In this thesis we analyze the buckling behavior of an infinitely long, thin, uniform, inextensible, elastic plate that has a steady current flowing along its length. We are concerned with the derivation of the nonlinear equations of motion using nonlinear continuum mechanics, and subsequent analysis of the buckling behavior of the plate under electromagnetic self-forces. In particular, we concentrate on how the body-forces that result from the applied current determine the buckled configurations. We derive both analytical and numerical results, and in the process develop a novel boundary value problem solver for integro-differential equations in addition to a predictor-corrector algorithm to continue solutions with respect to the control parameters. We take a relatively complex problem in magneto-solid mechanics and elasticity theory and form a realistic model that sheds light on the bifurcation and buckling behavior resulting from the electromagnetic-field- induced self-forces that are derived in their full, exact form using Biot-Savart Law. / Ph. D.

Circadian rhythms are periodic physiological cycles that recur about every 24 hours, by means of which organisms integrate their physiology and behavior to the daily cycle of light and temperature imposed by the rotation of the earth. Circadian derives from the Latin word circa "about" and dies "day". Circadian rhythms have three noteworthy properties. They are endogenous, that is, they persist in the absence of external cues (in an environment of constant light intensity, temperature, etc.). Secondly, they are temperature compensated, that is, the nearly 24 hour period of the endogenous oscillator is remarkably independent of ambient temperature. Finally, they are phase shifted by light. The circadian rhythm can be either advanced or delayed by applying a pulse of light in constant darkness. Consequently, the circadian rhythm will synchronize to a periodic light-dark cycle, provided the period of the driving stimulus is not too far from the period of the endogenous rhythm. A window on the molecular mechanism of 24-hour rhythms was opened by the identification of circadian rhythm mutants and their cognate genes in Drosophila, Neurospora, and now in other organisms. Since Konopka and Benzer first discovered the period mutant in Drosophila in 1971 (Konopka and Benzer, 1971), there have been remarkable developments. Currently, the consensus opinion of molecular geneticists is that the 24-hour period arises from a negative feedback loop controlling the transcription of clock genes. However, a better understanding of this mechanism requires an approach that integrates both mathematical and molecular biology. From the recent discoveries in molecular biology and through a mathematical approach, we propose that the mechanism of circadian rhythm is based upon the combination of both negative and positive feedback. / Master of Science

The focus of this thesis is the dynamical behaviour of linear arrays of laser oscillators with nearest-neighbour coupling. In particular, we study how laser dynamics are influenced by laser-coupling strength, $\kappa$, the natural frequencies of the uncoupled lasers, $\tilde{\Omega}_j$, and the coupling between the magnitude and phase of each lasers electric field, $\alpha$. Equivariant bifurcation analysis, combined with Lyapunov exponent calculations, is used to study different aspects of the laser dynamics. Firstly, codimension-one and -two bifurcations of relative equilibria determine the laser coupling conditions required to achieve stable phase locking. Furthermore, we find that global bifurcations and their associated infinite cascades of local bifurcations are responsible for interesting locking-unlocking transitions. Secondly, for large $\alpha$, vast regions of the parameter space are found to support chaotic dynamics. We explain this phenomenon through simulations of $\alpha$-induced stretching-and-folding of the phase space that is responsible for the creation of horseshoes. A comparison between the results of a simple {\it coupled-laser model} and a more accurate {\it composite-cavity mode model} reveals a good agreement, which further supports the use of the simpler model to study coupling-induced instabilities in laser arrays. Finally, synchronisation properties of the laser array are studied. Laser coupling conditions are derived that guarantee the existence of synchronised solutions where all the lasers emit light with the same frequency and intensity. Analytical stability conditions are obtained for two special cases of such laser synchronisation: (i) where all the lasers oscillate in-phase with each other and (ii) where each laser oscillates in anti-phase with its direct neighbours. Transitions from complete synchronisation (where all the lasers synchronise) to optical turbulence (where no lasers synchronise and each laser is chaotic in time) are studied and explained through symmetry breaking bifurcations. Lastly, the effect of increasing the number of lasers in the array is discussed in relation to persistent optical turbulence.

