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Understanding how the dynamics of a neural network is shaped by the networkstructure, and consequently how the network structure facilitates the functionsimplemented by the neural system, is at the core of using mathematical modelsto elucidate brain functions. This study investigates the tracking dynamics ofcontinuous attractor neural networks (CANNs). Due to the translationalinvariance of neuronal recurrent interactions, CANNs can hold a continuousfamily of stationary states. They form a continuous manifold in which theneural system is neutrally stable. We systematically explore how this propertyfacilitates the tracking performance of a CANN, which is believed to have clearcorrespondence with brain functions. By using the wave functions of the quantumharmonic oscillator as the basis, we demonstrate how the dynamics of a CANN isdecomposed into different motion modes, corresponding to distortions in theamplitude, position, width or skewness of the network state. We then develop aperturbative approach that utilizes the dominating movement of the network'sstationary states in the state space. This method allows us to approximate thenetwork dynamics up to an arbitrary accuracy depending on the order ofperturbation used. We quantify the distortions of a Gaussian bump duringtracking, and study their effects on the tracking performance. Results areobtained on the maximum speed for a moving stimulus to be trackable and thereaction time for the network to catch up with an abrupt change in thestimulus.

A Moving Bump in a Continuous Manifold: A Comprehensive Study of the Tracking Dynamics of Continuous Attractor Neural Networks