- Email: [email protected]
Chaotic synchronization of two coupled neurons via nonlinear control in external electrical stimulation
Chaos, Solitons and Fractals 27 (2006) 1272–1278 www.elsevier.com/locate/chaos
Chaotic synchronization of two coupled neurons via nonlinear control in external electrical stimulation Wang Jiang *, Deng Bin, Fei Xiangyang School of Electrical and Automation Engineering, Tianjin University, 300072 Tianjin, PR China Accepted 19 April 2005
Abstract The chaotic synchronization to two electrical coupled neurons via nonlinear control is investigated. The coupled model is based on the nonlinear cable model and the two neurons are coupled with gap junction. If the controller were not applied, the synchronization would occur only when the couple strength of gap junction satisfied some condition. Using techniques from modern control theory, a nonlinear controller can be obtained that result in two of coupled neurons being synchronized with each other without needing to consider the couple strength of gap junction. The detailed derivation that leads to the nonlinear controller and numerical results that verify the controllerÕs ability to synchronize the two neurons together are included. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Over last decade, many new types of synchronization have appeared. Since the discovery of chaos synchronization [1,2], there has been tremendous interest in studying the synchronization of chaotic systems. Recently, synchronization of coupled chaotic systems has received considerable attention [3–6]. Especially, a typical study of synchronization is the coupled chaotic identical chaotic systems [7]. Synchronization in physical and biological system is a fascinating subject that has attracted a lot of renewed attention [8]. Theoretical studies in this direction mainly focus on the synchronization of coupled oscillatory subsystem [9,10]. Because of its importance, chaotic synchronization has been studied extensively in recent years. Various modern control methods, such as adaptive control [11], back-stepping design [12], active control [13], and nonlinear control [14] have been successfully applied to chaos synchronization. The selective synchronization of neural activity has been suggested as a mechanism for binding spatially distributed into a coherent object and too much synchrony may cause dynamical disease [15]. The synchronization may play an important role in revealing communication pathways in neural system. Many cells are linked to each other by special intercellular pathways known as gap junctions [16], and the influence of gap junction on the synchronization of two coupled neurons is investigated in [17]. According to Wang et al. [17], if the controller were not applied, the synchronization would occur only when the couple strength of gap junction satisfied some condition. In this paper, a nonlinear controller is designed to synchronize two coupled neurons without needing to consider the couple strength of gap *
Corresponding author. Tel.: +86 22 2740 2293; fax: +86 22 2740 1101. E-mail address: [email protected] (J. Wang).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.102
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
1273
junction. The model of two gap junction coupled neurons is reviewed in Section 2. In Section 3, the development of a nonlinear controller for the synchronization of two coupled neurons is given. Section 4 contains the concluding comments.
2. The model of gap junction coupled neurons In [17], the influence of gap junction on the synchronization of two coupled neurons is investigated and the model of gap junction coupled neuron is given. The circuit diagram of the model is shown in Fig. 1. The model of gap junction coupled neurons is described by 8 dX 1 > > ¼ X 1 ðX 1 1Þð1 rX 1 Þ Y 1 gðX 1 X 2 Þ þ I 0 ; > > > dt > > > > > dY > > 1 ¼ bX 1 ; < dt ð1Þ > > > dX 2 ¼ X 2 ðX 2 1Þð1 rX 2 Þ Y 2 gðX 2 X 1 Þ þ I 0 ; > > > dt > > > > > dY > : 2 ¼ bX 2 ; dt where Xi, Yi (i = 1, 2) are status variables, provided that orbit is close enough to the basin of attraction, g presents the coupling strength of gap junction, and I 0 ðtÞ ¼ xA cos xt is the external stimulation. According to the result of [17], if then individual neuron were chaotic, then the synchronization would occur only when the couple strength of gap junction satisfied some condition. The case is shown in Figs. 2 and 3, respectively. As shown in Figs. 2 and 3, when the frequency of the external stimulation x = 127.1 Hz, the individual neuron without coupling is chaotic. If the coupling strength of the gap junction g = 0.1 < 0.5, the synchronization cannot occur; the synchronization occurs when g = 1 > 0.5.
