Biomedical Engineering Reference
In-Depth Information
Closed
Closed
m
m
m
m
h
m
m
h
Closed
Open
m
m
m
m
h
m
h
m
Na
+
Figure 3.4:
Schematic of Sodium channel dynamics.
probability of randomly picking an open channel. Next they next assumed that the rate at which channels
opened (on average) was not necessarily the same as the rate at which channels closed. This situation can
be represented kinematically by
α(V
m
)β(V
m
)C
where rate constants,
α(V
m
)
and
β(V
m
)
, would be functions of
V
m
. Using this formulation they could
write down a differential equation to describe the dynamics of the
O
variable.
O
dO
dt
=
α(V
m
)(
1
−
O)
−
βO(V
m
).
(3.5)
Given the variable
O
, they defined the Sodium current as:
I
Na
=
g
Na
×
[
V
m
−
E
Na
]
O(V
m
)
(3.6)
where
g
na
1, i.e., every sodium channel
is open), the maximum possible current will flow.
O
, on the other hand, can vary between a probability
of 0 and 1. When Hodgkin and Huxley fit parameters for
α
and
β
using the voltage clamp, they found
that the channel was more complicated than the simple
O
variable could capture. They assumed that the
Sodium channel was composed of four parts (Fig. 3.4) and for the channel to be fully open, all four parts
needed to be in the right configuration. They therefore assumed
I
Na
to take the form of:
is the
maximum conductance
. The meaning of
g
na
is that if
O
=
I
Na
=
g
Na
O
1
(V
m
)
O
2
(V
m
)O
3
(V
m
)O
4
(V
m
)
[
V
m
−
E
Na
]
(3.7)
