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 neuronsvia 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 Ghost-burster neurons under external electrical stimulation. Firstly, the dynamical behavior of the nonlinear Ghostburstermodel responding to various external electrical stimulations is studied. Then, using the results of the analysis, the activecontrol strategy is designed for global synchronization of the two unidirectional coupled neurons and stabilizing thechaotic bursting trajectory of the slave system to desired tonic firing of the master system. Numerical simulations dem-onstrate 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 manyother 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 ofneurons. In experiments, the synchronization of two coupled living neurons can be achieved when depolarized by anexternal DC current [10,11]. Synchronization control [12–14] which has been intensively studied during last decade isfound to be useful or has great potential in many domains such as in collapse prevention of power systems, biomedicalengineering 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 stimulationsand the various dynamical behavior of the Ghostburster model. With the development of the control theory, variousmodern control methods, such as feedback linearization control [17], backstepping design [18,19], fuzzy adaptive con-trol [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 orbitsembedded 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

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

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

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1214 L. Sun et al. / Chaos, Solitons and Fractals 40 (2009) 1213–1220

designed on the basis of these conditions to synchronize two unidirectional coupled neurons and convert the chaoticmotion 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 electricalstimulation is studied. In Section 3, the global synchronization of two unidirectional coupled Ghostbursters neuronsunder external electrical stimulation is derived and numerical simulations are done to validate the proposed synchro-nization 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 producehigh-frequency burst discharge with constant depolarizing current [15]. We investigate a two-compartment model ofan ELL pyramidal cell in Fig. 1, where one-compartment represents the somatic region, and the other as the entireproximal apical dendrite.

In Fig. 1, Both the soma and dendrite contain fast inward Na+ current, INa,s and INa,d, and outward delayedrectifying 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 compartmentsis assumed to be through simple electrotonic diffusion giving currents from soma to dendrite Is/d, or vice versa Id/s. Intotal, the dynamical system comprises six nonlinear differential equations, Eqs. (1)–(6), we refer to Eqs. (1)–(6) as theGhostburster model.Soma:

dV s

dt¼ I s þ gNa;s � m2

1;sðV sÞ � ð1� nsÞ � ðV Na � V sÞ þ gDr;s � n2s � ðV K � V sÞ þ

gc

k� ðV d � V sÞ þ gleak � ðV l � V sÞ ð1Þ

dns

dt¼ n1;sðV sÞ � ns

sn:s

ð2Þ

Dendrite:

dV d

dt¼ gNa;d � m2

1;dðV dÞ � hd � ðV Na � V dÞ þ gDr;d � n2d � pd � ðV K � V dÞ þ

gc

1� k� ðV s � V dÞ þ gleak � ðV l � V dÞ ð3Þ

dhd

dt¼ h1;dðV dÞ � hd

sh:d

ð4Þ

dnd

dt¼ n1;dðV dÞ � nd

sn:d

ð5Þ

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

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

Current gmax V1/2 K s

INa,s(m1,s(Vs)) 55 �40 3 N/AIDr,s(ns(Vs)) 20 �40 3 0.39INa,d(m1,d(Vd)/hd(Vd)) 5 �40/�52 5/�5 N/A/1IDr,d(nd(Vd)/pd(Vd)) 15 �40/�65 5/�6 0.9/5

L. Sun et al. / Chaos, Solitons and Fractals 40 (2009) 1213–1220 1215

dpd

dt¼

p1;dðV dÞ � pd

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;sðV sÞ ¼ 1

1þe�ðV s�V 1=2 Þ=k, and a channels time constant

s (in units of ms). x/y corresponds to channels with both activation (x) and inactivation (y), N/A means the channelactivation 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:

_xm ¼ Axm þ f ðxmÞ;_xs ¼ Axs þ f ðxsÞ 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 themaster systems and the slave ones, respectively. The synchronization problem is how to design the controller u(t), whichwould synchronize the states of both the master and the slave systems. If we define the error vector as e = y � x, thedynamic equation of synchronization error can be expressed as:

_e ¼ Aeþ f ðxsÞ � f ðxmÞ 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 thedynamical 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 choosethe active control function u(t) as:

uðtÞ ¼ �Be� f ðxsÞ þ f ðxmÞ; ð9Þ

where B 2 Rn is a constant feedback gain matrix.Then the error dynamical system (8) can be rewritten as

_e ¼ Me; ð10Þ

where M = A � B and M 2 Rn.

