Structure-Based Models of NO Diffusion in the Nervous System - Computational Neuroscience: A Comprehensive Approach - page 113

is therefore by definition singular. The singular nature of the solution represents a
fundamental problem to modelling sources with morphology which can be appreci-
ated by examining the steady-state solution for the 3D point source used by Wood
and Garthwaite [47]:

exp r ln 2

Dt 1 / 2

S
r

C ( r )= − (4.52)

where r is the distance from the source and S is a constant determined by the pro-
duction rate of the source.

The first thing to note is that the concentration at the

source r 0 is infinite. Although the central concentration itself could be ignored,

the ramifications of having a singularity at the heart of the solution causes many
complications and unrealistic results [34]. Firstly one must decide at what distance
from the centre the model is deemed to be 'correct', necessarily a somewhat arbitrary
choice. The approach taken by Wood and Garthwaite [47] was to use the surface of
the neuron and only consider points outside the cell. This highlights a second prob-
lem, namely that the internal concentration is indeterminate which in turn means that
obtaining a meaningful solution for hollow structures is impossible [34]. Finally, as
the concentration in Equation 4.52 is dependent on the distance from the source only,
this model cannot be used to address the impact of different source morphologies as
sources with the same value of S but different shapes and sizes will yield identical
results.
In compartmental models, on the other hand, one can include some notion of the
source morphology. However, while such models do give valid insights into the over-
all role of a diffusing messenger they are a form of explicit finite difference model
[1] and are thus hampered by the limit on the duration of the time step employed
given in Equation 4.34 [30]. This limitation necessitates the use of relatively large
compartments leading to gross approximations. In view of this, we believe a more so-
phisticated form of numerical approximation, such as the one presented here, should
be employed when the complexity of the morphology makes an analytical solution
impractical.

=

4.3.1 Diffusion from a typical neuron

We first examine the solution for a simple symmetrical structure representing, for
example, a neuronal cell body in which NO is synthesized in the cytoplasm but
not in the nucleus. We have therefore examined the solution for a hollow spherical
source of inner radius 50 m (cell body). These

dimensions, though large for many neurons especially in vertebrates, do correspond
to the dimensions for some identified giant molluscan neurons whose cell bodies
synthesise NO and have been shown to mediate volume signaling [33].
Of course we are not suggesting that neurons are perfectly spherical but rather
that hollow spheres are a useful approximation for neurons. They can, for exam-
ple, tell us about the importance of morphological irregularities. For instance, if
one had a cell which was mainly spherical but had a lot of small-scale variability

µ m (the nucleus) and outer radius 100 µ