Vol. 07  No. 12 ( 2018 ), Article ID: 28227 , 10 pages
10.12677/AAM.2018.712189

Stability and Hopf Bifurcation Analysis of eHR Neuron Model with Time-Delay

Zhengyu Lu, Huanhuan Yu, Wenjing Wang

School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou Gansu

Email: 18293132546@163.com

Received: Nov. 24th, 2018; accepted: Dec. 20th, 2018; published: Dec. 27th, 2018

ABSTRACT

In order to study the complex dynamic behavior of time-delayed neuron system, the time-delay term is introduced on the basis of eHR neuron system. By analyzing the characteristic equation of the linearized eHR model system at the unique equilibrium point, a critical value is obtained, so that Hopf bifurcation occurs when the value exceeds it, and the system is asymptotically stable when the value is less than it. In addition, the stability and bifurcation direction of the bifurcation periodic solution are given by the central manifold theorem and other theories. Finally, some numerical simulations are given to verify the conclusions.

Keywords:Hindmarsh-Rose Neuron, Hopf Bifurcation, Time-Delay

Email: 18293132546@163.com

1. 引言

$\left\{\begin{array}{l}\stackrel{˙}{x}=y-a{x}^{3}+b{x}^{2}-z+{I}_{ext}\\ \stackrel{˙}{y}=c-d{x}^{2}-y\\ \stackrel{˙}{z}=r\left[s\left(x-{x}_{0}\right)-z\right]\end{array}$ (1)

$\left\{\begin{array}{l}\stackrel{˙}{x}=y-a{x}^{3}+b{x}^{2}-z+{I}_{ext}\\ \stackrel{˙}{y}=c-d{x}^{2}-y-ew\\ \stackrel{˙}{z}=r\left[s\left(x-{x}_{0}\right)-z\right]\\ \stackrel{˙}{w}=h\left[f\left(y+g\right)-pw\right]\end{array}$ (2)

2. 模型介绍

$\left\{\begin{array}{l}\stackrel{˙}{x}=y\left(t-\tau \right)-a{x}^{3}+b{x}^{2}-z+{I}_{ext}\\ \stackrel{˙}{y}=c-d{x}^{2}-y\left(t-\tau \right)-ew\\ \stackrel{˙}{z}=r\left[s\left(x-{x}_{0}\right)-z\right]\\ \stackrel{˙}{w}=h\left[f\left(y\left(t-\tau \right)+g\right)-pw\right]\end{array}$ (3)

3. 平衡点分析和Hopf分岔存在性稳定性分析

$\left\{\begin{array}{l}\stackrel{˙}{x}=y\left(t-\tau \right)-a{x}^{3}+\left(b-3c{x}^{*}\right){x}^{2}+\left(2b{x}^{*}-3c{x}^{*2}\right)x-z\\ \stackrel{˙}{y}=-d{x}^{2}-2d{x}^{*}x-y\left(t-\tau \right)-ew\\ \stackrel{˙}{z}=rsx-rz\\ \stackrel{˙}{w}=hfy\left(t-\tau \right)-hpw\end{array}$ (4)

$\left\{\begin{array}{l}\stackrel{˙}{x}=y\left(t-\tau \right)+\left(2b{x}^{*}-3a{x}^{*2}\right)x-z\\ \stackrel{˙}{y}=-2d{x}^{*}x-y\left(t-\tau \right)-ew\\ \stackrel{˙}{z}=rsx\left(t-{\tau }_{1}\right)-rz\\ \stackrel{˙}{w}=hfy\left(t-{\tau }_{2}\right)-hpw\end{array}$ (5)

$A=\left(\begin{array}{cccc}2b{x}^{*}-3a{x}^{*2}& {e}^{-\lambda \tau }& -1& 0\\ -2d{x}^{*}& -{e}^{-\lambda \tau }& 0& -e\\ rs& 0& -r& 0\\ 0& hf{e}^{-\lambda \tau }& 0& -hp\end{array}\right)$

