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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 that have arisen in the work of Sneyd and collaborators. A common feature of the models is that they support solitary travelling pulse solutions which lie(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)
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 one-dimensional unstable manifold and a two-dimensional stable manifold while the stable and unstable manifolds of the periodic orbit are both two-dimensional. A codimension-two(More)
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)
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)
A variety of bursting and spiking patterns arise in models for respiratory neurons [1] and other neurons associated with hormone release [2]. These models often feature quantities evolving on distinct time scales, such as fast voltage and slower ion current activation or inac-tivation variables. Furthermore, such systems may combine multiple interacting(More)
Robust heteroclinic cycles are known to change stability in resonance bifurcations, which occur when an algebraic condition on the eigenvalues of the system is satisfied and which typically result in the creation or destruction of a long-period periodic orbit. Resonance bifurcations for heteroclinic networks are potentially more complicated because(More)