Planar radial spots in a three-component FitzHugh-Nagumo system

Localized planar patterns arise in many reaction-diffusion models. Most of the paradigm equations that have been studied so far are two-component models. While stationary localized structures are often found to be stable in such systems, travelling patterns either do not exist or are found to be unstable. In contrast, numerical simulations indicate that localized travelling structures can be stable in three-component systems. As a first step towards explaining this phenomenon, a planar singularly perturbed three-component reaction-diffusion system that arises in the context of gas-discharge systems is analysed in this paper. Using geometric singular perturbation theory, the existence and stability regions of radially symmetric stationary spot solutions are delineated and, in particular, stable spots are shown to exist in appropriate parameter regimes. This result opens up the possibility of identifying and analysing drift and Hopf bifurcations, and their criticality, from the stationary spots described here.

Front interactions in a three-component system

The three-component reaction-diffusion system introduced in [C. P. Schenk et al., Phys. Rev. Lett., 78 (1997), pp. 3781–3784] has become a paradigm model in pattern formation. It exhibits a rich variety of dynamics of fronts, pulses, and spots. The front and pulse interactions range in type from weak, in which the localized structures interact only through their exponentially small tails, to strong interactions, in which they annihilate or collide and in which all components are far from equilibrium in the domains between the localized structures. Intermediate to these two extremes sits the semistrong interaction regime, in which the activator component of the front is near equilibrium in the intervals between adjacent fronts but both inhibitor components are far from equilibrium there, and hence their concentration profiles drive the front evolution. In this paper, we focus on dynamically evolving N-front solutions in the semistrong regime. The primary result is use of a renormalization group method to rigorously derive the system of N coupled ODEs that governs the positions of the fronts. The operators associated with the linearization about the N-front solutions have N small eigenvalues, and the N-front solutions may be decomposed into a component in the space spanned by the associated eigenfunctions and a component projected onto the complement of this space. This decomposition is carried out iteratively at a sequence of times. The former projections yield the ODEs for the front positions, while the latter projections are associated with remainders that we show stay small in a suitable norm during each iteration of the renormalization group method. Our results also help extend the application of the renormalization group method from the weak interaction regime for which it was initially developed to the semistrong interaction regime. The second set of results that we present is a detailed analysis of this system of ODEs, providing a classification of the possible front interactions in the cases of $N=1,2,3,4$, as well as how front solutions interact with the stationary pulse solutions studied earlier in [A. Doelman, P. van Heijster, and T. J. Kaper, J. Dynam. Differential Equations, 21 (2009), pp. 73–115; P. van Heijster, A. Doelman, and T. J. Kaper, Phys. D, 237 (2008), pp. 3335–3368]. Moreover, we present some results on the general case of N-front interactions.

Pulse dynamics in a three-component system : existence analysis

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781–3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

Pulse dynamics in a three-component system : stability and bifurcations

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction-diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions. A mathematical analysis of pulse interactions is based on detailed information on the existence and stability of isolated pulse solutions. The existence of these isolated pulse solutions is established in previous work. Here, the pulse solutions are studied by an Evans function associated to the linearized stability problem. Evans functions for stability problems in singularly perturbed reaction-diffusion models can be decomposed into a fast and a slow component, and their zeroes can be determined explicitly by the NLEP method. In the context of the present model, we have extended the NLEP method so that it can be applied to multi-pulse and multi-front solutions of singularly perturbed reaction-diffusion equations with more than one slow component. The brunt of this article is devoted to the analysis of the stability characteristics and the bifurcations of the pulse solutions. Our methods enable us to obtain explicit, analytical information on the various types of bifurcations, such as saddle-node bifurcations, Hopf bifurcations in which breathing pulse solutions are created, and bifurcations into travelling pulse solutions, which can be both subcritical and supercritical.

Coexistence of stable spots and fronts in a three-component FitzHugh-Nagumo system

We investigate regions of bistability between different travelling and stationary structures in a planar singularly-perturbed three-component reaction-diffusion system that arises in the context of gas discharge systems. In previous work, we delineated the existence and stabil-ity regions of stationary localized spots in this system. Here, we complement this analysis by establishing the stability regions of planar travelling fronts and stationary stripes. Taken together, these results imply that stable fronts and spots can coexist in three-component systems. Numerical simulations indicate that the stable fronts never move towards stable spots but instead move away from them.

