Biomedical Engineering Reference
In-Depth Information
10 CHAPTER 2. PASSIVEMEMBRANES
2.2 STIMULATINGTHE PASSIVEMEMBRANE
If an external current (
I
stim
) is applied to the membrane of a cell such that positive ions are forced into the
cell, the membrane voltage will depolarize. For now we will assume that there are no synapses (
I
syn
=
0),
so
dV
m
dt
1
C
m
=
[
−
I
ion
+
I
stim
]
.
(2.8)
The membrane, although composed primarily of lipids, has “leaky” channels that will allow some current
to pass. Experimentally, it has been observed that when
V
m
is close to
V
rest
m
, the leakage of ions is
proportional to
V
m
. Therefore, we can approximate
I
ion
using Ohm's Law:
φ
i
−
φ
e
R
m
V
m
R
m
I
ion
=
=
(2.9)
where
R
m
is the
specific membrane resistivity
(
kcm
2
) to current flow and measures the “leakiness” of the
membrane. Given our assumption of linear
R
m
,
I
stim
.
dV
m
dt
1
C
m
V
m
R
m
+
=
−
(2.10)
Rearranging Eq. (2.10)
R
m
C
m
dV
m
dt
+
V
m
=
R
m
I
stim
.
(2.11)
It has become customary rewrite Eq. (2.11) as:
dV
m
dt
τ
m
+
V
m
=
V
∞
(2.12)
where
τ
m
=
R
m
C
m
(2.13)
is the subthreshold
membrane time constant
measured in
msec
and
V
∞
=
R
m
I
stim
(2.14)
is the steady-state voltage as time
.
When
I
stim
is applied,
V
m
will charge up the capacitor to
V
∞
→∞
at a rate governed by
τ
m
. If the
current remains on, the voltage will remain at
V
∞
. Solving Eq. (2.12) during stimulation,
