  Experimental research has shown that the brain's fast electrochemical
dynamics, or neurodynamics (ND), is strongly stochastic, chaotic, and instanton
(neuroavalanche)-dominated. It is also partly scale-invariant which has been
loosely associated with critical phenomena. It has been recently demonstrated
that the supersymmetric theory of stochastics (STS) offers a theoretical
framework that can explain all of the above ND features. In the STS, all
stochastic models possess a topological supersymmetry (TS), and the
"criticality" of ND and similar stochastic processes is associated with
noise-induced, spontaneous breakdown of this TS (due to instanton condensation
near the border with ordinary chaos in which TS is broken by
non-integrability). Here, we propose a new approach that may be useful for the
construction of low-energy effective theories of ND. Its centerpiece is a
coarse-graining procedure of neural networks based on simplicial complexes and
the concept of the "enveloping lattice." It represents a neural network as a
continuous, high-dimensional base space whose rich topology reflects that of
the original network. The reduced one-instanton state space is determined by
the de Rham cohomology classes of this base space, and the effective ND
dynamics can be recognized as interactions of the instantons in the spirit of
the Segal-Atiyah formalism.
