39 research outputs found
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
Exponential integrators for a Markov chain model of the fast sodium channel of cardiomyocytes
The modern Markov chain models of ionic channels in excitable membranes are
numerically stiff. The popular numerical methods for these models require very
small time steps to ensure stability. Our objective is to formulate and test
two methods addressing this issue, so that the timestep can be chosen based on
accuracy rather than stability.
Both proposed methods extend Rush-Larsen technique, which was originally
developed to Hogdkin-Huxley type gate models. One method, "Matrix Rush-Larsen"
(MRL) uses a matrix reformulation of the Rush-Larsen scheme, where the matrix
exponentials are calculated using precomputed tables of eigenvalues and
eigenvectors. The other, "hybrid operator splitting" (HOS) method exploits
asymptotic properties of a particular Markov chain model, allowing explicit
analytical expressions for the substeps.
We test both methods on the Clancy and Rudy (2002) INa Markov chain model.
With precomputed tables for functions of the transmembrane voltage, both
methods are comparable to the forward Euler method in accuracy and
computational cost, but allow longer time steps without numerical instability.
We conclude that both methods are of practical interest. MRL requires more
computations than HOS, but is formulated in general terms which can be readily
extended to other Markov Chain channel models, whereas the utility of HOS
depends on the asymptotic properties of a particular model.
The significance of the methods is that they allow a considerable speed-up of
large-scale computations of cardiac excitation models by increasing the time
step, while maintaining acceptable accuracy and preserving numerical stability.Comment: 9 pages, 5 figures main text + 14 pages, 1 figure appendix, as
submitted in final form to IEEE TBME 2014/11/11. Copyright IEEE (2014
Conditions for propagation and block of excitation in an asymptotic model of atrial tissue
Detailed ionic models of cardiac cells are difficult for numerical
simulations because they consist of a large number of equations and contain
small parameters. The presence of small parameters, however, may be used for
asymptotic reduction of the models. Earlier results have shown that the
asymptotics of cardiac equations are non-standard. Here we apply such a novel
asymptotic method to an ionic model of human atrial tissue in order to obtain a
reduced but accurate model for the description of excitation fronts. Numerical
simulations of spiral waves in atrial tissue show that wave fronts of
propagating action potentials break-up and self-terminate. Our model, in
particular, yields a simple analytical criterion of propagation block, which is
similar in purpose but completely different in nature to the `Maxwell rule' in
the FitzHugh-Nagumo type models. Our new criterion agrees with direct numerical
simulations of break-up of re-entrant waves.Comment: Revised manuscript submitted to Biophysical Journal (30 pages incl.
10 figures
Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone
Abnormal electrical activity from the boundaries of ischemic cardiac tissue
is recognized as one of the major causes in generation of ischemia-reperfusion
arrhythmias. Here we present theoretical analysis of the waves of electrical
activity that can rise on the boundary of cardiac cell network upon its
recovery from ischaemia-like conditions. The main factors included in our
analysis are macroscopic gradients of the cell-to-cell coupling and cell
excitability and microscopic heterogeneity of individual cells. The interplay
between these factors allows one to explain how spirals form, drift together
with the moving boundary, get transiently pinned to local inhomogeneities, and
finally penetrate into the bulk of the well-coupled tissue where they reach
macroscopic scale. The asymptotic theory of the drift of spiral and scroll
waves based on response functions provides explanation of the drifts involved
in this mechanism, with the exception of effects due to the discreteness of
cardiac tissue. In particular, this asymptotic theory allows an extrapolation
of 2D events into 3D, which has shown that cells within the border zone can
give rise to 3D analogues of spirals, the scroll waves. When and if such scroll
waves escape into a better coupled tissue, they are likely to collapse due to
the positive filament tension. However, our simulations have shown that such
collapse of newly generated scrolls is not inevitable and that under certain
conditions filament tension becomes negative, leading to scroll filaments to
expand and multiply leading to a fibrillation-like state within small areas of
cardiac tissue.Comment: 26 pages, 13 figures, appendix and 2 movies, as accepted to PLoS ONE
2011/08/0
BeatBox - HPC simulation environment for biophysically and anatomically realistic cardiac electrophysiology
The BeatBox simulation environment combines flexible script language user
interface with the robust computational tools, in order to setup cardiac
electrophysiology in-silico experiments without re-coding at low-level, so that
cell excitation, tissue/anatomy models, stimulation protocols may be included
into a BeatBox script, and simulation run either sequentially or in parallel
(MPI) without re-compilation. BeatBox is a free software written in C language
to be run on a Unix-based platform. It provides the whole spectrum of multi
scale tissue modelling from 0-dimensional individual cell simulation,
1-dimensional fibre, 2-dimensional sheet and 3-dimensional slab of tissue, up
to anatomically realistic whole heart simulations, with run time measurements
including cardiac re-entry tip/filament tracing, ECG, local/global samples of
any variables, etc. BeatBox solvers, cell, and tissue/anatomy models
repositories are extended via robust and flexible interfaces, thus providing an
open framework for new developments in the field. In this paper we give an
overview of the BeatBox current state, together with a description of the main
computational methods and MPI parallelisation approaches.Comment: 37 pages, 10 figures, last version submitted to PLOS ON
A simple mathematical model of gradual Darwinian evolution: Emergence of a Gaussian trait distribution in adaptation along a fitness gradient
We consider a simple mathematical model of gradual Darwinian evolution in
continuous time and continuous trait space, due to intraspecific competition
for common resource in an asexually reproducing population in constant
environment, while far from evolutionary stable equilibrium. The model admits
exact analytical solution. In particular, Gaussian distribution of the trait
emerges from generic initial conditions.Comment: 21 pages, 2 figures, as accepted to J Math Biol 2013/03/1