Biomedical Engineering Reference
In-Depth Information
To describe the behavior of an RC circuit, consider K
+
ion conductance in series
with its Nernst potential (
ΔΦ
rest
). A positive stimulus current,
I
inj
, is injected from
time
t
0
to
t
. Then using Kirchoff's current rule, the equivalent circuit of a cell is
written as
III
+− =
0
Ci
inj
where
I
C
is current through the capacitor and
I
i
is the ionic current through the
resistor. The injected current is subtracted from the other two. This is based on the
convention that the current is positive when positive charge flows out of the cell
membrane. Substituting (3.14) and (3.18),
d
ΔΦ
ΔΦ
− ΔΦ
C
m
+
m
rest
−
I
=
0
(3.20)
m
inj
dt
R
m
Due to injected current, the membrane reaches a new state potential,
ΔΦ
SS
. At
steady state
d
ΔΦ
m
/
dt
=
0, (3.20) reduces to
ΔΦ
=
IR
+ ΔΦ
SS
inj
m
rest
Rearranging and integrating with the limits
ΔΦ = ΔΦ
o
when
Δ
t
= Δ
t
o
and
ΔΦ =
ΔΦ
when
Δ
t
= Δ
t
(
)
ΔΦ
− ΔΦ
tt
τ
−
SS
0
ln
=
0
(
)
ΔΦ
− ΔΦ
SS
where
τ
is the product of
R
m
and
C
m
and has the units of time. In generally,
τ
is
called the time constant. An alternative form is
⎛
tt
−
⎞
0
(3.21)
(
)
⎜
⎟
⎝
τ
⎠
ΔΦ = ΔΦ
+ ΔΦ
− ΔΦ
e
SS
0
SS
Equation (3.21) is used for both charging and discharging of an RC circuit for
any ionic species. In general, the majority of experiments are conducted by inject-
ing a current as a step input to understand the property of a cellular membrane
with respect to a particular ionic species. A frequently measured property is
to
understand the resistance and capacitance properties of cells. At the instant when
the current is first turned on (
t
τ
=
0) the first term in square brackets
ΔΦ
SS
is zero
and
ΔΦ = ΔΦ
rest
. For
t
>>
0, the second term is nearly zero and
ΔΦ = ΔΦ
SS
.
