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Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control
Chaos, Solitons and Fractals 29 (2006) 182–189 www.elsevier.com/locate/chaos
Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control Deng Bin, Wang Jiang *, Fei Xiangyang School of Electrical and Automation Engineering, Tianjin University, 300072 Tianjin, PR China Accepted 16 August 2005
Abstract Backstepping design is a recursive procedure that combines the choice of a Lyapunov function with the design of a controller. In this paper, the backstepping control is used to synchronize two coupled chaotic neurons in external electrical stimulation. The coupled model is based on the nonlinear cable model and only one state variable can be controlled in practice. The backstepping design needs only one controller to synchronize two chaotic systems and it can be applied to a variety of chaotic systems whether they contain external excitation or not, so the two coupled chaotic neurons in external electrical stimulation can be synchronized perfectly by backstepping control. Numerical simulations demonstrate the effectiveness of this design. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Chaos control and synchronization have been intensively studied during last decade. There has been tremendous interest in studying the synchronization of chaotic systems and the synchronization of coupled chaotic systems has received considerable attention [1–4]. Especially, a typical synchronization study is about the coupled chaotic identical chaotic systems [5]. Synchronization in physical and biological system is a fascinating subject that has attracted a lot of renewed attention [6]. Theoretical studies in this direction mainly focus on the synchronization of coupled oscillatory subsystem [7,8]. Various modern control methods, such as adaptive control [9], backstepping design [10], active control [11], and nonlinear control [12] have been successfully applied to chaos synchronization in recent years. In the 1980s, some Japanese scholars studied the repetitive firing of the action potential in sinusoidal current stimulated squid giant axons experimentally, different kinds of motions including periodic, quasi-periodic and chaotic were found with coincidence to those from theoretical computations [13,14]. Since then, a few models of neuron have been developed to study the chaos of neurons in external stimulation [15,16]. Many cells are linked to each other by special intercellular pathways known as gap junctions [17]. They allow the direct transfer of ions and small molecules, including
*
Corresponding author. Tel.: +86 22 27402293; fax: +86 22 27401101. E-mail address: [email protected] (W. Jiang).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.027
D. Bin et al. / Chaos, Solitons and Fractals 29 (2006) 182–189
183
second messenger molecules, between cells without leakage to the extracellular fluid. As the gap junctions play an important role in the process of information transmitting among the coupled neurons, they become a major focus of study in neuron system [18–22]. The selective synchronization of neural activity has been suggested as a mechanism for binding spatially distributed object into a coherent one and too much synchrony may cause dynamical disease [23]. The synchronization may play an important role in revealing communication pathways in neural system. The influence of gap junction on the synchronization of two coupled neurons is investigated in [24]. In this paper, the control of the synchronization between two neurons electrically coupled with gap junctions in external electrical stimulation is the main focus. The coupled model is based on the nonlinear cable model and only one state variable can be controlled in practice. The backstepping design needs only one controller to synchronize the two chaotic systems and it can be applied to a variety of chaotic systems whether they contain external excitation or not, so the two coupled chaotic neurons in external electrical stimulation can be perfectly synchronized using backstepping design. The rest of this paper is organized as follows: In Section 2, the dynamics of two neurons electrically coupled with gap junctions in external electrical stimulation are reviewed. In Section 3, the backstepping design is proposed to synchronize the two neurons on the base of the single neuron model described in [16]. Numerical simulations to demonstrate the effectiveness of this design are also included in Section 3. Finally, conclusions are drawn in Section 4.
2. The dynamics of two neurons electrically coupled with gap junctions In [24], 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 neuron 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 > > > > > dt ¼ X 2 ðX 2 1Þð1 rX 2 Þ Y 2 gðX 2 X 1 Þ þ I 0 > > > > > > 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 represents the coupling strength of gap junction, and I 0 ðtÞ ¼ xA cos xt is the external stimulation. According to the result of [24], if the individual neurons were chaotic, then the synchronization would occur only when the couple strength of gap junction satisfied some condition. The case is shown in Fig. 2 and Fig. 3.
g I0
f(V)
Lm
I0
Cm
V1
f(V)
Lm
Fig. 1. The circuit diagram of two coupled neurons.
