HOPF BIFURCATION IN A SYNAPTICALLY COUPLED

FHN NEURON MODEL WITH TWO DELAYS

Liping Zhang

College of Science, Nanjing University of Aeronautics and Astronautics

2010/07/30

什么是生物数学？ 生物数学是生物学与数学之间的边缘学科，用数学

方法研究和解决生物学问题，也对与生物学有关的数学方法进行理论研究。

对于今天的生物学者，数学的价值更应该体现在建立在数量化基础上的 " 模型化 " 。通过数学模型的构建，可以将看上去杂乱无章的实验数据整理成有序可循的数学问题，将问题的本质抽象出来。

两个最近的事例

SARS 在对 SARS 的研究中，生物数学就发挥了作

用。 2003 年春 SARS 暴发时，在有效的疫苗和抗病毒药物研制出来之前，科学家最关心的是 SARS 流行的特征。两个国际合作的研究小组使用了 "SEIR" 数学模型，对 SARS的传播趋势进行分析和预测，给有关部门提供了参考意见。

Avian Influenza Bird flu

Avian influenza is a disease of birds caused by influenza viruses closely related to human influenza viruses.

Transmission to humans in close contact with poultry or other birds occurs rarely and only with some strains of avian influenza. The potential for transformation of avian influenza into a form that both causes severe disease in humans and spreads easily from person to person is a great concern for world health.

Avian Influenza Bird flu

生物数学几个领域的基本介绍 种群动力学 : 种群的相互作用 生物资源管理和综合害虫控制 流行病动力学 药物动力学 生物数学中的斑图 生物信息学

生物数学已有一百年多年的历史：

•1798 年 Malthus 人口增长模型 •1908 年遗传学的 Hardy-Weinbe“ 平衡原

理” •1925 年 Volterra 捕食与被捕食模型 •1927 年 KM 传染病模型 •1973 年许多著名的生物学杂志相继创刊

• 现如今“生物信息学”的诞生是 生物数学发展的里程碑

时滞 时滞对生物种群的影响一直是生物学家关心的问题，

时滞经常出现在生物的活动中。例如我们日常生活中遇到的视觉和听觉的时滞现象、动物血液再生原理，森林再生原理等。

考虑到种群密度变化对于增长率的影响都不是瞬间发生的 , 而是与过去的生活状态有关 , 即有时间滞后的，还有动物消化食物也需要一定的时间。在生物数学模型中如果引入时滞，相应的动力系统就变成了带时滞的非线性动力系统。

由于时滞生物动力系统的演化不仅依赖于系统的当前状态，还依赖于系统过去某一时刻或若干时刻的状态，其运动方程要用泛函微分方程来描述，和常微分方程系统所描述的系统不同，时滞对系统的动态性质有很大的影响，时滞动力系统一般有无穷多个特征值，解空间是无限维的，其理论分析往往很困难。

目前，对于非线性时滞动力系统尚没有针对性特别强的研究方法，讨论非线性常微分方程的方法，大多可以 经过改造用于非线性时滞微分方程的研究。例如研究生物动力系统平衡点存在唯一性方法有：不动点定理、 M-矩阵和重合度理论等；平衡点局部稳定性分析最基本的方法仍是考察特征方程根的变化，例如无害时滞不改变系统正平衡位置的渐近稳定性，所以利用时滞为零时系统的渐近性去研究时滞不为零时系统正平衡位置的局部稳定性，即用线性近似法研究研究平衡点的局部稳定性问题。对小时滞模型用平均法，对常数时滞以及连续时滞模型的全局稳定性主要用 Lyapunov 方法。研究分岔现象的常见方法有：中心流形法、规范形理论、 Lyapunov-Schmidt 方法、摄动法和多尺度法等。

1.assumptions

The basic model makes the following assumptions:

(H) The model is given by the following system:

021 bbD 0)()( 21 bbCEBA

0)]()()[( 2121 bbCEBAbbD

)1(

).()()(

)),(tanh()()()()(

),()()(

)),(tanh()()()()(

.

423

.

4

.

2124333

.

3

21

.

1

.

2

1312131

.

1

txbtxtx

txCtxtaxtxtx

txbtxtx

txCtxtaxtxtx

Fan D, Hong L. Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays.

Commun Nonlinear Sci Numer Simulat (2009), doi:10.1016/j.cnsns.2009.07.025

2.Stability for FHN neuron model with one delay

Obviously, E (0,0,0,0) is an equilibrium of system (1), linear

izing it gives

)2(

).()()(

)),(()()()()(

),()()(

)),(()()()()(

.

423

.

4

.

2124333

.

3

21

.

1

.

2

1312131

.

