  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.
