Bifurcations and Limit Cycles in Models of Biological Systems - Advanced Models of Neural Networks - page 65

where coefficients c n.2jm/ with j D 1;2; ;n and m D 1;2; ;n become
zero if n .2j m/ < 0 or if n .2j m/ > n.
The Routh-Hurwitz determinants show the existence of eigenvalues which are
conjugate imaginary numbers. A series of matrices containing the coefficients c j
are defined as follows:

1 D c n1

c n1 c n
c n3 c n2

2 D

(3.48)

0 1

c n1 c n 0
c n3 c n2 c n1
c n5 c n4 c n3

@ A

3 D

and so on. Each matrix j ;j D 1; ;n1 is square and the first column contains
every other coefficient of the characteristic polynomial c n1 ;c n3 ; . The roots of
the characteristic polynomial p./ have negative real parts if and only if det. j />
0 for j D 1; ;n. The following two remarks can be made:

(i) Since det. n / D c 0 det. n1 / this means that a necessary condition for the
roots of p./ to have negative real part is c 0 >0.Ifc 0 D 0, then there is a zero
eigenvalue.
(ii) If det. j />0for all j D 1;2; ;n 2 and det. n1 / D 0, then
the characteristic polynomial contains imaginary roots. Provided that the rest
of the roots of the characteristic polynomial have negative real part then
Hopf bifurcations appear. An additional condition for the appearance of Hopf
bifurcations is that the derivative of the determinant det. n1 / D 0 with respect
to the bifurcating parameter is not zero. That is d det. n1 / ¤ 0.
According to the above criteria (i) and (ii) one can determine where possible saddle-
node bifurcations (eigenvalue 0) and Hopf bifurcations (imaginary eigenvalues)
appear. Therefore, one can formulate conditions about the values of the bifurcating
parameter that result in a particular type of bifurcations.
Equivalently, conditions about the appearance of Hopf bifurcations can be
obtained from the Routh table according to Sect. 3.2 . The zeroing of certain rows
of the Routh table gives an indication that the characteristic polynomial contains
imaginary conjugate eigenvalues. By requiring the elements of these rows of the
Routh matrix to be set equal to zero one finds the values of the bifurcating parameter
that result in the appearance of Hopf bifurcations.
For the previous characteristic polynomial of the circadian cells one has

C Πv K d C K 1 C K 2 C

v M

C ΠK 2 v d
K d K M . v K d C K 1 C K 2 / C

v M v M

K M

K 2 v d
K d
(3.49)

p./D 3

K M 2

C

which after the substitution of numerical values for its parameters becomes