The Alpha and Beta Components of the Hodgkin-Huxley Model

gray curve

α[n]

10

orange curve

β[n]

80

green curve

α[m]

25

blue curve

β[m]

18

black curve

α[h]

20

red curve

β[h]

30

GHK function

filling

The Hodgkin–Huxley (HH) neuron model is based on the following equations:

C

dV

dt

=

g

K

(V-

V

K

)+

g

Na

(V-

V

Na

)+I(t)

,

g

K

=

g

K,max

4

n

,

g

Na

=

g

Na,max

3

m

h

,

dn

dt

=

α

n

(1-n)-

β

n

,

dm

dt

=

α

m

(1-m)-

β

m

,

dh

dt

=

α

h

(1-h)-

β

h

, where

V

K

,

V

Na

are the Nernst voltages for sodium and potassium [in mV];

V

is the membrane voltage [in mV];

C

is the membrane capacitance per unit area [F/cm²];

I

is the membrane current per unit area [mA/cm²].

The model consists of four current components: the current carried by sodium ions; the current carried by potassium ions; the current carried by other ions (designated leakage current, mainly from chloride ions); and the capacitive (displacement) current. In the standard Hodgkin–Huxley model there are only three types of channels: a sodium channel

g

Na

, a potassium channel

g

K

, and a nonspecific leakage channel with resistance

R

. The probability that a channel is open is described by additional variables

m

,

n

, and

h

called gating variables, which evolve according to the differential equations given. The combined action of

m

and

h

controls the

+

Na

channels. The

+

K

gates are controlled by

n

.