边界条件下的FizhughNagumo模型斑图动力学研究

郑前前，申建伟; Zheng Qianqian; Shen Jianwei

引 en

边界条件下的Fizhugh-Nagumo模型斑图动力学研究^*

郑前前
，申建伟

1. 许昌学院数学与统计学院,许昌 461000；

STUDIES ON PATTERN DYNAMICS OF FIZHUGH-NAGUMO MODEL UNDER BOUNDARY CONDITIONS ^∗

Zheng Qianqian
， Shen Jianwei

1. School of Mathematics and Statistics,Xuchang University,Xuchang 461000 ,China；

DOI:10.6052/1672-6553-2020-002

全文
评论
参考文献
出版信息

参考文献 1

Abbasian A H,Fallah H,Razvan M R.Symmetric burst-ing behaviors in the generalized FitzHugh-Nagumo model.Biological Cybernetics,2013,107(4):465~ 476

查找原文

参考文献 2

Ringqvist M.On dynamical behaviour of FitzHugh-Nagumo systems.Research reports in mathematics,2006

查找原文

参考文献 3

Fitzhugh R.Impulse and physiological states in theoretical models of nerve membrane.Biophysical Journal,1961,1(6):445~ 466

查找原文

参考文献 4

马军,靳伍银,易鸣,等.时变反应扩散系统中螺旋波和湍流的控制.物理学报,2008,57(5):2832~ 2841(Ma J,Jin W Y,Yi M,et al.Control of spiral wave and turbu-lence in the time-varied reaction-diffusion system.Acta Physica Sinica,2008,57(5):2832~ 2841(in Chinese))

查找原文

参考文献 5

薛凌云,段会龙,向学勤,等.基于 FitzHugh-Naguno 神经元随机共振机制的图像复原.浙江大学学报(工学版),2010,44(6):1103 ~ 1107(Xue L Y,Duan H L,Xiang X Q,et al.Image restoration based on stochastic resonance mechanism of FitzHugh-Nagumo neuron.Jour-nal o f Zhejiang University(Engineering Edition),2010,44(6):1103~ 1107(in Chinese))

查找原文

参考文献 6

蒋磊,陈博文,张群,等.α 稳定分布噪声下基于FHN模型的 UWB-IR 信号检测.系统工程与电子技术,2018,40(7):1423 ~ 1428(Jiang L,Chen B W,Zhang Q,et al.UWB-IR signal detection based on FHN model in α stable distribution noise.Systems Engineering and E-lectronics,2018,40(7):1423~ 1428(in Chinese))

查找原文

参考文献 7

Fu Y X,Kang Y M,Xie Y.Subcritical Hopf bifurcation and stochastic resonance of electrical activities in neuron under electromagnetic induct.Frontiers in Computational Neuroscience,2018,12:1~ 10

查找原文

参考文献 8

康宏春,陈冲,刘启宽.一类简化 Fitzhugh-Nagumo 方程的定性分析.河北科技大学学报,2011,32(3):208 ~ 211(Kang H C,Chen C,Liu Q K.Qualitative analysis of a simplified Fitzhugh-Nagumo equation.Journal o f Hebei U-niversity of Science and Technology,2011,32(3):208 ~ 211(in Chinese)

查找原文

参考文献 9

申伟杰,柴玉珍.具有周期边界的 Fitzhugh-Nagumo 方程的初边值问题.中北大学学报,2012,33(3):228 ~ 231(Shen W J,Chai Y Z.Initial value problem for Fitzhugh-Nagumo equations with periodic boundary.Journal of North University of China,2012,33(3):228 ~ 231(in Chinese))

查找原文

参考文献 10

Guo Y F,Wang L J,Wei F,et al.Dynamical behavior of simplified FitzHugh-Nagumo neural system driven by Lévy noise and Gaussian white noise.Chaos,Solitons and Frac-tals,2019,127:118~ 126.

