Biomedical Engineering Reference
In-Depth Information
4.4 NUMERICAL METHODS: THE FINITE AND DISCRETE
CABLE
In all but the most simple situations, the continuous cable model can only be solved numerically using
a computer. It is therefore required that the cable be of finite length and divided into small patches of
membrane. Beginning with the continuous Eq. (4.11) the
finite difference
approximation may be made
for the second derivative in space.
r
e
)
c
m
V
k
−
1
m
2
V
m
+
V
k
+
1
m
−
dV
m
dt
V
m
r
m
=
(r
i
−
+
(4.32)
(r
i
−
r
e
)dx
2
where
k
is a variable to represent any node in the middle of the cable. Note that the equation able is
similar to Eq. (4.10). Rearranging
V
k
−
1
m
2
V
m
+
V
K
+
1
m
dV
m
dt
1
c
m
−
V
m
r
m
=
−
(4.33)
(r
i
−
r
e
)dx
2
and using the Euler method of Sec. 2.5
V
k
−
1
m
2
V
m
+
V
k
+
1
m
dt
c
m
−
V
m
r
m
dV
m
=
−
(4.34)
r
e
)dx
2
(r
i
−
V
new
V
old
=
+
dV
m
(4.35)
m
m
an every
V
m
is not discrete in both time and space. Note that in active propagation the gating variable
must also be integrated in time, however, they do not require information from their neighbors.Therefore,
the simple Euler method may be used.
A problem is encountered, however, with the above formulation. Consider evaluating the left most
node on the cable (e.g.,
k
0 which does
not exist. The same problem occurs at the right most node of the cable. To overcome the problem at the
endpoints, we must enforce a boundary condition. The most common assumption in neural propagation
is that no current can leave either end of the cable. This is called a
sealed-end
boundary condition.
Mathematically,
=
1). An update of the end node requires information at
k
=
dφ
i
dx
=−
I
i
r
i
=
0
(4.36)
dφ
e
dx
=−
I
e
r
e
=
0
.
(4.37)
The most straightforward way to enforce this boundary condition is to define a
ghost
node that extends
past the end of the cable (see Fig. 4.8). Then, at the two cable ends
φ
gleft
i,e
=
φ
i,e,
(4.38)
φ
gright
i,e,
φ
n
−
1
i,e
=
(4.39)
