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An absorbing set for the Chialvo map

Paweł Pilarczyk, Grzegorz Graff

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.

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.
Paper Structure (19 sections, 17 theorems, 64 equations, 4 figures)

This paper contains 19 sections, 17 theorems, 64 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Counterexamples to the statements about $D^+$
  4. New construction of an absorbing set for the Chialvo map
  5. Region $\hat{A}$.
  6. Region $\hat{B}$.
  7. Regions $\hat{C}_1$ and $\hat{D}_1$.
  8. Region $\hat{E}_1$.
  9. Region $\hat{D}_1^+$.
  10. New construction of a weakly absorbing set for the Chialvo map
  11. Region $\hat{C}_2$.
  12. Regions $\hat{D}_2$ and $\hat{E}_2$.
  13. Region $\hat{D}_2^+$.
  14. Numerical simulations
  15. Advantage of $\hat{D}_2^+$ over $\hat{D}_1^+$ and practical construction of an absorbing set from $\hat{D}_2^+$.
  16. ...and 4 more sections

Key Result

Proposition 3

For every $(x,y) \in {\mathbb R}^2$, we have $(\bar{x},\bar{y}) \notin \hat{A}$. Moreover, if $(x,y) \notin \hat{A}$ then $\bar{x} > k$.

Figures (4)

  • Figure 1: Three hundred iterates of a seemingly chaotic trajectory observed in the Chialvo model for $a=0.9$, $b=0.6$, $c=0.6$, and $k=0.02$ after initial $100{,}000$ iterates that started at $(x_0,y_0)=(1,1)$. Consecutive points on the trajectory are joined by thin straight lines. The set $D^+$ calculated in \ref{['eq:absorbing1']} according to \ref{['AbsorbingSet']} is shown in brown.
  • Figure 2: Regions in the phase space discussed in the paper. The actual values of $u$, $v$ and $w$ are indicated in the text.
  • Figure 3: An outer approximation of the weakly absorbing set $\hat{D}_2^+$ and part of the absorbing set $\hat{P} := \bigcup_{n\geq 0} f^n(P)$, shown in grey, constructed for the Chialvo map with $a=0.9$, $b=0.01$, $c=0.9$ and $k=0.1$. The rectangles that form the set $\hat{P}$ are shown enlarged to make them clearly visible, also behind the lines. The "arms" of the set $\hat{P}$ actually extend downwards to $y \approx -430$ and rightwards almost to $x = 4400$.
  • Figure 4: A rigorously computed outer bound for the absorbing set (gray), a numerical simulation of a trajectory on an attractor (black), and the constants that bound the regions $\hat{D}_1^+$ and $\hat{D}_2^+$ for the Chialvo map with $a=0.89$, $b=0.18$, $c=0.28$ and $k=0.025$.

Theorems & Definitions (31)

  • Definition 1: absorbing set
  • Definition 2: weakly absorbing set
  • Proposition 3: Region $\hat{A}$
  • Lemma 4: moving downward in Region $\hat{B}$
  • proof
  • Lemma 5: no return to Region $\hat{B}$
  • proof
  • Proposition 6: Region $\hat{B}$
  • Proposition 7: Regions $\hat{C}_1$ and $\hat{D}_1$
  • Lemma 8: moving upward in Region $\hat{E}_1$
  • ...and 21 more