Non-linear 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, Turing patterns oscillate in time, a phenomenon which is expected to occur only in a three morphogen system. When varying a single parameter, a series of bifurcations lead to period doubling, quasi-periodic and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examined the Turing conditions for obtaining a diffusion driven instability and discovered that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. All this results demonstrates the limitations of the linear analysis for reaction-diffusion systems

Declaration of medical writing assistance in international peer-reviewed publications

Medical researchers have an ethical and scientific obligation to publish, but between one third and two thirds of research may remain unpublished. A major reason for nonpublication is lack of time, which may lead researchers to seek medical writing assistance. Guidelines from journal editors and medical writers encourage authors to acknowledge medical writers. We quantified the proportion of articles from international, peer-reviewed, high-ranking journals that reported medical writing assistance

Heterogeneity induces spatiotemporal oscillations in reaction-diffusions systems

We report on a novel instability arising in activator-inhibitor reaction-diffusion (RD) systems with a simple spatial heterogeneity. This instability gives rise to periodic creation, translation, and destruction of spike solutions that are commonly formed due to Turing instabilities. While this behavior is oscillatory in nature, it occurs purely within the Turing space such that no region of the domain would give rise to a Hopf bifurcation for the homogeneous equilibrium. We use the shadow limit of the Gierer-Meinhardt system to show that the speed of spike movement can be predicted from well-known asymptotic theory, but that this theory is unable to explain the emergence of these spatiotemporal oscillations. Instead, we numerically explore this system and show that the oscillatory behavior is caused by the destabilization of a steady spike pattern due to the creation of a new spike arising from endogeneous activator production. We demonstrate that on the edge of this instability, the period of the oscillations goes to infinity, although it does not fit the profile of any well known bifurcation of a limit cycle. We show that nearby stationary states are either Turing unstable, or undergo saddle-node bifurcations near the onset of the oscillatory instability, suggesting that the periodic motion does not emerge from a local equilibrium. We demonstrate the robustness of this spatiotemporal oscillation by exploring small localized heterogeneity, and showing that this behavior also occurs in the Schnakenberg RD model. Our results suggest that this phenomenon is ubiquitous in spatially heterogeneous RD systems, but that current tools, such as stability of spike solutions and shadow-limit asymptotics, do not elucidate understanding. This opens several avenues for further mathematical analysis and highlights difficulties in explaining how robust patterning emerges from Turing's mechanism in the presence of even small spatial heterogeneity

Science and RE teachers' perspectives on the purpose of RE on the secondary school curriculum in England

Renewed interest in curriculum in English schooling over the past decade has emanated from a particular focus on the place and role of knowledge in the classroom. Significant changes in policy and examination specifications have led to changes in religious education (RE). However, little is known about teachers' perspectives on the purpose of RE. We asked teachers of science and RE what they understood as the purpose of RE on the school curriculum. Data from 10 focus groups and a survey with 276 secondary teachers demonstrated that many secondary teachers of science have a different understanding to RE teachers of the purpose of RE on the school curriculum. Findings also show a lack of consensus from RE teachers on the purpose of RE, suggesting the impact of the knowledge turn in RE is not as strong as the Ofsted Research Review implies. Findings are significant as little is known about how knowledge works across disciplinary boundaries in schools. If students are to come to a full understanding of how knowledge works, teachers need to have some understanding of how knowledge is being constructed and utilised in other curriculum subjects. Knowledge of the intended purpose of RE is important for respectful co-existence of subjects on the curriculum and essential when RE is declining as a subject in secondary schools

Rotational Effects of Twisted Light on Atoms Beyond the Paraxial Approximation

The transition probability for the emission of a Bessel photon by an atomic system is calculated within first order perturbation theory. We derive a closed expression for the electromagnetic potentials beyond the paraxial approximation that permits a systematic multipole approximation . The matrix elements between center of mass and internal states are evaluated for some specially relevant cases. This permits to clarify the feasibility of observing the rotational effects of twisted light on atoms predicted by the calculations. It is shown that the probability that the internal state of an atom acquires orbital angular momentum from light is, in general, maximum for an atom located at the axis of a Bessel mode. For a Gaussian packet, the relevant parameter is the ratio of the spread of the atomic center of mass wave packet to the transversal wavelength of the photon.Comment: 10 pages, no figure

From One Pattern into Another: Analysis of Turing Patterns in Heterogeneous Domains via WKBJ

Pattern formation from homogeneity is well-studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is nontrivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction-diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing's original thesis to a far wider and more realistic class of systems.Comment: 23 pages, 7 Figure

The Distribution of Nearby Stars in Velocity Space Inferred from Hipparcos Data

(abridged) The velocity distribution f(v) of nearby stars is estimated, via a maximum- likelihood algorithm, from the positions and tangential velocities of a kinematically unbiased sample of 14369 stars observed by the HIPPARCOS satellite. f(v) shows rich structure in the radial and azimuthal motions, v_R and v_phi, but not in the vertical velocity, v_z: there are four prominent and many smaller maxima, many of which correspond to well known moving groups. While samples of early-type stars are dominated by these maxima, also up to 25% of red main-sequence stars are associated with them. These moving groups are responsible for the vertex deviation measured even for samples of late-type stars; they appear more frequently for ever redder samples; and as a whole they follow an asymmetric-drift relation, in the sense that those only present in red samples predominantly have large |v_R| and lag in v_phi w.r.t. the local standard of rest (LSR). The question arise, how these old moving groups got on their eccentric orbits. A plausible mechanism, known from solar system dynamics, which is able to manage a shift in orbit space involves locking into an orbital resonance. Apart from these moving groups, there is a smooth background distribution, akin to Schwarzschild's ellipsoidal model, with axis ratio of about 1:0.6:0.35 in v_R, v_phi, and v_z. The contours are aligned with the $v_r$ direction, but not w.r.t. the v_phi and v_z axes: the mean v_z increases for stars rotating faster than the LSR. This effect can be explained by the stellar warp of the Galactic disk. If this explanation is correct, the warp's inner edge must not be within the solar circle, while its pattern rotates with frequency of about 13 km/s/kpc or more retrograde w.r.t. the stellar orbits.Comment: 16 pages LaTeX (aas2pp4.sty), 6 figures, accepted by A