1、n 应用数学方法建立神经元膜上动作电位产生和在传导的模型应用数学方法建立神经元膜上动作电位产生和在传导的模型。n Hodgkin and Huxley Model was published in 1952 described a series of experiments that allowed the development of a model of the action potential. n This work was awarded a Nobel Prize in 1963.n Partial Differential Equation (PDE): function of
2、time and length Ordinary Differential Equation (ODE): function of timen Neural system is consisted of millions of neurons（1012）, neuron is the basic unit of the neural system. 前言前言BackgroundGiant axon of squid 9.1 9.1 动作电位产生和传导的神经生物学机制动作电位产生和传导的神经生物学机制9.1.1 神经细胞与神经信息传递神经细胞与神经信息传递9.1.2 神经细胞膜的离子通道特性神经
3、细胞膜的离子通道特性9.1.3 动作电位与神经细胞膜离子通道特性动作电位与神经细胞膜离子通道特性9.1.1 神经细胞与神经信息传递神经细胞与神经信息传递nsoma 胞体胞体nsynapse 突触突触ndendrite 树突树突nAxon 轴突轴突Diagram of typical neuron神经细胞神经细胞神经细胞各个部分在信息传递中的作用神经细胞各个部分在信息传递中的作用nThe neurons cell body is connected to a number of branches The neurons cell body is connected to a number of
4、branches called dendritescalled dendrites（树突）（树突）and long tube called the axonand long tube called the axon（轴突）轴突）that connects the cell body to the presynapticthat connects the cell body to the presynaptic terminals (synapse terminals (synapse 突触突触). ). nDendrites are the receptive surfaces of the
5、neuron that receive Dendrites are the receptive surfaces of the neuron that receive signals from thousands of other neurons; signals from thousands of other neurons; nThe single axon is served as an transmission line to move The single axon is served as an transmission line to move information from
6、one neuron to anothersinformation from one neuron to anothers. . 9.1.2 神经细胞膜的离子通道特性神经细胞膜的离子通道特性 细胞膜把细胞内外分隔，细胞膜两侧的离子成分和浓度不同：细胞膜把细胞内外分隔，细胞膜两侧的离子成分和浓度不同： 细胞内细胞内 正离子：主要是钾离子正离子：主要是钾离子 负离子：有机小分子负离子：有机小分子 细胞外细胞外 正离子：主要是钠离子正离子：主要是钠离子 负离子：氯离子负离子：氯离子 在这两种方向相反的作用下达到平衡在这两种方向相反的作用下达到平衡时，没有离子通过细胞膜；时，没有离子通过细胞膜； 达到
7、平衡状态时，细胞膜内外的电位差称为该类离子的达到平衡状态时，细胞膜内外的电位差称为该类离子的Nernst电位电位V*； 不同类离子的不同类离子的Nernst电位不同。电位不同。神经细胞膜电活动神经细胞膜电活动由于细胞内外离子的浓度差，由于细胞内外离子的浓度差，某一种离子要从高浓度侧向低某一种离子要从高浓度侧向低浓度侧扩散。浓度侧扩散。由于离子带电，扩散到另一侧的由于离子带电，扩散到另一侧的离子积累起来的电荷排斥将扩散离子积累起来的电荷排斥将扩散过来的带有同一类电荷的离子。过来的带有同一类电荷的离子。平衡平衡离子通过离子通道的驱动力离子通过离子通道的驱动力设：细胞的膜电位为设：细胞的膜电位为V，
8、 某个离子的某个离子的Nernst电位为电位为V*；那么那么 （VV*）是这种离子通过该离子通道的驱动力。是这种离子通过该离子通道的驱动力。e.g. （VVNa）是钠离子通过细胞膜钠离子通道的驱动力。是钠离子通过细胞膜钠离子通道的驱动力。（VVk） 是钾离子通过细胞膜钾离子通道的驱动力。是钾离子通过细胞膜钾离子通道的驱动力。Neurons, like all living cells, are surrounded（包围）（包围） by a plasma membrane that is impermeable （不可渗透（不可渗透 ）to ions. This property allows
9、 a neuron to maintain different concentrations of ions between the inside and outside of the cell.神经细胞膜内外不同的离子浓度神经细胞膜内外不同的离子浓度In a typical mammalian neuron, there is a large difference in the concentration of ions, such as sodium (Na+) and potassium (K+), between the intracellular and the extracellu
10、lar environments. In addition, the interior of the neuron has a high concentration of large negatively charged proteins.神经细胞膜内外不同的钾离子浓度、钠离子浓度神经细胞膜内外不同的钾离子浓度、钠离子浓度The plasma membrane is composed of a lipid bilayer (脂脂类类层层 ). Its hydrophobic （疏水）（疏水）nature prevents the diffusion of ions across the mem
11、brane.神经细胞膜神经细胞膜的构成和离子跨膜扩散的构成和离子跨膜扩散The only way ions can move across the lipid bilayer is by passing through specialized channels. These channels are transmembrane pores that permit the movement of particular ions while excluding others. Such channels can be in an open or closed state.神经细胞膜上不同类的离子通
12、道神经细胞膜上不同类的离子通道When a neuron is at rest, most ion channels are closed. However, some potassium channels (blue) are open, permitting potassium ions to diffuse out of the cell down their concentration gradient. Note that sodium channels are normally closed, and thus sodium ions cannot cross the membra
13、ne when the neuron is at rest.神经细胞膜静息：大部分通道关闭部分钾离子通道开启神经细胞膜静息：大部分通道关闭部分钾离子通道开启In a typical neuron, the internal concentration of potassium is higher than the external concentration. K+ ions are pulled by two opposing forces： First, a DIFFUSION FORCE drives K+ ions along their concentration gradient
14、towards the exterior of the cell.钾离子浓度细胞内高于细胞外钾离子浓度细胞内高于细胞外, , K K+ + ions ions 受扩散力推向细胞外受扩散力推向细胞外The movement of potassium ions out of the cell increases the internal negative charge. The positively charged potassium ions are attracted to the internal negative charge, and this ELECTRICAL FORCE pull
15、s potassium ions back into the cell.钾离子钾离子受受电相斥力推回细胞内电相斥力推回细胞内The diffusion and electrical forces eventually come into balance, and an electrical potential, or voltage, is reached at which the electrical force exactly balances the diffusion force. At this point, there is no NET movement of potassium
16、 ions into or out of the cell.两类相反的作用力达到平衡两类相反的作用力达到平衡, , 此时没有离子通过细胞膜此时没有离子通过细胞膜The electrical potential across the membrane can be measured by inserting an electrode into the cell. A neuron at rest has a voltage difference of about -70 -60 mv across the membrane.神经元的静息膜电压神经元的静息膜电压n 离子通道是分布在细胞膜上的跨膜蛋
17、白。离子通道是分布在细胞膜上的跨膜蛋白。 Ion channels are membrane protein complexes . They play an essential role in the diffusion of ions across cell membrane.n 具有复杂的门控（开关）特性，离子电导性决定神经元放电模式具有复杂的门控（开关）特性，离子电导性决定神经元放电模式 。 Gating: The conformational change between open and closed . open the conformation in which channel
