Spatially discrete Nagumo equations have widespread physical applications, including modeling electrical impulses traveling through a demyelinated axon, an environment typical in multiple scle- rosis. We construct steady-state, single front solutions by employing a piecewise linear reaction term. Using a combination of Jacobi-Operator theory and the Sherman-Morrison formula we de- rive exact solutions in the cases of homogeneous and inhomogeneous diffusion. Solutions exist only under certain conditions outlined in their construction. The range of parameter values that satisfy these conditions constitutes the interval of propagation failure, determining under what circumstances a front becomes pinned in the media. Our exact solutions represent a very specific solution to the spatially discrete Nagumo equation. For example, we only consider inhomogeneous media with one defect present. We created an original script in MATLAB which algorithmically solves more general cases of the equation, including the case for multiple defects. The algorithmic solutions are then compared to known exact solutions to determine their validity.

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In this thesis, we study minimizers of the energy functional 𝐽 (𝑢,Ω) = ∫_Ω |∇𝑢|²/2 + 𝑊(𝑢) 𝑑𝑥 for two different potentials 𝑊(𝑢). In the first part we consider the Allen-Cahn energy, where 𝑊(𝑢) = (1 − 𝑢²)² is a doublewell potential which is relevant in the theory of phase transitions and minimal interfaces. We investigate the rigidity properties of global minimizers in low dimensions. In particular we extend a result of Savin on the De Giorgi’s conjecture to include minimizers that are not necessarily bounded, and that can have subquadratic growth at infinity. In the second part we consider potentials of the type 𝑊(𝑢) = 𝑢⁺ which appear in obstacletype free boundary problems. We establish higher order estimates and the analyticity of the regular part of the free boundary. Our method relies on developing higher order boundary Harnack estimates iteratively and deducing them from Schauder estimates for certain elliptic equations with degenerate weights. Finally we consider similar regularity questions of the free boundary in the Signorini problem which also known as the thin obstacle problem. We develop 𝐶²^𝛼 estimates of the free boundary under sharp assumptions on the coefficients and the data.

Le travail de cette thèse a porté sur l'étude mathématique et numérique de quelques modèles multi-échelles issus de la physique des matériaux. La première partie de ce travail est consacrée à l'homogénéisation mathématique d'un problème elliptique avec une petite échelle. Nous étudions le cas particulier d'un matériau présentant une structure périodique avec un défaut. En adaptant la théorie classique d'Avellaneda et Lin pour les milieux périodiques, on démontre qu'on peut approximer finement la solution d'un tel problème, notamment à l'échelle microscopique. Nous obtenons des taux de convergence dépendant de l'étalement du défaut. On démontre aussi quelques propriétés des fonctions de Green d'un problème elliptique périodique avec conditions de bord périodiques. Les dislocations sont des lignes de défaut de la matière responsables du phénomène de plasticité. Les deuxième et troisième parties de ce mémoire portent sur la simulation de dislocations, d'abord en régime stationnaire puis en régime dynamique. Nous utilisons le modèle de Peierls, qui couple échelle atomique et échelle mésoscopique. Dans le cadre stationnaire, on obtient une équation intégrodifférentielle non-linéaire avec un laplacien fractionnaire: l'équation de Weertman. Nous en étudions les propriétés mathématiques et proposons un schéma numérique pour en approximer la solution. Dans le cadre dynamique, on obtient une équation intégrodifférentielle à la fois en temps et en espace. Nous en faisons une brève étude mathématique, et comparons différents algorithmes pour la simuler. Enfin, dans la quatrième partie, nous étudions la limite macroscopique d'une chaîne d'atomes soumis à la loi de Newton. Des arguments formels suggèrent que celle-ci devrait être décrite par une équation des ondes non-linéaires. Or, nous démontrons --sous certaines hypothèses-- qu'il n'en est rien lorsque des chocs apparaissent / In this thesis we study mathematically and numerically some multi-scale models from materials science. First, we investigate an homogenization problem for an oscillating elliptic equation. The material under consideration is described by a periodic structure with a defect at the microscopic scale. By adapting Avellaneda and Lin's theory for periodic structures, we prove that the solution of the oscillating equation can be approximated at a fine scale. The rates of convergence depend upon the integrability of the defect. We also study some properties of the Green function of periodic materials with periodic boundary conditions. Dislocations are lines of defects inside materials, which induce plasticity. The second part and the third part of this manuscript are concerned with simulation of dislocations, first in the stationnary regime then in the dynamical regime. We use the Peierls model, which couples atomistic and mesoscopic scales and involves integrodifferential equations. In the stationary regime, dislocations are described by the so-called Weertman equation, which is nonlinear and involves a fractional Laplacian. We study some mathematical properties of this equation and propose a numerical scheme for approximating its solution. In the dynamical regime, dislocations are described by an equation which is integrodifferential in time and space. We compare some numerical methods for recovering its solution. In the last chapter, we investigate the macroscopic limit of a simple chain of atoms governed by the Newton equation. Surprisingly enough, under technical assumptions, we show that it is not described by a nonlinear wave equation when shocks occur

