神經(jīng)元興奮動(dòng)力學(xué)課件第3章_圖文

1、Dynamics of Neuronal Excitability神經(jīng)元的數(shù)學(xué)物理模型 神經(jīng)元興奮動(dòng)力學(xué)內(nèi)容回顧Hodgkin-Huxley神經(jīng)元方程的模擬與計(jì)算介紹了Hodgkin和Huxley最初的幾篇論文實(shí)驗(yàn)、模型、模擬介紹了HH方程在當(dāng)前前沿科學(xué)中被廣泛使用分析了HH方程特點(diǎn),介紹了怎樣利用歐拉積分方法求解HH方程給出了模擬HH神經(jīng)元發(fā)放的Matlab程序模擬了單個(gè)動(dòng)作電位、不同刺激強(qiáng)度下的動(dòng)作電位時(shí)間演化、刺激強(qiáng)度與動(dòng)作電位發(fā)放頻率 如左圖所示,烏賊巨軸突的一小片膜接受圖(A 刺激得到的結(jié)果,要求編寫(xiě)程序畫(huà)出左圖中的所有曲線:A 刺激脈沖為矩形的電流;B 膜電位隨時(shí)間的演化;C 鈉電

2、導(dǎo)(實(shí)線和鉀電導(dǎo)(虛線隨時(shí)間演化D 鈉電流(實(shí)線和鉀電流(虛線隨時(shí)間演化提示:(1刺激矩形脈沖幅度位53 A/cm 2,持續(xù)0.2ms (2第二次脈沖刺激的到達(dá)時(shí)間延遲15ms (3靜息態(tài)膜電位位-60mV ,溫度為T(mén) = 6.3C.作業(yè):神經(jīng)元的其他數(shù)學(xué)物理模型在APS數(shù)據(jù)庫(kù)里輸入neuron 你會(huì)發(fā)現(xiàn):大量關(guān)于神經(jīng)元模型的論文神經(jīng)元的數(shù)學(xué)物理模型Integrate-and-fire neuronsRate modelsMcCulloch and PittsHodgkin-HuxleyFitzHugh-NagumoWilson-CowanMorris-LecarHindmarsh-RoseP

3、hase 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

4、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 al

5、l his research".法國(guó)科學(xué)家Louis Lapicque1907年提出來(lái)的Integrate-and-fire neurons v(t是膜電壓, theta是產(chǎn)生動(dòng)作電位峰的閾值,I_ext是外界的刺激電流,I_syn是所有突觸輸入的電流之和,tau_1和tau_2是確定突觸電流的時(shí)間常數(shù)。一個(gè)動(dòng)作電位發(fā)生僅當(dāng)神經(jīng)元的膜電位達(dá)到閾值theta;之后神經(jīng)元的膜電位被重置到靜息態(tài)/courses/cse590rr/03au/homework3.html A. IF神經(jīng)元模型動(dòng)作電位發(fā)放的時(shí)間序列,刺激為僅有直流刺

6、激且大小為I_0 = 1.5. 膜電位u(t 在這里相對(duì)閾值做了歸一化(u(t/theta.B. IF神經(jīng)元的發(fā)放頻率隨著刺激電流變化。實(shí)線為標(biāo)準(zhǔn)IF神經(jīng)元發(fā)放頻率;虛線為在IF模型中加入4 ms不應(yīng)期的結(jié)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 +

7、(I/Celseref = ref - 1;V = 0.2*V_th; % reset voltageendif (V > V_thV = 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_thV = 50; % emit spikeref = abs_ref; % set refractory counter

8、endV_trace = V_trace V;endplot(V_trace;xlabel('time'ylabel('menbrane potential'leak Integrate-and-fire neurons的matlab程序simulation results Rate models模型 Lotka, A. J., 1925, Elements of Physical Biology W illiams &Wilkins Co., Baltimore .是第i 個(gè)神經(jīng)元或者神經(jīng)元集群發(fā)放的速率;p_ij 是連接 矩陣;發(fā)放;F 、G 、Q

