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gate is open. It is noted that independently from the model derived by Hodgkin and
Huxley it is possible to obtain the values of the parameters of the neuron membrane's
model through nonlinear estimation and Kalman Filtering methods [
84
,
99
,
175
].
The change in time of parameters m, h, and n in time is given by
dn
dt
D a
n
.V/.1 n/ ˇ
n
.V/n D .n
1
.V/ n/=
n
.V/
dm
dt
(1.49)
D a
m
.V/.1 m/ ˇ
m
.V/m D .m
1
.V/ m/=
m
.V/
dh
dt
D a
h
.V/.1 h/ ˇ
h
.V/h D .h
1
.V/ h/=
h
.V/
For the boundary conditions of the aforementioned differential equations one has
(denoting the general variable X D n, m or h) that
a
x
.V/
a
x
.V/Cˇ
x
.V/
X
.V/ D
1
a
x
.V/Cˇ
x
.V/
X
1
.V/ D
(1.50)
Moreover, the following parameter values have been experimentally used [
16
]
D 120 mS/cm
3
g
K
D 36 mS/cm
3
g
L
D 0:3 mS/cm
3
g
Na
E
Na
D 50 mV E
K
D77 mV E
L
D 54:4 mV
a
n
.V/ D 0:01.V C 55/=.1 exp..V C 55/=10//
ˇ
n
.V/ D 0:125exp..V C 65/=80/
a
m
.V/ D 0:1.V C 40/=.1 exp..V C 40/=10//
ˇ
m
.V/ D 4exp..V C 65/=18/
a
h
.V/ D 0:07exp..V C 65/=20/
ˇ
h
.V/ D 1=.1 C exp..V C 35/=10//
(1.51)
1.4.2
Outline of the Hodgkin-Huxley Equations
By considering that there is no spatial variation of V.x;t/, i.e. that
@V.x;t/
@t
D 0, and
moreover that the conductances g
k
, g
Na
, and g
L
vary in time, the Hodgkin-Huxley
equation becomes an ordinary differential equation
c
M
d
dt
Dg
Na
m
3
h.V E
Na
/ g
K
n
4
.V E
K
/ g
L
.V E
L
/
dn
dt
D Œa
n
.V/.1 n/ ˇ
n
.V/n
(1.52)
dm
dt
D Œa
m
.V/.1 m/ ˇ
m
.V/m
dh
dt
D Œa
h
.V/.1 h/ ˇ
h
.V/h
Coefficient varies with the system's temperature according to the relation
D Q
10
.T T
base
/=10
where T
base
D 6:3
ı
C and Q
10
D 3
(1.53)
