Problems
1. Quantitative Description of Membrane Potentials 2. Hodgkin – Huxley Model 3. Action Potential Propagation

Reference
1. A. L. Hodgkin and A. F. Huxley (1952), A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation, J. Physiol. 117, 500-544
2. A. L. Hodgkin and W. A. H. Rushton (1946), The electrical constants of a Crustants of a Crustacean Nerve Fibre, Proc Royal, Soci (B:bio). 133, 444-479

Section 1

Quantitative Description of Membrane Potentials

Generic Cell Models
Na-K

INa, L 2K
K

3Na INa

_[c ]i
[cNa]o [cNa]i

[cK]o

IK, L

+

IK

Membrane Hypothesis for the origin of the resting potential and the action potential in nerve (1902, 1912)

1 selectively

2 diffusing
3 temporarily
Julius Bernstein

Diffusion and Drift

Fick’s Law of Diffusion

Einstein Relation

Ohm’s Law of Drift

J diff

dc  D dx

RT D  zF

J diff

dV   C dx

NERNST-PLANCK EQUATION

 dck   dV  jk  Dk     k ck  dx   ckV    dx  diffusion drift convection

NP equation
J  J diff  J drif dc dV   RT  zcF dx dx xi At equilibrium, J = 0, the Nernst equation is derived as ci RT dc dV E  Vi  Ve   dx   dx xe dx ce zFc dx

F is Faraday’s constant, zkF is charge per mole, R is Boltzmann's constant, T is temperature (Kelvin)

RT ce 58 ce  ln  ln zF ci z ci

GOLDMAN-HODGKIN-KATZ EQUATION
Vm V  x d c  Ae zFVm x RTd

dc dV J   RT  zcF dx dx zFVm   PF 2Vm  ci  ce e RT J zFVm  RT   1  e RT 

Jd  zFVm

    

Assuming steady-state conditions (e.g., at rest)

JK + JNa + JCl  0
RT  PK cK e  PNa cNa e  PCl cCl i  Vm  ln   F  PK cK i  PNa cNa i  PCl cCl e 

Squid Giant Axon
 110  0.03460  0.140  Vrest  58 ln    70mV 1400  0.0350  0.1540
Ion
K+ Na+

Cytoplasm (mM) 400
50

Extracellular (mM) 10
460

Ratio of permeabilities 1
0.03

Nernst potential (mV) 93
56

Cl

40

540

0.1

66

Section 2

Hodgkin – Huxley Model

Questions
• Studied giant squid axons
– Electrical stimulation – Measurements of ion currents

• Mathematical model of action potential
– Equivalent electric circuit of transmembrane processes – Four first order differential equations • Voltage rate of change • Rate of change of Na and K ion conductance

Circuit Models of Cell Membrane (1)
Electromotive Force Properties

Circuit Models of Cell Membrane (2)
Capacitive Properties

Lipid bilayer as insulating material

Circuit Models of Cell Membrane (3)
Resistive Properties

Conductance is related to membrane permeability

Circuit Models of Cell Membrane (4)
Resistive and Capacitive Properties

dV 1 I C  V dt R

Passive membrane: Equivalent Circuit

The Hodgkin Huxley Model
"for their discoveries concerning the ionic mechanisms involved in excitation and inhibition in the peripheral and central portions of the nerve cell membrane"

Alan L. Hodgkin

Andrew F. Huxley

The Hodgkin-Huxley Formalism
Basic Assumptions

The Hodgkin-Huxley Formalism
Ohmic Currents
Currents are linearly related to the driving potential Vm

I Na t   g Na VM , t   VM  E Na 
Ohm’s law

I K t   g K VM , t   VM  EK  I L t   g L VM , t   VM  EL 

The Nernst potential, here for Na+, K+, Cl, gives the reversal potential ENa, EK, ECl or the ionic battery – it is a function of the intra- and extracellular concentrations of the ion RT Nao E Na  ln zF Na i The Nernst Equation

EK  ECl 

RT K o ln zF K i

RT Cl i ln zF Cl o

The Hodgkin-Huxley Formalism
Basic Assumptions

dVm I t   Cm  g Na (Vm  E Na )  g K (Vm  EK )  g L (Vm  EL ) dt

The Hodgkin-Huxley Formalism
Gating Particles

g Na  g Na  m3h g K  g K  n4
Gating particles (m,h,n, etc.) were introduced to describe the dynamics of the conductances (time- and voltage-dependent) and scale a maximal conductance. They can be activating or inactivating.

The values range from 0 to 1 and (knowing what we know today with respect to ion channels) can be thought of as the percentage of channels in the activated or inactivated state.