${\lambda }^{4}+{k}_{3}{\lambda }^{3}+{k}_{2}{\lambda }^{2}+{k}_{1}\lambda +\left({\lambda }^{3}+{n}_{2}{\lambda }^{2}+{n}_{1}\lambda +{n}_{0}\right){e}^{-\lambda \tau }=0$ (6)

$\begin{array}{l}{k}_{3}=3a{x}^{2}-2bx+hp+r,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}_{2}=3\left(ahp+ar\right){x}^{2}-2\left(bhpx+br\right)x+hpr+rs,\\ {k}_{1}=3ahpr{x}^{2}-2bhprx+hprs,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{n}_{2}=3a{x}^{2}+2\left(d-b\right)x+hfe+hp+r,\\ {n}_{1}=3\left(aefh+ahp+ar\right){x}^{2}-2\left(befh+bhp-dhp+br-dr\right)x+efhr+hpr+rs,\\ {n}_{0}=3\left(aefhr+ahpr\right){x}^{2}-2\left(befhr-dhpr\right)x-2bhpr+efhrs+hprs\end{array}$

${\lambda }^{4}+{q}_{13}{\lambda }^{3}+{q}_{12}{\lambda }^{2}+{q}_{11}\lambda +{q}_{10}=0$ (7)

${q}_{13}={k}_{3}+1,\text{\hspace{0.17em}}{q}_{12}={k}_{2}+{n}_{2},\text{\hspace{0.17em}}{q}_{11}={k}_{1}+{n}_{1},\text{\hspace{0.17em}}{q}_{10}={n}_{0}$

(H1) ${q}_{13}>0,{q}_{12}{q}_{13}>{q}_{11},{q}_{11}{q}_{12}{q}_{13}>{q}_{10}{q}_{13}^{2}+{q}_{11}^{2}$

$\lambda =i\omega$ 是方程(6)的根，则有

${\omega }^{4}-i{k}_{3}{\omega }^{3}-{k}_{2}{\omega }^{2}+i{k}_{1}\omega +\left(-i{\omega }^{3}-{n}_{2}{\omega }^{2}+i{n}_{1}\omega +{n}_{0}\right)\left[\mathrm{cos}\left(\omega \tau \right)-i\mathrm{sin}\left(\omega \tau \right)\right]=0$ (8)

$\left\{\begin{array}{l}\left({n}_{0}-{n}_{2}{\omega }^{2}\right)\mathrm{cos}\left(\omega \tau \right)+\left({n}_{1}\omega -{\omega }^{3}\right)\mathrm{sin}\left(\omega \tau \right)={k}_{2}{\omega }^{2}-{\omega }^{4}\\ \left({n}_{1}\omega -{\omega }^{3}\right)\mathrm{cos}\left(\omega \tau \right)+\left({n}_{2}{\omega }^{2}-{n}_{0}\right)\mathrm{sin}\left(\omega \tau \right)={k}_{3}{\omega }^{3}-{k}_{1}\omega \end{array}$

$\left\{\begin{array}{l}\mathrm{sin}\left(\omega {\tau }_{1}\right)=\frac{{\omega }^{7}+{A}_{2}{\omega }^{5}+{A}_{1}{\omega }^{3}+{A}_{0}\omega }{{\omega }^{6}+{B}_{2}{\omega }^{4}+{B}_{1}{\omega }^{2}+{B}_{0}}\\ \mathrm{cos}\left(\omega {\tau }_{1}\right)=\frac{{A}_{5}{\omega }^{6}+{A}_{4}{\omega }^{4}+{A}_{3}{\omega }^{2}}{{\omega }^{6}+{B}_{2}{\omega }^{4}+{B}_{1}{\omega }^{2}+{B}_{0}}\end{array}$ (9)

