Our understanding of complex systems rests on our ability to characterise how they perform distributed computation and integrate information. Advances in information theory have introduced several quantities to describe complex information structures, where collective patterns of coordination emerge from higher-order (i.e. beyond-pairwise) interdependencies. Unfortunately, the use of these approaches to study large complex systems is severely hindered by the poor scalability of existing techniques. Moreover, there are relatively few measures specifically designed for multivariate time series data. Here we introduce a novel measure of information about macroscopic structures, termed M-information, which quantifies the higher-order integration of information in complex dynamical systems. We show that M-information can be calculated via a convex optimisation problem, and we derive a robust and efficient algorithm that scales gracefully with system size. Our analyses show that M-information is resilient to noise, indexes critical behaviour in artificial neuronal populations, and reflects states of consciousness and task performance in real-world macaque and mouse neuroimaging data. Furthermore, M-information can be incorporated into existing information decomposition frameworks to reveal a comprehensive taxonomy of information dynamics. Taken together, these results help us unravel collective computation in large complex systems.

翻译：对复杂系统的理解依赖于我们表征其执行分布式计算和整合信息的能力。信息论的进展引入了若干描述复杂信息结构的量，这些结构中涌现的集体协调模式源于高阶（即超越两两）相互依赖。遗憾的是，现有技术较差的扩展性严重阻碍了这些方法在大规模复杂系统研究中的应用。此外，专门针对多元时间序列数据的度量方法相对较少。本文提出了一种关于宏观结构信息的新型度量，称为M-信息，它量化了复杂动力系统中信息的高阶整合。我们证明M-信息可通过凸优化问题计算，并推导出一种鲁棒且高效的算法，该算法能优雅地随系统规模扩展。分析表明，M-信息对噪声具有弹性，能指示人工神经元群体中的临界行为，并在真实世界的猕猴和小鼠神经影像数据中反映意识状态与任务表现。此外，M-信息可被纳入现有信息分解框架，以揭示信息动力学的全面分类学。综合来看，这些结果有助于阐明大规模复杂系统中的集体计算。