Biomedical Engineering Reference
In-Depth Information
which is the differential equation relating membrane potential to net membrane
current that forms the basis of all cardiac cell models. Hodgkin and Huxley
demonstrated that
g
m
can be separated into sodium and potassium components, which
are both functions of voltage and time. In their analysis they introduced the concept of
activation and inactivation 'gates' and provided equations governing the time and
voltage dependence of these gates. In its simplest form, with a single gate controlling
the state of each channel, the ionic current flowing through 'gated' ion channels is
given by
I
ion
=
x
×
g
max
× (
V
m
-
E
ion
),
(2)
Where
g
max
is the maximal conductance (all channels open),
x
the proportion of
g
max
actually available, and
E
ion
the reversal potential. The activation 'gating variable'
x
,
i.e. the fraction of active (open) gates (ranging between 0 and 1), changes with time as
d
x
/d
t
=
Į
× (1
-
x
)
-
ȕ
×
x
, (3)
where
Į
and
ȕ
are first-order 'rate constants' that are both functions of membrane
potential. The link between theory and voltage-clamp experiment is that the rate
constants can be derived from the experimental steady-state activation curve (
x
)
and
time constant of activation (IJ
x
) through
x
1
¼ a=ða þ bÞ
(4)
and
s
x
¼
1
=ða þ bÞ:
(5)
Fig. 1.
Electrical equivalent of the cardiac cell membrane. The lipid bilayer acts as a capacitor
with capacitance
C
m
. The current charging the capacitor (
I
C
) is related to the voltage across the
cell membrane (
V
m
) through
I
C
=
C
m
× d
V
m
/d
t
. The ion channel in the cell membrane acts as a
resistor with resistance
R
m
, or conductance
g
m
, which is expressed in 'mho' or 'siemens' (S).
The current flowing through the ion channel (
I
m
) is related to
V
m
through
I
m
=
g
m
× (
V
m
-
E
m
),
where
E
m
is the Nernst or reversal potential for the ions passing through this particular channel.