$\begin{array}{l}{A}_{0}={n}_{0}{k}_{1},{A}_{1}={n}_{1}{k}_{2}-{n}_{0}{k}_{3}-{n}_{2}{k}_{1},{A}_{2}={n}_{2}{k}_{3}-{n}_{1}-{k}_{2},\\ {A}_{3}={n}_{0}{k}_{2}-{n}_{1}{k}_{2},{A}_{4}={n}_{1}{k}_{3}-{n}_{2}{k}_{2}+{k}_{1}-{n}_{0},{A}_{5}={n}_{2}-{k}_{3},\\ {B}_{0}={n}_{0}^{2},{B}_{1}={n}_{1}^{2}-2{n}_{0}{n}_{2},{B}_{2}={n}_{2}^{2}-2{n}_{1}\end{array}$

${\omega }^{8}+{b}_{3}{\omega }^{6}+{b}_{2}{\omega }^{4}+{b}_{1}{\omega }^{2}+{b}_{0}=0$ (10)

$\begin{array}{l}{b}_{3}={k}_{3}^{2}-2{k}_{2}-1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{2}={k}_{2}^{2}-2{k}_{1}{k}_{3}-{n}_{2}^{2}+2{n}_{1},\\ {b}_{1}={k}_{1}^{2}+2{n}_{0}{n}_{2}-{n}_{1}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{0}=-{n}_{0}^{2}\end{array}$

$\eta ={\omega }^{2}$ ，则方程(10)变为：

${\eta }^{4}+{b}_{3}{\eta }^{3}+{b}_{2}{\eta }^{2}+{b}_{1}\eta +{b}_{0}=0$ (11)

$f\left(\eta \right)={\eta }^{4}+{b}_{3}{\eta }^{3}+{b}_{2}{\eta }^{2}+{b}_{1}\eta +{b}_{0}$

${\tau }_{2k}^{\left(i\right)}=\frac{1}{{\omega }_{k}}\left\{\mathrm{arccos}\left(\frac{{A}_{25}{\omega }^{6}+{A}_{24}{\omega }^{4}+{A}_{23}{\omega }^{2}}{{\omega }^{6}+{B}_{22}{\omega }^{4}+{B}_{21}{\omega }^{2}+{B}_{20}}\right)+2i\text{π}\right\},k=1,2,3,4;i=0,1,2,\cdots$

$\lambda \left(\tau \right)=\alpha \left(\tau \right)+i\omega \left(\tau \right)$ 是特征方程(6)在时滞 $\tau ={\tau }_{20}$ 附近满足 $\alpha \left({\tau }_{20}\right)=0$$\omega \left({\tau }_{20}\right)={\omega }_{20}$ 的根，可以得到如下横截性条件。

${\eta }_{k}={\omega }_{k}^{2}$${h}^{\prime }\left({z}_{k}\right)\ne 0$ ，则 ${\left(\frac{\text{d}\left(\mathrm{Re}\lambda \right)}{\text{d}\tau }\right)}_{\tau ={\tau }_{2k}^{\left(i\right)}}\ne 0$${\left(\frac{\text{d}\left(\mathrm{Re}\lambda \right)}{\text{d}\tau }\right)}_{\tau ={\tau }_{2k}^{\left(i\right)}}$${f}^{\prime }\left({\eta }_{k}\right)$ 有相同的符号。

ii) 若 ${f}^{\prime }\left({\eta }_{k}\right)\ne 0$ ，在临界值 ${\tau }_{2}={\tau }_{2k}^{\left(i\right)}$ 时，系统(3)在平衡点 ${E}^{*}=\left({x}^{*},{y}^{*},{z}^{*},{w}^{*}\right)$ 处发生Hopf分岔，即一组非常数周期解会从平衡点分岔出来。

4. Hopf分岔的方向及稳定性

$\stackrel{˙}{x}\left(t\right)={L}_{\mu }\left({x}_{t}\right)+F\left(\mu ,{x}_{t}\right)$

