Special Topic: Neurodynamics May 2014 Vol.57 No.5: 872–878
• Article • doi: 10.1007/s11431-014-5535-z
Bifurcation analysis for Hindmarsh-Rose neuronal model with time-delayed feedback control and application to chaos control
WANG HaiXia1*, WANG QingYun2 & ZHENG YanHong3
1 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China; 2 Department of Dynamics and Control, Beihang University, Beijing 100191, China;
3 School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China
Received January 18, 2014; accepted March 3, 2014
This paper is concerned with bifurcations and chaos control of the Hindmarsh-Rose (HR) neuronal model with the time-delayed feedback control. By stability and bifurcation analysis, we find that the excitable neuron can emit spikes via the subcritical Hopf bifurcation, and exhibits periodic or chaotic spiking/bursting behaviors with the increase of external current. For the purpose of control of chaos, we adopt the time-delayed feedback control, and convert chaos control to the Hopf bifur-cation of the delayed feedback system. Then the analytical conditions under which the Hopf bifurcation occurs are given with an explicit formula. Based on this, we show the Hopf bifurcation curves in the two-parameter plane. Finally, some numerical simulations are carried out to support the theoretical results. It is shown that by appropriate choice of feedback gain and time delay, the chaotic orbit can be controlled to be stable. The adopted method in this paper is general and can be applied to other neuronal models. It may help us better understand the bifurcation mechanisms of neural behaviors.
hopf bifurcation, time-delayed feedback control, chaos control, neuronal model
Citation: Wang H X, Wang Q Y, Zheng Y H. Bifurcation analysis for hindmarsh-rose neuronal model with time-delayed feedback control and application to chaos control. Sci China Tech Sci, 2014, 57: 872878, doi: 10.1007/s11431-014-5535-z
1 Introduction
Bifurcation is one of the most important dynamical proper-ties in chaotic neuronal systems or other dynamical systems, which can demonstrate the mechanisms of excitability [1,2], bursting nomination [1,3], synchronization transitions [4], and the route to chaos is also associated with bifurcations. Bifurcations of co-dimension 2 can classify various bursting types in the two-parameter plane [5]. Bifurcations led by the coupling strength and time delay can induce firing behavior and form different synchronous transitions [6]. Bifurcation analysis can also help us better understand the properties of equilibria in the coupled neuronal model [7]. So the inves-
tigation of bifurcation is of vital importance for us to under-stand the cognitive activities of neural systems.
Since the discovery of Lorenz chaotic attractor in 1963, there has been much attention paid to study of chaos. In nature and human society, the fact of existence of chaos is generally accepted, and how to apply the chaos researches to the human services has become an important task in the development of nonlinear science. In some fields, chaos is useful, such as in secure communications, but in other fields, chaos may be harmful, for example, chaos in fluid system can lead to damage of the coherent structure. Because of the sensitivity to initial conditions and unpredictability, chaos control has become a key point in applications of chaos in many fields [8–11]. In a wide sense of the word, chaos synchronization belongs to chaos control. Synchronization, as a collective behavior in the coupled neural systems, has
Wang H X, et al. Sci China Tech Sci May (2014) Vol.57 No.5 873
been suggested as a mechanism for information binding, neural information processing and transmission, neurologi-cal diseases like Parkinson's disease or epilepsy [12].
In 1989, H ubler [13] first published an article about chaos control. In 1990, Ott et al. [14] proposed the famous OGY (Ott, Grelogi, Yorke) control method. In recent dec-ades, chaos control and chaos synchronization have attracted more and more attention. Many researchers have proposed various chaos control methods and synchronization schemes [15–19].The existing control methods can be classified, mainly, into two categories: feedback control and non-feed- back control, among which, the time-delayed feedback con-trol (DFC) method, proposed by Pyragas [15], has been a powerful tool for chaos control in different dynamical sys-tems [20–25]. This type of control employs a continuous feedback signal which is proportional to the difference be-tween the current state variable ( )x t and its delayed value:
( ) [ ( ) ( )],u t k x t x t
where is the time delay and k is the feedback gain. Using DFC method, we may appropriately choose the
time delay and feedback gain to realize the control of chaos. In refs. [20] and [21], the researchers gave a theoretical analysis on chaos control of the three-dimensional continu-ous dynamical systems, and obtained conditions under which a sequence of Hopf bifurcations occurred. Song et al. [22] investigated the effect of time delay on the dynamics of Chen's system with delayed feedback control. By the nor-mal form theory and center manifold argument, they de-rived the explicit formulas determining the stability and direction of bifurcating periodic solutions. Their results show that when the delay passes through certain values, chaotic oscillation is converted into a stable periodic orbit. Due to its simplicity, R o ssler system has become a bench-mark to test the effectiveness of the chaos control strategy. For example, Ding et al. [23] and Xu et al. [24] studied the time-delayed feedback controlled R o ssler system with the single delay and the multiple delays, respectively. Similar results about the existence, stability and direction of the Hopf bifurcation to ref. [22] were obtained.
