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Pinning synchronization of unilateral coupling neuron network with stochastic noise
Applied Mathematics and Computation 232 (2014) 1242–1248
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Pinning synchronization of unilateral coupling neuron network with stochastic noise Xuerong Shi a,⇑, Lixin Han a, Zuolei Wang b, Keming Tang c a
Computer and Information Engineering College, Hohai University, Nanjing 210098, China School of Mathematical Sciences, Yancheng Teachers University, Yancheng 224002, China c College of Information Science and Technology, Yancheng Teachers University, Yancheng 224002, China b
a r t i c l e
i n f o
Keywords: Pinning synchronization Unilateral coupling network Hindmarsh–Rose neuron model Stochastic noise
a b s t r a c t In this paper, pinning synchronization of unilateral coupling time delay neuron network with stochastic noise is investigated. Based on Lyapunov stability theory, by designing appropriate controller and particular Lyapunov function, pinning synchronization of unilateral coupling Hindmarsh–Rose network with stochastic noise is obtained. This method needs only one single controller. Simulation results are given to verify the effectiveness of the proposed scheme. From the simulations, the relation between the time needed to achieve pinning synchronization of unilateral coupling neuron network with stochastic noise and the number of nodes, the noise intensity, the coupling intensity is illustrated. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Synchronization is ubiquitous in nature and plays an important role in many fields, such as biology, ecology [1,2]. Various synchronizations are observed in biological experiments and numerical simulations, for example, mutual synchronization [3], entrainment and chaotic synchronization [4]. With the development of nonlinear dynamics, the classical concept of synchronization has been extended from the phase locking of periodic oscillators to that of chaotic oscillators. Many kinds of synchronizations are described, e.g., complete synchronization, phase synchronization, generalized synchronization, etc. [5–7]. So far, many approaches have been proposed for synchronization of chaotic systems, for instance, linear and nonlinear feedback synchronization method [8,9], adaptive synchronization method [10], time-delay feedback approach [11], backstepping design method [12], sliding mode control method [13], impulsive synchronization method [14], etc. Most of the existing methods can synchronize two identical or different low-dimensional chaotic systems. In the past decade, complex network attracted more and more attention of researchers. It is partially due to the fact that any large-scale and complicated system in real world can be modeled by a complex network, in which nodes can be seen as the elements of the system and edges can be considered as interactions between nodes. For example, the WWW, the Internet, neural network, social network and cited network are all complex networks. To master the complicated nature of complex network, more and more people began to investigate dynamics of complex network, such as robustness [15], pinning synchronization [16,17].
⇑ Corresponding author. E-mail address: [email protected] (X. Shi). http://dx.doi.org/10.1016/j.amc.2014.01.126 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
X. Shi et al. / Applied Mathematics and Computation 232 (2014) 1242–1248
1243
As a complex network, neural network exhibits collective dynamics, which give us insight into the mechanism of information processing and transfer. This fact makes it interesting to study the coupled neurons in recent years. The phenomenological neuron model proposed by Hindmarsh and Rose [18] may be seen either as a generalization of the Fitzhugh equations [19] or as a simplification of the physiologically realistic model proposed by Hodgkin and Huxley [20]. Hindmarsh–Rose (HR) model is a single compartment model and is able to reproduce all the dynamical behaviors of neural network. To understand it, many people explored the dynamics of neural networks [21,22]. Recently, kinds of synchronization in the model of HR neurons have been extensively investigated, such as robust synchronization [23], spike phase synchronization [24], adaptive synchronization [25]. Inspired by above work, pinning synchronization of unilateral coupling complex neural network with stochastic noise is investigated in this paper. Other parts of the paper are arranged as follows. HR neuron model is presented in Section 2. In Section 3, schemes are given to realize the pinning synchronization between the given reference signal and time-delay HR neuron system with stochastic noise. To verify the theoretical results, numerical simulations are given in Section 4. Some conclusions are drawn in Section 5.
2. Unilateral coupling Hindmarsh–Rose (HR) neural network with time delay and stochastic noise In this paper, HR neuronal model with time-delay is considered as follows: 3 x_ 1 ¼ ax21 bx1 þ y1 z1 ðt sÞ þ Iext ; 2 y_ 1 ¼ c dx1 y1 ; z_ 1 ¼ rðSðx1 þ kÞ z1 Þ;
ð1Þ
where s > 0 is the time delay. When s = 0, model (1) is HR neuronal model proposed by Hindmarsh and Rose as a mathematical representation of the firing behavior of neurons [18]. It was originally introduced to give a bursting type with long interspike intervals of real neurons. In real neuron system, time-delay always exists when signals are communicated among neurons, even in the same neuron. Therefore, it is necessary to investigate the dynamics of HR system with s > 0. When s > 0, system (1) with time delay can be regarded as self-feedback neuron system. In system (1), the variables x1, y1, and z1 represent the membrane potential of the neuron, the recovery variable, and the adaptation current, respectively. The current Iext represents an external influence on the system. a, b, c, d, r, S, k are real constants. Model (1) may describe regular bursting or chaotic bursting for certain domains of the parameters. When the time delay s is chosen as 1, and other parameters are chosen as a = 3.0, b = 1.0, c = 1.0, d = 5.0, r = 0.006, S = 4.0, k = 1.6, system (1) can show various complex dynamical behaviors with the changing of Iext. For instance, when Iext = 1.3, system (1) has periodic solution (see Fig. 1). When Iext = 2.6, regular bursting of system (1) is depicted in Fig. 2. Chaotic bursting can be observed in Fig. 3 with Iext = 3.1. Since the existence of stochastic noise in the information processing of large ensembles of neurons, it is necessary and important to take into account the stochastic noise. There have been some results concerning stochastic noises on neural networks [26,27]. In this paper, both the stochastic noise and the time delay are all considered in neural network. Assume that the network involves N nodes and each node is a HR system, the complex unilateral coupling time delay neural network stochastic noise is expressed as follows:
Fig. 1. Periodic solution of system (1) when Iext = 1.3. (a) Phase portrait, (b) time series.