1

txbtxtx

txCtxtaxtxtx

txbtxtx

txCtxtaxtxtx

The characteristic equation associated with system (2) is given by

1 2( )4 3 21 2( )( ) 0A B C D E b b e

1 2 2A b b a

21ccE

Where

,

,

2

1 2 1 22 ( ) 2B bb a b b a

abbbabbbaC 22)( 2121212

21212 1 ababbbaD

(3)

For and , Eq.(3) becomes 01 02

4 3 21 2( )( ) 0A B C D E b b (4)

By Routh-Hurwitz criterion we know that if (H) is satisfied then all roots of Eq.(3) have negative real parts.

3.Bifurcation for FHN neuron model with one delay

Obviously, iv(v>0) is a root of Eq.(4) if and only if

4 3 21 2 1 1( )( )(cos sin ) 0v Av i Bv Cvi D E vi b vi b v i (5)

Separating the real and imaginary parts gives (

.sincos

,cossin3

113

11124

vvbvEvCvAv

vEbvEvDvv

(6)

Taking square on the both sides of the equations of (6) and summing them up ,and let y=v2 , which leads to:

where

0234 sryqypyy (7)

62 Ap DACq 229 22 6 EDCr 221

2 EbDs

Denote sryqypyyyh 234)(

Then we have

rqypyyyh 234)( 23'

(8)

(9)

Set

.0234 23 rqypyy

(10)

Let 4

pyz , then (10) becomes

0113 qzpz (11)

where 2

1 16

3

2p

qp ,

4832

2

1

rpqpq

Define

3121 )3

()2

(pq

2

31

(12)

Without loss of generality, we assume that Eq.(7) has four positive roots, denoted by ,, and , respectively. Then Eq.(6) has the four positive roots *

kk zv )4,3,2,1( k we have

.)]()[(

)()()()(sin

,)]()[(

)()())((cos

221

221

213

2124

221

221

2132

2124

kk

kkkkkkk

kk

kkkkkkk

vbbvbbE

bbvCvAvbbvDBvvv

vbbvbbE

bbvCvAvvbbDBvvv

(13)

Denote

,

)]()[(

)()())((

,)]()[(

)()()()(

221

221

2132

2124

*

221

221

213

2124

*

kk

kkkkkk

kk

kkkkkk

vbbvbbE

bbvCvAvvbbDBvvb

vbbvbbE

bbvCvAvbbvDBvva

0),2arccos2(1

0),2(arccos1

**

**

ajbv

ajbv

k

kjk

,2,1,0jkivWhere k=1,2,3,4; Then is a pair of purely imaginary roots

of Eq.(4) with 0, 211 jk Similar to the proves of [8] we know that

Eq.(7) has more than one positive roots. Then the stability switch may exist.

Summarizing the above discussions we can ensure the stability interval.

(14)

Theorem 3.1 Suppose that (H) is satisfied and

1. If the conditions (a) (b) , , and △(c) , , and there exists a △

such that and are not satisfied, then the zero solutions of system ( 1) is asymptotically stable for all .

2. If one of the conditions (a),(b) and (c) of (1) is satisfied, then the zero solution of system (1) is asymptotically stable when

3. If one of the conditions (a),(b) and (c) of (1) is satisfied,and , then the system (1) undergos a Hopf bifurcation at (0,0,0,0) when

02

0r 0r 00r 0 321

* ,, yyyy

0* y 0)( * yh

01

],0[ 01

0)( *' kyh

).,2,1,0(,11 jjk

4.Stability and Hopf bifurcation for FHN neuron model with two delay

Now let ,*11 ,02 )0( wiw be a root of Eq.(2) Then we get

.0sincos

,0sincos

21223

222124

wEFwEFCwAw

wEFwEFDBww (16)

Where ],sin)(cos)[( *

121*121

21 wwbbwbbwF

].sin)(cos)[( *121

2*1212 wbbwwwbbF

Taking square on the both sides of the equations of (14), we get (15)

02)22()2( 22

221

2222342628 FEFEDwCBDwwACDBwBAw(15)

If Eq.(15) has positive root, without loss of generality , we assume Eq.(15) has N positive roots, denoted by 。 Notice Eq.（ 12） we get

2,1,0,

0sin),2)arccos(cos2(1

0sin),2)s(arccos(co1

22

22

2

j

wjww

wjww

iiiii

iiiiij

i

(16)

Define .Let be the root of Eq.(4) Satisfying . By computation, we get

}{min )0(2

},...2,1{

)0(2

02 i

Nii

00 i )()()( 222 iv

iji

ji ww )(,0)( 221

120

22

20

21

40

02

*10021

302121

5021

02

*10

202121

402121

602

)(21

02

)(21

23

)(210

2'

)2(

)(sin]3)(2[)](43[

)(cos)](2[]42)(3[4

)))(()2(234

))((Re()(

02

*1

02

*1

02

*1

Ewbwbw

wwbCbwCbAbbbBwbbA

wwbbCbBbwbbBbbAw

ebbEebbECBA

ebbE

Where Summarizing the discussions above, we hav

e the following conclusions.

