🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.001706s)

  1. Complexes of Tournaments, Directionality Filtrations and Persistent Homology (2020)

    Dejan Govc, Ran Levi, Jason P. Smith
    Abstract Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph \$\mathcal\G\\$, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of \$\mathcal\G\\$, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neocortex.

  3. Cliques of Neurons Bound Into Cavities Provide a Missing Link Between Structure and Function (2017)

    Michael W. Reimann, Max Nolte, Martina Scolamiero, Katharine Turner, Rodrigo Perin, Giuseppe Chindemi, Paweł Dłotko, Ran Levi, Kathryn Hess, Henry Markram
    Abstract The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence towards peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities.