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.

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