${L}_{\mu }\left(\varphi \right)=\left({\tau }_{0}+\mu \right)B\left(\begin{array}{c}{\varphi }_{1}\left(0\right)\\ {\varphi }_{2}\left(0\right)\\ {\varphi }_{3}\left(0\right)\\ {\varphi }_{4}\left(0\right)\end{array}\right)+\left({\tau }_{0}+\mu \right)C\left(\begin{array}{c}{\varphi }_{1}\left(-\frac{{\tau }_{2*}}{{\tau }_{0}}\right)\\ {\varphi }_{2}\left(-\frac{{\tau }_{2*}}{{\tau }_{0}}\right)\\ {\varphi }_{3}\left(-\frac{{\tau }_{2*}}{{\tau }_{0}}\right)\\ {\varphi }_{4}\left(-\frac{{\tau }_{2*}}{{\tau }_{0}}\right)\end{array}\right)+\left({\tau }_{0}+\mu \right)D\left(\begin{array}{c}{\varphi }_{1}\left(-1\right)\\ {\varphi }_{2}\left(-1\right)\\ {\varphi }_{3}\left(-1\right)\\ {\varphi }_{4}\left(-1\right)\end{array}\right)$

$F\left(\mu ,\varphi \right)=\left({\tau }_{0}+\mu \right)\left(\begin{array}{c}\left(b-3a{x}^{*}\right){\varphi }_{1}^{2}\left(0\right)-a{\varphi }_{1}^{3}\left(0\right)\\ -d{\varphi }_{1}^{2}\left(0\right)\\ 0\\ 0\end{array}\right)$

$B=\left(\begin{array}{cccc}2bx-3a{x}^{2}& 0& -d& 0\\ -2dx& 0& 0& -e\\ 0& 0& -r& 0\\ 0& 0& 0& -hp\end{array}\right),C=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),D=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 0\\ rs& 0& 0& 0\\ 0& hf& 0& 0\end{array}\right)$

${L}_{\mu }\varphi ={\int }_{-1}^{0}\text{d}\eta \left(\theta ,\mu \right)\varphi \left(\theta \right)\text{\hspace{0.17em}},\text{\hspace{0.17em}}\varphi \in C\left(\left[-1,0\right],{R}^{4}\right)$

$\eta \left(\theta ,\mu \right)=\left({\tau }_{0}+\mu \right)\left(\begin{array}{cccc}2bx-3a{x}^{2}& 0& -d& 0\\ -2dx& 0& 0& -e\\ 0& 0& -r& 0\\ 0& 0& 0& -hp\end{array}\right)\delta \left(\theta \right)-\left({\tau }_{0}+\mu \right)\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& -1& 0& 0\\ rs& 0& 0& 0\\ 0& hf& 0& 0\end{array}\right)\delta \left(\theta +1\right)$

$A\left(\mu \right)\varphi \left(\theta \right)=\left\{\begin{array}{ll}\frac{\text{d}\varphi \left(\theta \right)}{\text{d}\theta }，\hfill & \theta \in \left[-1,0\right)\hfill \\ {\int }_{-1}^{0}\text{d}\eta \left(\theta ,\mu \right)\varphi \left(\theta \right)，\hfill & \theta =0\hfill \end{array}$

$R\left(\mu \right)\varphi \left(\theta \right)=\left\{\begin{array}{ll}0,\hfill & \theta \in \left[-1,0\right)\hfill \\ F\left(\mu ,\varphi \right)，\hfill & \theta =0\hfill \end{array}$

${\stackrel{˙}{x}}_{t}=A\left(\mu \right){x}_{t}+R\left(\mu \right){x}_{t}$

${A}^{*}\left(\mu \right)\psi \left(s\right)=\left\{\begin{array}{ll}-\frac{\text{d}\psi \left(s\right)}{\text{d}s},\hfill & s\in \left(0,1\right]\hfill \\ {\int }_{-1}^{0}\text{d}\eta \left(t,0\right)\psi \left(-t\right)，\hfill & s=0\hfill \end{array}$

