Learning interpretable representations of neural dynamics at a population level is a crucial first step to understanding how observed neural activity relates to perception and behavior. Models of neural dynamics often focus on either low-dimensional projections of neural activity, or on learning dynamical systems that explicitly relate to the neural state over time. We discuss how these two approaches are interrelated by considering dynamical systems as representative of flows on a low-dimensional manifold. Building on this concept, we propose a new decomposed dynamical system model that represents complex non-stationary and nonlinear dynamics of time series data as a sparse combination of simpler, more interpretable components. Our model is trained through a dictionary learning procedure, where we leverage recent results in tracking sparse vectors over time. The decomposed nature of the dynamics is more expressive than previous switched approaches for a given number of parameters and enables modeling of overlapping and non-stationary dynamics. In both continuous-time and discrete-time instructional examples we demonstrate that our model can well approximate the original system, learn efficient representations, and capture smooth transitions between dynamical modes, focusing on intuitive low-dimensional non-stationary linear and nonlinear systems. Furthermore, we highlight our model's ability to efficiently capture and demix population dynamics generated from multiple independent subnetworks, a task that is computationally impractical for switched models. Finally, we apply our model to neural "full brain" recordings of C. elegans data, illustrating a diversity of dynamics that is obscured when classified into discrete states.

翻译：在群体水平上学习神经动力学的可解释表征，是理解观测到的神经活动如何与感知和行为相关联的关键第一步。神经动力学模型通常关注神经活动的低维投影，或学习与神经状态随时间变化明确相关的动力系统。我们通过将动力系统视为低维流形上的流形映射，探讨了这两种方法之间的相互关联。基于此概念，我们提出一种新的分解动力系统模型，将时间序列数据的复杂非平稳和非线性动力学表示为更简单、更可解释成分的稀疏组合。该模型通过字典学习过程进行训练，其中我们利用了追踪时变稀疏向量的最新研究成果。与先前的切换式方法相比，该分解动力学结构在给定参数数量下更具表达力，并能有效建模重叠与非平稳动力学。在连续时间和离散时间的教学示例中，我们证明该模型能很好逼近原始系统、学习高效表征，并捕捉动力学模式间的平滑过渡——尤其聚焦于直观的低维非平稳线性与非线性系统。此外，我们突出展示了模型高效捕获并分离多个独立子网络生成的群体动力学的能力，而这一任务对切换模型而言在计算上不可行。最终，我们将模型应用于秀丽隐杆线虫的神经“全脑”记录数据，揭示了在离散状态分类中会被掩盖的动力学多样性。