Biomedical Engineering Reference
In-Depth Information
28 CHAPTER 3. ACTIVEMEMBRANES
3.1.6 Steady-State andTime Constants
An alternative, and possibly more intuitive, way of writing the gating differential equations is
dm
dt
=
m
∞
−
m
(3.20)
τ
m
α
m
α
m
+
m
∞
=
(3.21)
β
m
1
α
m
+
τ
m
=
(3.22)
β
m
6WHDG\6WDWH
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7LPH&RQVWDQW
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Figure 3.5:
Steady-state and time constant for Hodgkin-Huxley gates.
with similar equations to describe the
h
and
n
variables. Plots of the steady-state and time constant curves
for
m
(red),
h
(blue), and
n
(green) are shown in Fig. 3.5. The reason for writing the equations this way is
that the steady state (
∞
-terms) and time constants (
τ
-terms) have a physical interpretation.The solution
to Eq. (3.20) is
m
0
)e
−
t/τ
m
=
m
∞
−
(m
∞
−
m(t)
(3.23)
where
m
0
is the initial value of
m
. In the context of the voltage clamp, consider that
V
m
is equal to a
value
A
and has been there for a long time, i.e.,
t
(A)
which could be read
directly from Fig. 3.5. Next, consider that the membrane potential is suddenly changed to
V
m
=
B
. Since
the gating variable,
m
, cannot change instantaneously, the initial condition
m
0
is equal to
m
∞
(A)
. The
steady-state and time constants, however, do change instantaneously. Therefore, to compute
m(t)
at any
point after
V
m
→∞
). Therefore,
m
→
m
∞
is clamped to
B
