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**Computation of the topological entropy in chaotic biophysical bursting models for excitable cells.**
*(English)*
Zbl 1106.92012

Summary: One of the interesting complex behaviors in many cell membranes is bursting, in which a rapid oscillatory state alternates with phases of relative quiescence. Although there is an elegant interpretation of many experimental results in terms of nonlinear dynamical systems, the dynamics of bursting models is not completely described.

We study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics of the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.

We study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics of the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.

### MSC:

92C20 | Neural biology |

37N25 | Dynamical systems in biology |

92C37 | Cell biology |

37B40 | Topological entropy |

37B10 | Symbolic dynamics |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

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\textit{J. Duarte} et al., Discrete Dyn. Nat. Soc. 2006, No. 3, Article ID 60918, 18 p. (2006; Zbl 1106.92012)

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