Study Of Analysis Method And Numerical Simulation On Nonlinear Mathematical Model Of Electrical Activity Of Neuron

The research of neuron discharge activity, a some-what new subject intersecting nonlinear science and nerve science, raises many scholars concerns in the fields of mathematical science and life science. It is also a marginal and mutual scientific research field.The rapid development of nonlinear dynamics creates a chance for the development of related sciences, which is shown especially obviously in the biological and medical science fields, because the research objects and system in the very fields generally have strong nonlinear character. The introduction of the method of nonlinear dynamics makes people have a fully new understanding of large quantities of messy and irregular biological and medical signals observed and recorded in the past researches. They are no longer regarded as random processes, but on the contrary, they may surely be chaos movement caused by qualitative mechanisms. Now, the complex movement forms, surrounding neuron discharge patterns, heart ratio variation and brain functions, arouse great interests and wide attention among scientific workers.The nerve system is an important modulating system in the human physiological enginery. The external stimulus are taken by the neurons of the nerve system, and coded, propagated and decoded by means of various patterns of neuron discharge. Different patterns of discharge reflect the difference of the external stimulus information and induce diverse physiological effects. Therefore, a deep research into the quantitive relationship between external stimulus and nerve discharge pattern has important theological and practical meanings for the better usage of electric stimulus to cure diseases. The electrophysiological function of nerve system contains very complex nonlinear dynamical behaviors.This paper, taking Hodgkin-Huxley(HH) model, FitzHugh-Nagumo(FHN) model and nerve propagation equation as the research objects, analyzes theologically the models themselves with the theories such as stability and quality. Many variables and strong coupling nonlinear, due to HH models adopting such math expression as high dimensional differential coefficient equation groups, bring a huge difficulty for analytic work. This paper educes the practical rational simplified model of HH model, makes an analysis of them, studies the neuron discharge conditions under stimulus of rectangle impulse and exponent attenuation wave mode impulse, using nonlinear dynamics theory and method, and gets the following results:1. We have given a practical rational simplified model of HH model, their solitary wave solutions and the solutions exited conditions.2. We give a new specific analysis of the simplified HH model, FHN model and nerve propagation equation, by using the qualitative theory as a series of differential equations. We give the mathematical proof which simplified HH equations; FHN equations may describe neuron discharge mechanism under different condition.3. We have constructed a Lyapunov function and given a stability analysis of FHN model and nerve propagation equation. We proved series stability theorems and given the full conditions for stability existence.4. We get the solitary wave is of the kink mode waves and its solution of nerve wave propagation equation by using the expansion methods of hyperbolic sine, hyperbolic tangent and others at first. We observed the breather in the small amplitude state.5. We have given the numerical simulation of the neuron discharge condition under external stimulus.