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The Institute of Mathematical Problems of Biology RAS, Russia
1. Autowave phenomena
Nowaday the importance of autowave processes is widely recognised, and the special attention is paid to them in cardiology . One of the most typical autowave solution in mathematical cardiology is a reverberator, that is an autowave vortex in 2D medium, or, in other words, an autowave rotating with its free tip. Usually, an autowave like this takes the form of a rotating spiral. The differences between an autowave reverberator and a spiral autowave are discussed sufficiently in . Typical reverberator behaviour appears as meander, that is to say a spontaneous drift of the reverberator in homogeneous medium. Recently a new type of reverberator behaviour, the autowave lacet, was revealed [4, 5]. The lacet appears as spontaneous braking of the spontaneous drift of the reverberator in homogeneous medium up to the drift ceases completely.
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 . 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  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 bifurcation memory (BM). BM is believed to be a specific transition process observed in the system behaviour near the bifurcation boundary [2, 3]. Although there is a deal of different opinions about the nature of the phenomenon, nevertheless, it is generally accepted that BM play a substantial part in practice because controllability of a dynamic system decreases dramatically in this case . Such a region of reduced controllability in phase portrait referred to as phase spot. Therefore, it is important from the practical standpoint to find the size of the bifurcation boundary zone in the model parameter space where the bifurcation memory can be observed. We will refer to a number of points located along a bifurcation boundary in parameter space where the phenomenon of BM ceases as a line of exit from bifurcation boundary zone. The bifurcation boundary zone is regarded to be a number of points contained between the bifurcation boundary and the line of exit from bifurcation boundary zone. Unfortunately, the used before method of quantitative analysis of spontaneous drift of reverberator in 2D-medium  do not give a chance to reveal the width of the bifurcation boundary zone.
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
We use used the Aliev-Panfilov mathematical model (AP-model) of excitable medium :
For more details, see elsewhere [Mathematical biology & bioinformatics, 2007; Chaos, Solitons and Fractals]
2.2. Analysis of the Reverberator Movement
Parameterization 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  dependence of the curvature radius, Rкр, on the time, t, were approximated by the function:
On graph of the dependence of Rкр on a, one can discover an extremum close a = 0.135, which is at some distance from the bifurcation boundary. While there being no well-founded objection, the most reasonable assumption is that the minimum along the bifurcation boundary corresponds just to the line of exit from bifurcation boundary zone.
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).
1. 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 a = 0.170. Thin line shows the curvature radius of the tip trajectory, Rcur, obtained experimentally.
Fig. 2. Dependence of average curvature radius of the tip trajectory (black circles), <Rcur>, and the characteristic times of the lacet drift halt (white triangles), τ1, on the the model parameter a of the Aliev–Panfilov model.