具非線性連接之Hindmarsh-Rose神經元耦合系統的同步化研究 - 政大學術集成

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(1)國立政治大學應用數學系碩士學位論文. 立. 政治大. ‧ 國. 學. 具非線性連接之 Hindmarsh-Rose 神經元耦合系統的同步化研究. ‧. Synchronization of nonlinearly coupled systems of Hindmarsh-Rose neurons with time delays. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. 指導教授：曾睿彬. v. 博士. 研究生：陳柏艾撰中華民國 109 年 1 月. DOI:10.6814/NCCU202000086.

(2) 中文摘要在此論文，我們研究 Hindmarsh-Rose 神經元耦合系統的同步化，我們所考慮的模型之耦合結構可以相等的一般性。模型所具備的耦合函數可以是非線性的，耦合. 政治大. 矩陣可容許非零的非對角元素能有不同的正負號，並且我們也考慮耦合時間延遲。藉由 [33] 的同步化理論，我們推導出與時間延遲相關的同步化條件。我們提供兩個. 立. 數值例子來表現本論文同步化理論之效用。. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. i. DOI:10.6814/NCCU202000086.

(3) Abstract In this thesis, we investigate the synchronization of coupled systems of Hindmarsh-Rose neurons. The coupling scheme under consideration is general.. 政治大. The coupling functions could be non-linear. The connection matrix could have non-zero and non-diagonal entries with different signs. We also consider the. 立. transmission delays in the coupling terms of the coupled systems. We derive a. ‧ 國. 學. delay-dependent criterion that leads to the synchronization of coupled neurons. Two examples with numerical simulations are illustrated to show the effectiveness of. ‧. theoretical result.. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. ii. DOI:10.6814/NCCU202000086.

(4) Contents 中文摘要. i. Abstract. 立. Contents. ‧ 國 al. n 4 Numerical examples 5 Conclusion Bibliography. Ch. engchi. er. io. 3 Synchronization of Hindmarsh-Rose neurons. sit. y. Nat. 2 Preliminaries. iii. ‧. 1 Introduction. ii. 學. List of Figures. 政治大. i n U. v. iv 1 4 11 20 32 33. DOI:10.6814/NCCU202000086.

(5) List of Figures 4.1. Simulation for the solution of the system considered in Example 1, with τ = 0.00001: evolution of components xi,k (t), i = 1, 2, 3, and k = 1, 2, 3. . . . . . .. 4.2. 政治大 0.00001: evolution of components (x (t), x (t), x 立. Simulation for the solution of the system considered in Example 1, with τ = i,1. i,2. i,3 (t)),. i = 1, 2, 3. . . . . .. 學. 0.00001: evolution of components zi,k (t), i = 1, 2, and k = 1, 2, 3. . . . . . . . 4.4. ‧. sit. y. Nat. er. io. n. al. Ch. i n U. v. e n gsystem c h iconsidered in Example 2, with τ Simulation for the solution of the. 29. =. 0.00002: evolution of components zi,k (t), i = 1, 2, and k = 1, 2, 3. . . . . . . . 4.8. 28. Simulation for the solution of the system considered in Example 2, with τ = 0.00002: evolution of components (xi,1 (t), xi,2 (t), xi,3 (t)), i = 1, 2, 3. . . . . .. 4.7. 25. Simulation for the solution of the system considered in Example 2, with τ = 0.00002: evolution of components xi,k (t), i = 1, 2, 3, and k = 1, 2, 3. . . . . . .. 4.6. 24. Simulation for the solution of the system considered in Example 1, with τ = 0.05: evolution of components zi,k (t), i = 1, 2, and k = 1, 2, 3. . . . . . . . . .. 4.5. 23. Simulation for the solution of the system considered in Example 1, with τ =. ‧ 國. 4.3. 22. 30. Simulation for the solution of the system considered in Example 2, with τ = 0.1: evolution of components zi,k (t), i = 1, 2, and k = 1, 2, 3 . . . . . . . . . .. 31. DOI:10.6814/NCCU202000086.

(6) Chapter 1 Introduction 政治大. Synchronization is an important phenomenon in several biological, physiological complex. 立. networks systems [14,31]. For instance, there has been an observation that neurons synchronize. ‧ 國. 學. each other by coupled dynamic neural network [2, 8, 22, 23, 27, 32, 34, 37, 38]. The process of conveying neural information in brain is conducted through mutual interaction of neural In addition, the information delivering process in the neural systems,. ‧. populations [6].. synchronization plays a key role in association and memory [26]. As the basic behavior of. y. Nat. sit. neuron, synchronization is an expression of neuron discharge, widely existing in the neural. er. io. system like the visual cortex [25, 35]. However, it is shown that too much synchronization. al. n. v i n C h based on severalUphysiological experiments [15, 24, 39]. Parkinson’s disease, and schizophrenia engchi. will do harm to the organism and cause such brain diseases as Alzheimer’s disease, epilepsy,. Since those illnesses have a lot to do with the abnormal synchronization of neuron systems, it is worth to study the synchronization in coupled dynamic networks. To obtain a deep understanding of neural network dynamics, lots of neural models such as the Hodgkin-Huxley model [21, 28], the FitzHugh-Nagumo model [22, 33], the Morris-Lecar model [1, 3] and the Hindmarsh-Rose model [17, 36] have been used by numerous researches for biological application. Among these models, Hindmarsh-Rose neurons were discovered from neuron cells in a pond snail which burst after it is depolarized by a pulse of short current [4, 36]. What’s more, it has been shown that the Hindmarsh-Rose neuron model, a system of three ordinary differential equations, is able to produce abundant neural behavior like spiking, bursting, and chaotic behavior [7, 11, 13]. In this paper, we will investigate the synchronization of the coupled systems of Hindmarsh-Rose neurons.. 1. DOI:10.6814/NCCU202000086.

(7) The dynamics of an isolated single Hindmarsh-Rose neuron could be depicted by the following system of ordinary differential equations [10]:                 . x˙ 1 (t) = x2 (t) − a(x1 (t))3 + b(x1 (t))2 − x3 (t) + I, (1.1). x˙ 2 (t) = 1 − dx1 (t)2 − x2 (t), x˙ 3 (t) = r(s(x1 (t) + x0 ) − x3 (t)),. where x1 represents the membrane potential of the neuron, x2 the recovery variable, and x3 the adaptation current. The external input current is represented by I to determine the output mode. 政治大 which occurs 立 in the propagation of action potentials along the axon,. of the neuron. The parameters a, b, d, r, s, x0 are all positive constants. Time delay,. ‧ 國. 學. transferring signals across the synapse or other artificial units, is an important factor in the coupled neural systems [5,12]. It is essential to tackle nonlinear systems under delayed coupling.. ‧. Therefore, in this paper, we shall consider the following coupled network consisting of N identical Hindmarsh-Rose neurons with coupling time delay:. y. sit. er. x˙ i,1 (t) = xi,2 (t) − axi,1 (t)3 + bxi,1 (t)2 − xi,3 (t) + I. io. a l ∑ aij (g(xj,1(t − τ )) − g(xi vi,1(t − τ ))), j∈NC ,j̸=i hengchi Un 2. n.              . Nat.               . +c. (1.2). x˙ i,2 (t) = 1 − dxi,1 (t) − xi,2 (t),. x˙ i,3 (t) = r(s(xi,1 (t) + x0 ) − xi,3 (t)),. for i ∈ N := {1, 2, ..., N }, where c ≥ 0 is the coupling strength ; A = [aij ]N ×N , with aii := −. N ∑. aij ,. (1.3). j=1,j̸=i. is the connection matrix representing the topological structure of the network; g : R → R is the coupling function; τ ≥ 0 is the transmission delay. We note that system (1.2) is linearly coupled if g is linear, otherwise it is non-linearly coupled. Among the existing investigations on coupled Hindmarsh-Rose neurons, some of the investigations considered with linearly coupling 2. DOI:10.6814/NCCU202000086.

