Vadim N. Biktashev

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We analyze the asymptotic structure of two classical models of mathematical biology, the models of electrical action by Hodgkin-Huxley (1952) for a giant squid axon and by Noble (1962) for mammalian Purkinje fibres. We use the procedure of parametric embedding to formally introduce small parameters in these experiment-based models. Although one of the(More)
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(More)
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(More)
Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological nature. In the presence of a small perturbation, the spiral wave's center of rotation and fiducial phase may change over time, i.e., the spiral wave drifts. In linear approximation, the velocity of the drift is proportional to(More)
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(More)
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(More)
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(More)
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(More)