Stochastic reaction-diffusion simulations has become an efficient approach for modelling spatial aspects of intracellular biochemical reaction networks. By accounting for intrinsic noise due to low copy number of chemical species, stochastic reaction-diffusion simulations have the ability to more accurately predict and model biological systems. As with many simulations software, exploration of the parameters associated with the model can be needed to yield new knowledge about the underlying system. The exploration can be conducted by executing parameter sweeps for a model. However, with little or no prior knowledge about the modelled system, the effort for practitioners to explore the parameter space can get overwhelming. To account for this problem we perform a feasibility study on an explorative behavioural analysis of stochastic reaction-diffusion simulations by applying spatial-temporal data mining to large parameter sweeps. By reducing individual simulation outputs into a feature space involving simple time series and distribution analytics, we were able to find similar behaving simulations after performing an agglomerative hierarchical clustering.

Numerous macroscale models of arteries have been developed, comprised of populations of discrete coupled Endothelial Cells (EC) and Smooth Muscle Cells (SMC) cells, an example of which is the model of Shaikh et al. (2012), which simulates the complex biochemical processes responsible for the observed propagating waves of Ca2+ observed in experiments. In a 'homogenised' model however, the length scale of each cell is assumed infinitely small while the population of cells are assumed infinitely large, so that the microscopic spatial dynamics of individual cells are unaccounted for. We wish to show in our study, our hypothesis that the homogenised modelling approach for a particular system can be used to replicate observations of the discrete modelling approach for the same system. We may do this by deriving a homogenised model based on Goldbeter et al. (1990), the simplest possible physiological system, and comparing its results with those of the discrete Shaikh et al. (2012), which have already been validated with experimental findings. We will then analyse the mathematical dynamics of our homogenised model to gain a better understanding of how its system parameters influence the behaviour of its solutions. All our homogenised models are essentially formulated as partial differential equations (PDE), specifically they are of type reaction diffusion PDEs. Therefore before we begin developing the homogenised Goldbeter et al. (1990), we will first analyse the Brusselator PDE with the goal that it will help us to understand reaction diffusion systems better. The Brusselator is a suitable preliminary study as it shares two common properties with reaction diffusion equations: oscillatory solutions and a diffusion term.

