The Study On Control Of Pinned Spiral Waves In The FitzHugh-Nagumo Model

Spiral waves are a kind of non-equilibrium pattern that exists widely in nature.They can be observed in stimulating media,oscillating media,and bistable media.Myocardial tissue is a typical excitable medium,which supports the formation and propagation of spiral waves.There are blood vessels and scar tissue in the heart,and many factors can also cause changes in the local excitability and conductivity of the myocardium.These local heterogeneities play an important role in the formation of spiral waves and the spontaneous breaking process.The existence of spiral wave state and small spiral wave turbulence state in the heart is closely related to heart diseases such as arrhythmia and ventricular fibrillation.Therefore,eliminating these states has important practical application value.The widely used methods to terminate arrhythmia and ventricular fibrillation are electric shock defibrillation and artificial cardiac pacemakers.These methods usually have greater negative effects,and it is necessary to find safer and more effective low-energy,operable control methods based on these conditions.This paper studies the low-energy control method of pinning spiral waves.The main work is divided into three parts:First part: The FHN model is used to study the interaction between the internal traveling wave pulse train generated by the full boundary disturbance and the external pinning spiral wave.The internal traveling wave pulse chain continuously competes with the pinning spiral wave,and the mechanism of the pinning spiral wave being eliminated and controlled is analyzed.The wave chain constantly competes with the pinned spiral wave in the process of showing some laws,and the mechanism by which the pinned spiral wave is eliminated and controlled.We select two different disturbance regions near the boundary for periodic disturbance.In one case,a circular pulse that propagates inward can be produced continuously,and in the other case,a square pulse that propagates inward can be produced continuously.These two pulse chains compete with the pinning spiral wave in the inward propagation.Under certain conditions,the inwardly propagating pulse train will gradually invade,and the pinning spiral wave area will gradually become smaller and eventually disappear at the obstacle boundary.We explored the relationship between the period,excitability parameters,obstacle size and other factors that can be eliminated and controlled by the pinning spiral wave.When the excitability parameter,the disturbance period and the obstacle radius are fixed,the circular pulse perturbation takes shorter time than the square pulse perturbation control to eliminate the pinning spiral wave,and when the excitability parameter and the obstacle radius are fixed,circular pulses have a larger range of perturbation period than square pulses.It is proved that the former is easier to control the pinning spiral wave than the latter under the same conditions.The second part: The FHN model is used to study the influence of local feedback on the dynamic behavior of the unpinning spiral wave,and the phase diagrams of various parameters that can be pinned are given,and the mechanism of the separation of the spiral wave head and the obstacle is analyzed.Before the local feedback is activated,the pinning spiral wave rotates counterclockwise around the obstacle boundary.Then the fast variable u value at a certain point in the system is derived,which acts as a feedback signal after a certain time delay and acts on the entire space.After the feedback is started,when the larger part of the feedback signal acts,a new excited state will be formed in front of the wave front to interact with the pinning spiral wave,which will cause the pinning wave head to temporarily separate.The maximum distance between the temporarily separated wave head and the obstacle increases with the increase of the number of times the large value part of the feedback signal acts.When the maximum distance increases to a certain value,the wave head will be completely separated from the obstacle.Under normal circumstances,after the wave head is separated from the obstacle,the wave head motion unit will gradually drift to the vicinity of the feedback collection point and rotate around the limit cycle centered on the feedback collection point.When the location of the feedback collection point is close to the right boundary,the wave head separates from the obstacle and drifts to the boundary and disappears from there,thereby achieving the unpinning of the pinning spiral wave.We give a phase diagram of feedback gain and obstacle size when the excitability parameter is constant.When the feedback gain is constant,the phase diagram of the excitability parameter and the obstacle size.The relationship between the feedback gain and the position of the feedback signal collection point is analyzed when the excitability parameter and the size of the obstacle are constant.All three have parameter ranges that can realize unpinning control.The third part: The FHN model is used to study the control and elimination of the pinning spiral wave by the traveling wave pulse chain,and the limitations of the conventional traveling wave pulse chain in controlling the pinning spiral wave are analyzed,and an improved traveling wave pulse chain control method with a wider application range is proposed.The mechanism by which the pinning spiral wave is controlled and the various relationships needed in the control process are given in the article.The conventional traveling wave pulse train is effective for the control of the free spiral wave,but for the pinned spiral wave,the effect is not so.Our improved method is to create a defect point on the left boundary,and the traveling wave pulse train with the defect propagates to the right to interact with the pinning spiral wave.When the topological charge at the obstacle boundary is conserved,the pinning is successful.We draw a phase diagram of the size of the obstacle and the position of the defect when the excitability parameter and the disturbance period are constant.When the parameter value is in the controllable area of the phase diagram,the pinning spiral wave is controllable.