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- 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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