1. The paper studies the effect of dynamic chemical and electrical synapses on phase synchronization in coupled bursting neurons. It considers a model where neurons are coupled through both delayed electrical (linear) and delayed chemical (nonlinear) interactions. 2. Chemical synapses are modeled using a sigmoid function representing steady-state synaptic activation. Electrical synapses provide instantaneous coupling between neurons. 3. The study finds that both the delay in chemical synapses and the relative strength of electrical and chemical coupling can induce phase transitions between synchronized and desynchronized states in the neuronal network. Understanding these effects provides insights into information processing in neuronal circuits involving multiple synaptic interactions.

1. UNIVERSITE DE DSCHANG
************
ECOLE DOCTORALE
************
UNITE DE FORMATION DOCTORALE
SCIENCES FONDAMENTALES ET
TECHNOLOGIES
*************
UNIVERSITY OF DSCHANG
************
POSTGRADUATE SCHOOL
************
DOCTORAL TRAINING UNIT
FUNDAMENTAL SCIENCES AND
TECHNOLOGIES
*************
DEPARTEMENT DE PHYSIQUE
DEPARTMENT OF PHYSICS
LABORATOIRE D’ELECTRONIQUE ET DE TRAITEMENT DU SIGNAL
LABORATORY OF ELECTRONICS AND SIGNAL PROCESSING (LETS)
THESIS
Presented for the achievement of the grade of
Doctorat / Ph.D degree in Physics
Option: Electronics
By
MEGAM NGOUONKADI Elie Bertrand
Registration number: 02S099
M. Sc. in Physics, Option: Electronics
Under the supervision of:
FOTSIN Hilaire Bertrand
Associate Professor
2014-2015
Dynamics, control and synchronization of
some models of neuronal oscillators

2. Dynamics, control and
synchronization of some models
of neuronal oscillators
Title
2/42

3. Outline
General Introduction
1. Bifurcation and multistability in the
extended Hindmarsh-Rose neuronal
oscillator
2. Phase synchronization of bursting neural
networks
General conclusion and outlook
3/42
3. Robust synchronization of a small
pacemaker neuronal ensemble via
nonlinear controller: electronic circuit
design

4. General introduction
Figure 1: Electrical phenomena are the main information processing in the brain
General introduction 5/42
Neurosciences are one the main research topics of the century. The human brain is one of
the most complex systems in science and understanding how it works is as old a question as
mankind. The expression "computational neuroscience" reflects the possibility of
generating theory of brain function in terms of the information-processing properties of
structures that make up the nervous system. It implies that we ought to be able to exploit
the conceptual and technical resources of computational research to help find explanations
of how neuronal structures achieve their effects, what functions are executed by neuronal
structures, and the nature of the states represented by the nervous system.

5. Neuronal activity modeling and network models
(a) (b)
Figure 2: Complex structure of neuronal networks
General introduction 5/42
A typical neuron has four morphologically defined regions: the cell body, the dendrites
the axon and the presynaptic terminals. Each of these regions has a distinct role in the
generation of signals and the communication between nerve cells. It has been shown in
recent works that depending on the area or the subpopulation, interneurons can
communicate through electrical synapse or chemical synapse alone or via both type of
interactions.

6. Neuronal activity modeling and network models
FitzHugh-Nagumo model: Fitzhugh and Nagumo 1961
Morris-Lecar model: Morris and Lecar 1981
Hindmarsh-Rose model : Hindmarsh J. L. et al. 1982, 1984
Extended Hindmarsh-Rose model : Selverston A. I.et al. 2000, Pinto R. et al. 2000
Why The extended Hindmarsh-Rose (eHR) neuronal model?
 Several details of the shape of spiking-bursting activity, can be adjusted with the help
of this extended model.
It can describe the calcium exchange between intracellular warehouse and the
cytoplasm, to completely produce the chaotic behavior of the stomatogastric ganglion
neurons.
 A better adjustment of the behavior of electronics neurons, when connected to its
living counterpart, is better represented by the fourth order HR model.
General introduction 6/42
The integate and fire model: Lapicque L. 1907
Using the Hodking-Huxley model is biophysically prohibitive, since we can only simulate a
handful of neurons in real time.
In contrast, using the integrate-and-fire model is computationally effective, but the model is
simple and unable to produce the rich spiking and bursting dynamics exhibited by cortical
neurons.