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Fig. 2. Regular bursting of system (1) when Iext = 2.6. (a) Phase portrait, (b) time series.
Fig. 3. Chaotic bursting of system (1) when Iext = 3.1. (a) Phase portrait, (b) time series. 3 x_ 1 ¼ ax21 bx1 þ y1 z1 ðt sÞ þ Iext ; 2 y_ 1 ¼ c dx1 y1 ; z_ 1 ¼ rðSðx1 þ kÞ z1 Þ: 3
dxi ¼ ½ax2i bxi þ yi zi ðt sÞ þ Iext þ Hi ðxi1 xi Þdt þ hðxi1 xi ÞddðtÞ;
ð2Þ
2
dyi ¼ ½c dxi yi dt; dzi ¼ ½rðSðxi þ kÞ zi Þdt;
ði ¼ 2; . . . ; NÞ;
where h is noise intensity. Hi (i = 2, . . ., N) are coupling intensity of nodes. d(t) is a one-dimension Brownian motion satisfying e{dd(t)} = 0, e{dd2(t)} = dt, where e{} is the expectation operator. 3. Control scheme In this section, with the aid of appropriate controller, the pinning synchronization of complex HR neuronal network is investigated. For any given reference signal s(t), pinning synchronization between system (2) and s(t) can be obtained. To this end, the controlled unilateral coupling neural network with time delay and stochastic noise is expressed as follows 3 x_ 1 ¼ ax21 bx1 þ y1 z1 ðt sÞ þ Iext ; 2 y_ 1 ¼ c dx1 y1 þ u; z_ 1 ¼ rðSðx1 þ kÞ z1 Þ: 3
dxi ¼ ½ax2i bxi þ yi zi ðt sÞ þ Iext þ Hi ðxi1 xi Þdt þ hðxi1 xi ÞddðtÞ; 2
dyi ¼ ½c dxi yi dt; dzi ¼ ½rðSðxi þ kÞ zi Þdt;
ði ¼ 2; . . . ; NÞ;
ð3Þ
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where u is external control input to be designed. Our aim is to make the output signal x1 of system (3) synchronize any given signal s(t) via designing appropriate controller. s(t) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Let e = x1 s(t), the following Theorem 1 can be obtained. Theorem 1. For a given reference signal s(t), synchronization will occur between s(t) and the output signal x1 of system (3) under controller u in (4) 2
2
u ¼ 3bx1 x_ 1 2ax1 x_ 1 þ dx1 þ y1 c þ rðSðx1 ðt sÞ þ kÞ z1 ðt sÞÞ þ €sðtÞ ae 3be_ 2b2 e:
ð4Þ
That is
lim e ¼ lim ½x1 sðtÞ ¼ 0:
t!1
ð5Þ
t!1
Proof. The positive and changeable Lyapunov function could be measured by 2 V 1 ¼ ae2 þ ðe_ þ beÞ ;
ð6Þ
where a and b are positive gain coefficients, the dot over e denotes the differential variable e to time. The differential of V1 to time t is approached by
_ ¼ 2bV 1 þ 2bV 1 þ 2aee_ þ 2ðe_ þ beÞð€e þ beÞ _ V_ 1 ¼ 2aee_ þ 2ðe_ þ beÞð€e þ beÞ 2
_ ¼ 2bV 1 þ 2b½ae2 þ ðe_ þ beÞ þ 2aee_ þ 2ðe_ þ beÞð€e þ beÞ 2
_ ¼ 2bV 1 þ 2ð€e þ 2be_ þ ae þ b2 eÞðe_ þ beÞ: ¼ 2bV 1 þ 2aeðe_ þ beÞ þ 2bðe_ þ beÞ þ 2ðe_ þ beÞð€e þ beÞ
Fig. 4. Stabilization of errors ei (i = 1, 2, . . ., N) with one controller in (4). Hi = 20, h = 2.0.