Cm
V2
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(f)
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Fig. 2. x = 127.1 Hz, g = 0.1 < 0.5: (a) X1Y1 phase plane diagram, (b) X2Y2 phase plane diagram, (c) e1 = X1X2, (d) e2 = Y1Y2, (e) X1X2 phase plane diagram and (f) Y1Y2 phase plane diagram.
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 will not occur; the synchronization occurs when g = 1 > 0.5.
3. Synchronize two coupled neurons using backstepping design For the convenience of designing the controller via backstepping method, the model of the coupled neurons is based on the single neuron model described in [16], having just a little bit difference as reviewed in Section 2. Only the state of the membrane potential can be controlled in practice via external electrical stimulation, so the controlled system is in the form of:
D. Bin et al. / Chaos, Solitons and Fractals 29 (2006) 182–189 2
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Fig. 3. x = 127.1, g = 1 > 0.5: (a) X1Y1 phase plane, (b) X2Y2 phase plane (c) e1 = X1X2, (d) e2 = Y1Y2, (e) X1X2 phase plane and (f) Y1Y2 phase plane.
8 dX 1 > > > dt ¼ X 1 ðX 1 1Þð1 rX 1 Þ Y 1 gðX 1 X 2 Þ þ I 0 > > > > > > dY 1 > > ¼ bðX 1 cY 1 Þ < dt > dX 2 > > ¼ X 2 ðX 2 1Þð1 rX 2 Þ Y 2 gðX 2 X 1 Þ þ I 0 þ u > > > dt > > > > > : dY 2 ¼ bðX cY Þ 2 2 dt
ð2Þ
Eq. (2) is a more general coupled model incorporating the recovery variable than that in Section 2, and u 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:
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e_ 1 ¼ X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ e2 2ge1 þ u e_ 2 ¼ bðe1 ce2 Þ
ð3Þ
The problem to realize the synchronization between two neurons is now transformed to a-problem on how to choose a control law u to make ei (i = 1, 2) generally converge to zero with time increasing. Here backstepping design is used to achieve the objective. First we consider the stability of system (4) e_ 2 ¼ bðe1 ce2 Þ
ð4Þ
where e1 is regarded as a controller. Choose Lyapunov function V1 as follows: 1 V 1 ðe2 Þ ¼ e22 2
ð5Þ
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Fig. 4. The controller is not applied and the synchronization is not achieved: (a) X1Y1 phase plane, (b) X2Y2 phase plane, (c) e1 = X1X2, (d) e2 = Y1Y2, (e) X1X2 phase plane and (f) Y1Y2 phase plane.
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The derivative of V1 is V_ 1 ¼ e_ 2 e2 ¼ bce22 þ be1 e2
ð6Þ
Assume controller e1 = a1(e2), Eq. (6) can be rewritten as V_ 1 ¼ e_ 2 e2 ¼ bce22 þ be2 a1 ðe2 Þ
ð7Þ
If a1(e2) = 0, then V_ 1 ¼ bce2 < 0
ð8Þ
2
Make system (4) asymptotically stable. Function a1(e2) is an estimative function when e1 is considered as a controller. The error between e1and a1(e2) is w1 ¼ e1 a1 ðe2 Þ
ð9Þ 2
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1
(f)
0
0.5
1
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2
Y1
Fig. 5. The controller is applied and the synchronization is achieved: (a) X1Y1 phase plane, (b) X2Y2 phase plane, (c) e1 = X1X2, (d) e2 = Y1Y2, (e) X1X2 phase plane diagram and (f) Y1Y2 phase plane.
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Study the (e2, w1) system represented by (10) e_ 2 ¼ bðw1 ce1 Þ w_ 1 ¼ ½X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ e2 2gw1 þ u
ð10Þ
Choose Lyapunov function V2 as follows: 1 V 2 ðe2 ; w1 Þ ¼ V 1 ðe2 Þ þ w21 2
ð11Þ
f ðX 1 ; X 2 Þ ¼ X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ
ð12Þ
Let
The derivation of V2 is V_ 2 ¼ V_ 1 þ w_ 1 w1 ¼ bce22 2gw21 þ w1 ðbe2 þ f ðX 1 ; X 2 Þ e2 þ uÞ
ð13Þ
Let the controller to be u ¼ ð1 bÞe2 f ðX 1 ; X 2 Þ
ð14Þ
Then V_ 2 can be described as V_ 2 ¼ bce22 2gw21 < 0
ð15Þ
The system is negative definite. As for an arbitrary initial error between two neurons, when the controller were applied, the initial error will converge to zero asymptotically and synchronization between the two coupled neurons will be achieved after a finite period of time. Take the parameters r = 10, b = 1, A = 0.1, x = 127.1, g = 0.05, c = 0.001, as is used in [16,24]. Fig. 4 shows the case that when the controller was not switched, the synchronization will not be achieved. The case of applying the controller is shown in Fig. 5.
4. Conclusion In this paper, backstepping design has been used to synchronize two neurons electrically coupled with gap junction in external electrical stimulation. The backstepping method is a systematic procedure for synchronizing chaotic systems and it can be applied to a variety of chaotic systems whether it contains external excitation of not. As this method needs only one controller to realize synchronization no matter how much dimensions the chaotic system contains and only the state of the membrane potential in the coupled neurons model can be controlled in practice, the backstepping design can synchronize the two coupled chaotic neurons perfectly. Further study is to use this method to synchronize n (n P 3) coupled chaotic neurons.
Acknowledgements The authors gratefully acknowledge the NSFC (no. 50177023) for supporting this research.
References [1] Chiu C, Lin W, Peng C. Asymptotic synchronization in lattices of coupled nonlinear Lorenz equations. Int J Bifur Chaos 2000;10:2717–28. [2] Wang C, Ge SS. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos, Solitons & Fractals 2001;12:1199–206. [3] Chua LO, Itah M, Kosarev L, Eckert K. Chaos synchronization in ChuaÕs circuits. J Cricuits Syst Comput 1993;3:93–108. [4] Agiza HN, Yassen MT. Synchronization of Rossler and Chen chaotic dynamical systems using active control. Phys Lett A 2001;278:191–7. [5] Hu G, Xiao J, Zheng Z. Chaos control. China: Shanghai Scientific and Technological Education Publishing House; 2000. [6] Pikovsky A, Rosenblum M, Kurths J. Synchronization: a universal concept in nonlinear sciences. New York: Cambridge University Press; 2001. [7] Mirollo RE, Strogatz SH. Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 1990;50:1645–62.
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Nischwitz A, GluK nder H. Local lateral inhibition-a key to spike synchronization. Biol Cybernet 1995;73:389. Chen SH, Lu J. Synchronization of an uncertain unified system via adaptive control. Chaos, Solitons & Fractals 2002;14:643–7. Tan X, Zhang J, Yang Y. Synchronizing chaotic systems using backstepping design. Chaos, Solitons & Fractals 2003;16:37–45. Ho MC, Hung YC. Synchronization two different systems by using generalized active control. Phys Lett A 2002;301:424–8. Huang L, Feng R, Wang M. Synchronization of chaotic systems via nonlinear control. Phys Lett A 2004;320:271–5. Matsumoto G, Aihara K, Utsunomiya T. A spatially ordered pacemaker observed in squid giant axons. J Phys Soc Jpn 1982;51:942. Aihara K, Matsumoto G, Ichiwaka M. An alternating periodic-chaotic sequence observed in neural oscillations. Phys Lett A 1985;111:251–5. Fumihiro F Shoji, Hee-Hyol Lee. On a response characteristics in the Hodgkin–Huxley model and muscle fiber to a periodic stimulation. IEEE 2000;3:2035–41. Thompson CJ, Bardos DC, Yang YS, Joyner KH. Nonlinear cable models for cells exposed to electric fields I. General theory and space—clamped solutions. Chaos, Solutions & Fractals 1999;10:1825–42. Bennett MVL, Verselis VK. Biophysics of gap junction. Seminars Cell Biol. 1992;3:29–47. Stephen B, Jaroslav S, Anne W. Convergent dynamics of two cells coupled by a nonlinear gap junction. Nonlinear Anal. 2001;47:257–68. Stephen B, Jaroslav S, Anne W. Modeling the effect of gap junction nonlinearities in systems of coupled cells. J Theor Biol 1997;186:223–39. Rolf V, Robert W. The electrophysiology of gap junctions and gap junction channels and their mathematical modeling. Biol Cell 2002;94:501–10. Stephen B. Cells coupled by voltage-dependent gap junctions: the asymptotic dynamical limit. BioSystems 2003;68:213–22. Rolf D. Gap junction wiring: a ÔnewÕ principle in cell–cell communication in the nervous system. Brain Res Rev 1998;26:176–83. Jian-Wei Shuai, Dominique MD. Phase synchronization in two coupled chaotic neurons. Phys Lett A 1999;264:289–97. Wang Jing, Deng Bin, Tsang KM. Chaotic synchronization of neurons coupled with gap junction under external electrical stimulation. Chaos, Solitons & Fractals 2004;22:469–76.
Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control Deng Bin, Wang Jiang *, Fei Xiangyang School of Electrical and Automation Engineering, Tianjin University, 300072 Tianjin, PR China Accepted 16 August 2005
Abstract Backstepping design is a recursive procedure that combines the choice of a Lyapunov function with the design of a controller. In this paper, the backstepping control is used to synchronize two coupled chaotic neurons in external electrical stimulation. The coupled model is based on the nonlinear cable model and only one state variable can be controlled in practice. The backstepping design needs only one controller to synchronize two chaotic systems and it can be applied to a variety of chaotic systems whether they contain external excitation or not, so the two coupled chaotic neurons in external electrical stimulation can be synchronized perfectly by backstepping control. Numerical simulations demonstrate the effectiveness of this design. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Chaos control and synchronization have been intensively studied during last decade. There has been tremendous interest in studying the synchronization of chaotic systems and the synchronization of coupled chaotic systems has received considerable attention [1–4]. Especially, a typical synchronization study is about the coupled chaotic identical chaotic systems [5]. Synchronization in physical and biological system is a fascinating subject that has attracted a lot of renewed attention [6]. Theoretical studies in this direction mainly focus on the synchronization of coupled oscillatory subsystem [7,8]. Various modern control methods, such as adaptive control [9], backstepping design [10], active control [11], and nonlinear control [12] have been successfully applied to chaos synchronization in recent years. In the 1980s, some Japanese scholars studied the repetitive firing of the action potential in sinusoidal current stimulated squid giant axons experimentally, different kinds of motions including periodic, quasi-periodic and chaotic were found with coincidence to those from theoretical computations [13,14]. Since then, a few models of neuron have been developed to study the chaos of neurons in external stimulation [15,16]. Many cells are linked to each other by special intercellular pathways known as gap junctions [17]. They allow the direct transfer of ions and small molecules, including
*
Corresponding author. Tel.: +86 22 27402293; fax: +86 22 27401101. E-mail address: [email protected] (W. Jiang).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.027
D. Bin et al. / Chaos, Solitons and Fractals 29 (2006) 182–189
183
second messenger molecules, between cells without leakage to the extracellular fluid. As the gap junctions play an important role in the process of information transmitting among the coupled neurons, they become a major focus of study in neuron system [18–22]. The selective synchronization of neural activity has been suggested as a mechanism for binding spatially distributed object into a coherent one and too much synchrony may cause dynamical disease [23]. The synchronization may play an important role in revealing communication pathways in neural system. The influence of gap junction on the synchronization of two coupled neurons is investigated in [24]. In this paper, the control of the synchronization between two neurons electrically coupled with gap junctions in external electrical stimulation is the main focus. The coupled model is based on the nonlinear cable model and only one state variable can be controlled in practice. The backstepping design needs only one controller to synchronize the two chaotic systems and it can be applied to a variety of chaotic systems whether they contain external excitation or not, so the two coupled chaotic neurons in external electrical stimulation can be perfectly synchronized using backstepping design. The rest of this paper is organized as follows: In Section 2, the dynamics of two neurons electrically coupled with gap junctions in external electrical stimulation are reviewed. In Section 3, the backstepping design is proposed to synchronize the two neurons on the base of the single neuron model described in [16]. Numerical simulations to demonstrate the effectiveness of this design are also included in Section 3. Finally, conclusions are drawn in Section 4.
