Bioelectric Phenomena - Introduction to Biomedical Engineering - page 768

field and then combined using space charge neutrality to complete the derivation of the
Goldman equation.

Potassium Ions
The flow equation for

K þ with mobility m

is

K

K dK ½

J K ¼ KT
q

K Z K K ½ dv
dx

m

dx

m ð 12 : 17 Þ

Under a constant electric field,

dv

dx ¼ D v D x ¼ d

ð 12 : 18 Þ

Substituting Eq. (12.18) into (12.17) with Z K ¼ 1 gives

d ½ K þ
dx

J K ¼ KT
q

K ½ K þ d

m m ð 12 : 19 Þ

K

K þ ,

Let the permeability for P K , equal

m

K KT
d q ¼ D d

P K ¼ ð 12 : 20 Þ

Substituting Eq. (12.20) into (12.19) gives

V ½ K þ P K d d ½ K þ
dx

J K ¼ P K q
KT

ð 12 : 21 Þ

Rearranging the terms in Eq. (12.21) yields

d ½ K þ

dx ¼ ð 12 : 22 Þ

P K d qV ½ K þ

J K

KT d

Taking the integral of both sides, while assuming that J K is independent of x , gives

Z d
0 dx ¼

Z ½ K þ o

d ½ K þ

ð 12 : 23 Þ

P K d qV ½ K þ

J K

½ K þ i

KT d

resulting in

K ½ o

P K d þ qV K ½

0 ¼ KT

d
qV

J K

d

x ln ð 12 : 24 Þ

KT d

K ½ i

and

0 1

P K d þ qV ½ K þ o

J K

@ A

d
qV

¼ KT

KT d

d ln ð 12 : 25 Þ

P K d þ qV ½ K þ i

J K

KT d

KT
qV

Removing d from both sides of Eq. (12.25), bringing the term to the other side of the

equation, and then taking the exponential of both sides yields