Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Volume 32, Issue 8, Pages 2879-2912Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcds.2012.32.2879
Keywords
MMOs; bifurcations; geometric singular perturbation theory; canards; return maps
Categories
Funding
- NSF [DMS 0917664]
- University of Sydney
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0917664] Funding Source: National Science Foundation
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Mixed mode oscillations (MMOs) are complex oscillatory waveforms that naturally occur in physiologically relevant dynamical processes. MMOs were studied in a model of electrical bursting in a pituitary lactotroph [34] where geometric singular perturbation theory and bifurcation analysis were combined to demonstrate that the MMOs arise from canard dynamics. In this work, we extend the analysis done in [34] and consider bifurcations of canard solutions under variations of key parameters. To do this, a global return map induced by the flow of the equations is constructed and a qualitative analysis given. The canard solutions act as separatrices in the return maps, organising the dynamics along the Poincare section. We examine the bifurcations of the return maps and demonstrate that the map formulation allows for an explanation of the different MMO patterns observed in the lactotroph model.
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