arXiv subjects

V. N. Biktashev

Publications and source records attributed to V. N. Biktashev.

25 records · Page 2Linked to original sources

arXiv2008

An analytical approach to initiation of propagating fronts

We consider the problem of initiation of a propagating wave in a one-dimensional excitable fibre. In the Zeldovich-Frank-Kamenetsky equation, a.k.a. Nagumo equation, the key role is played by the "critical nucleus'' solution whose stable manifold is the threshold surface separating initial conditions leading to initiation of propagation and to decay. Approximation of this manifold by its tangent linear space yields an analytical criterion of initiation which is in a good agreement with direct numerical simulations.

nlin.PS↗

arXiv2007

Generation and escape of local waves from the boundary of uncoupled cardiac tissue

We aim to understand the formation of abnormal waves of activity from myocardial regions with diminished cell-to-cell coupling. In route to this goal, we studied the behavior of a heterogeneous myocyte network in which a sharp coupling gradient was placed under conditions of increasing network automaticity. Experiments were conducted in monolayers of neonatal rat cardiomyocytes using heptanol and isoproterenol as means of altering cell-to-cell coupling and automaticity respectively. Experimental findings were explained and expanded using a modified Beeler-Reuter numerical model. The data suggests that the combination of a heterogeneous substrate, a gradient of coupling and an increase in oscillatory activity of individual cells creates a rich set of behaviors associated with self-generated spiral waves and ectopic sources. Spiral waves feature a flattened shape and a pin-unpin drift type of tip motion. These intercellular waves are action-potential based and can be visualized with either voltage or calcium transient measurements. A source/load mismatch on the interface between the boundary and well-coupled layers can lock wavefronts emanating from both ectopic sources and rotating waves within the inner layers of the coupling gradient. A numerical approach allowed us to explore how: i) the spatial distribution of cells, ii) the amplitude and dispersion of cell automaticity, iii) and the speed at which the coupling gradient moves in space, affects wave behavior, including its escape into well-coupled tissue.

q-bio.TO↗

arXiv2007

Critical fronts in initiation of excitation waves

We consider the problem of initiation of propagating wave 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.

nlin.PS↗

arXiv2005

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.

nlin.PS↗

arXiv2005

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.

nlin.PS↗

arXiv2004

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.

nlin.PS↗

arXiv2004

Pursuit-evasion predator-prey waves in two spatial dimensions

We consider a spatially distributed population dynamics model with excitable predator-prey dynamics, where species propagate in space due to their taxis with respect to each other's gradient in addition to, or instead of, their diffusive spread. Earlier, we have described new phenomena in this model in one spatial dimension, not found in analogous systems without taxis: reflecting and self-splitting waves. Here we identify new phenomena in two spatial dimensions: unusual patterns of meander of spirals, partial reflection of waves, swelling wavetips, attachment of free wave ends to wave backs, and as a result, a novel mechanism of self-supporting complicated spatio-temporal activity, unknown in reaction-diffusion population models.

nlin.PS↗