Biomedical Engineering Reference
In-Depth Information
Although these equations are not compact, they are easy to compute numerically. Because
R
is also
dependent upon
dx
, it is simple to extend our result to compartments of different lengths.
5.2.3 Branches
It is clear fromFig. 5.1 that we alsomust describe the points at which the dendrite branches. For simplicity,
we will consider a single branch, as shown in Fig. 5.7, and assume that
dx
and
a
remain constant. The
node labeled
b
is known as a
branch
node. There is no membrane at the branch node because it is not at
the center of a compartment and is only used to make computation easier. At Node
b
,
V
m
−
V
m
(
R
V
m
−
V
m
(
R
V
m
−
V
m
(
R
R
)
−
R
)
−
R
)
=
0
.
(5.6)
+
+
+
To simplify, we can multiply through by the denominator:
V
m
−
V
m
−
V
m
+
V
m
−
V
m
+
V
m
=
0
(5.7)
V
m
+
V
m
+
V
m
V
m
=
.
(5.8)
3
Note that if
dx
or
a
were not the same everywhere, this equation would become much more complicated
but still computable. We can now use
V
m
to derive equations for Nodes 1, 2, and 3:
V
m
−
V
m
2
R
V
m
−
V
m
2
R
−
−
I
m
=
0
(5.9)
V
m
−
2
V
m
+
V
m
=
I
m
.
(5.10)
2
R
Similarly:
V
m
−
2
V
m
+
V
m
2
R
=
I
m
(5.11)
V
m
−
2
V
m
+
V
m
=
I
m
.
(5.12)
2
R
The remainder of the extension of these branches (e.g., Nodes 4, 5, 6,
...
) can be formulated as uniform
cables. Therefore, the key to writing the equations for a branch is the branch node.
5.2.4 The Soma
Although the Soma is considered to be the center of the neuron, it is oftenmodeled as a single large passive
compartment. The special role of the soma, however, is to integrate all impulses entering the dendritic
tree. It is also directly connected to the axon hillock which
is
capable of generating an action potential.
So, if the dendrites charge up the soma above threshold, the axon hillock will fire and send an action
potential propagating down the axon. The soma therefore acts as a relay center between the dendrites
and axon. Figure 5.8 shows a simple model of a neuron composed of two active axon compartments of
