Contractive piecewise continuous maps modeling networks of inhibitory neurons

TL;DR

This paper demonstrates that generic networks of three or more inhibitory neurons exhibit stable periodic behavior with finite limit cycles, modeled by piecewise continuous, locally contractive Poincare transformations.

Contribution

It establishes that such inhibitory neuron networks generically have persistent periodic orbits under small structural perturbations, using a novel piecewise continuous contraction framework.

Findings

Networks have finite limit cycles that persist under perturbations.

Periodic behavior is topologically generic in the network space.

The model uses a piecewise continuous, locally contractive Poincare map.

Abstract

We prove that a topologically generic network (an open and dense set of networks) of three or more inhibitory neurons have periodic behavior with a finite number of limit cycles that persist under small perturbations of the structure of the network. The network is modeled by the Poincare transformation which is piecewise continuous and locally contractive on a compact region B of a finite dimensional manifold, with the separation property: it transforms homeomorphically the different continuity pieces of B into pairwise disjoint sets.