In control problems and basic scientific modeling, it is important to compare observations with dynamical simulations. For example, comparing two neural systems can shed light on the nature of emergent computations in the brain and deep neural networks. Recently, Ostrow et al. (2023) introduced Dynamical Similarity Analysis (DSA), a method to measure the similarity of two systems based on their recurrent dynamics rather than geometry or topology. However, DSA does not consider how inputs affect the dynamics, meaning that two similar systems, if driven differently, may be classified as different. Because real-world dynamical systems are rarely autonomous, it is important to account for the effects of input drive. To this end, we introduce a novel metric for comparing both intrinsic (recurrent) and input-driven dynamics, called InputDSA (iDSA). InputDSA extends the DSA framework by estimating and comparing both input and intrinsic dynamic operators using a variant of Dynamic Mode Decomposition with control (DMDc) based on subspace identification. We demonstrate that InputDSA can successfully compare partially observed, input-driven systems from noisy data. We show that when the true inputs are unknown, surrogate inputs can be substituted without a major deterioration in similarity estimates. We apply InputDSA on Recurrent Neural Networks (RNNs) trained with Deep Reinforcement Learning, identifying that high-performing networks are dynamically similar to one another, while low-performing networks are more diverse. Lastly, we apply InputDSA to neural data recorded from rats performing a cognitive task, demonstrating that it identifies a transition from input-driven evidence accumulation to intrinsically-driven decision-making. Our work demonstrates that InputDSA is a robust and efficient method for comparing intrinsic dynamics and the effect of external input on dynamical systems.

翻译：在控制问题与基础科学建模中，将观测结果与动力学仿真进行比较至关重要。例如，比较两个神经系统有助于揭示大脑和深度神经网络中涌现计算的性质。最近，Ostrow等人（2023）提出了动力学相似性分析（DSA），这是一种基于系统循环动力学而非几何或拓扑结构来衡量两个系统相似性的方法。然而，DSA未考虑输入如何影响动力学，这意味着两个相似系统若驱动方式不同，可能被归类为相异系统。由于现实世界中的动力学系统极少是自治的，因此必须考虑输入驱动的影响。为此，我们提出了一种用于比较内在（循环）动力学与输入驱动动力学的新度量方法，称为InputDSA（iDSA）。InputDSA通过基于子空间识别的带控制动态模态分解（DMDc）变体来估计并比较输入与内在动态算子，从而扩展了DSA框架。我们证明，InputDSA能够成功地从噪声数据中比较部分观测的、输入驱动的系统。研究表明，当真实输入未知时，可使用替代输入进行替换，而不会显著降低相似性估计的准确性。我们将InputDSA应用于通过深度强化学习训练得到的循环神经网络（RNNs），发现高性能网络在动力学上彼此相似，而低性能网络则表现出更大的多样性。最后，我们将InputDSA应用于记录自执行认知任务的大鼠神经数据，证明该方法能够识别从输入驱动的证据积累到内在驱动的决策过程的转变。我们的工作表明，InputDSA是一种稳健且高效的方法，可用于比较内在动力学以及外部输入对动力学系统的影响。