Biomedical Engineering Reference
In-Depth Information
It follows that the A
n
are given by
hV
ss
;
n
i
h
n
;
n
i
:
This approach was used to solve for three models of non uniform capaci-
tance: (1) point change in capacitance, (2) step change in capacitance, and
(3) exponentially graded capacitance. The three models are diagrammed in
the inset of Fig. 2.
A
n
=
4.2. Denition of Fundamental Point and Stepped Models
The solutions to point capacitance and stepped capacitance models were
rst determined as a reference point for the nal exponential model. If
C
m
= C
m
(L) and C
s
= C
m
(0) (i.e., C
s
is capacitance at the soma), then
C
s
if 0XZ
C
m
if Z < XL
for any Z between 0 and L: By previous methods
7;10;11
, the Laplace trans-
form of the transient at X = 0 is
C
m
(X) =
I
stim
G
1
sb
s
b
s
+ b
m
tanh (b
s
Z) tanh (b
m
(LZ))
b
m
tanh (b
m
(LZ)) + b
s
tanh (b
s
L)
;
W (0) =
(7)
p
p
where b
s
=
sR
m
C
m
+ 1: If we assume that =
G
1
tanh (L) =Z is constant with respect to Z; then (7) is given by
W (0) =
I
stim
b
s
sR
m
C
s
+ 1 and b
m
=
b
s
+ b
m
tanh (b
s
Z) tanh (b
m
(LZ))
G
1
b
m
tanh (b
m
(LZ)) + b
s
tanh (b
s
Z) coth (L) =Z
:
It is easy to show that the point membrane capacity model (in Fig. 2)
dened by (7) is a limiting case of the varying capacity model when C
m
(X)
is the step function. In the limit as Z!0; the transient W = W (0) becomes
I
in
p
p
W =
sR
m
C
m
+ 1 + coth L (sR
m
C
s
+ 1)
which is the solution for the voltage for a membrane cylinder with a dierent
point membrane capacity at the origin.
G
1
sR
m
C
m
+ 1 tanh L
4.3. Denition of an Exponentially-Graded Model
The weakness of fundamental models in which membrane capacitance
changes are at a point or are stepped, of course, is that the discontinuity at
the point of change that is unlikely to be biologically realistic. An exponen-
tially graded model, on the other hand, has a smooth, continuous change
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