Yu.E. Elkin*), A.V. Moskalenko**), C.F. Starmer***)
Autowave processes, particularly waves of excitation, are typical for a lot of physical, chemical and biological systems. For example, some regimes of excitation wave spread in myocardium support normal functioning of the heart. Other regimes could lead to life-threatened cardiac arrhythmias. These reason determine importance of investigation of excitation waves.
In two-dimensional excitable medium, one of the typical autowave processes is spiral wave, which is a curved half-wave. A break of the half-wave bears the name "spiral wave tip". A trajectory of the spiral wave tip describes the autowave process .
Until now three kinds of spiral wave tip behaviour were known in homogeneous two-dimensional excitable medium :
For the first time, we succeeded in finding a autowave regime when spiral wave tip draws a trajectory liked to cycloid (as it is during classic meander), but cycloid loops gradually takes its places more and more closely until the tip trajectory turn into a circle. As a result, spontaneous halt of spiral wave drift is observed.
The new kind of spiral wave drift was obtained with use of the Aliev-Panfilov mathematical model of homogeneous two-dimensional excitable medium.
We use used the Aliev-Panfilov mathematical model (AP-model) of excitable medium :
The AP-model is a modification of classic FitzHue-Nagumo model , with it suiting the heart tissue properties in the best way among FHN-class models. The parameters in the equations were adjusted  to accurately reflect cardiac tissue: k=8.0, e0=0.01, m1=0.2, m2=0,3 и a=0,15. We carried out our simulations with 0.11<a<0.23 with the step Da=0.01. Note that the parameter, a, specifies threshold of excitation. The simulations were carried out in 2D excitable media (128 elements along each dimension) with von Neumann boundary conditions. The explicit Euler's scheme of integration was used with time step Dt= 0.01 and space step Dx= 0.50. The spiral wave tip location was considered as crossing of two isolines u=0.89 and v=0.50.
The next two figures show the results of simulation with a=0.18. The first figure represents the location of a spiral wave in the instant t=979.87t.u. as well as the trajectory of the spiral wave tip in interval from t=100.00t.u to t=979.87t.u.
The next figure embodies the trajectory of the spiral wave tip in interval from t=510.12t.u. to t=1510.12t.u. (the trajectory is segregated in two parts because of its partial overriding itself)
In the figures, one can see that the distance between the adjacent cycloid loops gradually decrease with time until the drift of the spiral wave spontaneously halts and the tip trajectory turn into a circle.
The results prove to be identical for both 128х128 and 200х200 media. It allows us to exclude the assumption about boundary influence. The described effect is also reproduced when the space step Dx is twice decreased.
Such a kind of spiral wave drift we named as "lacet" to distinguish it from the classic two-periodic meander. We succeeded in observing the lacet only with the AP-model parameter, a, less than 0.1804. The trajectories of spiral wave tip appear as the classic meander when a>0.1804. In the range 0.1100<a<0.1803, we observed gradual increase of the time of the spontaneous halt of the spiral wave drift. When a<0.15, the drift deceleration happens in a few (2-4) loops. Confident observation of the lacet became possible only when a>0.175.
The phenomenon observed in this work, the lacet, gives birth to multitude of new questions supplying the matters for further investigation.
First, a question arose whether such drift deceleration in some further unobserved interval of time when a>0.1810. The question is of fundamental importance because our observation prejudices the existence of classical meander as fully two-periodical spiral wave tip movement. It is possible that any cycloid-liked trajectory of spiral wave tip always became into simple circle sooner or later.
The next question is whether the lacet is caused by some peculiarities of AP-model or the phenomenon is inherent to all models of FHN-class.
The answers on the question are quite important both for pure science and for practical use. As the Aliev-Panfilov model seems to suit the heart tissue properties in the best way among FHN-class models, the new kind of spiral wave behavior could help to comprehend biophysical mechanisms of life-threatened reentrant cardiac arrhythmias.
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