0)(,0])([ 02

'20

22

21

40

22

21

602 wbbwbbwE

Theorem 4.1 Suppose that (H), hold and Eq.(14) has positive roots. and have the same meaning as last definition. We get

(1) All root of Eq.(4) have negative real parts for and the equilibrium of syste

m (2) is asymptotically stable for . (2) If hold , then system (2) undergos a H

opf bifurcation at the equilibrium E, when .

I *11

02 )( 0

2'

),0( 022 )0,0,0,0(E

0)( 02

' 022

5.Stability and direction of the Hopf bifurcation

In the previous section, we obtained conditions for Hopf bifurcation to occur when

. In this section we study the direction of the Hopf bifurcation and the stability

of the bifurcation periodic solutions when , using techniques from normal form and center manifold theory.

022

022

We assume 02

*1 Letting R ,0

22

and dropping the bars for simplification

),(dt

)(d tt XFXLtX

（ 17）

Where CtXX t )()( and 3: RCL

3: RCRF

),()()0( 022

*111 tttt XBXBXAXL

(18) Wher

e TT

tttt txtxtxtxxxxx ))(),(),(),(())(),(),(),(( 43214321

.

0

)(3

)0(

0

)(3

)0(

0

)(3

)(

0

)(3

)(

),(

,

0000

000

0000

0000

,

0000

0000

0000

000

,

100

100

001

001

231

233

*1

33

131

231

233

*1

33

131

22

1

1

2

11

tt

tt

t

xc

x

xc

x

txc

tx

txc

tx

XF

cB

c

B

b

a

b

a

A

From the discussion in Section 2, we know that system (10) undergos a Hopf bifurcation at (0,0,0) when , and the associated characteristic equation of system (10) with

has a pair of simple imaginary roots .

0

0

0i

Result

Based on the above analysis, we can see that each gij can be determined by the parameters. Thus we compute the following quantities:

).33(2

,0020

*10

*

22

12*

21

021120

iwiw ecceccccccDg

ggg

(29)

2)

32(

2)0( 21

2

022

1120110

1

ggggg

iC

.))(Im())0(Im(

)),0(Re(2

,))(Re(

))0(Re(

0

02

'21

2

12

02

'1

2

CT

C

C

Theorem 5.1. In (29), determines the direction of Hopf bifurcation; if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcation periodic solution exist for ; determines the stability of the bifurcation period solution; bifurcating periodic solution are stable(unstable) if ;and determines the period of the bifurcating solution: the period increases (decreases) if .

22 20( 0)

)( 022

022 2

2 0( 0)

2 20( 0)T T

normal form and central manifold theory Numerical examples

2.0,1.0,48.0,47.0,33.0 2121 ccbba

)9916,17,0969.14()3808.11,9932.6()7700.4,0[1

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t

x3

-0.5

0

0.5

-0.5

0

0.5-0.4

-0.2

0

0.2

0.4

x1x2

x4

021 ）（ 2,0..5,0.4,0.30

0 100 200 300 400 500 600 700 800 900 1000-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x1

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x2

-0.5

0

0.5

-0.5

0

0.5-0.4

-0.2

0

0.2

0.4

x1x2

x3

0,5 21 ）（ 2,0..5,0.4,0.30

0 20 40 60 80 100 120 140 160 180 200-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x1

0 20 40 60 80 100 120 140 160 180 200-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

x2

0 20 40 60 80 100 120 140 160 180 200-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t

x3

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x3

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x4

0,10 21 ）（ 2,0..5,0.4,0.30

0 20 40 60 80 100 120 140 160 180 200-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x1

0 20 40 60 80 100 120 140 160 180 200-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

x2 -0.4

-0.20

0.20.4

0.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x3

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x4

0 20 40 60 80 100 120 140 160 180 200-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t

x3

)9916,17,0969.14()3808.11,9932.6()7700.4,0[1

3*11

)9916.14,0969.11()3808.8,9932.3()7700.1,0[2

7700.102

1,3 21 ）（ 2,0..5,0.4,0.30

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x3

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x1

0 100 200 300 400 500 600 700 800 900 1000-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

x2

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

t

x3

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x4

2,3 21 ）（ 2,0..5,0.4,0.30

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x1

0 100 200 300 400 500 600 700 800 900 1000-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

x2 -0.4

-0.20

0.20.4

0.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x3

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

t

x3

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x4

1,10 21

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x1

0 100 200 300 400 500 600 700 800 900 1000-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

x2

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x3

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

t

x3

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x4

2,10 21

0 100 200 300 400 500 600 700 800 900 1000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x1

0 100 200 300 400 500 600 700 800 900 1000-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

x2

-0.4-0.2

00.2

0.40.6

-0.5

0

0.5

1-0.4

-0.2

0

0.2

0.4

x1x2

x3

Future work

The synchronization of fractional-order Coupled HR neuron systems

Auastasio T j 1994 Bio.cybern. 72 67