神经元兴奋动力学课件第3章

Dynamics of Neuronal Excitability神经元的数学物理模型神经元兴奋动力学内容回顾Hodgkin-Huxley神经元方程的模拟与计算●介绍了Hodgkin和Huxley最初的几篇论文实验、模型、模拟●介绍了HH方程在当前前沿科学中被广泛使用●分析了HH方程特点，介绍了怎样利用欧拉积分方法求解HH方程●给出了模拟HH神经元发放的Matlab程序●模拟了单个动作电位、不同刺激强度下的动作电位时间演化、刺激强度与动作电位发放频率如左图所示，乌贼巨轴突的一小片膜接受图（A ）刺激得到的结果，要求编写程序画出左图中的所有曲线：A 刺激脉冲为矩形的电流；B 膜电位随时间的演化；C 钠电导（实线）和钾电导（虚线）随时间演化D 钠电流（实线）和钾电流（虚线）随时间演化提示：（1）刺激矩形脉冲幅度位53 μA/cm 2，持续0.2ms （2）第二次脉冲刺激的到达时间延迟15ms （3）静息态膜电位位-60mV ，温度为T = 6.3◦C.作业：神经元的其他数学物理模型在APS数据库里输入neuron 你会发现：大量关于神经元模型的论文神经元的数学物理模型Integrate-and-fire neuronsRate modelsMcCulloch and PittsHodgkin-HuxleyFitzHugh-NagumoWilson-CowanMorris-LecarHindmarsh-RosePhase oscillator modelsMap modelsIntegrate-and-fire neurons （IF ）模型Lapicque, L., 1907, J. Physiol. Pathol. Gen. 9, 620One of his main contributions was to propose the integrate-and-firemodel of the neuron in a seminal article published in 1907.Today, this model of the neuron is still one of the most popular models in computational neuroscience for both cellular and neural networks studies, as well as in mathematical neuroscience because of its simplicity. ).His wife, [[Jean Valkean ]], was also a neurophysiologist.[4] Louis Lapicque "insisted on the importance of his wife as equal co-worker in all his research".法国科学家Louis Lapicque1907年提出来的Integrate-and-fire neuronsv(t)是膜电压， \theta是产生动作电位峰的阈值，I_ext是外界的刺激电流，I_syn是所有突触输入的电流之和，tau_1和tau_2是确定突触电流的时间常数。

一个动作电位发生仅当神经元的膜电位达到阈值\theta；之后神经元的膜电位被重置到静息态/courses/cse590rr/03au/homework3.htmlA. IF神经元模型动作电位发放的时间序列，刺激为仅有直流刺激且大小为I_0 = 1.5. 膜电位u(t) 在这里相对阈值做了归一化（u(t)/theta）.B. IF神经元的发放频率随着刺激电流变化。

实线为标准IF神经元发放频率；虚线为在IF模型中加入4 ms不应期的结果。

abs_ref = 5; % absolute refractory period ref = 0; % absolute refractory period counterV_trace = []; % voltage trace for plotting V_th = 10; % spike thresholdfor t = 1:tstopif ~refV = V - (V/(R*C)) + (I/C)elseref = ref - 1;V = 0.2*V_th; % reset voltageendif (V > V_th)V = 50; % emit spikeref = abs_ref; % set refractory counter endV_trace = [V_trace V];end plot(V_trace);xlabel('time');ylabel('menbrane potential');if (V > V_th)V = 50; % emit spikeref = abs_ref; % set refractory counter endV_trace = [V_trace V];endplot(V_trace);xlabel('time');ylabel('menbrane potential');leak Integrate-and-fire neurons的matlab程序simulation resultsRate models模型Lotka, A. J., 1925, Elements of Physical Biology W illiams &Wilkins Co., Baltimore .是第i 个神经元或者神经元集群发放的速率；p_ij 是连接矩阵；发放；F 、G 、Q 均为多项式函数。

根据实际需要选择速率模型中的多项式方程。

例如：x 3 + 2xyz 2 −yz + 1.McCulloch and Pittstheta是发放阈值，x_j(n)是分立时间n上的突触输入；x_i(n)是输出；因而输入输出均为二进制的（0或者1）；这里突触输入g_ij为1，-1，三种取值。

