We consider a ring of identical neurons with delayed nearest neighborhood inhibitory interaction. Under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay di erential equation with negative feedback. Despite the fact that the slowly oscillatory periodic solution of the scalar equation is stable, we show that the associated synchronous solution is unstable if the size of the network is large

Chen, Yuming

Huang, Ying Sue

Wu, Jianhong

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Desynchronization of Large Scale Delayed Neural Networks

Wilfrid Laurier University

Wilfrid Laurier University Scholars Commons @ Laurier Mathematics Faculty Publications Mathematics 2000 Desynchronization of Large Scale Delayed Neural Networks Yuming Chen Wilfrid Laurier University, ychen@wlu.ca Ying Sue Huang Pace University Jianhong Wu York University Follow this and additional works at: https://scholars.wlu.ca/math_faculty Recommended Citation Chen, Yuming; Huang, Ying Sue; and Wu, Jianhong, "Desynchronization of Large Scale Delayed Neural Networks" (2000). Mathematics Faculty Publications. 22. https://scholars.wlu.ca/math_faculty/22 This Article is brought to you for free and open access by the Mathematics at Scholars Commons @ Laurier. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact scholarscommons@wlu.ca. PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 128, Number 8, Pages 2365–2371S 0002-9939(00)05635-5Article electronically published on February 25, 2000DESYNCHRONIZATION OF LARGE SCALEDELAYED NEURAL NETWORKSYUMING CHEN, YING SUE HUANG, AND JIANHONG WU(Communicated by Hal L. Smith)Abstract. We consider a ring of identical neurons with delayed nearest neigh-borhood inhibitory interaction. Under general conditions, such a network hasa slowly oscillatory synchronous periodic solution which is completely charac-terized by a scalar delay differential equation with negative feedback. Despitethe fact that the slowly oscillatory periodic solution of the scalar equation isstable, we show that the associated synchronous solution is unstable if the sizeof the network is large.1. IntroductionWe consider a ring of identical neurons with delayed nearest neighborhood in-hibitory interaction. The dynamics of such a network is described by the followingsystem of delay differential equationsx˙i(t) = −µxi(t)− 12[f(xi−1(t− r)) + f(xi+1(t− r))] ,(1)where i (mod n) for a given positive integer n, µ and r are positive real constantsand the activation function f : R→ R is bounded and C1-smooth with f(0) = 0 andf ′(x) > 0 for all x ∈ R. Such a system is a special case of the well-known Hopfield’smodel for a network of amplifiers (neurons) which, as an electronic device, wasdesigned as a hardware implementation of biological neural networks (Hopfield [6]).The delay was first explicitly introduced by Marcus and Westervelt [9] to accountfor the finite switching speed of amplifiers. It should be mentioned that the constantr here is not the absolute size of the time lag required for the communication andresponse among neurons. In fact, system (1) is obtained after some rescaling andreparametrization, and the constant r represents the ratio of the absolute size ofthe delay over the relaxation time of the system (see, for example, Belair, Campbelland van den Driessche [1], Marcus and Westervelt [9] and Wu [11]). Hence, thisconstant can be relatively large, and in such a case the dynamics of system (1)can be significantly different from that of the corresponding ordinary differentialequation model.Received by the editors September 19, 