The Phase Singularities Of The Helical Waves And The Singular Lines Of The Wrapping Waves Are Marked Based On The Experimental Results

Spatiotemporal patterns emerged under thermodynamic nonequilibrium conditions exist widely in physical,biological,and chemical systems.Pattern formation is one of the most important topics in the nonlinear interdisciplinary field.The most typical pattern is the spiral wave and its threedimensional form,scroll wave.A spiral wave in a two-dimensional medium rotates around phase singularity.The rotation center of scroll wave in the three-dimensional(3D)medium is a filament Its shape could be straight,curved,or even circular.In the Belousov-Zhabotinsk nonequilibrium chemical reaction,the collective behavior of the biological community of viscous mold,and the electrical activities that control cardiac function,the spiral wave(or scroll wave in 3D)and their dynamics have much research interest.For example,clinical studies have found that spiral waves(or scroll waves in 3D)are closely related to sudden cardiac deaths of human which occurs frequently in recent years.The main symptom of sudden cardiac death are tachycardia and cardiac fibrillation,which correspond to the pattern of cardiac electrical activity in the shape of spiral waves(or scroll waves in 3D)and their turbulence.The latest research focus in medicine to diagnose and treat sudden cardiac deaths is how to accurately locate it and directly treat on the phase singularity of the spiral wave(or filament of scroll wave in 3D).The existing method of locating the phase singularity is to obtain the phase distribution φ(x,y,t)according to the time series of cardiac membrane potential at each location,by using "empirical mode decomposition" and"Hilbert transform".The point where the phase does not converge in the entire space is marked as the phase singularity.However,this method requires data for a long period of time before and after the current time,in order to accurately locate the center of rotation.Thus,it is not a real-time location method in clinical experiments.So,we hope to find a new method to locate the phase singularity in real time.It could be extended to the nonintrusive treatment such as using ultrasound to locate and eliminate the phase singularity of spiral wave(or filament of scroll wave in 3D)in real time,so as to treat tachycardia and cardiac fibrillation in clinic.This will have great and far-reaching implications for clinical treatment and other real systems.The main research contents of this article include:In the first part,for the problem of locating the phase singularity of the spiral wave in a twodimensional medium,we use the spiral wave pre-processed image formed by the Dictyostelium discoideum amoebae.The data is provided by our cooperation group.Bilateral filtering,binarization,gaussian smoothing filter and other image processing methods are performed for noise reduction,and improveing contrast on the experimental image.Then we use the Jacobian-determinant method to mark the phase singularity of the spiral wave in real time.The Jacobian-determinant method we use only requires the data of the current time and after a short delay.Therefore,compared with the existing methods,it can achieve the requirement of real-time location of the phase singularity of the spiral wave.In the second part,we extend our method to locate the filament of the three-dimensional scroll wave,namely the three-dimensional Jacobian-determinant method.By using the LuoRudy91 model for mammalian ventricular myocytes,we verified that our method can locate the position of stable long straight filament of the scroll wave in real time.Our method is also applicable to locate the stable curved filament of scroll wave in another model,the Fenton-Karma four-variable model for human ventricular cells.Compared with the existing methods,our method is model independent,capable of locating various types of the scroll wave filament in real time under different models,and simple to apply.Unlike the existing method which needs to superpose of spiral rotation centers in twodimensional slices in a well-chosen slicing direction to be the scroll wave filament in threedimensional space,our method directly calculate the three-dimensional Jacobian-determinant and mark its extrema as the filament.