(8) function: g(x) = x, cf. [36]; while some others considered non-linearly coupling function: g(x) = tanh(x), cf. [17]. Moreover, the work in [36] considered linearly coupled systems of two Hindmarsh-Rose neurons with time delays (i.e. τ ̸= 0); the work in [17] considered nonlinearly coupled Hindmarsh-Rose neurons without delay (i.e. τ = 0), cf. [17]. It is worth noting that most of the previous investigations on the synchronization of coupled systems required that the non-zero and off-diagonal entries of connection matrix have the same signs, cf. [9, 16–20, 30]; moreover, most of the previous investigations on the synchronization of coupled systems considered linear couplings, cf. [2, 27]. In the following, (x1 (t), . . . , xN (t)) denotes an arbitrary solution of system (1.2) and (xt1 , . . . , xtN ) is the corresponding evolution of system (1.2), where xti ∈ C([−τ, 0]; RK ), i ∈ N ,. 政治大. written as xti (θ) = xi (t + θ) for θ ∈ [−τ, 0]. It is said that the system (1.2) achieves global. 立. synchronization if. ‧ 國. 學. xi,k (t) − xj,k (t) → 0, as t → ∞, for all i, j ∈ N , k ∈ {1, 2, 3},. ‧. for every solution (x1 (t), . . . , x3 (t)), where xi (t) = (xi,1 (t), xi,2 (t), xi,3 (t)). In this paper, we. sit. y. Nat. shall utilize the approach developed in [33] to investigate the global synchronization of coupled. io. n. al. er. system (1.2) with the coupling function g in the following class:. i n U. v. {g ∈ C 1 : δ := g ′ (0) > g ′ (x) > 0, x ̸= 0}.. Ch. engchi. (1.4). We emphasize that the connection matrix A = [aij ]N ×N , considered in this thesis, is allowed to have off-diagonal entries with different signs. The remainder of this thesis is organized as follows. In chapter 2, we introduce the synchronization theory developed in [33]. In chapter 3, we establish the synchronization of nonlinearly coupled systems of Hindmarsh-Rose neurons based on the theory in [33]. In chapter 4, we demonstrate two numerical examples to support our theory. In chapter 5, we give some discussions and conclusions.. 3. DOI:10.6814/NCCU202000086.

(9) Chapter 2 Preliminaries 政治大. In this chapter, we shall introduce the synchronization theory developed in [33], the model. 立. x˙ i (t) = F(xi (t), t) + c. ∑. 學. ‧ 國. considered in [33] is as follows:. aij (t)G(xj (t − τ (t))), i ∈ N , t ≥ t0 ,. (2.1). j∈N. ‧. where N = {1, . . . , N }, xi (t) = (xi,1 (t), . . . , xi,K (t)) ∈ RK , F = (F1 , . . . , FK ) is a smooth. y. Nat. sit. function describing the intrinsic dynamics of each subsystem, c ≥ 0 is the coupling strength,. n. al. er. io. and aij (t), i, j ∈ N , are bounded functions of t. Matrix A(t) := [aij (t)]1≤i,j≤N is referred to as. i n U. v. the connection matrix and is assumed to satisfy the condition: ∑. Ch. engchi. aij (t) = κ(t), for all i ∈ N and t ≥ t0 .. (2.2). j∈N. The function G = (G1 , . . . , GK ) is assumed to satisfy Gk (xj (t − τ (t))) = gk (xj,k (t − τ (t))), for all i, j ∈ K and t ≥ t0 ,. (2.3). 4. DOI:10.6814/NCCU202000086.

(10) where K := {1, . . . , K}, gk is a non-decreasing and differentiable function, and τ (t) ∈ [0, τM ] stands for the time-dependent transmission delay. For later use, set κ ¯ = sup{|κ(t)| : t ≥ t0 },. (2.4). κ ˇ = inf{κ(t) : t ≥ t0 },. (2.5). κ ˆ = sup{κ(t) : t ≥ t0 },. (2.6). a ¯ij = sup{|aij (t)| : t ≥ t0 },. (2.7). τ¯ = sup{τ (t) : t ≥ t0 }.. (2.8). 政治大. In this section, (x1 (t), . . . , xN (t)) denotes an arbitrary solution of system (2.1), and (xt1 , . . . , xtN ) is the corresponding evolution of system (2.1), where xti ∈ C([−τM , 0]; RK ), i ∈ N , are. 立. defined as xti (θ) = xi (t + θ) for θ ∈ [−τM , 0]. System (2.1) is said to attain global (identical). ‧ 國. 學. synchronization, if. ‧. xi,k (t) − xj,k (t) → 0, as t → ∞, for all i, j ∈ N , k ∈ K,. Nat. al. Let us introduce two assumptions as follows:. er. io. sit. y. for every solution (x1 (t), . . . , xN (t)), where xi (t) = (xi,1 (t), . . . , xi,K (t)).. n. v i n C h (2.1) eventuallyUenter and then remain in some compact Assumption (D): All solutions of system engchi set QN := Q × · · · × Q ⊂ RN K , where Q := [ˇ q1 , qˆ1 ] × · · · × [ˇ qK , qˆK ] ⊂ RK . Define CQ := {(Φ1 , . . . , ΦN ) : Φi = (ϕi,1 , . . . , ϕi,K ) ∈ C([−τM , 0]; RK ),. (2.9). ϕi,k (θ) ∈ [ˇ qk , qˆk ], θ ∈ [−τM , 0], i ∈ N , k ∈ K}. ˜ t) as Decompose Fk (E, t) − Fk (E, ˜ t) = hk (xk , x˜k , t) + wk (E, E, ˜ t), Fk (E, t) − Fk (E, where t ≥ t0 , E = (x1 , . . . , xK ), and E˜ = (E˜1 , . . . , E˜K ). 5. DOI:10.6814/NCCU202000086.

