Biomedical Engineering Reference
In-Depth Information
the technology exists to create three-dimensional images of the branching structures of real dendrites.
You may wish to think about how the analysis above needs to be changed to allow for a 3D structure.
5.2.5 Axon Collaterals and Semi-Active Dendrites
To simplify the discussion above we have made a number of assumptions that may not be true in a
real neuron. First, some specialized neurons have either multiple axon projections or a single axon that
branches many times. The geometry of a branching axon can be modeled in the same way as a branching
dendrite with the exception that the ionic current,
I
ion
will be more complex.
Recent data has also shown that dendrites are not simply passive cables. In other words, there are
some ion channels embedded in themembrane that have nonlinear behavior so
I
m
=
V
m
R
m
.Tomodel a
semi-
active
dendrite we can simply replace the passive
I
ion
term with the appropriate nonlinear currents. The
presence of these channels allows currents and potentials to
back propagate
in the orthodromic direction.
Back propagation may impact future pulses in the antidromic direction by changing the concentrations
of ions (in particular
Ca
2
+
]
i
) and transiently change
R
m
of the dendrite.
Lastly, anatomical studies have shown that small protrusions, called
spines
, appear on some den-
drites. Although it is not clear what role spines play, there have been a number of suggestions. One
possibility is that spines increase the amount of membrane area for synaptic inputs from other neurons.
Another view is that spines store
[
Ca
2
+
]
i
and other ions which modulate the strength of synaptic inputs.
Lastly, some consider the spines to serve an electrical role by modulating the attenuation of a voltage
headed toward the soma.
[
5.3 NUMERICALMETHODS: MATRIX FORMULATION
A typical node,
k
, in an unbranched cable of length
n
with uniform membrane properties is connected
to two neighbors. The governing equation for this node is therefore of the form:
V(k
−
1
)
−
2
V(k)
+
V(k)
=
I
m
(k) .
(5.13)
R
At the ends of the cable we can assume seals ends as in Sec. 4.3
−
2
V(
1
)
+
2
V(
2
)
=
I
m
(
1
)
(5.14)
R
−
−
2
V(n
1
)
2
V (n)
=
I
m
(n) .
(5.15)
R
Using matrices we can write this system of equations as
⎡
⎤
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
2
R
2
R
...
V(
1
)
V(
2
)
V(
3
)
.
V(n
I
m
(
1
)
I
m
(
2
)
I
m
(
3
)
.
1
R
−
2
R
1
R
⎣
⎦
...
1
R
−
2
R
1
R
0
...
.
.
.
.
.
.
=
1
R
−
2
R
1
R
−
2
)
I
m
(n
−
2
)
...
0
V(n
−
1
)
I
m
(n
−
1
)
1
R
−
2
R
1
R
...
V (n)
I
m
(n)
2
R
−
2
R
...
