Structure-Based Models of NO Diffusion in the Nervous System - Computational Neuroscience: A Comprehensive Approach - page 103

vanishingly small makes the approximation in Equations
4.11 and 4.13 exact and we can assume that the contribution to the concentration at
a point, y

Making r , δ θ and δ φ

=( r , 0 , 0 ) , from the volume X is as if from a point source at the point:

=( , θ , φ )

x . Inspecting Equation 4.7 we see that the concentration depends on the

time after synthesis and the distance between x and y ,

r

−

. Substituting this
and our strength/second term, S X , from Equation 4.13 into Equation 4.7 we obtain
the concentration of NO at y due to x at a time t after synthesis:

x y

ex p −

e − λ

r 2 sin

2

Q drd d x − y

t

C P ( x − y , t )= (4.14)

8 ( π Dt )

3 / 2

4 Dt

Now, to get the concentration at y due to the whole sphere we must sum up the
contributions from all the points x =( r , θ , φ ) inside the sphere, M , (i.e., 0 ≤ r ≤

a ;0 ≤ θ ≤ π and 0 ≤ φ ≤ 2 ) as shown below:

)= ∑ x ∈ M C P ( x − y , t )

C S ( a , r , t (4.15)

exp −

(4.16)

a π 2π

r 2 sin

2

Q x − y

=

8 ( π Dt )

3 / 2

4 Dt 0 0 0

e − λ

t drd

· d (4.17)

Using:

2

r 2 r 2 x − y = + − 2 rr cos (4.18)

and noting that there is radial symmetry so that the concentration at any point z =

( r , θ , φ ) at a distance of r from the origin is equal to the concentration at y =( r , 0 , 0 ) ,

we therefore obtain:

t 1
2

erf a

erf a

+ −

r
2 √ Dt

r
2 √ Dt

e − λ C S ( a , r , t )= Q +

Dt
π

exp (

exp (

2 2

1
r

a − r ) a + r )

− − (4.19)

4 Dt 4 Dt

where:

x

exp − u 2 du

2
√ π

erf ( x )= (4.20)

0

for the concentration at a distance r from the centre of a solid sphere of radius a at
a time t after synthesis. This leads naturally to the solution for a hollow sphere of
inner radius a and outer radius b :

C H ( a , b , r , t )= C S ( b , r , t ) − C S ( a , r , t ) (4.21)