Information Technology Reference
In-Depth Information
one also has
z
1
z
1
v
1
v
2
D
(4.71)
where the control inputs
v
1
and
v
2
are chosen as
v
1
D
z
1d
k
d
.
z
1d
z
1
/k
p
.
z
1d
z
1
/
and
v
2
D
z
1d
k
d
.
z
1d
z
1
/ k
p
.
z
1d
z
1
/.
Therefore, the associated control signal is
u
D G
m
.
v
D
m
.x//.
4.11
Linearization of Coupled FitzHugh-Nagumo Neurons
Using Differential Flatness Theory
4.11.1
Differential Flatness of the Model of the Coupled
Neurons
It can be proven that the model of the coupled FitzHugh-Nagumo neurons described
in Eq. (
4.50
) is a differentially flat one. The following flat outputs are defined
y
1
D h
1
.x/ D x
1
y
2
D h
2
.x/ D x
3
(4.72)
From the first row of the dynamical model of the coupled neurons one has
y
1
C
y
1
(4.73)
y
1
Dy
1
C x
2
)x
2
D
Similarly, from the third row of the dynamical model of the coupled neurons one has
y
2
C
y
2
y
2
Dy
2
C x
4
)x
4
D
(4.74)
Thus all state variables x
i
;iD 1; ;4 can be written as functions of the flat
outputs and their derivatives.
From the second row of the dynamical model of the coupled neurons one has
u
1
Dx
2
x
2
.x
2
a/.1 x
2
/ C x
1
g.x
2
x
1
/)
u
1
D f
1
.y
1
; y
1
:y
2
; y
2
/
(4.75)
Similarly, from the second row of the dynamical model of the coupled neurons
one has
u
2
Dx
4
x
4
.x
4
a/.1 x
4
/ C x
3
g.x
4
x
2
/)
u
2
D f
2
.y
1
; y
1
:y
2
; y
2
/
(4.76)