Synchronization of Circadian Neurons and Protein Synthesis Control - Advanced Models of Neural Networks - page 116

.K m C x 1 / 2 v s K i nx n1

K m

.K i C x 3 / 2 x 3 v m ΠK m 2.K m C x 1 /

x 1
K m C x 1

3

v m x 1 v m

.K m C x 1 / 4

K m
.K m C x 1 / 2 v m

K m
.K m C x 1 / 2 x 1

C v m (5.22)

By substituting the relations about the derivatives x 1 , x 2 , and x 3 and after performing

intermediate operations one finally obtains

v m v s

K i
K i C x 3 v m

K m 2.K m C x 1 /
.K m C x 1 / 4

x 1
K m C x 1

x .3/
1

D

v s

K i
K i C x 3

K m
.K m C x 1 / 2 v m

K m
.K m C x 1 / 2

C v m

v s K i nx n1

.K i C x 3 / 2 K 1 K d C x 2 K 1 x 2 C K 2 x 3

x 1
K m C x 1

x 2

3

v m v d

v m

.K m C x 1 / 2 v s K i nx n1

K m

3
.K i C x 3 / 2
v s K i n.n 1/x n 3 .K i C x 3 / K i nx n 3 2.K i C x 3 /
.K i C x 3 / 4

C

.K 1 x 2 K 2 x 3 /

.K i C x 3 / 2 K 2 ŒK 1 x 2 K 2 x 3

C v s K i nx n1

3

v s K i nx n1

.K i C x 3 / 2 K 1 x 1 K s

3

(5.23)

By defining the control input u D K s and the following two functions

n v m K m 2.K m C x 1 /
.K m C

h v s i

x 1 / 2 oh v s

i

K i
K i C

K i
K i C

x 1
K m C

K m
.K m C

K m
.K m C

f.x/D

x 3 v m

C v m

x 1 / 2 v m x 3 x 1 / 4 x 1

n v s K i nx n 1

x 3 / 2 K 1 oh x 2 K 1 x 2 C K 2 x 3 i

x 1
K m C

x 2
K d C

v m

x 1

3
.K i C

v d

n v m

.K m C x 1 / 2 v s K i nx n 1

K m

C

3
.K i C x 3 / 2
v s K i n.n 1/x n 3 .K i C x 3 / K i nx n 3 2.K i C x 3 /
.K i C

.K 1 x 2 K 2 x 3 /

x 3 / 4

x 3 / 2 K 2 o ŒK 1 x 2 K 2 x 3

C v s K i nx n 1

3
.K i C

(5.24)

v s K i nx n 1

.K i C x 3 / 2 K 1 x 1

3

g.x/ D (5.25)