Synchronization of Circadian Neurons and Protein Synthesis Control - Advanced Models of Neural Networks - page 109

Previous results on the control of circadian oscillators have been presented
in [ 14 , 53 , 102 , 142 , 182 , 188 - 190 , 219 ]. It appears that this control problem is
a nontrivial one when taking into account uncertainty and stochasticity of the
oscillators model. Up to now the problem of robust synchronization of coupled
circadian oscillators has not been solved efficiently. The approach followed in this
chapter for succeeding control of the protein synthesis procedure in the circadian
cells as well as for synchronizing coupled circadian cells when performing such
processes is based on transformation of the dynamical model of the circadian
oscillators into the linear canonical (Brunovsky) form. This transformation makes
use of differential flatness theory [ 58 , 101 , 109 , 172 ].
The problem of synchronization of coupled circadian oscillators becomes more
difficult when there is uncertainty about the parameters of the dynamical model of
the circadian cells and when the parameters of the dynamical models of these cells
are uneven. This chapter is concerned with proving differential flatness of the model
of the coupled circadian oscillators and its resulting description in the Brunovksy
(canonical) form [ 125 ]. By defining specific state variables as flat outputs an
equivalent description of the coupled circadian oscillators in the Brunovksy (linear
canonical) form is obtained. It is shown that for the linearized model of the
coupled oscillators it is possible to design a feedback controller that succeeds their
synchronization. At a second stage, the novel Kalman Filtering method proposed in
Chap. 4 , that is the Derivative-free nonlinear Kalman Filter, is used as a disturbance
observer, thus making possible to estimate disturbance terms affecting the model of
the coupled oscillators and to use these terms in the feedback controller. By avoiding
linearization approximations, the proposed filtering method improves the accuracy
of estimation and results in smooth control signal variations and in minimization of
the synchronization error [ 158 , 159 , 170 ].

5.2 Modelling of Circadian Oscillators Dynamics

5.2.1 The Functioning of the Circadian Oscillators

Circadian neurons are a small cluster of 210 4 neurons in the ventral hypothalamus
and provide the body with key time-keeping signals, known as circadian rhythms.
Circadian neurons mainly receive input from the retina, can be coupled through
neurotransmitters, and can stimulate other cells in the body also exhibiting an
oscillatory behavior that provides time keeping for other functions of the body.
Circadian oscillators affect cell division. A model that describes the cell cycle is
given in [ 32 ]. The model comprises several stages of the cell division cycle each one
denoted by a different index i (1iI), whereas the primary variable is denoted as
i D i .t;a/ and represents the density of cells of age a in phase i at time instant t.
During each phase, the density of cells varies either due to spontaneous death rate
d i D d i .t;a/ or due to a transition rate K iC1;i .t;a/ from phase i to phase i C 1.