Synchronization of chaotic neurons coupled with gap junction with time delays in external electrical stimulation

Chaos, Solitons and Fractals 35 (2008) 512–518 www.elsevier.com/locate/chaos

Synchronization of chaotic neurons coupled with gap junction with time delays in external electrical stimulation Jiang Wang *, Bin Deng, Xiangyang Fei School of Electrical and Automation Engineering, Tianjin University, 300072, Tianjin, PR China Accepted 22 May 2006

Abstract The synchronization of chaotic neurons coupled with gap junction with time delays is investigated. In this paper, the coupled model is based on the nonlinear cable model of neuron. The influences of the strength of gap junction and the time delay on the synchronization are discussed in detail. The so called delay-dependent criteria for the synchronization of two coupled neurons which contain the information of the time delay and the coupling strength is given.  2006 Elsevier Ltd. All rights reserved.

1. Introduction Many cells are linked to each other by specialized intercellular pathways known as gap junctions [1]. Gap junctions are clusters of aqueous channels that connect the cytoplasm of adjoining cells. They allow the direct transfer of ions and small molecules, including second messenger molecules, between cells without leakage to the extra-cellular 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 [2–6]. Over the last decade, many new types of synchronization have appeared. Since the discovery of chaos synchronization [7,8], there has been tremendous interest in studying the synchronization of chaotic systems. Recently, synchronization of coupled chaotic systems has received considerable attention [9–13]. Especially, a typical study of synchronization is the coupled chaotic identical chaotic systems [14]. Synchronization in physical and biological systems is a fascinating subject that has attracted a lot of renewed attention [15]. Theoretical studies in this direction mainly focus on the synchronization of coupled oscillatory subsystem [16,17]. In fact, there are many synchronous systems existing in the real world where their synchronization has been effected by time delays. Time delays may be caused by transportation retard or by the tolerance of some reaction itself in the communication and epidemic, respectively. It is well known that delays appear in many dynamic systems and have important impacts on the dynamics [18]. Many authors have paid attention to the investigation of systems with delays [19–21]. When time delays are considered, the dynamics of systems generally become more complicated and the stability problem becomes more interesting. In [22], the influence of the coupling strength of gap junction on the synchronization of chaotic neurons is investigated. In this paper, the facts of the coupling strength and the time delay are both considered. *

Corresponding author. Tel.: +86 22 27402293; fax: +86 22 27401101. E-mail address: [email protected] (J. Wang).

0960-0779/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.056

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The nonlinear cable model of individual neuron in external stimulation [23] is reviewed in Section 2. In Section 3, the so called delay-dependent criteria for the synchronization of two coupled neurons which contain the information of the time delay and the coupling strength is given. The conclusion is given in Section 4.

2. The nonlinear cable model of single neuron In [23], the nonlinear cable model of cylindrical cells with external electrical stimulation is described. The Fitzhugh– Nagumo recovery variable is introduced in this model. The membrane conductance of the cell is expected to be a nonlinear function of the membrane voltage and time. The circuit diagram is shown in Fig. 1. The nonlinear cable model is described by ( dX ¼ X ðX  1Þð1  rX Þ  Y þ I 0 ðtÞ; dt ð1Þ dY ¼ bX ; dt where X and Y are the membranes voltage V and recovery variable W rescaled by Vp, the peak of the active potential, respectively: X ¼

V ; Vp

Y ¼

W : Vp

ð2Þ

The variable r¼

Vp ; Vt

ð3Þ

Vt is the threshold value of active potential. The I0(t) represents the external electrical stimulation which is described as I 0 ðtÞ ¼

A cos xt: x

ð4Þ

Fix r = 10, b = 1, A = 0.1 and vary the frequency x with the initial condition X(0) = Y(0) = 0.1, the complex behavior including chaos can be observed as shown in Fig. 2.

3. The synchronization of coupled neurons with coupling delay First, let us investigate the stability properties of systems described by linear differential difference equation x_ ðtÞ ¼ AxðtÞ þ Bxðt  sÞ; n·n

ð5Þ

n

where A, B 2 R , x(t) 2 R , s > 0. In [24], the delay-dependent stability conditions are given as

I0

f (V )

Lm

Cm

V

Fig. 1. The circuit diagram of the nonlinear cable model.

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(a)

(b)

(c)

(d)

Fig. 2. The X  Y phase plane diagram: (a) x = 35 Hz, 1-periodic oscillation, (b) x = 78 Hz, 2-periodic oscillation, (c) x = 127 Hz, 3-periodic oscillation, (d) x = 127.1 Hz, chaotic attractor.

Theorem 1 [24]. Let l1 = l(A) + kBk, l2 = l (jA) + kBk, j2 = 1, if l1 P 0, then the following condition assures the stability of system (5) Re ki ðA þ B expðssÞÞ < 0;

i ¼ 1; 2; . . . ; n:

ð6Þ

And where s takes the value in the ranges given by s ¼ jx; 0 6 x 6 l2 ; s ¼ l1 þ jx; 0 6 x 6 l2 ;

ð7Þ ð8Þ

s ¼ r þ jl2 ;

ð9Þ

0 6 r 6 l1 ;

l(X) is the matrix measure for X 2 Rn·n which can be defined through l(X) = lim(kI + eXk  1)/e, e ! 0+, kXk denotes a matrix norm of X. The calculation of l(X) is given in [24] as the following: For any complex square matrices lðX Þ ¼

1 max ki ðX T þ X Þ; 2 i

ð10Þ

ki(X) denotes the eigenvalue of X and Re ki(X), Im ki(X) are the real and imaginary parts, respectively. Theorem 1 which carries information of the delays is referred to as delay-dependent criteria. Notice that condition (6) is a stability problem of the differential equation with complex coefficient, so the stability criteria for linear differential equations with complex coefficients is given as Theorem 2 [25]. The necessary and sufficient stability condition for the characteristic equation f ðkÞ ¼ kn þ ðaRn1 þ jaIn1 Þkn1 þ    þ ðaR1 þ jaI1 Þk þ ðaR0 þ jaI0 Þ ¼ 0; D01 ; D02 ; . . . ; D0n

is that all ‘north-westerly’ minors aRn1 aIn1  aIn2 r21 ¼ aRn1 ; r31 ¼ ; aRn1

of even order of the Bilharz matrix are positive, where aRn1 aRn2 þ aRn3 ri1;1 ri2;jþ1  ri2;1 ri1;jþ1 r32 ¼ ; rij ¼ ; aRn1 ri1;j

ð11Þ

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and D01 ¼ r21 ; D02 ¼ r21  r31  r41 ; D0n ¼ r21  r31      r2n1 : Now, we consider two neurons in the same external electrical stimulation coupled with gap junction with time delays described by 8 dX 1 ¼ X 1 ðX 1  1Þð1  rX 1 Þ  Y 1  gðX 1  X 2 ðt  sÞÞ þ I 0 ðtÞ; > dt > > > < dY 1 ¼ bX 1 ; dt

dX 2 > ¼ X 2 ðX 2  1Þð1  rX 2 Þ  Y 2  gðX 2  X 1 ðt  sÞÞ þ I 0 ðtÞ; > dt > > : dY 2 ¼ bX 2 ; dt

ð12Þ

where Xi, Yi, i = 1, 2 are status variables, provided that the orbit is close enough to the basin of attraction, g presents the coupling strength of gap junction and s is the time delay. On the base of Theorems 1 and 2, and the delay coupled model (12), Theorem 3 which is the delay-dependent criteria of the two neurons coupled with time delays can be given as the following: Theorem 3 (Delay-dependent criteria). The sufficient condition for the synchronization of the time delay coupled model (12) is gesr cos sx  f 0 ðsÞ þ g > 0: And variables r, x take the value in the ranges determined by the parameters in model (12). Proof. In fact, considering the symmetry of (12), the synchronization can always occur. However, the synchronization may be unstable under some conditions. Let eX = X1  X2, eY = Y1  Y2, the error dynamical system of the two coupled neurons can be expressed by  e_ X ¼ f ðX 1 Þ  f ðX 2 Þ  eY  geX  geX ðt  sÞ; ð13Þ e_ Y ¼ beX ; where f(x) = x(x  1)(1  rx). The linear equation of (13) in the equilibrium solution of (12) is  e_ X ¼ ðf 0 ðsÞ  gÞeX  eY  geX ðt  sÞ; e_ Y ¼ beX ;

