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3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching RECAP: GHK vs LINEAR I-V RELATIONSHIP 1 concentration flux electric potential flux concentration 1 1 2 2 GHK: linear: long channel short channel I * (V )= P F 2 RT · V · c i - c e e - VF RT 1 - e - VF RT I * (V )= P F 2 RT c e - c i ln c e /c i g · V - RT F ln c e c i V 0

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Transcript of RECAP: GHK vs LINEAR I-V RELATIONSHIPlmate/teaching/3G2_2008/02_membpot...RECAP: GHK vs LINEAR I-V...

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RECAP: GHK vs LINEAR I-V RELATIONSHIP

1

concentration

flux

electricpotential

flux

concentration

1

1

2

2

GHK:

linear:long channel →

short channel → I!(V ) = PF 2

RT· V · ci ! ce e"

V FRT

1! e"V FRT

I!(V ) = PF 2

RT

ce ! ci

ln ce/ci! "# \$g

·%V ! RT

Fln

&ce

ci

'(

! "# \$V0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RECAP: GHK vs LINEAR I-V RELATIONSHIP

1

concentration

flux

electricpotential

flux

concentration

1

1

2

2

GHK:

linear:long channel →

short channel →

I!

V (mV)

GHK

linear

I!(V ) = PF 2

RT· V · ci ! ce e"

V FRT

1! e"V FRT

I!(V ) = PF 2

RT

ce ! ci

ln ce/ci! "# \$g

·%V ! RT

Fln

&ce

ci

'(

! "# \$V0

V0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RECAP: GHK vs LINEAR I-V RELATIONSHIP

1

same current at• ← Nernst potential• ← Nernst-Planck eq

V = V0

V = 0

concentration

flux

electricpotential

flux

concentration

1

1

2

2

GHK:

linear:long channel →

short channel →

I!

V (mV)

GHK

linear

I!(V ) = PF 2

RT· V · ci ! ce e"

V FRT

1! e"V FRT

I!(V ) = PF 2

RT

ce ! ci

ln ce/ci! "# \$g

·%V ! RT

Fln

&ce

ci

'(

! "# \$V0

V0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RECAP: GHK vs LINEAR I-V RELATIONSHIP

1

same current at• ← Nernst potential• ← Nernst-Planck eq

V = V0

V = 0

concentration

flux

electricpotential

flux

concentration

1

1

2

2

GHK:

linear:long channel →

short channel →

I!

V (mV)

GHK

linear

I!(V ) = PF 2

RT· V · ci ! ce e"

V FRT

1! e"V FRT

I!(V ) = PF 2

RT

ce ! ci

ln ce/ci! "# \$g

·%V ! RT

Fln

&ce

ci

'(

! "# \$V0

V0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RECAP: GHK vs LINEAR I-V RELATIONSHIP

1

same current at• ← Nernst potential• ← Nernst-Planck eq

V = V0

V = 0

concentration

flux

electricpotential

flux

concentration

1

1

2

2

GHK:

linear:long channel →

short channel →

I!

V (mV)

GHK

linear

I!(V ) = PF 2

RT· V · ci ! ce e"

V FRT

1! e"V FRT

I!(V ) = PF 2

RT

ce ! ci

ln ce/ci! "# \$g

·%V ! RT

Fln

&ce

ci

'(

! "# \$V0

V0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RECAP: GHK vs LINEAR I-V RELATIONSHIP

1

same current at• ← Nernst potential• ← Nernst-Planck eq

V = V0

V = 0

current

membranevoltage

GHK:

linear:long channel →

short channel →

I!

V (mV)

GHK

linear

I!(V ) = PF 2

RT· V · ci ! ce e"

V FRT

1! e"V FRT

I!(V ) = PF 2

RT

ce ! ci

ln ce/ci! "# \$g

·%V ! RT

Fln

&ce

ci

'(

! "# \$V0

V0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RECAP: GHK vs LINEAR I-V RELATIONSHIP

1

same current at• ← Nernst potential• ← Nernst-Planck eq

V = V0

V = 0

current

membranevoltage

separation oftime-scales

GHK:

linear:long channel →

short channel →

I!

