Global synchronization of two Ghostburster neurons via active control

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Chaos, Solitons and Fractals 40 (2009) 1213–1220 www.elsevier.com/locate/chaos

Global synchronization of two Ghostburster neurons via active control Li Sun, Jiang Wang *, Bin Deng School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China Accepted 31 August 2007

Abstract In this paper, active control law is derived and applied to control and synchronize two unidirectional coupled Ghostburster neurons under external electrical stimulation. Firstly, the dynamical behavior of the nonlinear Ghostburster model responding to various external electrical stimulations is studied. Then, using the results of the analysis, the active control strategy is designed for global synchronization of the two unidirectional coupled neurons and stabilizing the chaotic bursting trajectory of the slave system to desired tonic firing of the master system. Numerical simulations demonstrate the validity and feasibility of the proposed method.  2007 Elsevier Ltd. All rights reserved.

1. Introduction Motivated by potential applications in physics, biology, electrical engineering, communication theory and many other fields, the synchronization of chaotic systems has received an increasing interest [1–6]. Experimental studies [7–9] have pointed out that the synchronization is significant in the information processing of large ensembles of neurons. In experiments, the synchronization of two coupled living neurons can be achieved when depolarized by an external DC current [10,11]. Synchronization control [12–14] which has been intensively studied during last decade is found to be useful or has great potential in many domains such as in collapse prevention of power systems, biomedical engineering application to the human brain and heart and so on. Ghostburster model [15,16] is a two-compartment of pyramidal cells in the electrosensory lateral line lobe (ELL) from weakly electric fish. In this paper, we will investigate the relationship between the external electrical stimulations and the various dynamical behavior of the Ghostburster model. With the development of the control theory, various modern control methods, such as feedback linearization control [17], backstepping design [18,19], fuzzy adaptive control [20] and active control [21], have been successfully applied to neuronal synchronization theoretically in recent years. These control methods have been investigated with the objective of stabilizing equilibrium points or periodic orbits embedded in chaotic attractors [22]. In this paper, based on Lyapunov stability theory and Routh–Hurwitz criteria, some criteria for globally asymptotical chaos synchronization are established. The active controller can be easily

*

Corresponding author. Tel./fax: +86 22 27402293. E-mail address: [email protected] (J. Wang).

0960-0779/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.086

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designed on the basis of these conditions to synchronize two unidirectional coupled neurons and convert the chaotic motion of the slave neuron into the tonic firing as the master neuron. The rest of the paper has been organized as follows. In Section 2, dynamics of the two neurons in external electrical stimulation is studied. In Section 3, the global synchronization of two unidirectional coupled Ghostbursters neurons under external electrical stimulation is derived and numerical simulations are done to validate the proposed synchronization approach. Finally, conclusions are drawn in Section 4.

2. Dynamics of nonlinear Ghostburster model for individual neuron Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish have been observed to produce high-frequency burst discharge with constant depolarizing current [15]. We investigate a two-compartment model of an ELL pyramidal cell in Fig. 1, where one-compartment represents the somatic region, and the other as the entire proximal apical dendrite. In Fig. 1, Both the soma and dendrite contain fast inward Na+ current, INa,s and INa,d, and outward delayed rectifying K+ current, respectively IDr,s and IDr,d. In addition, the Ileak is somatic and dendritic passive leak currents, the Vs is somatic membrane potential, the Vd is dendritic membrane potential. The coupling between the compartments is assumed to be through simple electrotonic diffusion giving currents from soma to dendrite Is/d, or vice versa Id/s. In total, the dynamical system comprises six nonlinear differential equations, Eqs. (1)–(6), we refer to Eqs. (1)–(6) as the Ghostburster model. Soma: dV s g ¼ I s þ gNa;s  m21;s ðV s Þ  ð1  ns Þ  ðV Na  V s Þ þ gDr;s  n2s  ðV K  V s Þ þ c  ðV d  V s Þ þ gleak  ðV l  V s Þ dt k

ð1Þ

dns n1;s ðV s Þ  ns ¼ dt sn:s

ð2Þ

Dendrite: dV d g ¼ gNa;d  m21;d ðV d Þ  hd  ðV Na  V d Þ þ gDr;d  n2d  pd  ðV K  V d Þ þ c  ðV s  V d Þ þ gleak  ðV l  V d Þ dt 1k

ð3Þ

dhd h1;d ðV d Þ  hd ¼ dt sh:d

ð4Þ

dnd n1;d ðV d Þ  nd ¼ dt sn:d

ð5Þ

Fig. 1. Schematic of two-compartment model representation of an ELL pyramidal cell.