ð7Þ
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Substituting (3) and (4) into (7), it can be gotten that
dV 1 2 ¼ 2bV 1 2bðe_ þ beÞ < 0: dt
ð8Þ
According to Lyapunov stability theory, the error of corresponding variables will stabilize to a certain threshold. As a result, the reference signal s(t) and the output signal x1 of system (3) will reach synchronization. Namely,
lim e ¼ lim ½x1 sðtÞ ¼ 0:
t!1
t!1
Proof is complete.h If xi (i = 2, . . ., N) can pinning synchronize x1, while x1 of system (3) follows the reference signal s(t), pinning synchronization between the reference signal s(t) and system (3) can be obtained ultimately. Let
ei ¼ xi x1
ði ¼ 1; . . . ; NÞ;
ð9Þ
and we have the following Theorem 2. Theorem 2. xi (i = 2, . . ., N) with stochastic noise can pinning synchronize x1 in system (3) under the controller u in (4). Proof. According to (9) and system (3), we have
dei ¼ ½aðxi þ x1 Þei bðx2i þ xi x1 þ x21 Þei þ ðyi y1 Þ ðzi ðt sÞ z1 ðt sÞÞ þ Hi ðei1 ei Þdt þ hðei1 ei ÞddðtÞ ði ¼ 2; . . . ; NÞ;
ð10Þ
which can be written as
E_ ¼ AE þ u1 u2
ð11Þ T
T
2 6 6 6 6 A¼6 6 6 4
M 2 H2 h H2 þ h
0
...
0
0
M 3 H3 h . . .
0
0
0 .. .
0 .. .
0 .. .
H3 þ h .. .
0 M i ¼ aðxi þ x1 Þ
bðx2i
x21 Þ
... .. .
3 7 7 7 7 7; 7 7 5
. . . HN1 þ h M N HN h
0 þ xi x1 þ
u2 ¼ ðz2 ðt sÞ z1 ðt sÞ; z3 ðt sÞ z1 ðt sÞ; . . . ;
u1 ¼ ðy2 y1 ; y3 y1 ; . . . ; yN y1 Þ ,
where E ¼ ðe2 ; e3 ; . . . ; eN Þ , zN ðt sÞ z1 ðt sÞÞT ;
ði ¼ 2; . . . ; NÞ: Consider Lyapunov function as following
T
V ¼ E E:
ð12Þ
Obviously, the Lyapunov function V in (12) is a positive function. Making use of (11), time derivative of V in (12) can be expressed as
V_ ¼ E_ T E þ ET E_ ¼ ðAE þ u1 u2 ÞT E þ ET ðAE þ u1 u2 Þ ¼ ET ðAT þ AÞE þ ðu1 u2 ÞT E þ ET ðu1 u2 Þ ¼ ET ðAT þ AÞE þ 2ET ðu1 u2 Þ:
ð13Þ
Since a chaotic system has bounded trajectories, there exists a positive constant L, such that |xi| < L, |yi| < L, |zi| < L, |zi (t s)| < L (i = 1, 2, . . ., N), therefore u1, u2 and Mi (i = 2, . . ., N) are bounded. If Hi (i = 2, . . ., N) are sufficiently large, AT + A is negative definite, thus V_ < 0. According to Lyapunov stability theory, we know that error system (10) or (11) is asymptotically stable. Therefore, the unilateral coupled time delay HR neuron system (3) with stochastic noise achieves pinning synchronization.h According to Theorems 1 and 2, for any given signal, unilateral coupling neural network can pinning synchronize it under appropriate control. 4. Simulation In this section, simulations are given to illustrate the effectiveness of the proposed method. In simulations, coupling intensity Hi (i = 2, . . ., N) is taken as the same value. The original values are taken as constants. Each node is a HR system. The reference signal s(t) is taken as the behavior of an isolated HR neural node. The parameters are chosen as a = 3.0, b = 1.0, c = 1.0, d = 5.0, r = 0.006, S = 4.0, k = 1.6, Iext = 3.1, and the time delay is chosen as s = 1, with which the HR system
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Fig. 5. Synchronization of errors ei (i = 1, 2, . . ., 20) with one controller in (4), Hi = 20.
Fig. 6. Synchronization of errors ei (i = 1, 2, . . ., 20) with one controller in (4), h = 2.0.
is chaotic bursting. To analyze the pinning synchronization process, numerical simulations are carried out from following three aspects. Firstly, we consider the stabilization process of error system (10) or (11) with node increasing. For this reason, complex neural networks with 5, 10, 20 nodes are simulated, respectively, in which Hi = 20, h = 2.0. Pinning synchronization errors are illustrated in Fig. 4, from which it is obviously to see that the neural network can pinning synchronize the given signal s(t). At the same time, we know that, more time is needed to realize the pinning synchronization of a complex neural network with the number of nodes increasing. Secondly, the convergence processes of error systems with noise intensity varying are showed in Fig. 5, where Hi = 20, N = 20, and h is chosen as 0.5, 2.0, respectively. From the numerical results in Fig. 5, we can predict that the greater the noise is, the more time is needed to realize pinning synchronization of a complex neural network with stochastic noise. Finally, when the number of nodes and the noise intensity are constants, the evolution of the errors (10) or (11) with coupling intensity changing are depicted (see Fig. 6, where N = 20, h = 2.0, Hi = 20, 40). It is found that the less time is required to realize pinning synchronization with larger coupling intensity Hi. 5. Conclusion In this paper, based on Lyapunov stability theory, scheme to achieve the pinning synchronization of unilateral coupling neural network with time delay and stochastic noise is proposed. We not only test the validity of the proposed scheme from theoretical analysis, but also verify the effectiveness of the scheme from numerical simulations. Furthermore, from the simulations, some results can be known as follows: (1) When coupling intensity and noise intensity are constants, the time of convergence is positively correlated with the number of nodes.
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(2) When coupling intensity and the number of nodes are constants, the time needed to stabilize error system is positively correlated with noise intensity. (3) When noise intensity and the number of nodes are constants, the time required to achieve pinning synchronization is negatively correlated with the coupling intensity. Because the considered HR neural network takes into account time-delay and stochastic noise, it is attractive and practical in understanding the dynamic behavior of neural network. To achieve pinning synchronization in unilateral coupling neural network, coupling strength should be large enough. Acknowledgements This work is supported by National Natural Science Foundation of China (Grant Nos. 11102180, 61379064 and 61273106), the Natural Science Foundation of the Jiangsu Province of China (Grant No. BK2012672), and the Qing Lan Project. References [1] R.E. Mirollo, S.H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math. 50 (1990) 1645–1662. [2] B. Blasius, L. Stone, Chaos and phase synchronization in ecological system, Int. J. Bifurcation Chaos 10 (2000) 2361–2380. [3] I. Baruchi, V. Volman, N. Raichman, M. Shein, E. Ben-Jacob, The emergence and properties of mutual synchronization in vitro coupled cortical networks, Eur. J. Neurosci. 28 (2008) 1825–1835. [4] R.C. Elson, A.I. Selverston, R. Huerta, N.F. Rulkov, M.I. Rabinovich, H.D. Abarbanel, Synchronous behavior of two coupled biological neurons, Phys. Rev. Lett. 81 (1998) 5692–5695. [5] J. Ma, F. Li, L. Huang, W.Y. Jin, Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 3770–3785. [6] H. Fujisaka, T. Yamada, G. Kinoshita, T. Kono, Chaotic phase synchronization and phase diffusion, Phys. D Nonlinear Phenom. 205 (2005) 41–47. [7] J.Z. Yang, G. Hu, Three types of generalized synchronization, Phys. Lett. A 361 (2007) 332–335. [8] F. Liu, Y. Ren, X.M. Shan, Z.L. Qiu, A linear feedback synchronization theorem for a class of chaotic systems, Chaos, Solitons Fractals 13 (2002) 723–730. [9] M.Y. Chen, Z.Z. Han, Controlling and synchronizing chaotic Genesio system via nonlinear feedback control, Chaos, Solitons Fractals 17 (2003) 709–716. [10] M.T. Yassen, Adaptive control and synchronization of a modified Chua’s circuit system, Appl. Math. Comput. 135 (2003) 113–118. [11] J.A. Lu, X.Q. Wu, X.P. Han, J.H. Lü, Adaptive feedback synchronization of a unified chaotic system, Phys. Lett. A 329 (2004) 327–333. [12] Y.G. Yu, S.C. Zhang, Controlling uncertain Lü system using backstepping design, Chaos, Solitons Fractals 15 (2003) 897–902. [13] K. Konishi, M. Hirai, H. Kokame, Sliding mode control for a class of chaotic systems, Phys. Lett. A 245 (1998) 511–517. [14] R. Kilic, Experimental study on impulsive synchronization between two modified Chua’s circuits, Nonlinear Anal. Real World Appl. 7 (2006) 1298– 1303. [15] D. Heide, M. Schäfer, M. Greiner, Robustness of networks against fluctuation-induced cascading failures, Phys. Rev. E 77 (2008) 056103. [16] C. Hu, J. Yu, H.J. Jiang, Z.D. Teng, Pinning synchronization of weighted complex networks with variable delays and adaptive coupling weights, Nonlinear Dyn. 67 (2012) 1373–1385. [17] L.P. Deng, Z.Y. Wu, Q.C. Wu, Pinning synchronization of complex network with non-derivative and derivative coupling, Nonlinear Dyn. 73 (2013) 775– 782. [18] J.L. Hindmarsh, R.M. Rose, A model of the nerve impulse using two first-order differential equations, Nature 296 (1982) 162. [19] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961) 445–446. [20] A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952) 500–544. [21] W. Wu, T.P. Chen, Global synchronization criteria of linearly coupled neural network systems with time-varying coupling, Neural Networks 19 (2008) 319–332. [22] J.D. Cao, J.Q. Lu, Adaptive synchronization of neural networks with or without time-varying delay, Chaos: An Interdisciplinary J. Nonlinear Sci. 16 (2006). 013133–013133-6. [23] C. Zheng, J.D. Cao, Robust synchronization of dynamical network with impulsive disturbances and uncertain parameters, Int. J. Control Autom. 11 (2013) 657–665. [24] M. Jalili, Spike phase synchronization in delayed-coupled neural networks: uniform vs. non-uniform transmission delay, Chaos: An Interdisciplinary J. Nonlinear Sci. 23 (2013). 013146–013146-6.. [25] X.R. Shi, Z.L. Wang, Adaptive synchronization of time delay Hindmarsh–Rose neuron system via self-feedback, Nonlinear Dyn. 69 (2012) 2147–2153. [26] Y. Sun, J.D. Cao, Stabilization of stochastic delayed neural networks with Markovian switching, Asian J. Control 10 (2008) 327–340. [27] L. Wan, J.H. Sun, Mean square exponential stability of stochastic delayed Hopfield neural networks, Phys. Lett. A 343 (2005) 306–318.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Pinning synchronization of unilateral coupling neuron network with stochastic noise Xuerong Shi a,⇑, Lixin Han a, Zuolei Wang b, Keming Tang c a
Computer and Information Engineering College, Hohai University, Nanjing 210098, China School of Mathematical Sciences, Yancheng Teachers University, Yancheng 224002, China c College of Information Science and Technology, Yancheng Teachers University, Yancheng 224002, China b
a r t i c l e
i n f o
Keywords: Pinning synchronization Unilateral coupling network Hindmarsh–Rose neuron model Stochastic noise
a b s t r a c t In this paper, pinning synchronization of unilateral coupling time delay neuron network with stochastic noise is investigated. Based on Lyapunov stability theory, by designing appropriate controller and particular Lyapunov function, pinning synchronization of unilateral coupling Hindmarsh–Rose network with stochastic noise is obtained. This method needs only one single controller. Simulation results are given to verify the effectiveness of the proposed scheme. From the simulations, the relation between the time needed to achieve pinning synchronization of unilateral coupling neuron network with stochastic noise and the number of nodes, the noise intensity, the coupling intensity is illustrated. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Synchronization is ubiquitous in nature and plays an important role in many fields, such as biology, ecology [1,2]. Various synchronizations are observed in biological experiments and numerical simulations, for example, mutual synchronization [3], entrainment and chaotic synchronization [4]. With the development of nonlinear dynamics, the classical concept of synchronization has been extended from the phase locking of periodic oscillators to that of chaotic oscillators. Many kinds of synchronizations are described, e.g., complete synchronization, phase synchronization, generalized synchronization, etc. [5–7]. So far, many approaches have been proposed for synchronization of chaotic systems, for instance, linear and nonlinear feedback synchronization method [8,9], adaptive synchronization method [10], time-delay feedback approach [11], backstepping design method [12], sliding mode control method [13], impulsive synchronization method [14], etc. Most of the existing methods can synchronize two identical or different low-dimensional chaotic systems. In the past decade, complex network attracted more and more attention of researchers. It is partially due to the fact that any large-scale and complicated system in real world can be modeled by a complex network, in which nodes can be seen as the elements of the system and edges can be considered as interactions between nodes. For example, the WWW, the Internet, neural network, social network and cited network are all complex networks. To master the complicated nature of complex network, more and more people began to investigate dynamics of complex network, such as robustness [15], pinning synchronization [16,17].
⇑ Corresponding author. E-mail address: [email protected] (X. Shi). http://dx.doi.org/10.1016/j.amc.2014.01.126 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
X. Shi et al. / Applied Mathematics and Computation 232 (2014) 1242–1248
1243
As a complex network, neural network exhibits collective dynamics, which give us insight into the mechanism of information processing and transfer. This fact makes it interesting to study the coupled neurons in recent years. The phenomenological neuron model proposed by Hindmarsh and Rose [18] may be seen either as a generalization of the Fitzhugh equations [19] or as a simplification of the physiologically realistic model proposed by Hodgkin and Huxley [20]. Hindmarsh–Rose (HR) model is a single compartment model and is able to reproduce all the dynamical behaviors of neural network. To understand it, many people explored the dynamics of neural networks [21,22]. Recently, kinds of synchronization in the model of HR neurons have been extensively investigated, such as robust synchronization [23], spike phase synchronization [24], adaptive synchronization [25]. Inspired by above work, pinning synchronization of unilateral coupling complex neural network with stochastic noise is investigated in this paper. Other parts of the paper are arranged as follows. HR neuron model is presented in Section 2. In Section 3, schemes are given to realize the pinning synchronization between the given reference signal and time-delay HR neuron system with stochastic noise. To verify the theoretical results, numerical simulations are given in Section 4. Some conclusions are drawn in Section 5.
2. Unilateral coupling Hindmarsh–Rose (HR) neural network with time delay and stochastic noise In this paper, HR neuronal model with time-delay is considered as follows: 3 x_ 1 ¼ ax21 bx1 þ y1 z1 ðt sÞ þ Iext ; 2 y_ 1 ¼ c dx1 y1 ; z_ 1 ¼ rðSðx1 þ kÞ z1 Þ;
ð1Þ
where s > 0 is the time delay. When s = 0, model (1) is HR neuronal model proposed by Hindmarsh and Rose as a mathematical representation of the firing behavior of neurons [18]. It was originally introduced to give a bursting type with long interspike intervals of real neurons. In real neuron system, time-delay always exists when signals are communicated among neurons, even in the same neuron. Therefore, it is necessary to investigate the dynamics of HR system with s > 0. When s > 0, system (1) with time delay can be regarded as self-feedback neuron system. In system (1), the variables x1, y1, and z1 represent the membrane potential of the neuron, the recovery variable, and the adaptation current, respectively. The current Iext represents an external influence on the system. a, b, c, d, r, S, k are real constants. Model (1) may describe regular bursting or chaotic bursting for certain domains of the parameters. When the time delay s is chosen as 1, and other parameters are chosen as a = 3.0, b = 1.0, c = 1.0, d = 5.0, r = 0.006, S = 4.0, k = 1.6, system (1) can show various complex dynamical behaviors with the changing of Iext. For instance, when Iext = 1.3, system (1) has periodic solution (see Fig. 1). When Iext = 2.6, regular bursting of system (1) is depicted in Fig. 2. Chaotic bursting can be observed in Fig. 3 with Iext = 3.1. Since the existence of stochastic noise in the information processing of large ensembles of neurons, it is necessary and important to take into account the stochastic noise. There have been some results concerning stochastic noises on neural networks [26,27]. In this paper, both the stochastic noise and the time delay are all considered in neural network. Assume that the network involves N nodes and each node is a HR system, the complex unilateral coupling time delay neural network stochastic noise is expressed as follows:
Fig. 1. Periodic solution of system (1) when Iext = 1.3. (a) Phase portrait, (b) time series.