$〈\psi \left(s\right),\varphi \left(\theta \right)〉=\stackrel{¯}{\psi }\left(0\right)\varphi \left(0\right)-{\int }_{-1}^{0}{\int }_{\xi =0}^{\theta }{\stackrel{¯}{\psi }}^{\text{T}}\left(\xi -\theta \right)\text{d}\eta \left(\theta \right)\varphi \left(\xi \right)\text{d}\xi$

${g}_{20}={g}_{11}={g}_{02}=2\stackrel{¯}{P}{\tau }_{{2}_{0}}\left(b-3a{x}^{*}-d\stackrel{¯}{{v}_{1}^{*}}\right)$

${g}_{21}=2\stackrel{¯}{P}{\tau }_{{2}_{0}}\left[\left(b-3a{x}^{*}\right)\left({W}_{20}^{\left(1\right)}+2{W}_{11}^{\left(1\right)}\right)-3a-f\left({W}_{20}^{\left(1\right)}+2{W}_{11}^{\left(1\right)}\right)\stackrel{¯}{{v}_{1}^{*}}\right]$

${W}_{20}\left(\theta \right)=\frac{i{g}_{20}}{{\omega }_{0}{\tau }_{0}}q\left(0\right){e}^{i{\omega }_{0}{\tau }_{0}\theta }+\frac{i{\stackrel{¯}{g}}_{02}}{3{\omega }_{0}{\tau }_{0}}\stackrel{¯}{q}\left(0\right){e}^{-i{\omega }_{0}{\tau }_{0}\theta }+{E}_{1}{e}^{2i{\omega }_{0}{\tau }_{0}\theta }$

${W}_{11}\left(\theta \right)=-\frac{i{g}_{11}}{{\omega }_{0}{\tau }_{0}}q\left(0\right){e}^{i{\omega }_{0}{\tau }_{0}\theta }+\frac{i{\stackrel{¯}{g}}_{11}}{{\omega }_{0}{\tau }_{0}}\stackrel{¯}{q}\left(0\right){e}^{-i{\omega }_{0}{\tau }_{0}\theta }+{E}_{2}$

${E}_{1}={\left(\begin{array}{cccc}2i{\omega }_{0}{\tau }_{20}-\left(2b{x}^{*}-3a{x}^{*2}\right)& -{e}^{-2i{\omega }_{0}{\tau }_{2}^{*}}& d& 0\\ 2d{x}^{*}& 2i{\omega }_{0}{\tau }_{20}+{e}^{-2i{\omega }_{0}{\tau }_{2}^{*}}& 0& e\\ -rs{e}^{-2i{\omega }_{0}{\tau }_{20}}& 0& 2i{\omega }_{0}{\tau }_{20}+r& 0\\ 0& -hf{e}^{-2i{\omega }_{0}{\tau }_{2}^{*}}& 0& 2i{\omega }_{0}{\tau }_{20}+hp\end{array}\right)}^{-1}\left(\begin{array}{c}2\left(b-3a{x}^{*}\right)\\ -2d\stackrel{¯}{{v}_{1}^{*}}\\ 0\\ 0\end{array}\right)$

${E}_{2}=-{\left(\begin{array}{cccc}2b{x}^{*}-3a{x}^{*2}& a& -d& 0\\ -2d{x}^{*}& -1& 0& -e\\ rs& 0& -r& 0\\ 0& hp& 0& -hp\end{array}\right)}^{-1}\left(\begin{array}{c}2\left(b-3a{x}^{*}\right)\\ -2d\stackrel{¯}{{v}_{1}^{*}}\\ 0\\ 0\end{array}\right)$

