Biomedical Engineering Reference
In-Depth Information
is plotted as a function of time in part (b) of the figure. In this new loop
Δφ
C
=
Δφ
1
−
Δφ
2
, however,
I
=
I
R
−
I
C
.
1
I
0
I
C
I
R
I
S
I
()
Δφ
Δφ
R
C
R
C
0
t
()
2
(a)
(b)
d
Δ
φ
1
I
(a) Show that:
for
t
≥
0 where
Δφ
=
Δφ
1
−
Δφ
2
.
+Δ=
φ
dt
RC
C
(b) Assuming
Δφ
=
0 at
t
=
0, solve the differential equation above for
t
≥
0 to
get an expression for
Δφ
(
t
). What is
Δφ
as
t
→ ∞
?
(c) Plot
Δφ
as a function of time.
(d) Show that at
t
=
τ
=
RC
,
V
will have reached 63.8% of its final value
defined as
Δφ
final
= Δφ
(
t
→ ∞
).
(e) Suppose that the circuit represents the membrane of a neuron receiving an
input
I
from another neuron. Then the voltage
Δφ
represents the membrane
potential of the neuron modeled by the circuit. How will decreasing
τ
, with
τ
=
RC
, affect the speed of the neuron's response to the current input
I
[personal communication with Dr. Nada Bowtany, Rutgers University,
2004]?
3.28
Hodgkin and Huxley applied a step voltage across the membrane and mea-
sured the current in their voltage clamp experiment. To measure conductance
as a function of time, they measured current.
R
is replaced with a time varying
resistor
R
(
t
) and the current source is replaced with a voltage source consisting
of
, and a resistor
r
m
. The voltage clamp is achieved by stepping the voltage
up and holding it at a value
Δφ
Δφ
m
across the membrane.
r
v
is sufficiently small
to assume that
Δφ
= −Δφ
C
= Δφ
1
− Δφ
2
= −Δφ
R
.
(a) Plot the voltage
Δφ
(
=Δφ
1
− Δφ
2
) across the membrane as a function of
time.
(b) What is the current in the capacitor before the clamping voltage is applied,
at
t
< 0?