7. Model’s equations:
Here, a, b, c, d, e, f , g, μ, s, h, v, k, r and l, are constants which express the
current and conductance based dynamics.
 IDC represents the injected current.
 x represents the membrane voltage, and y a fast current. z is a slow current, w is
a slow dynamical process.
The parameters μ and v play a very important role in neuron activity. The first
represents the ratio of time scales between fast and slow fluxes across the
neuron’s membrane and the second controls the speed of variation of the slow
current.
 
 
2 3
2
- -
- - -
-
-
DCx ay bx cx dz I
y e fx y gw
z z s x h
w v kw r y l

   



     
      
&
&
&
&
(1)
General introduction 7/42

8. Problems and objectives
1- From a nonlinear dynamical systems point of view, does the eHR neuronal
oscillator behavior bring out how neurons respond to stimulus? Does the
model present the multistability mechanism?
2- How the dynamic chemical synapse, particularly the neurotransmitters
binding time constant influences the time delayed interactions of bursting
neurons, since one knows that chemical synapses and neurons are dynamical
nonlinear devices? Are there time-delay induced phase-flip transitions to or out
of synchrony when the chemical and electrical synapses are taken
simultaneously into account?
3- The third goal of this work is to study the synchronized behavior of an
external neuron and a complex network constituted of the pacemaker group
neurons of the lobster's pyloric CPG.
General introduction 8/42
In 1948, Hodgkin found that, injecting a DC-current of different amplitude in isolated
axons, results in the production of repetitive spiking and inhibition with different
frequencies.
These observations were investigated a few decades later by Rinzel and Ermentrout. They
show that the observed behaviors are due to different bifurcation mechanisms.

9. Nonlinear Physics formalisms and assumptions
Phase space reconstruction
Fractal dimensions
The Lyapunov exponents
Symbolic dynamics
Autocorrelation and cross correlation functions
Mutual information
General introduction 9/42
1- The coupled individual dynamical
systems are all identical
2- The same function of the components
from each dynamical system is used to
couple networks’ nodes.
3- The synchronization manifold is an
invariant manifold.
4- The couplings are linear and
nonlinear.
5- A modification of the Watts-Strogatz
network suggested by Monasson Will be
considered.
→ Assumptions 1 and 3 guarantee the
existence of a unified synchronization
hyperplane.
→ Assumption 2, allows us to make the
stability diagram specific to the different
choice of dynamical systems.
→ Assumptions 4 and 5, help to choose a
large class of coupling structures and a
specified network model, which
themselves include many real-world
applications.
ImplicationsAssumptions
Nonlinear formalisms

10. 1. Bifurcation and multistability in
the Extended Hindmarsh-Rose
Neuronal Oscillator
10/42
E. B. Megam Ngouonkadi et al. , Chaos Solitons and Fractals, (2016) 85

11. Stationary points (1)
Nullclines:
3 2
2
( )
( )
( )
    
 


         
DCcx ds x h bx I
y F x
a
k grl
y e fx G x
k gr k
Figure 3: Nullclines as a function of f (a) and b (b)
(a) (b)
(2)
Bifurcation and multistability in the extended Hindmarsh-Rose neuronal Oscillator 11/42
Nullclines intersections are given by black and red dots. We observe that varying the value of f
(respectively the value of b) shifts the y-nullcline (x-nullcline), whose effect is to reduce
(increase) the number of equilibria from 1 to 3 (from 3 to 1).

12. Fixed points stability
Figure 4: Membrane potential as a
function of f (a) and b (b-c).
(a) (b)
(c)
Bifurcation and multistability in the extended Hindmarsh-Rose neuronal Oscillator 12/42
Stationary points (2)
We observe three quantitative changes or bifurcations as both f and b are varied. For
example at point "bifurcation3", a saddle-node bifurcation occurs since we observed
coalescence and disappearance of two equilibria. We observe in figure 4b that at point
marked by "bifurcation 1", the equilibrium switches stability giving birth to a transcritical
bifurcation.