V (mV)

GHK

linear

I!(V ) = PF 2

RT· V · ci ! ce e"

V FRT

1! e"V FRT

I!(V ) = PF 2

RT

ce ! ci

ln ce/ci! "# \$g

·%V ! RT

Fln

&ce

ci

'(

! "# \$V0

V0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

V

membranew/ channels

I1c1ec1

i

φ∗inside outside

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

Im(V ) =!

i

Ii(V )V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

Im(V ) =!

i

Ii(V )

Im(Vrest) = 0V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

Im(V ) =!

i

Ii(V )

Im(Vrest) = 0V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

none of the individual currentsare necessarily zero!

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

Im(V ) =!

i

Ii(V )

Im(Vrest) = 0

GHK: Vrest = !RT

Fln

!"j|zj=!1 Pj cj

e +"

j|zj=+1 Pj cji

"j|zj=!1 Pj cj

i +"

j|zj=+1 Pj cje

#

= !RT

Fln

!PNa+

\$Na+

%i+ PK+ [K+]i + PCl!

\$Cl!

%e

PNa+

\$Na+

%e+ PK+ [K+]e + PCl!

\$Cl!

%i

#

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

none of the individual currentsare necessarily zero!

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

Im(V ) =!

i

Ii(V )

Im(Vrest) = 0

linear:

GHK: Vrest = !RT

Fln

!"j|zj=!1 Pj cj

e +"

j|zj=+1 Pj cji

"j|zj=!1 Pj cj

i +"

j|zj=+1 Pj cje

#

= !RT

Fln

!PNa+

\$Na+

%i+ PK+ [K+]i + PCl!

\$Cl!

%e

PNa+

\$Na+

%e+ PK+ [K+]e + PCl!

\$Cl!

%i

#

Vrest =!

j gj Vj!j gj

=gNa+ VNa+ + gK+ VK+ + gCl! VCl!

gNa+ + gK+ + gCl!

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

none of the individual currentsare necessarily zero!

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

Im(V ) =!

i

Ii(V )

Im(Vrest) = 0

linear:

GHK: Vrest = !RT

Fln

!"j|zj=!1 Pj cj

e +"

j|zj=+1 Pj cji

"j|zj=!1 Pj cj

i +"

j|zj=+1 Pj cje

#

= !RT

Fln

!PNa+

\$Na+

%i+ PK+ [K+]i + PCl!

\$Cl!

%e

PNa+

\$Na+

%e+ PK+ [K+]e + PCl!

\$Cl!

%i

#

Vrest =!

j gj Vj!j gj

=gNa+ VNa+ + gK+ VK+ + gCl! VCl!

gNa+ + gK+ + gCl!

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗

depends on channel model

inside outside

none of the individual currentsare necessarily zero!

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

RESTING MEMBRANE POTENTIAL

2

when all channel currents cancel

current

membranevoltage

current

1

2

Im(V ) =!

i

Ii(V )

Im(Vrest) = 0V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗

depends on channel model

inside outside

none of the individual currentsare necessarily zero!

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE MEMBRANE AS A CAPACITOR

3

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside current

membranevoltage

current

1

2

+++

---

Cm

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE MEMBRANE AS A CAPACITOR

3

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside current

membranevoltage

current

1

2

+++

---

Cm

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE MEMBRANE AS A CAPACITOR

3

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside current

membranevoltage

current

1

2

+++

---

opposite charges accumulatedon opposing sides of an insulator (lipid bilayer)

membrane acts as a capacitor

Cm

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE MEMBRANE AS A CAPACITOR

3

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside current

membranevoltage

current

1

2

+++

---

opposite charges accumulatedon opposing sides of an insulator (lipid bilayer)

membrane acts as a capacitor

Cm

Cm V = Q

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE MEMBRANE AS A CAPACITOR

3

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside current

membranevoltage

current

1

2

+++

---

opposite charges accumulatedon opposing sides of an insulator (lipid bilayer)

membrane acts as a capacitor

Cm

Cm V = Q

Cmd

dtV (t) =

d

dtQ(t) = IC(t)

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm+++

---

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside+++

---

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside+++

---

I1

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside+++

---

I2I1

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside

IC

+++

---

I2I1

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside

Kirchoff’s 1st law: currents flowing in and out of a nodemust be equal

IC

+++

---

I2I1

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside

Im = I1 + I2

Kirchoff’s 1st law: currents flowing in and out of a nodemust be equal

IC

+++

---

I2I1

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside

Im = I1 + I2

Kirchoff’s 1st law: currents flowing in and out of a nodemust be equal

IC

+++

---

IC(t) + Im(t) = 0

I2I1

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside

Im = I1 + I2

Kirchoff’s 1st law: currents flowing in and out of a nodemust be equal

IC

current balance eq:

+++

---

IC(t) + Im(t) = 0

Cmd

dtV (t) = !Im(t)

I2I1

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE EQUIVALENT CIRCUIT MODEL

4

current

membranevoltage

current

1

2

V

membranew/ channels

V

I1

I2c2ec2

i

c1ec1

i

φ∗inside outside

Cm

V

Cm

inside

outside

Im = I1 + I2

Kirchoff’s 1st law: currents flowing in and out of a nodemust be equal

IC

current balance eq:

+++

---

IC(t) + Im(t) = 0

Cmd

dtV (t) = !Im(t)

I2I1

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE CELL AS AN RC CIRCUIT

5

current

membranevoltage

current

1

2

V

Cm

inside

outside

IC INa+ IK+ ICl!

Im = INa+ + IK+ + ICl!

gNa+ gK+ gCl!

VCl!VK+VNa+

current balance eq: Cmd

dtV (t) = !INa+(t)! IK+(t)! ICl!(t)

!INa+(t) = gNa+ · (VNa+ ! V (t))!IK+(t) = gK+ · (VK+ ! V (t))!ICl!(t) = gCl! · (VCl! ! V (t))! "# \$

driving force

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE CELL AS AN RC CIRCUIT

5

current

membranevoltage

current

1

2

V

Cm

inside

outside

IC INa+ IK+ ICl!

Im = INa+ + IK+ + ICl!

gNa+ gK+ gCl!

VCl!VK+VNa+

Na+ +55 0.01

K+ -75 0.20

Cl- -69 0.05

1.0

VX

(mV)gX!mScm2

"

Cm!µFcm2

"

current balance eq: Cmd

dtV (t) = !INa+(t)! IK+(t)! ICl!(t)

!INa+(t) = gNa+ · (VNa+ ! V (t))!IK+(t) = gK+ · (VK+ ! V (t))!ICl!(t) = gCl! · (VCl! ! V (t))! "# \$

driving force

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE CELL AS AN RC CIRCUIT

5

current

membranevoltage

current

1

2

V

Cm

inside

outside

IC INa+ IK+ ICl!

Im = INa+ + IK+ + ICl!

gNa+ gK+ gCl!

VCl!VK+VNa+

Na+ +55 0.01

K+ -75 0.20

Cl- -69 0.05

1.0

VX

(mV)gX!mScm2

"

Cm!µFcm2

"

current balance eq: Cmd

dtV (t) = !INa+(t)! IK+(t)! ICl!(t)

!INa+(t) = gNa+ · (VNa+ ! V (t))!IK+(t) = gK+ · (VK+ ! V (t))!ICl!(t) = gCl! · (VCl! ! V (t))! "# \$

driving force

mScm2

µAcm2

mV

µAcm2

µFcm2

1ms

mV

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE CELL AS AN RC CIRCUIT

6

V

Cm

inside

outside

gNa+ gK+ gCl!

VCl!VK+VNa+

Vrest =gNa+ VNa+ + gK+ VK+ + gCl! VCl!

gNa+ + gK+ + gCl!

Rm =1

gNa+ + gK+ + gCl!

Cmd

dtV (t) = !INa+(t)! IK+(t)! ICl!(t)

= gNa+ (VNa+ ! V (t)) + gK+ (VK+ ! V (t)) + gCl! (VCl! ! V (t))

=1

Rm(Vrest ! V (t))

membrane resistance

resting potential

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE CELL AS AN RC CIRCUIT

6

V

Cm

inside

outside

Vrest =gNa+ VNa+ + gK+ VK+ + gCl! VCl!

gNa+ + gK+ + gCl!

Rm =1

gNa+ + gK+ + gCl!

Vrest

1/Rm

Cmd

dtV (t) = !INa+(t)! IK+(t)! ICl!(t)

= gNa+ (VNa+ ! V (t)) + gK+ (VK+ ! V (t)) + gCl! (VCl! ! V (t))

=1

Rm(Vrest ! V (t))

membrane resistance

resting potential

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE CELL AS AN RC CIRCUIT

6

V

Cm

inside

outside

Vrest =gNa+ VNa+ + gK+ VK+ + gCl! VCl!

gNa+ + gK+ + gCl!

Rm =1

gNa+ + gK+ + gCl!

