Multi-Phase Patterns in Periodically Forced Oscillatory Systems

Periodic forcing of an oscillatory system produces frequency locking bands within which the system frequency is rationally related to the forcing frequency. We study extended oscillatory systems that respond to uniform periodic forcing at one quarter of the forcing frequency (the 4:1 resonance). These systems possess four coexisting stable states, corresponding to uniform oscillations with successive phase shifts of $\pi/2$. Using an amplitude equation approach near a Hopf bifurcation to uniform oscillations, we study front solutions connecting different phase states. These solutions divide into two groups: $\pi$-fronts separating states with a phase shift of $\pi$ and $\pi/2$-fronts separating states with a phase shift of $\pi/2$. We find a new type of front instability where a stationary $\pi$-front ``decomposes'' into a pair of traveling $\pi/2$-fronts as the forcing strength is decreased. The instability is degenerate for an amplitude equation with cubic nonlinearities. At the instability point a continuous family of pair solutions exists, consisting of $\pi/2$-fronts separated by distances ranging from zero to infinity. Quintic nonlinearities lift the degeneracy at the instability point but do not change the basic nature of the instability. We conjecture the existence of similar instabilities in higher 2n:1 resonances (n=3,4,..) where stationary $\pi$-fronts decompose into n traveling $\pi/n$-fronts. The instabilities designate transitions from stationary two-phase patterns to traveling 2n-phase patterns. As an example, we demonstrate with a numerical solution the collapse of a four-phase spiral wave into a stationary two-phase pattern as the forcing strength within the 4:1 resonance is increased

On the Origin of Traveling Pulses in Bistable Systems

The interaction between a pair of Bloch fronts forming a traveling domain in a bistable medium is studied. A parameter range beyond the nonequilibrium Ising-Bloch bifurcation is found where traveling domains collapse. Only beyond a second threshold the repulsive front interactions become strong enough to balance attractive interactions and asymmetries in front speeds, and form stable traveling pulses. The analysis is carried out for the forced complex Ginzburg-Landau equation. Similar qualitative behavior is found in the bistable FitzHugh-Nagumo model.Comment: 5 pages, RevTeX. Aric Hagberg: http://t7.lanl.gov/People/Aric/; Ehud Meron:http://www.bgu.ac.il/BIDR/research/staff/meron.htm

Boundary effects on localized structures in spatially extended systems

We present a general method of analyzing the influence of finite size and boundary effects on the dynamics of localized solutions of non-linear spatially extended systems. The dynamics of localized structures in infinite systems involve solvability conditions that require projection onto a Goldstone mode. Our method works by extending the solvability conditions to finite sized systems, by incorporating the finite sized modifications of the Goldstone mode and associated nonzero eigenvalue. We apply this method to the special case of non-equilibrium domain walls under the influence of Dirichlet boundary conditions in a parametrically forced complex Ginzburg Landau equation, where we examine exotic nonuniform domain wall motion due to the influence of boundary conditions.Comment: 9 pages, 5 figures, submitted to Physical Review

A Phase Front Instability in Periodically Forced Oscillatory Systems

Multiplicity of phase states within frequency locked bands in periodically forced oscillatory systems may give rise to front structures separating states with different phases. A new front instability is found within bands where $\omega_{forcing}/\omega_{system}=2n$ ($n>1$). Stationary fronts shifting the oscillation phase by $\pi$ lose stability below a critical forcing strength and decompose into $n$ traveling fronts each shifting the phase by $\pi/n$. The instability designates a transition from stationary two-phase patterns to traveling $n$-phase patterns

The evolution and comparative neurobiology of endocannabinoid signalling

CB(1)- and CB(2)-type cannabinoid receptors mediate effects of the endocannabinoids 2-arachidonoylglycerol (2-AG) and anandamide in mammals. In canonical endocannabinoid-mediated synaptic plasticity, 2-AG is generated postsynaptically by diacylglycerol lipase alpha and acts via presynaptic CB(1)-type cannabinoid receptors to inhibit neurotransmitter release. Electrophysiological studies on lampreys indicate that this retrograde signalling mechanism occurs throughout the vertebrates, whereas system-level studies point to conserved roles for endocannabinoid signalling in neural mechanisms of learning and control of locomotor activity and feeding. CB(1)/CB(2)-type receptors originated in a common ancestor of extant chordates, and in the sea squirt Ciona intestinalis a CB(1)/CB(2)-type receptor is targeted to axons, indicative of an ancient role for cannabinoid receptors as axonal regulators of neuronal signalling. Although CB(1)/CB(2)-type receptors are unique to chordates, enzymes involved in biosynthesis/inactivation of endocannabinoids occur throughout the animal kingdom. Accordingly, non-CB(1)/CB(2)-mediated mechanisms of endocannabinoid signalling have been postulated. For example, there is evidence that 2-AG mediates retrograde signalling at synapses in the nervous system of the leech Hirudo medicinalis by activating presynaptic transient receptor potential vanilloid-type ion channels. Thus, postsynaptic synthesis of 2-AG or anandamide may be a phylogenetically widespread phenomenon, and a variety of proteins may have evolved as presynaptic (or postsynaptic) receptors for endocannabinoids