ð14Þ

s denotes the equilibrium solution of (12). Notice that (14) has the equilibrium point (0, 0), then the question of the synchronization of two coupled neurons described in (12) is transformed to the stability of the error dynamical system. Eq. (14) has the form of (5) with  0    g 0 f ðsÞ  g 1 ; B¼ : ð15Þ A¼ b 0 0 0 Employing the matrix measure derived from Euclidean matrix norm with the parameters used in [22], we can easily obtain  0 f ðsÞ  g; f 0 ðsÞ  g P 0; lðAÞ ¼ ð16Þ 0; f 0 ðsÞ  g 6 0; lðjAÞ ¼ 0; kBk ¼ g:

ð17Þ ð18Þ

We notice that l(A) and g are always P0, then l1 = l(A) + kBk = l(A) + g P 0 and, l2 = l(jA) + kBk = g. According to Theorem 1, we only need to test the stability of the characteristic equation of the error system (14) f ðkÞ ¼ k2 þ ðgess  f 0 ðsÞ þ gÞk þ b ¼ 0; where s takes the value in the ranges given by

ð19Þ

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s ¼ jx; 0 6 x 6 g; s ¼ ðlðAÞ þ gÞ þ jx; 0 6 x 6 g; s ¼ r þ jg; 0 6 r 6 lðAÞ þ g:

ð20Þ ð21Þ ð22Þ

For the convenience of calculation, we assume that s ¼ r þ jx:

ð23Þ

Then (19) can be transformed as f ðkÞ ¼ k2 þ ððgesr cos sx  f 0 ðsÞ þ gÞ þ jðgesr sin sxÞÞk þ b ¼ 0:

ð24Þ

Then applying Theorem 2, we can get that the necessary and sufficient stability condition for Eq. (24) is D01 ¼ r21 ¼ gesr cos sx  f 0 ðsÞ þ g > 0;

ð25Þ

D02 ¼ r21  r31  r41 ¼ ðgesr cos sx  f 0 ðsÞ þ gÞ  ðgesr sin sxÞ 

gesr cos sx  f 0 ðsÞ þ g gesr sin sx

¼ ðgesr cos sx  f 0 ðsÞ þ gÞ2 > 0:

ð26Þ

As s takes the value in the ranges given by (20), (21) and (23), we can give the delay-dependent criteria for the synchronization of two coupled neurons which contain the information of the time delay and the coupling strength as gesr cos sx  f 0 ðsÞ þ g > 0:

ð27Þ

And variables r,x take the value in the ranges given by (20), (21) and (23).

(a)

(c)

h

(b)

(d)

Fig. 3. g = 0.2, s = 1: (a) X1  Y1 phase plane diagram, (b) X2  Y2 phase plane diagram, (c) the error e1 = X1  X2, (d) the error e2 = Y1  Y2.

J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 512–518

(a)

(b)

(c)

(d)

517

Fig. 4. g = 0.25, s = 1: (a) X1  Y1 phase plane diagram, (b) X2  Y2 phase plane diagram, (c) the error e1 = X1  X2, (d) The error e2 = Y1  Y2.

Now let the parameters in (12) take the values used in [22] r ¼ 10;

b ¼ 1;

A ¼ 0:1;

x ¼ 127:1:

ð28Þ

When g = 0.2, s = 1, the stability condition of the synchronization (27) is not satisfied, so the synchronization can not be achieved as shown in Fig. 3, and the synchronization is achieved by taking the value g = 0.25, s = 1, as shown in Fig. 4.

4. Conclusion In this paper, the synchronization of two chaotic neurons coupled with gap junction with time delays is investigated. The coupled model is based on the nonlinear cable model of neuron. Two stability criteria are applied to get the delaydependent criteria for the synchronization of two coupled neurons which contain the information of the time delay and the coupling strength. This paper is the first step to investigate the synchronization of neurons in complex conditions such as neurons coupled with different time delays and so on.

Acknowledgement The authors gratefully acknowledge the NSFC (no. 50537030) for supporting this research.

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