2. The dynamics of two neurons electrically coupled with gap junctions In [24], 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 neuron 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 > > > > > dt ¼ X 2 ðX 2 1Þð1 rX 2 Þ Y 2 gðX 2 X 1 Þ þ I 0 > > > > > > 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 represents the coupling strength of gap junction, and I 0 ðtÞ ¼ xA cos xt is the external stimulation. According to the result of [24], if the individual neurons were chaotic, then the synchronization would occur only when the couple strength of gap junction satisfied some condition. The case is shown in Fig. 2 and Fig. 3.
g I0
f(V)
Lm
I0
Cm
V1
f(V)
Lm
Fig. 1. The circuit diagram of two coupled neurons.
Cm
V2
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Fig. 2. x = 127.1 Hz, g = 0.1 < 0.5: (a) X1Y1 phase plane diagram, (b) X2Y2 phase plane diagram, (c) e1 = X1X2, (d) e2 = Y1Y2, (e) X1X2 phase plane diagram and (f) Y1Y2 phase plane diagram.
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 will not occur; the synchronization occurs when g = 1 > 0.5.
3. Synchronize two coupled neurons using backstepping design For the convenience of designing the controller via backstepping method, the model of the coupled neurons is based on the single neuron model described in [16], having just a little bit difference as reviewed in Section 2. Only the state of the membrane potential can be controlled in practice via external electrical stimulation, so the controlled system is in the form of:
D. Bin et al. / Chaos, Solitons and Fractals 29 (2006) 182–189 2
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Fig. 3. x = 127.1, g = 1 > 0.5: (a) X1Y1 phase plane, (b) X2Y2 phase plane (c) e1 = X1X2, (d) e2 = Y1Y2, (e) X1X2 phase plane and (f) Y1Y2 phase plane.
8 dX 1 > > > dt ¼ X 1 ðX 1 1Þð1 rX 1 Þ Y 1 gðX 1 X 2 Þ þ I 0 > > > > > > dY 1 > > ¼ bðX 1 cY 1 Þ < dt > dX 2 > > ¼ X 2 ðX 2 1Þð1 rX 2 Þ Y 2 gðX 2 X 1 Þ þ I 0 þ u > > > dt > > > > > : dY 2 ¼ bðX cY Þ 2 2 dt
ð2Þ
Eq. (2) is a more general coupled model incorporating the recovery variable than that in Section 2, and u 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:
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e_ 1 ¼ X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ e2 2ge1 þ u e_ 2 ¼ bðe1 ce2 Þ
ð3Þ
The problem to realize the synchronization between two neurons is now transformed to a-problem on how to choose a control law u to make ei (i = 1, 2) generally converge to zero with time increasing. Here backstepping design is used to achieve the objective. First we consider the stability of system (4) e_ 2 ¼ bðe1 ce2 Þ
ð4Þ
where e1 is regarded as a controller. Choose Lyapunov function V1 as follows: 1 V 1 ðe2 Þ ¼ e22 2
ð5Þ
2
1.5
1.5
1
1 Y2
Y1
2
0.5
0.5
0
0
-0.5 -0.4 -0.2
0
(a)
0.2 0.4 X1
0.6
0.8
-0.5 -0.4 -0.2
1
0
(b)
0.2 0.4 X2
0.6
0.8
1
2
1
1.5 1 e2=Y2-Y1
e1=X2-X1
0.5 0 -0.5
0.5 0 -0.5 -1
-1
-1.5 -2
-1.5 0
2000
(c)
4000
2000
0
6000
4000 time
6000
time
(d) 2
1 0.8
1.5
0.6 1 Y2
X2
0.4 0.2
0.5
0 0
-0.2 -0.4 -0.4 -0.2
(e)
0
0.2
0.4 X1
0.6
0.8
-0.5 -0.5
1
(f)
0
1
0.5
1.5
2
Y1
Fig. 4. The controller is not applied and the synchronization is not achieved: (a) X1Y1 phase plane, (b) X2Y2 phase plane, (c) e1 = X1X2, (d) e2 = Y1Y2, (e) X1X2 phase plane and (f) Y1Y2 phase plane.
