🍩 Database of Original & Non-Theoretical Uses of Topology

(found 5 matches in 0.003677s)
  1. The Growing Topology of the C. Elegans Connectome (2020)

    Alec Helm, Ann S. Blevins, Danielle S. Bassett
    Abstract Probing the developing neural circuitry in Caenorhabditis elegans has enhanced our understanding of nervous systems. The C. elegans connectome, like those of other species, is characterized by a rich club of densely connected neurons embedded within a small-world architecture. This organization of neuronal connections, captured by quantitative network statistics, provides insight into the system's capacity to perform integrative computations. Yet these network measures are limited in their ability to detect weakly connected motifs, such as topological cavities, that may support the systems capacity to perform segregated computations. We address this limitation by using persistent homology to track the evolution of topological cavities in the growing C. elegans connectome throughout neural development, and assess the degree to which the growing connectomes topology is resistant to biological noise. We show that the developing connectome topology is both relatively robust to changes in neuron birth times and not captured by similar growth models. Additionally, we quantify the consequence of a neurons specific birth time and ask if this metric tracks other biological properties of neurons. Our results suggest that the connectomes growing topology is a robust feature of the developing connectome that is distinct from other network properties, and that the growing topology is particularly sensitive to the exact birth times of a small set of predominantly motor neurons. By utilizing novel measurements that track biological features, we anticipate that our study will be helpful in the construction of more accurate models of neuronal development in C. elegans
  2. The Importance of the Whole: Topological Data Analysis for the Network Neuroscientist (2019)

    Ann E. Sizemore, Jennifer E. Phillips-Cremins, Robert Ghrist, Danielle S. Bassett
    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., For the network neuroscientist, this exposition aims to communicate both the mathematics and the advantages of using tools from applied topology for the study of neural systems. Using data from the mouse connectome, electrical and chemical synapses in C. elegans, and chromatin interaction data, we offer example computations and applications to further demonstrate the power of topological data analysis in neuroscience. Finally, we expose the reader to novel developments in applied topology and relate these developments to current questions and methodological difficulties in network neuroscience.
  3. Spatial Embedding Imposes Constraints on Neuronal Network Architectures (2018)

    Jennifer Stiso, Danielle S. Bassett
    Abstract Recent progress towards understanding circuit function has capitalized on tools from network science to parsimoniously describe the spatiotemporal architecture of neural systems. Such tools often address systems topology divorced from its physical instantiation. Nevertheless, for embedded systems such as the brain, physical laws directly constrain the processes of network growth, development, and function. We review here the rules imposed by the space and volume of the brain on the development of neuronal networks, and show that these rules give rise to a specific set of complex topologies. These rules also affect the repertoire of neural dynamics that can emerge from the system, and thereby inform our understanding of network dysfunction in disease. We close by discussing new tools and models to delineate the effects of spatial embedding.
  4. Knowledge Gaps in the Early Growth of Semantic Feature Networks (2018)

    Ann E. Sizemore, Elisabeth A. Karuza, Chad Giusti, Danielle S. Bassett
    Abstract Understanding language learning and more general knowledge acquisition requires the characterization of inherently qualitative structures. Recent work has applied network science to this task by creating semantic feature networks, in which words correspond to nodes and connections correspond to shared features, and then by characterizing the structure of strongly interrelated groups of words. However, the importance of sparse portions of the semantic network—knowledge gaps—remains unexplored. Using applied topology, we query the prevalence of knowledge gaps, which we propose manifest as cavities in the growing semantic feature network of toddlers. We detect topological cavities of multiple dimensions and find that, despite word order variation, the global organization remains similar. We also show that nodal network measures correlate with filling cavities better than basic lexical properties. Finally, we discuss the importance of semantic feature network topology in language learning and speculate that the progression through knowledge gaps may be a robust feature of knowledge acquisition.
  5. Cliques and Cavities in the Human Connectome (2018)

    Ann E. Sizemore, Chad Giusti, Ari Kahn, Jean M. Vettel, Richard F. Betzel, Danielle S. Bassett
    Abstract Encoding brain regions and their connections as a network of nodes and edges captures many of the possible paths along which information can be transmitted as humans process and perform complex behaviors. Because cognitive processes involve large, distributed networks of brain areas, principled examinations of multi-node routes within larger connection patterns can offer fundamental insights into the complexities of brain function. Here, we investigate both densely connected groups of nodes that could perform local computations as well as larger patterns of interactions that would allow for parallel processing. Finding such structures necessitates that we move from considering exclusively pairwise interactions to capturing higher order relations, concepts naturally expressed in the language of algebraic topology. These tools can be used to study mesoscale network structures that arise from the arrangement of densely connected substructures called cliques in otherwise sparsely connected brain networks. We detect cliques (all-to-all connected sets of brain regions) in the average structural connectomes of 8 healthy adults scanned in triplicate and discover the presence of more large cliques than expected in null networks constructed via wiring minimization, providing architecture through which brain network can perform rapid, local processing. We then locate topological cavities of different dimensions, around which information may flow in either diverging or converging patterns. These cavities exist consistently across subjects, differ from those observed in null model networks, and – importantly – link regions of early and late evolutionary origin in long loops, underscoring their unique role in controlling brain function. These results offer a first demonstration that techniques from algebraic topology offer a novel perspective on structural connectomics, highlighting loop-like paths as crucial features in the human brain’s structural architecture.