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a conclusion about the stability features of larger sets of fixed points where each set
is associated with an interval from which the bifurcation parameter takes its values.
A solution to this problem can be obtained from interval polynomials theory and
Kharitonov's theorem [
162
,
192
].
Aiming at understanding how the dynamics of neurons is modified due to exter-
nal stimuli and parametric variations the topic of fixed point bifurcations in such
biological models has been extensively studied during the last years [
116
,
204
,
214
].
Similar results and methods have been extended to artificial neural networks, such
as Hopfield associative memories [
76
,
178
]. Additionally, studies on bifurcations
appearing in biological models such as circadian cells show how the dynamics
and the stability of such models can be affected by variations of external inputs
and internal parameters [
128
,
193
]. A subject of particular interest has been also
the control of bifurcations appearing in biological systems which is implemented
with state feedback and which finally succeeds to modify the biological oscillator's
dynamics and its convergence to attractors [
49
,
50
,
54
,
131
,
132
,
199
,
205
].
In this chapter, as a case study, bifurcations of fixed points in two particular types
of biological models will be examined. The first model is the FitzHugh-Nagumo
neuron. The voltage of the neurons membrane exhibits oscillatory variations after
receiving suitable external excitation either when the neuron is independent from
neighboring neural cells or when the neuron is coupled to neighboring neural cells
through synapses or gap junctions, as shown in Chap.
1
. The FitzHugh-Nagumo
model of biological neurons is a simplified model (second order) of the Hodgkin-
Huxley model (fourth order), where the latter is considered to reproduce efficiently
in several cases the neuron's dynamics. The FitzHugh-Nagumo model describes the
variations of the voltage of the neuron's membrane as a function of the ionic currents
that get through the membrane and of an external current that is applied as input to
the neuron [
5
,
16
,
161
,
220
]. The second model is that of Neurospora circadian cells.
Circadian cells perform protein synthesis within a feedback loop. The concentration
of the produced proteins exhibits periodic variations in time. The circadian cells
regulate the levels of specific proteins' concentration through an RNA transcription
and translation procedure [
102
,
104
,
189
]. Circadian neurons in the hypothalamus
are also responsible for synchronization and periodic functioning of several organs
in the human body.
Typically, the stages for analyzing the stability features of the fixed points that
lie on bifurcation branches comprise (i) the computation of fixed points as functions
of the bifurcation parameter and (ii) the evaluation of the type of stability for each
fixed point through the computation of the eigenvalues of the Jacobian matrix that
is associated with the system's nonlinear dynamics model. Stage (ii) requires the
computation of the roots of the characteristic polynomial of the Jacobian matrix.
This problem is nontrivial since the coefficients of the characteristic polynomial
are functions of the bifurcation parameter and the latter varies within intervals [
36
,
113
,
129
]. To obtain a clear view about the values of the roots of the characteristic
polynomial and about the stability features they provide to the system the use of
interval polynomials theory and particularly of Kharitonov's stability theorem is
proposed [
162
,
192
]. In this approach the study of the stability of a characterized