Vrest

1/Rm

Cmd

dtV (t) = !INa+(t)! IK+(t)! ICl!(t)

= gNa+ (VNa+ ! V (t)) + gK+ (VK+ ! V (t)) + gCl! (VCl! ! V (t))

=1

Rm(Vrest ! V (t))

K+ Na+

-75 mV +55 mVVrest

gNa+gK+

membrane resistance

resting potential

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE CELL AS AN RC CIRCUIT

7

V

Cm

inside

outside

Vrest

1/Rm

K+ Na+

-75 mV +55 mVVrest

gNa+gK+

d

dtV (t) =

Vrest ! V (t)!m

, !m = Cm · Rm

membranetime const

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE CELL AS AN RC CIRCUIT

7

V

Cm

inside

outside

Vrest

1/Rm

K+ Na+

-75 mV +55 mVVrest

gNa+gK+

V (t) = Vrest ! (Vrest ! V0) e! t

!m

V (0) = V0

d

dtV (t) =

Vrest ! V (t)!m

, !m = Cm · Rm

membranetime const

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

THE CELL AS AN RC CIRCUIT

7

V

Cm

inside

outside

Vrest

1/Rm

V

Vrest

V0

t!m

K+ Na+

-75 mV +55 mVVrest

gNa+gK+

V (t) = Vrest ! (Vrest ! V0) e! t

!m

V (0) = V0

d

dtV (t) =

Vrest ! V (t)!m

, !m = Cm · Rm

membranetime const

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

THE CELL AS AN RC CIRCUIT

7

V

Cm

inside

outside

Vrest

1/Rm

V

Vrest

V0

t

1e

(Vrest ! V0)

!m

K+ Na+

-75 mV +55 mVVrest

gNa+gK+

V (t) = Vrest ! (Vrest ! V0) e! t

!m

V (0) = V0

d

dtV (t) =

Vrest ! V (t)!m

, !m = Cm · Rm

membranetime const

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

THE CELL AS AN RC CIRCUIT

7

V

Cm

inside

outside

Vrest

1/Rm

V

Vrest

V0

t

1e

(Vrest ! V0)

!m

K+ Na+

-75 mV +55 mVVrest

gNa+gK+

•Na+ and K+ conductances setthe resting membrane potential

•membrane potential tends towardsthe resting membrane potentialexponentially

•speed of convergence depends onthe overall membrane conductance

V (t) = Vrest ! (Vrest ! V0) e! t

!m

V (0) = V0

d

dtV (t) =

Vrest ! V (t)!m

, !m = Cm · Rm

membranetime const

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE ACTION POTENTIAL

8

V(mV)

2 ms

Hodgkin & Huxley, 1939

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE ACTION POTENTIAL

8

restingmembranepotential

depo

lari

sed

hype

rpol

aris

ed

V(mV)

2 ms

membranepotentialregimes:

Hodgkin & Huxley, 1939

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE ACTION POTENTIAL

8

restingmembranepotential

depo

lari

sed

hype

rpol

aris

ed

V(mV)

2 ms

membranepotentialregimes:

parts of the action potential:

Hodgkin & Huxley, 1939

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE ACTION POTENTIAL

8

restingmembranepotential

depo

lari

sed

hype

rpol

aris

ed

V(mV)

2 ms

membranepotentialregimes:

parts of the action potential:

depolarisation(upstroke)

Hodgkin & Huxley, 1939

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE ACTION POTENTIAL

8

restingmembranepotential

depo

lari

sed

hype

rpol

aris

ed

V(mV)

2 ms

membranepotentialregimes:

parts of the action potential:

depolarisation(upstroke)

repolarisation(downstroke)

Hodgkin & Huxley, 1939

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE ACTION POTENTIAL

8

restingmembranepotential

depo

lari

sed

hype

rpol

aris

ed

V(mV)

2 ms

membranepotentialregimes:

parts of the action potential:

depolarisation(upstroke)

repolarisation(downstroke)

hyperpolarisation

Hodgkin & Huxley, 1939

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE ACTION POTENTIAL

8

restingmembranepotential

depo

lari

sed

hype

rpol

aris

ed

V(mV)

