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Fig. 1.8
Membrane's circuit
currents
1.4
The Hodgkin-Huxley Model
1.4.1
Derivation of the Hodgkin-Huxley Model
The Hodgkin-Huxley model is obtained from a modification of the cable's PDE,
which describes the change of the voltage V.x;t/ along dendrites' axis. It holds
that [
15
,
39
]
I
L
D I
cap
C I
ion
(1.45)
Cable's equation comprises as inputs the currents which are developed in the ions'
channels
@
2
V
M
@
2
x
2
˛
2r
L
D c
M
@V
M
@t
C I
K
C I
Na
C I
L
(1.46)
which can be also written in the following form
@
2
V
M
@
2
x
2
c
M
@V
M
@t
˛
2r
L
g
K
.V
M
E
K
/ g
N
.V
M
E
Na
/ g
L
.V
M
E
L
/
(1.47)
D
where g
k
is the conductance of the K
C
channel, g
Na
is the conductance of the Na
C
channel, and g
L
is the conductance of the leakage channel (Fig.
1.8
).
Through an identification procedure, Hodgkin and Huxley derived specific
relations for the conductances in the ionic channels K
C
and Na
C
.
g
K
Dg
K
n
4
g
Na
Dg
Na
m
3
h
(1.48)
where g
K
and g
Na
are the maximum values for the conductances and n, m, h
are gating variables which take values between 0 and 1. Thus, n
4
represents the
probability channel K
C
to be open. Moreover, m
3
represents the probability that the
sodium activation gate is open, and h is the probability that the sodium inactivation
