Complex dynamics of the simplest neuron model: Singular chaotic Shilnikov attractor as specific oscillatory neuron activity

Abstract

We study complex dynamics of the Chialvo model that is the simplest neuron-type model in form of a four-parameter family of two-dimensional noninvertible maps (endomorphisms). Main elements of bifurcation diagram in the plane of two parameters have been constructed in which regions corresponding to both quasi-periodic and chaotic oscillations are selected. We also indicate special regions corresponding to singular discrete chaotic Shilnikov attractors that we consider as a new type of the so-called snap-back repellers (over an unstable focus). The study of time series was carried out in which there were classified patterns of specific oscillatory activities in the cases when homoclinic orbits to the unstable focus exists and, when such orbits were not yet formed but a strange attractor already exists. New dynamical characteristics are proposed, with the help of which it is possible to assess the level of distinctness of atypical oscillatory activity.