2 ms

membranepotentialregimes:

parts of the action potential:

depolarisation(upstroke)

repolarisation(downstroke)

hyperpolarisation

Hodgkin & Huxley, 1939

synonyms:•action potential•spike•firing

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

HODGKIN & HUXLEY

9

Alan L. Hodgkin & Andrew F. Huxley:Nobel prize in Physiology and Medicine, 1963

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

shape

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

shape

def: Vrest = 0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

def: Vrest = 0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

def: Vrest = 0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

def: Vrest = 0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

def: Vrest = 0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

def: Vrest = 0

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

refractory period

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

refractory period

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

refractory period

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

refractory period

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

refractory period

•relative

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

refractory period

•relative

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

refractory period

•relative

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

PROPERTIES OF THE ACTION POTENTIAL

10

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 200

5

10

15

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

!20

0

20

40

60

80

100

120

V [m

V]

0 5 10 15 20 25 30 35 40 45 500

100

200

I [µ

A/cm

2 ]

t [ms]

shape all-or-none

refractory period

•relative

•absolute

def: Vrest = 0 threshold

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE HODGKIN-HUXLEY MODEL

11

V

Cm

inside

outside

INa+ IK+ ICl!

gNa+ gK+ gCl!

VCl!VK+VNa+

current balance eq: Cmd

dtV (t) = !INa+(t)! IK+(t)! IL(t) + Iext(t)

conductances

membranevoltage

driving forces

!INa+(t) = gNa+(t) · (VNa+ ! V (t))!IK+(t) = gK+(t) · (VK+ ! V (t))!IL(t) = gL · (VCl! ! V (t))

-75 mV +55 mV

gNa+gK+

V

K+ Na+

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE HODGKIN-HUXLEY MODEL

11

V

Cm

inside

outside

INa+ IK+ ICl!

gNa+ gK+ gCl!

VCl!VK+VNa+

current balance eq: Cmd

dtV (t) = !INa+(t)! IK+(t)! IL(t) + Iext(t)

conductances

membranevoltage

driving forces

!INa+(t) = gNa+(t) · (VNa+ ! V (t))!IK+(t) = gK+(t) · (VK+ ! V (t))!IL(t) = gL · (VCl! ! V (t))

-75 mV +55 mV

gNa+gK+

V

K+ Na+

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE HODGKIN-HUXLEY MODEL

11

V

Cm

inside

outside

INa+ IK+ ICl!

gNa+ gK+ gCl!

VCl!VK+VNa+

current balance eq: Cmd

dtV (t) = !INa+(t)! IK+(t)! IL(t) + Iext(t)

conductances

membranevoltage

driving forces

!INa+(t) = gNa+(t) · (VNa+ ! V (t))!IK+(t) = gK+(t) · (VK+ ! V (t))!IL(t) = gL · (VCl! ! V (t))

-75 mV +55 mV

gNa+gK+

V

K+ Na+

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE HODGKIN-HUXLEY MODEL

11

V

Cm

inside

outside

INa+ IK+ ICl!

gNa+ gK+ gCl!

VCl!VK+VNa+

current balance eq: Cmd

dtV (t) = !INa+(t)! IK+(t)! IL(t) + Iext(t)

conductances

membranevoltage

driving forces

!INa+(t) = gNa+(t) · (VNa+ ! V (t))!IK+(t) = gK+(t) · (VK+ ! V (t))!IL(t) = gL · (VCl! ! V (t))

-75 mV +55 mV

gNa+gK+

V

K+ Na+

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

12

current no currentno current

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

12

reaction schemefor channel

current no currentno current

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

12

reaction schemefor channel

current no currentno current

g(S00)g(S01)g(S10)g(S11)g(S20)g(S21)

channelconductance

depends on state

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

12

reaction schemefor channel

current no currentno current

!(V )"(V )#(V )\$(V )

g(S00)g(S01)g(S10)g(S11)g(S20)g(S21)

rate constantsdepend on

membrane potential

channelconductance

depends on state

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

13

• each channel is composed of gates• a channel is only open if all its gates are open• gates open independently

the simplified view

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

13

• each channel is composed of gates• a channel is only open if all its gates are open• gates open independently

the simplified view

g(t) = g!

x

x(t)

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

13

• reaction scheme for a gate:

• each channel is composed of gates• a channel is only open if all its gates are open• gates open independently

the simplified view

C O!x(V )

!x(V )

g(t) = g!

x

x(t)

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

13

• reaction scheme for a gate:

• each channel is composed of gates• a channel is only open if all its gates are open• gates open independently

the simplified view

C O!x(V )

!x(V )

d

dtx(t) = !x(V (t)) (1! x(t))! "x(V (t)) x(t)

g(t) = g!

