| The study on spiral wave has been an important task of the pattern kinetics, which is subject to non-linear crossing subject. Spiral wave was observed in physics system, chemistry system, biology system.. such as Ising-Block phase change in crystal, Rayleigh-Benrd convection in fluid flows, the aggregating slime-mold cell, the contraction of heart muscles, the CO oxidation on platinum surface, the chemical wave in reaction-diffusion media and so on. The study on spiral waves has important practical perpurse. Experiment of physiology shows that: a kind of phenomena called of reentrant tachycardia are observed in the cardiopathy patient, which may be caused by spiral waves occurring in the contraction of heart muscles . the death caused by ventricular tachycardia have consanguineous ties in the losing of spiral wave stability. The heart tissue is an excitable system that is studied by the modified Greenberg-Hasting model, which is derived from a cellular automaton model. Plane wave, spiral wave and some complex phenomena are showed in the simulation. Plane wave front and wave back are formed inn the same speed. The wave front and wave back inosculate in the tip and change into special spiral wave tip. The main contribution of this article lies in four aspects: First, as r=2, studying the character of modified Greenberg-Hasting Model, we find out that six kinds of excitable behavior are in heterogeneous excitable systems: spiral wave, ring wave, plane wave, single travel wave, self-sustainment, unsuccessful excitement. The value of parameter θis the crucial factor that decides whether or not the systems produce spiral wave. First, the spiral wave will come to being, only if the value θis between 0.2 and 0.5 and there are the proper parameters E and R. Second, as θis between 0.1 and 0.4, the ring wave will arise, i.e., as θ=0.1 and E/R=1, the ring wave can be produced and the value of E/R increases as θincrease. Third, the plane wave and the single traveling wave can be produced as θis between 0.1 and 0.4. As θincreasing, the range of producing the plane wave and the single θtraveling wave become less and less. As θ>0.4, there are no plane waves and single traveling waves arising in the system. When θ>0.3 and E>R , there is self-sustainment, if θ=0.6,the range of phase is the biggest. Second: The deviation amplitude and average deviation of a cellular are introduced in the modified Greenberg-Hasting model for excitable media. Influence of the parameter on the produce of spiral waves are studied. The results show that: the probability of producing spiral wave rise as the increase of range of cell. Which shows that: it is easy if the range bigger; on the same condition, when the departure is not more than 0.5, the probability of producing spiral wave rises along with the addition of departure of cells. If the departure is more than 0.5, the probability of producing spiral wave trails off sharply, when departure is equal to 0.7 it almost drops to zero. Third: Considering the spatially discrete and temporally continuous properties of biological tissue . The modified Greenberg-Hastings model is applied in excitable media in the article . Computation simulation results show that: there are no significant effects of negative restitution as long as the slope of the negative restitution curve is bigger than –1; In media with slope less small than –1, the dynamics of spiral wave can change significantly. The spiral waves were broken up into fragments; Stable spiral waves could be made into turbulent patterns in the media with negative parabolic restitution curve.. |
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