Le système de FitzHugh-Nagumo stochastique est un modèle qualitatif pour la propagation de l'influx nerveux dans un neurone. Ce système lent-rapide s'écrit εdxt = (xt - xt3 + yt) dt + √εσ1 dWt(1), dyt = (a - bxt - cyt) dt + σ2 dwt(2) où a, b et c sont des réels, ε est un petit réel positif, σ1 et σ2 sont deux réels positifs représentant l'intensité du bruit, Wt(1) et Wt(2) sont deux mouvements browniens standards indépendants. Dans cette thèse, nous étudions d'abord le système déterministe associé (σ1 = σ2 = 0) et montrons qu'il est excitable. Nous regardons ensuite le cas particulier où b = 0. Dans ce cas, le comportement au voisinage du point d'équilibre est le même que celui d'un autre modèle, celui de Morris-Lecar. Nous étudions alors la loi du temps de sortie de ce voisinage. Dans le cas général, après avoir mis en évidence trois principaux régimes, nous montrons des résultats généraux sur la distribution du nombre de petites oscillations N entre deux spikes consécutifs en introduisant une chaîne de Markov. Puis nous étudions le cas particulier du régime de bruit faible.

Made available in DSpace on 2014-06-12T18:06:54Z (GMT). No. of bitstreams: 3 arquivo7763_1.pdf: 6132198 bytes, checksum: 9583aed73df9b715c6ac388fd2960d11 (MD5) arquivo7763_2.pdf: 8554510 bytes, checksum: 5c593d6759a4c5587c110585188fab4d (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2006 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / A atividade de um conjunto de neurônios interligados é um problema de atual interesse que pode ser abordado com uma descrição detalhada dos neurônios da população ou, estudando a dinâmica da resposta dessas populações sim descrever em detalhe o comportamento individual dos neurônios. O modelo de Wilson-Cowan consiste em equações para as taxas de disparo de subpopulações localizadas de neurônios excitatórios e inibitórios. A principal suposição para chegar nas equações está baseada no alto grau de redundância local (ou seja, a suposição de que neurônios vizinhos respondem da mesma forma a estímulos similares) e a aleatoriedade das conexões locais. A vantagem destas equações consiste em reduzir a atividade de um número grande de neurônios a uma descrição de duas variáveis, com o que se consegue simpli ficar consideravelmente o problema. Particularmente, elas prevêem a existência de atividade de ciclo-limite em resposta a um estímulo constante usando uma auto-interação mais forte na subpopulação excitatória que na inibitória. Nós analisamos se uma rede aleatória de neurô- nios de FitzHugh-Nagumo que tenta reproduzir a hipótese de Wilson-Cowan tem de fato esse comportamento dinâmico de baixa dimensionalidade. Os neurônios são conectados com sinapses químicas excitatórias e inibitórias que se descrevem usando modelos de Markov de dois estados. As sinapses são distribuídas aleatoriamente, gerando assim quatro grafos dirigidos de Erdos-Rényi: cada um dos NE(NI) neurônios excitatórios (inibitórios), recebe, em média, KEE(KEI) sinapses excitatórias da subpopulação excitatória, e KIE(KII) sinapses inibitórias da subpopulação inibitória. Os resultados mostram a existência de ciclos-limite e pontos fixos quando projetamos nosso sistema no plano de fase de Wilson-Cowan. Particularmente, o comportamento bidimensional de ciclo-limite é mais claro quando pelo menos uma das subpopulações (geralmente a popula ção excitatória) está aproximadamente sincronizada (sincronização perfeita não é observada devido à desordem própria da conectividade sináptica). Entretanto, quando as conectividades médias são pequenas, os neurônios se comportam de maneira diferente e a projeção no plano de Wilson-Cowan sugere uma descrição num espaço de fase com dimensão mais alta. Para quanti ficar essa alta dimensionalidade, calculamos a dimensão de imersão (embedding) necessária para desdobrar o atrator que descreve o sistema

Reliable signal transmission among cells is important for long-range coordination. While higher organisms have designated structures for signal transmission, such as axons, it remains unclear how simpler communities of cells are organized to relay signals. Furthermore, many biological systems exhibit spatial heterogeneity, which can interrupt signal propagation. In this thesis, we investigate this problem by modeling the spatial organization and dynamics of electrochemical signaling, and we compare our results to experiments from our collaborators on Bacillus subtilis bacterial biofilms. The experiments show that only a fraction of cells participates in signal propagation and that these cells are spatially clustered with a size distribution that follows a power-law decay. These observations suggest that the fraction of participating cells is just at the tipping point between a disconnected and a fully connected conduit for signal transmission. We utilize percolation theory and a minimal FitzHugh-Nagumo-type excitable dynamics model to test this hypothesis, and genetically modified biofilms with altered structure and dynamics to validate our modeling. Our results suggest that the biofilm is organized near the critical percolation point in order to negotiate the benefit and cost of long-range signal transmission. Then, more detailed experiments show that the participation probability is correlated from cell to cell and varies in space. We use these observations to develop an enhanced percolation model, and show using simulations and a renormalization argument that the main conclusions are unaffected by these features. Finally, we use our dynamic model to investigate the effects of heterogeneity beyond the radial wave regime and into the spiral wave regime. We find that spatial correlations in the heterogeneity promote or suppress spiraling depending on the parameters, a surprising feature that we explain by demonstrating that these spirals form by distinct mechanisms. We characterize the dependence of the spiral period on the heterogeneity using techniques from percolation theory. Taken together, our results reveal that the spatial structure of cell-to-cell heterogeneity can have important consequences for signal propagation in cellular communities.<br>