$\left\{\begin{array}{l}{c}_{1}\left(0\right)=\frac{i}{{\omega }_{0}{\tau }_{0}}\left({g}_{11}{g}_{20}-2{|{g}_{11}|}^{2}-\frac{{|{g}_{02}|}^{2}}{3}\right)+\frac{{g}_{21}}{2}\\ {\mu }_{2}=-\frac{\mathrm{Re}\left({c}_{1}\left(0\right)\right)}{\mathrm{Re}\left({{\lambda }^{\prime }}_{0}\left({\tau }_{0}\right)\right)}\\ {\beta }_{2}=2\mathrm{Re}\left({c}_{1}\left(0\right)\right)\\ {T}_{2}=-\frac{\mathrm{Im}\left({c}_{1}\left(0\right)\right)+{\mu }_{2}\mathrm{Im}\left({{\lambda }^{\prime }}_{0}\left({\tau }_{0}\right)\right)}{{\omega }_{0}{\tau }_{0}}\end{array}$

$\tau >{\tau }_{{2}_{0}}\left(\tau <{\tau }_{{2}_{0}}\right)$ 时，分岔周期解存在(不存在)；若 ${T}_{2}>0\left({T}_{2}<0\right)$ ，周期解的周期增加(减小)； ${\beta }_{2}>0\left({\beta }_{2}<0\right)$ ，在此中心流形上，周期解是渐进稳定的(不稳定的)。

5. 数值模拟

$\left\{\begin{array}{l}\stackrel{˙}{x}=y\left(t-\tau \right)-{x}^{3}+3{x}^{2}-z+2.924976\\ \stackrel{˙}{y}=1.01-5.0128{x}^{2}-y\left(t-\tau \right)-0.0278w\\ \stackrel{˙}{z}=0.12\left[3.966\left(x-1.605\right)-z\right]\\ \stackrel{˙}{w}=0.009\left[3\left(y\left(t-\tau \right)+1.619\right)-0.9573w\right]\end{array}$ (2)

$\begin{array}{l}{q}_{13}=1.949051>0,{q}_{12}{q}_{13}=13.211406>{q}_{11}=1.231740,\\ {q}_{11}{q}_{12}{q}_{13}=16.273022>{q}_{10}{q}_{13}^{2}+{q}_{11}^{2}=1.553386\end{array}$

$\tau \ne 0$ 时，由情况2的理论方法计算得 ${\tau }_{{2}_{0}}=0.140347$ ，由引理1可得满足横截性条件。当 $\tau =0.18>{\tau }_{{2}_{0}}$ 时，系统是不稳定的，当 $\tau$ 穿过每一个临界值 ${\tau }_{{2}_{j}}\left(j=0,1,2,\cdots \right)$ 是，系统(3)的一个稳定周期

Figure 1. τ = 0, The time serise and phase diagrams of x(t), y(t), z(t), w(t) at the initial values (0.3, 0.3, 3.0, 0.05) show that the equilibrium point P (0.881210, 2.781651, 2.870551, 3.643691) are asymptotically stable

Figure 2. $\tau =0.18>{\tau }_{{2}_{0}}$ ,The time serise and phase diagrams of $x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)$ at the initial values (0.3, 0.3, 3.1, 0.1) show that The unique equilibrium point P (0.881210, 2.781651, 2.870551, 3.643691) is unstable and a stable periodic solution bifurcates from P

6. 结束语

Figure 3. $\tau =0.007<{\tau }_{{2}_{0}}$ , The time serise and phase diagrams of $x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)$ at the initial values (0.3, 0.3, 6.1, 0.1) show that the equilibrium point P (0.881210, 2.781651, 2.870551, 3.643691) are asymptotically stable

Stability and Hopf Bifurcation Analysis of eHR Neuron Model with Time-Delay[J]. 应用数学进展, 2018, 07(12): 1616-1625. https://doi.org/10.12677/AAM.2018.712189

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