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one also has
z 1
z 1
v 1
v 2
D
(4.71)
where the control inputs v 1 and v 2 are chosen as v 1 D z 1d k d . z 1d z 1 /k p . z 1d z 1 /
and v 2 D z 1d k d . z 1d z 1 / k p . z 1d z 1 /.
Therefore, the associated control signal is u D G m . v D m .x//.
4.11
Linearization of Coupled FitzHugh-Nagumo Neurons
Using Differential Flatness Theory
4.11.1
Differential Flatness of the Model of the Coupled
Neurons
It can be proven that the model of the coupled FitzHugh-Nagumo neurons described
in Eq. ( 4.50 ) is a differentially flat one. The following flat outputs are defined
y 1 D h 1 .x/ D x 1
y 2 D h 2 .x/ D x 3
(4.72)
From the first row of the dynamical model of the coupled neurons one has
y 1 C y 1
(4.73)
y 1 Dy 1 C x 2 )x 2 D
Similarly, from the third row of the dynamical model of the coupled neurons one has
y 2 C y 2
y 2 Dy 2 C x 4 )x 4 D
(4.74)
Thus all state variables x i ;iD 1; ;4 can be written as functions of the flat
outputs and their derivatives.
From the second row of the dynamical model of the coupled neurons one has
u 1 Dx 2 x 2 .x 2 a/.1 x 2 / C x 1 g.x 2 x 1 /) u 1 D f 1 .y 1 ; y 1 :y 2 ; y 2 /
(4.75)
Similarly, from the second row of the dynamical model of the coupled neurons
one has
u 2 Dx 4 x 4 .x 4 a/.1 x 4 / C x 3 g.x 4 x 2 /) u 2 D f 2 .y 1 ; y 1 :y 2 ; y 2 /
(4.76)
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