查找原文

参考文献 11

吕晓莉,彭建华,刘延柱.延时 FitzHugh-Nagumo 神经网络的时空编码.上海交通大学学报,2005,39(10):1664 ~ 1667(Lv X L,Peng J H,Liu Y Z.Spatiotemporal cod-ing in FitzHugh-Nagumo neural network with time delay.Journal of Shanghai Jiao Tong University,2005,39(10):1664~ 1667(in Chinese))

查找原文

参考文献 12

Hu D L,Yang Y H,Liu X B.Delay-induced vibrational multiresonance in FitzHugh-Nagumo system.Communica-tions in Nonlinear Science and Numerical Simulation,2012,17:1031~ 1035

查找原文

参考文献 13

欧阳颀.非线性科学与斑图动力学导论.北京:北京大学出版社,2010(Ouyang Q.Nonlinear science and intro-duction to pattern dynamics.Beijing:Peking University Press,2010(in Chinese))

查找原文

参考文献 14

Zheng Q Q,Shen J W.Pattern formation in the FitzHugh-Nagumo model.Computers and Mathematics with Applica-tions,2016,70(5):1~ 31

查找原文

参考文献 15

Asgari Y,Ghaemi M,Mahjani M G.Pattern formation of the FitzHugh-Nagumo model:cellular automata approach.Iranian Journal of Chemistry & Chemical Engineering-in-ternational English Edition,2011,3(1):135~ 142

查找原文

参考文献 16

Sebastian E,et al.Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units.Physical Review E,2019,99(4):042207(9)

查找原文

参考文献 17

Hafting T,et al.Microstructure of a spatial map in the en-torhinal cortex.Nature,2005,436:801~ 806

查找原文

参考文献 18

Marianne F,et al.Spatial representation in the entorhinal cortex.Science,2004,305:1258~ 1264

查找原文

摘要

本文主要研究边界条件下 FN(Fizhugh-Nagumo)模型的神经定位动力学行为,首先针对带有扩散的 FN 模型进行了理论研究;然后通过离散拉普拉斯算子的方法,得到了 FN 模型带有边界的常微分方程;接着分析了边界条件对系统斑图形态的具体影响,对混合边界不同参数取值进行了研究,得到了混合边界值和矩阵谱倒数的关系;最后通过数值模拟的方法验证了已有实验结果,解释了上述系统的斑图机制,给出了斑图形态的调控方法..

Abstract

In this paper,the neural localization dynamics behavior of FN (fizhugh-nagumo) model under bound- ary conditions is studied.The theoretical study is carried out on the FN model with diffusion,and the normal differ- ential equation with boundary is obtained by the method of discrete Laplace operator,then the specific effect of boundary conditions on the system is analyzed.Also,the different parameter of the mixed boundary is studied and the relationship between a mixed boundary value and matrix spectral is obtained.Finally,the results are verified by numerical simulation,the new pattern mechanism of FN model about grid cell is proposed,and the regulation method of pattern formation is present.

关键词

FN模型；边界条件；离散；拉普拉斯算子；斑图；欧拉法

Keywords

FNmodel ； pattern ； formation ； Laplace ； operator ； Euler ； method ； discrete

0 引言
FN模型是一个经典的反应扩散方程,常被应用于生物神经电传导及神经元动力学行为研究等方面,是HH(Hodgkin-Huxley) 模型的简化^{^[1-3]}.研究者也从不同方面和层次进行了应用和理论研究.在基本模型应用上,研究者通过对FN模型的动力学行为分析,试图解释了心颤致死的过程与螺旋波失稳之间存在的作用关系^{^[4]},且在FN模型的基础上借助噪声随机共振能量实现图像复原,提高图像输出质量^{^[5]}.另外在改进FN模型的原理上提出了超宽带冲激无线电信号检测方法^{^[6]},研究了磁场对神经系统动力学行为的影响^{^[7]}.在理论上,学者们研究了FN模型的稳定性,关于解存在性和唯一性的初边值问题系统^[8,9],在Levy噪声和高斯白噪声驱动下的概率密度函数及首次穿越时间^{^[10]},考虑了具有延时效应的FN神经元组成的神经元网络^{^[11]},以及时滞引起的振动^{^[12]}.总之,FN系统不管在应用还是理论研究上都被做了大量研究.
斑图是一种由系统内部决定的､自发的对称性破缺引起系统重新自组织的空间结构,能够准确描述不同系统间共同存在的､具有普遍指导意义的空间动力学行为,从而被广泛地应用在生物医学､物理学以及材料控制工程等研究领域,是非线性动力学的一个前沿方向^{^[13]}.同时斑图动力学也用于FN模型的研究^{^[14-16]},但是对其边界条件对斑图形态影响的理论却研究甚少.
边界问题作为一种系统的可控条件,在一定程度上也可以是外界的一种干扰手段.而外界干扰对神经系统的信息处理及各项功能都有一定的影响.对于生命而言,位置的感知和在空间的巡航能力, 无疑是至关重要的,而大脑内嗅皮层的网格细胞对空间的认知有着决定作用^[17,18],其在一定程度上受到目标物边界的影响.但边界对于网格细胞定位的影响鲜有研究.鉴于以上分析,本文以FN模型的边界条件为出发点,就神经系统的定位能力进行了理论和数值研究.
1 FN模型
一般情况下,带有扩散项的FN模型可以表示为:

