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where
L f h 2 .x/ D ./.x 3 C x 4 / C Œx 4 .x 4 a/.1 x 4 / x 3 C g.x 4 x 2 /
L g 1 L f h 2 .x/ D 0L g 2 L f h 2 .x/ D
(4.83)
By defining the state variables z 1 D y 1 , z 1 Dy 1 , z 1 D y 2 and z 1 Dy 2 one obtains a
description of the coupled neurons in the state-space form
L f h 1 .x/
L f h 2 .x/
!
z 2
z 2
L g 1 L f h 1 .x/ L g 2 L f h 1 .x/
L g 1 L f h 2 .x/ L g 2 L f h 2 .x/
u 1
u 2
D
C
(4.84)
z D D m C G m u
(4.85)
Denoting
v 1 D L f h 1 .x/ C L g 1 L f h 1 .x/ u 1 C L g 2 L f h 1 .x/ u 2
v 2 D L f h 2 .x/ C L g 1 L f h 2 .x/ u 1 C L g 2 L f h 2 .x/ u 2
one also has
z 1
z 1
v 1
v 2
D
(4.86)
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//.
4.12
State and Disturbances Estimation
with the Derivative-Free Nonlinear Kalman Filter
4.12.1
Kalman and Extended Kalman Filtering
It will be shown that a new nonlinear filtering method, the so-called Derivative-
free nonlinear Kalman Filter, can be used for estimating wave-type dynamics
in the neuron's membrane. This can be done through the processing of noisy
measurements and without knowledge of boundary conditions. Other results on the
application of Kalman Filtering in the estimation of neuronal dynamics can be found
in [ 77 , 99 , 134 , 161 ]
An overview of Kalman and Extended Kalman Filtering is given first. In the
discrete-time case a dynamical system is assumed to be expressed in the form of a
discrete-time state model [ 166 ]:
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