Information Technology Reference
In-Depth Information
@f
1
@x
1
!
1 3x
2
C 2.1 C a/x
2
a
@f
1
@x
2
J D
D
(3.25)
@f
2
@x
1
@f
2
@x
2
Stability at the fixed point x
1
D 0 and x
2
D 0: Substituting the values of x
1
and x
2
in Eq. (
3.27
) the obtained characteristic polynomial is p./ D
2
C .a C / C
.˛ C /.
Next, it is considered that a2Œa
min
;a
max
D Œ6:8;7:2, whereas D 1:5 and
D 0:2 are crisp numerical values. Then for the coefficients of the characteristic
polynomial p.s/ D
2
C k
1
C k
2
one has k
1
D .a C /
2
Œ7:1;7:5 and
k
2
D .a/ C 2 Œ3:54;3:66.
The application of the Routh-Hurwitz criterion gives
2
j 1k
2
1
j k
1
0
(3.26)
0
j k
2
Since k
1
>0and k
2
>0there is no change of sign in the first column of the Routh
matrix and the roots of the characteristic polynomial p./ are stable. Therefore, the
equilibrium .0;0/ is a stable one.
3.4.2
Condition for the Appearance of Limit Cycles
Finally, it is noted that the appearance of the limit cycles in the FitzHugh-Nagumo
neuron model is possible. For instance, if state feedback is implemented in the form
I Dqx
2
, then .x
1
;x
2
/ D .0;0/ is again a fixed point for the model while the
Jacobian matrix J at .x
1
;x
2
/ becomes
1 3x
2
C 2.1 C a/x
2
a q
J D
(3.27)
The characteristic polynomial of the above Jacobian matrix is p./ D
2
C .a C
q C C .q C //. Setting the feedback gain q Da the characteristic
polynomial becomes p./ D
2
C.
2
2
C˛/. Then, the condition .˛/ >
0 with a>0, >0and >0suffices for the appearance of imaginary eigenvalues.
In such a case the neuron model exhibits sustained oscillations (limit cycles).