3. Nonlinear controller for two coupled neurons First, a typical system for studies of chaotic synchronization between two coupled, identical oscillators can be expressed as u_ j ¼ f ðuj ; tÞ þ Cðui uj Þ;
i; j ¼ 1; 2; i 6¼ j;
n
ð2Þ n
n
where ui,j 2 R represent the state variables of the two oscillators, the function f : R ! R defines the dynamics of a single oscillator in the absence of coupling, and C is the coupling matrix. The controller U 2 Rn is added into the second system, and then the response system is given by u_ 2 ¼ f ðu2 ; tÞ þ Cðu1 u2 Þ þ U .
ð3Þ
The synchronization problem is to design a controller U, which synchronize the states of the coupled systems. The dynamics of synchronization errors can be expressed as
Fig. 1. The circuit diagram of two coupled neurons.
1274
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
Fig. 2. x = 127.1 Hz, g = 0.1 < 0.5.
e_ ¼ f ðu2 ; tÞ f ðu1 ; tÞ þ Cðu1 u2 Þ Cðu2 u1 Þ þ U ;
ð4Þ
where e = u2 u1. The aim of synchronization is to make limt!1ke(t)k = 0. The problem of synchronization between the coupled systems can be translated into a problem of how to realize the asymptotical stabilization of the system (4). So the objective is to design a controller U for stabilizing the error dynamical system (4) at origin.
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
1275
Fig. 3. x = 127.1, g = 1 > 0.5.
Then let Lyapunov error function be V ðeÞ ¼ 12 eT e, where V(e) is a positive definite function. Assuming that the parameters of the coupled systems are measurable, we may achieve the synchronization by selecting the controller U to make the derivative of V(e), i.e., V_ ðeÞ < 0. Then the states of the coupled systems are synchronized asymptotically globally. The controller U 2 R2 is added to the system of gap junction coupled neurons reviewed in Section 2, and the new system is in the form of
1276
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
8 dX 1 > > ¼ X 1 ðX 1 1Þð1 rX 1 Þ Y 1 gðX 1 X 2 Þ þ I 0 ; > > dt > > > > dY 1 > > ¼ bX 1 ; < dt ð5Þ > dX > > 2 ¼ X 2 ðX 2 1Þð1 rX 2 Þ Y 2 gðX 2 X 1 Þ þ I 0 þ u1 ; > > dt > > > > dY > : 2 ¼ bX 2 þ u2 ; dt where U = [u1, u2]T is the controller to be designed. Let e1 = X2 X1, e2 = Y2 Y1, the error dynamical system of the coupled neurons can be expressed by e_ 1 ¼ X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ e2 2ge1 þ u1 ; ð6Þ e_ 2 ¼ be1 þ u2 . Consider a Lyapunov function candidate as V ðeÞ ¼ 12eT e.
ð7Þ
Then we can get the first derivation of V(e): V_ ðeÞ ¼ ½X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ e1 e2 e1 2ge21 þ u1 e1 þ be2 e1 þ u2 e2 . Therefore, if we choose U as follows: u1 ¼ ½X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ þ e2 ; u2 ¼ be1 ;
Fig. 4. The controller is applied at the time t = 3000.
ð8Þ
ð9Þ
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
1277
Fig. 5. The controller is unavailable at the time t = 3000.
then V_ ðeÞ ¼ 2ge21 .
ð10Þ
Since g > 0 all the time, V_ ðeÞ < 0 is satisfied. V_ ðeÞ is a negative-definite function and V(e) is a positive-definite function, so the error states lim keðtÞk ¼ 0.
t!1
ð11Þ
Therefore, the coupled neurons are globally synchronized asymptotically. Choose the value of the parameters to be r = 10, b = 1, A = 0.1, x = 127.1, g = 0.1 as appear in [17]. This choice of value cannot make the coupled neurons synchronized if the controller were not applied, as shown in Fig. 2. When a nonlinear controller defined in (9) is applied at the time t = 3000, the coupled neurons are synchronized, as shown in Fig. 4. If the controller was applied at beginning and it was unavailable at the time t = 3000, the synchronization of the coupled neurons will disappear at once, as shown in Fig. 5.
4. Conclusion In this paper, we have obtained a nonlinear controller that can be used to synchronize two neurons coupled with gap junction without needing to consider the coupling strength as in [17]. Numerical simulations are also given to value the proposed synchronization approach. The simulation results show that the states of two coupler neurons are globally
1278
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
asymptotically synchronized. Since more and more people are interested in the synchronization in the neural system, the synchronization procedure discussed in this study may have practical application in the future.
Acknowledgement The authors gratefully acknowledge the support of the NSFC (no. 50177023).