(11) Assumption (F): For each k ∈ K, there exist µ ˇk , µ ˆk ∈ R, ρw ¯kl ≥ 0, for k ≥ 0, and µ l ∈ K − {k}, such that for any E, E˜ ∈ Q, the following two properties hold for all t ≥ t0 :. (F-i):.   µ ˇk ≤ hk (xk , x˜k , t)/(xk − x˜k ) ≤ µ ˆk. if xk − x˜k ̸= 0,.  hk (xk , x˜k , t) = 0. if xk − x˜k = 0, ∑ ˜ t)| ≤ ρw , and |wk (E, E, ˜ t)| ≤ (F-ii): |wk (E, E, µ ¯kl |xl − x˜l |. k l∈K−{k}. For later use, define the following sets of indices:. 政治大. A := (N − {N }) × K and Ai,k := A − {i} × {k}, where (i, k) ∈ A.. 立. (2.10). ‧ 國. 學. Assume that (x1 (t), . . . , xN (t)), where xi (t) = (xi,1 (t), . . . , xi,K (t)), is an arbitrary solution of system (2.1). Setting. zi (t) := xi (t) − xi+1 (t), i ∈ N − {N },. (2.11). ‧. y. Nat. where zi (t) = (zi,1 (t), . . . , zi,K (t)), cf. (2.1) and (2.3), then (z1 (t), . . . , zN −1 (t)) satisfies the. al. er. io. sit. following difference-differential system corresponding to system (2.1):. n. z˙i,k (t) = Hi,k (xt1 , . . . , xtN , t), (i, k) ∈ A, t ≥ t0 , where. Ch. engchi U. v ni. (2.12). Hi,k (Φ1 , . . . , ΦN , t) := Fk (Φi (0), t) − Fk (Φi+1 (0), t) ∑ [aij (t) − a(i+1)j (t)]gk (ϕj,k (−τ (t))), +c. (2.13). j∈N. for Φj = (ϕj,1 , . . . , ϕj,K ) ∈ C([−τM , 0]; RK ), j ∈ N . Clearly, system (2.1) attains global synchronization if zi,k (t) → 0, as t → ∞, for every (i, k) ∈ A. ˜ = [˜ Via A(t) = [aij (t)]1≤i,j≤N , define matrix A(t) aij (t)]1≤i,j≤N , where. a ˜ij (t) =.   aii (t) − κ(t), if i = j ∈ N ,  aij (t),. (2.14). if i, j ∈ N and i ̸= j.. 6. DOI:10.6814/NCCU202000086.

(12) ¯ We introduce matrix A(t): T ¯ = [αij (t)]1≤i,j≤N −1 := CA(t)C ˜ A(t) (CCT )−1 ∈ R(N −1)×(N −1) ,. where.        C :=       . (2.15).  1. −1. 0. ···. 0. 0. 1. −1. ... .. .. .. .. .. .... .... .... 0.        ∈ R(N −1)×N .      . 0 1 −1 政治大 ¯ in (2.15) is well-defined, and satisfies Then, A(t) 立 ···. 學. ‧ 國. 0. ˜ = A(t)C, ¯ CA(t). Nat. y. ‧. for all t ≥ t0 . For later use, set. n. i n U. α ˆ ij = sup{αij (t) : t ≥ t0 },. Ch. engchi. (2.18). er. io. α ˇ ij = inf{αij (t) : t ≥ t0 },. (2.17). sit. α ¯ ij = sup{|αij (t)| : t ≥ t0 },. al. (2.16). v. (2.19). ¯ defined in (2.15). where αij (t), 1 ≤ i, j ≤ N − 1, are entries of A(t) Proposition 2.1. ( [33]) Consider system (2.1) which satisfies Assumptions (D) and (F). Then, functions Hi,k , (i, k) ∈ A, defined in (2.13), can be decomposed as Hi,k (Φ1 , . . . , ΦN , t) = hi,k (ϕi,k (0), ϕi+1,k (0), t) ˜ i,k (ϕi,k , ϕi+1,k , t) + wi,k (Φ1 , . . . , ΦN , t), +h hi,k (ϕi,k (0), ϕi+1,k (0), t) = hk (ϕi,k (0), ϕi+1,k (0), t), ˜ i,k (ϕi,k , ϕi+1,k , t) = c[κ(t) + αii (t)][gk (ϕi,k (−τ (t))) − gk (ϕi+1,k (−τ (t)))], h wi,k (Φ1 , . . . , ΦN , t) = wk (Φi (0), Φi+1 (0), t) ∑ +c αij (t)[gk (ϕj,k (−τ (t))) − gk (ϕj+1,k (−τ (t)))]. j∈N −{i,N }. 7. DOI:10.6814/NCCU202000086.

(13) Moreover, for all (i, k) ∈ A and all (Φ1 , . . . , ΦN ) ∈ CQ , where Φi = (ϕi,1 , . . . , ϕi,K ), i ∈ N , the following three properties hold for all t ≥ t0 :    h (ϕ (0),ϕi+1,k (0),t)   µ ˇk ≤ i,k[ϕi,ki,k ≤µ ˆk if ϕi,k (0) − ϕi+1,k (0) ̸= 0, (0)−ϕi+1,k (0)] (H-i):     h (ϕ (0), ϕ (0), t) = 0 if ϕ (0) − ϕ (0) = 0, i,k. i,k. i+1,k. i,k. i+1,k. ˜ i,k (ϕi,k , ϕi+1,k , t)| ≤ ρh , and (H-ii): |h ik     . βˇik ≤.    . ˜ i,k (ϕi,k , ϕi+1,k , t) = 0 h. ˜ i,k (ϕi,k ,ϕi+1,k ,t) h [ϕi,k (−τ (t))−ϕi+1,k (−τ (t))]. ≤ βîk. if ϕi,k (−τ (t)) − ϕi+1,k (−τ (t)) ̸= 0, if ϕi,k (−τ (t)) − ϕi+1,k (−τ (t)) = 0,. 政治大. (H-iii): |wi,k (Φ1 , . . . , ΦN , t)| ≤ ρw ik , and. 立. ‧ 國. ∑. (jl). {¯ µik |ϕj,l (0) − ϕj+1,l (0)|. 學. |wi,k (Φ1 , . . . , ΦN , t)| ≤. (j,l)∈Ai,k (jl). + β¯ik |ϕj,l (−τ (t)) − ϕj+1,l (−τ (t))|},. ‧. n. al. g w ρw ik ≥ ρk + 2cρk. and. er. io. ρhik ≥ 2c[κ(t) + αii (t)]ρgk. sit. y. Nat. for ρhik and ρw ik satisfying. respectively, and. Ch. engchi U. v ni. ∑. α ¯ ij ,. j∈N −{i,N }. 8. DOI:10.6814/NCCU202000086.

(14) βˇik.   ˇk c[κ(t) + αii (t)]L. =. βîk. =. µ ¯ik. (jl). =. (jl) β¯ik. =.  c[κ(t) + αii (t)]L ˆk   ˆk c[κ(t) + αii (t)]L. if κ(t) + αii (t) ≥ 0, if κ(t) + αii (t) < 0, if κ(t) + αii (t) ≥ 0,.  c[κ(t) + αii (t)]L ˇ k if κ(t) + αii (t) < 0,    µ ¯kl if i = j, k ̸= l,   0 otherwise,   ˆ k if j ̸= i, k = l, c¯ αij L. 治政大   0 otherwise,. 立. (2.21) (2.22). er. io. ˆ k := max{gk′ (xi ) : xi ∈ [ˇ L qk , qˆk ]} ≥ 0.. y. Nat. ˇ k := min{g ′ (xi ) : xi ∈ [ˇ L qk , qˆk ]} ≥ 0, k. (2.20). sit. ρgk := max{|gk (xi )| : xi ∈ [ˇ qk , qˆk ]} ≥ 0,. ‧. ‧ 國. 學. where. al. n. v i n Herein, κ ¯, κ ˇ , and κ ˆ are defined inC(2.6), AU ¯ ij , α ˇ ij , and α ˆ ij in (2.17), i,k in (2.10), α h e CnQ gin c(2.9), i h µ ˇ ,µ ˆ , ρw , µ ¯ , and functions h , w in Assumption (F). k. k. k. kl. k. k. (jl) (jl) With µ ˇk , µ ˆk , βˇik , βîk , µ ¯ik , and β¯ik , introduced in Proposition 2.1, and τ¯ defined in (2.8),. an associated matrix can be defined as follows:. M = [M (kl) ]1≤k,l≤K ,. (2.23). (kl). for each k, l ∈ K, M (kl) = [mij ]1≤i,j≤N −1 is an (N − 1) × (N − 1) matrix, and its entries are defined by (kl). mij =.   ηik ,. if i = j ∈ N − {N } and k = l ∈ K,.  −L ¯ (jl) , ik. (2.24). otherwise.. 9. DOI:10.6814/NCCU202000086.

(15) where β¯ik := max{|βˇik |, |βîk |},. (2.25). ηik := −ˆ µk − βîk + β¯ik τ¯(ˇ µk + µ ˆk + βˇik + βîk ),. (2.26). (jl) (jl) ¯ (jl) := µ L ¯ik + β¯ik . ik. (2.27). Let us introduce the following condition: Condition (S): For all (i, k) ∈ A, µ ˆk + βîk < 0 and µk + βîk )/[(ˇ µk + µ ˆk + βˇik + βîk )(3ρhik + ρw β¯ik τ¯ < 3ρhik (ˆ ik )].. 立. 政治大. ‧ 國. 學. We note that Condition (S) involves τ¯, and is thus delay-dependent.. ‧. Theorem 2.2. ( [33]) Consider system (2.1) which satisfies Assumptions (D) and (F). Then, the system globally synchronizes if Condition (S) holds, and the Gauss-Seidel iterations for the. Mv = 0,. n. al. (2.28). er. io. sit. y. Nat. linear system:. i n U. v. converge to zero, the unique solution of (2.28); or equivalently, λsyn :=. max. Ch. engchi. {|λσ | : λσ : eigenvalue of (DM − LM )−1 UM } < 1.. 1≤σ≤K×(N −1). (2.29). where M is defined in (2.23) and DM , −LM , and −UM represent the diagonal, strictly lowertriangular, and strictly upper-triangular parts of M, respectively.. 10. DOI:10.6814/NCCU202000086.

(16) Chapter 3 Synchronization of Hindmarsh-Rose neurons 立. 政治大. ‧ 國. 學. To apply the synchronization theory in [33], we first rewrite system (1.2) into the form of (2.1). Recalling (1.3), we rewrite the coupling part of (1.2) as follows:. ‧. ∑. ∑. j∈N −{i}. ∑. aij g(xj,1 (t − τ )) −. io. =c[. al. j∈N −{i}. sit. y. aij (g(xj,1 (t − τ )) − g(xi,1 (t − τ ))). j∈N −{i}. aij g(xi,1 (t − τ ))]. er. Nat. c. =c[. n. v∑ i n =c[ aij g(xC (t − τ )) − g(xi,1 (tU j,1h e n g c h i − τ )) j∈N −{i} aij ] j∈N −{i} ∑. ∑. (3.1). aij g(xj,1 (t − τ )) + aii g(xi,1 (t − τ ))]. j∈N −{i}. =c. ∑. aij g(xj,1 (t − τ )).. j∈N. 11. DOI:10.6814/NCCU202000086.

(17) By (3.1), system (1.2) can be written as follows :                             . x˙ i,1 (t) = xi,2 (t) − axi,1 (t)3 + bxi,1 (t)2 − xi,3 (t) + I +c. ∑. aij g(xj,1 (t − τ )), (3.2). j∈N. x˙ i,2 (t) = 1 − dxi,1 (t)2 − xi,2 (t), x˙ i,3 (t) = r(s(xi,1 (t) + x0 ) − xi,3 (t)),. for all i ∈ N . Notably, the terms κ, defined in (2.2), now satisfies the diffusive coupling. 政治大 κ=κ ˇ=κ ˆ = 0.. condition:. a ˜ij =.   aii − κ = aii ,. if i = j ∈ N ,.  aij ,. if i, j ∈ N and i ̸= j.. 學. Accordingly,. (3.3). ‧. ‧ 國. 立. y. Nat. sit. ˜ in (2.14) is now A(t) ˜ = A. Applying (2.15) yields that A(t) ¯ satisfies A(t) ¯ = A¯ = Thus, A(t). n. al. er. io. [αij ]1≤i,j≤N −1 := CACT (CCT )−1 ∈ R(N −1)×(N −1) . Hence, α ˇ ij , α ˆ ij , α ¯ ij in (2.17) are now. v i n Ch α ˇ =α ˆ =α , e n g ijc h i ij U ij α ¯ ij = |αij |,. (3.4) (3.5). for all i, j ∈ N − {N }. In addition, for system (3.2), the time delay τ (t) is now independent of t (i.e. τ (t) ≡ τ ) and the functions Fk (xi (t), t) and Gk (xj (t − τ (t))) defined in (2.1) and (2.3). 12. DOI:10.6814/NCCU202000086.

(18) are now     F1 (xi (t), t) = xi,2 (t) − axi,1 (t)3 + bxi,1 (t)2 − xi,3 (t) + I,    F2 (xi (t), t) = 1 − dxi,1 (t)2 − xi,2 (t),      F3 (xi (t), t) = r(s(xi,1 (t) + x0 ) − xi,3 (t)),     G1 (xj (t − τ (t))) = g(xj,1 (t − τ )),    G2 (xj (t − τ (t))) = 0,      G3 (xj (t − τ (t))) = 0,. (3.6). (3.7). 政治大 Let us introduce a condition for system (3.2), which plays the role as Condition (D) for 立. respectively.. system (2.1):. ‧ 國. 學. Condition (D)*: All solutions of system (3.2) eventually enter and then remain in some. ‧. compact set QN := Q∗ × · · · × Q∗ ⊂ R3N , where Q∗ := [−ρ∗1 , ρ∗1 ] × [−ρ∗2 , ρ∗2 ] × [−ρ∗3 , ρ∗3 ] ⊂ R3. er. io. sit. y. Nat. with ρ∗k ≥ 0, k = 1, 2, 3.. Next, let us establish Assumption (F) for system (3.2) under Condition (D)*.. al. n. v i n C h (D)* holds, then Proposition 3.1. Assume that Condition e n g c h i U system (3.2) satisfies Assumption (F) ∗ 2 ∗ 2 with µ ˇ1 = −3a(ρ1 ) − 2bρ1 , µ ˆ1 = b /3a, µ ˇ2 = µ ˆ2 = −1, µ ˇ3 = µ ˆ3 = −r, µ ¯12 = µ ¯13 = 1,. ∗ ∗ w ∗ 2 w ∗ µ ¯23 = µ ¯32 = 0, µ ¯21 = 2ρ∗1 d, µ ¯31 = rs, ρw 1 = 2(ρ2 + ρ3 ), ρ2 = 4d(ρ1 ) , ρ3 = 2rsρ1 , where. ρ∗k , k = 1, 2, 3, are defined in Condition (D)*. Proof. We first compute quantities µ ˇk , µ ˆk , µ ¯kl ρw k , where k, l ∈ {1, 2, 3} and k ̸= l. By (3.6), applying the mean value theorem yields that                 . ˜ t) = h1 (x1 , x˜1 , t) + w1 (E, E, ˜ t), F1 (E, t) − F1 (E, ˜ t) = h2 (x2 , x˜2 , t) + w2 (E, E, ˜ t), F2 (E, t) − F2 (E,. (3.8). ˜ t) = h3 (x3 , x˜3 , t) + w3 (E, E, ˜ t), F3 (E, t) − F3 (E,. 13. DOI:10.6814/NCCU202000086.

(19) for E = (x1 , x2 , x3 ) ∈ R3 , E˜ = (˜ x1 , x˜2 , x˜3 ) ∈ R3 , and t ≥ t0 , where h1 (x1 , x˜1 , t) = [−3as2 + 2bs](x1 − x˜1 ),. (3.9). ˜ t) = x2 − x˜2 − (x3 − x˜3 ), w1 (E, E,. (3.10). h2 (x2 , x˜2 , t) = −(x2 − x˜2 ),. (3.11). ˜ t) = −d(x21 − x˜21 ), w2 (E, E,. (3.12). h3 (x3 , x˜3 , t) = −(x3 − x˜3 ),. (3.13). ˜ t) = rs(x1 − x˜1 ), w3 (E, E,. (3.14). 政治大. and s is some number between x1 and x˜1 . Recall Assumption (F), where E = (x1 , x2 , x3 ), E˜ = (˜ x1 , x˜2 , x˜3 ) ∈ Q∗ = [−ρ∗1 , ρ∗1 ] × [−ρ∗2 , ρ∗2 ] × [−ρ∗3 , ρ∗3 ] which leads to s ∈ [−ρ∗1 , ρ∗1 ];. 立. moreover,. (3.15). ˜ t)| ≤ 2(ρ∗2 + ρ∗3 ), |w1 (E, E,. (3.16). ‧. ‧ 國. 學. −3a(ρ∗1 )2 − 2bρ∗1 ≤ −3as2 + 2bs ≤ b2 /3a,. (3.18). n. er. io. sit. ˜ t)| ≤ 2rsρ∗1 . |w3 (E, E,. al. (3.17). y. Nat. ˜ t)| ≤ 4d(ρ∗ )2 , |w2 (E, E, 1. i n U. v. Based on (3.9)-(3.18), we shall show that system (3.2) satisfies Assumption(F) for which quantities are chosen as follows:. Ch. engchi. By (3.9) and (3.15), we can choose µ ˇ1 = −3a(ρ∗1 )2 − 2bρ∗1 and µ ˆ1 = b2 /3a. By (3.11), we can choose µ ˇ2 = µ ˆ2 = −1. By (3.13), we can choose µ ˇ3 = µ ˆ3 = −r. ∗ ∗ By (3.10) and (3.16), we can choose µ ¯12 = µ ¯13 = 1 and ρw 1 = 2(ρ2 + ρ3 ). ∗ 2 ¯23 = 0 and ρw By (3.12) and (3.17), we can choose µ ¯21 = 2ρ∗1 d, µ 2 = 4d(ρ1 ) . ∗ By (3.14) and (3.18), we can choose µ ¯31 = rs, µ ¯32 = 0 and ρw 3 = 2rsρ1 .. From Proposition 3.1, system (3.2) satisfies Assumption (D) and (F) under Condition (D)*. Accordingly, the assumption for the assertion in Proposition 2.1 holds for system (3.2) under Condition (D)*. In addition, the quantities in the assertion of Proposition 2.1 can be chosen as those in the following proposition.. 14. DOI:10.6814/NCCU202000086.

(20) Proposition 3.2. Assume that Condition (D)* holds. The assertion in Proposition 2.1 holds with. ρhik. =. βˇik. =. βîk. =.   2c¯ αii ρg ,. =. (3.21). if k = 1,. 4d(ρ∗1 )2 , if k = 2,       2rsρ∗ , if k = 3, 1    1, if i = j, (k, l) = (1, 2) or (1, 3),       2ρ∗1 d, if i = j, (k, l) = (2, 1),. (3.22). y. al. n. =. sit. io. (jl) β¯ik. (3.20). ‧. Nat (jl). µ ¯ik. (3.19). 政治大. =. er. ρw ik.  v, if k = 2, 3,    ˇ if k = 1, αii ≥ 0, cαii δ,    cαii δ, if k = 1, αii < 0,      0, if k = 2, 3,     cαii δ if k = 1, αii ≥ 0,    cαii δˇ if k = 1, αii < 0,      0 if k = 2, 3,  ∑  ∗ ∗ g  2(ρ + ρ ) + 2cρ α ¯ ij ,  2 3    j∈N −{i,N } . 學. ‧ 國. 立. if k = 1,. v ni.    rs, if i = j, (k, l) = (3, 1),      0, otherwise,   c¯ αij δ, if j ̸= i, k = l = 1,. Ch  0,. engchi U. (3.23). (3.24). otherwise,. where δ = max{g ′ (xi ) : xi ∈ [−ρ∗1 , ρ∗1 ]} is defined in (1.4), α ¯ ii and α ¯ ij in (3.4), ρ∗k in Condition (D)*, µ ¯kl in Proposition 3.1; v is an arbitrary positive number and δˇ := min{g ′ (x) : x ∈ [−ρ∗1 , ρ∗1 ]}.. 15. DOI:10.6814/NCCU202000086.

(21) Proof. By (3.7), for E = (x1 , x2 , x3 ) ∈ R3 , we have                 . g1 (x1 ) = G1 (E) = g(x), (3.25). g1 (x2 ) = G2 (E) = 0, g1 (x3 ) = G3 (E) = 0.. ˇ L ˇ k and L ˆ k , k = 1, 2, 3, are now L ˇ 1 = δ, ˆ 1 = δ = max{g ′ (x) : By Proposition 2.1 and (3.25), L ˇ2 = L ˆ2 = L ˇ3 = L ˆ 3 = 0, where δ is defined in (1.4). By (3.25), ρg defined x ∈ [−ρ∗1 , ρ∗1 ]} and L k in (2.20), k = 1, 2, 3, can be chosen as. ρg1 = ρg ,. 學. ‧ 國. 立. 政治大 ρg2 = ρg3 = v,. ‧. ˇk, L ˆ k and ρg , k = 1, 2, 3, chosen above for an arbitrary v > 0. Combining those quantities of L k. y. Nat. as well as µ ˇk , µ ˆk , µ ¯kl and ρw k , for k, l ∈ {1, 2, 3} and k ̸= l, chosen in Proposition 3.1, the. al. er. io. sit. quantities in the assertion of Proposition 2.1 can be determined as those in (3.19)-(3.24).. n. Let us now introduce the following condition for system (3.2) which plays the role as Condition (S) for system (2.1):. Ch. engchi. i n U. v. Condition (S)*: b2 /3a + cαii δˇ < 0 and τ < τ˜i∗ , for all i ∈ {1, 2, . . . , N − 1}, where ˇ β¯i (3¯ τ˜i∗ := −3¯ ρhi (b2 /3a + cαii δ)/[ ρhi + ρ¯w i )], with ρ¯hi := 2c¯ αii ρg , ˇ β¯i := cαii δ[b2 /3a − 3a(ρ∗1 )2 − 2bρ∗1 + cαii (δ + δ)], ∑ ∗ ∗ ρ¯w α ¯ ij ρg . i := 2(ρ2 + ρ3 ) + 2c j∈N −{i,N }. Therein, the quantities δˇ and ρg are defined in Propositions 3.2; ρ∗k , k = 1, 2, 3, is defined in Condition (D)*; α ¯ ij and α ¯ ii are defined in (3.4). 16. DOI:10.6814/NCCU202000086.