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Table 1 Parameter values have been introduced in Eqs. (1)–(6) Current

gmax

V1/2

K

s

INa,s(m1,s(Vs)) IDr,s(ns(Vs)) INa,d(m1,d(Vd)/hd(Vd)) IDr,d(nd(Vd)/pd(Vd))

55 20 5 15

40 40 40/52 40/65

3 3 5/5 5/6

N/A 0.39 N/A/1 0.9/5

dpd p1;d ðV d Þ  pd ¼ dt sp:d

ð6Þ

In Table 1, each ionic current (INa,s; IDr,s; INa,d; IDr,d) is modeled by a maximal conductance gmax (in units of ms/cm2), infinite conductance curves involving both V1/2 and k parameters m1;s ðV s Þ ¼ ðV s1V 1=2 Þ=k , and a channels time constant 1þe

s (in units of ms). x/y corresponds to channels with both activation (x) and inactivation (y), N/A means the channel activation tracks the membrane potential instantaneously. Other parameters values are k = 0.4, VNa = 40 mV, VK = 88.5 mV, Vleak = 70 mV, gc = 1, gleak = 0.18, Cm = 1 lF/cm2. The values of all channel parameters are used in the simulations.

3. Global synchronization of two Ghostbursters systems using active control 3.1. Synchronization principle Consider a general master–slave unidirectional coupled chaotic system as following:  x_ m ¼ Axm þ f ðxm Þ; x_ s ¼ Axs þ f ðxs Þ þ uðtÞ;

ð7Þ

where xm(t), xs(t) 2 R is n-dimensional state vectors of the system, A 2 Rn is a constant matrix for the system parameter, f: Rn ! Rn is a nonlinear part of the system, and u(t) 2 Rn is the control input. The lower scripts m and s stand for the master systems and the slave ones, respectively. The synchronization problem is how to design the controller u(t), which would synchronize the states of both the master and the slave systems. If we define the error vector as e = y  x, the dynamic equation of synchronization error can be expressed as: e_ ¼ Ae þ f ðxs Þ  f ðxm Þ þ uðtÞ:

ð8Þ

Hence, the objective of synchronization is to make limx!1 ke(t)k = 0. The problem of synchronization between the master and the slave systems can be transformed into a problem of how to realize the asymptotical stabilization of the error system (8). So the aim is to design a controller u(t) to make the dynamical system (8) asymptotically stable at the origin. Following the active control approach of [23], to eliminate the nonlinear part of the error dynamics, we can choose the active control function u(t) as: uðtÞ ¼ Be  f ðxs Þ þ f ðxm Þ;

ð9Þ

n

where B 2 R is a constant feedback gain matrix. Then the error dynamical system (8) can be rewritten as e_ ¼ Me;

ð10Þ n

where M = A  B and M 2 R . 3.2. Synchronization of the Ghostburster neuron via active control In order to state the synchronization problem of two Ghostbursters neurons, let us redefine the equations of unidirectional coupled system based on the Ghostburster model which has been stated in Section 2.

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Master system: dV s;m g ¼ I s;m þ gNa;s  m21;s ðV s;m Þ  ð1  ns;m Þ  ðV Na  V s;m Þ þ gDr;s  n2s;m  ðV K  V s;m Þ þ c  ðV K  V s;m Þ dt k þ gleak  ðV l  V s;m Þ dns;m n1;s ðV s;m Þ  ns;m ¼ dt sn;s

ð11Þ ð12Þ

dV d;m gc ¼ gNa;d  m21;d ðV d;m Þ  hd;m  ðV Na  V d;m Þ þ gDr;d  n2d;m  pd;m  ðV K  V d;m Þ þ  ðV s;m  V d;m Þ dt ð1  kÞ þ gleak  ðV l  V d;m Þ

ð13Þ

dhd;m h1;d ðV d;m Þ  hd;m ¼ dt sh;d

ð14Þ

dnd;m n1;d ðV d;m Þ  nd;m ¼ dt sn;d

ð15Þ

dpd;m p1;d ðV d;m Þ  pd;m ¼ dt sp;d Slave system: dV s;s g ¼ I s;s þ gNa;s  m21;s ðV s;s Þ  ð1  ns;s Þ  ðV Na  V s;s Þ þ gDr;s  n2s;s  ðV K  V s;s Þ þ c  ðV K  V s;s Þ dt k þ gleak  ðV l  V s;s Þ þ u1 dns;s n1;s ðV s;s Þ  ns;s ¼ dt sn;s dV d;s gc ¼ gNa;d  m21;d ðV d;s Þ  hd;m  ðV Na  V d;s Þ þ gDr;d  n2d;s  pd;m ðV K  V d;s Þ þ  ðV s;s  V d;s Þ dt ð1  kÞ þ gleak  ðV l  V d;s Þ þ u2

ð16Þ

ð17Þ ð18Þ

ð19Þ

dhd;s h1;d ðV d;s Þ  hd;s ¼ dt sh;d

ð20Þ

dnd;s n1;d ðV d;s Þ  nd;s ¼ dt sn;d

ð21Þ

dpd;s p1;d ðV d;s Þ  pd;s ¼ dt sp;d

ð22Þ

The added term u in Eq. (8) is the control force (synchronization command). Let e1 = Vs,s  Vs,m, e2 = Vd,s  Vd,m, We define the nonlinear functions f1, f2, f3 and f4 in Eqs. 11, 13, 17, 19, respectively as follows: ð23Þ f1 ¼ gNa;s  m21;s ðV s;m Þ  ð1  ns;m Þ  ðV Na  V s;m Þ þ gDr;s  n2s;m  ðV K  V s;m Þ f2 ¼ gNa;d  m21;s ðV d;m Þ  hd  ðV Na  V d;m Þ þ gDr;d  n2d;m  pd  ðV K  V d;m Þ

ð24Þ

f3 ¼ gNa;s  m21;s ðV s;s Þ  ð1  ns;s Þ  ðV Na  V s;s Þ þ gDr;s  n2s;s  ðV K  V s;s Þ

ð25Þ

f4 ¼ gNa;d  m21;s ðV d;s Þ  hd  ðV Na  V d;s Þ þ gDr;d  n2d;s  pd  ðV K  V d;s Þ

ð26Þ

The error dynamical system of the coupled neurons can be expressed by ( e_ 1 ¼ I s;s  I s;m þ f3  f1 þ gkc  ðe1  e2 Þ  gleak  e2 þ u1 gc e_ 2 ¼ f4  f2 þ 1k  ðe2  e1 Þ  gleak  e1 þ u2

ð27Þ

L. Sun et al. / Chaos, Solitons and Fractals 40 (2009) 1213–1220

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We define the active control functions u1(t) and u2 (t) as follows:  u1 ¼ I s;m  I s;s  f3 þ f1 þ V 1 u2 ¼ f4 þ f2 þ V 2

ð28Þ

40

20

Vs,m;Vs,s

0

-20

-40

-60

-80 400

420

440

460

480

500 time(ms)

520

540

560

580

600

Fig. 2. Regular synchronized state of the action potentials Vs,m and Vs,s under controller (28).

10

0

-10

Vd,m;Vd,s

-20

-30

-40

-50

-60

-70 400

420

440

460

480

500 time(ms)

520

540

560

580

600

Fig. 3. Regular synchronized state of the action potentials Vd,m and Vd,s under controller (28).

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Hence the error system (27) becomes ( e_ 1 ¼ gkc  e1  ðgkc þ gleak Þ  e2 þ V 1 gc gc e_ 2 ¼ ð1k þ gleak Þ  e1 þ 1k  e2 þ V 2

ð29Þ

The system (29) describes the error dynamics and can be interpreted as a control problem, where the system to be controlled is a linear system with a control input V1(t) and V2(t) as functions of e1 and e2. As long as these feedbacks stabilize the system, e1 and e2 converge to zero as time t goes to infinity. This implies that two systems are synchronized with active control. There are many possible choices for the control V1(t) and V2(t). We choose     V 1 ðtÞ e1 ; ð30Þ ¼B V 2 ðtÞ e2

a

b

40

10 0

20 -10 0 Vd,m

Vs,m

-20 -20

-30 -40

-40 -50 -60

-60

-80 -80

-60

-40

-20 Vs,s

0

20

40

-70 -60

-50

-40

-30

-20

-10

0

10

Vd,s

Fig. 4. Vs,m  Vs,s and Vd,m  Vd,s phase plane, before the controller (28) is applied. (a) Vs,m  Vs,s phase plane, (b) Vd,m  Vd,s phase plane.

10

40

0 20 -10 0

Vd,s

Vs,s

-20 -20

-30 -40

-40 -50 -60 -60 -80 -80

-60

-40

-20 Vs,m

0

20

40

-70 -70

-60

-50

-40

-30 Vd,m

-20

-10

0

10

Fig. 5. Vs,m  Vs,s and Vd,m  Vd,s phase plane, after the controller (28) is applied. (a) Vs,m  Vs,s phase plane, (b) Vd,m  Vd,s phase plane.

L. Sun et al. / Chaos, Solitons and Fractals 40 (2009) 1213–1220



1219



k1 k2 is a 2 · 2 constant matrix. Hence the error system (29) can be rewritten as: where B ¼   k3 k4   e_ 1 e1 ¼ MðtÞ ; e_ 2 e2 ! k 1 þ gkc k 2  ðgkc þ gleak Þ  gc  is the coefficient matrix. where MðtÞ ¼ gc k 3  1k k 4 þ 1k þ gleak According to Lyapunov stability theory and Routh–Hurwitz criteria, if ( gc ðk 1 þ gkc Þ þ ðk 4 þ 1k Þ<0 g c   gc  ðk 2  k þ gleak Þ  ðk 3  1k þ gleak Þ < 0

ð31Þ

ð32Þ

Then the eigenvalues of the error system (31) must be negative real or complex with negative real parts. From Theorem 1, the error system will be stable and the two Ghostbursters systems are globally asymptotic synchronized. 3.3. Numerical simulations In this subsection, Numerical simulations were carried out for the above Ghostburster neuronal synchronization system. We choose the system of tonic firing behavior as the master system (at Is = 6.5) and the chaotic bursting behavior as the slave system (at Is = 9). All the parameters and the initial conditions are the same as them in Section 2. The con  3 0 . trol action was implemented at time t0 = 500 ms. And a particular form of the matrix B is given by B ¼ 2 2 For this particular choice, the conditions (32) are satisfied, thus leading to the synchronization of two Ghostburster systems. In Figs. 2 and 3, the initial state of the master (solid line) and the slave (dashed line) systems is tonic firing and chaotic bursting, respectively. After the controller (28) is applied, the slave system transforms from chaotic bursting state into tonic firing synchronizing with the master one. In Figs. 4 and 5, Vs,m  Vs,s and Vd,m  Vd,s phase plane diagram of the master and the slave system before and after the controller (28) is applied, respectively, which shows the globally asymptotic synchronization.

4. Conclusions In this paper, synchronization of two Ghostbursters neurons under external electrical stimulation via the active control method is investigated. Firstly, the different dynamical behavior of the ELL pyramidal cells based on the Ghostburst model responding to various external electrical stimulations is studied. The tonic firing or chaotic bursting state of the trans-membrane voltage is obtained, as shown in Fig. 2a and b, respectively. Next, based on Lyapunov stability theory and Routh–Hurwitz criteria, this paper offers some sufficient conditions for global asymptotic synchronization between two Ghostbursters systems by the active control method. The controller can be easily designed on the basis of these conditions to ensure the global chaos synchronization of the action potentials as shown in Figs. 3–5. Moreover, numerical simulations show that the proposed control methods can effectively stabilize the chaotic bursting trajectory of the slave system to tonic firing of the master system.

Acknowledgement The authors gratefully acknowledge the support of the NSFC (No. 50537030).

References [1] Caroll TL, Pecora LM. Synchronizing chaotic circuits. IEEE Trans Circ Syst I 1991;38:453–6. [2] Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons & Fractals 1998;9:1555–61. [3] Restrepo JG, Ott E, Hunt BR. Emergence of synchronization in complex networks of interacting dynamical systems. Phys D: Nonlinear Phenom 2006;224(1-2):114–22. [4] Yang JZ, Hu G. Three types of generalized synchronization. Phys Lett A 2007;361(4–5):332–5.

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L. Sun et al. / Chaos, Solitons and Fractals 40 (2009) 1213–1220

[5] David I, Almeida R, Alvarez J, Barajas JG. Robust synchronization of Sprott circuits using sliding mode control. Chaos, Solitons & Fractals 2006;30(1):11–8. [6] Cuomo KM, Oppenheim AV, Strogatz SH. Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans Circ Syst I 1993;40:626–33. [7] Meister M, Wong RO, Baylor DA, Shatz CJ. Synchronous bursts of action potentials in ganglion cells of the developing mammalian retina. Science 1991;252:939–43. [8] Kreiter AK, Singer W. Stimulus-dependent synchronization of neuronal responses in the visual cortex of the awake macaque monkey. J Neurons 1996;16:2381–96. [9] Garbo AD, Barbi M, Chillemi S. The synchronization properties of a network of inhibitory interneurons depend on the biophysical model. BioSystems 2007;88:216–27. [10] Elson RC, Selverston AI, Huerta R, Rulkov NF, Rabinovich MI, Abarbanel HDI. Synchronization behavior of two coupled biological neurons. Phys Rev Lett 1999;81:5692–5. [11] Gray CM, Konig P, Engel AK, Singer W. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 1989;338:334–7. [12] Yu Hongjie, Peng Jianhua. Chaotic synchronization and control in nonlinear-coupled Hindmarsh–Rose neural systems. Chaos, Solitons & Fractals 2006;29(2):342–8. [13] Lu Hongtao, van Leeuwen C. Synchronization of chaotic neural networks via output or state coupling. Chaos, Solitons & Fractals 2006;30(1):166–76. [14] Longtin A. Effect of noise on the tuning properties of excitable systems. Chaos, Solitons & Fractals 2000;11(12):1835–48. [15] Doiron B, Laing C, Longtin A. Ghostbursting: a novel neuronal burst mechanism. J Comput Neurosci 2002;12:5–25. [16] Laing CR, Doiron B, Longtin A, Maler L. Ghostburster: the effects of dendrites on spike patterns. Neurocomputing 2002;44– 46:127–32. [17] Liao XX, Chen GR. Chaos synchronization of general Lur’e systems via time-delay feedback control. Int J Bifurcat Chaos 2003;13(1):207–13. [18] Bowong S, Kakmeni FM. Synchronization of uncertain chaotic systems via backstepping approach. Chaos, Solitons & Fractals 2004;21:999–1011. [19] Wang C, Ge SS. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos, Solitons & Fractals 2001;12:1199–206. [20] Wang LX. Adaptive fuzzy system and control: design and stability analysis. Englewood Cliffs, NJ: Prentice-Hall; 1994. [21] Bai EW, Lonngran EE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. [22] Wang CC, Su JP. A new adaptive variable structure control for chaotic synchronization and secure communication. Chaos, Solitons & Fractals 2004;20(5):967–77. [23] Bai E, Lonngrn KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000;11:1041–4.