Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model.

*(English)*Zbl 1080.92505Summary: Interior crises are understood as discontinuous changes of the size of a chaotic attractor that occurs when an unstable periodic orbit collides with the chaotic attractor. We present here numerical evidence and theoretical reasoning which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous. This occurs in the Hindmarsh-Rose model of a neuron, at the transition point between the bursting and spiking dynamics, which are two different dynamic behaviors that this system is able to present. Moreover, besides the change in attractor size, other significant properties of the system undergoing the transitions do change in a relevant qualitative way. The mechanism for such transition is understood in terms of a simple one-dimensional map whose dynamics undergoes a crossover between two different universal behaviors.

##### MSC:

92C20 | Neural biology |

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

37N25 | Dynamical systems in biology |

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\textit{J. M. González-Miranda}, Chaos 13, No. 3, 845--852 (2003; Zbl 1080.92505)

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##### References:

[1] | DOI: 10.1103/PhysRevLett.79.4697 |

[2] | DOI: 10.1103/PhysRevE.61.718 |

[3] | DOI: 10.1103/PhysRevE.66.047202 |

[4] | DOI: 10.1016/S0006-3495(81)84782-0 |

[5] | DOI: 10.1006/bulm.1999.0095 · Zbl 1323.92085 |

[6] | DOI: 10.1063/1.165848 · Zbl 0900.92162 |

[7] | DOI: 10.1098/rspb.1984.0024 |

[8] | DOI: 10.1109/81.633889 |

[9] | DOI: 10.1103/PhysRevA.57.2527 |

[10] | DOI: 10.1103/PhysRevE.65.041915 |

[11] | DOI: 10.1209/epl/i1998-00423-y |

[12] | DOI: 10.1016/S0167-2789(01)00290-1 · Zbl 1026.92008 |

[13] | DOI: 10.1016/S0928-4257(00)01102-5 |

[14] | DOI: 10.1162/089892901564199 |

[15] | DOI: 10.1103/PhysRevLett.48.1507 |

[16] | DOI: 10.1103/PhysRevLett.76.708 |

[17] | Foss J., J. Neurophysiol. 84 pp 975– (2000) |

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