Observation of Crises and Bifurcations in the
Hodgkin-Huxley Neuron Model
Wuyin Jin1, Qian Lin2, Yaobing Wei1, and Ying Wu3
1 College of Mechano-Electronic Engineering,
Lanzhou University of Technology, Lanzhou 730050, China
jinwuyin@263.net
2 College of Petrochemical Technology, Lanzhou University of Technology,
Lanzhou 730050, China
3 College of Science, Lanzhou University of Technology,
Lanzhou 730050, China
Abstract. With the changing of the stimulus frequency, there are a lot
of firing dynamics behaviors of interspike intervals (ISIs), such as quasiperiodic,
bursting, period-chaotic, chaotic, periodic and the bifurcations
of the chaotic attractor appear alternatively in Hodgkin-Huxley (H-H)
neuron model. The chaotic behavior is realized over a wide range of
frequency and is visualized by using ISIs, and many kinds of abrupt undergoing
changes of the ISIs are observed in deferent frequency regions,
such as boundary crisis, interior crisis and merging crisis displaying alternately
along with the changes changes of external signal frequency,
too. And there are many periodic windows and fractal structures in ISIs
dynamics behaviors. The saddle node bifurcation resulted collapses of
chaos to period-12 orbit in dynamics of ISIs is identified.
From www.elsevier.com
Wednesday, January 30, 2008
Aperiodic flow-induced oscillations of collapsible tubes:
a critical reappraisalC.D. Bertram a,, J. Timmer b, T.G. Mu¨ ller b, T. Maiwald b, M. Winterhalder b,
H.U. Voss b
a Graduate School of Biomedical Engineering, University of New South Wales, Sydney, NSW 2052, Australia
b Freiburger Zentrum fu¨ r Datenanalyse und Modellbildung, Fakulta¨t fu¨ r Physik, Albert–Ludwigs–Universita¨ t Freiburg, Eckerstraße 1, 79 104
Freiburg, Germany
Received 7 April 2003; received in revised form 14 October 2003; accepted 21 November 2003
Abstract
The evidence for the aperiodic self-excited oscillations of flow-conveying collapsible tubes being mathematically chaotic is reexamined.
Many cases which powerfully suggest nonlinear deterministic behaviour have not been recorded over time-spans which
allow their exhaustive examination. The present investigation centred on a previously recorded robust and generic oscillation, but
more recent and more discerning tests were applied. Despite hints that a low embedding dimension might suffice, the data
appeared on most indices high-dimensional. A U-shaped return map was found and modelled using both radial basis functions
and polynomials, but lack of detailed structure in the map prevented effective parameter estimation. On the basis of power-law
rather than exponential divergence of nearby trajectories, and of inability to discriminate against behaviour which would also be
manifested by a surrogate consisting of a noise-perturbed nonlinear periodic oscillator, it is concluded that the data do not support
the idea that the aperiodicity in the particular oscillation examined is caused by deterministic chaos. There was evidence that
the distributed nature of the physical system might underlie aspects of the high dimensionality. We advocate equally searching
testing of any future candidate chaotic oscillations in the investigation of collapsed-tube flows.
# 2003 IPEM. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear dynamics; Deterministic chaos; Time-series analysis; Self-excited oscillation
From www.elsevior.com
a critical reappraisalC.D. Bertram a,, J. Timmer b, T.G. Mu¨ ller b, T. Maiwald b, M. Winterhalder b,
H.U. Voss b
a Graduate School of Biomedical Engineering, University of New South Wales, Sydney, NSW 2052, Australia
b Freiburger Zentrum fu¨ r Datenanalyse und Modellbildung, Fakulta¨t fu¨ r Physik, Albert–Ludwigs–Universita¨ t Freiburg, Eckerstraße 1, 79 104
Freiburg, Germany
Received 7 April 2003; received in revised form 14 October 2003; accepted 21 November 2003
Abstract
The evidence for the aperiodic self-excited oscillations of flow-conveying collapsible tubes being mathematically chaotic is reexamined.
Many cases which powerfully suggest nonlinear deterministic behaviour have not been recorded over time-spans which
allow their exhaustive examination. The present investigation centred on a previously recorded robust and generic oscillation, but
more recent and more discerning tests were applied. Despite hints that a low embedding dimension might suffice, the data
appeared on most indices high-dimensional. A U-shaped return map was found and modelled using both radial basis functions
and polynomials, but lack of detailed structure in the map prevented effective parameter estimation. On the basis of power-law
rather than exponential divergence of nearby trajectories, and of inability to discriminate against behaviour which would also be
manifested by a surrogate consisting of a noise-perturbed nonlinear periodic oscillator, it is concluded that the data do not support
the idea that the aperiodicity in the particular oscillation examined is caused by deterministic chaos. There was evidence that
the distributed nature of the physical system might underlie aspects of the high dimensionality. We advocate equally searching
testing of any future candidate chaotic oscillations in the investigation of collapsed-tube flows.
# 2003 IPEM. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear dynamics; Deterministic chaos; Time-series analysis; Self-excited oscillation
From www.elsevior.com
Chaotic Title
Self-Emergence of Chaos in Identifying Irregular Periodic Behavior
Oscar DE FEO
Laboratory of Nonlinear Systems
Swiss Federal Institute of Technology Lausanne
EL-E, EPFL-I&C-LANOS, CH-1015 Lausanne, Switzerland
Phone:+41-21-693-5683, Fax:+41-21-693-6700
Email: Oscar.DeFeo@epfl.ch
Abstract — In order to exploit generalized chaotic synchronization
phenomena for the solution of temporal pattern
recognition problems, a chaotic dynamical system representing
the class of signals that are to be recognized must be established.
This system can be determined by means of identification
techniques. Although the fulfillment of the chaotic
condition could appear as a constraint, it is shown here that,
for a very simple identification algorithm, chaos self-emerges
when a model, fitting unprecise periodic signals, is identified.
From www.elsevior.com
Self-Emergence of Chaos in Identifying Irregular Periodic Behavior
Oscar DE FEO
Laboratory of Nonlinear Systems
Swiss Federal Institute of Technology Lausanne
EL-E, EPFL-I&C-LANOS, CH-1015 Lausanne, Switzerland
Phone:+41-21-693-5683, Fax:+41-21-693-6700
Email: Oscar.DeFeo@epfl.ch
Abstract — In order to exploit generalized chaotic synchronization
phenomena for the solution of temporal pattern
recognition problems, a chaotic dynamical system representing
the class of signals that are to be recognized must be established.
This system can be determined by means of identification
techniques. Although the fulfillment of the chaotic
condition could appear as a constraint, it is shown here that,
for a very simple identification algorithm, chaos self-emerges
when a model, fitting unprecise periodic signals, is identified.
From www.elsevior.com
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