Asymptotic properties of mathematical models of excitability

We analyse small parameters in selected models of biological excitability, including Hodgkin-Huxley (1952) model of nerve axon, Noble (1962) model of heart Purkinje fibres, and Courtemanche et al. (1998) model of human atrial cells. Some of the small parameters are responsible for differences in the characteristic timescales of dynamic variables, as in the traditional singular perturbation approaches. Others appear in a way which makes the standard approaches inapplicable. We apply this analysis to study the behaviour of fronts of excitation waves in spatially-extended cardiac models. Suppressing the excitability of the tissue leads to a decrease in the propagation speed, but only to a certain limit; further suppression blocks active propagation and leads to a passive diffusive spread of voltage. Such a dissipation may happen if a front propagates into a tissue recovering after a previous wave, e.g. re-entry. A dissipated front does not recover even when the excitability restores. This has no analogy in FitzHugh-Nagumo model and its variants, where fronts can stop and then start again. In two spatial dimensions, dissipation accounts for break-ups and self-termination of re-entrant waves in excitable media with Courtemanche et al. (1998) kinetics.Comment: 15 pages, 8 figures, to appear in Phil Trans Roy Soc London

Asymptotic analysis and analytical solutions of a model of cardiac excitation.

The original publication is available at www.springerlink.com - http://link.springer.com/article/10.1007/s11538-007-9267-0Journal ArticleCopyright © SpringerWe describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317-352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an "Archetypal Model" with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable "caricature" of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the "caricature" right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model

Workplace Alignment: An evaluation of office worker flexibility and workplace provision

Purpose – The paper aims to explore the relationship between office occupier work activity and workplace provision. It tests the proposition that location-fixed office workers are not as well-supported in the working environment as location-flexible office workers. The research also explores the perceptions of the workplace provision based upon the types of tasks completed at the desk-location, whether this was collaborative or focused. Design/methodology/approach – The research adopts a cross-sectional approach using an online questionnaire to collect data from several offices in the Middles East. The dataset consists of 405 responses. One-way ANOVA was conducted to understand the relationship between location flexibility and perception of productivity. In addition, a series of T-Test were used to evaluate the relationship between work activities and office environment. Findings – The results show that those workers who were location-fixed perceived the workplace provision to have a more negative impact on their productivity than those who had a greater level of location-flexibility, particularly with regards to noise levels and interruptions. In terms of types of activities, those that undertook more collaborative tasks valued the facilitation of creativity and interaction from the workplace provision. Research limitations/implications – The research has limitations as data collection was at one-point in time and therefore lacks the opportunity to undertake longitudinal analysis. However, the research gives greater insights into the alignment of office environments based on flexibility and work activity. Practical implications – The paper identifies implications for the design and development of office environments by identifying the need for office occupier activity profiles. These profiles can underpin data led design which should promote a tailored choice appropriate work setting that can maximise productivity. Originality/value – This paper contributes to the research area of workplace alignment. It establishes that optimal workplace alignment requires a better understanding of office occupier needs based on location-flexibility and work activity

Analytically Solvable Asymptotic Model of Atrial Excitability

We report a three-variable simplified model of excitation fronts in human atrial tissue. The model is derived by novel asymptotic techniques \new{from the biophysically realistic model of Courtemanche et al (1998) in extension of our previous similar models. An iterative analytical solution of the model is presented which is in excellent quantitative agreement with the realistic model. It opens new possibilities for analytical studies as well as for efficient numerical simulation of this and other cardiac models of similar structure

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Asymptotic analysis and analytical solutions of a model of cardiac excitation

We describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317–352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an “Archetypal Model” with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable “caricature” of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the “caricature” right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model