Dynamics Of Excitable System Driven By Lévy Noise

With the development of science and technology, nonlinear problem occurs inmany subjects. Nonlinear dynamic contact to many subjects, such as mechanics,mathematics, physics, chemistry, and even some social science, so nonlineardynamics is comprehensive subject. Excitable medium is a special kind of medium.when there is a disturbance beyond a certain threshold, medium will produce a quickand violent campaign, and go back to balance after a cycle. Excitable system is akind of open systems. When it is driven by external factors such as noise, itspotential motion can be restored. The influence on the system of nonlinear dynamicsmade by noise and periodic signals has been a research hot spot. Under the influenceof the noise and periodic signals, excitable system will have some special dynamicbehaviors. The main research purpose of this paper is the influence of the noise andperiodic signals to the single-point and two dimension excitable medium. Here wechoice Lévy noise because of its trailing and the great fluctuation.For the research of single-point, we use the H-H neural model. First we add theperiodic stimulation of different amplitude and frequency to the system, and find thethreshold value of the system. Then add a Lévy noise in the periodic signal, andanalyze sequence, spacing distribution, variance coefficient and signal-to-noise ratio,and contrast the results with Gaussian noise. The results show that, if we add a noisein the signal under the threshold, and the strength of the noise and signal reach to thethreshold value of neurons, the neurons can have a action potential. The best responseof the system depends on the combination of the noise and the input signal. Increasingthe noise intensity in a certain range can improve the spike of the neurons. If thestrength of the noise is too big, the noise will hinder the response of the system to theoriginal input signal, and abate the spike of the neurons. Once the strength of the noiseexceed a boundary, the external input signal will be covered by the noise, and thespike of the neurons will disappear. Therefore, only when the intensity take anappropriate value, there can be a regular spike. When the intensity is best, theregularity of the spike is the best, and there will be a stochastic resonance in the system. If there is only a noise input without input signal, the system can also have aspike when the strength reaches a certain threshold. When the intensity take aappropriate value, the regularity of the spike will be best, and there will be acoherence resonance phenomenon in the system.For the research of the two dimension excitable medium, we use the FHN model.The object of study is the dynamics behavior of the spiral wave in the two dimensionexcitable medium. Periodic signal can make the trajectory of the spiral tip morecomplex, and product the resonance and the Arnold tongue, while the Lévy noise canproduct the stochastic resonance and coherence resonance in the system. We analyzethe influence made by the Lévy noise and periodic signal to the system by changingthe strength of the noise and the amplitude of the signal. We found that the spiralwaves track is more complex with the changing of the periodic signal, and themaximum radius increase with the increasing of the period at1:1resonance and2:1resonance. The average period increases with the increasing of the period of the signal,and changes obviously with the changing of other parameters in the middle range ofthe period. Noise and periodic signal can make the appearance of the stochasticresonance phenomenon, and product ladder phenomenon and Arnold tongue.