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With the application of differential geometric methods and differential flatness
theory it was shown that the nonlinear model of the FitzHugh-Nagumo neuron
can be written in the linear canonical form. It was also shown that by applying
differential geometric methods and by computing Lie derivatives exact linearization
of the model of the coupled FitzHugh-Nagumo neurons (connected through a gap
junction) can be succeeded. Moreover, it was proven that the model of the coupled
FitzHugh-Nagumo neuron is a differentially flat one and by defining appropriate flat
outputs it can be transformed to the MIMO linear canonical form. For the linearized
representation of the coupled neuron's model the design of a feedback control is
possible and synchronization between the two neurons can be attained.
Next, the problem of synchronization of the coupled neurons under external
perturbations and parametric uncertainties was examined. To obtain simultaneous
state and disturbances estimation a disturbance observer based on the Derivative-
free nonlinear Kalman Filter has been used. The Derivative-free nonlinear Kalman
Filter consists of the standard Kalman Filter recursion on the linearized equivalent
model of the coupled neurons and on computation of state and disturbance estimates
using the diffeomorphism (relations about state variables transformation) provided
by differential flatness theory. After estimating the disturbance terms in the neurons'
model their compensation has become possible.
The performance of the synchronizing control loop has been tested through
simulation experiments. It was shown that the proposed method assures that the
neurons will remain synchronized, despite parametric variations and uncertainties in
the associated dynamical model and despite the existence of external perturbations.
The method can be extended to multiple interconnected neurons, thus enabling to
succeed more complicated patterns in robot's motion.
