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In this work, we show theoretically and numerically that a one-dimensional reaction-diffusion system, near the Turing bifurcation, produces different number of stripes when, in addition to random noise, the Fourier mode of a prepattern used to initialize the system changes. We also show that the Fourier modes that persist are inside the Eckhaus stability regions, while those outside this region follow a wave number selection process not predicted by the linear analysis. To test our results, we use the Brusselator reaction-diffusion system obtaining an excellent agreement between the weakly nonlinear predictions of the real Ginzburg-Landau equations and the numerical solutions near the bifurcation. Although the persistence of patterns is not relevant as a simple generating mechanism of self-organization, it is crucial to understand the formation of patterns that occurs in multiple stages. In this work, we discuss the relevance of our results on the robustness and diversity of solutions in multiple-steps mechanisms of biological pattern formation and auto-organization in growing domains.
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We propose a simple model of the electrical activity of the heart that reproduces realistic healthy electrocardiogram (ECG) signals. The model consists of two RLC linear oscillators periodically kicked by impulses of the main pacemaker with the frequency rate of a real heart. In the proposed model, one oscillator represents the atria, another represents the ventricles, and an electrical cardiac conduction system is included using a coupling capacitor, which can be either unidirectional or bidirectional. The network of the two capacitively coupled oscillators is periodically kicked by the main pacemaker to introduce the periodic forcing of limit cycles into the system; a time delay is introduced to represent the electrical transport delay from atria to ventricles. In this manner, healthy synthetic ECG signals are obtained by combining the signals of the currents of the oscillators. We show that an analytical solution of the model can be obtained when a single impulse is applied. From this, by the superposition principle, a solution with an impulse train is obtained. Note that analytical treatment is a feature not available in current cardiac oscillator models.
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Potenciais de Ação , Relógios Biológicos , Eletrocardiografia , Sistema de Condução Cardíaco/fisiopatologia , Modelos Cardiovasculares , HumanosRESUMO
By using the principles behind phononic crystals, a periodic array of circular holes made along the polarization thickness direction of piezoceramic resonators are used to stop the planar resonances around the thickness mode band. In this way, a piezoceramic resonator adequate for operation in the thickness mode with an in phase vibration surface is obtained, independently of its lateral shape. Laser vibrometry, electric impedance tests and finite element models are used to corroborate the performances of different resonators made with this procedure. This method can be useful in power ultrasonic devices, physiotherapy and other external medical power ultrasound applications where piston-like vibration in a narrow band is required.
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An eutactic star, in a n-dimensional space, is a set of N vectors which can be viewed as the projection of N orthogonal vectors in a N-dimensional space. By adequately associating a star of vectors to a particular sea urchin, we propose that a measure of the eutacticity of the star constitutes a measure of the regularity of the sea urchin. Then, we study changes of regularity (eutacticity) in a macroevolutive and taxonomic level of sea urchins belonging to the Echinoidea class. An analysis considering changes through geological time suggests a high degree of regularity in the shape of these organisms through their evolution. Rare deviations from regularity measured in Holasteroida order are discussed.
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Filogenia , Ouriços-do-Mar/anatomia & histologia , Ouriços-do-Mar/genética , Anatomia Regional , Anatomia Veterinária , Animais , Tamanho Corporal/genética , Evolução Molecular , Expressão Gênica , Tamanho do Órgão/genética , Fenótipo , Dinâmica PopulacionalRESUMO
An average structure associated with a given quasilattice is a system composed of several average lattices that in reciprocal space produces strong main reflections. The average lattice of a quasicrystal is a useful concept closely related to the geometric description of the quasicrystal to crystal transformation and has been proved to be structurally significant. Here we calculate average structures for arbitrary two- and three-dimensional quasilattices using the dual generalized method. Additionally, closed analytical expressions for the coordinates of the average structure, the quasiperiodic lattice and its diffraction pattern are given.
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We explore numerically the formation of Turing patterns in a confined circular domain with small aspect ratio. Our results show that stable fivefold patterns are formed over a well defined range of disk sizes, offering a possible mechanism for inducing the fivefold symmetry observed in early development of regular echinoids. Using this pattern as a seed, more complex biological structures can be mimicked, such as the pigmentation pattern of sea urchins and the plate arrangements of the calyxes of primitive camerate crinoids.
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For many years Turing systems have been proposed to account for spatial and spatiotemporal pattern formation in chemistry and biology. We extend the study of Turing systems to investigate the rô1e of boundary conditions, domain shape, non-linearities, and coupling of such systems. We show that such modifications lead to a wide variety of patterns that bear a striking resemblance to pigmentation patterns in fish, particularly those involving stripes, spots and transitions between them. Using the Turing system as a metaphor for activator-inhibitor models we conclude that such a mechanism, with the aforementioned modifications, may play a rô1e in fish patterning.
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Padronização Corporal , Modelos Biológicos , Animais , Biologia do Desenvolvimento , Peixes , Dinâmica não Linear , Análise Numérica Assistida por ComputadorRESUMO
We address the problem of pattern formation on the surface of a sphere using Turing equations. By considering a generic reaction-diffusion model, we numerically investigate the patterns formed under different conditions on the parameter values. Our results show that a closed surface with curvature, as a sphere, imposes geometrical restrictions on the shape of the pattern. This is important in some biological systems where curvature plays an important role in guiding chemical, biochemical, and embryological processes.