13. Hopf Bifurcation (1)
0
2 1 0
0 0
0 0
e
e
a d
fx g
J
s
vr kv

 
 
 
   
 
 
 
Characteristic equation : 4 3 2
1 2 3 4 0a a a a       
with:
1
2
3
4
1
2 ( 1 )
2 2
( )
2 ( )
 
    
    
  
   
   
         

     
    

     
e
e e
e
a kv
a vrg kv kv sd fx a kv
a vrg kv sd kv kvafx sd fax
vrg kv kv
a vrg sd kv sd kv fax kv vrg
(3)
(4)
Bifurcation and multistability in the extended Hindmarsh-Rose neuronal Oscillator 13/42
Jacobian matrix at
equilibrium Se:
(5)
What happens when a pair of complex-conjugate characteristic exponents of an equilibrium
state crosses over the imaginary axis? The drain of stability is directly connected to the
disappearance or the birth of a periodic orbit. This bifurcation represents the most mechanism
for transition from a stationary regime to oscillations and can highlight proper interpretation
of numerous physical phenomena.

14. Hopf Bifurcation (2)
1.0; 3.0; 1.0; 0.99; 1.01; 5.0128;
0.0278; 3.966; 1.605; 0.0009;
0.9573; 3.0; 1.619; 3.024972.
     
   
   DC
a b c d e f
g s h v
k r l I
The unique equilibrium:
0.7553399395
1.8314834490
.
3.3697518000
0.6658835764
  
 
  
 
    
e
e
e
e
e
x
y
S
z
w
The critical value of the bifurcation parameter μ is:
0.1230628577 c
(6)
(7)
(8)
Parameter values , Hindmasrh et al. 1984; Selverston et al. 2000
Bifurcation and multistability in the extended Hindmarsh-Rose neuronal Oscillator 14/42

15. Bifurcation and birth of chaos
Figure 5: Bifurcation diagram showing
the coordinate x(t) and the corresponding
graph of the maximal Lyapunov exponent
versus IDC.
Figure 6: Bifurcation diagram in
(v, x) plane respectively with IDC =
3.0249 showing reverse period
doubling (RPD), Exterior crisis
(EC) and interior crisis (IC).
Bifurcation and multistability in the extended Hindmarsh-Rose neuronal Oscillator 15/42
Exterior Crisis: v=0.03207
The evolution of the asymptotic behavior of solutions is well described through the
bifurcation diagram, as a function of a single parameter.
The model presents Block structure, when IDC is varied .
A small change in the bifurcation parameter v shows up continuous crisis and reverse period
doubling.

16. Multistability
Figure 7: Bifurcation diagrams in (a) (v, x) and (b) (μ, x) planes respectively with IDC =
3.0249
(a) (b)
Bifurcation and multistability in the extended Hindmarsh-Rose neuronal Oscillator 16/42
The model presents multistability, which represents an essential inherent property of the
dynamics of neurons and neuronal networks.
It helps to understand short-term memory.

17. 2. Phase synchronization of
bursting neural networks
17/42

18. 2.1. The combined effect of dynamic chemical and
electrical synapses in time-delay-induced phase-transition
to synchrony in coupled bursting neurons
Synaptic models
Figure 8: Electrical coupling
1 2 1
2 1
( ( ) ( )),
( ) ( ).
 
 
EI g x t x t
I t I t
Figure 9: Chemical coupling: Destexhe et
al. 1994.
0
( ) ( )( ),
( ) ( )( )
.
( )
 

 



syn c rev post
pre
pre
I t g R t E x
R x R tdR t
dt R R x
is a sigmoid function which represents
the steady-state synaptic activation:
( )R x
(9)
(10)
(11)
Phase synchronization of bursting neural networks 18/42
( )
tanh
( ) ,
0, .

  
  
    


f
th
thslop
th
x x
R x x xx
x x
E. B. Megam Ngouonkadi et al. IJBC, 24, (2014).

19. In the case of delayed electrical (linear coupling) and delayed chemical (nonlinear coupling)
interactions, the whole system is described by the following equations:
1 2
1, 1,
0
( ) ( , , ) ( )( ).
( ) ( ) ( )
where , 1, 2 and .
( )
 

   


    

 

 &
N N
i i i E ij i j c ij j rev i
j i j j i j
j j j
j
X F X g G H x x g C R t E x
dR t R x R t
i j
dt R R x
 1 1( , , ) ( ) ( ) .i j j iH x x x t x t   
From physiological experiments, 1=0 and 2 0
(12)
(13)
Phase synchronization of bursting neural networks 19/42
Network equations
Figure 10: Electrical and chemical coupling
Xi: represents an m-dimensional vector of dynamical variables of neuron i.
Fi: is the velocity field.
Gij: Electrical connection matrix.
Cij: describes the way neurons are chemically coupled.
H: represents the electrical coupling between nodes i and j.

20. Phase:
Phase difference:
1 ( )
( ) tan
( )
i
i
i
x t
t
x t
   
  
 
%
1 1 2 1 2
1 2
1 2 1 2
( ) ( ) ( ) ( )
( ) ( ) tan
( ) ( ) ( ) ( )
x t x t x t x t
t t
x t x t x t x t
    
   
 
% %
% %
(15)
(16)
(17)
Phase synchronization of bursting neural networks 20/42
Statistical quantities
The synchronization of slow bursts between two coupled neurons is analyzed based on the
method described by Pinto et al. (2000). It uses the normalized maximal deviation.
The normalized maximal deviation:
 
max max min
1 1/f f f
N dx x x  
1 2( ) ( ) ( ) f f f
dx t x t x t
(14)
Phase-flip transitions to synchronization are analyzed using the instantaneous phase and
phase difference of the time series.
Pinto et al. (2000) ; Pikovsky et al. (2003)

21. Effects of neurotransmitter binding time constant β and
time delay τ2 .
Figure 11: Normalized maximal deviation computed after 20 Hz low-pass filtering in the
membrane potential of two coupled gC=0.8 and gE=0.0
(a) (b)
(d)(c)
Inhibitory chemical coupling Excitatory chemical coupling
Phase synchronization of bursting neural networks 21/42

22. Effects of gE and gC on the phase-flip-transition
Figure 12: Characteristics of induced phase transitions when electrical and chemical synapses
are simultaneously considered.
(a) (b)
(c) (d)
Excitatory chemical coupling Inhibitory chemical coupling
Phase synchronization of bursting neural networks 22/42
The induced phase-flip transitions are noted by the abrupt changes in the relative
phases between spikes and bursts.
Bursting neurons can go from in-phase to out-of-phase synchrony with phase difference
less than , observed near the bifurcation point.

23.  int 1ij c    
 
%
0
( ) ( ) ( )
.
( )
ij j ij
j
dR t R x R t
dt R R x
 




( ) ( ) ( )( )syn ij C ij ij rev iI t g R t E x   (19)
(18)
(20)
2.2. Phase synchronization of bursting neural networks
with electrical and delayed dynamic chemical couplings
Phase synchronization of bursting neural networks 23/42
E. B. Megam Ngouonkadi et al., European Physical Journal B, 88, (2015).
Many studies have confirmed that, large-scale brain has small-world property as
anatomical networks (Buzsaki G. 2006, Sporns O.et al. 2006).
We use a modification of the Watts-Strogatz network suggested by Monasson R.
In this configuration no rewiring takes place but additional long range links are added
randomly. This allows the network quantities mathematically easier accessible.
Delay is not still constant, but vary with some probabilistic law.
It is spatially distributed and its value depends on the distance between neurons.
Int[x] represents the integer part of x.
: Gaussian white noise with zero mean and unitary standard deviation.
: the distances fluctuations in realistic neural systems.c%

24. Small-world network topology
(a) (b)
Figure 13: Example of small-world network topology
N=8 N=16
Order parameter: measure of the spikes synchrony
( )
1 1
1 f
j
T N
i t
t jf
e
NT


 
  
1
1
( ) 2 , ,
j
j ji
j i ij j
i i
t T
t T t T
T T
  


  

j = 1,…,N and N is the number of nodes in the network.
ρ turns to unity for complete phase coherence and near zero for weak coherence among the
phases of spike trains.
(24)
(23)
Phase synchronization of bursting neural networks 24/42

25. Network size effect on synchronization
Figure 14: Order parameter for different pairs of coupling strengths gE and gC with τ =20
N=8 N=16
Phase synchronization of bursting neural networks 25/42
(a)
(c)
(b)
(d)
Inhibitory
coupling
Excitatory
coupling
In the case of excitatory coupling, both electrical and chemical synapses act in a combined
manner to favor synchronization.
In the inhibitory coupling, the larger the chemical strength is the larger the electrical
strength requires being to perform phase synchronization.
Synchronization is hampered as the network size increases; but there may exist some
threshold value of delay for which its enhancement is observed.

26. Diffusive delays effect
(a) (b)
(c) (d)
Excitatory chemical Inhibitory chemical
τ =6.0
τ =20.0
Phase synchronization of bursting neural networks 26/42
Figure 15: Order parameter for different pairs of coupling strengths gE and gC with N = 8
when distributed time delay (τij=int[τ(1+ ξ)] is considered.c%
In the case of excitatory coupling, diffusive time delays promote phase synchronization
In the inhibitory coupling, diffusive time delays also promote phase synchronization but
only for its highest mean value.
Both synapses (electrical and chemical) play a complementary role; sometimes promoting
phase synchronization or sometimes compete it.

27. 3. Robust synchronization of
a small pacemaker neuronal
ensemble via nonlinear
controller: electronic circuit
design
27/42
E. B. Megam Ngouonkadi et al., Cognitive Neurodynamics, In revision

28. Robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design 28/42
The lobster’s pyloric circuit
Figure 16: The electronic equivalent
diagram of the lobster’s pyloric circuit.
Figure 17: The pacemaker ensemble
of the lobster’s pyloric CPG.
1,
( ) ( , )
N
i i E ij i j
j i j
x F x g G H x x
 
  &
( , ) ( ( ) ( ))i j j iH x x x t x t 
(25)
(26)
Selverston et al. in 1987 described the
connectivities of the lobster’s pyloric CPG.
It contains a group named the pacemaker
(AB/PDs neurons).
The pacemaker group can be isolated from
other neurons using pharmacological tools.

29. The pacemaker network
1 01 11
2 02 12
0 1
3 03 13
4 04 14
E y y
E y y
E Y Y
E y y
E y y
 
  
   
 
  
Controlled network
Figure 18: Controlled network of pyloric
CPGs ensemble.
0 0( ),Y K Y&
3
1 1 1 1
2
( ) ( , ) ,E j j
j
Y K Y g G H Y Y Au

  &
3
1,
( ) ( , ), 2, 3i i E ij i j
j j i
Y K Y g G H Y Y i
 
  &
Control law
1
2
1 2 1 1 1 1 1 1
ˆ ˆ ˆ ˆ ˆsgn( ) ( ) ,Ep p L p p p p p u     &
2
2 2 1 1
ˆ ˆsgn( ),p L p p   &
 2 1 1
1
1
ˆ ˆ .
ˆ( )E
u p p
p
   
Error system
(28)
(27a)
(29a)
Robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design 29/42
(27b)
(27c)
(29b)
(29c)
A first step to experimental design is the analog simulation using software such as Pspice.
The obtained circuits in our case are:

30. Figure 19: Circuit diagram of neuron N0.
Pspice implementation of the synchronization scheme (1)
Robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design 30/42

31. Figure 20: Circuit diagram of the AB neuron.
Pspice implementation of the synchronization scheme (2)
Robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design 31/42

32. Figure 21: Circuit diagram of the PD1 neuron.
Pspice implementation of the synchronization scheme (3)
Robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design 32/42

33. Figure 22: Circuit diagram of the PD2 neuron.
Pspice implementation of the synchronization scheme (4)
Robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design 33/42

34. Figure 23: Circuit diagram of the controller.
Pspice implementation of the synchronization scheme (5)
Absolute value function
Sign function
Robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design 34/42

35. Figure 24: Time series of different neurons, time evolution of the errors (y01 − y11 ),
(y01 − y21 ), (y01 − y31 ) and the control law for the robust synchronization scheme.
(c) (d)
Numerical and analog simulations
(a) (b)
Numerical simulations Analog (Pspice) simulations
Robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design 35/42

36. General conclusion and outlook
2. It is clear that multistability has important implications for information processing and
dynamical memory in a neuron. It seems to be a major mechanism of operation in the area of
motor control, particularly in the operation of multifunctional central pattern generators.
These results furnish potentially useful information for enhancing our knowledge on the way
by which the neuronal system works and encodes information.
4. As the synchronization error depends on the control gain and the synchronization of the
network on the coupling strength, a compromise exists between both the coupling strength
and the control gain.
3. With the fact that, chemical synapses are responsible of the non-local nature of the
synapses, we have also shown that the distributed time-delays affect the phase
synchronization of the network cells.
1. Possible mechanisms to highlight how a nervous system give rapid response to stimulus are
described by the abrupt changes observed in the system’s dynamics named, the Hopf
bifurcation and the interior crisis.
Main results
General conclusion and outlook 36/42