1998.2000 Mathematics Subject Classification. Primary 34K15, 92B20.Key words and phrases. Synchronization, slow oscillation, delay, neural network.This research was partially supported by the Natural Sciences and Engineering Research Coun-cil of Canada.c©2000 American Mathematical Society23652366 YUMING CHEN, YING SUE HUANG, AND JIANHONG WUThe focus of this paper is the desynchronization of system (1) with large n. Forsystem (1), we say that a solution x = (x1, · · · , xn)T : [−r,∞)→ Rn is synchronousif x1(t) = · · · = xn(t) for all t ∈ [−r,∞), and asynchronous if otherwise. Similarly,we can speak of synchronous or asynchronous states in the phase space Cn =C([−r, 0];Rn).A synchronous solution is completely characterized by the following scalar delaydifferential equation with negative feedbackz˙(t) = −µz(t)− f(z(t− r))(2)which has been extensively investigated in the literature. In particular, it is shownthat for r in a certain range equation (2) has a slowly oscillatory periodic solution.Here and in what follows, a slowly oscillatory periodic solution of equation (2) is aperiodic solution p : R→ R of (2) such that distances of consecutive zeros are largerthan r, and the minimal period ω is the distance of 3 consecutive zeros. Clearly,ω > 2r. The stability of such a periodic solution of (2) was studied by Chowand Walther [3] (coupled with the technique developed in Ivanov, Lani-Wayda andWalther [7] since µ > 0) and by Xie [12, 14, 13], to name a few. Such a stabilityimplies the stability of the synchronized periodic solution ps = (p, · · · , p)T : R→ Rnof the network (1) under small synchronous perturbation. This paper, however,shows that the above periodic solution of (1) is always unstable (of course, underasynchronous perturbation) if the network is large. This shows, in terms of theaforementioned stability of the periodic solution p of (2), that the large scale andthe delayed inhibitation jointly lead to desynchronization in the considered networkof neurons.2. Main resultsLet p : R → R be a slowly oscillatory periodic solution of (2) of the minimalperiod ω > 2r. Let ps be the corresponding synchronous periodic solution of system(1). By linearizing system (1) around ps, we obtain the following linear periodicdelay differential system:y˙i(t) = −µyi(t)− 12f′(p(t− r))[yi−1(t− r) + yi+1(t− r)] ,(3)where i (mod n).Let CnC = C([−r, 0];Cn) and let MC : CnC → CnC be the monodromy operator ofsystem (3), namelyMC(φ)(s) = yφ(ω + s) for s ∈ [−r, 0], φ ∈ CnC ,where yφ is the solution y = (y1, · · · , yn)T : [−r,∞) → Cn of system (3) withy(s) = φ(s) for s ∈ [−r, 0]. Because ω > 2r, MC is linear, continuous and compact,and every point λ ∈ σ(MC) \ {0} is an eigenvalue of finite multiplicity and isisolated in σ(MC). These eigenvalues are called Floquet multipliers of ps, whichdetermine the linear stability of the synchronous periodic solution ps. See, forexample, Diekmann, van Gils, Verduyn Lunel and Walther [4].Our main result is as follows:Theorem. Let p : R→ R be a slowly oscillatory periodic solution of (2) and let psbe the corresponding synchronous periodic solution of system (1). Then for large n,ps is linearly unstable. More precisely, ps has a real Floquet multiplier larger than1.DESYNCHRONIZATION OF DELAYED NEURAL NETWORKS 23673. The proof and remarksLet σ(MC) be the spectrum of the monodromy operator of system (3). Ourgoal is to show that σ(MC) contains a real λ > 1 if n is large. For the sake ofsimplicity, we let Mb : CC = C1C → CC denote the linear compact operator given by(Mbϕ)(θ) = uϕ(ω + θ) for θ ∈ [−r, 0], where b : R→ R is a continuous map with aperiod ω > 2r, uϕ is the solution ofu˙(t) = −µu(t) + b(t)u(t− r)(4)with uϕ |[−r,0]= ϕ ∈ CC. Note that λ ∈ σ(Mb) \ {0} if and only if (4) has a nonzerosolution u : R→ C such that u(t+ ω) = λu(t) for all t ∈ R (see, for example, Haleand Verduyn Lunel [5], Diekmann, van Gils, Verduyn Lunel and Walther [4]).Letbk(t) = −f ′(p(t− r)) cos 2kpin, k ∈ {0, 1, · · · , n− 1} .