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 unidi-rectional coupled system based on the Ghostburster model which has been stated in Section 2.


Master system:

dV s;m

dt¼ I s;m þ gNa;s � m2

1;sðV s;mÞ � ð1� ns;mÞ � ðV Na � V s;mÞ þ gDr;s � n2s;m � ðV K � V s;mÞ þ

gc

k� ðV K � V s;mÞ

þ gleak � ðV l � V s;mÞ ð11Þ

dns;m

dt¼ n1;sðV s;mÞ � ns;m

sn;s

ð12Þ

dV d;m

dt¼ gNa;d � m2

1;dðV d;mÞ � hd;m � ðV Na � V d;mÞ þ gDr;d � n2d;m � pd;m � ðV K � V d;mÞ þ

gc

ð1� kÞ � ðV s;m � V d;mÞ

þ gleak � ðV l � V d;mÞ ð13Þ

dhd;m

dt¼ h1;dðV d;mÞ � hd;m

sh;d

ð14Þ

dnd;m

dt¼ n1;dðV d;mÞ � nd;m

sn;d

ð15Þ

dpd;m

dt¼

p1;dðV d;mÞ � pd;m

sp;d

ð16Þ

Slave system:

dV s;s

dt¼ I s;s þ gNa;s � m2

1;sðV s;sÞ � ð1� ns;sÞ � ðV Na � V s;sÞ þ gDr;s � n2s;s � ðV K � V s;sÞ þ

gc

k� ðV K � V s;sÞ

þ gleak � ðV l � V s;sÞ þ u1 ð17Þ

dns;s

dt¼ n1;sðV s;sÞ � ns;s

sn;s

ð18Þ

dV d;s

dt¼ gNa;d � m2

1;dðV d;sÞ � hd;m � ðV Na � V d;sÞ þ gDr;d � n2d;s � pd;mðV K � V d;sÞ þ

gc

ð1� kÞ � ðV s;s � V d;sÞ

þ gleak � ðV l � V d;sÞ þ u2 ð19Þ

dhd;s

dt¼ h1;dðV d;sÞ � hd;s

sh;d

ð20Þ

dnd;s

dt¼ n1;dðV d;sÞ � nd;s

sn;d

ð21Þ

dpd;s

dt¼

p1;dðV d;sÞ � pd;s

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, Wedefine the nonlinear functions f1, f2, f3 and f4 in Eqs. 11, 13, 17, 19, respectively as follows:

f1 ¼ gNa;s � m21;sðV s;mÞ � ð1� ns;mÞ � ðV Na � V s;mÞ þ gDr;s � n2

s;m � ðV K � V s;mÞ ð23Þ

f2 ¼ gNa;d � m21;sðV d;mÞ � hd � ðV Na � V d;mÞ þ gDr;d � n2

d;m � pd � ðV K � V d;mÞ ð24Þ

f3 ¼ gNa;s � m21;sðV s;sÞ � ð1� ns;sÞ � ðV Na � V s;sÞ þ gDr;s � n2

s;s � ðV K � V s;sÞ ð25Þ

f4 ¼ gNa;d � m21;sðV d;sÞ � hd � ðV Na � V d;sÞ þ gDr;d � n2

d;s � pd � ðV K � V d;sÞ ð26Þ
The error dynamical system of the coupled neurons can be expressed by
_e1 ¼ I s;s � I s;m þ f3 � f1 þ gc

k � ðe1 � e2Þ � gleak � e2 þ u1

_e2 ¼ f4 � f2 þ gc

1�k � ðe2 � e1Þ � gleak � e1 þ u2

(ð27Þ


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

400 420 440 460 480 500 520 540 560 580 600-80

-60

-40

-20

0

20

40

time(ms)

Vs,m

;Vs,

s

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

400 420 440 460 480 500 520 540 560 580 600-70

-60

-50

-40

-30

-20

-10

0

10

time(ms)

Vd,m

;Vd,

s

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


Hence the error system (27) becomes

Fig. 5.plane.

a

Fig. 4.plane.

