||In this thesis, there are three parts related to continuous attractor neural networks (CANNs). They are 1) continuous attractor neural networks with Mexican-hat coupling 2) two-dimensional (2D) continuous attractor neural networks with divisive inhibition, and 3) one-dimensional (1D) continuous attractor neural networks with divisive inhibition and dynamical synapse. In the first part, two kinds of Mexican-hat couplings are chosen. With this, some properties of CANNs with Mexican-hat couplings will be studied. With some reasonable neuronal response gain functions, the CANNs with Mexican-hat coupling will be unstable. In the second part, the study will be an extension of 1D CANNs with divisive inhibition. The perturbative approach for 1D CANNs will be applied on the its 2D counterpart. In the analysis, eigenfunctions of 2D harmonic oscillator in polar coordinate are chosen to be the basis to decompose the distortion around the local stable bump. This enables us to introduce the perturbative approach and predict the motion of the bump. Also, anisotropic eigenfunctions of the quantum harmonic oscillator are able to describe distortion modes such as those with cigar and triangular shapes. In the last part, the synaptic plasticity effect is introduced to the original 1D model with divisive inhibition. There are two kinds of synaptic plasticity, short-term depression (STD) and short-term facilitation (STF). STD may make the system relax to a spontaneous motion, but it can also improve the tracking performance. STF, as a opposite effect as STD, will slow down the tracking process. By using a perturbative analysis on the function modeling STD, the behavior of STD can be explained, including various phases on the parameter space.