18、 allow ions to pass； closed the conformation in which channel forbid ions to pass.n 离子通道的状态（关闭、开放和失活）与膜电位和时间有关，离子通道的状态（关闭、开放和失活）与膜电位和时间有关，n 某一个离子通道的状态可以通过测定通道的电导来评估。某一个离子通道的状态可以通过测定通道的电导来评估。Electrical properties of ion channels Ion channels can be classified according to their gating mechanism1. 电压门
19、控电压门控2. 配体门控配体门控细胞外配体细胞外配体 细胞内配体细胞内配体3. 机械门控机械门控根据门控机制的不同，离子通道可分为三类：根据门控机制的不同，离子通道可分为三类： 9.1.3 细胞膜离子通道特性与动作电位细胞膜离子通道特性与动作电位Action potentialn动作电位是神经元编码信息，及与其它神经元通讯的手段，动作电位是神经元编码信息，及与其它神经元通讯的手段，n动作电位的产生是由于神经细胞膜跨膜电位的变化，动作电位的产生是由于神经细胞膜跨膜电位的变化，n细胞膜上的钾离子通道、钠离子通道等离子通道电导特性，细胞膜上的钾离子通道、钠离子通道等离子通道电导特性， 在动作电位的产
20、生中起关键作用。在动作电位的产生中起关键作用。nH-H模型描述神经细胞膜上的离子通道电导状态产生和模型描述神经细胞膜上的离子通道电导状态产生和 沿着轴突传导动作电位。沿着轴突传导动作电位。nAction Potential: all-or-none: action potential does not decrease in amplitude as it is conducted away from its site of initiation;The action potential is due to voltage and time-dependent changes in condu
21、ctance.nResting potential: -70 -60mVnAction potential: +20 +30mVAction potential动作电位的动作电位的5个相位个相位n 静息电位（静息电位（ The resting potential ）：极化）：极化（polarization）n 阈值电位（阈值电位（threshold）n 上升（上升（The rising phase）：去极化（）：去极化（depolarization）n 下降（下降（The falling phase)：复极化（：复极化（repolarization）n 恢复期（恢复期（The recovery
22、 phase）When the neuron is at rest, only a small subset of potassium channels are open, permitting K+ ions to enter and exit the cell based on electrochemical forces. For each K+ ion that leaves the cell, another returns, maintaining the membrane potential at a constant value.openclosedThe resting po
23、tential静息状态（极化）静息状态（极化）处于静息电位处于静息电位（-70 -70 60mV60mV）As a depolarizing stimulus arrives at the segment of the membrane, a few Na+ channels open, permitting Na+ ions to enter the neuron. The increase in positive ions inside the cell depolarizes the membrane potential (making it less negative), and br
24、ings it closer to the threshold at which an action potential is generated.Threshold阈值电位阈值电位当细胞膜受到刺激，部分Na+通道打开，允许Na+进入膜内。由于Na+离子内流，细胞膜内电位上升并超过阈值电位。If the depolarization reaches the threshold potential, additional voltage gated sodium channels open. As positive Na+ ions rush into the cell, the voltage
25、 across the membrane rapidly reverses and reaches its most positive value.openThe rising phase膜电位超过阈值电位，导致更多钠离子通道开放，电导GNa 增加，使钠离子内流；内流的钠离子电流导致进一步的去极化，使钠离子电流进一步增加。At the peak of the action potential, two processes occur simultaneously. First many of the voltage-gated sodium channels begin to close. S
