Membrane Models and the Hodgkin/Huxley Model of the …moehlis/APC591/Lecture2.pdf · Membrane...
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1 Membrane Models and the Hodgkin/Huxley Model of the Action Potential 1. Passive Membrane 2. Resting Potential 3. Dendrites/Axons as Cables 4. Properties of Action Potential 5. Hodgkin/Huxley Model Passive Membrane Model 2 / 1 cm F C µ ≈
Transcript of Membrane Models and the Hodgkin/Huxley Model of the …moehlis/APC591/Lecture2.pdf · Membrane...
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Membrane Models and the Hodgkin/HuxleyModel of the Action Potential
1. Passive Membrane2. Resting Potential3. Dendrites/Axons as Cables4. Properties of Action Potential5. Hodgkin/Huxley Model
Passive Membrane Model
2/1 cmFC µ≈
2
R
VVI
dt
dVC rm
injm )( −+=
rV
RIV injr +
injI
t
3
int
ext
int
extBKK K
K
zF
RT
K
K
q
TkVE
][
][ln
][
][ln ===
Nernst Potential
int
extKK K
K
z
mVVE
][
][log
58==
[K]int>[K]ext
resting potentialis negative
Goldman Hodgkin Katz Equation
extClintNaintK
intClextNaextK
ClPNaPKP
ClPNaPKPV
][][][
][][][log58
++++=
ClNaK PPP :: relative permeabilities
ClNaK PPP ,>>
ClKNa PPP ,>>KNaCl PPP ,>>
KV
ClV
NaV
4
Spheres and Cables
Frog muscle fiber (Fatt and Katz, 1951)
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lobster axon,Hodgkin andRushton, 1946
mm3.1≈λ
Membrane potential decay from point ofconstant current injection
),(),(1
2
2txi
x
txV
r mm
a=
∂∂
),(),(
),( txIt
Vc
r
VtxVtxi inj
mm
m
restmm −
∂∂+−=
),()),((),(),(
2
22 txIrVtxV
t
txV
x
txVinjmrestm
mm
m −−+∂
∂=∂
∂ τλ
mmm cr=τ
a
m
r
r=λ
units:
rm: ohm-cm
ra: ohm/cm
im: A/cm
cm: F/cm
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)/exp()( 0 λxVxV −=
experiment
theory
temporal distributionat different locations
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refractory period
Adrian and Lucas, 1912
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Accomodation
Valibo, 1964Xenopus laevis nerve fiber
Squid Giant Axon
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mVVK 77−=
mVVNa 50+=
mVVL 4.54−=
Space Clamp/Voltage Clamp
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Cole, 1968
Bernstein TheoryPermeability to K at rest
Membrane breaks down transiently
Na+ permeabilitychange associated
with peak
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cm
mcmo
VK
KV
VVKVV
1
)(
+=
−==
Voltage Clamp
voltagecommandVc =
12
commandvoltages
separating the currentsby altering Nernst
potential
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)( KKK VVgI −=
gating driving force
)( NaNaNa VVgI −=
)( LLL VVgI −=
Ohmic for each stateV: mV
g : m mho/cm2
Ι : µA/cm2
units:
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))(()1)(( nVnVdt
dnnn βα −−=
)/exp()()( 0 ntnnntn τ−−−= ∞∞
nnn βα
τ+
= 1
nn
nnβα
α+
=∞
),(),( 4 tVngtVg KK =
α,β: msec-1
g : m mho/cm2
n : [0,1]
units:τ: msec
maximum conductance (constant)
)4exp()exp()(1 tgttnndt
dnn nKnn βββ −≈⇒−≈⇒−≈⇒≈
4)()(0 tgttndt
dnn nKnn ααα ≈⇒≈⇒≈⇒≈
)(4KKK VVngg −=
))(()1)(( nVnVdt
dnnn βα −−=
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))(()1)(( mVmVdt
dmmm βα −−=
),(),(),( 3 tVhtVmgtVg NaNa =
))(()1)(( hVhVdt
dhhh βα −−=
α,β: msec-1
g : m mho/cm2
m,h : [0,1]units:
m fast
m,n activate with depolarization
h inactivates with depolarization
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nn
n
τα ∞=
nn
n
τβ ∞−= 1
α,β: msec-1units: V: mV
]10/)55(exp[1
)55(01.0)(
+−−+=
V
VVnα
]10/)40(exp[1
)40(1.0)(
+−−+=
V
VVmα
]20/)65(exp[07.0)( +−= VVhα ]10/)35(exp[1
1)(
+−+=
VVhβ
]80/)65(exp[125.0)( +−= VVnβ
]18/)65(exp[4)( +−= VVmβ
nn
n
τα ∞=
nn
n
τβ ∞−= 1
α,β: msec-1units: V: mV
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]10/)55(exp[1
)55(01.0)(
+−−+=
V
VVnα
]10/)40(exp[1
)40(1.0)(
+−−+=
V
VVmα
]20/)65(exp[07.0)( +−= VVhα ]10/)35(exp[1
1)(
+−+=
VVhβ
]80/)65(exp[125.0)( +−= VVnβ
]18/)65(exp[4)( +−= VVmβ
)()()( 43LLKKNaNa VVgVVngVVhmgI
dt
dVC −−−−−−=
))(()1)(( mVmVdt
dmmm βα −−=
))(()1)(( hVhVdt
dhhh βα −−=
))(()1)(( nVnVdt
dnnn βα −−=
Space Clamped Hodgkin/Huxley Model
2/120 cmmmhogNa =2/36 cmmmhogK =2/3.0 cmmmhogL =
mVVNa 50=
mVVK 77−=
mVVL 4.54−=2/1 cmFC µ=
2/: cmAI µmVV :
numerical integration of H/H equations
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)()()( 43LLKKKNa VVgVVngVVhmg
t
VCI −+−+−+
∂∂=
2
2
2 x
V
R
aI
∂∂= cm)(kyresistivitlarintracelluspecific
(cm)radius
Ω==
R
a
)()()(2
432
2
2 LLKKKNa VVgVVngVVhmgdt
dVC
dt
Vd
R
a −+−+−+=θ
)(),(assume txVtxV θ−=
(cm/msec)velocitygpropagatin=θ
integrateynumericall,guessθ
2
2
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2 1
t
V
x
V
∂∂=
∂∂
θ
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Cooley & Dodge
PDEODE
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