Eckhaus selection: The mechanism of pattern persistence in a reaction-diffusion system.

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.

Generation of ECG signals from a reaction-diffusion model spatially discretized.

We propose a model to generate electrocardiogram signals based on a discretized reaction-diffusion system to produce a set of three nonlinear oscillators that simulate the main pacemakers in the heart. The model reproduces electrocardiograms from healthy hearts and from patients suffering various well-known rhythm disorders. In particular, it is shown that under ventricular fibrillation, the electrocardiogram signal is chaotic and the transition from sinus rhythm to chaos is consistent with the Ruelle-Takens-Newhouse route to chaos, as experimental studies indicate. The proposed model constitutes a useful tool for research, medical education, and clinical testing purposes. An electronic device based on the model was built for these purposes.

Periodically kicked network of RLC oscillators to produce ECG signals.

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.

Planar modes free piezoelectric resonators using a phononic crystal with holes.

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.

Nonlinear effects on Turing patterns: time oscillations and chaos.

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, the patterns oscillate in time. When varying a single parameter, a series of bifurcations leads to period doubling, quasiperiodic, and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examine the Turing conditions for obtaining a diffusion-driven instability and show that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. These results demonstrate the limitations of the linear analysis for reaction-diffusion systems.

Coincidence lattices in the hyperbolic plane.

The problem of coincidences of lattices in the space R(p,q), with p + q = 2, is analyzed using Clifford algebra. We show that, as in R(n), any coincidence isometry can be decomposed as a product of at most two reflections by vectors of the lattice. Bases and coincidence indices are constructed explicitly for several interesting lattices. Our procedure is metric-independent and, in particular, the hyperbolic plane is obtained when p = q = 1. Additionally, we provide a proof of the Cartan-Dieudonné theorem for R(p,q), with p + q = 2, that includes an algorithm to decompose an orthogonal transformation into a product of reflections.

Eutacticity in sea urchin evolution.

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.

Strong far-field coherent scattering of ultraviolet radiation by holococcolithophores.

By considering the structure of holococcoliths (calcite plates that cover holococcolithophores, a haploid phase of the coccolithophore life cycle) as a photonic structure, we apply a discrete dipolar approximation to study the light backscattering properties of these algae. We show that some holococcolith structures have the ability to scatter the ultraviolet radiation. This property may represent an advantage for holococcolithophores possessing it, by allowing them to live higher in the water column than other coccolithophores.

Quasicrystalline and rational approximant wave patterns in hydrodynamic and quantum nested wells.

The eigenfunctions of nested wells with an incommensurate boundary geometry, in both the hydrodynamic shallow water regime and quantum cases, are systematically and exhaustively studied in this Letter. The boundary arrangement of the nested wells consists of polygonal ones, square or hexagonal, with a concentric immersed, similar but rotated, well or plateau. A rich taxonomy of wave patterns, such as quasicrystalline states, their crystalline rational approximants, and some other exotic but well known tilings, is found in these mimicked experiments. To the best of our knowledge, these hydrodynamic rational approximants are presented here for the first time in a hydrodynamic-quantum framework. The corresponding statistical nature of the energy level spacing distribution reflects this taxonomy by changing the spectral types.

Simplified dynamic model for the motility of irregular echinoids.

Inspired by the locomotion mechanism of sea urchins, we study the locomotion of an irregular echinoid by means of a simplified dynamical model. We prove that if two conjectures are assumed, the geometrical arrangement of the five ambulacral petals of irregular echinoids should form a eutactic star in order to optimize motility. We firstly propose an adequate "measure" of eutacticity that allows us to to verify the statistical tendency to such a property for a representative collection of fossil specimens. Next, regarding dynamics, the biological advantage of eutactic stars is addressed as a minimal path problem. Finally, we study the stability of some eutactic stars under small perturbations.

Clifford algebra approach to the coincidence problem for planar lattices.

The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions.

Polyhedral truncations as eutactic transformations.

An eutactic star is a set of N vectors in Rn (N > n) that are projections of N orthogonal vectors in RN. First introduced in the context of regular polytopes, eutactic stars are particularly useful in the field of quasicrystals where a method to generate quasiperiodic tilings is by projecting higher-dimensional lattices. Here are defined the concepts of eutactic transformations (as mappings that preserve eutacticity) and of vector radiations (vectors that stem from the vectors of an eutactic star), which are used to describe and parameterize polyhedral truncations. The polyhedral truncations preserve eutacticity, a result of relevance to the faceting and habit-forming characteristics of quasicrystals.

Quasiperiodic Bloch-like states in a surface-wave experiment.

Bloch-like surface waves associated with a quasiperiodic structure are observed in a classic wave propagation experiment which consists of pulse propagation with a shallow fluid covering a quasiperiodically drilled bottom. We show that a transversal pulse propagates as a plane wave with quasiperiodic modulation, displaying the characteristic undulatory propagation in this quasiperiodic system and reinforcing the idea that analogous concepts to Bloch functions can be applied to quasicrystals under certain circumstances.

Turing patterns with pentagonal symmetry.

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.

A multigrid approach to the average lattices of quasicrystals.

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.

Regularity in irregular echinoids.

Using the mathematical concept of eutactic star, we prove that it is possible to define a morphospace for irregular echinoids by using a single parameter. In particular, we have found an extraordinary geometric property in the flower-like patterns of the five ambulacral petals of these animals. This property is fulfilled with great accuracy for a large collection of fossil specimens and provides new insights in the study of the viable skeletal designs of extinct and/or living organisms.

Ultrasonic wedges for elastic wave bending and splitting without requiring a full band gap.

We present a novel twinned-square periodic structure for ultrasonic wave bending and splitting that does not require the existence of a complete band gap and plays the role of an ultrasonic wedge. The device allows 45 degrees bending of waves and by adequately switching the twinned structure to an ultrasonic crystal 90 degrees bending is achieved. An extreme refraction law at the grain boundaries is experimentally observed.

Dynamics of the eye's wave aberration.

It is well known that the eye's optics exhibit temporal instability in the form of microfluctuations in focus; however, almost nothing is known of the temporal properties of the eye's other aberrations. We constructed a real-time Hartmann-Shack (HS) wave-front sensor to measure these dynamics at frequencies as high as 60 Hz. To reduce spatial inhomogeneities in the short-exposure HS images, we used a low-coherence source and a scanning system. HS images were collected on three normal subjects with natural and paralyzed accommodation. Average temporal power spectra were computed for the wave-front rms, the Seidel aberrations, and each of 32 Zernike coefficients. The results indicate the presence of fluctuations in all of the eye's aberration, not just defocus. Fluctuations in higher-order aberrations share similar spectra and bandwidths both within and between subjects, dropping at a rate of approximately 4 dB per octave in temporal frequency. The spectrum shape for higher-order aberrations is generally different from that for microfluctuations of accommodation. The origin of these measured fluctuations is not known, and both corneal/lenticular and retinal causes are considered. Under the assumption that they are purely corneal or lenticular, calculations suggest that a perfect adaptive optics system with a closed-loop bandwidth of 1-2 Hz could correct these aberrations well enough to achieve diffraction-limited imaging over a dilated pupil.

Turing patterns on a sphere.

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.

A two-dimensional numerical study of spatial pattern formation in interacting Turing systems.

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.