Biomedical Engineering Reference
In-Depth Information
the compartment) that has a membrane resistance equal to the cytoplasmic resist-
ance. Knowing the time constant and the length constant, the propagation speed of
the impulse is calculated. Consider two synaptic inputs represented by two current
steps. The voltage change from the second current step will add to the first current
step, if delivered within a short time interval after the end of the first current. If
is small, then the voltage decay following the first step is quick, and the two inputs
do not add. If
λ
is large, the voltage changes are more likely to add and the voltage
reaches threshold for an action potential. Another terminology used is the electrot-
onic length, which is the ratio of physical length to the space constant.
In steady-state condition (10.49) reduces to
λ
∂ΔΦ
2
∂ΔΦ
2
(10.50)
2
2
0
=
λ
− ΔΦ
or
λ
= ΔΦ
2
2
∂
x
∂
x
Integration of (10.50) twice will result in
x
x
−
−
(10.51)
ΔΦ =
Ae
+
Be
λ
λ
where
A
and
B
are constants.
EXAMPLE 10.6
Under steady-state conditions, the intracellular resistance (
r
i
) of a nerve cell in the axial
direction is 8 M
Ω
/cm. The resistance of the cell membrane in the radial direction is 2
×
10
4
Ω
cm (
r
m
) and the capacitance is 12 nF/cm. Calculate the characteristic length and time
constant.
Solution: Since the extracellular axial resistance is frequently negligible relative to
intracellular axial resistance
4
6
r
r
210
×
m
i
λ
=
λ
=
=
0.05 cm 500 m
=
μ
810
×
Time constant of the membrane is
[
]
4
−
9
s
τ
=
rc
=
210
×
Ω×
cm 12
×
10
F cm 240
=
μ
mmm
Appropriate boundary conditions needs to be specified for computing the
membrane potential. The boundary conditions specify what happens to the
membrane potential when the neuronal cable branches or terminates. The point
at which a cable branches out and the point where multiple cable segments join
is called a node. At a branching node, the potential must be continuous; that is,
the functions
V
(
x, t
)
defined along each segment must yield the same result when
evaluated at
x
value corresponding to the node. For obeying conservation of charge
