137 research outputs found
Fast-slow asymptotics for a Markov chain model of fast sodium current
We explore the feasibility of using fast-slow asymptotic to eliminate the
computational stiffness of the discrete-state, continuous-time deterministic
Markov chain models of ionic channels underlying cardiac excitability. We focus
on a Markov chain model of the fast sodium current, and investigate its
asymptotic behaviour with respect to small parameters identified in different
ways.Comment: 16 pages, 6 figures, as accepted to Chaos 2017/09/0
Dynamics of filaments of scroll waves
This has been written as a chapter for "Engineering Chemical Complexity II",
and as such does not have an abstract.Comment: 18 pages, 10 figure
Localization of response functions of spiral waves in the FitzHugh-Nagumo system
Dynamics of spiral waves in perturbed, e. g. slightly inhomogeneous or
subject to a small periodic external force, two-dimensional autowave media can
be described asymptotically in terms of Aristotelean dynamics, so that the
velocities of the spiral wave drift in space and time are proportional to the
forces caused by the perturbation. The forces are defined as a convolution of
the perturbation with the spiral's Response Functions, which are eigenfunctions
of the adjoint linearised problem. In this paper we find numerically the
Response Functions of a spiral wave solution in the classic excitable
FitzHugh-Nagumo model, and show that they are effectively localised in the
vicinity of the spiral core.Comment: 11 pages, 2 figure
Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity does not violate the Second Law
Solitons, defined as nonlinear waves which can reflect from boundaries or
transmit through each other, are found in conservative, fully integrable
systems. Similar phenomena, dubbed quasi-solitons, have been observed also in
dissipative, "excitable" systems, either at finely tuned parameters (near a
bifurcation) or in systems with cross-diffusion. Here we demonstrate that
quasi-solitons can be robustly observed in excitable systems with excitable
kinetics and with self-diffusion only. This includes quasi-solitons of fixed
shape (like KdV solitons) or envelope quasi-solitons (like NLS solitons). This
can happen in systems with more than two components, and can be explained by
effective cross-diffusion, which emerges via adiabatic elimination of a fast
but diffusing component. We describe here a reduction procedure can be used for
the search of complicated wave regimes in multi-component, stiff systems by
studying simplified, soft systems.Comment: 11 pages, 2 figures, as accepted to Scientific Reports on 2016/07/0
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
Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator-prey pursuit and evasion example
We have studied properties of nonlinear waves in a mathematical model of a
predator-prey system with pursuit and evasion. We demonstrate a new type of
propagating wave in this system. The mechanism of propagation of these waves
essentially depends on the ``taxis'', represented by nonlinear
``cross-diffusion'' terms in the mathematical formulation. We have shown that
the dependence of the velocity of wave propagation on the taxis has two
distinct forms, ``parabolic'' and ``linear''. Transition from one form to the
other correlates with changes in the shape of the wave profile. Dependence of
the propagation velocity on diffusion in this system differs from the
square-root dependence typical of reaction-diffusion waves. We demonstrate also
that, for systems with negative and positive taxis, for example, pursuit and
evasion, there typically exists a large region in the parameter space, where
the waves demonstrate quasisoliton interaction: colliding waves can penetrate
through each other, and waves can also reflect from impermeable boundaries.Comment: 15 pages, 18 figures, submitted to Physica
Critical fronts in initiation of excitation waves
Copyright © 2007 The American Physical SocietyJournal ArticleWe consider the problem of initiation of propagating waves in a one-dimensional excitable fiber. In the FitzHugh-Nagumo theory, the key role is played by "critical nucleus" and "critical pulse" solutions whose (center-) stable manifold is the threshold surface separating initial conditions leading to propagation and those leading to decay. We present evidence that in cardiac excitation models, this role is played by "critical front" solutions
Asymptotics of conduction velocity restitution in models of electrical excitation in the heart.
Copyright © Springer 2011Journal ArticleThe original publication is available at www.springerlink.com - http://link.springer.com/article/10.1007/s11538-010-9523-6We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh-Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified "caricature" of Noble (J. Physiol. Lond. 160:317-352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177-210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler-Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method
Orbital Motion of Spiral Waves in Excitable Media
Spiral waves in active media react to small perturbations as particle-like objects. Here we apply the asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation centre along circular orbits around the inhomogeneity. The stationary orbits have alternating stability and fixed radii, determined by the properties of the bulk medium and the type of inhomogeneity, while the drift speed along an orbit depends on the strength of the inhomogeneity. Direct simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment
Localization of response functions of spiral waves in the FitzHugh-Nagumo system
Preprint of an article submitted for consideration and subsequently published in International Journal of Bifurcation and Chaos © 2005 copyright World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ijbcDynamics of spiral waves in perturbed, e. g. slightly inhomogeneous or subject to a small periodic external force, two-dimensional autowave media can be described asymptotically in terms of Aristotelean dynamics, so that the velocities of the spiral wave drift in space and time are proportional to the forces caused by the perturbation. The forces are defined as a convolution of the perturbation with the spiral's Response Functions, which are eigenfunctions of the adjoint linearised problem. In this paper we find numerically the Response Functions of a spiral wave solution in the classic excitable FitzHugh-Nagumo model, and show that they are effectively localised in the vicinity of the spiral core
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