Department of Physics, Hong Kong University of Science and Technology, Hong Kong, China.
Neural Comput. 2010 Mar;22(3):752-92. doi: 10.1162/neco.2009.07-08-824.
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 perturbation 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 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.
理解神经网络的动态如何受到网络结构的影响，以及网络结构如何促进神经系统执行的功能，是使用数学模型阐明大脑功能的核心。本研究调查了连续吸引子神经网络 (CANN) 的跟踪动态。由于神经元递归相互作用的平移不变性，CANN 可以保持连续的静止状态族。它们形成了一个连续的流形，其中神经网络处于中性稳定状态。我们系统地探讨了这种特性如何促进 CANN 的跟踪性能，这被认为与大脑功能有明显的对应关系。通过使用量子谐振子的波函数作为基础，我们展示了 CANN 的动力学如何分解为不同的运动模式，对应于网络状态的幅度、位置、宽度或偏度的扭曲。然后，我们开发了一种微扰方法，该方法利用网络静止状态在状态空间中的主导运动。这种方法允许我们根据使用的微扰阶数，将网络动力学近似到任意精度。我们量化了跟踪过程中高斯峰的扭曲，并研究了它们对跟踪性能的影响。结果是在可跟踪的运动刺激的最大速度和网络赶上刺激突然变化的反应时间方面获得的。