MCP模型是最早的人工神经元计算模型；也被称之为线性阈值模型（linear threshold model）。

这个模型忽视了神经元动作的相对时间。

Hodgkin-Huxley这些非线性常微分方程组给出了神经元的点模型。

积分起来的步骤非常繁琐，但是该方程业已成为计算神经科学中的基本工具（principal tool ）。

该方程也可以被修改，例如，在等式的右边可以添加任何离子电流。

该方程表现出来的动力学分叉的行为也已经在实验中被观察到。

FitzHugh-Nagumo model Richard FitzHugh with analog computer atNIH, ca. 1960.received his PhD. in Biophysicsfrom Johns Hopkins University.FitzHugh R. (1961) Impulses andphysiological states in theoreticalmodels of nerve membrane.Biophysical J. 1:445-466FitzHugh R. (1962) Computation ofimpulse initiation and saltatoryconduction in a myelinated nervefiber. Biophys J. 2:11-21Richard FitzHugh during the preparation ofNagumo J., Arimoto S., and YoshizawaS. (1962) An active pulse transmissionline simulating nerve axon. Proc. IRE.50:2061–2070.J. Nagumo et al. whocreated the equivalentcircuit the following year,describes a prototype ofan excitable system (e.g.,a neuron).帮我找到Nagumo J.的照片FitzHugh-Nagumo modelis a two-dimensional simplification of the Hodgkin-Huxley model of spike generation in squid giant axons. Here,This system was suggested by FitzHugh (1961), who called it "Bonhoeffer-van der Pol model", and the equivalent circuit byNagumo et al. (1962; see Figure 1).Circuit diagram of thetunnel-diode nerve model of Nagumo et al.膜电位随时间演化FHN模型的相图。

蓝色线是FHN模型相空间轨迹，紫色和黄色的线分别是相空间中的三次零线和线性零线。

增加FHN神经元的外界刺激时，相图中轨迹、零线以及膜电位的变化Wilson-Cowan model{E(x,t),I(x,t)}是兴奋性和抑制性神经元在连续神经介质x 邻域内的数目密度（wee(x),wie(x),wei(x),wii(x)）是在神经细胞中连接性分布。

{Le ，Li}是反应不同神经元群体的阈值的非线性响应。

操作子为对连接分布的卷积。

这是第一个神经的平均场模型。

他尝试去描述一群神经元，并且通过平均噪声分布的方法，避免了单个神经元噪声作用下的相干动力学行为。

Morris-Lecar neuronMorris and Lecar, 1981v(t) 是膜电位，n(t)来描述钾电流的恢复活动；I 是外界刺激电流ML 模型是从HH 神经元模型中简化而来的模型，最早是用来描述天鹅的肌肉细胞的兴奋性。

Catherine Morris (b. 24 December 1949) is aCanadian biologist. She won a Commonwealthscholarship to study at Cambridge University,where she earned her PhD in 1977. She became a professor at the University of Ottawa in theearly 1980s. As of 2015, she is an emeritus professor at the University of Ottawa.http://www.ohri.ca/profile/CatherineEMorrisHistoryHarold Lecar (18 Oct 1935-4 Feb 2014) 2014)was an American professor of biophysics andneurobiology at the University of CaliforniaBerkeley. He graduated with his PhD inphysics from Columbia University in 1963.https:///news-and-events/department-news/harold-lecar-has-diedV : membrane potentialN : recovery variable: the probability that the K+ channel is conductinBifurcation in the Morris–Lecar model have been analyzedwith the applied current I, as the main bifurcation parameter and φ, gCa, V3, V4 as secondary parameters for phase plane analysis：Hopf BifurcationSNIC bifurcation Homoclinic bifurcationCurrent clamp simulations of the Morris–Lecar model. The injected current for the SNIC bifurcation and the homoclinic bifurcation is varied between 30 nA and 50 nA, while the current for the Hopf bifurcation is varied between 80nA and 100nAg_ca=4.0;g_k=8.0;g_l=2;v_ca=120;v_k=-84;v_l=-60;Q=1/15;V1=-1.2;V2=18;V3=12;V4=17.4;c_m=20;dt=0.1; t_span=220; t=0:dt:t_span;I_app=50; v=30.83; w=0.2;vt=zeros(1,length(t));wt=zeros(1,length(t));for it=1:length(t)tau_w=1/cosh((v-V3)/(2*V4));m_inf=0.5*(1+tanh((v-V1)/V2));w_inf=0.5*(1+tanh((v-V3)/V4));if t(it)>=80 && t(it)<=80.5II=480;elseII=0;endvt(it)=(-g_ca*m_inf*(v-v_ca)-g_k*w*(v-v_k)-g_l*(v-v_l)+I_app+II)*dt/c_m+v; wt(it)=Q*(w_inf-w)*dt/tau_w+w;v=vt(it);w=wt(it);endfigure(1)hold onplot(t,vt,'--','LineWidth',2);hold offHindmarsh-RoseHindmarsh and Rose, 1984x(t)是膜电压；y(t)用来描述快电流变量；z(t)来描述慢电流变量；I 是外界刺激电流； 该模型是利用多项式拟合简化HH 神经元模型得到的；该模型可以得到粗发放和混沌发放的模式该模型不能用来模拟发放后的超极化状态Bifurcation diagram and spiking patterns for an HR neuron. (a) Bifurcation diagram plotted as the interspike interval of the response spike train of the neuron vs. I ext . (b)-(f) are period-1, period-2, period-3, period-4 and chaotic burstings, respectively. The corresponding externalDC inputs were established as I ext = 1:37; 1:67; 2:23; 2:67; 3:11, respectively.Phase oscillator modelsCohen et al.,1982; Ermentrout and Kopell, 1984; Kuramoto,1984Map models作业：玩一下IF模型 ^_^依照本节课给出的程序：1）重复IF模型模拟的动作电位2）改变刺激强度大小，观察动作电位的变化3）试求出动作电位的发放频率随刺激强度变化规律。