Topology of nerves and formal concepts

2019.

Ayzenberg A.

The general goal of this paper is to gather and review several methods from homotopy and combinatorial topology and formal concepts analysis (FCA) and analyze their connections. FCA appears naturally in the problem of combinatorial simplification of simplicial complexes and allows to see a certain duality on a class of simplicial complexes. This duality generalizes Poincare duality on cell subdivisions of manifolds. On the other hand, with the notion of a topological formal context, we review the classical proofs of two basic theorems of homotopy topology: Alexandrov Nerve theorem and Quillen--McCord theorem, which are both important in the applications. A brief overview of the applications of the Nerve theorem in brain studies is given. The focus is made on the task of the external stimuli space reconstruction from the activity of place cells. We propose to use the combination of FCA and topology in the analysis of neural codes. The lattice of formal concepts of a neural code is homotopy equivalent to the nerve complex, but, moreover, it allows to analyse certain implication relations between collections of neural cells.

Keywords: анализ формальных понятий, neural code, нейронный код, Formal Concept Analysis, place cells, нейроны места, Quillen-McCord theorem, Nerve theorem, applied topology, nerve complexes, complete lattice, Hasse diagram, Теорема Квиллена-МакКорда, теорема о нерве, прикладная топология, нерв-комплексы, полная решетка, диаграмма Хассе

Publication based on the results of: Geometry and topology of the brain (2019)