2019/11/13 by Anton Ayzenberg, Ayzenberg, Anton
Computer Science · Mathematics · Neuroscience · #03G10 #05C20 #05E45 #06B30 #18B35 #52C45 #55P10 #55U10 #68R10 (Primary) 52B70 #68T30 #90B85 #92C20 #92C55 #97M60 (Secondary) #Algebraic Topology (math.AT) #Axon Guidance and Neuronal Signaling #Combinatorics (math.CO) #FOS: Mathematics #History and Overview (math.HO) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1911.05491
openalex publication_date 2019/11/13 · openalex created_date 2019/11/22 · openalex updated_date 2026/07/28
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.