Dynamical Analysis To Rulkov Neuron Models And Head Direction Neural Network

In this dissertation, the parameter space of the two dimensional Rulkov chaotic neuron model is taken into account by using the qualitative analysis, the co-dimension 2 bifurcation, the center manifold theorem, and the normal form. The goal is intend-ed to clarify analytically different dynamics and firing regimes of a single neuron in a two dimensional parameter space. Our research demonstrates the origin that there exist very rich nonlinear dynamics and complex biological firing regimes lies in dif-ferent domains and their boundary curves in the two dimensional parameter plane. We present the parameter domains of fixed points, the saddle-node bifurcation, the super-critical/subcritical Neimark-Sacker bifurcation, stability conditions of nonhyperbolic fixed points and quasiperiodic solutions. Based on these parameter domains, it is easy to know that the Rulkov chaotic neuron model can produce what kinds of firing regimes as well as their transition mechanisms.Secondly, stability and chaos of a simple system consisting of two identical Rulkov map-based neurons with the bidirectional electrical synapse are investigated in detail. On the one hand, as a function of control parameters and electrical coupling strength, the conditions for stability of fixed points of this system are obtained by using the qualitative analysis. On the other hand, chaos in the sense of Marotto is proved by a strict mathematical way.These results are very useful for building-up a large-scale neu-ron network with specific dynamics, rich biophysical phenomena, different biological functional roles and cognitive activities, especially in establishing some specific neuron network models of neurological diseases.Head direction neural network is made of head direction cells which are found in the limbic system in the brains of many mammals and they increase their firing rates above baseline levels only when the animal’s head points are in a specific direction. So head direction neurons are mostly orientation specific and location invariant.Here we get some results from the head direction neural network by analyzing a model of dy- namics of the head direction cell ensemble.By using mathematical analysis we found that under specific conditions all solutions of head direction neural network are bound-ed. That means that the average output is limited with any initial values. By Fourier analysis we get the relationship of Fourier series coefficient in head direction neural network, and then we solve the general form of equilibrium solutions. Meanwhile the stability of head direction neural network is proved strictly through building Lyapunov Function. This result shows that the average output of head direction neural network will finally converge to one equilibrium state as t tends to infinity independent of which kind of self-motion information for inertially based updating or what kind of visual land marks for calibration.At the same time we choose special synaptic weight and gain function to further explain these properties. We not only solved all equilibrium states exactly, but also proved the stability of these equilibrium states. This corresponds to the general properties of head direction neural network. To our surprise, we discov-ered an asymmetric equilibrium activity pattern even when the network connectivity pattern is strictly symmetric. These results have a significant impact in the field be-cause it raises the possibility of an alternative interpretation of the neurophysiological data, and provides a theoretical foundation for developing new data analysis methods and new experimental designs for distinguishing different plausible neural mechanisms.