Reasoning about component similarity in building product models from the construction perspective

Identifying the design features that impact construction is essential to developing cost effective and constructible designs. The similarity of building components is a critical design feature that affects method selection, productivity, and ultimately construction cost and schedule performance. However, there is limited understanding of what constitutes similarity in the design of building components and limited computer-based support to identify this feature in a building product model. This paper contributes a feature-based framework for representing and reasoning about component similarity that builds on ontological modelling, model-based reasoning and cluster analysis techniques. It describes the ontology we developed to characterize component similarity in terms of the component attributes, the direction, and the degree of variation. It also describes the generic reasoning process we formalized to identify component similarity in a standard product model based on practitioners' varied preferences. The generic reasoning process evaluates the geometric, topological, and symbolic similarities between components, creates groupings of similar components, and quantifies the degree of similarity. We implemented this reasoning process in a prototype cost estimating application, which creates and maintains cost estimates based on a building product model. Validation studies of the prototype system provide evidence that the framework is general and enables a more accurate and efficient cost estimating process.

Timed Up and Go Tests in cardiac rehabilitation : reliability and comparison with the 6-minute walk test

PURPOSE: To test the reliability of Timed Up and Go Tests (TUGTs) in cardiac rehabilitation (CR) and compare TUGTs to the 6-Minute Walk Test (6MWT) for outcome measurement. METHODS: Sixty-one of 154 consecutive community-based CR patients were prospectively recruited. Subjects undertook repeated TUGTs and 6MWTs at the start of CR (start-CR), postdischarge from CR (post-CR), and 6 months postdischarge from CR (6 months post-CR). The main outcome measurements were TUGT time (TUGTT) and 6MWT distance (6MWD). RESULTS: Mean (SD) TUGTT1 and TUGTT2 at the 3 assessments were 6.29 (1.30) and 5.94 (1.20); 5.81 (1.22) and 5.53 (1.09); and 5.39 (1.60) and 5.01 (1.28) seconds, respectively. A reduction in TUGTT occurred between each outcome point (P ≤ .002). Repeated TUGTTs were strongly correlated at each assessment, intraclass correlation (95% CI) = 0.85 (0.76–0.91), 0.84 (0.73–0.91), and 0.90 (0.83–0.94), despite a reduction between TUGTT1 and TUGTT2 of 5%, 5%, and 7%, respectively (P ≤ .006). Relative decreases in TUGTT1 (TUGTT2) occurred from start-CR to post-CR and from start-CR to 6 months post-CR of −7.5% (−6.9%) and −14.2% (−15.5%), respectively, while relative increases in 6MWD1 (6MWD2) occurred, 5.1% (7.2%) and 8.4% (10.2%), respectively (P < .001 in all cases). Pearson correlation coefficients for 6MWD1 to TUGTT1 and TUGTT2 across all times were −0.60 and −0.68 (P < .001) and the intraclass correlations (95% CI) for the speeds derived from averaged 6MWDs and TUGTTs were 0.65 (0.54, 0.73) (P < .001). CONCLUSIONS: Similar relative changes occurred for the TUGT and the 6MWT in CR. A significant correlation between the TUGTT and 6MWD was demonstrated, and we suggest that the TUGT may provide a related or a supplementary measurement of functional capacity in CR.

Application of theory in developing peer-support based intervention to improve self-management for cardiac patients with diabetes

The biosafety of carbon nanomaterial needs to be critically evaluated with both experimental and theoretical validations before extensive biomedical applications. In this letter, we present an analysis of the binding ability of two dimensional monolayer carbon nanomaterial on actin by molecular simulation to understand their adhesive characteristics on F-actin cytoskeleton. The modelling results indicate that the positively charged carbon nanomaterial has higher binding stability on actin. Compared to crystalline graphene, graphene oxide shows higher binding influence on actin when carrying 11 positive surface charge. This theoretical investigation provides insights into the sensitivity of actin-related cellular activities on carbon nanomaterial.