (j) dτ τ=τ. This complees he proof. By he well-nown Rouh-Hurwiz crieria, we can find he following se of condiions: a H 1 =de ( ( ) ) a1 1 a 3 a 2 a 1... a 1 >, H2 =de >,..., H a 3 a n =de >.... a n (37) To discuss he disribuion of he roos of he exponenial polynomial equaion (8), we need he following resul from Ruan and Wei [14]. Theorem 3.3. Consider he exponenial polynomial p(λ, e λτ1,..., e λτm )=λ n + p () 1 λn +.. +p () n λ+p() n +[p (1) 1 λn +...+p (1) n λ+p(1) n ]e λτ [p (m) 1 λ n p (m) n λ + p(m) n ]e λτ m, where τ i (i = 1, 2,..., m), and p (i) j (i =, 1, 2,..., m; j = 1, 2,..., n) are consans. As (τ 1, τ 2,..., τ m ) vary, he sum of he orders of he zeros of p(λ, e λτ 1,..., e λτ m ) on he open righ half plane can change only if a zero appears on or crosses he imaginary axis. Now, we can sae he following main heorem: Theorem 3.4. Suppose ha condiions (1), (5) and h (z ) hold, where h(z) is defined by (4) and z = ω2. Then, sysem (1) undergoes Hopf bifurcaion a he origin when τ = n i=1 τ i passes hrough τ, and i has a branch of periodic soluions bifurcaing from he zero soluion near τ = τ, where τ is defined by (31). Proof. By he well-nown Rouh-Hurwiz crieria, we can conclude ha when (37) holds, all he roos of equaion (8) a τ =, have negaive real pars.

11 Hopf bifurcaion in a general n-neuron ring newor wih n ime delays 25 Hence, by using heorem 3.3, we conclude ha he zero soluion of sysem (1) is asympoically sable when τ < τ. By using lemma 3.2, we can see ha he condiions of Hopf bifurcaion are saisfied a τ = τ in sysem (1), and so Hopf bifurcaion occurs a he origin. In addiion, a family of periodic soluions appear as τ passes hrough τ. 4 Numerical simulaions In his secion, numerical simulaions are presened o suppor our heoreical resuls. We will sudy sysem (1) for he case n = 7. Consider he following sysem ẋ 1 () = 2x 1 () + anh(x 1 ()) + 2anh(x 7 ( τ 7 )), ẋ 2 () = 2x 2 () + anh(x 2 ()) + anh(x 1 ( τ 1 )), ẋ 3 () = 2x 3 () + anh(x 3 ()) + 1.2anh(x 2 ( τ 2 )), ẋ 4 () = 2x 4 () + anh(x 4 ()) + anh(x 3 ( τ 3 )), ẋ 5 () = 2x 5 () + anh(x 5 ()).5anh(x 4 ( τ 4 )), ẋ 6 () = 2x 6 () + anh(x 6 ()) +.5anh(x 5 ( τ 5 )), ẋ 7 () = 2x 7 () + anh(x 7 ()) + 2anh(x 6 ( τ 6 )), (38) which has he origin as an equilibrium poin. By equaions (8) and (9), we obain he associaed characerisic equaion for n = 7: λ 7 + 7λ λ λ λ λ 2 + 7λ e λτ =. (39) By he equaions (3) and (31), we can compue τ = Choosing τ 1 =.8, τ 2 = 1, τ 3 =.6, τ 4 = 1, τ 5 = 1.1, τ 6 =.7 and τ 7 = 1, Figure 2 shows ha he origin is asympoically sable, and he phase porrais for sysem (38) are shown in Figure 4. When τ = 7 i=1 τ i passes hrough he criical value τ = 6.767, he origin loses is sabiliy and Hopf bifurcaion occurs, i.e., a family of periodic soluions bifurcae from he origin. Choosing τ 1 =.9, τ 2 = 1, τ 3 =.8, τ 4 = 1, τ 5 = 1.3, τ 6 =.7 and τ 7 = 1.1, Hopf bifurcaion happens a he zero soluion. In Figure 3, he bifurcaing periodic soluions are presened, and he phase porrais for sysem (38) are shown in Figure 5. Hopf bifurcaion is supercriical and he bifurcaing periodic soluions exis for τ > τ.

12 26 E. Javidmanesh and M. Khorshidi x1 x3 x x2 x4 x x Figure 2: The origin is asympoically sable while τ 1 =.8, τ 2 = 1, τ 3 =.6, τ 4 = 1, τ 5 = 1.1, τ 6 =.7 and τ 7 = 1 5 Concluions In his paper, we invesigaed he dynamics of a general class of ring newors wih n neurons and n ime delays. We have chosen τ = τ 1 +τ τ n as a bifurcaion parameer and analyzed he corresponding characerisic equaion. Then we have proved ha he zero soluion (he equilibrium poin of he

13 Hopf bifurcaion in a general n-neuron ring newor wih n ime delays 27 x1 x3 x x2 x4 x x Figure 3: A family of periodic soluions bifurcaes from he origin when τ 1 =.9, τ 2 = 1, τ 3 =.8, τ 4 = 1, τ 5 = 1.3, τ 6 =.7 and τ 7 = 1.1 sysem) loses is sabiliy and Hopf bifurcaion occurs. Therefore, a family of periodic soluions bifurcae from he zero soluion when τ passes hrough a criical value τ. Finally, he resuls have been validaed by numerical simulaions. A he end, we would lie o poin ou ha i is so significan o sudy he newors in general case, no for a special value of n. Alhough, in his

14 28 E. Javidmanesh and M. Khorshidi.5.5 x3 x x x2 x x1.5 x7 x x5 x5 x1 x2 Figure 4: The phase porrais while τ1 =.8, τ2 = 1, τ3 =.6, τ4 = 1, τ5 = 1.1, τ6 =.7 and τ7 = x7 x x6.4.4 x x2.5 x1.5 x7 x x5.5.5 x1 x5.5 x2 Figure 5: The phase porrais when τ1 =.9, τ2 = 1, τ3 =.8, τ4 = 1, τ5 = 1.3, τ6 =.7 and τ7 = 1.1 paper, we considered a general ind of ring newors, he mehods, we have proposed can be generalized o be applied for oher inds of neural newors. We leave his as he fuure research.

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17 بررسی انشعاب هاف در شبکه حلقه ای n نرونی با n تاخیر زمانی الهام جاویدمنش و محسن خورشیدی دانشگاه فردوسی مشهد دانشکده علوم ریاضی گروه ریاضی کاربردی چکیده : در این مقاله یک شبکه حلقه ای n نرونی با n تاخیر زمانی را در نظر می گیریم و به مطالعه ی انشعاب هاف سیستم حاصل از این شبکه می پردازیم. با طبقه بندی بر حسب باقیمانده های تقسیم n بر ۴ به مطالعه ی معادله مشخصه حاصل از این دسته بندی می پردازیم. مجموع تاخیرها را به عنوان پارامتر سیستم در نظر می گیریم. نشان می دهیم وقتی پارامتر تاخیر از یک مقدار بحرانی می گذرد انشعاب هاف رخ می دهد. در واقع زمانی که نقطه تعادل سیستم (جواب صفر) پایداری مجانبی اش را از دست می دهد یک خانواده از جواب های دوره ای از مبدا منشعب می شوند. کلمات کلیدی : شبکه حلقه ای پایداری جواب های دوره ای انشعاب هاف تاخیر زمانی.