Topology of nerves and formal concepts

TL;DR

This paper reviews methods from homotopy, combinatorial topology, and formal concepts analysis, exploring their connections and applications in neural code analysis and brain studies.

Contribution

It introduces a novel approach combining FCA and topology to analyze neural codes and their homotopy properties, extending classical theorems.

Findings

FCA provides a duality that generalizes Poincare duality.

The lattice of formal concepts is homotopy equivalent to the nerve complex.

Application to neural data helps analyze neural cell implication relations.

Abstract

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.