  Understanding how the dynamics of a neural network is shaped by the network
structure, and consequently how the network structure facilitates the functions
implemented by the neural system, is at the core of using mathematical models
to elucidate brain functions. This study investigates the tracking dynamics of
continuous attractor neural networks (CANNs). Due to the translational
invariance of neuronal recurrent interactions, CANNs can hold a continuous
family of stationary states. They form a continuous manifold in which the
neural system is neutrally stable. We systematically explore how this property
facilitates the tracking performance of a CANN, which is believed to have clear
correspondence with brain functions. By using the wave functions of the quantum
harmonic oscillator as the basis, we demonstrate how the dynamics of a CANN is
decomposed into different motion modes, corresponding to distortions in the
amplitude, position, width or skewness of the network state. We then develop a
perturbative approach that utilizes the dominating movement of the network's
stationary states in the state space. This method allows us to approximate the
network dynamics up to an arbitrary accuracy depending on the order of
perturbation used. We quantify the distortions of a Gaussian bump during
tracking, and study their effects on the tracking performance. Results are
obtained on the maximum speed for a moving stimulus to be trackable and the
reaction time for the network to catch up with an abrupt change in the
stimulus.
