# 具有时滞的eHR神经元模型稳定性及Hopf分岔分析Stability and Hopf Bifurcation Analysis of eHR Neuron Model with Time-Delay

DOI: 10.12677/AAM.2018.712189, PDF, HTML, XML, 下载: 442  浏览: 591

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 nu-merical simulations are given to verify the conclusions.

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

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