\begin{matrix} \frac{\partial u}{\partial t} = & c (u - \frac{u^{3}}{3} - V) + d_{1} \nabla^{2} u \\ \frac{\partial v}{\partial t} = c ( & a u - b v) + d_{2} \nabla^{2} v \end{matrix}

(1)

式中,u为电势,v为恢复量,a,b,c为非负参数,d₁, d₂ 为非负的扩散系数,∇² 为拉普拉斯算子.
本节首先考虑系统(1)在d₁=d₂=0 情况下的稳定性,在系统不动点(0,0)有雅克比矩阵
进而可得特征方程

λ^{2} + (b c - c) λ + a c^{2} - b c^{2} = 0

(2)

计算可证,当a>b>1 时,系统是稳定的.以下研究内容均在微分方程稳定的基础上进行研究的.
然后,同理可得带有扩散项的特征方程

\begin{matrix} λ^{2} + (d_{1} k^{2} + d_{2} k^{2} + b c - c) λ + a c^{2} - b c^{2} + d_{1} d_{2} k^{4} + \\ b c d_{1} k^{2} - c d_{2} k^{2} = 0 \end{matrix}

(3)

由于a>b>1,可知d₁ k ² +d₂ k ² +bc-c>0,那么系统(3)失稳的充分条件为
进而分析研究系统(1) 带有混合边界条件的情况(如微分方程 $\frac{\partial y}{\partial t} = f (t, x)$ ,某一部分边界条件满足y=f,另一部分满足条件 $c_{0} y + c_{1} \frac{\partial y}{\partial x} = g$ ).
下面对线性系统进行离散和矩阵分块.即线性系统

\frac{\partial p}{\partial t} = A p + D \nabla^{2} p

(4)

式中, $p = (\begin{matrix} u \\ v \end{matrix})$ , $D = (\begin{matrix} d_{1} & 0 \\ 0 & d_{2} \end{matrix})$ .
为了计算方便,本文采用n×n的网格进行离散研究,左右边界设为0(即在左右边界处y=0),即有变量矩阵
进而转化为变量数
$u_{1 \times n^{2}} = {(u_{11}, u_{21}, \dots, u_{n 1}, u_{12}, \dots, u_{n 2}, \dots, u_{n n})}^{T}$ ,
一般情况下,二维空间的拉普拉斯算子可以离散为
经过计算构造,拉普拉斯算子可以表示为
其中, u₀=(u₁₀,‧‧‧,u_n0),u_n+1=(u_1n+1,‧‧‧,u_nn+1)的取值由边界条件决定.另外
此时系统(4)可以改写为

\begin{matrix} \frac{\partial}{\partial t} (\begin{matrix} u_{1 \times (n + 2) 2} \\ v_{1 \times (n + 2) 2} \end{matrix}) = \\ (\begin{matrix} d_{1} \nabla_{1 \times (n + 2) 2}^{2} & 0 \\ 0 & d_{2} \nabla_{1 \times (n + 2) 2}^{2} \end{matrix}) (\begin{matrix} u_{1 \times (n + 2) 2} \\ v_{1 \times (n + 2) 2} \end{matrix}) + \\ (\begin{matrix} c I_{(n + 2) 2 \times (n + 2) 2} & - c I_{(n + 2) 2 \times (n + 2) 2} \\ a c I_{(n + 2) 2 \times (n + 2) 2} & - b c I_{(n + 2) 2 \times (n + 2) 2}) (u_{1 \times (n + 2) 2} \end{matrix}) \end{matrix}

(5)

