Information Technology Reference
In-Depth Information
By defining the state variables x
1
D
w
, x
2
D V and setting current I as the external
control input one has
dx
1
dt
Dx
1
C x
2
dx
2
dt
D x
2
.x
2
a/.1 x
2
/ x
1
C
u
(3.19)
y D h.x/ D x
1
Equivalently one has
x
1
x
2
x
1
C x
2
x
2
.x
2
a/.1 x
2
/ x
1
0
1
u
D
C
(3.20)
y D h.x/ D x
1
Next, the fixed points of the dynamic model of the FitzHugh-Nagumo neuron are
computed. It holds that
x
1
Dx
1
C x
2
x
2
D x
2
.x
2
˛/.1 x
2
/ x
1
C I
(3.21)
Setting I D 0, and setting
x
1
D 0 and
x
2
D 0 (nullclines), the associated fixed
points are found
x
2
Dx
1
x
1
.
3
x
1
C .a
2
(3.22)
C
2
/x
1
˛ 1/ D 0
The numerical values of the parameters are taken to be D 0:2 and a D 7. Then by
substituting the relation for x
2
from the first row of Eq. (
3.22
) into the second row
one obtains
x
1
Œ
3
x
1
C
2
.1 C a/x
1
C .a 1/ D 0
(3.23)
From the above conditions one has the first fixed point
x
;1
1
D 0x
;1
2
(3.24)
D 0
whereas, since the determinant D
4
.1 C a/ 2a
4
4
3
/<0no other real
valued solutions for .x
1
;x
2
/ can be found and consequently no other fixed points
can be considered.
Next, the type of stability that is associated with the fixed point x
;1
1
D 0; x
;
2
D
0 is examined. The Jacobian matrices of the right-hand side of Eq. (
3.21
)are
computed and the associated characteristic polynomials are found