Networks of planar neural organizing centers

Abstract

A codimension-three degenerate Bogdanov–Takens bifurcation at a cusp organizes Type I excitability of neural membranes and has parameter regions for Type II. We argue from the electrophysiology why this is expected. We derive and analyze a local canonical model for weakly connected neural networks of multiple such bifurcations. The canonical model suggests a dynamical form of the network adaptation conditions, which mimic an afterhyperpolarizing potential causing neuron spike frequency adaptation. We also simulate strongly connected neural oscillators and two-component models of mixed types constructed from this model, and bursting from periodic variation of parameters.