An absorbing set for the Chialvo map

• Published in: Communications in nonlinear science & numerical simulation Vol. 132; p. 107947
• Main Authors: Pilarczyk, Paweł, Graff, Grzegorz
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.05.2024
• Subjects: Absorbing set; Chialvo map; Conley index; Grid; Neuron model; Rigorous numerics; Neuron model; Absorbing set; Chialvo map; Rigorous numerics; Conley index; Grid
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2024.107947

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 (2010). 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. •An explicit formula is provided for an absorbing set for the Chialvo neuron model.•Earlier results existing in literature are corrected and improved.•The notion of a weakly absorbing set is introduced.•Numerical computations show advantage of the weakly absorbing set.•A formula is provided for a weakly absorbing set for the Chialvo model.