Biomedical Engineering Reference
In-Depth Information
Table 4.1:
Parameter Name
Unit
r
m
Membrane Resistance
cm
r
i
Axial Intracellular Resistance
/cm
r
e
Axial Extracellular Resistance
/cm
c
m
Axial Membrane Capacitance
μF /cm
i
m
Axial Membrane Current
μA/cm
dx
Spatial Step Size
cm
a
Cable radius
cm
Table 4.2:
Variable Name
Unit
Equivalence
πa
2
r
i
R
i
Specific Intracellular Resistivity
cm
πa
2
r
e
R
e
Specific Extracellular Resistivity
cm
cm
2
R
m
Specific Membrane Resistivity
2
πar
m
μF /cm
2
C
m
Specific Membrane Capacitance
c
m
/(
2
πa)
μA/cm
2
I
m
Specific Membrane Current
i
m
/(
2
πa)
4.1.4 An Applied Stimulus
A stimulus,
i
stim
, may be applied at a particular location on the cable for some particular duration.
Mathematically, a stimulus applied to a point can be represented using the Dirac delta function,
δ(s)
.If
the stimulus at this point is applied at
t
0 and remains on, the unit step function,
u(t)
, may be used. For
the simple case where a stimulus is applied to a point in the middle of an infinitely long cable beginning
at
t
=
=
0,
λ
2
∂
2
V
m
∂x
2
∂V
m
∂t
+
=
τ
m
V
m
±
r
m
i
stim
δ(x)u(t)
(4.17)
where the
±
is to take into account either an intracellular or extracellular stimulus.
4.1.5 Steady-State Solution
We can consider the steady state solution to Eq. (4.17) by assuming
∂V
m
∂t
=
0:
λ
2
d
2
V
m
dx
2
−
V
m
=−
r
m
i
stim
δ(x)
(4.18)
where we have assumed the stimulus will depolarize the membrane. Equation (4.18) is an ordinary
differential equation that no longer depends upon time. We can find the homogeneous solution by
finding the solution to:
