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V N Biktashev

Publications and source records attributed to V N Biktashev.

At least 19 recordsLinked to original sources
PubMed2006

Asymptotic properties of mathematical models of excitability.

We analyse small parameters in selected models of biological excitability, including Hodgkin-Huxley (Hodgkin & Huxley 1952 J. Physiol.117, 500-544) model of nerve axon, Noble (Noble 1962 J. Physiol.160, 317-352) model of heart Purkinje fibres and Courtemanche et al. (Courtemanche et al. 1998 Am. J. Physiol.275, H301-H321) model of human atrial cells. Some of the small parameters are responsible for differences in the characteristic time-scales 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 breakups and self-termination of re-entrant waves in excitable media with Courtemanche et al. kinetics.

Action Potentials↗
PubMed2005

Causodynamics of autowave patterns.

Perturbative dynamics of spiral and scroll waves involves the "response functions," i.e., critical eigenvectors of the adjoint linearized operator, dual to the Goldstone modes. A well known method of calculating the Goldstone modes is time integration of the linearized equation. We suggest that backward-time integration of the adjoint linearized equation, which we call causodynamics, can be used to calculate the response functions. This new method is more robust and easier to implement than existing methods. We illustrate how it works for propagating and rotating autowaves in reaction-diffusion systems. The method reveals unexpected qualitative difference between similarly looking regimes.

Journal Article↗
PubMed2004

Non-Tikhonov asymptotic properties of cardiac excitability.

Models of electric excitability of cardiac cells can be studied by singular perturbation techniques. To do this one should take into account parameters appearing in equations in nonstandard ways. The physical reason for this is near-perfect switch behavior of ionic current gates. This leads to a definition of excitability different from the currently accepted one. The asymptotic structure revealed by our analysis can be used to devise simplified caricature models, to obtain approximate analytical solutions, and to facilitate numerical simulations.

Action Potentials↗
PubMed2004

Half-soliton interaction of population taxis waves in predator-prey systems with pursuit and evasion.

In this paper, we use numerical simulations to demonstrate a half-soliton interaction of waves in a mathematical model of a "prey-predator" system with taxis when of two colliding waves, one annihilates and the other continues to propagate. We show that this effect depends on the "ages" or, equivalently, "widths" of the colliding waves. In two spatial dimensions we demonstrate that the type of interaction, i.e., annihilation, quasisoliton, or half-soliton, depends not only on curvature and width of the colliding waves, but also on the angle of the collision. When conditions of collision are varying in such a way that only a part of a wave survives the collision, then "taxitons," compact pieces of solitary waves, may form, which can exist for a significant time.

Adaptation, Physiological↗
PubMed2004

Phytoplankton blooms and fish recruitment rate: effects of spatial distribution.

We consider the spatio-temporal dynamics of a spatially-structured generalization of the phytoplankton-zooplankton-fish larvae model system proposed earlier (Biktashev et al., 2003, J. Plankton Res. 5, 21-33; James et al., 2003, Ecol. Model. 160, 77-90). In contrast to Pitchford and Brindley (2001, Bull. Math. Biol. 63, 527-546), who were concerned with small scale patchiness (i.e., 1-10 m), on which the (stochastic) raptorial behaviour of individual larvae is important, we address here the much larger scale 'patchy' problems (i.e., 10-100 km), on which both larvae and plankton may be regarded as passive tracers of the fluid motion, dispersed and mixed by the turbulent diffusion processes. In particular, we study the dependence of the fish recruitment on carrying capacities of the plankton subsystem and on spatio-temporal evolution of that subsystem with respect to the larvae hatching site(s). It is shown that the main features found both in the nonstructured and age-structured spatially uniform models are observed in the spatially structured case, but that spatial effects can significantly modify the overall quantitative outcome. Spatial patterns in the metamorphosed fish distribution are a consequence of quasi-local interaction of larvae with plankton, in which the dispersion of larvae by large scale turbulent eddies plays little part due to the relatively short timescale of the larvae development. As a result, in a strong phyto/zooplankton subsystem, with fast reproduction rate and large carrying capacity of phytoplankton and high conversion ratio of zooplankton, recruitment success depends only on the localization and timing of the hatching with respect to the plankton patches. In a weak phyto/zooplankton system, with slow reproduction rate and small carrying capacity of phytoplankton and low conversion ratio of zooplankton, the larvae may significantly influence the evolution of the plankton patches, which may lead to nontrivial cooperative effects between different patches of larvae. However, in this case, recruitment is very low.