(5)We haveLemma 1.n−1⋃k=0σ(Mbk) \ {0} ⊂ σ(MC) \ {0}.Proof. Assume λ ∈ σ(Mbk) \ {0} for some k ∈ {0, 1, · · · , n− 1}. Then there existsψ ∈ CC \ {0} such that the solution of (4) with b = bk and u |[−r,0]= ψ satisfiesu(t+ ω) = λu(t) for t ≥ −r. Let y = (y1, · · · , yn)T : [−r,∞)→ Cn be given byyj(t) = ei2pin kju(t), t ≥ −r, j = 1, · · · , n.Then for t ≥ 0, we havey˙j(t) = ei2pin kj [−µu(t) + bk(t)u(t− r)] = −µyj(t) + ei 2pin kjbk(t)u(t− r) .Note thatyj−1(t− r) + yj+1(t− r) = [ei 2pin k(j−1) + ei 2pin k(j+1)]u(t− r)= 2 cos2piknei2pin kju(t− r) .Therefore for t ≥ 0, we havey˙j(t) = −µyj(t)− 12f′(p(t− r))[yj−1(t− r) + yj+1(t− r)] .As y |[−r,0]= (ei 2pin k, ei 2pin (2k), · · · , ei 2pin (nk))Tψ ∈ CnC \ {0} andyj(t+ ω) = ei2pin kju(t+ ω) = λei2pin kju(t) = λyj(t), t ≥ −r, j = 1, · · · , n,we conclude that λ ∈ σ(MC). This completes the proof.The next result concerns the leading eigenvalue of M−b0 , where −b0 =f ′(p(t− r)). Recall that ω > 2r.Lemma 2. There exists a real simple eigenvalue λ > 1 of M−b0 with an associatedeigenvector φ ∈ C+ := {φ ∈ C([−r, 0];R); φ(s) > 0 for s ∈ [−r, 0]}.Proof. Due to the increasing property of f , it is easy to show that M−b0 |C : C =C1 → C is a positive linear operator. Therefore, the existence of a real eigenvalueλ > 0 of M−b0 associated with an eigenvector φ ∈ C+ is an immediate consequenceof the Krein-Rutman theorem (see, for example, Smith [10]). So it suffices to showthat λ > 1.2368 YUMING CHEN, YING SUE HUANG, AND JIANHONG WUWe are going to use the following simple coupling technique (used in the workof Chen, Krisztin and Wu [2]): (X,Y )T : R → C2 is a solution of the decoupledsystemX˙(t) = −µX(t) + b0(t)X(t− r)(6)andY˙ (t) = −µY (t)− b0(t)Y (t− r)(7)if and only if (U, V )T : R→ C2 given byU(t) = X(t) + Y (t), V (t) = −X(t) + Y (t)(8)is a solution of the coupled system{U˙(t) = −µU(t)− b0(t)V (t− r) ,V˙ (t) = −µV (t)− b0(t)U(t− r) .(9)In particular, (p˙,−p˙)T : R→ R2 is an ω-periodic solution of (9).Let F : C2 → C2 be the (real) monodromy operator of system (9). Namely,F (ψ)(θ) = (Uψ(ω + θ), V ψ(ω + θ))T , θ ∈ [−r, 0], ψ ∈ C2 ,where (Uψ, V ψ)T is the solution of (9) with (Uψ, V ψ)T |[−r,0]= ψ. LetK2 = {ψ = (ψ1, ψ2)T ∈ C2; ψi(s) ≥ 0 for all i = 1, 2 and s ∈ [−r, 0]} .It is easy to verify, due to ω > r and b0(t) < 0 for all t ∈ R, the following orderpreserving propertyF (K2) ⊂ K2, F 2(K2 \ {0}) ⊂ intK2.(10)We have noticed that (φ, φ)T ∈ intK2 is an eigenvector of F associated withsome real λ > 0. We now follow the trick in Krisztin, Walther and Wu [8] to showthat λ > 1. If λ < 1, then (φ, φ)T + ε(p˙0,−p˙0)T ∈ K2 for small ε > 0 and thusF j((φ, φ)T + ε(p˙0,−p˙0)T ) = λj(φ, φ)T + ε(p˙0,−p˙0)T → ε(p˙0,−p˙0)T ∈ C2 \K2as j → ∞. Therefore, for large j, we have F j((φ, φ)T + ε(p˙0,−p˙0)T ) /∈ K2, acontradiction to (10). We now exclude the case where λ = 1 by contradiction.Assume that λ = 1. There must be τ > 0 such that (φ, φ)T + τ(p˙0,−p˙0)T ∈C2 \ K2. Therefore, there exists s ∈ (0, τ) such that (φ, φ)T + β(p˙0,−p˙0)T ∈intK2 for β ∈ (0, s) and (φ, φ)T + s(p˙0,−p˙0)T ∈ K2 \ intK2. We clearly have(φ, φ)T + s(p˙0,−p˙0)T 6= 0 since (φ, φ)T ∈ intK2 and s(−p˙0,−p˙0)T /∈ K2. Hence,F 2((φ, φ)T + s(p˙0,−p˙0)T ) = (φ, φ)T + s(p˙0,−p˙0)T ∈ K2 \ intK2, a contradiction to(10). This shows λ > 1, completing the proof.The following result shows the persistence of the monodromy operator undersmall perturbation. For a general result regarding the persistence of the spectraof Poincare´-type mappings associated with delay differential equations, see Ivanov,Lani-Wayda and Walther [7].Lemma 3. Assume that ck : R → R are continuous and ω-periodic and ck(t) →b0(t) as k →∞, uniformly for t ∈ R. Then for large k, M−ck has a real eigenvalueλk > 1 with an associated eigenvector φk ∈ C+.DESYNCHRONIZATION OF DELAYED NEURAL NETWORKS 2369Proof. Let λ be given in Lemma 2 and define α = 1ω lnλ. Then there exists apositive ω-periodic C1-smooth map q : R→ R such that eαtq(t) is a corresponding(characteristic) solution of (4) with b(t) = −b0(t) for all t ∈ R. Thenq˙(t) + (α+ µ)q(t) + b0(t)e−αrq(t− r) = 0, t ∈ R .(11)LetH0ω denote the Banach space of ω-periodic continuous real mappings equippedwith the super-norm |·|H0ω and letH1ω={w ∈ H0ω; w is C1-smooth and |w˙|H0ω <∞}.Then H1ω, equipped with the norm |w|H1ω = |w|H0ω + |w˙|H0ω for w ∈ H1ω is a Banachspace. Define T = (T1, T2) : H1ω × R×H0ω → H0ω × R byT1(w, ν, b) = w˙(t) + (µ+ ν)w(t) + b(t)e−νrw(t − r)andT2(w, ν, b) =1ω∫ ω0w(t)dt − 1ω∫ ω0q(t)dt ,where (w, ν, b) ∈ H1ω × R × H0ω. Then T (q, α, b0) = 0. We will show thatD(w,ν)T |(q,α,b0) is bijective.First of all, we note thatD(w,ν)T |(q,α,b0) (u,Γ) = (u˙(t) + (α+ µ)u(t) + b0(t)e−αru(t− r)+ Γ(q(t)− b0(t)e−αrrq(t − r)), 1ω∫ ω0u(t)dt) .We claim that KerD(w,ν)T |(q,α,b0)= (0, 0). To verify this claim, we suppose thereexists (u,Γ) such thatD(w,ν)T |(q,α,b0) (u,Γ) = (0, 0) .(12)Thenu˙(t) + (α+ µ)u(t) + b0(t)e−αru(t− r) + Γ[q(t)− b0(t)e−αrrq(t− r)] = 0and1ω∫ ω0u(t)dt = 0 .If Γ 6= 0, then u 6= 0 and eαtq(t) and eαt[tq(t) + Γ−1u(t)] are both solutions ofequation (4) with b = −b0, a contradiction to the simplicity of the eigenvalueλ = eωα of M−b0 . If Γ = 0, thenu˙(t) + (µ+ α)u(t) + b0(t)e−αru(t− r) = 0 .Since λ as an eigenvalue of M−b0 is simple, we have that u = cq for some constantc. But 1ω∫ ω0 u(t)dt = 0 and∫ ω0 q(t)dt > 0. Therefore, c = 0 which gives u = 0.Next, we show that D(w,ν)T |(q,α,b0) is surjective. Since λ as an eigenvalue ofM−b0 is simple, the space {w ∈ H1ω; T1(w,α, b0) = 0} is one-dimensional. SodimKer(DwT1) = codim (RangeDwT1) = 1 .If D(w,ν)T1 |(q,α,b0) is not surjective, thenRange(DwT1) + Range(DνT1) 6= H0ω .Therefore,Range(DνT1) ⊂ Range(DwT1) .2370 YUMING CHEN, YING SUE HUANG, AND JIANHONG WUThus, there exists a function u ∈ H1ω such that u˙(t)+(µ+α)u(t)+b0(t)e−αru(t−r) =−[q(t) − b0(t)e−αrrq(t − r)], which implies that equation (4) has a characteristicsolution of form eαt[tq(t) + u(t)], and thus the multiplicity of λ as an eigenvalue ofM−b0 is at least two, a contradiction.Given any number m ∈ R and γ ∈ H0ω, from the above discussion there exists(u,Γ) ∈ H0ω × R such thatD(w,ν)T1 |(q,α,b0) (u,Γ) = γ .Since for any constant k, (u + kq,Γ) is also a solution of the above equation, wecan select k such that 1ω∫ ω0(u(t) + kq(t))dt = m. This shows the surjectivity ofD(w,ν)T |(q,α,b0).Then we can use the implicit function theorem to solve equationT (w, ν, b) = 0 .