Biomedical Engineering Reference
In-Depth Information
cable (Fig.
7c
). For the sake of simplicity, such morphology is further compartmen-
talized into a series of heterogeneous segments that are electrically coupled to each
other through the axial ion fl ow, occurring within the neuronal cytoplasm. Both pas-
sive and excitable electrical properties of each compartment are assumed to arise
from voltage-gated ion conductances (Dayan and Abbott
2001
; Traub and Miles
1991
) and modeled according to standard electrical equivalents, as originally pro-
posed by Hodgkin and Huxley (
1952
) (Fig.
7b, c
).
In the model of Fig.
7c
, the membrane potential
V
m, j
and the ion current at the
j
th
compartment can be expressed in terms of ion currents:
d
V
2
×
V
- -
V
V
mj
,
mj
,
mj
,
-
1
mj
,
+
1
(1)
i
=×
C
+ +++ +
i
i
...
i
.
mj
,
m
Na,
j
Kj
,
leak ,
j
d
t
R
a
Equation (
1
) follows from cable theory (Dayan and Abbott
2001
) and it is derived
imposing the charge conservation across a patch of neuronal membrane, character-
ized by capacitance
C
m
and axial cytoplasmic resistance
R
a
. We further indicated by
i
Na
,
i
K
, and
i
leak
the transmembrane ion currents. Each of these currents is selective to
different ion species (e.g., Na
+
, K
+
, Cl
−
) and it is characterized by maximal conduc-
tances (
g
Na
,
g
K
, and
g
leak
) and apparent reversal potentials (
E
Na
,
E
K
, and
E
leak
). Reversal
potentials are modeled as ideal voltage sources and their values satisfy the Nernst-
equilibrium relationship for the corresponding ion species in solution (Fig.
7b
)
(Dayan and Abbott
2001
). As in the Hodgkin-Huxley model (Hodgkin and Huxley
1952
) , fi rst-order kinetic schemes account for the instantaneous fractions of volt-
age-gated sodium and potassium channels:
(
)
3
i
=×
g
mh
× ×
VE
-
Na,
j
Na
j
j
m j
,
Na
(
)
4
i
=×
g
n
×
VE
-
K,
j
K
j
m j
,
K
(
)
i
=×
g
V
-
E
(2)
leak ,
j
leak
m j
,
leak
This model can be implemented and computer simulated by standard neuronal
simulators, such as the NEURON environment (Carnevale and Hines
2006
) , and
reproduces some of the electrophysiological features of real-cultured neurons, such
as the resting membrane potential at −70 mV, a membrane time constant in the
range 10-20 ms, an input resistance of 100-130 MW, a rheobase current of 30-50 pA,
a spike overshot of 20-30 mV, as well as the dendritic backpropagation of action
potentials (Schaefer et al.
2003
) .
Similarly to the ion fl ows occurring across the neuronal membrane, the electric
fi elds and current fl ows at the CNT-neuron interface can be described in terms of
equivalent circuit models (Fig.
8a-b
). The electrochemical model fi rst discussed in
Fig.
4c
, accounting for the CNT-electrolyte interface, was incorporated and the
values of the resistance
R
cnt
and capacitance
C
cnt
were varied to account for either
metal or CNT properties (Fig.
8c
). The (intimate) coupling between the substrate
and the neuronal membrane is modeled in terms of a “seal” resistance and of a