x

x(t)

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

13

• reaction scheme for a gate:

• each channel is composed of gates• a channel is only open if all its gates are open• gates open independently

the simplified view

C O!x(V )

!x(V )

d

dtx(t) = !x(V (t)) (1! x(t))! "x(V (t)) x(t)

g(t) = g!

x

x(t)

d

dtx(t) =

x!(V (t))! x (t)!x(V (t))

x!(V ) ="x(V )

"x(V ) + #x(V )

!x(V ) =1

"x(V ) + #x(V )

time constant:

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

13

• reaction scheme for a gate:

• each channel is composed of gates• a channel is only open if all its gates are open• gates open independently

the simplified view

C O!x(V )

!x(V )

d

dtx(t) = !x(V (t)) (1! x(t))! "x(V (t)) x(t)

g(t) = g!

x

x(t)

if voltage is kept fixed

0 t

x0

x

x!(V )

!x(V )

1e

(x!(V )! x0)

d

dtx(t) =

x!(V (t))! x (t)!x(V (t))

x!(V ) ="x(V )

"x(V ) + #x(V )

!x(V ) =1

"x(V ) + #x(V )

time constant:

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

14

in the Hodgkin-Huxley model

gNa+(t) = gNa+ m3(t) h(t)

gK+(t) = gK+ n4(t)

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

14

in the Hodgkin-Huxley model

gNa+(t) = gNa+ m3(t) h(t)

gK+(t) = gK+ n4(t)

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

14

in the Hodgkin-Huxley model

d

dtx(t) = !x(V (t)) (1! x(t))! "x(V (t)) x(t)

d

dtx(t) =

x!(V (t))! x (t)#x(V (t))

gNa+(t) = gNa+ m3(t) h(t)

gK+(t) = gK+ n4(t)

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

14

in the Hodgkin-Huxley model

d

dtx(t) = !x(V (t)) (1! x(t))! "x(V (t)) x(t)

d

dtx(t) =

x!(V (t))! x (t)#x(V (t))

!50 0 50 1000

0\$5

1

1\$5

2

2\$5

3

!' )1

*m,-

. )m.-!50 0 50 1000

0\$5

1

1\$5

2

2\$5

3

" ' )1*m

,-

. )m.-

gNa+(t) = gNa+ m3(t) h(t)

gK+(t) = gK+ n4(t)

!m!m

!h

!h!n

!n

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

14

in the Hodgkin-Huxley model

d

dtx(t) = !x(V (t)) (1! x(t))! "x(V (t)) x(t)

d

dtx(t) =

x!(V (t))! x (t)#x(V (t))

!50 0 50 1000

0\$5

1

1\$5

2

2\$5

3

!' )1

*m,-

. )m.-!50 0 50 1000

0\$5

1

1\$5

2

2\$5

3

" ' )1*m

,-

. )m.-

!50 0 50 1000

0.2

0.4

0.6

0.8

1

x ! [1

]

V [mV]!50 0 50 1000

2

4

6

8

10

" x [ms]

V [mV]

gNa+(t) = gNa+ m3(t) h(t)

gK+(t) = gK+ n4(t)

!m!m

!m

m!

h! !h

!h

!h!n

!n

!n

n!

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

VOLTAGE-GATED ION CHANNELS

14

in the Hodgkin-Huxley model

d

dtx(t) = !x(V (t)) (1! x(t))! "x(V (t)) x(t)

d

dtx(t) =

x!(V (t))! x (t)#x(V (t))

!50 0 50 1000

0\$5

1

1\$5

2

2\$5

3

!' )1

*m,-

. )m.-!50 0 50 1000

0\$5

1

1\$5

2

2\$5

3

" ' )1*m

,-

. )m.-

!50 0 50 1000

0.2

0.4

0.6

0.8

1

x ! [1

]

V [mV]!50 0 50 1000

2

4

6

8

10

" x [ms]

V [mV]

gNa+(t) = gNa+ m3(t) h(t)

gK+(t) = gK+ n4(t)

!m!m

!m

m!

h! !h

!h

!h!n

!n

!n

n!

• activating gate: opens with depolarisation

• inactivating gate: opens with hyperpolarisation

• de(in)activation:(in)activating gate closes

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

THE HODGKIN HUXLEY MODEL

15

V

Cm

inside

outside

INa+ IK+ ICl!

gNa+ gK+ gCl!