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Fig. 2. Regular bursting of system (1) when Iext = 2.6. (a) Phase portrait, (b) time series.
Fig. 3. Chaotic bursting of system (1) when Iext = 3.1. (a) Phase portrait, (b) time series. 3 x_ 1 ¼ ax21 bx1 þ y1 z1 ðt sÞ þ Iext ; 2 y_ 1 ¼ c dx1 y1 ; z_ 1 ¼ rðSðx1 þ kÞ z1 Þ: 3
dxi ¼ ½ax2i bxi þ yi zi ðt sÞ þ Iext þ Hi ðxi1 xi Þdt þ hðxi1 xi ÞddðtÞ;
ð2Þ
2
dyi ¼ ½c dxi yi dt; dzi ¼ ½rðSðxi þ kÞ zi Þdt;
ði ¼ 2; . . . ; NÞ;
where h is noise intensity. Hi (i = 2, . . ., N) are coupling intensity of nodes. d(t) is a one-dimension Brownian motion satisfying e{dd(t)} = 0, e{dd2(t)} = dt, where e{} is the expectation operator. 3. Control scheme In this section, with the aid of appropriate controller, the pinning synchronization of complex HR neuronal network is investigated. For any given reference signal s(t), pinning synchronization between system (2) and s(t) can be obtained. To this end, the controlled unilateral coupling neural network with time delay and stochastic noise is expressed as follows 3 x_ 1 ¼ ax21 bx1 þ y1 z1 ðt sÞ þ Iext ; 2 y_ 1 ¼ c dx1 y1 þ u; z_ 1 ¼ rðSðx1 þ kÞ z1 Þ: 3
dxi ¼ ½ax2i bxi þ yi zi ðt sÞ þ Iext þ Hi ðxi1 xi Þdt þ hðxi1 xi ÞddðtÞ; 2
dyi ¼ ½c dxi yi dt; dzi ¼ ½rðSðxi þ kÞ zi Þdt;
ði ¼ 2; . . . ; NÞ;
ð3Þ
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where u is external control input to be designed. Our aim is to make the output signal x1 of system (3) synchronize any given signal s(t) via designing appropriate controller. s(t) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Let e = x1 s(t), the following Theorem 1 can be obtained. Theorem 1. For a given reference signal s(t), synchronization will occur between s(t) and the output signal x1 of system (3) under controller u in (4) 2
2
u ¼ 3bx1 x_ 1 2ax1 x_ 1 þ dx1 þ y1 c þ rðSðx1 ðt sÞ þ kÞ z1 ðt sÞÞ þ €sðtÞ ae 3be_ 2b2 e:
ð4Þ
That is
lim e ¼ lim ½x1 sðtÞ ¼ 0:
t!1
ð5Þ
t!1
Proof. The positive and changeable Lyapunov function could be measured by 2 V 1 ¼ ae2 þ ðe_ þ beÞ ;
ð6Þ
where a and b are positive gain coefficients, the dot over e denotes the differential variable e to time. The differential of V1 to time t is approached by
_ ¼ 2bV 1 þ 2bV 1 þ 2aee_ þ 2ðe_ þ beÞð€e þ beÞ _ V_ 1 ¼ 2aee_ þ 2ðe_ þ beÞð€e þ beÞ 2
_ ¼ 2bV 1 þ 2b½ae2 þ ðe_ þ beÞ þ 2aee_ þ 2ðe_ þ beÞð€e þ beÞ 2
_ ¼ 2bV 1 þ 2ð€e þ 2be_ þ ae þ b2 eÞðe_ þ beÞ: ¼ 2bV 1 þ 2aeðe_ þ beÞ þ 2bðe_ þ beÞ þ 2ðe_ þ beÞð€e þ beÞ
Fig. 4. Stabilization of errors ei (i = 1, 2, . . ., N) with one controller in (4). Hi = 20, h = 2.0.
