Biomedical Engineering Reference
In-Depth Information
m
d
½
K
þ
¼
KT
q
Z
½
K
þ
dv
dx
0
dx
m
ð
12
:
8
Þ
With Z
¼þ
1, Eq. (12.8) simplifies to
dv
¼
KT
q
½
K
þ
d
½
K
þ
ð
12
:
9
Þ
Integrating Eq. (12.9) from outside the cell to inside yields
Z
v
i
Z
½
K
þ
i
d
½
K
þ
½
K
þ
dv
¼
KT
q
ð
12
:
10
Þ
v
o
½
K
þ
o
K
½
o
K
½
i
where
v
o
and
v
i
are the voltages outside and inside the membrane, and
and
are
the concentrations of potassium outside and inside the membrane. Thus,
¼
KT
q
K
½
i
K
½
o
K
½
o
K
½
i
v
i
v
o
¼
KT
q
ln
ln
ð
12
:
11
Þ
Equation (12.11) is known as the
Nernst equation
, named after German physical chemist
K
þ
. At room temperature,
Walter Nernst, and
E
K
¼
v
i
v
o
is known as the
Nernst potential
for
KT
q
¼
K
þ
becomes
26
mV
, and thus the Nernst equation for
26 ln
½
K
þ
o
E
K
¼
v
i
v
o
¼
½
K
þ
i
mV
ð
12
:
12
Þ
K
þ
, it can be easily derived for any permeable
ion. At room temperature, the Nernst potential for
While Eq. (12.12) is specifically written for
Na
þ
is
26 ln
½
Na
þ
o
E
Na
¼
v
i
v
o
¼
½
Na
þ
i
mV
ð
12
:
13
Þ
Cl
is
and the Nernst potential for
26 ln
½
Cl
o
26 ln
½
Cl
i
E
Cl
¼
v
i
v
o
¼
½
Cl
i
¼
½
Cl
o
mV
ð
12
:
14
Þ
Cl
.
The negative sign in Eq. (12.14) is due to Z
¼
1 for
12.4.3 Donnan Equilibrium
Consider a neuron at steady state that is permeable to more than one ion—for
example,
Cl
. The Nernst potential for each ion can be calculated
using Eqs. (12.12) to (12.14). The membrane potential,
K
þ
,
Na
þ
,and
V
m
¼
v
i
v
o
,however,isdueto
the presence of all ions and is influenced by the concentration and permeability of
each ion. In this section, the case in which two permeable ions are present is considered.
In the next section, the case in which any number of permeable ions are present is
considered.