Frequency Locking in Spatially Extended Systems

A variant of the complex Ginzburg-Landau equation is used to investigate the frequency locking phenomena in spatially extended systems. With appropriate parameter values, a variety of frequency-locked patterns including flats, $\pi$ fronts, labyrinths and $2\pi/3$ fronts emerge. We show that in spatially extended systems, frequency locking can be enhanced or suppressed by diffusive coupling. Novel patterns such as chaotically bursting domains and target patterns are also observed during the transition to locking

Poincare' normal forms and simple compact Lie groups

We classify the possible behaviour of Poincar\'e-Dulac normal forms for dynamical systems in $R^n$ with nonvanishing linear part and which are equivariant under (the fundamental representation of) all the simple compact Lie algebras and thus the corresponding simple compact Lie groups. The ``renormalized forms'' (in the sense of previous work by the author) of these systems is also discussed; in this way we are able to simplify the classification and moreover to analyze systems with zero linear part. We also briefly discuss the convergence of the normalizing transformations.Comment: 17 pages; minor corrections in revised versio

Four-phase patterns in forced oscillatory systems

We investigate pattern formation in self-oscillating systems forced by an external periodic perturbation. Experimental observations and numerical studies of reaction-diffusion systems and an analysis of an amplitude equation are presented. The oscillations in each of these systems entrain to rational multiples of the perturbation frequency for certain values of the forcing frequency and amplitude. We focus on the subharmonic resonant case where the system locks at one fourth the driving frequency, and four-phase rotating spiral patterns are observed at low forcing amplitudes. The spiral patterns are studied using an amplitude equation for periodically forced oscillating systems. The analysis predicts a bifurcation (with increasing forcing) from rotating four-phase spirals to standing two-phase patterns. This bifurcation is also found in periodically forced reaction-diffusion equations, the FitzHugh-Nagumo and Brusselator models, even far from the onset of oscillations where the amplitude equation analysis is not strictly valid. In a Belousov-Zhabotinsky chemical system periodically forced with light we also observe four-phase rotating spiral wave patterns. However, we have not observed the transition to standing two-phase patterns, possibly because with increasing light intensity the reaction kinetics become excitable rather than oscillatory.Comment: 11 page

Preventing and lessening exacerbations of asthma in school-age children associated with a new term (PLEASANT) : Study protocol for a cluster randomised control trial

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly citedBackground: Within the UK, during September, there is a pronounced increase in the number of unscheduled medical contacts by school-aged children (4-16 years) with asthma. It is thought that that this might be caused by the return back to school after the summer holidays, suddenly mixing with other children again and picking up viruses which could affect their asthma. There is also a drop in the number of prescriptions administered in August. It is possible therefore that children might not be taking their medication as they should during the summer contributing to them becoming ill when they return to school. It is hoped that a simple intervention from the GP to parents of children with asthma at the start of the summer holiday period, highlighting the importance of maintaining asthma medication can help prevent increased asthma exacerbation, and unscheduled NHS appointments, following return to school in September.Methods/design: PLEASANT is a cluster randomised trial. A total of 140 General Practices (GPs) will be recruited into the trial; 70 GPs randomised to the intervention and 70 control practices of "usual care" An average practice is expected to have approximately 100 children (aged 4-16 with a diagnosis of asthma) hence observational data will be collected on around 14000 children over a 24-month period. The Clinical Practice Research Datalink will collect all data required for the study which includes diagnostic, prescription and referral data.Discussion: The trial will assess whether the intervention can reduce exacerbation of asthma and unscheduled medical contacts in school-aged children associated with the return to school after the summer holidays. It has the potential to benefit the health and quality of life of children with asthma while also improving the effectiveness of NHS services by reducing NHS use in one of the busiest months of the year. An exploratory health economic analysis will gauge any cost saving associated with the intervention and subsequent impacts on quality of life. If results for the intervention are positive it is hoped that this could be adopted as part of routine care management of childhood asthma in general practice. Trial registration: Current controlled trials: ISRCTN03000938 (assigned 19/10/12) http://www.controlled-trials.com/ISRCTN03000938/.UKCRN ID: 13572.Peer reviewe