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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)

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)

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)

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)

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)

This paper uses Hamiltonian methods to nd and determine the stability of some new solution branches for an equivariant Hopf bifurcation on C 4. The normal form has a symmetry group given by the semi-direct product of D2 with T 2 S 1. The Hamiltonian part of the normal form is completely integrable and may be analyzed using a system of invariants. The idea… (More)

We describe an example of a structurally stable heteroclinic network for which nearby orbits exhibit irregular but sustained switching between the various sub-cycles in the network. The mechanism for switching is the presence of spiralling due to complex eigenvalues in the flow linearised about one of the equilibria common to all cycles in the network. We… (More)

- Martin Golubitsky, Jerrold E Marsden, Ian Stewart, Michael Dellnitz, M Golubitsky, J E Marsden +11 others
- 2008

This paper develops the Liapunov-Schmidt procedure for systems with additional constraints such as having a first integral, being Hamiltonian, or being a gradient system. Similar developments for systems with symmetry, including reversibility, are well known, and the method of this paper augments and is consistent with that approach. One of the results… (More)