Die FitzHugh-Nagumo-Gleichungen besitzen Puls-Lösungen zu unterschiedlichen Geschwindigkeiten c. In dieser Arbeit wird ein Überblick der Existenz und Stabilität dieser Pulse gegeben. Dazu werden die geometrische singuläre Störungstheorie (Fenichel-Theorie) und der Ansatz der Evans-Funktion in allgemeinerer Form dargestellt. Im eigentlichen Hauptteil der Arbeit werden dann zunächst die langsamen Pulse konstruiert und ein zu [14] alternativer Beweis ihrer Instabilität geführt. Die schnellen Pulse wurden in [25] durch Shilnikov-Koordinaten konstruiert. Dieser Existenzbeweis wird in der Arbeit dargestellt.

We study a system of spatially discrete FitzHugh-Nagumo equations, which are nonlinear differential-difference equations on an infinite one-dimensional lattice. These equations are used as a model of impulse propagation in nerve cells. We employ McKean's caricature of the cubic as our nonlinearity, which allows us to reduce the nonlinear problem into a linear inhomogeneous problem. We find exact solutions for standing waves, which are steady states of the system. We derive formulas for all 1-pulse solutions. We determine the range of parameter values that allow for the existence of standing waves. We use numerical methods to demonstrate the stability of our solutions and to investigate the relationship between the existence of standing waves and propagation failure of traveling waves.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2018-11-12T14:27:26Z No. of bitstreams: 1 raffaelbechararameh.pdf: 1503042 bytes, checksum: 87e66fa77937ca9a85aac3231b27ac84 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-11-23T13:13:22Z (GMT) No. of bitstreams: 1 raffaelbechararameh.pdf: 1503042 bytes, checksum: 87e66fa77937ca9a85aac3231b27ac84 (MD5) / Made available in DSpace on 2018-11-23T13:13:22Z (GMT). No. of bitstreams: 1 raffaelbechararameh.pdf: 1503042 bytes, checksum: 87e66fa77937ca9a85aac3231b27ac84 (MD5) Previous issue date: 2018-08-31 / Para representar diferentes fenômenos e sistemas modelos matemáticos são largamente utilizados. Muitos deles são fundamentados em sistemas de equações diferenciais ordinárias (EDOs), isto é, baseiam-se em conjuntos de igualdades que envolvem variáveis dependentes, suas derivadas de primeira ordem e a variável independente. Neste trabalho, estudamos a modelagem da geração do potencial de ação em células excitáveis, como os neurônios. Existem dois modelos tradicionais e pioneiros que se destacam nessa área: Hodgkin-Huxley e FitzHugh-Nagumo. O objetivo desta dissertação é avaliar a possibilidade de modelar a geração do potencial de ação via uma única equação diferencial com atraso. Equações diferenciais com atraso são importantes por sua capacidade em reproduzir uma grande diversidade de fenômenos. Porém, seu uso na modelagem do potencial de ação de células excitáveis é ainda incipiente. Nesta dissertação, o método usado para alcançar este objetivo se baseou no desenvolvimento, inicialmente, de uma equação integro-diferencial que aproxima o sistema de EDOs. Em seguida, desenvolvemos uma aproximação para as integrais que usa termos tanto no instante atual quanto em instante anteriores, i.e., atrasados no tempo. Dessa forma, mostramos que é possível aproximar cada um dos sistemas de EDOS dos modelos de Hodgkin-Huxley e FitzHugh-Nagumo por uma única equação diferencial com atraso. Por fim, estes novos modelos são comparados com os originais, e são apontadas direções para a continuidade desta pesquisa. / To represent different phenomena and systems mathematical models are widely used. Many of them are based on systems of ordinary differential equations (ODEs), that is, they are based on sets of equalities involving dependent variables, their derivatives of first order and the independent variable. In this work, we study the modeling of action potential generation in excitable cells, such as neurons. There are two traditional and pioneering models that stand out in this area: Hodgkin-Huxley and FitzHugh-Nagumo. The objective of this dissertation is to evaluate the possibility of modeling the generation of the action potential via a single differential equation with delay. Differential equations with delay are important because of their capacity to reproduce a great diversity of phenomena. However, its use in modeling the action potential of excitable cells is still incipient. In this dissertation, the method used to achieve this goal was based on the development, initially, of an integral-differential equation that approximates the ODE system. Next, we develop an approximation for integrals that uses terms at both the current instant and the previous instant, i.e., time delayed. Thus, we show that it is possible to approximate each of the ODEs systems of the Hodgkin-Huxley and FitzHugh-Nagumo models by a single differential equation with delay. Finally, these new models are compared with the original ones, and directions are indicated for future works.

Ce mémoire consiste en l’étude du comportement dynamique de deux oscillateurs FitzHugh-Nagumo identiques couplés. Les paramètres considérés sont l’intensité du courant injecté et la force du couplage. Juqu’à cinq solutions stationnaires, dont on analyse la stabilité asymptotique, peuvent co-exister selon les valeurs de ces paramètres. Une analyse de bifurcation, effectuée grâce à des méthodes tant analytiques que numériques, a permis de détecter différents types de bifurcations (point de selle, Hopf, doublement de période, hétéroclinique) émergeant surtout de la variation du paramètre de couplage. Une attention particulière est portée aux conséquences de la symétrie présente dans le système. / We study the dynamical behaviour of a pair of identical, coupled FitzHugh-Nagumo oscillators. We determine the parameter values leading to the existence of up to five equilibrium solutions, and analyze the asymptotic stability of each one. A combination of analytical and numerical techniques is used to analyze the numerous bifurcations (saddle-node, Hopf, period-doubling, heteroclinic) occurring as parameters, most notably the coupling strength, are varied, attention being paid to the rôle played by symmetries in the system.

Ce mémoire analyse et valide des applications possibles de méthodes de réduction de modèle pour la simulation directe, et la résolution de problèmes inverses d'estimation de paramètres sur des modèles complexes. Il se concentre sur la réduction par proper orthogonal decomposition (POD), et ses extensions. On démontre d'abord de nouvelles estimations a priori pour l'erreur de réduction sur des problèmes abstraits types (paraboliques et hyperboliques, linéaires ou avec non-linéarités lipschitziennes), validées dans de nombreux cas non linéaires. On évite notamment le problème de contrôle des termes d'ordre élevé par l'exploitation d'une suite spécifique de normes de projecteurs. Puis, pour couvrir les systèmes dépendant de paramètres, et par des résultats d'interpolation, on adapte la méthode précédente en réduction par multi-POD. On étend aussi, au prix d'un terme additif, les estimations a priori précédentes pour l'erreur maximum de réduction sur une plage paramétrique donnée. On illustre la puissance de la méthode sur le système électrophysiologique de FitzHugh-Nagumo, fortement sensible aux variations paramétriques. On valide enfin numériquement les versions réduites, toujours avec la réduction par multi-POD, de problèmes d'estimation de paramètres : de type variationnel avec le système de FitzHugh-Nagumo, et de type séquentiel (filtrage " kalmanien ") avec un modèle mécanique de coeur (multiéchelles, 3D, grandes déformations). En particulier, la méthode présente une efficacité et une robustesse similaires à celles obtenues pour les problèmes directs.

Ce mémoire consiste en l’étude du comportement dynamique de deux oscillateurs FitzHugh-Nagumo identiques couplés. Les paramètres considérés sont l’intensité du courant injecté et la force du couplage. Juqu’à cinq solutions stationnaires, dont on analyse la stabilité asymptotique, peuvent co-exister selon les valeurs de ces paramètres. Une analyse de bifurcation, effectuée grâce à des méthodes tant analytiques que numériques, a permis de détecter différents types de bifurcations (point de selle, Hopf, doublement de période, hétéroclinique) émergeant surtout de la variation du paramètre de couplage. Une attention particulière est portée aux conséquences de la symétrie présente dans le système. / We study the dynamical behaviour of a pair of identical, coupled FitzHugh-Nagumo oscillators. We determine the parameter values leading to the existence of up to five equilibrium solutions, and analyze the asymptotic stability of each one. A combination of analytical and numerical techniques is used to analyze the numerous bifurcations (saddle-node, Hopf, period-doubling, heteroclinic) occurring as parameters, most notably the coupling strength, are varied, attention being paid to the rôle played by symmetries in the system.

The aim of this master thesis is to create a simple 3D electro-anatomical model of cardiac tissue that will be able to simulate the electrical activation in both a healthy heart and a heart with arrhytmogenic substrate. The model of electrical activation is realized in the COMSOL Multiphysics, simulation software for modelling using the finite element method. The Fitzhugh-Nagumo equation was used to model the excitatory feature of the myocardium and 2D models of myocardial tissue describing the propagation of action potential in healthy tissue, ischemic tissue, spontaneous action potential formation in the SA node, and spiral wave formation were first developed based on appropriate parameters. Subsequently, simplified 3D models of the heart describing the spread of excitement in a healthy heart, in the presence of accessory pathway and in third-degree atrioventricular block were created. The simplified 3D heart model offers a compromise between computational load and model complexity and can be used as a diagnostic tool for tissue and whole heart adjustment with appropriate equation parameter settings.