Bifurcations and Limit Cycles in Models of Biological Systems - Advanced Models of Neural Networks - page 71

3 j 1 175:03

2 j55:12 88:17

(3.74)

1 j176:63 0

0 j88:17

From the changes of sign of the coefficients of the first column of the Routh tables
it can be concluded that polynomial p 2 ./ is unstable. The associated Routh table
for the third Kharitonov polynomial p 3 ./ is

3

j 1 163:82

2

j55:12 88:17

(3.75)

1 j165:42 0

0 j88:17

Since there is change of sign of the coefficients of the first column of the Routh
table it can be concluded that the characteristic polynomial p 3 ./ is an unstable one.
Finally, the associated Routh table for the fourth Kharitonov polynomial p 4 ./ is

3 j 1 175:03

2 j58:12 84:13

(3.76)

1

j176:48 0

0

j84:13

From the changes of sign of the coefficients of the first column of the Routh tables
it can be concluded that the characteristic polynomial p 4 ./ is unstable. From the
previous analysis it can be seen that all Kharitonov polynomials have unstable roots;
therefore, the fixed point .x ; 1 ;x ; 2 ;x ; 3 / of the circadian oscillator under feedback
gain c 2 2Œ270:2; 260:2 is an unstable one.

3.7 Simulation Tests

The following example is concerned with bifurcations of the fixed point of
the FitzHugh-Nagumo neuron. Assuming zero external current input and
known parameters of the model it has been found the associated fixed point is
.x ; 1 ;x ; 2 ;x ; 2 / D .0;0;0/. As confirmed by the results given in Fig. 3.2 this fixed
point is a stable one. The nominal values of the model's parameters were D 1:5,
D 0:1, and a D 7.
Next, examples on the stability of fixed points in the circadian oscillator model
are given under variation of specific parameters of the dynamical model. The
nominal values of the model's parameters were: v s D 0:02, K i D 0:9, v m D 1:5,
K m D 1:6, K s D 1, v d D 1:4, K 1 D 0:5, K 2 D 0:6, n D 4, while K d