Bifurcations and Limit Cycles in Models of Biological Systems - Advanced Models of Neural Networks - page 61

3.5 Bifurcation Analysis of Circadian Oscillators

3.5.1 Fixed Points Bifurcation Analysis Using

Kharitonov's Theory

The model of the circadian cells is given by

K i

x 1
K m C x 1

x 1 D v s

K i C x 3 v m

(3.28)

x 2
K d C x 2 K 1 x 2 C K 2 x 3 C x 1 K s

x 2 D v d (3.29)

x 3 D K 1 x 2 K 2 x 3 (3.30)

The model's parameters are defined as follows: v s denotes the rate of transcription
of the frq gene into FRQ protein inside the nucleus. K i represents the threshold
constant beyond which nuclear FRQ protein inhibits transcription of the frq gene. n
is the Hill coefficient showing the degree of cooperativity of the inhibition process,
v m is the maximum rate of frq mRNA degradation, K M is the Michaelis constant
related to the transcription process. Parameter K s defines the rate of FRQ synthesis
and stands for the control input to the model. Parameter K 1 denotes the rate of
transfer of FRQ protein from the cytoplasm to the nucleus and K 2 denotes the
rate of transfer of the FRQ protein from the nucleus into the cytoplasm. Parameter
v d denotes the maximum rate of degradation of the FRQ protein and K d is the
Michaelis constant associated with this process.
Fixed points are computed from the nullclines

x 1 D 0, x 2 D 0, and x 3 D 0.

From Eq. ( 3.28 ) one has

K 1

x 3 D

K 2 x 2

(3.31)

By substituting Eq. ( 3.31 ) and after intermediate operations one obtains x 1 also as a
function of x 2 that is

K 2 . v d C K 1 K d /x 2 C K 1 K 2 K d x 2 .K 2 K d C K 2 x 2 /K 1 x 2
K 2 .K s K d CK s x 2 /

(3.32)

x 1 D

By substituting Eqs. ( 3.31 ) and ( 3.32 ) into Eq. ( 3.28 ) and after intermediate

operations one obtains the equation

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
CK i K 1 K 2 K d x 2 v s K i K m K 2 K s K d

(3.33)