Animals↗
PubMed2004

Pursuit-evasion predator-prey waves in two spatial dimensions.

We consider a spatially distributed population dynamics model with excitable predator-prey kinetics, 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 wave tips, attachment of free wave ends to wave backs, and as a result, a novel mechanism of self-supporting complicated spatiotemporal activity, unknown in reaction-diffusion population models.

Adaptation, Physiological↗
PubMed2003

Quasisoliton interaction of pursuit-evasion waves in a predator-prey system.

We consider a system of partial differential equations describing two spatially distributed populations in a "predator-prey" interaction with each other. The spatial evolution is governed by three processes: positive taxis of predators up the gradient of prey (pursuit), negative taxis of prey down the gradient of predators (evasion), and diffusion resulting from random motion of both species. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the taxis and is entirely different from waves in a reaction-diffusion system. Unlike typical reaction-diffusion waves, which annihilate on collision, these "taxis" waves can often penetrate through each other and reflect from impermeable boundaries.

Animals↗
PubMed2003

Wave-particle dualism of spiral waves dynamics.

We demonstrate and explain a wave-particle dualism of such classical macroscopic phenomena as spiral waves in active media. That means although spiral waves appear as nonlocal processes involving the whole medium, they respond to small perturbations as effectively localized entities. The dualism appears as an emergent property of a nonlinear field and is mathematically expressed in terms of the spiral waves response functions, which are essentially nonzero only in the vicinity of the spiral wave core. Knowledge of the response functions allows quantitatively accurate prediction of the spiral wave drift due to small perturbations of any nature, which makes them as fundamental characteristics for spiral waves as mass is for the condensed matter.

Journal Article↗
PubMed2002

Dissipation of the excitation wave fronts.

An excitation wave in cardiac tissue will fail to propagate if the transmembrane voltage at its front rises too slowly and does not excite the tissue ahead of it. Then the sharp voltage profile of the front will dissipate, and the subsequent spread of voltage will be purely diffusive. This mechanism is impossible in FitzHugh-Nagumo type systems. Here a simplified mathematical model for this mechanism is suggested. The model has exact traveling front solutions, and gives conditions for the front dissipation. In particular, a front will dissipate if it is not allowed to propagate faster than a certain nonzero speed. This critical speed depends only on the properties of the fast sodium current.

Atrial Function↗
PubMed2001

Re-entrant cardiac arrhythmias in computational models of long QT myocardium.

The long QT syndrome (LQTS) is an inherited disorder in which repolarization of cardiac ventricular cells is prolonged. Patients with the LQTS are at an increased risk of ventricular cardiac arrhythmias. Two phenotypes of the inherited LQTS are caused by defects in K(+)channels (LQT1 and LQT2) and one by defects in Na(+)channels (LQT3). Patients with LQT1 are more likely to have self-terminating arrhythmias than those with LQT3. The aim of this computational study was to propose an explanation for this finding by comparing the vulnerability of normal and LQT tissue to re-entry, and estimating the likelihood of self-termination by motion of re-entrant waves to an inexcitable boundary in simulated LQT1, LQT2 and LQT3 tissue. We modified a model of mammalian cardiac cells to simulate LQT1 by reducing maximal I(K(s))conductance, LQT2 by reducing maximal I(K(r))conductance, and LQT3 by preventing complete inactivation of I(Na)channels. Each simulated phenotype was incorporated into a computational model of action potential propagation in one- and two-dimensional homogeneous tissue. Simulated LQT tissue was no more vulnerable to re-entry than simulated normal tissue, but the motion of re-entrant waves in simulated LQT1 tissue was between 2 and 5 times greater than the motion of re-entrant waves in simulated LQT2 and LQT3 tissue. These findings suggest that LQT arrhythmias do not result from increased vulnerability to re-entry, and that re-entry once initiated is more likely to self-terminate by moving to an inexcitable tissue boundary in LQT1 than in LQT2 and LQT3. This finding is consistent with clinical observations.

Action Potentials↗
PubMed1999

Spatiotemporal irregularity in an excitable medium with shear flow.

We consider an excitable medium moving with relative shear, subjected to a localized disturbance that in a stationary medium would produce a pair of spiral waves. The spiral waves so created are distorted and then broken by the motion of the medium. Such breaks generate new spiral waves, and so a "chain reaction" of spiral wave births and deaths is observed. This leads to a complicated spatiotemporal pattern, the "frazzle gas" [term suggested by Markus et al., Nature (London) 371, 402 (1994)], which eventually fills the whole medium. In this paper, we display and interpret the main features of the pattern.

