1

59

Goldman-Hodgkin-Katz (GHK) equations

major components of membrane current

assume:Ion flux within membrane obeys Nernst_Planck equationIons move across membrane independently (no interactions with one another)Electric field in the membrane is constant

then:

This is the GHK voltage equation. (There is also a GHK current equation, which we neglect for the moment.)

61

Permeability

assume [C] falls linearly within the membrane,

and constants

b is the water-membrane partition coefficient for ion i

m* is the mobility of ion i within the membrane

then:

is the diffusion coeff within membrane

62

I

V

g1

g 2

g

in

out

E K

g

in

out

i g

i g

I

V

g 2

g1

E K=�80mV

2

63 64

g iC=dV

dt

iC�i g= I out

CdV

dt�g �V�E L�= I out

I inj

g L=�dV

dt�V�E L

V=I inj

g�1�exp �

�t�

���E L

I inj

I inj=0 ,dV

dt=0

� V=E LV

I out= I inj ,dV

dt=0

� V=E L�I inj

g

I inj

g L

I outI inj

I out

in

out

65

Outward (applied), +ve (depolarize)

Inward (applied), -ve (hyperpolarize)

inside

outside

EK

gNaEK gNa

ENa ECl+ve

-ve

Cm

I m

I m=iC�iK�iNa� I Cl

Cm

dV

dt�g K �V�E K ��gNa �V�ENa ��gCl �V�ECl�

when I m=0,dv

dt=0

V=g K EK�g NaE Na�gCl EClg K�gNa�gCl

external source

66

I

V

EK=�80mV ENa=�30mV

I

V

in

out

outward, +ve inward, -ve

EK ENa

iK

iK

iNa

iNa

iNa

iK

g K

gNa

gNa

gNa

outward

inward

3

89

Voltage clamp configuration

90

91 92

I

V

E K=�80mV E Na=�30mV

I

V

inward, -ve

iK

iNa

iNa

iK

g K

g Na

g Na

outward

inward

in

out

outward, +ve

E K E Na

iK iNa

g Nag K

( When capactitave current = 0 )

4

95

g i�V ,��

E iV

�g �V , t �=g��V ��g �V ,t �

�

E i

insideg axial

E leak

g leakCm

outside

g i

V soma

96

g Na�V ,�Na�

ENa�30mV

EK�80mV

g K �V ,�K �

V

V

g Na

�Na

�K

g K

iNa

iK

depol hyperpol

iNa

g Na�V ,�Na�

g Na g K

iK-ve feedback+ve feedback V

5

97

normalized so that model range is [0 1]

98

An action potential

� gNa increases quickly, but then inactivation

kicks in and it decreases again.

� gK increases more slowly, and only decreases

once the voltage has decreased.

� The Na+ current is autocatalytic. An increase in V increases m, which increases the Na+ current, which increases V, etc.

� Hence, the threshold for action potential initiation is where the inward Na+ current exactly balances the outward K+ current.

99

Vclamp

Note that both the amplitude of the conductance change and its time constant change with Vclamp

100

Experimental data: K+ conductance

If voltage is stepped up and held fixed, gK increases

to a new steady level.

time constant

steady-state

four subunits

rate of rise gives τn

steady state gives n

�

gK=gK n4

dn

dt=�n�V ��1�n���n�V �n

�n �V �dn

dt=nss�V ��n

6

101

Experimental data: Na+ conductance

If voltage is stepped up and held fixed, gNa increases

and then decreases.

time constant

steady-state

Four subunits.Three switch on.One switches off.

gNa=gnam3h

dm

dt=�m�V ��1�m���m�V �m

�h �V �dh

dt=hss�V ��h

�m �V �dm

dt=mss �V ��m

102

Hodgkin-Huxley equations

�

�

�

generic leak

applied current

much smaller thanthe others

inactivation(decreases with V)

activation(increases with V)

CdV

dt�gK n

4�V�V K ��gNam

3h�V�V Na��g L�V�V L�� I inj=0

103

The functions n�(V), m

�(V), and h

�(V) determine whether gates serve to

activate channels (conventionally, open the channel with depolarization) or inactivate the channel (close the channel with depolarization). τm, τh, and τn are the time constants.

the m and n gatesopen with depolarization

the h gate closeswith depolarization

restingpotential

104

-10 0 10 20

0.5

Time re voltage step, ms

Vr = -60 mVV1 = -30 mV

n

n4

resting value

n (V )ﾰ 1

n (V )ﾰ 1

τν (ς 1)

n�(V)

n�(Vr)

The exponent 4 is chosen to make the rise of GK sigmoidal.

h�

m�

7

105 106

108

8

109 110

111