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- Alan R. Champneys, Vivien Kirk, Edgar Knobloch, Bart E. Oldeman, James Sneyd
- SIAM J. Applied Dynamical Systems
- 2007

This paper considers a hierarchy of mathematical models of excitable media in one spatial dimension, specifically the FitzHugh–Nagumo equation and several models of the dynamics of intracellular calcium. A common feature of the models is that they support solitary traveling pulse solutions which lie on a characteristic C-shaped curve of wave speed versus… (More)

- Dieter Armbruster, Emily Stone, Vivien Kirk
- 2003

The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching are found, depending on the details of the underlying deterministic dynamics: random switching between the heteroclinic cycles determined by the linear dynamics… (More)

- Alan R. Champneys, Vivien Kirk, Edgar Knobloch, Bart E. Oldeman, Jens D. M. Rademacher
- SIAM J. Applied Dynamical Systems
- 2009

Abstract. The dynamics occurring near a heteroclinic cycle between a hyperbolic equilibrium and a hyperbolic periodic orbit is analyzed. The case of interest is when the equilibrium has a onedimensional unstable manifold and a two-dimensional stable manifold while the stable and unstable manifolds of the periodic orbit are both two-dimensional. A… (More)

- Elan Gin, Vivien Kirk, James Sneyd
- Journal of theoretical biology
- 2006

Many mathematical models of calcium oscillations model buffering implicitly by using a rapid buffering approximation. This approximation assumes that separate time scales can be distinguished, with the buffer reactions occurring on a faster time scale than the other calcium fluxes. The rapid buffering approximation is convenient as it reduces the model to a… (More)

- Xingjiang Chen, Karl J Iremonger, Allan E Herbison, Vivien Kirk, James Sneyd
- Bulletin of mathematical biology
- 2013

Gonadotropin-releasing hormone (GnRH) neurons are hypothalamic neurons that control the pulsatile release of GnRH that governs fertility and reproduction in mammals. The mechanisms underlying the pulsatile release of GnRH are not well understood. Some mathematical models have been developed previously to explain different aspects of these activities, such… (More)

In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [J. Sneyd et al., "A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations," Proc. Natl. Acad. Sci. U.S.A. 103, 1675-1680 (2006)] that the… (More)

- Emily P. Harvey, Vivien Kirk, Martin Wechselberger, James Sneyd
- J. Nonlinear Science
- 2011

- Je-Chiang Tsai, Wenjun Zhang, Vivien Kirk, James Sneyd
- SIAM J. Applied Dynamical Systems
- 2012

- Pingyu Nan, Yangyang Wang, Vivien Kirk, Jonathan E. Rubin
- SIAM J. Applied Dynamical Systems
- 2015

- Wenjun Zhang, Vivien Kirk, James Sneyd, Martin Wechselberger
- Journal of mathematical neuroscience
- 2011

A major obstacle in the analysis of many physiological models is the issue of model simplification. Various methods have been used for simplifying such models, with one common technique being to eliminate certain 'fast' variables using a quasi-steady-state assumption. In this article, we show when such a physiological model reduction technique in a… (More)