Cette thèse traite de modèles mésoscopiques de cortex appelés champs neuronaux. Les équations des champs neuronaux décrivent l'activité corticale de populations de neurones, ayant des propriétés anatomiques/fonctionnelles communes. Elles ont été introduites dans les années 1950 et portent le nom d'équations de Wilson et Cowan. Mathématiquement, elles consistent en des équations intégro-différentielles avec retards, les retards modélisant les délais de propagation des signaux ainsi que le passage des signaux à travers les synapses et l'arbre dendritique. Dans la première partie, nous rappelons la biologie nécessaire à la compréhension de cette thèse et dérivons les équations principales. Puis, nous étudions ces équations du point de vue des systèmes dynamiques en caractérisant leurs points d'équilibres et la dynamique dans la seconde partie. Dans la troisième partie, nous étudions de façon générale ces équations à retards en donnant des formules pour les diagrammes de bifurcation, en prouvant un théorème de la variété centrale et en calculant les principales formes normales. Nous appliquons tout d'abord ces résultats à des champs neuronaux simples mono-dimensionnels qui permettent une étude détaillée de la dynamique. Enfin, dans la dernière partie, nous appliquons ces différents résultats à trois modèles de cortex visuel. Les deux premiers modèles sont issus de la littérature et décrivent respectivement une hypercolonne, /i.e./ l'élément de base de la première aire visuelle (V1) et un réseau de telles hypercolonnes. Le dernier modèle est un nouveau modèle de V1 qui généralise les deux modèles précédents tout en permettant une étude poussée des effets spécifiques des retards / This thesis deals with mesoscopic models of cortex called neural fields. The neural field equations describe the activity of neuronal populations, with common anatomical / functional properties. They were introduced in the 1950s and are called the equations of Wilson and Cowan. Mathematically, they consist of integro-differential equations with delays, the delays modeling the signal propagation and the passage of signals across synapses and the dendritic tree. In the first part, we recall the biology necessary to understand this thesis and derive the main equations. Then, we study these equations with the theory of dynamical systems by characterizing their equilibrium points and dynamics in the second part. In the third part, we study these delayed equations in general by giving formulas for the bifurcation diagrams, by proving a center manifold theorem, and by calculating the principal normal forms. We apply these results to one-dimensional neural fields which allows a detailed study of the dynamics. Finally, in the last part, we study three models of visual cortex. The first two models are from the literature and describe respectively a hypercolumn, i.e. the basic element of the first visual area (V1) and a network of such hypercolumns. The latest model is a new model of V1 which generalizes the two previous models while allowing a detailed study of specific effects of delays

La bifurcation coronaire est un site privilégié d’athérosclérose. Jusqu’alors aucune des techniques de stenting provisionnel percutanées avec juxtaposition de ballons n’a démontré de bénéfice clinique. Ces échecs peuvent être expliqués par le non-respect de la géométrie fractale des bifurcations qui pourtant doit toujours guider la revascularisation (correction de la malapposition et optimisation de l’ostium de la branche collatérale). Fort de ce constat, nous avons imaginé une nouvelle technique séquentielle et simple, en 3 temps, le rePOT, associant Proximal Optimizing Technique (POT) initial, ouverture de la branche collatérale et POT final. Son évaluations s’est voulue progressive en 4 étapes : 1) concept, 2) preuve expérimentale de concept, 3) confirmation des bénéfices mécaniques in vivo, et 4) validation clinique. Dans ce travail nous avons donc d’abord expérimentalement démontré la supériorité du résultat mécanique final du rePOT par rapport aux techniques non séquentielles de provisional stenting (manuscrit # 1). Ainsi, le rePOT effondre la malapposition globale du stent, conserve la circularité proximale physiologique et optimise l’obstruction ostiale résiduelle de la branche accessoire. Ces excellents résultats sont confirmés indépendamment du design ou de la composition des stents (manuscrits # 2, 4). Nous avons par ailleurs démontré l’importance de chacune des 3 étapes du rePOT: POT initial (manuscrit # 1), ouverture de SB et POT final (manuscrit # 3). Enfin, fort de ces démonstrations expérimentales, nous avons