lecture 4: more ion channels and their functions na + channels: persistent k + channels: a current,...

Lecture 4: more ion channels and their functions

• Na+ channels: persistent

• K+ channels: A current, slowly inactivating current, Ca-dependent K currents IC, IAHP

• Ca2+ channels: low-threshold IT and high-threshold IL, non-ohmic currents

• Refs: Dayan and Abbott, Ch 6; Gerstner and Kistler, Sect.2.3, T F Weiss. Cellular Biophysics (MIT Press) Ch 7.

General formalism: ohmic channels

extj

j IIdtdV

C General equation

General formalism: ohmic channels

extj

j IIdtdV

C

)( jqj

pjjj VVhmgI jj

General equation

Currents have form

General formalism: ohmic channels

extj

j IIdtdV

C

)( jqj

pjjj VVhmgI jj

General equation

Currents have form

m: activating variables h: inactivating variables

General formalism: ohmic channels

extj

j IIdtdV

C

)( jqj

pjjj VVhmgI jj

General equation

Currents have form

m: activating variables h: inactivating variables

HH Na channel:

Persistent (noninactivating) Na channel

)( NaNaPNaPNaP VVmgI

Persistent (noninactivating) Na channel

)( NaNaPNaPNaP VVmgI No h!

Persistent (noninactivating) Na channel

)( NaNaPNaPNaP VVmgI No h!

Persistent (noninactivating) Na channel

)( NaNaPNaPNaP VVmgI No h!

Increases neuronal excitability

K channels: “A currents”

)(3KAAAA VVhmgI (same form as HH Na channel)

K channels: “A currents”

)(3KAAAA VVhmgI (same form as HH Na channel)

fast

slow-inactivating current

K channels: “A currents”

)(3KAAAA VVhmgI (same form as HH Na channel)

fast

slow-inactivating current

2 kinds of each

Effect of A currents

h ~ 10-20 ms

Effect of A currents

h ~ 10-20 ms

Opposite direction from Na current: hyperpolarizes membrane

Effect of A currents

h ~ 10-20 ms

Opposite direction from Na current: hyperpolarizes membraneSlows spike initiation: have to wait for IA to inactivate:

Effect of A currents

h ~ 10-20 ms

Opposite direction from Na current: hyperpolarizes membraneSlows spike initiation: have to wait for IA to inactivate:

Type I and Type II neurons

Type I: arbitrarilyslow rate possible(fx with A current)

Type II: minimumfiring rate >0 (fxStandard HH)

Ca2+ -dependent K conductances (1): IC)( KCCC VVmgI

Ca2+ -dependent K conductances (1): IC)( KCCC VVmgI (persistent)

Ca2+ -dependent K conductances (1): IC)( KCCC VVmgI

CCC mm

dtdm )1(

(persistent)

Ca2+ -dependent K conductances (1): IC)( KCCC VVmgI

CCC mm

dtdm )1(

24/24/25 e1.0e]Ca[105.2 VV

(persistent)

Ca2+ -dependent K conductances (1): IC)( KCCC VVmgI

CCC mm

dtdm )1(

24/24/25 e1.0e]Ca[105.2 VV

(persistent)

Activation is [Ca2+]-dependent

Ca2+ -dependent K conductances (1): IC)( KCCC VVmgI

CCC mm

dtdm )1(

24/24/25 e1.0e]Ca[105.2 VV

[Ca2+] = 0.1, 0,2, 0.5, 1.0, 2.0, 5.0 mol/l

(persistent)

Activation is [Ca2+]-dependent

Ca2+ -dependent K conductances (1): IC)( KCCC VVmgI

CCC mm

dtdm )1(

24/24/25 e1.0e]Ca[105.2 VV

[Ca2+] = 0.1, 0,2, 0.5, 1.0, 2.0, 5.0 mol/lContributes to repolarization after spikes

(persistent)

Activation is [Ca2+]-dependent

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI After-hyperpolarization current

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

After-hyperpolarization current

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

001.0)),1(()01.0],Ca[min( 2 Occ

After-hyperpolarization current

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

001.0)),1(()01.0],Ca[min( 2 Occ Slow, no voltage dependence!

