Parra Lucas C, Silvan Aimar, Nentwich Maximilian, Madsen Jens, Parra Vera E, Babadi Behtash
Department of Biomedical Engineering, City College of New York, New York, NY, United States of America.
Institute of Bioelectronic Medicine, Northwell Health Feinstein Institutes for Medical Research, Manhasset, NY, United States of America.
PLoS One. 2025 Jan 9;20(1):e0313875. doi: 10.1371/journal.pone.0313875. eCollection 2025.
Complex systems, such as in brains, markets, and societies, exhibit internal dynamics influenced by external factors. Disentangling delayed external effects from internal dynamics within these systems is often difficult. We propose using a Vector Autoregressive model with eXogenous input (VARX) to capture delayed interactions between internal and external variables. Whereas this model aligns with Granger's statistical formalism for testing "causal relations", the connection between the two is not widely understood. Here, we bridge this gap by providing fundamental equations, user-friendly code, and demonstrations using simulated and real-world data from neuroscience, physiology, sociology, and economics. Our examples illustrate how the model avoids spurious correlation by factoring out external influences from internal dynamics, leading to more parsimonious explanations of these systems. For instance, in neural recordings we find that prolonged response of the brain can be explained as a short exogenous effect, followed by prolonged internal recurrent activity. In recordings of human physiology, we find that the model recovers established effects such as eye movements affecting pupil size and a bidirectional interaction of respiration and heart rate. We also provide methods for enhancing model efficiency, such as L2 regularization for limited data and basis functions to cope with extended delays. Additionally, we analyze model performance under various scenarios where model assumptions are violated. MATLAB, Python, and R code are provided for easy adoption: https://github.com/lcparra/varx.
诸如大脑、市场和社会等复杂系统呈现出受外部因素影响的内部动态。在这些系统中，将延迟的外部效应与内部动态区分开来往往很困难。我们建议使用带外生输入的向量自回归模型（VARX）来捕捉内部和外部变量之间的延迟相互作用。虽然该模型与格兰杰用于检验“因果关系”的统计形式主义一致，但两者之间的联系并未得到广泛理解。在这里，我们通过提供基本方程、用户友好的代码以及使用来自神经科学、生理学、社会学和经济学的模拟数据和真实世界数据进行演示来弥合这一差距。我们的示例说明了该模型如何通过从内部动态中排除外部影响来避免虚假相关性，从而对这些系统进行更简洁的解释。例如，在神经记录中，我们发现大脑的长时间反应可以解释为短暂的外部效应，随后是长时间的内部循环活动。在人体生理记录中，我们发现该模型恢复了诸如眼球运动影响瞳孔大小以及呼吸与心率的双向相互作用等已确定的效应。我们还提供了提高模型效率的方法，例如针对有限数据的L2正则化和用于处理延长延迟的基函数。此外，我们分析了在各种违反模型假设的情况下的模型性能。提供了MATLAB、Python和R代码以便于采用：https://github.com/lcparra/varx。