Tip growth is a ubiquitous process throughout the plant kingdom in which a single cell elongates in one direction in a self-similar manner. To sustain tip growth in plants, the cell must regulate the extensibility of the wall to promote growth and avoid turgor-induced rupture. This process is heavily dependent on the cytoskeleton, which is thought to coordinate the delivery and recycling of vesicles containing cell wall materials at the cell tip. Although significant work has been done to elucidate the various molecular players in this process, there remains a need for a more mechanistic understanding of the cytoskeletonÂ’s role in tip growth. For this reason, specific emphasis should be placed on understanding the dynamics of the cytoskeleton, its associated motors, and their cargo. Since the advent of fluorescence fusion technology, various quantitative fluorescence dynamics techniques have emerged. Among the most prominent of these techniques is fluorescence recovery after photobleaching (FRAP). Despite its prominence, it is unclear how to interpret fluorescence recoveries in confined cellular geometries such as tip-growing cells. Here we developed a digital confocal microscope simulation of FRAP in tip-growing cells. With this simulation, we determined that fluorescence recoveries are significantly influenced by cell boundaries. With this FRAP simulation, we then measured the diffusion of VAMP72-labeled vesicles in the moss Physcomitrella patens. Using finite element modeling of polarized cell growth, and the measured VAMP72-labeled vesicle diffusion coefficient, we were able to show that diffusion alone cannot support the required transport of wall materials to the cell tip. This indicates that an actin-based active transport system is necessary for vesicle clustering at the cell tip to support growth. This provides one essential function of the actin cytoskeleton in polarized cell growth. After establishing the requirement for actin-based transport, we then sought to characterize the in vivo binding interactions of myosin XI, vesicles, and filamentous actin. Particle tracking evidence from P. patens protoplasts suggests that myosin XI and VAMP72-labeled vesicles exhibit fast transient interactions. Hidden Markov modeling of particle tracking indicates that myosin XI and VAMP72- labeled vesicles move along actin filaments in short-lived linear trajectories. These fast transient interactions may be necessary to achieve the rapid dynamics of the apical actin, important for growth. This work advances the fieldÂ’s understanding of fluorescence dynamics, elucidates a necessary function of the actin cytoskeleton, and provides insight into how the components of the cytoskeleton interact in vivo.

A fibrilação, atrial ou ventricular, é caracterizada por uma desorganização da atividade elétrica do músculo. O coração, que normalmente contrai-se globalmente, em uníssono e uniforme, durante a fibrilação contrai-se localmente em várias regiões, de modo descoordenado. Para estudar qualitativamente este fenômeno, é aqui proposto um novo modelo matemático, mais simples do que os demais existentes e que, principalmente, admite uma representação singela na forma de circuito elétrico equivalente. O modelo foi desenvolvido empiricamente, após estudo crítico dos modelos conhecidos, e após uma série de sucessivas tentativas, ajustes e correções. O modelo mostra-se eficaz na simulação dos fenômenos, que se traduzem em padrões espaciais e temporais das ondas de excitação normais e patológicas, propagando-se em uma grade de pontos que representa o tecido muscular. O trabalho aqui desenvolvido é a parte básica e essencial de um projeto em andamento no Departamento de Engenharia Elétrica da EESC-USP, que é a elaboração de uma rede elétrica ativa, tal que possa ser estudada utilizando recursos computacionais de simuladores usualmente aplicados em projetos de circuitos integrados / Atrial and ventricular fibrillation are characterized by a disorganized electrical activity of the cardiac muscle. While normal heart contracts uniformly as a whole, during fibrillation several small regions of the muscle contracts locally and uncoordinatedly. The present work introduces a new mathematical model for the qualitative study of fibrillation. The proposed model is simpler than other known models and, more importantly, it leads to a very simple electrical equivalent circuit of the excitable cell membrane. The final form of the model equations was established after a long process of trial runs and modifications. Simulation results using the new model are in accordance with those obtained using other (more complex) models found in the related literature. As usual, simulations are performed on a two-dimensional grid of points (representing a piece of heart tissue) where normal or pathological spatial and temporal wave patterns are produced. As a future work, the proposed model will be used as the building block of a large active electrical network representing the muscle tissue, in an integrated circuit simulator

In this thesis, we study a class of reaction-diffusion equations of the form $\frac{\partial u}{\partial t} = \mathcal{L}u + \phi u - \tfrac{1}{k} u^{k+1}$ where $\mathcal{L}$ is the stochastic generator of a Markov process, $\phi$ is a function of the space variables and $k\in \mathbb{R}\backslash\{0\}$. An important example, in the case when $k > 0$, is equations of the FKPP-type. We also give an example from the theory of utility maximisation problems when such equations arise and in this case $k < 0$. We introduce a new representation, for the solution of the equation, as the optimal value of an optimal control problem. We also give a second representation which can be seen as a dual problem to the first optimisation problem. We note that this is a new type of dual problem and we compare it to the standard Lagrangian dual formulation. By choosing controls in the optimisation problems we obtain upper and lower bounds on the solution to the PDE. We use these bounds to study the speed of the wave front of the PDE in the case when $\mathcal{L}$ is the generator of a suitable Lévy process.