The Institute of Mathematical Problems of Biology RAS, Russia
## 1. Autowave phenomenaNowaday the importance of autowave processes is widely recognised, and the special attention is paid to them in cardiology [4]. One of the most typical autowave solution in mathematical cardiology is a All autowave models demonstrate very multifarious behaviour, with often a little modification of their parameters being enough for radical change of the model behaviour, as it was shown, for example, in recent works concerning investigation of the human coagulation system [3]. This circumstance makes us see building an accurate parametric portrait of the system under investigation as the first requisite step in the way of using an autowave model. However, it should be pointed out some oddity of the situation that exists now due to historical reason only: when autowave solution studying, the tradition of qualitative visual analysis of the derivable solutions predominates. Nevertheless, recent research has shown [5] that the use of quantitative methods in this field gives well-defined advantage. In this way, just the quantitative analysis of the reverberator drift furnishes evidence that the meander and the lacet are two essentially different autowave solutions, with the frontier separating meander and lacet areas in the model parameter space revealed quite precisely. The autowave lacet is a kind of the phenomenon of so called In this work, we propose a quantitative approach, which enables to study the peculiarities of the autowave model behaviour in the bifurcation boundary zone as well as to reveal the line of exit from bifurcation boundary zone ## 2. Methods and Results## 2.1. ModelWe use used the Aliev-Panfilov mathematical model (AP-model) of excitable medium [1]:
For more details, see elsewhere [Mathematical biology & bioinformatics, 2007; Chaos, Solitons and Fractals] ## 2.2. Analysis of the Reverberator MovementParameterization of a curve corresponding to measuring results is one of standard approach to analysis of experimental data. Before carrying out the parameterization of the trajectories of reverberator tip movement, we found it to be sensible to transform the dynamics of two space variables into dynamics of one variable, which is a curvature of the trajectory. Taking measurements of the curvature was fulfilled as it described here. For each point of the trajectory, its part close to the point was approximated with circular arc by technique of least squares. Each time we fulfilled the approximation in an approximation window that is the part of the trajectory corresponding to some fixed time interval. In this way, the estimation of curvature radius was obtained at each instant of time. Then corresponding to the lacet [5] dependence of the curvature radius, ## 3. DiscussionOn graph of the dependence of This work demonstrates that it is impossible in some cases to find the answer to a number of important scientific questions without using quantitative methods for analysis of autowave solution obtained. The combination of quantitative methods of estimation of spontaneous drift of the reverberator enables us to automatize the building an accurate parametric portrait of diferent autovave 2D systems. The partial support of Russian Foundation for Basic Research is acknowledged (the projects 08-07-00353 and 10-01-00609). ## References1. Aliev R., Panfilov A. A simple two-variable model of cardiac excitation. Chaos, Solutions & Fractals 1996; 7(3): 293-301. 2. Feigin M., Kagan M. Emergencies as a manifestation of effect of bifurcation memory in controlled unstable systems. International Journal of Bifurcation and Chaos, 2004; 14(7): 2439-2447. 3. Ataullakhanov F, Lobanova E, Morozova O, Shnol’ E, Ermakova E, Butylin A, et al. Intricate regimes of propagation of an excitation and self-organization in the blood clotting model. Physics–Uspekhi 2007;50(1):89–94. 4. Elkin Yu. E., Moskalenko A.V. Basic mechanisms of cardiac arrhythmias. In: Ardashev, A.V., editor. Clinical Arrhythmology. Moscow: MedPraktika-M; 2009, pp 45-74, 2009 (In Russian). 5. Elkin Yu. E., Moskalenko A.V., Starmer Ch.F. Spontaneous halt of spiral wave drift in homogeneous excitable media. Mathematical biology & bioinformatics, 2007; 2(1):1-9. 6. Moskalenko A.V., Elkin Yu. E. Is monomorphic tachycardia indeed monomorphic? Biophysics, 2007; 52(2):237-240. |

Fig. 1.
The result of the approximation of curvature radius of the tip trajectory at the Aliev–Panfilov model parameter |

Fig. 2.
Dependence of average curvature radius of the tip trajectory (black circles), < |