26、econd, many more potassium channels open, allowing positive charges to leave the cell. This causes the membrane potential to begin to shift back towards the resting membrane potential.At the peak of the action potential当膜电位达到最大值时（膜电压趋近钠离子的静息电位VNa）： Na+离子通道闭合，GNa逐渐减小至零； 部分K+离子通道开放，使得K+离子外流。As the mem
27、brane potential approaches the resting potential, voltage-gated potassium channels are maximally activated and open.The falling phase部分K+离子通道开放，使K+离子外流；K+离子通道电导GK增加，使膜电位下降。The membrane actually repolarizes beyond the resting membrane voltage. This undershoot occurs because more potassium channels ar
28、e open at this point than during the membranes resting state, allowing more positively charged K+ ions to leave the cell. The recovery phaseGk 的增加导致的增加导致K+离子外流，外流的离子外流，外流的K+离子离子电流引起进一步超极化，进而使电流引起进一步超极化，进而使K+离子电流离子电流进一步增加，促使膜电位进一步增加，促使膜电位V下降到趋近下降到趋近K+离子的离子的Nernst电位电位Vk，低于静息电位。，低于静息电位。The return to st
29、eady state continues as the additional potassium channels that opened during the action potential now close. The membrane potential is now determined by the subset of potassium channels that are normally open during the membranes resting state.The return to steady state膜电位逐渐恢复到静息电位膜电位逐渐恢复到静息电位细胞膜的离子
30、通道特性与动作电位细胞膜的离子通道特性与动作电位n极化（极化（polarization）：）：细胞处于静息电位（细胞处于静息电位（-70 -60mV）时称为极化。）时称为极化。n阈值电位（阈值电位（threshold）当细胞膜受到刺激，部分当细胞膜受到刺激，部分Na+离子通道打开，允许离子通道打开，允许Na+进入膜内。由于进入膜内。由于Na+离子内流，离子内流，细胞膜内电位上升并超过阈值电位。细胞膜内电位上升并超过阈值电位。n去极化（去极化（depolarization）：）： 膜电位超过阈值电位，导致更多钠离子通道开放，膜电位超过阈值电位，导致更多钠离子通道开放， 电导电导GNa增加，使钠离
31、子内流；增加，使钠离子内流； 内流的钠离子电流导致进一步的去极化，使钠离子电流进一步增加。内流的钠离子电流导致进一步的去极化，使钠离子电流进一步增加。当膜电位达到最大值时（膜电压当膜电位达到最大值时（膜电压V趋近钠离子的趋近钠离子的Nernst电位电位VNa），）， Na离子通道闭合，离子通道闭合，GNa逐渐减小至零；部分逐渐减小至零；部分K+离子通道开放，使得离子通道开放，使得K+外流。外流。n复极化（复极化（repolarization）：）： 部分部分K+ 离子通道开放，使得离子通道开放，使得K+外流；外流；K+ 离子通道电导离子通道电导GK增加，使得膜电位下降。增加，使得膜电位下降。
32、Gk 的增加导致的增加导致K+外流，外流的钾离子电流引起进一步超极化，进而使钾离子电流进一外流，外流的钾离子电流引起进一步超极化，进而使钾离子电流进一步增加，促使膜电位步增加，促使膜电位V下降到趋近钾离子的下降到趋近钾离子的Nernst电位电位Vk，低于静息电位。，低于静息电位。n恢复期（恢复期（recovery）：）：膜电位逐渐恢复到正常静息电位。膜电位逐渐恢复到正常静息电位。nExcitation transmission When a neuron sends an excitatory signal to another neuron, then this signal will be
33、 added to all of the other inputs of that neuron. If it exceeds a given threshold then it will cause the target neuron to fire an action potential, if it is below the threshold then no action potential occurs.n Transmission velocity can be reach to : 100m/sec神经细胞兴奋神经细胞兴奋（在轴突上的）（在轴突上的）传递传递 9.2 H-H 模型
34、的建立模型的建立描述动作电位产生和（在轴突上的）传递描述动作电位产生和（在轴突上的）传递H-H 方程9.2.1. Equivalent Circuit Model for the Cell MembraneNa-K PumpJVVkVLVNaCmJcJNaJLJkGkGLGNaVL - Leak equilibrium potentialVk- equilibrium potential of K+VNa- equilibrium potential of Na+JL - Leakage currentJk - Na+ currentJNa - Na+ currentJc - Capaciti
35、ve currentCm - Capacitance of the membraneV - Cell membrane potential (about -60 mV)GL - Leak ConductanceGk- Potassium ConductanceGNa- Sodium ConductanceNa-K PumpJVVkVLVNaCmJcJNaJLJkGkGLGNa膜电流密度膜电流密度 J 为为设设G 和和C 分别为单位面积的电导和电容，根据电路知识可得分别为单位面积的电导和电容，根据电路知识可得将上（将上（9.1）代入（）代入（9.2），可得），可得（9.1）小，可忽略小，可忽略（
36、9.2）（9.3）Not a constant, depends on V9.2.2 动作电位（在轴突上）传递动作电位（在轴突上）传递设神经纤维为管状结构，为了表示神经纤维上不同位置电位的大小，沿神经纤维建设神经纤维为管状结构，为了表示神经纤维上不同位置电位的大小，沿神经纤维建立一维坐标（立一维坐标（O-X）；神经纤维直径为）；神经纤维直径为d（截面积为（截面积为A=D2/4），长度为），长度为l，D l；神经纤维内部电导均匀分布，单位长度电阻系数为神经纤维内部电导均匀分布，单位长度电阻系数为R。在在 t 时刻时刻 x 处，膜电位为处，膜电位为v(x,t)。在在x，电阻为，电阻为在在 t 时刻
37、，从时刻，从 x 到到 x x 膜电位变化量为膜电位变化量为 设电流与设电流与 X 轴方向一致为正，通过横截面的电流平均值为轴方向一致为正，通过横截面的电流平均值为 i ，根据，根据Ohm定律可得定律可得取极限取极限 ，得到通过，得到通过 x 点神经纤维截面的点神经纤维截面的电流为电流为在在x 和和 x+ x 位置上，通过神经纤维横截面的电流分别是位置上，通过神经纤维横截面的电流分别是 i 和和 i+ i （ i 是由于是由于从从x 到到 x+ x 这一段神经纤维表面生物膜上有穿过膜表面的电流产生）这一段神经纤维表面生物膜上有穿过膜表面的电流产生）Def. 膜电流密度膜电流密度 ( J ) 穿
38、过膜表面穿过膜表面单位面积元单位面积元上的电流，设电流流入为负，流上的电流，设电流流入为负，流出为正。圆柱形膜表面面积为出为正。圆柱形膜表面面积为 ，电流的关系应该有，电流的关系应该有取极限取极限 ，并将（，并将（9.6）代入得）代入得（9.6）（9.8）9.2.3 H-H 方程方程npartial differential H-H equation （9.9）nordinary differential H-H equation（9.12） If C0 = 18.74 m/sec， H-H PDE ODE9.2.3 H-H 方程方程npartial differential H-H equa
39、tion （9.9）nordinary differential H-H equation（9.12） If C0 = 18.74 m/sec， H-H PDE ODE 9.3 H-HH-H方程中的离子通道电导方程中的离子通道电导 nG GNaNa, , G Gk k and and G G0 0 are not constants, there values are functions of membrane are not constants, there values are functions of membrane potential potential V Vm me.g. e.g
40、. At resting potential: At resting potential: V Vm m = -70 = -70 - 60mV high - 60mV high G Gk k , , low low G GNaNa 在动作电位的顶峰在动作电位的顶峰: : V Vm m = 20 30mV low = 20 30mV low G Gk k , , high high G GNaNanTechniquesTechniques（钳制膜电压，测量膜电流电流，获得电导）钳制膜电压，测量膜电流电流，获得电导）Voltage clamp Voltage clamp 电压钳（电压钳（测量局部膜
41、总电流测量局部膜总电流 膜片上总电导）膜片上总电导）Patch clamp Patch clamp 膜片钳（膜片钳（测量离子通道电流测量离子通道电流 离子通道电导）离子通道电导）nConductance estimationConductance estimationEstimation of Conductance in H H ModelVoltage Clamp 电压钳技术电压钳技术n Breakthroughs in scientific research often rely on the development of Breakthroughs in scientific rese
42、arch often rely on the development of new technologies. In the case of the action potential, detailed new technologies. In the case of the action potential, detailed understanding came only after of the invention of the voltage clamp understanding came only after of the invention of the voltage clam
