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CROSS-CURRENTS BETWEEN BIOLOGY AND MATHEMATICS: THE CODIMENSION OF PSEUDO-PLATEAU BURSTING
Hinke M Osinga1, Arthur Sherman, Krasimira Tsaneva-Atanasova
1Bristol Centre for Applied Nonlinear Mathematics Department of Engineering Mathematics University of Bristol, Queen's Building, University Walk Bristol BS8 1TR, UK.
Discrete and Continuous Dynamical Systems. Series A
|September 18, 2012
Summary
This study classifies bursting oscillations in fast-slow systems by analyzing bifurcations near singularities. Researchers identified a codimension-four singularity that generates most known bursting oscillation types.
Area of Science:
- Dynamical Systems Theory
- Computational Neuroscience
- Mathematical Biology
Background:
- Bursting oscillations, characterized by alternating spiking and quiescence, are crucial in modeling biological systems.
- Classification of bursting oscillations relies on analyzing bifurcations near singularities in fast-slow systems.
- Existing models suggest fold/homoclinic bursting occurs near codimension-three singularities.
Purpose of the Study:
- To determine the codimension of fold/subHopf bursting, a recently identified burst type.
- To investigate the relationship between fold/homoclinic and fold/subHopf bursting.
- To identify singularities that give rise to various bursting oscillation types.
Main Methods:
- Analysis of bifurcations in fast-slow systems.
- Examination of singularities in cubic Liénard systems.
- Investigation of a doubly-degenerate Bogdanov-Takens point unfolding.
Main Results:
- Fold/homoclinic bursting is associated with codimension-three singularities.
- Fold/subHopf bursting was hypothesized to have a higher codimension than previously assumed.
- A codimension-four singularity was identified that generates nearly all known bursting oscillation types.
Conclusions:
- The identified codimension-four singularity provides a unified framework for understanding diverse bursting oscillations.
- This finding advances the classification of bursting phenomena in dynamical systems.
- The study offers insights into the complexity and generation mechanisms of neural oscillations.
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