Although DFC is an effective method for chaos control in the single-time-scale systems, such as Lorenz system, R o ssler system, it is not easy to apply in to the neuronal systems because of the strong nonlinearity and mul-ti-time-scale. In ref. [25], Yu et al. considered chaos control of HR neuron via DFC with the help of interspike intervals (ISIs) bifurcation diagrams, and found that the chaotic bursting behavior could be turned into periodic bursting patterns in certain range of feedback gain or time delay. Schöll et al. [26] pointed out that the stochastic synchroni-zation could be deliberately controlled by local time- delayed feedback in two instantaneously coupled FHN neurons. From above, we find that DFC method has been applied to neural systems, even if there is lack of theoretical
analysis. So, in this paper, we will further examine this is-sue analytically. By using bifurcation theory of functional differential equation [27], the conditions that generate peri-odic solutions in the controlled system are obtained with an explicit formula. Based on which, we can show the Hopf bifurcation curves in the feedback gain and time delay plane. The adopted method in this paper is general and can be ap-plied to other neuronal models. It may help us better under-stand the bifurcation mechanisms of neural behaviors.
This paper is organized as follows: In Section 2 we out-line the neuronal model. Section 3 deals with Hopf bifurca-tions of the uncontrolled and controlled systems, and we give the conditions under which the Hopf bifurcation occurs. Some numerical simulations are carried out in Section 4 to support the theoretical results. Our conclusions are left to the last section.
2 Model description
It is well known that the Hindmarsh-Rose (HR) neuronal model is an alternative candidate for studying dynamics of neuronal systems since it has a relatively simple mathemat-ical form and can exhibit rich spiking/bursting firing pat-terns. The single HR neuronal model is described by the following three-dimensional continuous system:
3 2
2
,
,
[ ( ) ],
x y ax bx z I
y c x y
z r s x x z
d (1)
where the state variable x represents the membrane potential, y describes the exchange of ions across the neuron mem-brane through fast ionic channels, and z is a slowly chang-ing adaptation current. I mimics the external current for biological neurons, and r is a small parameter which con-trols the speed of variation of the slow variable z, x sets the resting potential of the system. And a, b, c, d, s, r are system parameters. Throughout the paper, we always fix
1a , 3b , 1c , 5d , 4s , 0.006r , 1.56x , when I varies, the system can exhibit rich dynamics, for example, when 3I , system (1) is chaotic bursting (see Figure 1).
In the present paper, we focus on control of HR chaotic
Figure 1 Chaotic attractor for HR neuron.
874 Wang H X, et al. Sci China Tech Sci May (2014) Vol.57 No.5
attractor. We adopt the method of Pyragas, adding a time-delayed difference feedback to the membrane potential of (1), then the controlled system is described by
3 2
2
[ ( ) ( )],
,
[ ( ) ],
x y ax bx z I k x t x t
y c dx y
z r s x x z
(2)
where k is the feedback gain and is the time delay. Ob-viously, when 0 , eq. (2) becomes (1). The effect of feedback gain on dynamics of eq. (2) can be seen in Figure 2.
From the figure, we can see that when we fix 0.3 , the period-doubling bifurcation occurs, after which, many periodic bursting behaviors appear.
