Structure-Based Models of NO Diffusion in the Nervous System - Computational Neuroscience: A Comprehensive Approach - page 100

referred to as the modified diffusion equation . A production term can also be added
to the right-hand side of Equation 4.5 though this is often factored into the solution
later via the initial conditions (see Section 4.2.2).
Under certain conditions and for some source morphologies, Equation 4.5 can be
solved analytically (the analytical solution), although this usually involves some nu-
merical integration. Since the more complex a system is, the more numerical integra-
tion is required, this approach is often impractical and, in general, radial symmetry
is required for tractability. If the analytical solution cannot be derived, a numeri-
cal approximation method must be used [26]. That is not to say that the numerical
solutions are somehow 'worse' than the analytic ones or that they are simply crude
approximations to the true solution [39]. Rather, they can usually be made as accu-
rate as desired, or as accurate as the situation warrants given the unavoidable errors
in empirical measurements of diffusion parameters. Indeed, as all the analytic so-
lutions presented here required numerical integration they are also approximate and
all results have been derived to the same degree of accuracy. However, a numerical
approximation is always an approximation to the analytical solution and so it seems
sensible to use the latter if its calculation is tractable. A more practical reason for do-
ing so is that when it is available, evaluating the analytical solution normally requires
much less computational power. Our approach therefore is to use the analytical so-
lution whenever possible and, when not, to employ finite difference methods. In the
next two sections, we discuss these techniques.

4.2.2 Analytic solutions to the diffusion equation

God does not care about our mathematical difficulties. He integrates

empirically. Albert Einstein.

In this section we discuss how analytic solutions to the diffusion equation are
generated. The solution for a point-source is given first and we then show how
solutions for other simple structures are derived from this. We next state the solutions
thus obtained for hollow spherical and tubular sources and finally give some details
of the numerical integration techniques used to calculate these solutions.

4.2.2.1 Modelling NO diffusion from a point-source

As stated earlier (Section 4.2.1), the dynamics of diffusion are governed by the mod-
ified diffusion equation. Assuming that there are no local NO sinks present and only
global decay is acting, this equation becomes:

C

2 C

t −

D = − λ C (4.6)

where C is concentration, D is the diffusion coefficient and the decay-rate [8]. We

first generate the instantaneous solution , that is, the solution for an instantaneous
burst of synthesis from a point source positioned at the origin of some co-ordinate
system. To do this we envision an amount S 0 of NO being deposited instantaneously