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The importance of the whole: topological data analysis for the network neuroscientist
Ann E. Sizemore, Jennifer Phillips-Cremins, Robert Ghrist, Danielle S. Bassett
TL;DR
Pairwise network models miss higher-order structure such as topological cavities, while the mathematics needed to analyze these features remains unfamiliar to many neuroscientists. The paper introduces persistent homology, demonstrates it on three biological datasets, and shows how algebraic topology can support interpretation of neural and genomic interaction structure.
Problem
Pairwise network representations omit higher-order interactions and topological features such as cavities, while topology’s mathematical foundations remain difficult for many neuroscientists to apply.
Method
The paper introduces persistent homology and demonstrates simplicial-complex analyses on mouse, C. elegans, and genomic interaction datasets.
Results
The three example datasets illustrate how to translate networks into simplicial complexes, apply persistent homology, and develop reasonable topological interpretations.
Takeaways & Limitations
Algebraic topology provides network scientists with additional tools for analyzing higher-order structure and generating questions about neural systems.
Takeaways & Limitations
Applying these methods requires understanding their fundamental assumptions, underlying mathematics, and the meaning of cavities in the system studied.
Abstract
from arXiv · showhide
The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems. Among many important contributions, networks have proven useful in identifying sets of node pairs that are densely connected and that collectively support brain function. Yet the restriction to pairwise interactions prevents us from realizing intrinsic topological features such as cavities within the interconnection structure that may be just as crucial for proper function. To detect and quantify these topological features we must turn to methods from algebraic topology that encode data as a simplicial complex built of sets of interacting nodes called simplices. On this substrate, we can then use the relations between simplices and higher-order connectivity to expose cavities within the complex, thereby summarizing its topological nature. Here we provide an introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as we move through a combinatorial object such as a weighted network. We detail the underlying mathematics and perform demonstrative calculations on the mouse structural connectome, electrical and chemical synapses in \textit{C. elegans}, and genomic interaction data. Finally we suggest avenues for future work and highlight new advances in mathematics that appear ready for use in revealing the architecture and function of neural systems.
Introduction
The paper introduces algebraic topology as a coordinate-free, noise-robust framework for studying higher-order organization in neural systems. It develops basic persistent homology and demonstrates its analyses across mouse, C. elegans, and genomic datasets.
- Motivation: Pairwise network models miss higher-order interactions among groups of biological units, which algebraic topology provides a language to encode and study.Such organization occurs across cellular and areal scales, including in the brain.
- Motivation: Algebraic topology characterizes shape through properties invariant under stretching and shrinking, offering a perspective distinct from geometric network topology.This perspective distinguishes structures such as circular, linear, and star-like forms that graph statistics may group together.
- Applications: Topological methods have been applied across neural scales to study place fields, neural codes, learning, parallel information processing, and drug-versus-placebo differences.Persistent homology detected cyclic motifs in structural networks, while cycles in functional networks supported studies of learning and treatment distinction.
- Scope: The paper motivates topology for data analysis by emphasizing its freedom from coordinates and robustness to noise, while noting that its mathematics may be unfamiliar to neuroscientists.The authors frame the work as an introduction intended to connect mathematical advances with biological questions.
- Scope: The paper introduces basic persistent homology and demonstrates related analyses on the mouse connectome, C. elegans synapses, and genomic interaction data.The datasets span multiple species and multiple spatial scales of inquiry.
When should we use topological data analysis?
Topological data analysis is most appropriate when higher-order interactions matter and when topological cavities have a meaningful interpretation and functional or scientific importance in the system under study.
- When should we use topological data analysis?: Assess whether higher-order interactions beyond pairwise relationships are important in the system before applying topological data analysis.The paper highlights higher-order interactions across spatial scales in neuroscience.
- When should we use topological data analysis?: Determine what a topological cavity represents in the system and why that feature might matter scientifically.Examples include axonal dropout in structural connectomes and coordinated but not collectively similar regional activity in functional imaging networks.
Pieces and parts of the simplicial complex
Topological data analysis represents network data as a simplicial complex, extending beyond pairwise graph relations to include higher-dimensional simplices. Its closure rule, k-skeleta, simplex distributions, and data-dependent constructions provide complementary views of system structure.