References [1] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [2] Pecora LM, Carroll TL. Driving systems with chaotic signal. Phys Rev A 1991;44:2374–83. [3] Chiu C, Lin W, Peng C. Asymptotic synchronization in lattices of coupled nonlinear Lorenz equations. Int J Bifur Chaos 2000;10:2717–28. [4] Wang C, Ge SS. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos, Solitons & Fractals 2001;12:1199–206. [5] Chua LO, Itah M, Kosarev L, Eckert K. Chaos synchronization in ChuaÕs circuits. J Circ Syst Comput 1993;3:93–108. [6] Agiza HN, Yassen MT. Synchronization of Ro¨ssler and Chen chaotic dynamical systems using active control. Phys Lett A 2001;278:191–7. [7] Hu G, Xiao J, Zheng Z. Chaos control. China: Shanghai Scientific and Technological Education Publishing House; 2000. [8] Pikovsky A, Rosenblum M, Kurths J. Synchronization: a universal concept in nonlinear sciences. New York: Cambridge University Press; 2001. [9] Mirollo RE, Strogatz SH. Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 1990;50:1645–62. [10] Nischwitz A, Glu¨nder H. Local lateral inhibition—a key to spike synchronization. Biol Cybernet 1995;73:389. [11] Chen SH, Lu J. Synchronization of an uncertain unified system via adaptive control. Chaos, Solitons & Fractals 2002;14:643–7. [12] Tan X, Zhang J, Yang Y. Synchronizing chaotic systems using backstepping design. Chaos, Solitons & Fractals 2003;16:37–45. [13] Ho MC, Hung YC. Synchronizing two different systems by using generalized active control. Phys Lett A 2002;301:424–8. [14] Huang L, Feng R, Wang M. Synchronization of chaotic systems via nonlinear control. Phys Lett A 2004;320:271–5. [15] Shuai J-W, Durand DM. Phase synchronization in two coupled chaotic neurons. Phys Lett A 1999;264:289–97. [16] Bennett MVL, Verselis VK. Biophysics of gap junction. Semin Cell Biol 1992;3:29–47. [17] Wang J, Deng B, Tsang KM. Chaotic synchronization of neurons coupled with gap junction under external electrical stimulation. Chaos, Solitons & Fractals 2004;22:469–76.
Chaotic synchronization of two coupled neurons via nonlinear control in external electrical stimulation Wang Jiang *, Deng Bin, Fei Xiangyang School of Electrical and Automation Engineering, Tianjin University, 300072 Tianjin, PR China Accepted 19 April 2005
Abstract The chaotic synchronization to two electrical coupled neurons via nonlinear control is investigated. The coupled model is based on the nonlinear cable model and the two neurons are coupled with gap junction. If the controller were not applied, the synchronization would occur only when the couple strength of gap junction satisfied some condition. Using techniques from modern control theory, a nonlinear controller can be obtained that result in two of coupled neurons being synchronized with each other without needing to consider the couple strength of gap junction. The detailed derivation that leads to the nonlinear controller and numerical results that verify the controllerÕs ability to synchronize the two neurons together are included. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Over last decade, many new types of synchronization have appeared. Since the discovery of chaos synchronization [1,2], there has been tremendous interest in studying the synchronization of chaotic systems. Recently, synchronization of coupled chaotic systems has received considerable attention [3–6]. Especially, a typical study of synchronization is the coupled chaotic identical chaotic systems [7]. Synchronization in physical and biological system is a fascinating subject that has attracted a lot of renewed attention [8]. Theoretical studies in this direction mainly focus on the synchronization of coupled oscillatory subsystem [9,10]. Because of its importance, chaotic synchronization has been studied extensively in recent years. Various modern control methods, such as adaptive control [11], back-stepping design [12], active control [13], and nonlinear control [14] have been successfully applied to chaos synchronization. The selective synchronization of neural activity has been suggested as a mechanism for binding spatially distributed into a coherent object and too much synchrony may cause dynamical disease [15]. The synchronization may play an important role in revealing communication pathways in neural system. Many cells are linked to each other by special intercellular pathways known as gap junctions [16], and the influence of gap junction on the synchronization of two coupled neurons is investigated in [17]. According to Wang et al. [17], if the controller were not applied, the synchronization would occur only when the couple strength of gap junction satisfied some condition. In this paper, a nonlinear controller is designed to synchronize two coupled neurons without needing to consider the couple strength of gap *
Corresponding author. Tel.: +86 22 2740 2293; fax: +86 22 2740 1101. E-mail address: [email protected] (J. Wang).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.102
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
1273
junction. The model of two gap junction coupled neurons is reviewed in Section 2. In Section 3, the development of a nonlinear controller for the synchronization of two coupled neurons is given. Section 4 contains the concluding comments.
