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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 Π3 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
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