ð7Þ
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Substituting (3) and (4) into (7), it can be gotten that
dV 1 2 ¼ 2bV 1 2bðe_ þ beÞ < 0: dt
ð8Þ
According to Lyapunov stability theory, the error of corresponding variables will stabilize to a certain threshold. As a result, the reference signal s(t) and the output signal x1 of system (3) will reach synchronization. Namely,
lim e ¼ lim ½x1 sðtÞ ¼ 0:
t!1
t!1
Proof is complete.h If xi (i = 2, . . ., N) can pinning synchronize x1, while x1 of system (3) follows the reference signal s(t), pinning synchronization between the reference signal s(t) and system (3) can be obtained ultimately. Let
ei ¼ xi x1
ði ¼ 1; . . . ; NÞ;
ð9Þ
and we have the following Theorem 2. Theorem 2. xi (i = 2, . . ., N) with stochastic noise can pinning synchronize x1 in system (3) under the controller u in (4). Proof. According to (9) and system (3), we have
dei ¼ ½aðxi þ x1 Þei bðx2i þ xi x1 þ x21 Þei þ ðyi y1 Þ ðzi ðt sÞ z1 ðt sÞÞ þ Hi ðei1 ei Þdt þ hðei1 ei ÞddðtÞ ði ¼ 2; . . . ; NÞ;
ð10Þ
which can be written as
E_ ¼ AE þ u1 u2
ð11Þ T
T
2 6 6 6 6 A¼6 6 6 4
M 2 H2 h H2 þ h
0
...
0
0
M 3 H3 h . . .
0
0
0 .. .
0 .. .
0 .. .
H3 þ h .. .
0 M i ¼ aðxi þ x1 Þ
bðx2i
x21 Þ
... .. .
3 7 7 7 7 7; 7 7 5
. . . HN1 þ h M N HN h
0 þ xi x1 þ
u2 ¼ ðz2 ðt sÞ z1 ðt sÞ; z3 ðt sÞ z1 ðt sÞ; . . . ;
u1 ¼ ðy2 y1 ; y3 y1 ; . . . ; yN y1 Þ ,
where E ¼ ðe2 ; e3 ; . . . ; eN Þ , zN ðt sÞ z1 ðt sÞÞT ;
ði ¼ 2; . . . ; NÞ: Consider Lyapunov function as following
T
V ¼ E E:
ð12Þ
Obviously, the Lyapunov function V in (12) is a positive function. Making use of (11), time derivative of V in (12) can be expressed as
V_ ¼ E_ T E þ ET E_ ¼ ðAE þ u1 u2 ÞT E þ ET ðAE þ u1 u2 Þ ¼ ET ðAT þ AÞE þ ðu1 u2 ÞT E þ ET ðu1 u2 Þ ¼ ET ðAT þ AÞE þ 2ET ðu1 u2 Þ:
ð13Þ
Since a chaotic system has bounded trajectories, there exists a positive constant L, such that |xi| < L, |yi| < L, |zi| < L, |zi (t s)| < L (i = 1, 2, . . ., N), therefore u1, u2 and Mi (i = 2, . . ., N) are bounded. If Hi (i = 2, . . ., N) are sufficiently large, AT + A is negative definite, thus V_ < 0. According to Lyapunov stability theory, we know that error system (10) or (11) is asymptotically stable. Therefore, the unilateral coupled time delay HR neuron system (3) with stochastic noise achieves pinning synchronization.h According to Theorems 1 and 2, for any given signal, unilateral coupling neural network can pinning synchronize it under appropriate control. 4. Simulation In this section, simulations are given to illustrate the effectiveness of the proposed method. In simulations, coupling intensity Hi (i = 2, . . ., N) is taken as the same value. The original values are taken as constants. Each node is a HR system. The reference signal s(t) is taken as the behavior of an isolated HR neural node. The parameters are chosen as a = 3.0, b = 1.0, c = 1.0, d = 5.0, r = 0.006, S = 4.0, k = 1.6, Iext = 3.1, and the time delay is chosen as s = 1, with which the HR system
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Fig. 5. Synchronization of errors ei (i = 1, 2, . . ., 20) with one controller in (4), Hi = 20.
Fig. 6. Synchronization of errors ei (i = 1, 2, . . ., 20) with one controller in (4), h = 2.0.
is chaotic bursting. To analyze the pinning synchronization process, numerical simulations are carried out from following three aspects. Firstly, we consider the stabilization process of error system (10) or (11) with node increasing. For this reason, complex neural networks with 5, 10, 20 nodes are simulated, respectively, in which Hi = 20, h = 2.0. Pinning synchronization errors are illustrated in Fig. 4, from which it is obviously to see that the neural network can pinning synchronize the given signal s(t). At the same time, we know that, more time is needed to realize the pinning synchronization of a complex neural network with the number of nodes increasing. Secondly, the convergence processes of error systems with noise intensity varying are showed in Fig. 5, where Hi = 20, N = 20, and h is chosen as 0.5, 2.0, respectively. From the numerical results in Fig. 5, we can predict that the greater the noise is, the more time is needed to realize pinning synchronization of a complex neural network with stochastic noise. Finally, when the number of nodes and the noise intensity are constants, the evolution of the errors (10) or (11) with coupling intensity changing are depicted (see Fig. 6, where N = 20, h = 2.0, Hi = 20, 40). It is found that the less time is required to realize pinning synchronization with larger coupling intensity Hi. 5. Conclusion In this paper, based on Lyapunov stability theory, scheme to achieve the pinning synchronization of unilateral coupling neural network with time delay and stochastic noise is proposed. We not only test the validity of the proposed scheme from theoretical analysis, but also verify the effectiveness of the scheme from numerical simulations. Furthermore, from the simulations, some results can be known as follows: (1) When coupling intensity and noise intensity are constants, the time of convergence is positively correlated with the number of nodes.