(22) In the following lemma, we shall show the matrix M in (2.23) in terms of the quantities shown in Proposition 3.2. Moreover, the matrix M in (2.23) is determined by quantities in Proposition 2.1. Basically, the following lemma comes from Proposition 3.1 and 3.2. Lemma 3.3. Assume that Condition (D)* holds. Then, the matrix M = [M (kl) ]1≤k,l≤K , where (kl) ˜ (kl) ]1≤k,l≤K , where M ˜ (kl) = ˜ = [M M (kl) = [mij ]1≤i,j≤N −1 , in (2.23) is now denoted by M (kl). [m ˜ ij ]1≤i,j≤N −1 , with. 立 ik. n. al. :=.    rs,      0,   βij∗ ,. Ch. if i = j, (k, l) = (1, 2) or (1, 3), if i = j, (k, l) = (2, 1),. y.    1,       2ρ∗1 d,. sit. ‧ 國 :=. io (jl) βik. otherwise,. ik. ‧. Nat. (jl). µik. (3.26). 學. where. = j and (k, l) = (3, 3), 政治 if i 大.    r,      −µ(jl) − β (jl) ,. if i = j, (k, l) = (3, 1), otherwise,. i n U. v. e n ifgjc̸=hi,i(k, l) = (1, 1),.  0,. (3.27). er. (kl). m ˜ ij :=.    −b2 /3a − cαii δˇ − τ β¯i , if i = j and (k, l) = (1, 1),       1, if i = j and (k, l) = (2, 2),. (3.28). otherwise.. Herein, βij∗ := c¯ αij δ, δˇ is defined in Proposition 3.2; β¯i is defined in Condition (S1)*; α ¯ ij is defined in (3.4); ρ∗1 is defined in Condition (D)*. Proof. From Propositions 3.1 and 3.2, system (3.2) satisfies Condition (D)* with Q∗ = [−ρ∗1 , ρ∗1 ] × [−ρ∗2 , ρ∗2 ] × [−ρ∗3 , ρ∗3 ], µ ˇk , µ ˆk , ρw ¯kl , k, l ∈ {1, 2, 3} and k ̸= l, k , for k = 1, 2, 3, and µ ˇ k , and L ˆ k , k = 1, 2, 3, defined in determined in Proposition 3.1. As seen from (3.7), the terms L Proposition 2.1, are now chosen as those in Proposition 3.2. Notably, Condition (S)* implies that αii < 0 for all i = 1, . . . , N − 1, because b2 /3a > 0, c > 0, and δˇ > 0. Based on Propositions 3.1 and 3.2, system (3.2) satisfies the assertion of Proposition 2.1, with µ ˇ1 = −3a(ρ∗1 )2 − 2bρ∗1 , (jl) (jl) ˇ βˇi2 = βî2 = βˇi3 = βî3 = 0, µ µ ˆ1 = b2 /3a, µ ˇ2 = µ ˆ2 = −1, βˇi1 = cαii δ, βî1 = cαii δ, ¯ik = µik ,. 17. DOI:10.6814/NCCU202000086.

(23) (jl) (jl) β¯ik = βik . In particular, τ¯ = τ for system (3.2), cf. (2.8). As seen from the definition of ηik. in (2.27), ηik is now ηik = η˜ik . Consider η˜ik satisfying ∗ η˜ik := −ˆ µk − βîk + βik τ¯(ˇ µk + µ ˆk + βˇik + βîk ),. (3.29). ∗ ∗ ∗ where βi1 = cαii δ and βi2 = βi3 = 0 by (2.25), (3.20), and (3.21). Thus, by (3.29),.     −b2 /3a − cαii δˇ − τ β¯i ,    η˜ik = 1,      r,. if k = 1, (3.30). if k = 2, if k = 3.. 政治大 ¯ , defined in (2.27), is now Moreover, by (3.23) and立 (3.24), L ¯ (jl) = µ(jl) + β (jl) L ik ik ik. 學. ‧ 國. (jl) ik. (3.31). ‧. (jl) (jl) ˜ (kl) = with µik and βik defined in (3.27) and (3.28). By (3.29) and (3.31), the matrix M (kl). (kl). al. y. if i = j ∈ N − {N } and k = l ∈ K,. n. m ˜ ij =. sit. io.   η˜ik ,. er. Nat. [m ˜ ij ]1≤i,j≤N −1 defined in (2.24), now satisfies.  −L ¯ (jl) ,. C h otherwise. U engchi. ik. v ni. (3.32). ˜ defined in (3.26) come from (3.27), (3.28), (3.30), and (3.32). Therefore, the entries of M Theorem 3.4. Assume that Conditions (D)* and (S)* hold. Then, the system (3.2) globally synchronizes if the Gauss-Seidel iterations for the linear system: ˜ =0 Mv. (3.33). converges to zero, the unique solution of (3.33); or equivalently, ˜ syn := λ. max. ˜σ | : λ ˜ σ : eigenvalue of (D ˜ − L ˜ )−1 U ˜ } < 1, {|λ M M M. 1≤σ≤K×(N −1). (3.34). ˜ is defined in Lemma 3.3, and DM˜ , −LM˜ , −UM˜ represent the diagonal, strictly lowerwhere M ˜ respectively. triangular and strictly upper-triangular parts of M, 18. DOI:10.6814/NCCU202000086.

(24) Proof. By Proposition 3.1, system (3.2) satisfies Assumption (D) and (F) under Condition (D)*. In addition, system (3.2) satisfies Condition (S) under Condition (S)*, and the matrix M in (2.28) ˜ in Lemma 3.3. By Theorem 2.2, system (3.2) achieves global synchronization, if the is now M Gauss-Seidel iterations for the linear system (3.33), converge to zero. Hence, we complete the proof.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 19. DOI:10.6814/NCCU202000086.

(25) Chapter 4 Numerical examples 政治大. In this chapter, we will illustrate two examples with numerical simulations to demonstrate. 立. the effectiveness of the theoretical result derived in this thesis.. ‧ 國. 學. Example 1. Consider three coupled Hindmarsh-Rose neurons (3.2) with a = 1, b = 3, I = 3.0, d = 5, r = 0.005, s = 4, x0 = 1.6, c = 50, g(x) = 10 tanh(x/10), τ = 0.00001 and. ‧. . n. Ch. 0.4. −0.8. −0.1. engchi. 0.4. y. 0.6. sit. io. al. −1.0. er. Nat. A = [aij ]1≤i,j≤3.     =   . i n U 0.6. v. 0.4. −0.5.      .   . (4.1). From (1.4), (2.6), and (2.8), we have κ ¯=κ ˇ=κ ˆ = 0, a ¯ = 2.0 and τ¯ = τ = 0.00001. By (2.15) and (4.1), we obtain    A¯ = [αij ]1≤i,j≤2 =  .  −1.4 0.5. 0 −0.9.   . . (4.2). By (4.2), the quantities defined in (2.17) are now α ˇ 11 = α ˆ 11 = −1.4, α ¯ 11 = 1.4, α ˇ 22 = α ˆ 22 = −0.9, α ¯ 22 = 0.9, α ¯ 12 = 0, α ¯ 21 = 0.5. By numerical simulation cf. Figure 4.1, we can observe that the system satisfies Condition (D)* with Q∗ = [−ρ∗1 , ρ∗1 ] × [−ρ∗2 , ρ∗2 ] × [−ρ∗3 , ρ∗3 ], where ρ∗1 = 2, ρ∗2 = 9, ρ∗3 = 3.5.. (4.3). 20. DOI:10.6814/NCCU202000086.

