Information Technology Reference
In-Depth Information
According to the Hodgkin-Huxley formulation, the aggregate current that goes
through an ion channel is given by a relation that has the generic form:
I
channel
D m
p
h
q
I
drive
.V/
(1.85)
where m and h are variables taking values between 0 and 1, p and q are non-negative
integers, and V is the membrane's potential. In general, the more the membrane's
potential grows, the more parameter h gets smaller (inactivation) and the more
parameter m grows (activation). Sodium channels of the Hodgkin-Huxley model
have both activation and inactivation.
Channel K
C
of the Hodgkin-Huxley model, and channels Ca
2C
and K
C
of the
Morris-Lecar model do not exhibit inactivation. In general, the drive current can
take the following form
I
lin
D g
max
.V V
max
/
RT
V
ŒC
in
ŒC
out
e
z
VF
(1.86)
I
cfe
D P
max
z
2
F
2
RT
1e
z
VF
RT
where the units of g
max
are Siemens/cm
2
and cfe means constant field equation. In
the equation about the channel's current Eq. (
1.85
), the variation of parameters m
and h is given by differential equations of the form
dx
dt
D a
x
.1 x/ b
x
x
dx
dt
(1.87)
D .x
1
x/=
x
while for the voltage-dependent parameters a
x
and b
x
, as well as for parameters x
1
and
x
the functions which approximate them have the form
F
e
.V;A;B;C/D Ae
.V B/=C
F
l
.V;A;B;C/D
A.V B/
1e
.V
B/=C
F
h
.V;A;B;C/D A=.1 C e
.V B/=C
/
(1.88)
whereas parameters x
1
and
x
are described by
1
.1Ce
.V
V
T
/=K
/
x
.V/ D
min
C
amp
=cosh.
V
V
max
x
1
.V/ D
(1.89)
/
A key parameter for the appearance and density of voltage pulses is the leak
potential E
L
. According to the value of this parameter, at points of the membrane,
the variation of the voltage V in time may take the form of spikes or bursts.
