Biomedical Engineering Reference
In-Depth Information
[Ca
2+
]
e
[Ca
2+
]
b
Ca
2+
Buffer
[Ca
2+
]
i
Figure 3.13:
Ionic buffering by intracellular compartments.
V
m
dV
m
dt
=
V
m
−
3
−
+
W
I
stim
(3.36)
dW
dt
=
a
[
V
m
+
b
−
cW
]
.
(3.37)
The model has a steady-state resting potential, fast all-or-none upstroke, and slower repolarization.
Furthermore, the parameters
a
,
b
, and
c
can be tuned to generate action potentials of different shapes
and durations.
3.5.2 Hindmarsh-Rose Model
The Fitzhugh-Naghumo model does not generate bursting behavior. In 1984, Hindmarsh and Rose
developed a set of three differential equations that would allow for the phenomenon of bursting:
dV
m
dt
aV
m
+
bV
m
+
=−
y
−
z
+
I
(3.38)
dy
dt
=−
dV
m
−
+
y
c
(3.39)
dz
dt
=
rsV
m
−
rz
−
rsV
rest
(3.40)
where
a
,
b
,
c
,
d
,
r
,
s
, and
V
rest
are constants and may be tuned to producing different bursting behavior.
3.5.3 Integrate and Fire Model
In 1907, long before Hodgkin and Huxley, Lapicque proposed that the firing of an action potential could
be modeled simply as a spike in voltage. As
I
stim
is applied,
V
m
will depolarize according to the familiar
passive model
dV
m
dt
1
C
m
=
[
−
G
L
(V
m
−
E
L
)
+
I
stim
]
.
(3.41)