43、p technique by Kenneth Cole in the 1940s. technique by Kenneth Cole in the 1940s. n This device is called a voltage clamp because it controls, or clamps, This device is called a voltage clamp because it controls, or clamps, membrane potential (or voltage) at any level desired by the experimenter.mem
44、brane potential (or voltage) at any level desired by the experimenter.Voltage ClampPatch Clamp 膜片钳技术膜片钳技术n 一种记录通过离子通道微小电流的技术。用微电极一种记录通过离子通道微小电流的技术。用微电极接触接触细胞膜，使细胞膜，使电极的尖开口处和细胞膜片电极的尖开口处和细胞膜片“封接封接”（与周围绝缘），产生（与周围绝缘），产生1010 以以上大阻抗，在固定膜电位时，测量该膜片上离子通道的离子电流上大阻抗，在固定膜电位时，测量该膜片上离子通道的离子电流(pA级级) 。n 膜片钳技术的建立，是生物
45、医学，特别是神经科学的一项重大技术膜片钳技术的建立，是生物医学，特别是神经科学的一项重大技术革命。记录单离子（或多个）通道的离子电流来研究细胞膜上单个革命。记录单离子（或多个）通道的离子电流来研究细胞膜上单个（或多个）离子通道的分子水平电活动，将电生理技术从细胞水平提（或多个）离子通道的分子水平电活动，将电生理技术从细胞水平提高到分子水平高到分子水平。Patch Clamp 操作过程：找细胞、形成封接、破膜等操作过程：找细胞、形成封接、破膜等Patch Clamp 膜片钳技术膜片钳技术n Patch Clamp is a technique capable of measuring the c
46、urrents flowing through single channels was devised in 1976 by Neher and Sakmann.nA glass pipette with a very small opening is used to make tight contact with a tiny area, or patch, of neuronal membrane. After the application of a small amount of suction to the back of the pipette,n The seal（封接）（封接）
47、 between pipette and membrane becomes so tight (1010 ） that no ions can flow between the pipette and the membrane. Thus, all the ions that flow when a single ion channel opens must flow into the pipette. The resulting electrical current, though small (pA) , can be measured with an ultrasensitive ele
48、ctronic amplifier connected to the pipette. Based on the geometry involved, this arrangement usually is called the cell-attached patch clamp recording method. 应用膜片钳技术获取电导膜电压关系曲线应用膜片钳技术获取电导膜电压关系曲线nHH方程中电导方程中电导G是膜电压的函数是膜电压的函数G(v)n应用膜片钳技术测取电导膜电压离散点应用膜片钳技术测取电导膜电压离散点n fitting（拟合）电导膜电压关系曲线（拟合）电导膜电压关系曲线Con
49、ductance EstimationConductance Estimation The conductance was estimated by ways of curve fittingPotassium Conductance（钾电导）（钾电导）式中，式中，n 服从微分方程服从微分方程 上两式中，上两式中，GK0是一常数，其量纲与单位面积的电导的量纲相同；是一常数，其量纲与单位面积的电导的量纲相同；n 和和n 是与电位有关而与时间无关的速率常数（是与电位有关而与时间无关的速率常数（9.15-9.16)，量纲，量纲时间时间-1；n 是一个无量纲变量，取值在是一个无量纲变量，取值在0到到1
50、之间。之间。 (9.13)(9.14)Conductance EstimationSodium Conductance （钠电导）（钠电导） m 和和 h 服从微分方程服从微分方程 式中，式中，GNa0是一常数，其量纲为是一常数，其量纲为电导电导长度长度-2；m ,m ,h ,h 是与电位有关而与时间无关的速率常数是与电位有关而与时间无关的速率常数 (9.20)-(9.23)，量纲量纲为为时间时间-1；m 和和h 是一个无量纲变量，取值在是一个无量纲变量，取值在0到到1之间。之间。 (9.17)(9.18)(9.19) 估计参数关系式：在不同钳位电压下，测量实验数据（各离子通道跨膜电流密度）估