(13)In particular, for any ε > 0, there exists an open neighborhood Ub0,ε in H0ω ofb0 such that for every b ∈ Ub0,ε the operator equation (13) has a unique solution(qb, αb) ∈ H1ω × R. One can see that when ε is small enough, αb > 0 and qb(t) > 0for all t ∈ R. This completes the proof.We can now prove the main result:Proof of the Main Theorem. For k = [n2 ],limn→∞ cos2pikn= −1 .Therefore, b[n2 ](t) → −b0(t) as n → ∞, uniformly for t ∈ R. Thus the theoremfollows from Lemma 3.Remark. The above argument applies to systems with arbitrary even n. So synchro-nous periodic solutions of network (1) arising from the slowly oscillatory periodicsolutions of (2) are always unstable for the network (1) with either large n or ar-bitrary even n. However, these synchronous periodic solutions can be stable underpossible perturbations if n is odd and small. Examples of this type will be given inthe case study of Wu, Faria and Huang [15].References[1] J. Be´lair, S. A. Campbell and P. van den Driessche, Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56 (1996), 245–255.MR 96j:92003[2] Y. Chen, J. Wu and T. Krisztin, Connecting orbits from synchronous periodic solutionsto phase-locked periodic solutions in a delay differential system, to appear in Journal ofDifferential Equations, 1999.[3] S. N. Chow and H. O. Walther, Characteristic multipliers and stability of symmetric periodicsolutions of x˙(t) = g(x(t−1)), Trans. Amer. Math. Soc. 307 (1988), 124-142. MR 89d:34115[4] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Func-tional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995. MR 97a:34001[5] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,Applied Mathematical Sciences, Vol. 99, Springer-Verlag, New York, 1993. MR 94m:34169[6] J. J. Hopfield, Neurons with graded response have collective computational properties liketwo-stage neurons, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 3088-3092.[7] A. Ivanov, B. Lani-Wayda and H. O. Walther, Unstable hyperbolic periodic solutions ofdifferential delay equations, World Sci. Ser. Appl. Anal. 1 (1992), 301-316. MR 93h:34125DESYNCHRONIZATION OF DELAYED NEURAL NETWORKS 2371[8] T. Krisztin, H. O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of anAttracting Set for Delayed Monotone Positive Feedback, Fields Institute Monographs, Vol.11, American Mathematical Society, Providence, 1999.[9] C. M. Marcus and R. M. Westervelt, Stability of analog neural networks with delay. Phys.Rev. A (3) 39 (1989), 347–359. MR 89m:92025[10] H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitiveand Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, Amer. Math.Soc., Providence, RI, 1995. MR 96c:34002[11] J. Wu, Symmetric functional-differential equations and neural networks with memory. Trans.Amer. Math. Soc. 350 (1998), 4799-4838. MR 99i:34104[12] X. W. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equationswith bounded nonlinearity, J. Dyna. Diff. Eqns. 3 (1991), 515-540. MR 93b:34093[13] X. W. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equationswith unbounded nonlinearity, J. Diff. Eqns. 103 (1993), 350-374. MR 94c:34115[14] X. W. Xie, The multiplier equation and its application to S-solutions of a delay equation, J.Diff. Eqns. 102 (1992), 259-280. MR 93f:34110[15] J. Wu, T. Faria and Y. S. Huang, Absolute synchronization and stable phase-locking in anetwork of neurons with memory, Mathematical and Computer Modeling 30 (1999), 117–138.Department of Mathematics and Statistics, York University, Toronto, Ontario,Canada M3J 1P3E-mail address: yumingc@mathstat.yorku.caDepartment of Mathematics, Pace University, Pleasantville, New York 10570E-mail address: huang@risc.dac.pace.eduDepartment of Mathematics and Statistics, York University, Toronto, Ontario,Canada M3J 1P3E-mail address: wujh@mathstat.yorku.ca

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Desynchronization of Large Scale Delayed Neural Networks

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