(26) We note that Figure 4.1 demonstrates the evolution for the solution of the considered system, evolved from (3.5, 0.3, −2.1, 3.6, 0.4, −2.2, 3.7, 0.5, −2.3) at t0 = 0. It appears that the solution eventually enters, and then remains in Q∗ × Q∗ × Q∗ , where Q∗ is defined in (4.3). From Propositions 3.1 and 3.2, Lemma 3.3, and (4.3), we can obtain b2 /3a = 3, δˇ ≈ 0.97104, ρ¯h1 ≈ ¯ ¯ 276.32545, ρ¯h2 ≈ 177.63779, ρ¯w ¯w 1 ≈ 25, ρ 2 ≈ 123.68766, β1 ≈ 11079.11062, β2 ≈ 4916.11204, ∗ ∗ ∗ ∗ ∗ ∗ τ˜1∗ ≈ 0.00563, τ˜2∗ ≈ 0.00664, β12 = β13 = β23 = β31 = β32 = 0, and β21 = 25. By the. ˜ quantities above and Lemma 3.3, we can further verify that Condition (S)* holds and matrix M in (3.33) is approximately  −1.0. 64.16221. 0. −25.0. 40.19777. −20.0. 0. −1.0. 0. 0. −20.0. 0. 1.0. 0. 0. 0. 0. 0. 0.005. 0. −0.02. 0. 0. sit. ‧ 國. 0. io. y. n. al. 0. 0.005.            .          . (4.4). er. 0. 0. 0. ‧. −0.02. 1.0. −1.0. 學. 0. 立. 0. 政治大 0 −1.0. Nat.                      . . i n U. v. ˜ syn ≈ 0.60, cf. (2.29). By the matrix in (4.4), we can compute the corresponding value λ. Ch. engchi. Hence, the system attains global synchronization by Theorem 2.2.. Figure 4.2 and 4.3. demonstrate that the evolution for the solution of the considered system, evolved from (3.5, 0.3, −2.1, 3.6, 0.4, −2.2, 3.7, 0.5, −2.3) at t0 = 0. It appears that the solution remains oscillatory. Figures 4.3(a), 4.3(b) and 4.3(c) show that the solution (x1 (t), x2 (t), x3 (t)), xi (t) = (xi,1 (t), xi,2 (t), xi,3 (t)), with zi,k = xi,k (t) − xi+1,k (t) converging to zero for i = 1, 2 and k = 1, 2, 3. This demonstrates that the solution synchronizes. If we consider large coupling delay τ = 0.05 instead of τ = 0.00001, then Condition (S)* does not hold. Figures 4.4(a), 4.4(b) and 4.4(c) show that each of the solution does not synchronize, and exhibits asynchronous oscillatory behavior. This shows that large delay may destroy synchronization.. 21. DOI:10.6814/NCCU202000086.

(27) 3 2. x i,1. 1 0 -1 -2 0. 100. 200. 政治大 (a) x. 300. 400. 500. 600. 700. 800. 900. 1000. 800. 900. 1000. 800. 900. 1000. time t. 立. 200. 300. 400. 500. (b) xi,2. al. n. 4. 700. time t. io 5. 600. y. 100. Nat. 0. sit. -10. er. -5. ‧. ‧ 國. x i,2. 學. 0. x i,3. i,1. 3. Ch. 2. engchi. i n U. v. 1 0 0. 100. 200. 300. 400. 500. 600. 700. time t. (c) xi,3. Figure 4.1: Simulation for the solution of the system considered in Example 1, with τ = 0.00001: evolution of components xi,k (t), i = 1, 2, 3, and k = 1, 2, 3.. 22. DOI:10.6814/NCCU202000086.

(28) 政治大 (a) (x1,1 , x1,2 , x1,3 ). 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. i n U. (b) (x2,1 , x2,2 , x2,3 ). engchi. v. (c) (x3,1 , x3,2 , x3,3 ). Figure 4.2: Simulation for the solution of the system considered in Example 1, with τ = 0.00001: evolution of components (xi,1 (t), xi,2 (t), xi,3 (t)), i = 1, 2, 3.. 23. DOI:10.6814/NCCU202000086.

(29) 0.1 z 1,1 z 2,1. z 1,1 ,z 2,1. 0.05. 0. -0.05. -0.1 0. 200. 400. 600. 800. 1000. time t z 政 (a)治大 i,1. 立. z 1,2. 學. z 1,2 ,z 2,2 ‧ 國. 1. z 2,2. 0.5. ‧ sit. Nat. y. 0. -0.5. 200. 600. time t. n. al. 400. Ch. (b) zi,2. engchi. 800. er. io. 0. i n U. v. 6. z 1,3 ,z 2,3. 1000. z 1,3 z 2,3. 4 2 0 0. 200. 400. 600. 800. 1000. time t (c) zi,3. Figure 4.3: Simulation for the solution of the system considered in Example 1, with τ = 0.00001: evolution of components zi,k (t), i = 1, 2, and k = 1, 2, 3.. 24. DOI:10.6814/NCCU202000086.

(30) z 1,1. 10. z 1,1 ,z 2,1. z 2,1. 0. -10. 990. 992. 994. 996. 998. 1000. time t. z 政 (a)治大 i,1. 立. 20. z ,z ‧ 國. 學. z 1,2. 10. 2,2. z 2,2. 1,2. ‧. 0. sit 994. n. al. 992. Ch. 996. time t (b) zi,2. engchi. 998. er. io. -20 990. y. Nat. -10. i n U. 1000. v. 7 z 1,3. z 1,3 ,z 2,3. 6. z 2,3. 5 4 3 2 990. 992. 994. 996. 998. 1000. time t (c) zi,3. Figure 4.4: Simulation for the solution of the system considered in Example 1, with τ = 0.05: evolution of components zi,k (t), i = 1, 2, and k = 1, 2, 3.. 25. DOI:10.6814/NCCU202000086.

(31) Example 2. Consider three coupled Hindmarsh-Rose neurons (3.2) with a = 1, b = 3, I = 3.0, d = 5, r = 0.005, s = 4, x0 = 1.6, c = 200, g(x) = [tanh(x) + x]/2, τ = 0.00002 and . A = [aij ]1≤i,j≤3.     =   .  −0.4. 0.3. 0.1. 0.2. −0.7. 0.5. 0.1. 0.8. −0.9.     .   . (4.5). From (1.4), (2.6), and (2.8), we have κ ¯=κ ˇ=κ ˆ = 0, a ¯ = 3.2 and τ¯ = τ = 0.00002. By (2.15) and (4.5), we obtain. 立. 政治大  −0.6 0.4. ‧ 國. 0.1. −1.4. 學.  A¯ = [αij ]1≤i,j≤2 =  .    . . (4.6). ‧. By (4.6), the quantities defined in (2.17) are now α ˇ 11 = α ˆ 11 = −0.6, α ¯ 11 = 0.6, α ˇ 22 = α ˆ 22 = −1.4, α ¯ 22 = 1.4, α ¯ 12 = 0.4, α ¯ 21 = 0.1. By numerical simulation cf. Figure 4.5, we can observe. Nat. n. al. er. io. sit. y. that the system satisfies Condition (D)* with Q∗ = [−ρ∗1 , ρ∗1 ] × [−ρ∗2 , ρ∗2 ] × [−ρ∗3 , ρ∗3 ], where ρ∗1 = 2, ρ∗2 = 9, ρ∗3 = 3.5.. Ch. engchi U. v ni. (4.7). We note that Figure 4.5 demonstrates the evolution for the solution of the considered system, evolved from (0.7; 2.5; −2.8; 1; 2.7; −2.5; 0.5; 2.9; −2.2) at t0 = 0. It appears that the solution eventually enters and then remains in Q∗ × Q∗ × Q∗ , where Q∗ is defined in (4.7). From Propositions 3.1 and 3.2, Lemma 3.3, and (4.7), we can obtain b2 /3a = 3, δˇ ≈ 0.53533, ρ¯h1 ≈ ¯ ¯ ¯w 444.60414, ρ¯h2 ≈ 1037.40965, ρ¯w 2 ≈ 99.10069, β1 ≈ 37694.82178, β2 ≈ 1 ≈ 321.40276, ρ ∗ ∗ ∗ = = 25.0, and β13 = 99.99999, β21 195427.36302, τ˜1∗ ≈ 0.00165, τ˜2∗ ≈ 91427.56468, β12 ∗ ∗ ∗ β23 = β31 = β32 = 0. By the quantities above and Lemma 3.3, we can further verify that. 26. DOI:10.6814/NCCU202000086.

