Biomedical Engineering Reference
In-Depth Information
φ
i
I
Stim
I
Cm
C
m
R
m
I
Rm
φ
e
Figure 2.4:
RC circuit analog of a passive cell membrane.
V
m
(t)
=
V
∞
(
1
−
e
−
t/τ
m
)
(2.15)
where
t
is the time after applying
I
stim
.
When
I
stim
is turned off,
V
m
will be at some initial voltage
V
0
due to charging. From this initial
value,
V
m
will return to
V
rest
, again at a rate governed by
τ
m
m
V
0
e
−
t/τ
m
.
V
m
(t)
=
(2.16)
2.2.1 Finding Membrane Properties from
V
m
(t)
The left-hand side of Fig. 2.5 is a plot of
V
m
(t)
over time as a current pulse is applied and then removed.
The rising phase is governed by Eq. (2.15) while the falling phase is governed by Eq. (2.16).The rising or
falling phase can be used to determine the membrane properties,
R
m
,
C
m
, and
τ
m
. The right-hand side
of Fig. 2.5 shows a stimulus that is applied only for a short time, so
V
m
does not have time to reach
V
∞
before the stimulus is turned off.
2.2.2 The Passive Membrane
To perform the analysis above, we assumed that the current flowing through the membrane was linearly
proportional to the membrane voltage. It turns out that this assumption is a good approximation as long
as
V
m
is below some
threshold
voltage,
V
t
m
. When
V
m
<V
t
m
, the membrane is called
passive
,
R
m
is a
constant and the membrane can be represented as the RC circuit in Fig. 2.4. In a real neuron,
V
t
m
is
typically between 5mV and 10mV higher than
V
res
m
.When
V
m
depolarizes above the threshold, however,
the membrane will become
active
and
R
m
will no longer be a constant. In Ch. 3 we will consider the
nonlinear relationship between
R
m
and
V
m
above the threshold.
