Biomedical Engineering Reference
In-Depth Information
0
Na
m
3
h
assumption made by Hodgkin and Huxley, which replaces
g
Na
and
g
K
with
g
0
K
n
4
, the equation for the membrane's capacitive current becomes
and
g
d
V
d
t
=−
g
0
Na
m
3
h
0
K
n
4
C
m
(
V
−
V
Na
)
−
g
(
V
−
V
K
)
−
g
L
(
V
−
V
L
)
+
I
ext
(2.23)
Here, the conductances for both sodium and potassium ions are expressed in terms
of some baseline values
g
0
Na
0
K
, respectively, and secondary variables
m
,
h
,
and
n
. The variables are hypothesized as potential-dependent gating variables, whose
dynamics are assumed to follow first-order kinetics, and the equation takes the fol-
lowing form:
and
g
d
s
d
t
=
τ
s
(
V
)
s
−
(
V
)
−
s
;
s
=
m
,
h
,
n
,
(2.24)
where
τ
s
(
are respectively the time constant and the rate constant
determined from experimental data. The above two equations, taken together, repre-
sent a four-dimensional dynamical system known as the simplified Hodgkin-Huxley
model, which provides a basis for qualitative explanation of the formation of action
potentials in the squid giant axon.
FitzHugh later sought to reduce the Hodgkin-Huxley model to a two-variable
model, for which phase plane analysis can be carried out reasonably easily. In the
Hodgkin-Huxley model, the gating variables
n
and
h
were found to have slow kinetics
relative to
m
, and that
n
V
)
and
s
−
(
V
)
8). As a
result of these two observations, a new two-variable model, referred to as the fast-
slow phase model, was proposed by FitzHugh for calculating the capacitive current
of the membrane, which is as follows:
+
h
assumes an approximately constant value (
∼
0
.
d
V
d
t
=−
g
0
Na
m
3
K
n
4
0
C
m
−
(
V
)(
0
.
8
−
n
)(
V
−
V
Na
)
−
g
(
V
−
V
K
)
−
g
L
(
V
−
V
L
)
+
I
ext
(2.25)
and
d
n
d
t
=
n
s
(
V
)
n
−
(
V
)
−
n
(2.26)
Here, a phase-space description of the action potential's formation and decay has been
provided. In these equations,
V
and
n
are the fast and slow variables, respectively.
The
V
nullcline can be defined by
C
m
d
d
t
=
0 and has a cubic shape, while the
n
nullcline is
n
, and it increases monotonically. Since it can be approximated by
a straight line, this suggests a polynomial model reduction of the form
−
(
V
)
d
d
t
=
v(v
−
β)(
1
−
v)
−
u
+
I
(2.27)
and
d
u
d
t
=
δ(v
−
γ
u
)
(2.28)
Search WWH ::
Custom Search