(18h) Quantifying Hierarchical Causality in Active Matter Systems

Active particles (i.e., agents, constituents) within active matter can intake and dissipate energy through motion at microscopic scales, which results in collective motion at macroscopic scales across the entire system [1]. This can be described as a hierarchical self-organization that is facilitated by dynamic interactions that occur within and across scales [2]. Active matter systems can exhibit varying collective phenomena such as self-organization, pattern formation, and phase separation that drive many physical and chemical processes [3]. This rich multi-scale complexity is exhibited by systems in many fields of science, such as biology, neuroscience, chemistry, material science, and living systems. Examples of such causal dynamics include flying flocks of birds where neighboring birds follow a leader, and bacteria secreting chemicals that serve as attractants for other bacteria [4,5].

A significant challenge in comprehending and modeling these complex phenomena lies in systematically identifying and quantifying interactions across various scales [6]. The framework of causal discovery allows us to characterize the strength and direction (i.e., cause and effect) of interactions within a dynamical system and is purely data driven. This provides an understanding of how quantities such as information or energy may flow through a system and can aid in modelling and design of effective control strategies [7]. However, many of the methods for causal discovery are focused on relationships between individual elements (e.g. particles) and do not directly consider the mesoscale and collective organization of a system. Thus, the direct application of most causal discovery frameworks will completely miss the hierarchy of structure and dynamics within active matter systems.

In response to this challenge, this talk proposes a novel approach rooted in topology for the systematic quantification of causality across hierarchical levels in active matter systems. Leveraging the simplicity and efficiency of the Euler characteristic—a fundamental topological descriptor—alongside a filtration technique, we present a non-parametric method capable of capturing the system's topology across multiple hierarchical levels [8]. The multi-scale, self-organized topology of the system is observed over time and represented as a dynamical system for which causal discovery can be directly applied. This framework facilitates the exploration and quantification of causal relationships between hierarchical self-organization levels within active and dynamical systems. We demonstrate the application of this technique on both simulated dynamical systems and real experimental data from a system of self-organizing bacteria in a quasi-2D geometry, in which we identify causal relationships across scales within bacterial suspensions [9].

[1] Ramaswamy, Sriram. "The mechanics and statistics of active matter." Annu. Rev. Condens. Matter Phys. 1.1 (2010): 323-345.

[2] Ziepke, Alexander, et al. "Multi-scale organization in communicating active matter." Nature communications 13.1 (2022): 6727.

[3] Shaebani, M. Reza, et al. "Computational models for active matter." Nature Reviews Physics 2.4 (2020): 181-199.

[4] Marchetti, M. Cristina, et al. "Hydrodynamics of soft active matter." Reviews of modern physics 85.3 (2013): 1143.

[5] Cavagna, Andrea, and Irene Giardina. "Bird flocks as condensed matter." Annu. Rev. Condens. Matter Phys. 5.1 (2014): 183-207.

[6] Chepizhko, Oleksandr, David Saintillan, and Fernando Peruani. "Revisiting the emergence of order in active matter." Soft Matter 17.11 (2021): 3113-3120.

[7] Glymour, Clark, Kun Zhang, and Peter Spirtes. "Review of causal discovery methods based on graphical models." Frontiers in genetics 10 (2019): 524.

[8] Smith, Alexander, and Victor M. Zavala. "The Euler characteristic: A general topological descriptor for complex data." Computers & Chemical Engineering 154 (2021): 107463.

[9] Ghosh, Dipanjan, and Xiang Cheng. "To cross or not to cross: Collective swimming of Escherichia coli under two-dimensional confinement." Physical Review Research 4.2 (2022): 023105.