confirmé in vivo avec mesures OCT itératives à la fois les excellents résultats expérimentaux et la bonne évolution clinique à moyen terme (manuscrit # 5). Ce travail démontre donc étape par étape, de l’expérimentale à la clinique, l’ensemble des bénéfices de cette nouvelle technique séquentielle de stenting provisionnel "rePOT", devenue une référence en Europe dans la revascularisation percutanée des bifurcations coronaires / Coronary bifurcations are a preferential location for atherosclerosis development. Until now, no technic with balloons juxtaposition demonstrated a clinical benefit in percutaneous coronary bifurcation revascularization by provisional stenting (with 1 stent). Successive failures could be explained by the absence of respect of the bifurcations fractal geometry, which need to be systematically followed during all revascularization (correction of the malapposition and optimization of the side branch ostium). Thus, we imagined a new technique, simple and sequential, in 3 steps, named rePOT. It is combining initial Proximal Optimizing Technique (POT), side-branch opening and final POT. We proposed a demonstration in 4 steps : 1) concept, 2) experimental proofs of concept, 3) confirmation of the clinical benefits in vivo, and 4) clinical validation. In this work, we experimentally demonstrated the superiority of the final mechanical results of the rePOT compared to all non-sequential provisional stenting (manuscript # 1). Thus, rePOT decreased stent global malapposition, maintained the initial proximal circularity and optimized the final ostial side branch obstruction. These excellent results were confirmed independently of stent design or material (manuscripts # 2, 4). Moreover, we demonstrated the specific benefits of each steps of the rePOT : initial POT (manuscript # 1), SB opening, and final POT (manuscript # 3). Finally, we confirmed in vivo, with serial OCT analysis, these excellent mechanical results and the good clinical outcome at mid-term. (manuscript # 5). Thank to this step by step demonstration, from experimental to clinic, we confirmed all benefits of this new provisional stenting sequential technique "rePOT". Thereby, before the last step of the demonstration, rePOT became a reference in Europe for the percutaneous revascularization of coronary bifurcations

Le travail présenté dans ce manuscrit est une contribution à l’étude des vibrations non-linéaires des poutres isotropes et en composite, en mouvement de rotation. Le modèle mathématique utilisé est basé sur la formulation intrinsèque et géométriquement exacte de Hodges, dédiée au traitement des poutres ayant des grands déplacements et de petites déformations. La résolution est faite dans le domaine fréquentiel suite à une discrétisation spatio-temporelle, en utilisant l’approximation de Galerkin et la méthode de l’équilibrage harmonique, avec des conditions aux limites correspondantes aux poutres encastrées-libres. Le systéme dynamique final est traité par des méthodes de continuation : la méthode asymptotique numérique et la méthode pseudo-longueur d’arc. Des algorithmes basés sur ces méthodes de continuation ont été développés et une étude comparative de convergence a été menée. Cette étude a cerné les aspects : statique, analyse modale linéaire, vibrations libres non-linéaires et les vibrations forcées non-linéaires des poutres rotatives. Ces algorithmes de continuations ont été testés pour le calculs des courbes de réponse sur des cas traités dans la littérature. La résonance interne et la stabilité des solutions obtenues sont étudiées / The work presented in this manuscript is a contribution to the non-linear vibrations of the isotropic beams and composite rotating beams study. The mathematical model used is based on the intrinsic formulation and geometrically exact of Hodges, developped for beams subjected to large displacements and small deformations. The resolution is done in the frequency domain after a spatial-temporal dicretisation, by using the Galerkin approximation and the the harmonic balance method, with boundary conditions corresponding to the clamped-free. The final dynamic system is treated by continuation methods : asymptotic numerical method and the pseudo-arc length method, whose algorithms based on these continuation methods were developed and a convergence study was carried out. This study surround the aspects : statics, linear modal analysis, non-linear free vibrations and the non-linear forced vibrations of the rotating beams. These continuation algorithms were tested for the response curves calculations on cases elaborated in the literature. Internal resonance and the stability of the solutions obtained are studied