The Hodgkin-Huxley Formalism
The ionic conductances

Curve A (V = -25 mV) gK0 gK n 0.09 m  mho/cm2 7.06 m  mho/cm2 0.75 msec

Curve B (V = 0) 7.06 m  mho/cm2 0.09 m  mho/cm2 1.1 msec

The Hodgkin-Huxley Formalism
Voltage-Dependence of Conductances

Experimentally recorded (circles) and theoretically calculated (smooth curves) changes in gNa and gK in the squid giant axon at 6.3C C during depolarizing voltage steps away from the resting potential (here set to 0). Inactivation is demonstrated by the decay of gNa following its initial rise. Reproduced from Hodgkin AL (1958) Ionic movements and electrical activity in giant nerve fibres, Proc R Soc Lond B 148:1-37

Activation and Inactivation Kinetics
Potassium Current IK
Non-inactivating current

I K  g K  n 4  Vm  EK 

Activation particle n

dn n  n  dn n   n (Vm )(1  n )   n (Vm )n dt dt n : [0. 1]

Time-dependent solution

nt   n  n  n0 e



t

n

 : msec g : m mho/cm2 ,  : msec1

g K  36 mS/cm 2
Hodgkin and Huxley’s Parameterization

10  Vm  n Vm   100  e (10Vm ) /10  1

 Vm   0.125  e V





m

/ 80

Activation and Inactivation Kinetics
Sodium Current INa
Activating and inactivating current

I Na  g Na  m3h  Vm  ENa  dh   h Vm 1  h    h Vm h dt inactivation Gating particles m and h

dm   m Vm 1  m    m Vm m dt activation Hodgkin and Huxley’s Parameterization m, h : [0. 1]  : msec g : m mho/cm2 ,  : msec1

g Na  120 mS/cm 2
25  Vm 10  e ( 25Vm ) /10  1

 m Vm  





 h Vm   0.07  e V
 h Vm  
1

m

/ 20

 m Vm   4  e

Vm / 18

e (30Vm ) /10  1

The Hodgkin-Huxley Formalism
Gating particles obey first order kinetics pi = probability (or fraction of) gate(s) i being in permissive state (1pi) = probability (or fraction of) gate(s) i being in non-permissive state

Steady state solution

dpi  i (V )(1  pi )  i (V ) pi dt i (V ) pi ,t  (V )  i (V )  i (V )
1  i (V )  i (V )  i (V )

Time constant

Activation and Inactivation Kinetics
Graphical Representation

m h n



m h n

Time constants (upper plot) and steady-state activation and inactivation (lower plot) as a function of the relative membrane potential V for sodium activation m (solid line) and inactivation h (long dashed line) and potassium activation n (short dashed line).

The Action Potential

Section 3

Voltage Clamp and Patch Clamp

Circuit Model of Cell Membrane (5)

The Voltage Clamp (1)
Cole et al.
Cole and colleagues developed a method for maintaining Vm at any desired voltage level Required monitoring voltage changes, feeding it through an amplifier to then drive current into or out of the cell to dynamically maintain the voltage while recording the current required to do so

The Voltage Clamp (2)
G. Marmont

Selective Inhibitors of Ion Channels
Tetrodotoxin (TTX) blocks sodium channels.
Tetraethylammonium (TEA) blocks potassium channels. After TTX treatment, currents elicited by voltage clamp resemble those seen by Hodgkin and Huxley in sodium-free media. After TEA treatment, the only remaining current is the inward, or sodium, current.

The mechanism of action potential generation (1)

The mechanism of action potential generation (2)

The Patch Clamp (1)
"for their discoveries concerning the function of single ion channels in cells"

Erwin Neher

Bert Sakmann

The Patch Clamp (2)

The Patch Clamp (3)

The Patch Clamp (4)

The Patch Clamp (5)

Section 4

Action Potential Propagation

Action Potential Propagation in Axon

Core Conductor Model (1)

Core Conductor Model (2)
Assumptions: – Inner and outer conductor are resistive – Membrane is black-box – unknown

Core Conductor Model (3)

边界条件：Vm = 0 ( x =  ) 初始条件：Vm = 0 ( -  < x < 0) I = I0

Core Conductor Model (4)

Core Conductor Model (5)

Action Potential Propagation Device

AP in different Position

Just knowing the “shape” of the AP, we know the membrane current density using:

AP Propagation in H-H Equation
I  Cm dVm  g Na m 3 hVm  E Na   g K n 4 Vm  E K   g L Vm  E L  dt a = radius (cm) R2 = specific intercellular resistivity (kcm)  = propagating velocity (cm/msec)

a  2Vm I 2 R2 x 2
Assume V(x, t) = V(x t)

 2Vm x
2



1  2Vm

 2 t 2

a

 2Vm

2 R2 2 t 2

dVm  Cm  g Na m 3 hVm  E Na   g K n 4 Vm  E K   g L Vm  E L  dt

Numerical integration of H-H Equations