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

Toward an Effective Theory of Neurodynamics: Topological Supersymmetry Breaking, Network Coarse-Graining, and Instanton Interaction