An absorbing set for the Chialvo map

Abstract

The classical Chialvo model, introduced in 1995, is one of the most important models that describe single neuron dynamics. In order to conduct effective numerical analysis of this model, it is necessary to obtain a rigorous estimate for the maximal bounded invariant set. We discuss this problem, and we correct and improve the results obtained by Courbage and Nekorkin [Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 1631-1651.] In particular, we provide an explicit formula for an absorbing set for the Chialvo neuron model. We also introduce the notion of a weakly absorbing set, outline the methodology for its construction, and show its advantage over an absorbing set by means of numerical computations.