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Sodium Channels Na C
1.8.2
Sodium channels are distinguished into (1) transient or fast sodium current channels,
(2) persistent or slow sodium channel currents. As an example of persistent sodium
channel currents one can consider the ones appearing in the so-called pre-Bötzinger
complex, that is neurons that coordinate respiration pulses. The associated model
has the form described in Eq. ( 1.52 ) of the Hodgkin-Huxley dynamics and is
given by
C m d dt Dg L .V E L / g k n 4 .V E k /
g Na m 1 .V/ 3 .1 /.V E Na / g Nap w 1 .V/h.V E Na /
(1.90)
d
dt
D .n 1 .V/ /= .V/
dh
dt
D .h 1 .V/ h/= .V/
The leak potential E L is a key parameter for introducing bursting in such a model.
Calcium Channels Ca 2C
1.8.3
The equation of the current of the associated channel is given by the constant
field equation, given in Eq. ( 1.86 ). Calcium channels are distinguished in two large
categories:
1. T-type calcium currents I Ca;T which exhibit inactivation with respect to parame-
ter h in the current equation I Channel D m p h q I drive .V/.
2. L-type calcium currents I Ca;L which do not exhibit inactivation with respect to
parameter h in the current equation I Channel D m p h q I drive .V/.
T -currents exhibit bursts and subthreshold oscillations. They are described by a set
of equations of the form
C d dt
D I o g L .V E L / I T
dh
dt D .h 1 .V/ h/= h .V/
I T D m 1 .V/ 2 hIcfe.V;ŒCa o ;ŒCa i /
(1.91)
m 1 .V/ D 1=.1 C exp..V C 59/=6:2//
h 1 .V/ D 1=.1 C exp..V C 83/=4//
h .V/ D 22:7 C 0:27=Œexp..V C 48/=4/ C exp..V C 407=50//
The response of this model depends on the activation variable m and on the
deactivation variable h.
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