Structure-Based Models of NO Diffusion in the Nervous System - Computational Neuroscience: A Comprehensive Approach - page 102

of radius a , the method is to sum the contributions to the concentration at a point in
space y , from all the points within the sphere as described below.

+

+

0 r r+ r a

Figure 4.1
The element X (see Equation 4.10) of a sphere of radius a containing the points ( r , θ

, φ )

where: r ≤ r ≤ r + δ

≤ θ + δ θ

and φ ≤ φ ≤ φ + δ φ r ; θ ≤ θ . X is outlined by solid lines

with dashed lines denoting radii and surface of the sphere.

r , θ

, φ )

Take a volume X within the sphere containing the points ( where:

r ≤

r ≤ r + δ r ; θ ≤ θ ≤ θ + δ θ ; φ ≤ φ ≤ φ + δ φ . (4.10)

as shown in Figure 4.1. If this element, X , is relatively small (i.e., if r , δ θ and δ φ

are sufficiently small), we can approximate its volume with:

r 2 sin

V X ≈

θ δ θ δ φ δ r (4.11)

with the error in the approximation getting smaller as the dimensions of the element
( ) are reduced and the error becoming zero in the limit of the dimen-

sions becoming vanishingly small. Now, the amount of NO produced per second in
a volume V of NO-producing tissue is:

r , δ θ and δ φ

S V = Q × N V (4.12)

where Q is the amount of NO produced per second from a single NO producing unit
and N V is the number of these units within V . This number is simply the product of
the volume of V and

, the density of units in V . Hence for the element, X (Figure
4.1), we have a strength/second term, S X , of:

r 2 sin S X = Q × N X = Q × ρ V X ≈ Q θ δ θ δ φ δ r (4.13)

In this equation, r , , the concentration

of NO produced per second, is independent of the particular shape of the structure
being studied and so can be determined by empirical experiments as in [47].

and are variables whilst the product Q