89 research outputs found
A model for the orientational ordering of the plant microtubule cortical array
The plant microtubule cortical array is a striking feature of all growing
plant cells. It consists of a more or less homogeneously distributed array of
highly aligned microtubules connected to the inner side of the plasma membrane
and oriented transversely to the cell growth axis. Here we formulate a
continuum model to describe the origin of orientational order in such confined
arrays of dynamical microtubules. The model is based on recent experimental
observations that show that a growing cortical microtubule can interact through
angle dependent collisions with pre-existing microtubules that can lead either
to co-alignment of the growth, retraction through catastrophe induction or
crossing over the encountered microtubule. We identify a single control
parameter, which is fully determined by the nucleation rate and intrinsic
dynamics of individual microtubules. We solve the model analytically in the
stationary isotropic phase, discuss the limits of stability of this isotropic
phase, and explicitly solve for the ordered stationary states in a simplified
version of the model.Comment: 15 pages, 5 figure
The effects of symmetry on the dynamics of antigenic variation
In the studies of dynamics of pathogens and their interactions with a host
immune system, an important role is played by the structure of antigenic
variants associated with a pathogen. Using the example of a model of antigenic
variation in malaria, we show how many of the observed dynamical regimes can be
explained in terms of the symmetry of interactions between different antigenic
variants. The results of this analysis are quite generic, and have wider
implications for understanding the dynamics of immune escape of other
parasites, as well as for the dynamics of multi-strain diseases.Comment: 21 pages, 4 figures; J. Math. Biol. (2012), Online Firs
On the generalised Ritt problem as a computational problem
The Ritt problem asks if there is an algorithm that tells whether one prime
differential ideal is contained in another one if both are given by their
characteristic sets. We give several equivalent formulations of this problem.
In particular, we show that it is equivalent to testing if a differential
polynomial is a zero divisor modulo a radical differential ideal. The technique
used in the proof of equivalence yields algorithms for computing a canonical
decomposition of a radical differential ideal into prime components and a
canonical generating set of a radical differential ideal. Both proposed
representations of a radical differential ideal are independent of the given
set of generators and can be made independent of the ranking.Comment: 9 page
Bifurcations of periodic orbits with spatio-temporal symmetries
Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare' return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal Floquet multiplier and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb (1996). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems
Heteroclinic Ratchets in a System of Four Coupled Oscillators
We study an unusual but robust phenomenon that appears in an example system
of four coupled phase oscillators. We show that the system can have a robust
attractor that responds to a specific detuning between certain pairs of the
oscillators by a breaking of phase locking for arbitrary positive detunings but
not for negative detunings. As the dynamical mechanism behind this is a
particular type of heteroclinic network, we call this a 'heteroclinic ratchet'
because of its dynamical resemblance to a mechanical ratchet
Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation
Copyright © 2011 Springer. The final publication is available at www.springerlink.comWe consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible “inflations” of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells
End-tidal carbon dioxide monitoring during bag valve ventilation: the use of a new portable device
<p>Abstract</p> <p>Background</p> <p>For healthcare providers in the prehospital setting, bag-valve mask (BVM) ventilation could be as efficacious and safe as endotracheal intubation. To facilitate the evaluation of efficacious ventilation, capnographs have been further developed into small and convenient devices able to provide end- tidal carbon dioxide (ETCO<sub>2</sub>). The aim of this study was to investigate whether a new portable device (EMMA™) attached to a ventilation mask would provide ETCO<sub>2 </sub>values accurate enough to confirm proper BVM ventilation.</p> <p>Methods</p> <p>A prospective observational trial was conducted in a single level-2 centre. Twenty-two patients under general anaesthesia were manually ventilated. ETCO<sub>2 </sub>was measured every five minutes with the study device and venous PCO<sub>2 </sub>(PvCO<sub>2</sub>) was simultaneously measured for comparison. Bland- Altman plots were used to compare ETCO<sub>2, </sub>and PvCO<sub>2</sub>.</p> <p>Results</p> <p>The patients were all hemodynamically and respiratory stable during anaesthesia. End-tidal carbon dioxide values were corresponding to venous gases during BVM ventilation under optimal conditions. The bias, the mean of the differences between the two methods (device versus venous blood gases), for time points 1-4 ranges from -1.37 to -1.62.</p> <p>Conclusion</p> <p>The portable device, EMMA™ is suitable for determining carbon dioxide in expired air (kPa) as compared to simultaneous samples of PvCO<sub>2</sub>. It could therefore, be a supportive tool to asses the BVM ventilation in the demanding prehospital and emergency setting.</p
Symmetry justification of Lorenz' maximum simplification
In 1960 Edward Lorenz (1917-2008) published a pioneering work on the `maximum
simplification' of the barotropic vorticity equation. He derived a coupled
three-mode system and interpreted it as the minimum core of large-scale fluid
mechanics on a `finite but unbounded' domain. The model was obtained in a
heuristic way, without giving a rigorous justification for the chosen selection
of modes. In this paper, it is shown that one can legitimate Lorenz' choice by
using symmetry transformations of the spectral form of the vorticity equation.
The Lorenz three-mode model arises as the final step in a hierarchy of models
constructed via the component reduction by means of symmetries. In this sense,
the Lorenz model is indeed the `maximum simplification' of the vorticity
equation.Comment: 8 pages, minor correction
Analysis of the shearing instability in nonlinear convection and magnetoconvection
Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced
Hyperbolic planforms in relation to visual edges and textures perception
We propose to use bifurcation theory and pattern formation as theoretical
probes for various hypotheses about the neural organization of the brain. This
allows us to make predictions about the kinds of patterns that should be
observed in the activity of real brains through, e.g. optical imaging, and
opens the door to the design of experiments to test these hypotheses. We study
the specific problem of visual edges and textures perception and suggest that
these features may be represented at the population level in the visual cortex
as a specific second-order tensor, the structure tensor, perhaps within a
hypercolumn. We then extend the classical ring model to this case and show that
its natural framework is the non-Euclidean hyperbolic geometry. This brings in
the beautiful structure of its group of isometries and certain of its subgroups
which have a direct interpretation in terms of the organization of the neural
populations that are assumed to encode the structure tensor. By studying the
bifurcations of the solutions of the structure tensor equations, the analog of
the classical Wilson and Cowan equations, under the assumption of invariance
with respect to the action of these subgroups, we predict the appearance of
characteristic patterns. These patterns can be described by what we call
hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of
the planforms that were used in [1, 2] to account for some visual
hallucinations. If these patterns could be observed through brain imaging
techniques they would reveal the built-in or acquired invariance of the neural
organization to the action of the corresponding subgroups.Comment: 34 pages, 11 figures, 2 table
- …