Journal Article↗
PubMed1999

Reentrant arrhythmias and their control in models of mammalian cardiac tissue.

We use detailed biophysical and simplified models of excitation propagation in heart muscle to study the properties of reentrant arrhythmias. Using a detailed model of excitation combined with a bidomain description of propagation and action of electric current, we have obtained a theoretical estimation for the defibrillation threshold consistent with experimental data. Reentry acts as a spiral wave, propagating around a region of block, the core. A series of properly timed low-voltage stimuli can cause directed "resonant" drift of this block and act as a low-voltage defibrillation strategy. Experimentally observed activation patterns in fibrillating tissue are more complicated than the simplest spiral wave patterns. This is due to complicated geometry, the 3-dimensional nature of the tissue, and its anisotropy and inhomogeneity. However, some fibrillation patterns can be produced by a single reentrant wave, modulated by inhomogeneous tissue properties and Wenckebach frequency division.

Animals↗
PubMed1998

Reentrant waves and their elimination in a model of mammalian ventricular tissue.

The vulnerability to reentrant wave propagation, its characteristics (period, meander, and stability), the effects of rotational transmural anisotropy, and the control of reentrant waves by small amplitude perturbations and large amplitude defibrillating shocks are investigated theoretically and numerically for models based on high order, stiff biophysically derived excitation equations.

Journal Article↗
PubMed1996

Re-entrant activity and its control in a model of mammalian ventricular tissue.

We characterize the meander of re-entrant excitation in a model of a sheet of mammalian ventricular tissue, and its control by resonant drift under feedback driven stimulation. The Oxsoft equations for excitability in a guinea pig single ventricular cell were incorporated in a two dimensional reaction-diffusion system to model homogeneous, isotropic tissue with a plane wave conduction velocity of 0.35 m s-1. Re-entrant spiral wave solutions have a spatially extended transient motion (linear core) that settles down into rotation with an irregular period of 100-110 ms around an irregular, multi-lobed spiky core. In anisotropic tissue this would appear as a linear conduction block. The typical velocity of drift of the spiral wave induced by low amplitude resonant forcing is 0.4 cm s-1.

Animals↗
PubMed1995

Control of re-entrant activity in a model of mammalian atrial tissue.

We evaluate the feasibility of using resonant drift under feedback driven stimulation to control re-entrant excitation in atrial muscle. We simulate a two-dimensional sheet of atrial tissue, where the local kinetics are described by the Earm-Hilgemann-Noble equations for a rabbit atrial cell, and the effects of small amplitude spatially uniform forcing of the whole sheet are computed. Repetitive forcing can induce a drift of a spiral wave in the two-dimensional model, with a drift velocity of up to 10 cm s-1. For a 4 cm x 4 cm atrial surface this resonant drift can move the re-entrant spiral to the inexcitable boundaries, eliminating re-entry in less than 10 s when the amplitude of the repetitive stimulation is 10% that of the single shock defibrillation threshold.

Animals↗
PubMed1994

Design principles of a low voltage cardiac defibrillator based on the effect of feedback resonant drift.

We explore the possibility of exploiting the phenomenon of the resonant drift of autowave vortices in the design of low-voltage cardiac defibrillators. The problem of reflection of resonantly drifting vortices by inexcitable boundaries is overcome by using feedback. Numerical studies show that the voltage needed for defibrillation by this method may be substantially less (by an order of magnitude) than that used in conventional single-shock defibrillation.

Defibrillators, Implantable↗
PubMed1993

Vulnerability in an excitable medium: analytical and numerical studies of initiating unidirectional propagation.

Cardiac tissue can display unusual responses to certain stimulation protocols. In the wake of a conditioning wave of excitation, spiral waves can be initiated by applying stimuli timed to occur during a period of vulnerability (VP). Although vulnerability is well known in cardiac and chemical media, the determinants of the VP and its boundaries have received little theoretical and analytical study. From numerical and analytical studies of reaction-diffusion equations, we have found that 1) vulnerability is an inherent property of Beeler-Reuter and FitzHugh-Nagumo models of excitable media; 2) the duration of the vulnerable window (VW) the one-dimensional analog of the VP, is sensitive to the medium properties and the size of the stimulus field; and 3) the amplitudes of the excitatory and recovery processes modulate the duration of the VW. The analytical results reveal macroscopic behavior (vulnerability) derived from the diffusion of excitation that is not observable at the level of isolated cells or single reaction units.

Animals↗