分数阶FHN模型在外加刺激下的动力学特性

发布时间：2018-04-16 14:26

本文选题：分数阶FitzHug-Nagumo模型 + 激发周期　；参考：《河北师范大学》2017年硕士论文

【摘要】：生理学的实验中表明,在心脏病中观察到的一类叫做在进入性心率过速的现象,是由于心肌电信号出现的螺旋波而引起的。而心颤至死的过程和螺旋波的失稳有着密切联系。目前治疗心颤时比较好的方法是给患者心脏加一个6000V左右的瞬时电压将心脏暂时打停,使得螺旋波信号消失,之后再使得心脏博起。这种方法虽然简单有效,但它给患者带来莫大痛苦。通过动力学研究发现,如果能掌握螺旋波的运动规律,5mV左右的电压就可以将心脏中的螺旋波引出。在组成螺旋波的中心是一个时空拓扑缺陷,相当于一个奇点,在很小的区域内反应扩散方程已经不能准确地描述其动理学行为,研究此类时空缺陷深受科学家们的青睐。因此,开发新的治疗心律失常的方法和心脏活动的动力学特性的理论,有利于非线性科学家对行波规律深入了解。本文采取分数阶微积分,模拟神经元动力在分数阶FHN模型下的动力学特性以及模拟神经元在外加刺激条件下表现出的新的特性。首先研究了分数阶FHN模型神经元的放电特性及其两个神经元同步行为。通过对不同参数及初始值下单个神经元放电活动的数值模拟,得到了神经元周期放电对应的外加电流范围,分析了神经元进行放电的机制。通过与整数阶FHN模型结果比较,发现了分数阶比整数阶外加电流范围小。并且在外周期刺激下,发现了在一些范围内,外加周期和激发周期呈现锁频行为。即外加刺激周期和激发周期的比值关系有3:1,2:1,1:1等的线性关系。并且随着分数阶FHN模型的阶数变小,激发周期和外加刺激周期比值相同时的外加刺激周期初始取值变大,外加刺激周期范围变大。对于两个相互耦合分数阶FHN模型神经元,可以发现同步放电活动。耦合系数越大,两个神经元同步速度越快。分数阶阶数越小,神经元同步速度越快。其次研究了一维分数阶FHN模型中行波及行波链的动力学行为,给出了相应的色散关系等随分数阶阶数的变化关系。在行波传递过程研究中发现行波速度随着分数阶阶数变大反而变小。在外加刺激下,行波激发周期和外加周期呈现锁频行为。行波脉冲链的传播速度和外加刺激周期不是直接有关的线性关系,行波速度和激发周期呈单调递增的关系。并且随着外加刺激周期不断变大,行波速度增长率不断减小,外加周期达到一定值时,行波速度不再变化。
[Abstract]:Physiological experiments have shown that a kind of phenomenon called entering heart rate overvelocity observed in heart disease is caused by the helical wave of electrocardiogram.The process of cardiac fibrillation to death is closely related to the instability of spiral waves.At present, a better way to treat cardiac fibrillation is to temporarily stop the heart by adding a transient voltage of about 6,000 V to the patient's heart, so that the spiral wave signal disappears, and then the heart starts to swell.Although this method is simple and effective, it brings great pain to patients.It is found that the helical wave in the heart can be induced by a voltage of about 5 MV if we can master the motion law of the helical wave.In the center of the helical wave is a space-time topological defect, which is equivalent to a singularity. In a very small region, the reaction diffusion equation can no longer accurately describe its kinetic behavior, so the study of this kind of space-time defect is favored by scientists.Therefore, the development of new methods for the treatment of arrhythmias and the theory of the dynamic characteristics of cardiac activity are beneficial to the understanding of traveling wave law by nonlinear scientists.In this paper, fractional calculus is used to simulate the dynamic characteristics of neuron dynamics under fractional FHN model and the new characteristics of simulated neurons under the condition of external stimulation.The discharge characteristics of fractional order FHN model neurons and the synchronization behavior of two neurons were studied.Based on the numerical simulation of the firing activity of neurons with different parameters and initial values, the range of external current corresponding to the periodic discharge of neurons is obtained, and the mechanism of firing is analyzed.By comparing with the results of integer order FHN model, it is found that the fractional order is smaller than the integer order external current range.In some range, the frequency locking behavior of external cycle and excitation cycle is also found.That is, there is a linear relation of 3: 1: 2: 1: 1: 1 and so on.With the order of fractional FHN model becoming smaller, the initial value of the external stimulus cycle and the range of the external stimulus cycle become larger when the ratio of the excitation cycle to the external stimulus cycle is the same.Synchronous discharge activity can be found for two fractional order FHN model neurons coupled with each other.The greater the coupling coefficient, the faster the synchronization speed of the two neurons.The smaller the fractional order, the faster the synchronization speed of neurons.Secondly, the dynamic behavior of traveling wave chain in one-dimensional fractional order FHN model is studied, and the relationship between dispersion and fractional order is given.In the study of traveling wave transfer process, it is found that the traveling wave velocity decreases with the increase of fractional order.Under external stimulation, the excitation period and the applied period of traveling wave exhibit frequency locking behavior.The propagating velocity of the traveling wave pulse chain is not directly related to the external stimulus period, but the traveling wave velocity is monotonously increasing with the excitation period.With the increasing of the external stimulus period, the growth rate of traveling wave velocity decreases, and the traveling wave velocity does not change when the applied period reaches a certain value.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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