The disturbances and abnormalities occurring in the components of the Acetylcholine (ACh) neurocycle are considered one of the main features of cholinergic sicknesses like Parkinson’s and Alzheimer’s diseases. A fundamental understanding of the ACh neurocycle is therefore very critical in order to design drugs that keep the ACh concentrations in the normal physiological range. In this dissertation, a novel two-enzyme-two-compartment model is proposed in order to explore the bifurcation, dynamics, and chaotic characteristics of the ACh neurocycle. The model takes into consideration the physiological events of the choline uptake into the presynaptic neuron and the ACh release in the postsynaptic neuron. In order to approach more realistic behavior, two complete kinetic mechanisms for enzymatic processes pH-dependent are built: the first mechanism is for the hydrolysis reaction catalyzed by the acetylcholinesterase (AChE) and the other is for the synthesis reaction catalyzed by the cholineacetyltransferase (ChAT). The effects of hydrogen ion feed concentrations, AChE activity, ChAT activity, feed ACh concentrations, feed choline concentrations, and feed acetate concentrations as bifurcation parameters, on the system performance are studied. It was found that hydrogen ions play an important role, where they create potential differences through the plasma membranes. The concentrations of ACh, choline and acetate in compartments 1 and 2 are affected by the activity of AChE through a certain range of their concentrations, where the activity of AChE is inhibited completely after reaching certain values. A detailed bifurcation analysis over a wide range of parameters is carried out in order to uncover some important features of the system, such as hysteresis, multiplicity, Hopf bifurcation, period doubling, chaotic characteristics, and other complex dynamics. The effects of the feed choline concentrations and the feed acetate concentrations as bifurcation parameters are studied in this dissertation. It is found that the feed choline concentrations play an important role and have a direct effect on the ACh neurocycle through a certain important range of the parameters. However, the feed acetate concentrations have less effect. It is concluded from the results that the feed choline is a more important factor than the feed acetate in ACh processes. The effects of ChAT activity and the choline recycle ratio as bifurcation parameters, on the system performance are investigated. It was found that as the ChAT activity increases, ACh concentrations in compartments 1 and 2 increase continuously. The effect of the choline recycle ratio shows that choline reuptake plays a very critical role in the synthesis of ACh in compartment 1, where it supplies the choline as a substrate for the synthesis reaction by ChAT. The concentrations of ACh, choline and acetate in compartments 1 and 2 are affected by the choline recycle ratio through a certain range of the choline recycle ratio; then, they become constant as the choline recycle ratio increases further. It is concluded from our results that choline uptake is the rate limiting step in the ACh processes in both compartments in comparison to ChAT activity. Based on partial dissociation of the acetic acid in compartments 1, and 2 of the ACh cholinergic system, the two-parameter continuation technique has been applied to investigate the pH range to be closer to physiological ranges of pH values. In addition, static/dynamic solutions of the ACh cholinergic neurocycle system based on feed choline concentration as the main bifurcation parameter in both compartments have been investigated. The findings of the above studies are related to the real phenomena occurring in the neurons, like periodic stimulation of neural cells and non-regular functioning of ACh receptors. It was found that ACh, choline, acetate, and pH exist inside the physiological range associated with taking into consideration the partial dissociation of the acetic acid. The disturbances and irregularities (chaotic attractors) occurring in the ACh cholinergic system may be good indications of cholinergic diseases such as Alzheimer’s and Parkinson’s diseases. The results have been compared to the results of physiological experiments and other published models. As there is strong evidence that cholinergic brain diseases like Alzheimer’s disease and Parkinson’s disease are related to the concentration of ACh, the present findings are useful for uncovering some of the characteristics of these diseases and encouraging more physiological research.

