Systems Theory for the Analysis of Biological Neuron Dynamics - Advanced Models of Neural Networks - page 42

Fig. 2.13 Phase diagram of
state variables x 1 , x 2 of a
second order linear
autonomous system with
identical stable eigenvalues

10

8

6

4

2

0

−2

−4

−6

−8

−10

−15 −10 −5 0 5 10 15

x 1

Fig. 2.14 Phase diagram of
state variables x 1 , x 2 of a
second order nonlinear
oscillator that exhibits
multiple equilibria

6

4

2

0

−2

−4

−6

−30 −20 −10 0 10 20 30

x 1

where

@f 1 .x 1 ;x 2 /
@x 1

@f 1 .x 1 ;x 2 /
@x 2

˛ 11 D j x 1 Dp 1 ;x 2 Dp 2 ˛ 12 D j x 1 Dp 1 ;x 2 Dp 2

(2.54)

@f 2 .x 1 ;x 2 /
@x 1

@f 2 .x 1 ;x 2 /
@x 2

˛ 21 D j x 1 Dp 1 ;x 2 Dp 2 ˛ 22 D j x 1 Dp 1 ;x 2 Dp 2

For the equilibrium it holds:

f 1 .p 1 ;p 2 / D 0
f 2 .p 1 ;p 2 / D 0

(2.55)

Next, by defining the new variables y 1 D x 1 p 1 and y 2 D x 2 p 2 one can rewrite
the state-space equation