Bifurcations and Limit Cycles in Models of Biological Systems - Advanced Models of Neural Networks - page 67

Feedback can change the oscillation characteristics of the circadian model. It can
change the type of stability characterizing the system's fixed points. Additionally
fixed points bifurcations can be altered through state feedback control. The previ-
ously analyzed Eq. ( 3.33 ) is considered again for computing fixed points

v m k 1 .K d 1/x 2 C Πv m K i K 1 K 2 K d v m K i K 1 K 2 C v m K 1 v d v s K i K 1 K 2 K d
C v s K i K 1 K 2 x 2 C Πv m K i K 2 v d v s K i K m K s K 2 v s K 1 K 2 . v d C K 1 K d /
C v s K i K 1 K 2 K d x 2 v s K i K m K 2 K s K d

(3.55)

Next, the nominal value of K d will be considered known and will be substituted
in the characteristic polynomial, while the input K s will be considered equal to the
feedback law of the form K s D c 2 x 2 . In such a case the equation for computing x 2
becomes

v m k 1 .K d 1/x 2
CΠv m K i K 1 K 2 K d v m K i K 1 K 2 C v m K 1 v d v s K i K 1 K 2 K d C v s K i K 1 K 2
v s K i K m c 2 K 2 x 2 C Πv m K i K 2 v d v s K 1 K 2 . v d C K 1 K d / C v s K i K 1 K 2 K d
v s K i K m K 2 c 2 K d x 2

(3.56)

Substituting numerical values for the model's parameters and taking K d D 1:4 the
equation for computing x 2 becomes

x 2 f0:15x 2 C .1:1665 C 0:0126c 2 /x 2 C .0:8157 C 0:0176c 2 /gD0 (3.57)

The fixed points for the model of the circadian cells are found to be:
First fixed point : .x ; 1 ;x ; 2 ;x ; 3 / D .0;0;0/ whereas from the computation
of the associated determinant one obtains D ..1:6665 C 0:0126c 2 / 2 C

0:6.0:8157 C 0:0176c 2 // which is positive if c 2 takes values in the intervals
c 2 < 251:15 and c 2 > 82:39. Thus, for >0one has the fixed points
expressed as functions of the feedback gain c 2
Second fixed point :

K 1 .K d 1/x 2 ;2

x ;2
1

C v d

D

c 2 .k d C1/

.1:665 C 0:0126c 2 / C p 1:58 10 4 c 2 C 0:0527c 2 C 3:2694
0:3

(3.58)

x ;2
2

D

x ;2
3

K 1

K 2 x 2 ;2

D