Information Technology Reference
In-Depth Information
By denoting as
v
D x
1
C x
2
)y
1
D.x
1
C x
2
/ C Œx
2
.x
2
a/.1
x
2
/x
1
C
u
, one arrives again at a state space description of the system in the linear
canonical form. Thus setting
z
1
D y
1
and
z
2
Dy
1
D y
2
one gets Eq. (
4.40
)
z
1
z
2
01
00
z
1
z
2
0
1
u
D
C
(4.46)
For the linearized neuron model one can define a feedback control signal using that
y D
v
.
v
Dy
d
k
d
. y y
d
/ k
p
.y y
d
/)
(4.47)
and using that L
f
h.x/ C L
g
L
f
h.x/
u
D
v
one has
u
D .
v
L
f
h.x//=L
g
L
f
h.x/
(4.48)
4.10
Linearization of Coupled FitzHugh-Nagumo Neurons
Using Differential Geometry
The following system of coupled neurons is considered
dV
1
dt
D V
1
.V
1
˛/.1 V
1
/
w
1
C g.V
1
V
2
/ C I
1
dw
1
dt
D .V
1
w
1
/
(4.49)
dV
2
dt
D V
2
.V
2
˛/.1 V
2
/
w
2
C g.V
2
V
1
/ C I
2
dw
2
dt
D .V
2
w
2
/
The following state variables are defined x
1
D
w
1
, x
2
D V , x
3
D
w
2
, x
4
D
V
2
,
u
1
D I
1
and
u
2
D I
2
(Fig.
4.8
). Thus, one obtains the following state-space
description
dx
1
dt
Dx
1
C x
2
dx
2
dt
D x
2
.x
2
a/.1 x
2
/ x
1
C g.x
2
x
1
/ C
u
1
(4.50)
dx
3
dt
Dx
3
C x
4
dx
4
dt
D x
4
.x
4
a/.1 x
4
/ x
3
C g.x
4
x
2
/ C
u
2
while the measured output is taken to be
y
1
D h
1
.x/ D x
1
y
2
D h
2
.x/ D x
3
(4.51)