3 Hopf bifurcation analysis and chaos control
In this section, we first investigate the effect of I on the dynamical behavior of system (1). The equilibrium of the single neuron satisfies the following equations:
3 2
2
0,
0,
( ) 0,
y ax bx z I
c dx y
s x x z
(3)
then
3 2
2
( ) ( ) 0,
,
( ).
ax d b x sx sx c I
y c dx
z s x x
(4)
By numerical computation, we can get that when 1.2I , eq. (4) has only one real solution, i.e. equilibrium
( 1.3059, 7.5269,1.0164)H , and the corresponding char-acteristic equation has a pair of purely imaginary roots
0.0409i , so the Hopf bifurcation occurs in system (1) as 1.2I . If we further take 1.19I and 1.21I , then
the only equilibrium of system (1) is stable and unstable, respectively, meanwhile, the single neuron emits spikes with period-1. By calculating the first Lyapunov efficient
Figure 2 ISIs bifurcation diagram of the membrane potential versus the feedback gain k with = 0.3 for system (2).
1 (0)l of the Hopf bifurcation point with the bifurcation
theory [7,28], we can get 1(0) 0l , so system (1) under-
goes the subcritical Hopf bifurcation at point H for 1.2I . As I becomes larger, the HR neuron can exhibit
various periodic or chaotic spiking/bursting behaviors, and the bifurcation diagram of ISIs of the membrane potential with respect to I is shown in Figure 3. We can find the route from the period-adding bifurcation and the inverse period-doubling bifurcation to chaos with the increase of I . If we choose 3I , the single HR neuron exhibits chaotic bursting behavior (see the inserted diagram of Figure 3), and there is only one unstable equilibrium 0 0 0 0( , , )e x y z
( 0.7288, 1.6557,3.3248) in system (1).
In the following, we will investigate under what condi-tion the parameters and k can satisfy chaos control. From the mathematical point of view, the above problem can be converted to the Hopf bifurcation of the delayed feedback system (2). Clearly, system (2) has the same equi-librium as system (1).
By the linear transform,
0
0
0
,
,
.
x x x
y y y
z z z
(5)
Substituting eq. (5) into (2), we get the linearizing equa-tions:
20 0
0
3 2 [ ( ) ( )],
2 ,
( ).
x y ax x bx x z k x t x t
y dx x y
z r sx z
(6)
By introducing T( , , ) ,X x y z eq. (6) can be rewritten
as the following vector form:
( ) ( ) ( ),X t BX t KX t (7)
where
Figure 3 ISIs bifurcation diagram of the membrane potential versus the external current I for system (1), the inserted small figure is the time evolu-tion with I = 3.
Wang H X, et al. Sci China Tech Sci May (2014) Vol.57 No.5 875
20 0
0
20 0
0
3 2 1 1
2 1 0
0
3 2 1 1 0 0
2 1 0 0 0 0
0 0 0 0
,
ax bx k
B dx
rs r
ax bx k
dx
rs r
A K
A is the Jacobian matrix evaluated at 0e .
The associated characteristic equation of eq. (7) is
e( ) 0,h I B K
i.e.
e2
0 0
0
( 3 6 )( 1)( )
4 ( 1) 10 ( ) 0.
