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Consequently, all state variables x
i
;iD 1; ;4and all control inputs
u
i
i D 1;2
of the model of the coupled FitzHugh-Nagumo neurons can be written as functions
of the flat outputs y
i
i D 1;2 and their derivatives. Therefore, the model of the
coupled neurons is a differentially flat one.
4.11.2
Linearization of the Coupled Neurons Using
Differential Flatness Theory
The problem of linearization and decoupling of the interconnected FitzHugh-
Nagumo neural oscillators can be efficiently solved using differential flatness theory.
For the model of the coupled FitzHugh-Nagumo neurons described in Eq. (
4.50
)the
flat outputs y
1
D h
1
.x/ D x
1
and y
2
D h
2
.x/ D x
2
have been defined. It holds that
y
1
Dx
1
)y
1
D f
1
)y
1
Dx
1
C x
2
(4.77)
and
@f
1
@x
4
x
4
)
y
1
D./x
1
C x
2
C 0x
3
C 0x
4
)
y
1
D./f
1
C .f
2
C
u
1
/)
y
1
D ./.x
1
C x
2
/ C Œx
2
.x
2
a/.1 x
2
/ x
1
C g.x
2
x
4
/ C
u
1
)
(4.78)
@f
1
@x
1
x
1
C
@f
1
@x
2
x
2
C
@f
1
@x
3
x
3
C
y
1
D
or equivalently
y
1
D L
f
h
1
.x/ C L
g
1
L
f
h
1
.x/
u
1
C L
g
2
L
f
h
1
.x/
u
2
(4.79)
where
L
f
h
1
.x/ D ./.x
1
C x
2
/ C Œx
2
.x
2
a/.1 x
2
/ x
1
C g.x
2
x
4
/
L
g
1
L
f
h
1
.x/ D L
g
2
L
f
h
1
.x/ D 0
(4.80)
Equivalently, one computes
y
2
Dx
3
)y
2
D f
3
)y
2
Dx
3
C x
4
(4.81)
and
@f
3
@f
3
@f
3
@f
3
@x
4
x
4
)
y
2
D 0x
1
C 0x
2
C ./x
3
C x
4
)
y
2
D ./f
3
C .f
4
C
u
2
/)
y
2
D ./.x
3
C x
4
/ C Œx
4
.x
4
a/.1 x
4
/ x
3
C g.x
4
x
2
/ C
u
2
)
y
2
D L
f
h
2
.x/ C L
g
1
L
f
h
2
.x/
u
1
C L
g
2
L
f
h
2
.x/
u
2
y
2
D
@x
1
x
1
C
@x
2
x
2
C
@x
3
x
3
C
(4.82)
