Synchronization of Circadian Neurons and Protein Synthesis Control - Advanced Models of Neural Networks - page 122

Since x 2 , x 4 , x 5 , and x 6 are functions of the flat output and of its derivatives it holds
that K s 2 D f 6 .y 1 ; y 1 ;y 2 ; y 2 /.
From the above analysis, for the model of the coupled circadian oscillators it
holds that all state variables and the control inputs are functions of the flat output
and of its derivatives. Therefore, this system is a differentially flat one.
As in the case of the isolated circadian neuron here it holds

K i

x 1
K M Cx 1

x 1 D v 3

K i Cx 3 v m

(5.64)

and deriving with respect to time gives

x 1 D v 3 K i nx n1

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

v s

K i K m

.K M C x 1 / 2

x 1
K m C x 1

v m

K i C x 3 v m

(5.65)

while the relation for x .3/
1

is modified by including in it the coupling term K c .x 2

x 5 / which appears now in x 2 as described in Eq. ( 5.52 ). This results into

x .3/
1

D f A .y 1 ; y 1 ;y 2 ; y 2 /
Cg A 1 .y 1 ; y 1 ;y 2 ; y 2 / u 1 C g A 2 .y 1 ; y 1 ;y 2 ; y 2 / u 2 (5.66)

u 1 D K s 1 and u 2 D K s 2 .Forf A .y 1 ; y 1 ;y 2 ; y 2 /, g A 1 .y 1 ; y 1 ;y 2 ; y 2 /, and

g A 2 .y 1 ; y 1 ;y 2 ; y 2 / one has

f A .y 1 ; y 1 ;y 2 ; y 2 /
D

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

K m Cx 1 i .K m Cx 1 / 2 o

K i

x 1 K m K m

K i Cx 3 v m

C v m

.K m Cx 1 / 2 v m .K m Cx 1 / 4

h v s

K m Cx 1 i

K i

x 1

K i Cx 3 v m

n v s K i nx n 1

.K i Cx 3 / 2 K 1 oh K d Cx 2 K 1 x 2 C K 2 x 3 C K c .x 2 x 5 / i

x 2 3

v d

(5.67)

n v m

.K m Cx 1 / 2 v s K i nx n 1

K m

C

3
.K i Cx 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 Cx 3 / 4

.K 1 x 2 K 2 x 3 /

.K i Cx 3 / 2 K 2 o ŒK 1 x 2 K 2 x 3

C v s K i nx n 1

3

g A 1 .y 1 ; y 1 ;y 2 ; y 2 / D v s K i nx .n1/

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

(5.68)

g A 2 .y 1 ; y 1 ;y 2 ; y 2 / D 0 (5.69)