The disturbances and abnormalities occurring in the components of the Acetylcholine (ACh) neurocycle are considered one of the main features of cholinergic sicknesses like Parkinson’s and Alzheimer’s diseases. A fundamental understanding of the ACh neurocycle is therefore very critical in order to design drugs that keep the ACh concentrations in the normal physiological range. In this dissertation, a novel two-enzyme-two-compartment model is proposed in order to explore the bifurcation, dynamics, and chaotic characteristics of the ACh neurocycle. The model takes into consideration the physiological events of the choline uptake into the presynaptic neuron and the ACh release in the postsynaptic neuron. In order to approach more realistic behavior, two complete kinetic mechanisms for enzymatic processes pH-dependent are built: the first mechanism is for the hydrolysis reaction catalyzed by the acetylcholinesterase (AChE) and the other is for the synthesis reaction catalyzed by the cholineacetyltransferase (ChAT). The effects of hydrogen ion feed concentrations, AChE activity, ChAT activity, feed ACh concentrations, feed choline concentrations, and feed acetate concentrations as bifurcation parameters, on the system performance are studied. It was found that hydrogen ions play an important role, where they create potential differences through the plasma membranes. The concentrations of ACh, choline and acetate in compartments 1 and 2 are affected by the activity of AChE through a certain range of their concentrations, where the activity of AChE is inhibited completely after reaching certain values. A detailed bifurcation analysis over a wide range of parameters is carried out in order to uncover some important features of the system, such as hysteresis, multiplicity, Hopf bifurcation, period doubling, chaotic characteristics, and other complex dynamics. The effects of the feed choline concentrations and the feed acetate concentrations as bifurcation parameters are studied in this dissertation. It is found that the feed choline concentrations play an important role and have a direct effect on the ACh neurocycle through a certain important range of the parameters. However, the feed acetate concentrations have less effect. It is concluded from the results that the feed choline is a more important factor than the feed acetate in ACh processes. The effects of ChAT activity and the choline recycle ratio as bifurcation parameters, on the system performance are investigated. It was found that as the ChAT activity increases, ACh concentrations in compartments 1 and 2 increase continuously. The effect of the choline recycle ratio shows that choline reuptake plays a very critical role in the synthesis of ACh in compartment 1, where it supplies the choline as a substrate for the synthesis reaction by ChAT. The concentrations of ACh, choline and acetate in compartments 1 and 2 are affected by the choline recycle ratio through a certain range of the choline recycle ratio; then, they become constant as the choline recycle ratio increases further. It is concluded from our results that choline uptake is the rate limiting step in the ACh processes in both compartments in comparison to ChAT activity. Based on partial dissociation of the acetic acid in compartments 1, and 2 of the ACh cholinergic system, the two-parameter continuation technique has been applied to investigate the pH range to be closer to physiological ranges of pH values. In addition, static/dynamic solutions of the ACh cholinergic neurocycle system based on feed choline concentration as the main bifurcation parameter in both compartments have been investigated. The findings of the above studies are related to the real phenomena occurring in the neurons, like periodic stimulation of neural cells and non-regular functioning of ACh receptors. It was found that ACh, choline, acetate, and pH exist inside the physiological range associated with taking into consideration the partial dissociation of the acetic acid. The disturbances and irregularities (chaotic attractors) occurring in the ACh cholinergic system may be good indications of cholinergic diseases such as Alzheimer’s and Parkinson’s diseases. The results have been compared to the results of physiological experiments and other published models. As there is strong evidence that cholinergic brain diseases like Alzheimer’s disease and Parkinson’s disease are related to the concentration of ACh, the present findings are useful for uncovering some of the characteristics of these diseases and encouraging more physiological research.

