The importance of the whole: Topological data analysis for the network neuroscientist

Abstract

Data analysis techniques from network science have fundamentally improved our understanding of neural systems and the complex behaviors that they support. Yet the restriction of network techniques to the study of pairwise interactions prevents us from taking into account intrinsic topological features such as cavities that may be crucial for system function. To detect and quantify these topological features, we must turn to algebro-topological methods that encode data as a simplicial complex built from sets of interacting nodes called simplices. We then use the relations between simplices to expose cavities within the complex, thereby summarizing its topological features. 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 mathematics and perform demonstrative calculations on the mouse structural connectome, synapses in C. elegans, and genomic interaction data. Finally, we suggest avenues for future work and highlight new advances in mathematics ready for use in neural systems.

Topological Data Analysis reveals robust alterations in the whole-brain and frontal lobe functional connectomes in Attention-Deficit/Hyperactivity Disorder

“…Algebraic information extracted from this topological space are called Betti numbers, particularly, the Betti-0 number (B 0 ) accounts for the number of components, i.e. the number of isolated nodes or sets of nodes connected by a sequence of edges; Betti-1 number refers to the number of cavities in the 2-dimensional space between nodes, and so on (for more extensive reviews on topological data analysis see Edelsbrunner, 2000;and Sizemore et al, 2019). In this work, we focus exclusively on B 0 .…”

“…Recently, topological data analysis (TDA), has been adopted in neuroimaging as a tool to quantify and visualize the evolution of the brain network at different thresholds (Lee et al, 2011(Lee et al, , 2017Sizemore et al, 2018Sizemore et al, , 2019Expert et al, 2019). The main objective of this method is to model the network as a topological space instead of a graph (Edelsbrunner et al, 2000;Zomorodian and Carlsson, 2005), allowing the assessment of the functional connectivity matrix as a topological process instead of a static threshold-dependent representation of the network.…”

Stability of spontaneous, correlated activity in mouse auditory cortex

“…It would be interesting in future to consider maximum entropy models as an alternative method to estimate connections between units [91], both for its sensitivity to underlying structure [92], and for its ability to assess higher-order interactions [93]. Approaches that could then take advantage of the richer assessment of higher order interactions in these data include emerging tools from algebraic topology [94,95], which have already proven relevant for understanding structure-function relationships at both large and small scales in neural systems [96,97].…”

Network topology of neural systems supporting avalanche dynamics predicts stimulus propagation and recovery

“…Future work is needed to better understand the rules by which neurons connect to one another, and to determine whether those rules serve to increase the memory capacity of cortical networks. It would also be interesting in the future to determine whether higher-order structural motifs, such as those accessible to tools from algebraic topology [19,61], might also play a role in the relationships between topology, dynamics, and computation [53,60].…”