The phenomenon of spiral wave propagation in Belousov-Zhabotinsky media is a remarkable example of self-organisation. This distinctive waveform arises in a variety of excitable systems. The primary objective of the work described in this thesis is the construction and analysis of deterministic reaction-diffusion models in terms of partial differential equations, to explain the local and global geometry of the spiral pattern. The secondary objective is to design experiments that enable observation and recording of evolving chemical waves in three dimensional Belousov-Zhabotinsky media. A mathematical formulation of the one dimensional A-w system based on a hierarchy of trial phase functions is introduced. A Schr8dinger type boundary value problem in an eigen sub-domain is established; an algebraic formula for the wave number spectrum and an analytical representation for the concentration amplitude are derived. This formulation suggests a piece-wise linear approach to X-w systems in higher dimensions. The concentrations are expressible in terms of real combinations of solutions to the Helmholtz equation with complex wave number and the solutions are matched by using continuity, differentiability and threshold conditions. A detailed analysis of the existence of solutions to piece-wise linear A-w systems in two dimensions is presented; existence is demonstrated by solving the matching equations. A stability analysis completes the discussion. Plots of the concentration contours characterised by the matching parameters are included. These contours simulate the cross-section of the scroll wave observed in experiments carried out in three dimensional media. The experimental design allows direct observation of undistorted three dimensional chemical waves in situ. The kinematics, dynamics and transformations of a variety of three dimensional scroll-based structures are recorded. The dominant waveforms are simple scroll waves. In addition, transient but significant events such as fission of a complex structure are recorded.

Rotating wave solutions to evolution equations have been shown to govern many important biological and chemical processes. Much of the rigorous mathematical investigations of rotating waves rely on the model exhibiting a continuous Euclidean symmetry, which is only present in an idealized situation. Here we investigate the existence of rotationally propagating solutions in a discrete spatial setting, in which typical symmetry methods cannot be applied, thus presenting an unique perspective on rotating waves. Our goal in this thesis is to demonstrate the existence and potential stability of rotating wave solutions to a spatially discretized infinite systems of coupled differential equations. This goal is achieved by considering so-called Lambda-Omega systems, which have frequently been used to model typical oscillatory dynamics. Our work is broken into three major components: 1. An infinite system of coupled phase equations is investigated and we demonstrate that under some mild assumptions the system exhibits a phase-locked rotating wave solution. The phase system is derived from a limiting case of the original Lambda-Omega system, and therefore solutions of the phase equation will be useful in finding rotating wave solutions to the full Lambda-Omega system. 2. We examine the stability of the rotating wave solution found in the coupled phase equations. This is achieved by providing a link with an underlying graph-theoretic geometry endowed by the spatially discretized system. We use results from random walks on infinite graphs to provide a general stability theorem for coupled phase equations. 3. We use the rotating wave solution of the phase equations to extend to a rotating wave solution of the full Lambda-Omega system. This result is achieved using a non-standard Implicit Function Theorem, since we show that typical implicit function arguments cannot be applied to our present situation.

Spiral waves occur in several natural phenomena, including reaction fronts in two-dimension excitable media. In this thesis we attempt to characterize the motion of the spiral tip of a rigidly rotating wave and a linearly travelling wave in the context of a lattice perturbation. This system can be reduced to its center manifold, which allows us to describe the system as ordinary differential equations. This in turn means dynamical systems methods are appropriate to describe the motion of the tip. It is in such a context that we work on spiral waves. We study perturbed rotating waves and travelling waves using standard techniques from dynamical systems theory.

Spiral waves occur in several natural phenomena, including reaction fronts in two-dimension excitable media. In this thesis we attempt to characterize the motion of the spiral tip of a rigidly rotating wave and a linearly travelling wave in the context of a lattice perturbation. This system can be reduced to its center manifold, which allows us to describe the system as ordinary differential equations. This in turn means dynamical systems methods are appropriate to describe the motion of the tip. It is in such a context that we work on spiral waves. We study perturbed rotating waves and travelling waves using standard techniques from dynamical systems theory.

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

No description available.

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

No description available.