The work we present in this thesis is a series of studies of how heterogeneities in coupling affect the synchronization of coupled neural oscillators. We begin by examining how heterogeneity in coupling strength affects the equilibrium phase difference of a pair of coupled, spiking neurons when compared to the case of identical coupling. This study is performed using pairs of Hodgkin-Huxley and Wang-Buzsaki neurons. We find that heterogeneity in coupling strength breaks the symmetry of the bifurcation diagrams of equilibrium phase difference versus the synaptic rate constant for weakly coupled pairs of neurons. We observe important qualitative changes such as the loss of the ubiquitous in-phase and anti-phase solutions found when the coupling is identical and regions of parameter space where no phase locked solution exists. Another type of heterogeneity can be found by having different types of coupling between oscillators. Synaptic coupling between neurons can either be exciting or inhibiting. We examine the synchronization dynamics when a pair of neurons is coupled with one excitatory and one inhibitory synapse. We also use coupled pairs of Hodgkin-Huxley neurons and Wang-Buzsaki neurons for this work. We then explore the existance of 1:n coupled states for a coupled pair of theta neurons. We do this in order to reproduce an observed effect called quantal slowing. Quantal slowing is the phenomena where jumping between different $1:n$ coupled states is observed instead of gradual changes in period as a parameter in the system is varied. All of these topics fall under the general heading of coupled, non-linear oscillators and specifically weakly coupled, neural oscillators. The audience for this thesis is most likely going to be a mixed crowd as the research reported herein is interdisciplinary. Choosing the content for the introduction proved far more challenging than expected. It might be impossible to write a maximally useful introductory portion of a thesis when it could be read by a physicist, mathematician, engineer or biologist. Undoubtedly readers will find some portion of this introduction elementary. At the risk of boring some or all of my readers we decided it was best to proceed so that enough of the mathematical (biological) background is explained in the introduction so that a biologist (mathematician) is able to appreciate the motivations for the research and the results presented. We begin with a introduction in nonlinear dynamics explaining the mathematical tools we use to characterize the excitability of individual neurons, as well as oscillations and synchrony in neural networks. The next part of the introductory material is an overview of the biology of neurons. We then describe the neuron models used in this work and finally describe the techniques we employ to study coupled neurons.

L'épilepsie est l’un des désordres neurologiques les plus courants; environ 1% de la population mondiale en est atteinte. Elle affecte le fonctionnement des neurones qui s'exprime par une survenue de décharges rapides, appelées crises.Les crises peuvent parfois résister aux médicaments antiépileptiques. Cet état de crise, nommé refractory status epilepticus (RSE), se définit par une survenue des décharges de façon continue, qui sont difficiles à arrêter. Malheureusement, un patient en état RSE risque même de mourir.Le dysfonctionnement des neurones peut parfois se propager lentement vers une dépression corticale envahissante, qui se caractérise par une dépolarisation lente des neurones entraînant une baisse transitoire de l’activité cérébrale.Les mécanismes qui initient ces activités pathologiques restent encore mal connus. Mettant en œuvre une approche mathématique, nous présentons une analyse qualitative d’un modèle dit Epileptor, qui reproduit l’activité épileptique, et décrivons l’évolution temporelle de la crise jusqu’au moment où la personne revient à la normale. La transition vers l’état normal est interprétée comme une bifurcation. Nous démontrons qu’il en existe plusieurs types en variant certains paramètres du modèle, permettant ainsi de classer les crises.Une caractérisation de ces activités est nécessaire pour s’en échapper. Pour ce faire, nous explorons un espace des paramètres permettant de conclure que ces activités coexistent dans le cerveau et surviennent de plusieurs façons. Cet espace des paramètres propose plusieurs voies, qui sont validées expérimentalement, pour éviter ces activités pathologiques et retourner à la normale, d’où son importance. / Epilepsy is one of the most common serious neurological disorders affecting about 1% of the population in the world. It is a condition of the nervous system in which neuronal populations manifest as abnormal excessive discharges, called seizures.On rare occasions, a seizure follows another in a series without recovery, and does not respondto antiepileptic drugs. This state of continuous seizure activity is called refractory status epilepticus, which are difficult to treat. Patients suffering from this state are unfortunately at an increased risk of death.The neuronal dysfunction can sometimes spread slowly towards a spreading depression, which corresponds to a slowly propagating depolarization wave (or depolarization block DB) in neuronal networks, followed by a shut down of brain activity.The mechanisms underlying the genesis of these activities remain unknown. Using a mathematical approach, we present a qualitative analysis of a so-called Epileptor model, which generate epileptic dynamics, and describe how seizures evolve toward termination. The transition to the normal state occurs through a bifurcation. We demonstrate that many types of the bifurcation exist, depending on the values of some parameters. As a consequence, we can classify seizures.To escape from these activities, a characterization is going to be necessary. To this, we explore a parameter space of the model which demonstrate that these activities coexist in the brain, and under some ways. In addition, the parameter space can provide pathways to switch between these activities. Interestingly, we could propose how to return to the ’normal’ brain activity.