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Chapter 9
Synchronization of Chaotic and Stochastic
Neurons Using Differential Flatness Theory
Abstract This chapter presents a control method for neuron models that have the
dynamics of chaotic oscillators. The proposed chaotic control method makes use
of a linearized model of the chaotic oscillator which is obtained after transforming
the oscillators dynamic model into a linear canonical form through the application
of differential flatness theory. The chapter also analyzes a synchronizing control
method (flatness-based control) for stochastic neuron models which are equivalent
to particle systems and which can be modeled as coupled stochastic oscillators. It is
explained that the kinematic model of the particles can be derived from the model of
the quantum harmonic oscillator (QHO). It is shown that the kinematic model of the
particles is a differentially flat system. It is also shown that after applying flatness-
based control the mean of the particle system can be steered along a desirable path
with infinite accuracy, while each individual particle can track the trajectory within
acceptable accuracy levels.
9.1
Chaotic Neural Oscillators
9.1.1
Models of Chaotic Oscillators
Chaotic systems exhibit dynamics which are highly dependent on initial conditions.
Since future dynamics are defined by initial conditions the evolution in time of
chaotic systems appears to be random. The output or the state vector of a chaotic
oscillator has been used to model the dynamics of biological neurons. For example,
the Duffing oscillator has been used to model the EEG signal received that is
from brain neuronal groups, as well as to describe the dynamics of variants of
the FitzHugh-Nagumo neuron [ 63 , 181 , 184 , 207 , 214 , 216 ]. Main types of chaotic
oscillators are described in the sequel:
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