9、均為多項(xiàng)式函數(shù)。根據(jù)實(shí)際需要選擇速率模型中的多項(xiàng)式方程。例如:x 3 + 2xyz 2 yz + 1.McCulloch and Pitts theta是發(fā)放閾值,x_j(n是分立時(shí)間n上的突觸輸入;x_i(n是輸出;因而輸入輸出均為二進(jìn)制的(0或者1;這里突觸輸入g_ij為1, -1,三種取值。MCP模型是最早的人工神經(jīng)元計(jì)算模型;也被稱(chēng)之為線性閾值模型(linear threshold model。這個(gè)模型忽視了神經(jīng)元?jiǎng)幼鞯南鄬?duì)時(shí)間。Hodgkin-Huxley 這些非線性常微分方程組給出了神經(jīng)元的點(diǎn)模型。積分起來(lái)的步驟非常繁瑣,但是該方程業(yè)已成為計(jì)算神經(jīng)科學(xué)中的基本工具(principa

10、l tool 。該方程也可以被修改,例如,在等式的右邊可以添加任何離子電流。 該方程表現(xiàn)出來(lái)的動(dòng)力學(xué)分叉的行為也已經(jīng)在實(shí)驗(yàn)中被觀察到。FitzHugh-Nagumo model Richard FitzHugh with analog computer at NIH, ca. 1960.received his PhD. in Biophysicsfrom Johns Hopkins University.FitzHugh R. (1961 Impulses andphysiological states in theoreticalmodels of nerve membrane.Bioph

11、ysical J. 1:445-466FitzHugh R. (1962 Computation ofimpulse initiation and saltatoryconduction in a myelinated nervefiber. Biophys J. 2:11-21 Richard FitzHugh during the preparation of Nagumo J., Arimoto S., and YoshizawaS. (1962 An active pulse transmissionline simulating nerve axon. Proc. IRE.50:20

12、612070. J. Nagumo et al. whocreated the equivalentcircuit the following year,describes a prototype ofan excitable system (e.g.,a neuron.幫我找到Nagumo J.的照片F(xiàn)itzHugh-Nagumo model is a two-dimensional simplification of the Hodgkin-Huxley model of spike generation in squid giant axons. Here,This system was

13、 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 the tunnel-diode nerve model of Nagumo et al. 膜電位隨時(shí)間演化FHN模型的相圖。藍(lán)色線是FHN模型相空間軌跡,紫色和黃色的線分別是相空間中的三次零線和線性零線。 增加FHN神經(jīng)元的外界刺激時(shí),相圖中軌跡、零線以及膜電位

14、的變化 Wilson-Cowan model E(x,t,I(x,t是興奮性和抑制性神經(jīng)元在連續(xù)神經(jīng)介質(zhì)x 鄰域內(nèi)的數(shù)目密度(wee(x,wie(x,wei(x,wii(x是在神經(jīng)細(xì)胞中連接性分布。 Le ,Li是反應(yīng)不同神經(jīng)元群體的閾值的非線性響應(yīng)。 操作子為對(duì)連接分布的卷積。這是第一個(gè)神經(jīng)的平均場(chǎng)模型。他嘗試去描述一群神經(jīng)元,并且通過(guò)平均噪聲分布的方法,避免了單個(gè)神經(jīng)元噪聲作用下的相干動(dòng)力學(xué)行為。Morris-Lecar neuron Morris and Lecar, 1981v(t 是膜電位,n(t來(lái)描述鉀電流的恢復(fù)活動(dòng);I 是外界刺激電流ML 模型是從HH 神經(jīng)元模型中簡(jiǎn)化而來(lái)的模型

15、,最早是用來(lái)描述天鵝的肌肉細(xì)胞的興奮性。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 the early 1980s. As of 2015, she is an emeritus professor at t

16、he University of Ottawa.http:/www.ohri.ca/profile/CatherineEMorrisHistoryHarold Lecar (18 Oct 1935-4 Feb 2014 2014was an American professor of biophysics andneurobiology at the University of CaliforniaBerkeley. He graduated with his PhD inphysics from Columbia University in 1963.https:/mcb.berkeley.

17、edu/news-and-events/department-news/harold-lecar-has-died V : membrane potentialN : recovery variable: the probability that the K+ channel is conductinBifurcation in the MorrisLecar model have been analyzedwith the applied current I, as the main bifurcation parameter and , gCa, V3, V4 as secondary p

18、arameters for phase plane analysis: Hopf Bifurcation SNIC bifurcation Homoclinic bifurcationCurrent clamp simulations of the MorrisLecar 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 i

19、s 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(ttau_w=1/cosh(v-V3/(2*V4;m_inf=0.5*(1+tanh(v-V1/V2;w_inf=0.5*(1+

20、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(1hold onplot(t,vt,'-','LineWidth',2;hold off Hindmarsh-Rose Hindmarsh and Rose, 1984x(t是膜電壓;y(t用來(lái)描述快電流變量;z(t來(lái)描