Biomedical Engineering Reference
In-Depth Information
dV
C
dt
=
m
G
()(
t
E
V
())
t
+
I
,
[1]
m
j
j
m
ext
j
where
C
m
is the membrane capacitance,
V
m
is the membrane voltage,
dV
m
/
dt
is
the time derivative of
V
m
,
I
ext
is an externally applied current,
G
j
(
t
) is some ionic
conductance,
E
j
is the reversal potential of conductance
j
, and
j
runs over the
various species of conductances present in the membrane.
G
j
may vary as a re-
sult of dependence on voltage or on the concentration of some chemical species,
such as a neurotransmitter, or may represent a constant leak conductance. In
principle, any type of neuronal membrane can be modeled with this equation by
a suitable selection of relevant
G
j
functions.
The most basic function of the neuronal membrane that we must be able to
simulate is the propagation of an action potential down an axon. Our earliest full
understanding of how counterflowing sodium and potassium ions generate the
action potential came from the
w
ork of Hodgkin and Huxley (2), who pro
p
osed
a sodium conductance,
3
4
,
and a leakage conductance,
G
L
, where
N
G
,
G
, and
G
L
are constant maximum
conductances, and
h
,
m
, and
n
are variables specified by equations that capture
the voltage dependencies of the Na
+
and K
+
conductances in a semiempirical
fashion. To generate accurate models of ion channel function, it may be neces-
sary, for at least two reasons, to allow for varying intracellular and possibly ex-
tracellular ion concentrations in the model. First, the reversal potential,
E
j
, in the
treatment above, depends on the ratio of the two concentrations in accord with
the Nernst equation:
GGmh
=
, a po
ta
ssium conductance,
GGn
=
Na
Na
K
K
E
j
= (
RT
/
z
j
F
) ln([
X
j
]
e
/[
X
j
]
i
),
[2]
where
E
j
is the reversal potential of the ion
X
j
(
X
= Na
+
, K
+
, etc.) associated with
conductance
j
;
R
is the gas constant;
T
is the absolute temperature;
z
is the
charge on
X
j
;
F
is Faraday's constant; [
X
j
]
e
is the extracellular concentration of
X
j
; and [
X
j
]
i
is the intracellular concentration of
X
j
(3, p. 84). While for many
purposes
E
j
may be taken as constant, because ion concentrations are homeo-
statically regulated, ions with particularly large variations in concentration, such
as calcium, may require a more detailed treatment. Second, the opening of some
channels, notably calcium-dependent potassium channels, is controlled by ion
concentration, and, obviously, changes in concentration must be modeled if
these channels are to be included in the model. Unfortunately, ion concentra-
tions are notoriously difficult to model accurately due to interactions of ions
with proteins or other molecules that may buffer their concentrations, poorly
known volumes of the spaces to which the ions have diffusional access, and
active pumping. In a simple treatment, the ion currents needed to support the
electric currents found for channels passing ion
X
in Eq. [1] can be calculated
