Dynamics And Control Of Scroll Waves In Three-dimensional Excitable Media

Spiral wave dynamics in excitable media is one of the most important research topics in the nonlinear interdisciplinary.Its research covers various fields such as physics,chemistry and biology.Spiral waves have been observed from the chemical waves in the Belousov-Zhabotinsky(BZ)reactions to the waves of CO oxidation on platinum surfaces,as well as from the cyclic adenosine monophosphate(cAMP)waves during amoeba morphogenesis to the electrical waves in cardiac tissue.Three-dimensional spiral waves,also known as scroll waves,can be considered as an extension of two-dimensional spiral waves in three-dimensional space.Spiral waves rotate around cores,while scroll waves rotate around filaments,which can not only move in space,but also change shape.The filaments of scroll waves can be straight,curved,or closed into a circular loop known as a scroll ring.Compared to spiral waves,the scroll waves have more degrees of freedom and more dynamical behaviors.This thesis mainly studied the dynamics and control of scroll waves in three-dimensional excitable media under external fields by numerical simulation and theoretical analysis.Recently,experimental observations of scroll rings in a BZ reaction have shown that,the external direct current(DC)field can slow down the rate at which the ring shrinks,and under certain conditions the DC electric field can cause the ring to expand.To interpret these experimental findings,we combined the evolution equations of the unperturbed scroll rings with the drift formula of spiral waves due to the effects of the DC electric field,and then proposed a kinematical model for scroll rings under the DC electric field.It was found that,the kinematical model can not only interpret the above experimental phenomena well,but also can further predict that the applied DC electric field is able to slow down the drift velocity of scroll rings along their symmetry axes,and even in some cases the scroll rings can reverse their drift direction.The theoretical results are in quantitative agreement with that of the direct numerical simulations based on the reaction diffusion equation.Furthermore,we also studied the dynamics and control of scroll rings under an alternating(AC)electric field.At first,the direct numerical simulations showed that when the frequency of the AC electric field is twice that of the scroll wave rotation,the scroll rings can resonate with the AC electric field and exhibit various dynamical behaviors,for example,their reversals,shrinkage or growths.These dynamical behaviors are sensitive to the initial phase of the AC electric field as well as the initial phase of the scroll rings.Then,by combining the evolution equations of the free scroll rings with the drift formula of the spiral due to the effects of the AC electric field,we proposed a kinematical model characterizing the drift velocity of scroll rings along their radial direction as well as the drift velocity of scroll rings along their symmetry axes.The proposed kinematical model can effectively account for the numerical observations and predict the behaviors of scroll rings.Besides,the kinematical model could make a quantitative prediction that one scroll ring can be in a stationary state under the AC electric field when the strength and the initial phase of the electric field are appropriately set.Direct numerical simulations further validate such a theoretical prediction.Cardiac tissue is another typical three-dimensional excitable system.Experimental studies of the heart have shown that tachycardia and fibrillation are closely related to the occurrence of spiral and scroll waves in the heart.Based on the newly proposed far-field stimulation mechanism,we systematically studied the unpinning of scroll waves in three-dimensional cardiac tissue under DC pulsed electric fields,AC electric fields,and circularly polarized electric fields(CPEF).It was shown that the removal of pinned scroll waves can only be successful under the CPEF.After analysis,we found that the main reasons are as follows:in the resting state of the three-dimensional cardiac tissue,three forms of external electric fields can induce spherical waves from the heterogeneities,but only the frequency of the spherical waves induced by the CPEF can be higher than that of the pinned scroll waves.Hence,only the CPEF can successfully unpin and remove the pinned scroll waves out of the cardiac tissue.