Biomedical Engineering Reference
In-Depth Information
A.1.3
Neuronal Currents
Electric currents in biological tissue are primarily due to ions, e.g., K+, Na+, Cl
,
Ca2+, etc. These ions flow in response to the local electric field, according to Ohm's
law, but also in response to their local concentration gradient, according to Fick's
law [1]. In the resting state of the membrane, the concentration gradients and electric
field are due to ion channel pumps, which use energy acquired from ATP to move
ions across the membrane against their diffusion gradient.
To a good approximation, the concentration of each ion inside and outside the
membrane may be assumed constant in time. The transmembrane voltage, however,
changes radically in time, the strongest example being the action potential. Thus for
the purposes of discussing neural activity, we take the transmembrane potential
V
m
as the primary dynamic state variable to be considered. By convention,
V
m
is defined
as the potential inside relative to that outside, i.e.,
V
m
=
−
V
o
. The current flowing
across the membrane maybe be viewed as a function of
V
m
, and therefore as the basis
of the extracellular fields detected by MEG and EEG.
In the resting state of the neuron, the balance between electric and diffusive flows
determine the resting membrane potential of the neuron [2]. To consider this balance,
we define two related quantities: the charge current density,
J
[C/m
2
s], and the ionic
flux
j
[mol/m
2
s]. For a single ionic species, these are trivially related by:
J
V
i
−
=
zF
j
,
where
z
e
is the signed integer number of charges carried by an ion, and
F
is the
Faraday's constant (96,500 C/mol). The flux
j
has two contributions, arising from
the local concentration gradient
=
q
/
∇
C
and the local electric field
E
, where
C
indicates
the ionic concentration for different ions defined as the number of ions (in mol) per
unit volume. Fick's law states that ions diffuse down their concentration gradient
according to the linear relation
j
C
, where
D
is the diffusion coefficient.
Furthermore, an ion within an applied electric field
E
accelerates initially and
reaches a terminal velocity
v
given by
v
=−
D
∇
accounts
for the fact that negative ions travel in the opposite direction of the electric field. The
quantity
=
μ(
z
/
|
z
|
)
E
where the factor
(
z
/
|
z
|
)
RT
|
μ
is called the mobility and the diffusion coefficient,
D
=
F
μ
, where
z
|
R
314 J/(mol K) is the ideal gas constant. Counting flow of particles through a
parallelepiped, the ionic flux is
=
8
.
z
E
. This equation is related to the more common
way of expressing the flow of charges in an electric field via Ohm's law, which in a
volume conductor is written as
J
μ
C
|
z
|
E
. Therefore, the total ionic flux is the linear
sum of its diffusive and electrical contributions:
=
˃
RT
F
∇
j
E
=−
μ
j
=
j
S
+
C
+
zC
∇
ʦ
,
(A.1)
|
z
|
ʦ
=−∇
ʦ
where the electric potential
. This equation can be used
to derive the Nernst equation, and the Goldman-Hodgkin-Katz equation [3], for the
resting potential of a neuron, and also form the basis for modeling synaptic activity.
is defined as
E