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(2) When coupling intensity and the number of nodes are constants, the time needed to stabilize error system is positively correlated with noise intensity. (3) When noise intensity and the number of nodes are constants, the time required to achieve pinning synchronization is negatively correlated with the coupling intensity. Because the considered HR neural network takes into account time-delay and stochastic noise, it is attractive and practical in understanding the dynamic behavior of neural network. To achieve pinning synchronization in unilateral coupling neural network, coupling strength should be large enough. Acknowledgements This work is supported by National Natural Science Foundation of China (Grant Nos. 11102180, 61379064 and 61273106), the Natural Science Foundation of the Jiangsu Province of China (Grant No. BK2012672), and the Qing Lan Project. References [1] R.E. Mirollo, S.H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math. 50 (1990) 1645–1662. [2] B. Blasius, L. Stone, Chaos and phase synchronization in ecological system, Int. J. Bifurcation Chaos 10 (2000) 2361–2380. [3] I. Baruchi, V. Volman, N. Raichman, M. Shein, E. Ben-Jacob, The emergence and properties of mutual synchronization in vitro coupled cortical networks, Eur. J. Neurosci. 28 (2008) 1825–1835. [4] R.C. Elson, A.I. Selverston, R. Huerta, N.F. Rulkov, M.I. Rabinovich, H.D. Abarbanel, Synchronous behavior of two coupled biological neurons, Phys. Rev. Lett. 81 (1998) 5692–5695. [5] J. Ma, F. Li, L. Huang, W.Y. Jin, Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 3770–3785. [6] H. Fujisaka, T. Yamada, G. Kinoshita, T. Kono, Chaotic phase synchronization and phase diffusion, Phys. D Nonlinear Phenom. 205 (2005) 41–47. [7] J.Z. Yang, G. Hu, Three types of generalized synchronization, Phys. Lett. A 361 (2007) 332–335. [8] F. Liu, Y. Ren, X.M. Shan, Z.L. Qiu, A linear feedback synchronization theorem for a class of chaotic systems, Chaos, Solitons Fractals 13 (2002) 723–730. [9] M.Y. Chen, Z.Z. Han, Controlling and synchronizing chaotic Genesio system via nonlinear feedback control, Chaos, Solitons Fractals 17 (2003) 709–716. [10] M.T. Yassen, Adaptive control and synchronization of a modified Chua’s circuit system, Appl. Math. Comput. 135 (2003) 113–118. [11] J.A. Lu, X.Q. Wu, X.P. Han, J.H. Lü, Adaptive feedback synchronization of a unified chaotic system, Phys. Lett. A 329 (2004) 327–333. [12] Y.G. Yu, S.C. Zhang, Controlling uncertain Lü system using backstepping design, Chaos, Solitons Fractals 15 (2003) 897–902. [13] K. Konishi, M. Hirai, H. Kokame, Sliding mode control for a class of chaotic systems, Phys. Lett. A 245 (1998) 511–517. [14] R. Kilic, Experimental study on impulsive synchronization between two modified Chua’s circuits, Nonlinear Anal. Real World Appl. 7 (2006) 1298– 1303. [15] D. Heide, M. Schäfer, M. Greiner, Robustness of networks against fluctuation-induced cascading failures, Phys. Rev. E 77 (2008) 056103. [16] C. Hu, J. Yu, H.J. Jiang, Z.D. Teng, Pinning synchronization of weighted complex networks with variable delays and adaptive coupling weights, Nonlinear Dyn. 67 (2012) 1373–1385. [17] L.P. Deng, Z.Y. Wu, Q.C. Wu, Pinning synchronization of complex network with non-derivative and derivative coupling, Nonlinear Dyn. 73 (2013) 775– 782. [18] J.L. Hindmarsh, R.M. Rose, A model of the nerve impulse using two first-order differential equations, Nature 296 (1982) 162. [19] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961) 445–446. [20] A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952) 500–544. [21] W. Wu, T.P. Chen, Global synchronization criteria of linearly coupled neural network systems with time-varying coupling, Neural Networks 19 (2008) 319–332. [22] J.D. Cao, J.Q. Lu, Adaptive synchronization of neural networks with or without time-varying delay, Chaos: An Interdisciplinary J. Nonlinear Sci. 16 (2006). 013133–013133-6. [23] C. Zheng, J.D. Cao, Robust synchronization of dynamical network with impulsive disturbances and uncertain parameters, Int. J. Control Autom. 11 (2013) 657–665. [24] M. Jalili, Spike phase synchronization in delayed-coupled neural networks: uniform vs. non-uniform transmission delay, Chaos: An Interdisciplinary J. Nonlinear Sci. 23 (2013). 013146–013146-6.. [25] X.R. Shi, Z.L. Wang, Adaptive synchronization of time delay Hindmarsh–Rose neuron system via self-feedback, Nonlinear Dyn. 69 (2012) 2147–2153. [26] Y. Sun, J.D. Cao, Stabilization of stochastic delayed neural networks with Markovian switching, Asian J. Control 10 (2008) 327–340. [27] L. Wan, J.H. Sun, Mean square exponential stability of stochastic delayed Hopfield neural networks, Phys. Lett. A 343 (2005) 306–318.