Bifurcations and Limit Cycles in Models of Biological Systems - Advanced Models of Neural Networks - page 68

Third fixed point :

K 1 .K d 1/x 2 ;3

x ;3
1

C v d

D

c 2 .k d C1/

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

(3.59)

x ;3
2

D

x ;3
3

K 1

K 2 x 2 ;3

D

The bifurcation diagram of the fixed points considering as bifurcating parameter the
feedback control gain c 2 is given in Fig. 3.1 b. Next, from the right part of Eq. ( 3.54 )
the system's Jacobian is computed

0 1

v s K i nx n 1

K m
.K m Cx 1 / 2

v m 0

3
.K i Cx 3 / 2

@ A

J D (3.60)

K d

c 2 x 2 v d

.K d Cx 2 / 2 K 1 C c 2 x 1

K 2

0 K 1 K 2

From the system's Jacobian a generic form of the characteristic polynomial is
computed at the fixed point .x 1 ;x 2 ;x 2 /. After intermediate operations one obtains

nh k 2 C

x 2 2 K 1 c 2 x 1 i

h v m C

x 1 2 io 2

v d k d
K d

K m
K m

det.I J/D 3 C C

nh

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

h v m C

.K m C x 1 / 2 ih K 2 C

C

.K d C x 2 / 2 K 1 c 2 x 1 i

C

.K i C x 3 / 2 o

v s K i nx n 1
3 v d k d K m v d K d

C C

nh v m C

x 1 2 ih v d K d

.K d C x 2 / 2 K 1 c 2 x 1

K 1 K 2 i

.K i C x 3 / 2 v d K d

C K 1 c 2 x 2 o
(3.61)

v s K i nx n 1
3 K m

K m C

C

x 2 2

K d C

or equivalently

det.I J/D 3 C c f 2 2

C c f 1 C c f 0 (3.62)

where

v m C

K m C x 1 2

.K d C x 2 / 2 K 1 c 2 x 1

K 1 K 2

K m v d K d

c f 0 D

v d K d

K d C x 2 2 C K 1 c 2 x 2

v s K i nx n1

3
.K i C x 3 / 2

(3.63)

.K d C x 2 / 2 K 1 K 1 K 2

v m C

.K m C x 1 / 2

v d k d K m

c f 1 D C

K 2 C v d K d

.K d C x 2 / 2 K 1 c 2 x 1

v s K i nx n1

3
.K i C x 3 / 2

(3.64)