连续流形上的移动凸点：连续吸引子神经网络跟踪动力学的综合研究
A Moving Bump in a Continuous Manifold: A Comprehensive Study of the Tracking Dynamics of Continuous Attractor Neural Networks
摘要：
理解神经网络的动态如何由网络结构塑造，以及网络结构如何促进神经系统实现的功能，是使用数学模型阐明大脑功能的核心。本文研究了连续吸引子神经网络（CANNs）的跟踪动力学。由于神经元周期性相互作用的平移不变性，CANN可以保持连续的稳态家族。它们形成了一个连续的流形，其中神经系统是中性稳定的。我们系统地探讨了这种属性如何促进CANN的跟踪性能，这被认为与大脑功能有明确的对应关系。通过使用量子谐振子的波函数作为基础，我们演示了如何CANN的动力学分解成不同的运动模式，对应于网络状态的振幅，位置，宽度或偏斜度的失真。然后，我们开发了一种微扰的方法，利用网络的静态在状态空间中的主导运动。这种方法使我们能够近似的网络动态到任意精度取决于所使用的扰动的顺序。我们量化的高斯凹凸跟踪过程中的失真，并研究其对跟踪性能的影响。结果得到的最大速度为一个移动的刺激是可跟踪的和反应时间的网络赶上一个突然变化的刺激。
英文摘要：
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.
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