37. Outlook
1. An interesting point to consider, is to carry the dynamics of extended HR excitable systems
with delayed coupling.
2. An extension of the master stability function towards more than one delay time is desirable,
and when both, electrical and chemical synapses are taken into account.
3. A system of memristive coupled neurons can show information creation and recovery,
expressed quantitatively by the information recovery inequality, in distinction to properties
established for passive communication channels. Instead, these aspects of nonlinear activity
should provide an interesting framework for understanding the rich properties of memristor
synapses and realistic neuronal networks.
General conclusion and outlook 37/42
General conclusion and outlook

38. Personal references (1)
 Articles coming from the thesis
4. Implementing a Memristive Van Der Pol Oscillator Coupled to a Linear Oscillator:
Synchronization and Application to Secure Communication, E. B. Megam Ngouonkadi, H.
Fotsin, P. Louodop Fotso, Physica Scripta, 89, (2014).
3. The combined effect of dynamic chemical and electrical synapses in time-delay-induced
phase-transition to synchrony in coupled bursting neurons, E. B. Megam Ngouonkadi, H.
Fotsin, P. Louodop Fotso, International Journal of Bifurcation and Chaos, 24, (2014).
2. Phase synchronization of bursting neural networks with electrical and delayed dynamic
chemical couplings, E. B. Megam Ngouonkadi, M. Kabong Nono, V. Kamdoum Tamba and H.
B. Fotsin, European Physical Journal B, 88, (2015).
1. Bifurcation of Periodic Solutions and multistability in the Extended Hindmarsh-Rose
Neuronal Oscillator, E. B. Megam Ngouonkadi, H. B. Fotsin, P. Louodop Fotso, V. Kamdoum
Tamba and Hilda A. Cerdeira, Chaos Solitons and Fractals, (2016) 85.
5. Noise effects on robust synchronization of a small pacemaker neuronal ensemble, via
nonlinear controller: electronic circuit design, E. B. Megam Ngouonkadi, H. B. Fotsin, M.
Kabong Nono and Louodop Fotso Patrick Herve, Cognitive Neurodynamics, In revision
Personal references 38/42

39.  Other articles
1. Emergence of complex dynamical behavior in Improved Colpitts oscillators:
antimonotonicity, chaotic Bubbles, coexisting attractors and transient chaos, V. Kamdoum
Tamba, H. B. Fotsin, J. Kengne, E. B. Megam Ngouonkadi, and P.K. Talla, International
Journal of Dynamics and Control, (2016) 1-12.
2. Finite-time synchronization of tunnel diode based chaotic oscillators, P. Louodop, H. Fotsin,
M. Kountchou, E. B. Megam Ngouonkadi, Hilda A. Cerdeira and S. Bowong, Physical
Review E, (2014) 89.
3. Effective Synchronization of a Class of Chua’s Chaotic systems Using an Exponential
Feedback Coupling, P. Louodop, H. Fotsin, E. B. Megam Ngouonkadi, S. Bowong and Hilda
A. Cerdeira, Journal of Abstract and Applied Analysis, (2013).
4. Dynamics, analysis and implementation of a new multiscroll memristor based chaotic circuit,
N. Henry Alombah, H. B. Fotsin, E. B. Megam Ngouonkadi, Tekou Nguazon, International
Journal of Bifurcation and Chaos, In revision.
5. Dynamics and indirect finitie-time stability of modified relay-coupled chaotic systems, P.
Louodop, E. B. Megam Ngouonkadi, H. Fotsin, S. Bowong and H. A. Cerdeira, Physical
Review E, In revision.
Personal references 39/42
Personal references (2)

40. Acknowledgements
 I thank the University of Dschang for all facilities that they gave us,
 the Abdus Salam International Center for Theoretical Physics (ICTP) which
permit us to present some parts of this work during conferences.
 I also thank Professor Hilda Cerdeira for collaboration and moral support,
which resulted in the publication of some works related to this thesis.
 I thank all the jury's members who kindly accepted to review and evaluate
this work.
 I thank my family for moral and financial supports.
 I thank the LETS (Laboratory of Electronics and Signal Processing)
members for their collaborations.
 I also thank the public for their kind attentions.
Acknowledgements 40/42

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