  We introduce an analytically solvable model of two-dimensional continuous
attractor neural networks (CANNs). The synaptic input and the neuronal response
form Gaussian bumps in the absence of external stimuli, and enable the network
to track external stimuli by its translational displacement in the
two-dimensional space. Basis functions of the two-dimensional quantum harmonic
oscillator in polar coordinates are introduced to describe the distortion modes
of the Gaussian bump. The perturbative method is applied to analyze its
dynamics. Testing the method by considering the network behavior when the
external stimulus abruptly changes its position, we obtain results of the
reaction time and the amplitudes of various distortion modes, with excellent
agreement with simulation results.
