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This study explores how uncertainty in time-varying parameter estimates obtained using nonlinear filtering algorithms with parameter tracking affects corresponding model output predictions. Results are demonstrated on a numerical example estimating the time-varying external voltage parameter in the FitzHugh-Nagumo system for modeling the spiking dynamics of neurons.
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Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci
A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a canard explosion. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.
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- Award ID(s):
- 2204758
- PAR ID:
- 10336958
- Date Published:
- Journal Name:
- Discrete & Continuous Dynamical Systems - S
- Volume:
- 0
- Issue:
- 0
- ISSN:
- 1937-1632
- Page Range / eLocation ID:
- 0
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3934/dcdss.2022036