VCl!VK+VNa+

current balance eq:

conductances

membranevoltage

driving forces

gating eqs:

-75 mV +55 mV

K+ Na+

gNa+gK+

V

Cmd

dtV (t) = gNa+ m3(t) h(t) (VNa+ ! V (t)) +

+gK+ n4(t) (VK+ ! V (t)) + gL (VL ! V (t)) + Iext(t)

d

dtm(t) =

m!(V (t))!m(t)!m(V (t))

d

dth(t) =

h!(V (t))! h(t)!h(V (t))

d

dtn(t) =

n!(V (t))! n(t)!n(V (t))

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

50

100

V [m

V]

0

0.5

1

gatin

g va

rs [1

]

0

20

g X [mS/

cm2 ]

!1000

0

1000

I X [µA/

cm2 ]

0 5 10 15 200

10

20

I ext [µ

A/cm

2 ]

t [ms]

THE HODGKIN HUXLEY MODEL

16

in action

mh n

gK+

IL

gL

gNa+

IK+

INa+

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

50

100

V [m

V]

0

0.5

1

gatin

g va

rs [1

]

0

20

g X [mS/

cm2 ]

!1000

0

1000

I X [µA/

cm2 ]

0 5 10 15 200

10

20

I ext [µ

A/cm

2 ]

t [ms]

THE HODGKIN HUXLEY MODEL

16

in action

I. depolarisation:• injected current depolarises V• depolarised V activates m• activated m further depolarises V

mh n

gK+

IL

gL

gNa+

IK+

INa+

I.

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

50

100

V [m

V]

0

0.5

1

gatin

g va

rs [1

]

0

20

g X [mS/

cm2 ]

!1000

0

1000

I X [µA/

cm2 ]

0 5 10 15 200

10

20

I ext [µ

A/cm

2 ]

t [ms]

THE HODGKIN HUXLEY MODEL

16

in action

I. depolarisation:• injected current depolarises V• depolarised V activates m• activated m further depolarises V

fastpositivefeedback

mh n

gK+

IL

gL

gNa+

IK+

INa+

I.

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

50

100

V [m

V]

0

0.5

1

gatin

g va

rs [1

]

0

20

g X [mS/

cm2 ]

!1000

0

1000

I X [µA/

cm2 ]

0 5 10 15 200

10

20

I ext [µ

A/cm

2 ]

t [ms]

THE HODGKIN HUXLEY MODEL

16

in action

I. depolarisation:• injected current depolarises V• depolarised V activates m• activated m further depolarises V

II. repolarisation:• depolarised V

activates n and inactivates h• activated n and inactivated h

repolarises V• repolarising V deactivates m

fastpositivefeedback

mh n

gK+

IL

gL

gNa+

IK+

INa+

II.I.

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

50

100

V [m

V]

0

0.5

1

gatin

g va

rs [1

]

0

20

g X [mS/

cm2 ]

!1000

0

1000

I X [µA/

cm2 ]

0 5 10 15 200

10

20

I ext [µ

A/cm

2 ]

t [ms]

THE HODGKIN HUXLEY MODEL

16

in action

I. depolarisation:• injected current depolarises V• depolarised V activates m• activated m further depolarises V

II. repolarisation:• depolarised V

activates n and inactivates h• activated n and inactivated h

repolarises V• repolarising V deactivates m

fastpositivefeedback

slownegativefeedback

mh n

gK+

IL

gL

gNa+

IK+

INa+

II.I.

3G2: Mathematical electrophysiology - Membrane potential, 6 Feb 2008 http://www.gatsby.ucl.ac.uk/~lmate/teaching

0

50

100

V [m

V]

0

0.5

1

gatin

g va

rs [1

]

0

20

g X [mS/

cm2 ]

!1000

0

1000

I X [µA/

cm2 ]

0 5 10 15 200

10

20

I ext [µ

A/cm

2 ]

t [ms]

THE HODGKIN HUXLEY MODEL

16

in action

I. depolarisation:• injected current depolarises V• depolarised V activates m• activated m further depolarises V

II. repolarisation:• depolarised V

activates n and inactivates h• activated n and inactivated h

repolarises V• repolarising V deactivates m

III. hyperpolarisation:• n deactivates and h deinactivates• V returns to resting membrane potential

fastpositivefeedback

slownegativefeedback

mh n

gK+

IL

gL

gNa+

IK+

INa+

III.II.I.