即有

\frac{\partial}{\partial t} (\begin{matrix} u_{1 \times (n + 2)}^{2} \\ v_{1 \times (n + 2)}^{2} \end{matrix}) = (\begin{matrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix}) (\begin{matrix} u_{1 \times (n + 2) 2} \\ v_{1 \times (n + 2) 2} \end{matrix})

(6)

其中,
以上过程把反应扩散系统(1) 转化为具有耦合项的常微分方程(6),并且根据边界条件的不同,矩阵B也会发生变化.而在实际模拟过程中B_n+2变为B_n(剔除矩阵边界n行n列),然后考虑边界条件并加入矩阵B_n 中.
2 边界条件
混合边界的一般表达式为y=f(其中一部分边界), $c_{0} y + c_{1} \frac{\partial y}{\partial x} = g_{1}$ (另外一部分边界),黎曼边界条件为 $\frac{\partial y}{\partial t} = g_{2}$ .在本章中,系统变量 $u_{1}$ 受到边界条件的影响 $\frac{\partial u_{1}}{\partial t} = C u_{0} + D u_{1} + C u_{2}$ ,根据边界条件可使 $u_{0}$ 表示为关于 $u_{2}$ 的函数,此时方程(6)可简化为
其中
为含有边界条件的矩阵,bdy₁,bdy₂ 的取值分别由u₀,u₂ 及u_n,u_n-2的关系确定,且D=D_n,C=C_n.离散的混合边界条件可表示为
即有 $y (0, i) = \frac{y (2, i) + g_{1} h_{1}}{c_{0} h_{1} + c_{1}}$
进一步结合黎曼边界条件
可得线性关系为

y (0, i) = g y (2, i) + c o n s

(7)

