Analysis And Control Of Nonlinear Dynamics Of Neural Systems

The neural system is a complex nonlinear system and its characteristics are studied from a nonlinear theory perspective in this dissertation.The Hodgkin-Huxley (HH) model is a four-dimensional differential equation which quantitatively describes the dynamical process of trans-membrane voltage and ionic current that observed in experiments. In this dissertation, the algebra criterion in high-dimensional equations is employed to perform the bifurcation analysis of HHM model in which the leakage conductance and the anti-electromotive of sodium ion channel are chosen to be the bifurcation parameters. The conclusion reveals that the Hopf bifurcation will occur in the HHM model when the leakage conductance and the anti-electromotive sodium ion channel changes. That is, the membrane produces continuous action potentials in response to external stimulation, which is consistent with the pathological phenomena observed in medical experiments. The multi-parameters bifurcation analysis of HHM model exposed to external electrical field shows that the change of parameters of external electrical field will cause the bifurcation of the model, which might be the reason that electromagnetism radiation leads to disease.Analytical solution to high-dimensional nonlinear system like Hodgkin-Huxley model is impossible to be obtained, and the numerical analysis is also very complex. The asymptotic methods can solve this problem through describing the fast and slow process of the system in the same time. This dissertation investigates the modified HH model by employing the Tikhonov asymptotic theory and the analysis method of Zeeman model, and proves that important characteristics of HH model such as excitability are still preserved.In this dissertation, the firing patterns of HH model exposed to ELF external electric field are studied from the view of ISI (interspikes interval). Various firing patterns such as period-n spiking, fast/slow burst, amplitude modulation firing, multi-sinusoidal firing and chaotic burst are found through numerical simulation. The tuning fork bifurcation, a route to chaos of the neuron, has also been found. By introducing different time constants into the HH model exposed to external electrical field, the four-dimensional nonlinear system is analyzed in terms of fast/slow system. A special firing pattern, slow spiking, is obtained near the Hopf bifurcation point, and it is proved that this firing pattern is chaotic.An adaptive fuzzy control method based on functional electric stimulation is employed to realize the synchronization in HH models and FHN models. This control method which combines the advantages of adaptive control and fuzzy control can approach the unknown nonlinear part as well as the disturbances. The results of simulation demonstrate the availability of the design. This new method provides a route to treatment of diseases such as epilepsy.In this dissertation, based on the computer digital control technique, a discrete variable universe fuzzy control strategy, which avoids the disadvantage that all the consequent parameters must be tuned on-line in the traditional fuzzy controller based on T-S model, is employed to control the discrete chaos system. This control strategy realizes continuous changes of the universe and dynamical fuzzy rules by on-line tuning a single parameter -- the contraction and expansion factor. Thus the performance of control is improved. The simulation results give the perfect control of Lorenz chaos system.Theories such as continuous wavelet transform, discrete wavelet transform, power spectrum and nonlinear dynamics are employed to investigate the electrical signals of neural system recorded in the experiment of acupuncture and it is demonstrated that the signals induced by different acupuncture method have different quantitative characteristics. This attempt gives elementary study on quantifying of the electrical information of acupuncture.