x x k k r
r x r
We might as well rewrite the characteristic equation as
e3 2 21 1 1 2 2 2( ) 0,p q r p q r (8)
where
21 0 01 3 6 ( ),p k r x x k Tr A
2
1 0 0 0
1 1 2 3
5 10 (3 6 )( 1)
( ) ,
q r x x x k r
kTr A A A A
21 0 0 0
1
[ (3 6 ) 4 10 ]
,
r k x x x r
k A A
2 ,p k
2 1( 1) ( ),q k r kTr A
2 1 ,r kr k A
here ( )Tr denotes the trace of matrix, and represents the
determinant, ( 1,2,3)iA i denotes the cofactor of ele-
ment iia in matrix ( ).ijA a
It is obvious that when 0 , eq. (8) has no zero root, so fold bifurcation does not occur to 0e . When 0 , if
Hopf bifurcation occurs, i.e. the characteristic equation has a pair of conjugate purely imaginary roots, we can substitute
( 0)i into eq. (8), separate the real and imaginary parts, and get
1 2 1
2 1 2
cos sin 0,
cos sin 0,
b k b k c
b k b k c
(9)
where 2
1 ,b r
2
21 0
32 0
20 0
( 1),
( 1 ) 4 10 ,
[ ( 1) 4 10 ] ,
3 6 ,
b r
c p r k kr rp r x r
c k r r rp r p x
p x x
solving eq. (9), we obtain
2 22 2 0
2
2 2
2 22 2 2 0
2
2 2
10 ( )4 ( )
1sin ,( )
10 ( )4 ( )( )
1cos .( )
x rr r
k r
x rr p k r
k r
From the identities
sin
tancos
and
2 2 2 2 21 2 1 2( ),c c k b b
we get
2 22 2 0
2
2 22 2 2 0
2
10 ( )4 ( )
1tan ,10 ( )
4 ( )( )1
x rr r
x rr p k r
(10)
and
6 4 23 3 3 0,p q r (11)
where
23 1 1 2 32[ ( ) ] [ ( )] 2 ( ),p kTr A A A A Tr A kTr A
3 1
21 1 2 3 1 2 3
2[ ( ) ( )
( )( )] ( ) ,
q kTr A A A Tr A k A
kTr A A A A A A A
3 1( 2 ) .r A k A A
Let 2 0 , then eq. (11) is transformed to be
3 23 3 3 0.p q r (12)
eq. (12) has at least one positive real root if the following theorem is satisfied.
Theorem 3.1. [22] Assume 3 23 3( )h p q
3 ,r 31 3
p
is the stationary point of function h,
where 23 33p q , we have the following results:
(i) If 3 0r , then eq. (12) has at least one positive real
root. (ii) If 3 0r and 0 , then eq. (12) has no positive
876 Wang H X, et al. Sci China Tech Sci May (2014) Vol.57 No.5
real root. (iii) If 3 0r and 0 , then eq. (12) has positive real
roots if and only if 1 0 and 1( ) 0h .
Without loss of generality, we assume that eq. (12) has three positive roots, denoted by 1 , 2 and 3 , respec-
tively. Then eq. (11) has three positive roots 1 1 ,
2 2 and 3 3 . From eq. (10), we have
( ) 1ji
i
2 22 2 0
2
2 22 2 2 2 0
0 0 2
10 ( )4 ( )
1arctan ,
10 ( )4 (3 6 )( )
1
i ii i i
i
ii
i
x rr r
jx r
r x x k r
(13)
where 1,2,3; 0,1,2, .i j
Substituting ( ) into eq. (8) and taking the derivative with respect to , we get
dd
1 21 1 2 2
2 22 2 2 2 2 2
(3 2 ) 2,
( ) ( )
p q e p q
p q r p q r
(14)
Enlightened by the proof of Lemma 8 in ref. [22], we obtain
dd ( )
1
2 4 2 2 22 2 2
( ).( )
ji
ii
i i i
hq r p
Re (15)
Since 0,i ( )
1
ji
d
d
Re and ( )ih have the
same sign.
4 Numerical simulations
In this section, we will apply the theoretical results obtained in the previous section to system (2) for the purpose of con-trol of chaos. Based on the Hopf bifurcation analysis for the controlled system (2), we know that under certain critical conditions about k and , a family of periodic solutions bi-furcate from the unstable equilibrium 0e , then chaos may
vanish. For the equilibrium 0e of the controlled system (2), if
we substitute 0.006r , 0 0.7288x into eqs. (8) and
(12), we will have the corresponding characteristic equa-tions as
3 2
2
( 6.9722) (1.006 1.256) (0.006
0.0161) ( 1.006 0.006 ) 0
k k k
k k k e
(16)
and
3 23 3 3 0,p q r (17)
where 3 51.1242 11.9324 ,p k 3 1.3529 2.6428 ,q k 3r
0.0161(0.0161 0.012 ).k
According to Theorem 3.1, if 3 0r , i.e. 1.3417k ,
then eq. (12) has at least one positive real root. On the other hand, from 3 0r , we can obtain 1.3417k which
guarantees 2 23 33 142.3822 1228 2609.6 0p q k k .