- Representation: A simplicial complex represents network data as nodes and simplices, enabling topological methods beyond recording only pairwise relations.Unlike a binary graph, this representation organizes interacting node sets into simplices of multiple dimensions.
- Representation: The closure rule requires every subset of a simplex to belong to the complex, so each higher-dimensional simplex contains its constituent faces.A 2-simplex must include its edges and vertices; analogous requirements apply in higher dimensions.
- Representation: The k-skeleton X_k is the collection of all simplices with dimension at most k, allowing the complex to be examined layer by layer.Separating dimensions distinguishes simplices, chains, and cycles across the complex.
- Structural summaries: Simplex distributions across k-skeleta can reveal functional-unit size and help distinguish system structure from noise.This distribution provides an initial structural summary after the simplicial complex is constructed.
- Data encodings: When only binary pairwise relations are known, the clique complex assigns a k-simplex to each (k+1)-clique in the graph.Other constructions include nerve complexes from overlapping node groups and Vietoris–Rips complexes from point clouds.
Topological feature compression
Topological feature compression extracts cavities from simplicial complexes by distinguishing cycles that enclose cavities from cycles that are boundaries of higher-dimensional simplices. Homology groups provide a compressed representation whose dimensions count cavities of each dimension, including connected components.
- Chain complexes: Boundary operators ∂k encode relations among k-simplices and (k−1)-simplices, including the cycles and boundaries of the simplicial complex.Each ∂k is a matrix mapping chains of k-simplices to their boundaries.
- Cycles and boundaries: The k-cycle subspace Zk = ker ∂k contains cycles surrounding cavities, while the k-boundary subspace Bk contains cycles that bound higher-dimensional chains.Because Bk is a subspace of Zk, not every cycle represents a distinct cavity.
- Homology: Adding a boundary cycle to a cycle does not change which cavity it surrounds, so equivalent cycles are grouped into homology classes.This quotient removes information in the cycle space that varies only along the boundary subspace.
- Homology: The dimension of the k-th homology group counts the number of topological cavities enclosed by k-cycles, with H0 counting connected components.Homology groups are the compressed version of the chain complex.
Homology from complex to complex: Persistent homology
Persistent homology converts a weighted simplicial complex into a filtration and tracks cavities as simplices are added. The resulting barcodes record when topological features arise, persist, and disappear, providing a global signature of evolving structure.
- Constructing the filtration: A weighted graph yields a filtration by ordering edges from greatest to least weight and adding them successively to form nested binary graphs.This produces a sequence G0, G1, … with each Gi ⊆ Gi+1.
- Tracking homology: Maps between successive chain complexes induce maps between homology groups, allowing equivalence classes of cycles to be followed across the filtration.A cavity may be newly formed, remain non-trivial, or become a boundary as additional simplices are added.
- Tracking homology: Persistent homology records each cavity’s birth and death as the filtration evolves, with surviving equivalence classes collectively defining the persistent homology.Figure 5 illustrates a cycle born in one complex, persisting through the next, and dying when it is filled.
- Interpreting barcodes: Barcodes in each dimension summarize how topological features evolve, showing their existence, number, emergence, and overlap rather than only a summary statistic.These signatures provide insight into local-to-global organization.
- Applications: Persistent homology was computed for the mouse connectome, C. elegans electrical and chemical synapses, and genomic Hi-C data.The mouse interareal connectome showed a few long-lived 1-cavities and little higher-dimensional persistent homology.
Conclusion
The conclusion presents algebraic topology and persistent homology as powerful tools for network analysis, while emphasizing that continued methodological development and interdisciplinary discussion can open further avenues for biological questions.
- Conclusion: Algebraic topology provides network scientists with powerful tools for translating networks into simplicial complexes and applying persistent homology.The paper illustrates this process using three example datasets and discusses results and reasonable interpretations.
- Conclusion: Applied topology can address increasingly complicated and detailed questions beyond the methods demonstrated in this paper.The conclusion points to systems with weighted nodes and time-varying networks as examples of extensions.
- Conclusion: Translating data into algebraic topology opens additional avenues for analysis and insight.Scientists who understand this language can add powerful tools to their analytical arsenal.
- Conclusion: Continued discussion between mathematicians developing topological tools and scientists applying them can spur methodological advances driven by biological questions.The conclusion frames interdisciplinary exchange as a continuing source of progress.