Information Technology Reference
In-Depth Information
0
@
1
A
0
@
1
A
0
@
1
A
0
@
1
A
z
1
z
2
z
3
z
4
z
5
z
6
010000
001000
000000
000010
000001
000000
z
1
z
2
z
3
z
4
z
5
z
6
00
00
10
00
00
01
v
1
v
2
D
C
(5.47)
0
1
z
1
z
2
z
3
z
4
z
5
z
6
@
A
z
m
1
z
m
2
100000
000110
D
(5.48)
The control law that allows FRQ protein concentration converge to desirable levels
is given by
v
1
D x
.3/
1;d
K
1
.x
1
x
1
/ K
2
.x
1
x
1
/ K
3
.x
1
x
1
/
v
2
D x
.3/
(5.49)
1;d
K
1
.x
4
x
4
/ K
2
.x
4
x
4
/ K
3
.x
4
x
4
/
By knowing the control inputs .
v
1
;
v
2
/
T
one can compute the control inputs .
u
1
;
u
2
/
T
that are finally applied to the system of the coupled circadian neurons. By setting
L
f
h
1
.x/
L
f
h
2
.x/
!
L
g
1
L
f
h
1
.x/ L
g
2
L
f
h
1
.x/
L
g
1
L
f
h
2
.x/ L
g
2
L
f
h
2
.x/
!
f D
G D
(5.50)
5.6
Robust Synchronization of Coupled Circadian
Oscillators Using Differential Flatness Theory
In case that there is coupling between circadian oscillators (neurons) it is also
possible to find a linearizing and decoupling control law based on differential
flatness theory. Without loss of generality the model of two coupled circadian
neurons is considered. The state vector of this model now comprises the following
state variables: x
1
: mRNA concentration for the first neuron, x
2
: concentration of the
FRQ protein in the cytoplasm of the first neuron, x
3
: concentration of FRQ protein
in the nucleus of the first neuron, x
4
: concentration of mRNA of the second neuron,
x
5
: concentration of the FRQ protein in the cytoplasm of the second neuron, x
6
:
concentration of FRQ protein in the nucleus of the second neuron.
K
i
K
i
C x
3
v
m
x
1
K
M
C x
1
x
1
D
v
s
(5.51)