2. The model of gap junction coupled neurons In [17], the influence of gap junction on the synchronization of two coupled neurons is investigated and the model of gap junction coupled neuron is given. The circuit diagram of the model is shown in Fig. 1. The model of gap junction coupled neurons is described by 8 dX 1 > > ¼ X 1 ðX 1 1Þð1 rX 1 Þ Y 1 gðX 1 X 2 Þ þ I 0 ; > > > dt > > > > > dY > > 1 ¼ bX 1 ; < dt ð1Þ > > > dX 2 ¼ X 2 ðX 2 1Þð1 rX 2 Þ Y 2 gðX 2 X 1 Þ þ I 0 ; > > > dt > > > > > dY > : 2 ¼ bX 2 ; dt where Xi, Yi (i = 1, 2) are status variables, provided that orbit is close enough to the basin of attraction, g presents the coupling strength of gap junction, and I 0 ðtÞ ¼ xA cos xt is the external stimulation. According to the result of [17], if then individual neuron were chaotic, then the synchronization would occur only when the couple strength of gap junction satisfied some condition. The case is shown in Figs. 2 and 3, respectively. As shown in Figs. 2 and 3, when the frequency of the external stimulation x = 127.1 Hz, the individual neuron without coupling is chaotic. If the coupling strength of the gap junction g = 0.1 < 0.5, the synchronization cannot occur; the synchronization occurs when g = 1 > 0.5.
3. Nonlinear controller for two coupled neurons First, a typical system for studies of chaotic synchronization between two coupled, identical oscillators can be expressed as u_ j ¼ f ðuj ; tÞ þ Cðui uj Þ;
i; j ¼ 1; 2; i 6¼ j;
n
ð2Þ n
n
where ui,j 2 R represent the state variables of the two oscillators, the function f : R ! R defines the dynamics of a single oscillator in the absence of coupling, and C is the coupling matrix. The controller U 2 Rn is added into the second system, and then the response system is given by u_ 2 ¼ f ðu2 ; tÞ þ Cðu1 u2 Þ þ U .
ð3Þ
The synchronization problem is to design a controller U, which synchronize the states of the coupled systems. The dynamics of synchronization errors can be expressed as
Fig. 1. The circuit diagram of two coupled neurons.
1274
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
Fig. 2. x = 127.1 Hz, g = 0.1 < 0.5.
e_ ¼ f ðu2 ; tÞ f ðu1 ; tÞ þ Cðu1 u2 Þ Cðu2 u1 Þ þ U ;
ð4Þ
where e = u2 u1. The aim of synchronization is to make limt!1ke(t)k = 0. The problem of synchronization between the coupled systems can be translated into a problem of how to realize the asymptotical stabilization of the system (4). So the objective is to design a controller U for stabilizing the error dynamical system (4) at origin.
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
1275
Fig. 3. x = 127.1, g = 1 > 0.5.
Then let Lyapunov error function be V ðeÞ ¼ 12 eT e, where V(e) is a positive definite function. Assuming that the parameters of the coupled systems are measurable, we may achieve the synchronization by selecting the controller U to make the derivative of V(e), i.e., V_ ðeÞ < 0. Then the states of the coupled systems are synchronized asymptotically globally. The controller U 2 R2 is added to the system of gap junction coupled neurons reviewed in Section 2, and the new system is in the form of
1276
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
8 dX 1 > > ¼ X 1 ðX 1 1Þð1 rX 1 Þ Y 1 gðX 1 X 2 Þ þ I 0 ; > > dt > > > > dY 1 > > ¼ bX 1 ; < dt ð5Þ > dX > > 2 ¼ X 2 ðX 2 1Þð1 rX 2 Þ Y 2 gðX 2 X 1 Þ þ I 0 þ u1 ; > > dt > > > > dY > : 2 ¼ bX 2 þ u2 ; dt where U = [u1, u2]T is the controller to be designed. Let e1 = X2 X1, e2 = Y2 Y1, the error dynamical system of the coupled neurons can be expressed by e_ 1 ¼ X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ e2 2ge1 þ u1 ; ð6Þ e_ 2 ¼ be1 þ u2 . Consider a Lyapunov function candidate as V ðeÞ ¼ 12eT e.