After-hyperpolarization current

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

001.0)),1(()01.0],Ca[min( 2 Occ

Ca2+ enters (through other channels) during action potentials

Slow, no voltage dependence!

After-hyperpolarization current

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

001.0)),1(()01.0],Ca[min( 2 Occ

Ca2+ enters (through other channels) during action potentialsEach spike bigger

Slow, no voltage dependence!

After-hyperpolarization current

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

001.0)),1(()01.0],Ca[min( 2 Occ

Ca2+ enters (through other channels) during action potentialsEach spike bigger , bigger m

Slow, no voltage dependence!

After-hyperpolarization current

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

001.0)),1(()01.0],Ca[min( 2 Occ

Ca2+ enters (through other channels) during action potentialsEach spike bigger , bigger m slows down spiking

Slow, no voltage dependence!

After-hyperpolarization current

Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

001.0)),1(()01.0],Ca[min( 2 Occ

Ca2+ enters (through other channels) during action potentialsEach spike bigger , bigger m slows down spiking

Slow, no voltage dependence!

After-hyperpolarization current

Ca2+ currents (1): low-threshold IT

)(2CaTTTT VVhmgI

Ca2+ currents (1): low-threshold IT

)(2CaTTTT VVhmgI (ohmic approximation here, but see later)

Ca2+ currents (1): low-threshold IT

)(2CaTTTT VVhmgI (ohmic approximation here, but see later)

Ca2+ currents (1): low-threshold IT

)(2CaTTTT VVhmgI (ohmic approximation here, but see later)

Closed at rest because h nearly 0(channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0(channel is “not activated”)

Ca2+ currents (1): low-threshold IT

)(2CaTTTT VVhmgI (ohmic approximation here, but see later)

Closed at rest because h nearly 0(channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0(channel is “not activated”)

Consequences:

(1) “Post-inhibitory rebound”;fires “Ca spike” on releasefrom hyperpolarization

Ca2+ currents (1): low-threshold IT

)(2CaTTTT VVhmgI (ohmic approximation here, but see later)

Closed at rest because h nearly 0(channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0(channel is “not activated”)

Consequences:

(1) “Post-inhibitory rebound”;fires “Ca spike” on release from hyperpolarization

(2) Ca spikes can lead toNa spikes

Ca2+ currents (2): high-threshold IL

)(2 CaLLL VVmgI in ohmic approximation

Ca2+ currents (2): high-threshold IL

)(2 CaLLL VVmgI

Persistent:

in ohmic approximation

Ca2+ currents (2): high-threshold IL

)(2 CaLLL VVmgI

Persistent:

in ohmic approximation

Lets in some Ca2+ with each action potential

Ca2+ currents (2): high-threshold IL

)(2 CaLLL VVmgI

Persistent:

in ohmic approximation

Lets in some Ca2+ with each action potentialThis activates Ca-dependent K current

Ca2+ currents (2): high-threshold IL

)(2 CaLLL VVmgI

Persistent:

in ohmic approximation

Lets in some Ca2+ with each action potentialThis activates Ca-dependent K current

CaCaCa

Idt

d

]Ca[]Ca[ 22

Ca2+ dynamics:

Non-ohmic Ca currentsCurrent through membrane:

Non-ohmic Ca currents

xDJdiff

Current through membrane:

Diffusive part: = ion density

Non-ohmic Ca currents

xDJdiff

2

2lD

Current through membrane:

Diffusive part:

diffusion constant

= ion density

Non-ohmic Ca currents

xDJdiff

xV

ze

F

vJ drift

2

2lD

Current through membrane:

Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity

Non-ohmic Ca currents

xDJdiff

xV

ze

F

vJ drift

2

2lD

Current through membrane:

Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity

= mobility, F = force

Non-ohmic Ca currents

xDJdiff

xV

ze

F

vJ drift

2

2lD

Current through membrane:

Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity

= mobility, F = force

z = valence, e = proton charge,V = electrostatic potential

Non-ohmic Ca currents

xDJdiff

xV

ze

F

vJ drift

2

2lD

xV

zex

DJ

Current through membrane:

Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity

= mobility, F = force

z = valence, e = proton charge,V = electrostatic potential

Total current:

Non-ohmic Ca currents

xDJdiff

xV

ze

F

vJ drift

2

2lD

xV

zex

DJ

Current through membrane:

Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity

= mobility, F = force

z = valence, e = proton charge,V = electrostatic potential

Total current: Nernst-Planck equation

Nernst-Planck equation

xV

zex

TkJ B

Can also be written

Nernst-Planck equation

xV

zex

TkJ B

TkD B

Can also be written

using Einstein relation

Nernst-Planck equation

xV

zex

TkJ B

TkD B

xJ

~

Can also be written

using Einstein relation

or

Nernst-Planck equation

xV

zex

TkJ B

TkD B

xJ

~

log~ TkzeV B

Can also be written

using Einstein relation

or

where

Nernst-Planck equation

xV

zex

TkJ B

TkD B

xJ

~

log~ TkzeV B

Can also be written

using Einstein relation

or

where

is the electrochemical potential

Steady state: J = const

)/1( TkxV

zex

TkxV

zex

DJ BB

Nernst-Planck equation:

Steady state: J = const

)/1( TkxV

zex

TkxV

zex

DJ BB

:e )(xzeV

Nernst-Planck equation:

Use integrating factor

Steady state: J = const

)/1( TkxV

zex

TkxV

zex

DJ BB

:e )(xzeV

)()( e)(e xzeVB

xzeV xx

TkJ

Nernst-Planck equation:

Use integrating factor

Steady state: J = const

)/1( TkxV

zex

TkxV

zex

DJ BB

:e )(xzeV

)()( e)(e xzeVB

xzeV xx

TkJ

Nernst-Planck equation:

Use integrating factor

Integrate from x0 to x1:

Steady state: J = const

)/1( TkxV

zex

TkxV

zex

DJ BB

:e )(xzeV

)()( e)(e xzeVB

xzeV xx

TkJ

1

0

01

)(

)(0

)(1

e

e)(e)(x

x

xzeV

xzeVxzeV

B

dx

xxTkJ

Nernst-Planck equation:

Use integrating factor

Integrate from x0 to x1:

Goldman-Hodgkin-Katz equation: assume constant field in membrane

V = membrane potential, d = membrane thickness

dxdVxxV 0/)(

Goldman-Hodgkin-Katz equation: assume constant field in membrane

V = membrane potential, d = membrane thickness

can integrate denominatorx1 = 0, x2 = d

dxdVxxV 0/)(

Goldman-Hodgkin-Katz equation: assume constant field in membrane

)1e(e0

/ zeVd

dzeVx

zeVd

dx

V = membrane potential, d = membrane thickness

can integrate denominatorx1 = 0, x2 = d

dxdVxxV 0/)(

Goldman-Hodgkin-Katz equation: assume constant field in membrane

)1e(e0

/ zeVd

dzeVx

zeVd

dx

1ee

zeV

zeVinout

dzeV

J

V = membrane potential, d = membrane thickness

can integrate denominatorx1 = 0, x2 = d

Result:

dxdVxxV 0/)(

Goldman-Hodgkin-Katz equation: assume constant field in membrane

)1e(e0

/ zeVd

dzeVx

zeVd

dx

1ee

zeV

zeVinout

dzeV

J

V = membrane potential, d = membrane thickness

can integrate denominatorx1 = 0, x2 = d

Result:

vanishes at reversal potential, by definition

dxdVxxV 0/)(

Ohmic limit

rzeVoutin

ein

outBr ze

TkV

logUsing i.e.,

Ohmic limit

rzeVoutin

ein

outBr ze

TkV

log

1ee1 )(

zeV

VVzeout

r

d

zeVJ

Using i.e.,

Ohmic limit

rzeVoutin

ein

outBr ze

TkV

log

1ee1 )(

zeV

VVzeout

r

d

zeVJ

Using i.e.,

Now expand in V-Vr:

Ohmic limit

rzeVoutin

ein

outBr ze

TkV

log

1ee1 )(

zeV

VVzeout

r

d

zeVJ

)()(

1

)(

1])(1[1

22

VVTdkVzeVV

TdkVze

eVVze

dzeV

J

rinout

inout

B

rr

B

outr

zeVroutr

in

out

r

Using i.e.,

Now expand in V-Vr: