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


extj

j IIdtdV

C

)( jqj

pjjj VVhmgI jj

General equation

Currents have form


extj

j IIdtdV

C

)( jqj

pjjj VVhmgI jj

General equation

Currents have form

m: activating variables h: inactivating variables


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


)( NaNaPNaPNaP VVmgI No h!


)( NaNaPNaPNaP VVmgI No h!

Increases neuronal excitability

K channels: “A currents”

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



fast

slow-inactivating current



fast

slow-inactivating current

2 kinds of each

Effect of A currents

h ~ 10-20 ms


h ~ 10-20 ms

Opposite direction from Na current: hyperpolarizes membrane


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)


CCC mm

dtdm )1(

(persistent)


CCC mm

dtdm )1(

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

(persistent)


CCC mm

dtdm )1(

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

(persistent)

Activation is [Ca2+]-dependent


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)



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)


Ca2+ -dependent K conductances (2): IAHP

)( KAHPAHPAHP VVmgI After-hyperpolarization current


)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

After-hyperpolarization current


)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

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



)( KAHPAHPAHP VVmgI

AHPAHPAHP mm

dtdm )1(

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



)( 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!



)( 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




)( 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




)( 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



Ca2+ currents (1): low-threshold IT

)(2CaTTTT VVhmgI


)(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




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


)(2 CaLLL VVmgI

Persistent:

in ohmic approximation


)(2 CaLLL VVmgI

Persistent:


Lets in some Ca2+ with each action potential


)(2 CaLLL VVmgI

Persistent:


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


)(2 CaLLL VVmgI

Persistent:


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


xDJdiff

2

2lD


Diffusive part:

diffusion constant

= ion density


xDJdiff

xV

ze

F

vJ drift

2

2lD


Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity


xDJdiff

xV

ze

F

vJ drift

2

2lD


Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity

= mobility, F = force


xDJdiff

xV

ze

F

vJ drift

2

2lD


Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity


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


xDJdiff

xV

ze

F

vJ drift

2

2lD

xV

zex

DJ


Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity



Total current:


xDJdiff

xV

ze

F

vJ drift

2

2lD

xV

zex

DJ


Diffusive part:

diffusion constant

Drift in field:

= ion density

v = velocity



Total current: Nernst-Planck equation

Nernst-Planck equation

xV

zex

TkJ B

Can also be written


xV

zex

TkJ B

TkD B

Can also be written

using Einstein relation


xV

zex

TkJ B

TkD B

xJ

~

Can also be written


or


xV

zex

TkJ B

TkD B

xJ

~

log~ TkzeV B

Can also be written


or

where


xV

zex

TkJ B

TkD B

xJ

~

log~ TkzeV B

Can also be written


or

where

is the electrochemical potential

Steady state: J = const

)/1( TkxV

zex

TkxV

zex

DJ BB

Nernst-Planck equation:


)/1( TkxV

zex

TkxV

zex

DJ BB

:e )(xzeV


Use integrating factor


)/1( TkxV

zex

TkxV

zex

DJ BB

:e )(xzeV

)()( e)(e xzeVB

xzeV xx

TkJ




)/1( TkxV

zex

TkxV

zex

DJ BB

:e )(xzeV

)()( e)(e xzeVB

xzeV xx

TkJ



Integrate from x0 to x1:


)/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



Integrate from x0 to x1:

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

V = membrane potential, d = membrane thickness

dxdVxxV 0/)(



can integrate denominatorx1 = 0, x2 = d

dxdVxxV 0/)(


)1e(e0

/ zeVd

dzeVx

zeVd

dx



dxdVxxV 0/)(


)1e(e0

/ zeVd

dzeVx

zeVd

dx

1ee

zeV

zeVinout

dzeV

J



Result:

dxdVxxV 0/)(


)1e(e0

/ zeVd

dzeVx

zeVd

dx

1ee

zeV

zeVinout

dzeV

J



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:

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

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