Multiscale geometrical and topological learning in the analysis of soft matter collective dynamics

Understanding the behavior and evolution of a dynamical many-body system by analyzing patterns in their experimentally captured images is a promising method relevant for a variety of living and non-living self-assembled systems. The arrays of moving liquid crystal skyrmions studied here are a representative example of hierarchically organized materials that exhibit complex spatiotemporal dynamics driven by multiscale processes. Joint geometric and topological data analysis (TDA) offers a powerful framework for investigating such systems by capturing the underlying structure of the data at multiple scales. In the TDA approach, we introduce the $Ψ$-function, a robust numerical topological descriptor related to both the spatiotemporal changes in the size and shape of individual topological solitons and the emergence of regions with their different spatial organization. The geometric method based on the analysis of vector fields generated from images of skyrmion ensembles offers insights into the nonlinear physical mechanisms of the system’s response to external stimuli and provides a basis for comparison with theoretical predictions. The methodology presented here is very general and can provide a characterization of system behavior both at the level of individual pattern-forming agents and as a whole, allowing one to relate the results of image data analysis to processes occurring in a physical, chemical, or biological system in the real world.

💡 Research Summary

This paper presents a comprehensive framework that combines topological data analysis (TDA), a novel topological descriptor called the Ψ‑function, L p,q‑norm based image distance matrices, and multidimensional scaling (MDS) to quantitatively characterize the spatiotemporal dynamics of electrically driven liquid‑crystal skyrmion (toron) ensembles. The authors begin by motivating the need for multiscale analysis of pattern‑forming systems, noting that traditional persistence homology and Betti number approaches capture global topological features but often miss time‑dependent changes in the size and shape of individual solitonic structures. To address this gap, they introduce the Ψ‑function, which integrates the birth–death information of 0‑dimensional (clusters) and 1‑dimensional (loops) features obtained from a pixel‑intensity filtration into a single scalar that is highly sensitive to periodic variations in soliton morphology while remaining robust against noise.

Experimentally, a chiral nematic liquid‑crystal mixture (ZLI2806 doped with CB‑15 and CTAB) is sandwiched between ITO‑coated glass plates with strong perpendicular anchoring. An AC voltage (U = 20 V, f = 2 Hz) is applied and switched off repeatedly, inducing electrohydrodynamic instability that spontaneously generates toron solitons. By controlling the number of voltage cycles, the authors create ensembles ranging from dilute random distributions to tightly packed pseudo‑crystalline domains. High‑speed optical videomicroscopy records the evolution of these ensembles, producing RGB video frames that are subsequently converted to grayscale for analysis.

For each selected frame (every fifth frame in some datasets, every second frame in others), the authors compute L 2,2‑norms and L 2,2‑distances between all pairs of images. The L p,q‑norm formulation treats each pixel as a unit‑mass element and aggregates pixel‑wise differences across the entire image, providing a flexible metric that works for both grayscale and color data. In parallel, image gradients are calculated using central differences, and the gradient magnitude |∇X| is subjected to the same L 2,2‑norm procedure. Gradient‑based distances emphasize edges and regions of rapid intensity change, thereby highlighting soliton boundaries and defect lines.

The resulting distance matrices encode the global similarity structure of the video sequence. Applying classical (metric) multidimensional scaling to these matrices yields low‑dimensional Euclidean embeddings that visualize the temporal trajectory of the entire ensemble. Distinct clusters in the embedding correspond to different dynamical regimes: random dispersal, intermediate ordering, and near‑crystalline arrangements. The authors demonstrate that changes in the embedding correlate with specific voltage protocols, revealing how external fields modulate the nonlinear repulsive interactions that drive self‑organization.

Simultaneously, the Ψ‑function is evaluated for each frame by constructing a filtration over pixel intensities, extracting persistence diagrams, and integrating the lifetimes of the most persistent 0‑ and 1‑dimensional features. Peaks in the Ψ‑time series align with moments when solitons expand, contract, or undergo shape transitions, providing a concise scalar signature of the underlying topological dynamics. By comparing Ψ‑values with the MDS trajectory, the authors show that local topological events (e.g., a soliton splitting or merging) are reflected in global geometric shifts, thereby linking microscopic topology to macroscopic geometry.

The paper also discusses the mathematical underpinnings of the L p,q‑norm, the construction of distance matrices on discrete measure spaces, and the stability properties of the Ψ‑function under noise and illumination variations. Supplemental material includes code snippets in Python (NumPy, SciPy) for gradient computation, distance matrix assembly, and MDS implementation, facilitating reproducibility.

Overall, the study demonstrates that a combined geometric–topological learning pipeline can simultaneously capture fine‑scale soliton morphology and coarse‑scale collective behavior in a soft‑matter system. The methodology is general: any image‑based time series—ranging from cellular migration assays to phase‑separating polymer blends—could be processed through the same pipeline to extract multiscale dynamical signatures. By bridging experimental imaging, rigorous topological descriptors, and unsupervised dimensionality reduction, the work opens new avenues for quantitative, data‑driven exploration of complex, hierarchical materials.