(32) ˜ in (3.33) is approximately Condition (S)* holds and matrix M                       .  60.74648. −80.0. −1.0. 0. −1.0. 0. −20.0. 144.36613. 0. −1.0. 0. −1.0. −20.0. 0. 1.0. 0. 0. 0. 0. −20.0. 0. 1.0. 0. 0. −0.02. 0. 0. 0. 0.005. 0. 0. −0.02. 0. 0.005. 立. 政0 治 0大.            .          . (4.8). ≈. 0.64,. ˜ syn By the matrix in (4.8), we can compute the corresponding value λ. ‧ 國. 學. cf. (2.29).. Hence, the system attains global synchronization by Theorem 2.2.. Figures. 4.5 and 4.6 demonstrate that the evaluation for the solution of the considered system,. ‧. evolved from (0.7, 2.5, −2.8, 1, 2.7, −2.5, 0.5, 2.9, −2.2) at t0 = 0.. It shows that the. y. Nat. solution remains oscillatory. Figures 4.7(a), 4.7(b) and 4.7(c) illustrate that the solution. io. sit. (x1 (t), x2 (t), x3 (t)), xi (t) = (xi,1 (t), xi,2 (t), xi,3 (t)), with zi,k = xi,k (t) − xi+1,k (t) converging. n. al. er. to zero for i = 1, 2 and k = 1, 2, 3. This demonstrates that the solution synchronizes.. i n U. v. If we consider large coupling delay τ = 0.1 instead of τ = 0.00002, then Condition. Ch. engchi. (S)* does not hold. Figures 4.8(a), 4.8(b) and 4.8(c) show that each of the solution does not synchronize, and exhibits asynchronous oscillatory behavior. This shows that large coupling delay may lead to asynchrony. Remark 4.1. Among the existing studies on synchronization of coupled systems, the synchronization theories in [9, 16–20, 30] required that all non-zero and off-diagonal entries of the connection matrix have the same sign. The connection matrices considered in Examples 1 and 2 have offdiagonal entries with the mixed signs, and do not satisfy the circulant condition required in [29]. Therefore, the synchronization of systems considered in Examples 1 and 2 can not be treated by previous approaches in [9, 16–20, 29, 30].. 27. DOI:10.6814/NCCU202000086.

(33) x i,1. 2 0 -2 0. 200. 立. 400. 600. 治t 政 (a)time x 大. 400. 600. time t (b) xi,2. n. al. x i,3. Ch. engchi. 1000. sit. io 4. 800. y. 200. er. Nat. 0. ‧. ‧ 國. x i,2. 學. -10. 1000. i,1. 0 -5. 800. i n U. v. 2 0 0. 200. 400. 600. 800. 1000. time t (c) xi,3. Figure 4.5: Simulation for the solution of the system considered in Example 2, with τ = 0.00002: evolution of components xi,k (t), i = 1, 2, 3, and k = 1, 2, 3.. 28. DOI:10.6814/NCCU202000086.

(34) (a) (x ,治 x ,x ) 政大 1,1. 立. 1,2. 1,3. ‧. ‧ 國. 學. n. (b) (x2,1 , x2,2 , x2,3 ). Ch. engchi. er. io. sit. y. Nat. al. i n U. v. (c) (x3,1 , x3,2 , x3,3 ). Figure 4.6: Simulation for the solution of the system considered in Example 2, with τ = 0.00002: evolution of components (xi,1 (t), xi,2 (t), xi,3 (t)), i = 1, 2, 3.. 29. DOI:10.6814/NCCU202000086.

(35) 0.2 z 1,1 z 2,1. z 1,1 ,z 2,1. 0.1 0 -0.1 -0.2 0. 200. 400. 600. 800. 1000. time t (a) zi,1. 1.5. 立. 2,2 1,2. 0. z 2,2. ‧. 0.5. z 1,2. 學. ‧z ,z 國. 1. 政治大. y. sit. Nat. -0.5 -1. 200. 600. time t. n. al. 400. Ch. (b) zi,2. engchi. 800. er. io. 0. i n U. 1000. v. z 1,3. 6. z 1,3 ,z 2,3. z 2,3. 4 2 0 0. 200. 400. 600. 800. 1000. time t (c) zi,3. Figure 4.7: Simulation for the solution of the system considered in Example 2, with τ = 0.00002: evolution of components zi,k (t), i = 1, 2, and k = 1, 2, 3. 30. DOI:10.6814/NCCU202000086.

(36) 30 z 1,1. 20. z 2,1. z 1,1 ,z 2,1. 10 0 -10 -20 -30 990. 992. 994. 996. 998. 1000. time t. (a) z 政治大 i,1. 立. 500. z 1,2 z 2,2. z 1,2 ,z 2,2. 300. ‧. ‧ 國. 學. 400. io. 990. -5. 994. 996. time t. n. al. 992. Ch. 998. sit. 100. 1000. er. Nat. y. 200. (b) zi,2. engchi. i n U. v. z 1,3. z 1,3 ,z 2,3. z 2,3. -6. -7. -8 990. 992. 994. 996. 998. time t. (c) zi,3. Figure 4.8: Simulation for the solution of the system considered in Example 2, with τ = 0.1: evolution of components zi,k (t), i = 1, 2, and k = 1, 2, 3. 31. DOI:10.6814/NCCU202000086.

(37) Chapter 5 Conclusion 政治大. In the literature, there has been some investigations which addressed the global synchronization. 立. of coupled systems of Hindmarsh-Rose neurons. Among these investigations, most of them. ‧ 國. 學. considered linear coupling functions and did not consider coupling time-delays. In addition, these studies commonly required that all nonzero and non-diagonal entries of the connection. ‧. matrix have the same sign. In this thesis, we establish the global synchronization of non-linearly coupled systems of Hindmarsh-Rose neurons based on the theory in [33]. The coupling terms. y. Nat. sit. could be with time delays, the coupling function could be nonlinear, and the connection matrix. er. io. could be with both negative and positive off-diagonal entries. By applying the synchronization. al. n. v i n C h be treated byUthe previous methods, cf. Hindmarsh-Rose neurons, which cannot engchi. criterion derived in this thesis, we can investigate the synchronization of systems of coupled Remark 4.1. and Examples 1, and 2.. 32. DOI:10.6814/NCCU202000086.

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(42) [38] Shi Xia and Lu Qi-Shao.. Firing patterns and complete synchronization of coupled. hindmarsh–rose neurons. Chinese Physics, 14(1):77–85, dec 2004. [39] T. Yang, Z. Meng, G. Shi, Y. Hong, and K. H. Johansson. Network synchronization with nonlinear dynamics and switching interactions. IEEE Transactions on Automatic Control, 61(10):3103–3108, October 2016.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 37. DOI:10.6814/NCCU202000086.

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