同理可得y(n,i)=gy(n-2,i)+cons
3 数值模拟
本文中的数值均在MATLAB中模拟,时间步长和空间步长分别为0.01,0.5,初始值为u(i,j)=1,v(i,j)=2,并且定义矩阵谱半径的倒数为空间信息率.其中谱半径为
生物系统模型参数的取值为
a=2,b=1.5,c=1,d₁=0.1,d₂=1
边界条件采用如方程(7)线性关系并且cons=0.从方程(2)(3)可知系统(1)在没有扩散项的情况下是稳定的(图1),带有扩散项时发生失稳(图2).
下面针对矩阵A₁ 中不同边界取值进行进究, 图3 展示了边界条件的取值与矩阵A₁ 谱倒数之间的关系符合正态分布,与文献^{^[18]}中空间信息率保持一致.并且其斑图形态也会随之发生变化,如图4 的斑图呈现出严重两极分化形态,使神经系统无法做出正确定位.随着g(信号强度先增后减)的增大发生变化斑图形态发生变化,并且形成规则的斑图( 图5), 但是并没有形成鲜明的对比.当g=-115时,系统形成了明显的斑图形态(图6),但由于中间斑点斑图的拉长,不能形成神经定位唯一的目标物.而当信号强度 λ^-1达到最大时,会形成单一的斑点图(图7),此时在一定程度上意味着神经系统定位成功.在相关文献^{^[18]} 中也有此种现象,但是斑点斑图也会随着g值的继续增大而破裂(图8-9),并最终失去目标物(图10).
图1 系统稳态演化
Fig.1 The system is stable
图2 R的取值情况
Fig.2 The value of R
图3 矩阵的谱和g的关系
Fig.3 The relationship of matrix spectrum and g
图4 当g=-116 时的斑图形态
Fig.4 The pattern formation when g=-116
图5 当g=-115.5 时的斑图形态
Fig.5 The pattern formation when g=-115.5
图6 当g=-115 时的斑图形态
Fig.6 The pattern formation when g=-115
图7 当g=0 时的斑图形态
Fig.7 The pattern formation when g=0
图8 当g=4 时的斑图形态
Fig.8 The pattern formation when g=4
图9 当g=4.5 时的斑图形态
Fig.9 The pattern formation when g=4.5
图10 当g=5 时的斑图形态
Fig.10 The pattern formation when g=5
4 结论
本文首先给出了系统的稳定性分析,然后通过FN模型边值问题的研究,提出了一种特殊边值问题矩阵谱研究方法,确定了空间信息率和边界值的关系,阐述了神经定位过程中目标物周围环境(边界条件)变化对系统定位的影响,如空间信息率与g之间的关系图(图3)说明了信息在g=0 时最为集中,对应的斑图也形成了单一的环形态(图7), 与现实中集中定位保持一致性.在g≠0 时,空间信息迅速降低,而对应的斑图不再存在唯一的环状斑图,即定位发生了变化,要么定位目标不唯一,要么定位目标消失.这也是新的FN模型斑图机制,为神经网格细胞(定位)的研究提供了理论指导.
参考文献
- [1] Abbasian A H,Fallah H,Razvan M R.Symmetric burst-ing behaviors in the generalized FitzHugh-Nagumo model.Biological Cybernetics,2013,107(4):465~ 476
- [2] Ringqvist M.On dynamical behaviour of FitzHugh-Nagumo systems.Research reports in mathematics,2006
- [3] Fitzhugh R.Impulse and physiological states in theoretical models of nerve membrane.Biophysical Journal,1961,1(6):445~ 466
- [4] 马军,靳伍银,易鸣,等.时变反应扩散系统中螺旋波和湍流的控制.物理学报,2008,57(5):2832~ 2841(Ma J,Jin W Y,Yi M,et al.Control of spiral wave and turbu-lence in the time-varied reaction-diffusion system.Acta Physica Sinica,2008,57(5):2832~ 2841(in Chinese))
- [5] 薛凌云,段会龙,向学勤,等.基于 FitzHugh-Naguno 神经元随机共振机制的图像复原.浙江大学学报(工学版),2010,44(6):1103 ~ 1107(Xue L Y,Duan H L,Xiang X Q,et al.Image restoration based on stochastic resonance mechanism of FitzHugh-Nagumo neuron.Jour-nal o f Zhejiang University(Engineering Edition),2010,44(6):1103~ 1107(in Chinese))
- [6] 蒋磊,陈博文,张群,等.α 稳定分布噪声下基于FHN模型的 UWB-IR 信号检测.系统工程与电子技术,2018,40(7):1423 ~ 1428(Jiang L,Chen B W,Zhang Q,et al.UWB-IR signal detection based on FHN model in α stable distribution noise.Systems Engineering and E-lectronics,2018,40(7):1423~ 1428(in Chinese))
- [7] Fu Y X,Kang Y M,Xie Y.Subcritical Hopf bifurcation and stochastic resonance of electrical activities in neuron under electromagnetic induct.Frontiers in Computational Neuroscience,2018,12:1~ 10
- [8] 康宏春,陈冲,刘启宽.一类简化 Fitzhugh-Nagumo 方程的定性分析.河北科技大学学报,2011,32(3):208 ~ 211(Kang H C,Chen C,Liu Q K.Qualitative analysis of a simplified Fitzhugh-Nagumo equation.Journal o f Hebei U-niversity of Science and Technology,2011,32(3):208 ~ 211(in Chinese)
- [9] 申伟杰,柴玉珍.具有周期边界的 Fitzhugh-Nagumo 方程的初边值问题.中北大学学报,2012,33(3):228 ~ 231(Shen W J,Chai Y Z.Initial value problem for Fitzhugh-Nagumo equations with periodic boundary.Journal of North University of China,2012,33(3):228 ~ 231(in Chinese))
- [10] Guo Y F,Wang L J,Wei F,et al.Dynamical behavior of simplified FitzHugh-Nagumo neural system driven by Lévy noise and Gaussian white noise.Chaos,Solitons and Frac-tals,2019,127:118~ 126.
- [11] 吕晓莉,彭建华,刘延柱.延时 FitzHugh-Nagumo 神经网络的时空编码.上海交通大学学报,2005,39(10):1664 ~ 1667(Lv X L,Peng J H,Liu Y Z.Spatiotemporal cod-ing in FitzHugh-Nagumo neural network with time delay.Journal of Shanghai Jiao Tong University,2005,39(10):1664~ 1667(in Chinese))
- [12] Hu D L,Yang Y H,Liu X B.Delay-induced vibrational multiresonance in FitzHugh-Nagumo system.Communica-tions in Nonlinear Science and Numerical Simulation,2012,17:1031~ 1035
- [13] 欧阳颀.非线性科学与斑图动力学导论.北京:北京大学出版社,2010(Ouyang Q.Nonlinear science and intro-duction to pattern dynamics.Beijing:Peking University Press,2010(in Chinese))
- [14] Zheng Q Q,Shen J W.Pattern formation in the FitzHugh-Nagumo model.Computers and Mathematics with Applica-tions,2016,70(5):1~ 31
- [15] Asgari Y,Ghaemi M,Mahjani M G.Pattern formation of the FitzHugh-Nagumo model:cellular automata approach.Iranian Journal of Chemistry & Chemical Engineering-in-ternational English Edition,2011,3(1):135~ 142
- [16] Sebastian E,et al.Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units.Physical Review E,2019,99(4):042207(9)
- [17] Hafting T,et al.Microstructure of a spatial map in the en-torhinal cortex.Nature,2005,436:801~ 806
- [18] Marianne F,et al.Spatial representation in the entorhinal cortex.Science,2004,305:1258~ 1264

基本信息

DOI: 10.6052/1672-6553-2020-002

基金信息

国家自然科学基金资助项目(11772291)､河南省青年人才托举工程和许昌学院科研项目(2020HYTP012,2020ZD012)；
The project supported by the National Natural Science Foundation of China(11772291)､Young Talent Support Project of Henan and Research Pro - ject of Xuchang University(2020HYTP012,2020ZD012 )；

引用信息

稿件历史

收稿日期: 2019-08-30

参考文献

[1] Abbasian A H,Fallah H,Razvan M R.Symmetric burst-ing behaviors in the generalized FitzHugh-Nagumo model.Biological Cybernetics,2013,107(4):465~ 476
[2] Ringqvist M.On dynamical behaviour of FitzHugh-Nagumo systems.Research reports in mathematics,2006
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[5] 薛凌云,段会龙,向学勤,等.基于 FitzHugh-Naguno 神经元随机共振机制的图像复原.浙江大学学报(工学版),2010,44(6):1103 ~ 1107(Xue L Y,Duan H L,Xiang X Q,et al.Image restoration based on stochastic resonance mechanism of FitzHugh-Nagumo neuron.Jour-nal o f Zhejiang University(Engineering Edition),2010,44(6):1103~ 1107(in Chinese))
[6] 蒋磊,陈博文,张群,等.α 稳定分布噪声下基于FHN模型的 UWB-IR 信号检测.系统工程与电子技术,2018,40(7):1423 ~ 1428(Jiang L,Chen B W,Zhang Q,et al.UWB-IR signal detection based on FHN model in α stable distribution noise.Systems Engineering and E-lectronics,2018,40(7):1423~ 1428(in Chinese))
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[10] Guo Y F,Wang L J,Wei F,et al.Dynamical behavior of simplified FitzHugh-Nagumo neural system driven by Lévy noise and Gaussian white noise.Chaos,Solitons and Frac-tals,2019,127:118~ 126.
[11] 吕晓莉,彭建华,刘延柱.延时 FitzHugh-Nagumo 神经网络的时空编码.上海交通大学学报,2005,39(10):1664 ~ 1667(Lv X L,Peng J H,Liu Y Z.Spatiotemporal cod-ing in FitzHugh-Nagumo neural network with time delay.Journal of Shanghai Jiao Tong University,2005,39(10):1664~ 1667(in Chinese))
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[16] Sebastian E,et al.Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units.Physical Review E,2019,99(4):042207(9)
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边界条件下的Fizhugh-Nagumo模型斑图动力学研究^*

STUDIES ON PATTERN DYNAMICS OF FIZHUGH-NAGUMO MODEL UNDER BOUNDARY CONDITIONS ^∗

摘要

Abstract

关键词

Keywords

0 引言

1 FN模型

(1)

(2)

(3)

(4)

(5)

(6)

2 边界条件

(7)

3 数值模拟

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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边界条件下的Fizhugh-Nagumo模型斑图动力学研究*

STUDIES ON PATTERN DYNAMICS OF FIZHUGH-NAGUMO MODEL UNDER BOUNDARY CONDITIONS ∗

摘要

Abstract

关键词

Keywords

0 引言

1 FN模型

(1)

(2)

(3)

(4)

(5)

(6)

2 边界条件

(7)

3 数值模拟

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

微信公众号二维码

手机版网站二维码

边界条件下的Fizhugh-Nagumo模型斑图动力学研究^*

STUDIES ON PATTERN DYNAMICS OF FIZHUGH-NAGUMO MODEL UNDER BOUNDARY CONDITIONS ^∗