D. Bin et al. / Chaos, Solitons and Fractals 29 (2006) 182–189
187
The derivative of V1 is V_ 1 ¼ e_ 2 e2 ¼ bce22 þ be1 e2
ð6Þ
Assume controller e1 = a1(e2), Eq. (6) can be rewritten as V_ 1 ¼ e_ 2 e2 ¼ bce22 þ be2 a1 ðe2 Þ
ð7Þ
If a1(e2) = 0, then V_ 1 ¼ bce2 < 0
ð8Þ
2
Make system (4) asymptotically stable. Function a1(e2) is an estimative function when e1 is considered as a controller. The error between e1and a1(e2) is w1 ¼ e1 a1 ðe2 Þ
ð9Þ 2
1.5
1.5
1
1
Y1
Y2
2
0.5
0.5
0
0
-0.5 -0.4 -0.2
0
0.6
0.8
1
-0.5 -0.4 -0.2 (b)
0.15
0.15
0.1
0.1
0.05
0.05
0
0
e2=Y2-Y1
e1=X2-X1
(a)
0.2 0.4 X1
-0.05 -0.1 -0.15
0.2 0.4 X2
0.6
0.8
1
-0.05 -0.1 -0.15
-0.2 0
(c)
0
2000
-0.2
6000
4000 time
0
(d)
1
2000
4000 time
6000
2
0.8 1.5 1
0.4
Y2
X2
0.6
0.2
0.5
0 0
-0.2 -0.4 -0.4 -0.2
(e)
0
0.2
0.4 X1
0.6
0.8
-0.5 -0.5
1
(f)
0
0.5
1
1.5
2
Y1
Fig. 5. The controller is applied and the synchronization is achieved: (a) X1Y1 phase plane, (b) X2Y2 phase plane, (c) e1 = X1X2, (d) e2 = Y1Y2, (e) X1X2 phase plane diagram and (f) Y1Y2 phase plane.
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D. Bin et al. / Chaos, Solitons and Fractals 29 (2006) 182–189
Study the (e2, w1) system represented by (10) e_ 2 ¼ bðw1 ce1 Þ w_ 1 ¼ ½X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ e2 2gw1 þ u
ð10Þ
Choose Lyapunov function V2 as follows: 1 V 2 ðe2 ; w1 Þ ¼ V 1 ðe2 Þ þ w21 2
ð11Þ
f ðX 1 ; X 2 Þ ¼ X 2 ðX 2 1Þð1 rX 2 Þ X 1 ðX 1 1Þð1 rX 1 Þ
ð12Þ
Let
The derivation of V2 is V_ 2 ¼ V_ 1 þ w_ 1 w1 ¼ bce22 2gw21 þ w1 ðbe2 þ f ðX 1 ; X 2 Þ e2 þ uÞ
ð13Þ
Let the controller to be u ¼ ð1 bÞe2 f ðX 1 ; X 2 Þ
ð14Þ
Then V_ 2 can be described as V_ 2 ¼ bce22 2gw21 < 0
ð15Þ
The system is negative definite. As for an arbitrary initial error between two neurons, when the controller were applied, the initial error will converge to zero asymptotically and synchronization between the two coupled neurons will be achieved after a finite period of time. Take the parameters r = 10, b = 1, A = 0.1, x = 127.1, g = 0.05, c = 0.001, as is used in [16,24]. Fig. 4 shows the case that when the controller was not switched, the synchronization will not be achieved. The case of applying the controller is shown in Fig. 5.
4. Conclusion In this paper, backstepping design has been used to synchronize two neurons electrically coupled with gap junction in external electrical stimulation. The backstepping method is a systematic procedure for synchronizing chaotic systems and it can be applied to a variety of chaotic systems whether it contains external excitation of not. As this method needs only one controller to realize synchronization no matter how much dimensions the chaotic system contains and only the state of the membrane potential in the coupled neurons model can be controlled in practice, the backstepping design can synchronize the two coupled chaotic neurons perfectly. Further study is to use this method to synchronize n (n P 3) coupled chaotic neurons.
Acknowledgements The authors gratefully acknowledge the NSFC (no. 50177023) for supporting this research.
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