Then there exist positive real roots in eq. (12) if and only if
1 0 and 1( ) 0.h By computation, 0.0026k .
Thus, we obtain the range of k , which leads to the Hopf bifurcation of system (2). For the purpose of control of cha-os, we consider 0.0026k . In particular, we take 1k , and calculate that eq. (12) has two positive real roots:
1 0.000357 , 2 0.0201 . Thus, 1 0.0189 , 2
0.1418 . From eq. (13), we get
( )1 52.9101(1.5236 ), 0,1,2, ,j j j
( )2 7.05221(1.3991 ), 0,1,2, ,j j j
Furthermore, we can test 1( ) 1.2449 0h , 2( )h
1.2462 0. Due to eq. (15), we have
dd ( )
1
1
0j
Re , dd ( )
2
1
0j
Re ,
which are the transversality condition for the Hopf bifurca-tion.
Note that
(0) (1)2 2
(0) (1)1 1
9.8667 32.0219
80.6138 246.8358
Thus, according to eqs.(17) and (13), we get the Hopf bifurcation curves in the two-parameter plane shown in Figure 4. The detailed steps are as follows. First, for a fixed value of k , for example 1k , we substitute k into eq. (17), and solve it, then we get two positive solutions 1 2, ,
and 1 1 , 2 2 are obtained. Second, substitut-
ing k and 1,2 into eq. (13), we get the expressions ( )1,2
j with 0,1,2, .j Third, drawing the points ( )1,2( , )jk
for different j , we can give the Hopf bifurcation curves in
the k plane . In fact, we can even calculate numerically that when
0.62k , the positive real roots exist in eq. (17), so the Hopf bifurcation takes place beyond the line 0.63k shown in Figure 4 with dash line. Also we can find that the originating Hopf bifurcation curves are a group of open arcs on which the Hopf bifurcation of equilibrium 0e occurs,
and the periodic orbits bifurcate from the interior of the
Wang H X, et al. Sci China Tech Sci May (2014) Vol.57 No.5 877
Figure 4 Bifurcation diagram of the feedback gain k and time delay , where the curves denote Hopf bifurcation curves, the inserted dash line denotes k = 0.63.
open arcs for each j. In order to see clearly the Hopf bifurcation curves with
the variation of j, we take 0,1,2j and show the corre-
sponding curves in Figure 5. The neuronal system is a typical multi-time-scale system
with two firing rhythm patterns: spiking and bursting. Spik-ing corresponds to a limit cycle attractor, while bursting is associated with two important bifurcations: 1) bifurcation of an equilibrium resulting in transition to spiking state, and 2) bifurcation of a limit cycle attractor resulting in transition to resting state. For the controlled chaotic HR neuron, in terms of 1k , we can choose some values of time delay to
demonstrate the effect of control in Figure 6. We take = 0.01, 0.1, 0.2, 0.4, respectively, and find that the chaotic bursting orbit can be controlled to be periodic bursting orbit with different periods.
In this paper, chaos control in the time-delayed feedback HR neuronal model is studied with the theory of functional differential equation. By means of bifurcation theory, we find that the single neuron can be induced to fire via the subcritical Hopf bifurcation and exhibits rich periodic or chaotic spiking/bursting behaviors. For the purpose of con-trol of chaos, we adopt the time-delayed feedback control method, and analytically investigate the conditions for the Hopf bifurcation of the controlled system. Furthermore, an explicit formula about the time delay and feedback gain which determines the Hopf bifurcation is derived, and the corresponding Hopf bifurcation curves based on the formula are given in the two-parameter k plane. Illustrating with numerical simulations, we find that the time-delayed feedback control is an effective method for chaos control.
Figure 6 Time evolutions (first line) and phase portraits (second line) from left column to right column being = 0.01, = 0.1, = 0.2, = 0.4 for the fixed k = 1.
878 Wang H X, et al. Sci China Tech Sci May (2014) Vol.57 No.5
By appropriate choice of feedback gain and time delay, the chaotic bursting orbit can be controlled to be stable periodic bursting orbit with different periods. The adopted method in this paper is general and can be applied to other neuronal models. It may help us better understand the bifurcation mechanisms of neural behaviors.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11002073, 11172017, 11102041) .
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