Decoding of neural data using cohomological feature extraction

Rybakken, Erik; Baas, Nils A.; Dunn, Benjamin

doi:10.48550/arxiv.1711.07205

Cited by 2 publications

(2 citation statements)

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“…Is the system time-varying with either unweighted [10,14] or weighted edges [19,46,68]? Would one perhaps wish to obtain circular coordinates for the data [23,56]? Would equivalence classes of paths instead of cycles be useful [17]?…”

Section: Discussionmentioning

confidence: 99%

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

Sizemore

Phillips-Cremins

Ghrist

et al. 2019

Network Neuroscience

175

139

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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.

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Section: Discussionmentioning

confidence: 99%

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

Sizemore

Phillips-Cremins

Ghrist

et al. 2019

Network Neuroscience

175

139

View full text Add to dashboard Cite

show abstract

“…4 Cohomology theory has also been applied to persistent homology analysis. A 1-dimensional cohomology was used to assign circular values to the input data associated to a homology generator 18 which further led to applications in several fields including the analysis of neural data 38 and the study of periodic motion. 39 Persistent cohomology in higher dimensions has been used to produce coordinate representations which reduces dimensionality while retaining the topological property of data.…”

Section: Introductionmentioning

confidence: 99%

Persistent Cohomology for Data With Multicomponent Heterogeneous Information

Cang¹,

Wei²

2020

SIAM Journal on Mathematics of Data Science

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Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and reveal certain interaction patterns in terms of topological invariants. However, in addition to the geometric information, there is a wide variety of non-geometric information of molecular structures, such as element types, atomic partial charges, atomic pairwise interactions, and electrostatic potential function, that is not described by persistent homology. Although element specific homology and electrostatic persistent homology can encode some non-geometric information into geometry based topological invariants, it is desirable to have a mathematical framework to systematically embed both geometric and non-geometric information, i.e., multicomponent heterogeneous information, into unified topological descriptions. To this end, we propose a mathematical framework based on persistent cohomology. In our framework, nongeometric information can be either distributed globally or resided locally on the datasets in the geometric sense and can be properly defined on topological spaces, i.e., simplicial complexes. Using the proposed persistent cohomology based framework, enriched barcodes are extracted from datasets to represent heterogeneous information. We consider a variety of datasets to validate the present formulation and illustrate the usefulness of the proposed persistent cohomology. It is found that the proposed framework using cohomology boosts the performance of persistent homology based methods in the protein-ligand binding affinity prediction on massive biomolecular datasets.

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Decoding of neural data using cohomological feature extraction

Cited by 2 publications

References 0 publications

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

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

Persistent Cohomology for Data With Multicomponent Heterogeneous Information

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