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Hodgkin-Huxley Model Formulation
Hodgkin and Huxley developed an empirical kinetic
description of the excitable membrane, simple enough to make
practical calculations of electrical responses, yet sufficiently
good as to predict correctly the major features of excitability
such as action potential shape, conduction velocity, and
threshold dynamics. They started with the cable equation and
the parallel-conductance model of the axon membrane
2
2,
4
m mm ion app
i
dV VdI I
dt R xc
where d (cm) is the diameter of the axon, Ri ( cm) is the
volume resistivity of the intracellular medium, Iapp (A/cm2 ) is
the applied current per unit area, and Iion is defined as
( ) ( ) ( ).ion Na m Na K m K L m LI g V E g V E g V E{ { {INa IK IL
Leakage current due to small, relatively
voltage-independent background
conductance of undetermined ionic basis.
Key questions to be resolved from
experiments on the squid giant axon:
(1) Can Na+ and K+ currents be separated ?
(2) Does the relation between ionic current and membrane
potential at constant permeability obey Ohm’s law* ?
*Side note: Charge-density-wave (CDW) materials are "super-dielectrics" which do not obey Ohm's Law -- the current is
not proportional to the voltage -- because at very small voltages the CDW can become depinned and slide through the
crystal. When the CDW slides, dc currents produce ac voltages. At the depinning voltage, the materials exhibit dramatic
"electromechanical" (the elastic constants change) and "electro-optic" (the infrared properties change) effects.
That is, for fixed permeabilities is it correct to represent the ionic
conductances as
,( )
NaNa
m Na
Ig
V E ?
( )
KK
m K
Ig
V E
(3) Contingent upon question (2), can gNa and gK be found experimentally ?
No feedback
(constant current)
Vm
i
Vm
i
Intermediate
feedback
Vm
i
Strong feedback
(voltage clamp)
Voltage-Clamp experiment V0
Vi
Current-injecting
electrode Iapp
Vm
i
Strong feedback
(voltage clamp)
Voltage-Clamp experiment
For strong feedback, the membrane potential is kept constant (voltage-clamp), and
V0
Vi
Current-injecting
electrode Iapp
2
2,
4
m mm ion app
i
dV VdI I
dt R xc
Current is applied uniformly over the length of the electrode (space-clamp):
Vm
i
Strong feedback
(voltage clamp)
Voltage-Clamp experiment
For strong feedback, the membrane potential is kept constant (voltage-clamp), and
V0
Vi
2
2,
4
m mm ion app
i
dV VdI I
dt R xc
Current is applied uniformly over the length of the electrode (space-clamp):
0 0
Current-injecting
electrode Iapp
Vm
i
Strong feedback
(voltage clamp)
Voltage-Clamp experiment
For strong feedback, the membrane potential is kept constant (voltage-clamp), and
V0
Vi
app ionI I
Current is applied uniformly over the length of the electrode (space-clamp):
Current-injecting
electrode Iapp
( ) ( ) ( ).Na m Na K m K L m Lg V E g V E g V E
I io
nm
A/c
m2
Time (msec)
inou
t
Currents measured with
voltage-clamp of squid axon.
Inward currents indicated by
downward deflections.
Membrane held at about -60 mV
(near resting potential), then
stepped to potentials shown.
(After Hodgkin et al, 1952.)
I io
nm
A/c
m2
Vm (mV)
8 msec
0.5 msec
Current-voltage relation for t=0.5 and t=8 msec.
I io
nm
A/c
m2
Vm (mV)
8 msec
0.5 msec
Current-voltage relation for t=0.5 and t=8 msec.
(a)
(b)
(c)
(a) 0 leak conductance low (g 0)ionL
m
I
V
(b) largeion
m
I
VThese late currents, and
hence the conductance, are
greatly increased. This is
called delayed rectification.
(c) The I-V curve here is biphasic (falls and
then rises). Where the slope of the I-V
curve is negative is sometimes referred to
as the region of negative resistance
From (a),
( ) ( ).ion Na m Na K m KI g V E g V E
(1) Can Na+ and K+ currents be separated ?
Normal Na+ free (IK )
Iion
Time (msec)
Currents measured in normal
(left) and sodium-free solutions
(right). Membrane potential held
at -60 mV, then stepped to
potentials shown on right
(After Hodgkin and Huxley, 1952)
Most of the NaCl of the external
medium was replaced by choline
chloride*. Choline maintains
normal osmotic pressure of the
external fluid but the molecule is too
big to go through the membrane.
*Nowadays there are dozens of compounds that selectively block different currents, many derived from natural
toxins. Tetrodotoxin (TTX), a toxin from the Pacific puffer fish, is used to block Na+ channels.
Next slide
Separation of IK and INa when voltage stepped from -60 -4 mV
(Compare previous slide)
Voltage-clamp currents in squid axon measured in normal (INa +IK ) and
sodium free (IK ) solutions.
normal sodiumfree
( ) ( )Na Na K KI I I ICURRENTS CAN BE
SEPARATED !
(2) Does the relation between ionic current and membrane
potential at constant permeability obey Ohm’s law ?
First: Depolarize the axon long enough to allow the permeability to
reach a steady state.
Second: Step the voltage to other levels, but measure the current within
10-30 sec, before the permeability has a chance to change.
Result: The relation between current and voltage is linear! That is, ( ).I g V E
(3) Find gNa and gK :
( )( ) ,
( )
NaNa
m Na
I tg t
V E
( )( ) ,
( )
KK
m K
I tg t
V E
Vm
Time courses gNa(t) and gK(t)
obtained during a depolarizing voltage
step from -65 -9 mV, and then a
repolarizing step from -9 -65 mV.
Conductances gNa(t) and gK(t) are
calculated using the equations
and the time course of the separated
currents IK and INa found experimentally.
Note: 1. gK(t) is a sigmoidal saturating exponential in response to the depolarizing step,
but decays exponentially in response to the repolarizing step.
2. gNa(t) activates and then inactivates in response to the depolarizing step,
but decays exponentially in response to the repolarizing step.
The dynamics of gK :
m
h
n
m
h
n
4
( ) ( )
,
where ( ).
K K
n V Vm m
g g n
dnn n
dt
Curve fit from voltage-clamp experiment
The dynamics of gNa :
m
h
n
m
h
n
3
( ) ( )
( ) ( )
,
where ( )
and ( )
Na Na
m
h
V Vm m
V Vm m
g g m h
dmm m
dt
dhh h
dt
Curve fit from voltage-clamp experiment
23 4
m Na K K L L2
( )
( )
( )
( )
( )
( )
( ) ( ) ( )4
( )
( )
( ),
m mm Na m m
Vm m
Vm m
Vm m
V
V
V
V VdC g m h V V g n V V g V V
t R xi
mm m
m t
hh h
h t
nn n
n t
Putting it all together, the Hodgkin-Huxley Cable equation is
With appropriate boundary and initial conditions.
Examples of solutions:
References:
1. G.B. Ermentrout and D.H. Terman, Mathematical Foundations of Neuroscience,
Springer, New York, 2010 .
2. B. Hille, Ionic Channels of Excitable Membranes, Sinauer associates, Inc.,
Sunderland, Mass., 1984.
3. D. Junge, Nerve and Muscle Excitation, 3rd edition, Sinauer associates, Inc.,
Sunderland, Mass., 1992.
http://www.sfn.org/index.aspx?pagename=HistoryofNeuroscience_classicpapers
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