_e1 ¼ gc

k � e1 � ðgc

k þ gleakÞ � e2 þ V 1

_e2 ¼ �ð gc

1�k þ gleakÞ � e1 þ gc

1�k � 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 con-trolled is a linear system with a control input V1(t) and V2(t) as functions of e1 and e2. As long as these feedbacks sta-bilize the system, e1 and e2 converge to zero as time t goes to infinity. This implies that two systems are synchronizedwith active control. There are many possible choices for the control V1(t) and V2(t). We choose

V 1ðtÞV 2ðtÞ

� �¼ B

e1

e2

� �; ð30Þ

-80 -60 -40 -20 0 20 40-80

-60

-40

-20

0

20

40

Vs,m

Vs,s

-70 -60 -50 -40 -30 -20 -10 0 10-70

-60

-50

-40

-30

-20

-10

0

10

Vd,m

Vd,s

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

-80 -60 -40 -20 0 20 40-80

-60

-40

-20

0

20

40

Vs,s

Vs,m

-60 -50 -40 -30 -20 -10 0 10-70

-60

-50

-40

-30

-20

-10

0

10

Vd,s

Vd,m

b

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


where B ¼ k1 k2

k3 k4

� �is a 2 · 2 constant matrix. Hence the error system (29) can be rewritten as:� � � �

_e1

_e2

¼ MðtÞe1

e2

; ð31Þ

where MðtÞ ¼k1 þ gc

k k2 � ðgck þ gleakÞ

k3 � gc

1�k þ gleak

� �k4 þ gc

1�k

!is the coefficient matrix.

According to Lyapunov stability theory and Routh–Hurwitz criteria, if

ðk1 þ gc

k Þ þ ðk4 þ gc

1�kÞ < 0

ðk2 � gc

k þ gleak

� �Þ � ðk3 � gc

1�k þ gleak

� �Þ < 0

(ð32Þ

Then the eigenvalues of the error system (31) must be negative real or complex with negative real parts. From Theorem1, 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 sys-tem. We choose the system of tonic firing behavior as the master system (at Is = 6.5) and the chaotic bursting behavioras the slave system (at Is = 9). All the parameters and the initial conditions are the same as them in Section 2. The con-

trol action was implemented at time t0 = 500 ms. And a particular form of the matrix B is given by B ¼ �3 02 �2

� �.

For this particular choice, the conditions (32) are satisfied, thus leading to the synchronization of two Ghostburstersystems.

In Figs. 2 and 3, the initial state of the master (solid line) and the slave (dashed line) systems is tonic firing and cha-otic bursting, respectively. After the controller (28) is applied, the slave system transforms from chaotic bursting stateinto tonic firing synchronizing with the master one. In Figs. 4 and 5, Vs,m � Vs,s and Vd,m � Vd,s phase plane diagram ofthe master and the slave system before and after the controller (28) is applied, respectively, which shows the globallyasymptotic synchronization.

4. Conclusions

In this paper, synchronization of two Ghostbursters neurons under external electrical stimulation via the active con-trol method is investigated. Firstly, the different dynamical behavior of the ELL pyramidal cells based on the Ghost-burst model responding to various external electrical stimulations is studied. The tonic firing or chaotic bursting state ofthe trans-membrane voltage is obtained, as shown in Fig. 2a and b, respectively. Next, based on Lyapunov stabilitytheory and Routh–Hurwitz criteria, this paper offers some sufficient conditions for global asymptotic synchronizationbetween two Ghostbursters systems by the active control method. The controller can be easily designed on the basis ofthese 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 ofthe slave system to tonic firing of the master system.

Acknowledgement

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

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global synchronization of two ghostburster neurons via active control

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