Biomedical Engineering Reference
In-Depth Information
where O
1
,
2
,
3
,
4
were the four parts. Further analysis showed that three of the parts functioned in the same
way, i.e.,
α
and
β
functions were identical). These channel parts were each called
m
and the remaining
part was called
h
. Their model of the Sodium current was described by
g
Na
m
3
h(V
m
)
I
Na
=
[
V
m
−
E
Na
]
(3.8)
dm
dt
=
α
m
(
1
−
m)
−
β
m
m
(3.9)
dh
dt
=
−
−
α
h
(
1
h)
β
h
h
(3.10)
where
m
and
h
are known as
gating
variable because they control how ions are gated through the channel.
The only difference between the
h
and
m
variables is that the functions of
α
and
β
are different.
Using their experimental data and some curve fitting, Hodgkin Huxley found the following fits
for the
α
's an
β
's:
v
m
e
(
25
−
v
m
)/
10
25
−
α
m
=
0
.
1
(3.11)
−
1
4
e
−
v
m
/
18
βm
=
(3.12)
0
.
07
e
v
m
/
20
α
h
=
(3.13)
1
e
(
30
−
v
m
)/
10
β
h
=
(3.14)
+
1
is scaled by the resting potential such that
v
m
=
V
m
−
V
rest
where
v
m
.
m
3.1.5 The PotassiumCurrent
The Potassium current may be assumed to be of the nonlinear form
I
K
=
[
V
m
−
E
K
]
g
K
O(V
m
)
.
(3.15)
Recall, however, that the first assumption was that each current was independent of the other currents
present. So, simply performing the voltage clamp as described above would yield data on
I
K
+
I
Na
.To
separate the two currents, Hodgkin and Huxley used tetrodotoxin (TTX), a Sodium channel blocker, to
isolate
I
K
. They then performed the same experiment without TTX to yield
I
K
+
I
Na
and by simple
subtraction they isolated
I
Na
. Using this two-step procedure, they found that
O(V
m
)
for
I
K
was the
product of four gating variables that were all identical:
I
K
=
g
K
n
4
[
V
m
−
E
K
]
(3.16)
dn
dt
=
α
n
(
1
−
n)
−
β
n
n.
(3.17)
Experimental data for the
n
gating variable was fit to the following
α
and
β
functions:
−
v
m
e
(
10
−
v
m
)/
10
10
α
n
=
0
.
01
(3.18)
−
1
0
.
125
e
−
v
m
/
80
.
β
n
=
(3.19)