ð7Þ
Then we can get the first derivation of V(e): V_ ðeÞ ¼ ½X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ e1 e2 e1 2ge21 þ u1 e1 þ be2 e1 þ u2 e2 . Therefore, if we choose U as follows: u1 ¼ ½X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ þ e2 ; u2 ¼ be1 ;
Fig. 4. The controller is applied at the time t = 3000.
ð8Þ
ð9Þ
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
1277
Fig. 5. The controller is unavailable at the time t = 3000.
then V_ ðeÞ ¼ 2ge21 .
ð10Þ
Since g > 0 all the time, V_ ðeÞ < 0 is satisfied. V_ ðeÞ is a negative-definite function and V(e) is a positive-definite function, so the error states lim keðtÞk ¼ 0.
t!1
ð11Þ
Therefore, the coupled neurons are globally synchronized asymptotically. Choose the value of the parameters to be r = 10, b = 1, A = 0.1, x = 127.1, g = 0.1 as appear in [17]. This choice of value cannot make the coupled neurons synchronized if the controller were not applied, as shown in Fig. 2. When a nonlinear controller defined in (9) is applied at the time t = 3000, the coupled neurons are synchronized, as shown in Fig. 4. If the controller was applied at beginning and it was unavailable at the time t = 3000, the synchronization of the coupled neurons will disappear at once, as shown in Fig. 5.
4. Conclusion In this paper, we have obtained a nonlinear controller that can be used to synchronize two neurons coupled with gap junction without needing to consider the coupling strength as in [17]. Numerical simulations are also given to value the proposed synchronization approach. The simulation results show that the states of two coupler neurons are globally
1278
J. Wang et al. / Chaos, Solitons and Fractals 27 (2006) 1272–1278
asymptotically synchronized. Since more and more people are interested in the synchronization in the neural system, the synchronization procedure discussed in this study may have practical application in the future.
Acknowledgement The authors gratefully acknowledge the support of the NSFC (no. 50177023).
References [1] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [2] Pecora LM, Carroll TL. Driving systems with chaotic signal. Phys Rev A 1991;44:2374–83. [3] Chiu C, Lin W, Peng C. Asymptotic synchronization in lattices of coupled nonlinear Lorenz equations. Int J Bifur Chaos 2000;10:2717–28. [4] Wang C, Ge SS. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos, Solitons & Fractals 2001;12:1199–206. [5] Chua LO, Itah M, Kosarev L, Eckert K. Chaos synchronization in ChuaÕs circuits. J Circ Syst Comput 1993;3:93–108. [6] Agiza HN, Yassen MT. Synchronization of Ro¨ssler and Chen chaotic dynamical systems using active control. Phys Lett A 2001;278:191–7. [7] Hu G, Xiao J, Zheng Z. Chaos control. China: Shanghai Scientific and Technological Education Publishing House; 2000. [8] Pikovsky A, Rosenblum M, Kurths J. Synchronization: a universal concept in nonlinear sciences. New York: Cambridge University Press; 2001. [9] Mirollo RE, Strogatz SH. Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 1990;50:1645–62. [10] Nischwitz A, Glu¨nder H. Local lateral inhibition—a key to spike synchronization. Biol Cybernet 1995;73:389. [11] Chen SH, Lu J. Synchronization of an uncertain unified system via adaptive control. Chaos, Solitons & Fractals 2002;14:643–7. [12] Tan X, Zhang J, Yang Y. Synchronizing chaotic systems using backstepping design. Chaos, Solitons & Fractals 2003;16:37–45. [13] Ho MC, Hung YC. Synchronizing two different systems by using generalized active control. Phys Lett A 2002;301:424–8. [14] Huang L, Feng R, Wang M. Synchronization of chaotic systems via nonlinear control. Phys Lett A 2004;320:271–5. [15] Shuai J-W, Durand DM. Phase synchronization in two coupled chaotic neurons. Phys Lett A 1999;264:289–97. [16] Bennett MVL, Verselis VK. Biophysics of gap junction. Semin Cell Biol 1992;3:29–47. [17] Wang J, Deng B, Tsang KM. Chaotic synchronization of neurons coupled with gap junction under external electrical stimulation. Chaos, Solitons & Fractals 2004;22:469–76.