×

zbMATH — the first resource for mathematics

Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model. (English) Zbl 1080.92505
Summary: 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
PDF BibTeX XML Cite
Full Text: DOI
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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.