Competition between co-existing heteroclinic cycles that have a c o m o n hetemclinic connection is considered. A simple model problem, consisting of a system of ordinary 'difimtial equations in R4 with 2; symmetry, is analysed. The differential equations possess four hyperbolic fixedpoints el. &. tj, and 54, with heteroclinic mnnections joining pairs of fixed points to form a 'hetemclinic network'. The network contains two heteroclinic cycles tt + 52 + b + PI and 61 + & -+ 64 + el, each of which is strucnually stable with respect to perturbations that p m m e the 2 ; symmetry of the problem. Local analysis, valid in the vicinity of the heteroclinic cycles, shows that while neither cycle can be asymptotically stable, there are conditions under which both cycles have strong attractivity properties simultaneously.For example, it is possible for both cycles to have the property that trajectories that pass ihrough an open neighbourhwd of one or more (but not all) of the heteroclinic comedons in the given cycle are asymptotic to that cycle. The stability results depend on the strengths of the contracting and expanding eigenvalues of the flow lin& about each of the fixed points and on the validity of certain nondegeneracy conditions. The possible stability propelries of the network and the cycles within it are determined.

The task of combining toolbox components into an overall model is neither simple nor easy. Firstly, it cannot be emphasised too strongly that the type of model to be constructed depends critically on the question which one wishes to study with that model. For some questions, a simple well-mixed cell model -a set of ordinary differential equations -is sufficient. For other questions, it might be necessary to construct a stochastic partial differential equation in three dimensions, using a finite element model based on anatomical measurements. In most cases, it is simply not true to say that one type of model is "better" than another. A model may be more complicated, or less complicated, but more complication does not mean that the model is more accurate, or more realistic; such judgements are highly dependent on the context.As a corollary to this, we must also always keep in mind that the purpose of a model is not to include, without discrimination, every single possible physiological complication, and thus reproduce, in silico, a miniature cell in all its complexity. Even if we wanted to do this, the task would be impossible. Something must always be left out. It is our job as modellers to decide, given the question under consideration, what must be included in our model, and what can be safely omitted.Neither is the sole purpose of a model to reproduce, with impressive accuracy, a particular experimental curve. As a general rule, the purpose of a model is to predict the results of experiments that have not yet been performed, rather than reproduce the results of those that have been. Obviously, models need to be validated by comparison to existing results, but such validation by itself is of limited use. It is not until a model can tell the experimentalist something they did not already know that a model has been truly useful.In this chapter we discuss some of the basic principles of how deterministic models of Ca 2+ signalling are constructed, and the different types of models that are commonly used. Stochastic models are treated similarly in the next chapter. It is not our purpose here to present detailed models of particular cell types (this task is left for subsequent chapters) and thus the discussion here has a more theoretical bent, with less reference to experimental data. Nevertheless, the reader should keep firmly in mind the necessity for comparison to, and prediction of, experimental data. Types of Models Models of Ca2+ signalling -as with most models of the real world -are divided into four major groups. The first major division is between deterministic and stochastic models; the second major division is between spatially homogeneous and spatially distributed models.At the most fundamental level, Ca 2+ signalling is driven by stochastic events. IPR, for example, exist in a number of discrete states, some open, some closed, and move from state to state in random fluctuations driven by thermal noise. When IPR open they release a small amount of Ca 2+ into the cytoplasm, which can then open neighbouring...

Abstract. 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 parameter. This C lies to the left of a Ushaped locus of Hopf bifurcations that corresponds to the onset of small-amplitude linear waves. The central question addressed is how the Hopf and solitary wave (homoclinic orbit in a moving frame) bifurcation curves interact in these "CU systems." A variety of possible codimension-two mechanisms is reviewed through which such Hopf and homoclinic bifurcation curves can interact. These include Shil'nikov-Hopf bifurcations and the local birth of homoclinic chaos from a saddle-node/Hopf (Gavrilov-Guckenheimer) point. Alternatively, there may be barriers in phase space that prevent the homoclinic curve from reaching the Hopf bifurcation. For example, the homoclinic orbit may bump into another equilibrium at a so-called T-point, or it may terminate by forming a heteroclinic cycle with a periodic orbit. This paper presents the results of detailed numerical continuation results on different CU systems, thereby illustrating various mechanisms by which Hopf and homoclinic curves interact in CU systems. Owing to a separation of time scales in these systems, considerable care has to be taken with the numerics in order to reveal the true nature of the bifurcation curves observed.

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Oscillations in the concentration of free cytosolic Ca 2+ are an important and ubiquitous control mechanism in many cell types. It is thus correspondingly important to understand the mechanisms that underlie the control of these oscillations and how their period is determined. We show that Class I Ca 2+ oscillations (i.e., oscillations that can occur at a constant concentration of inositol trisphosphate) have a common dynamical structure, irrespective of the oscillation period. This commonality allows the construction of a simple canonical model that incorporates this underlying dynamical behavior. Predictions from the model are tested, and confirmed, in three different cell types, with oscillation periods ranging over an order of magnitude. The model also predicts that Ca 2+ oscillation period can be controlled by modulation of the rate of activation by Ca 2+ of the inositol trisphosphate receptor. Preliminary experimental evidence consistent with this hypothesis is presented. Our canonical model has a structure similar to, but not identical to, the classic FitzHugh-Nagumo model. The characterization of variables by speed of evolution, as either fast or slow variables, changes over the course of a typical oscillation, leading to a model without globally defined fast and slow variables. ) are a ubiquitous signaling mechanism, occurring in many cell types and controlling a wide array of cellular functions (1-6). In many cases, the signal is carried by the oscillation frequency; for example, Ca 2+ oscillation frequency is known to control contraction of pulmonary and arteriole smooth muscle (7,8), as well as gene expression and differentiation (9-11). Although there are cell types where the frequency of Ca 2+ oscillation appears to be less important than the mean [Ca 2+ ] (12), an understanding of how Ca 2+ oscillation frequency is controlled remains critical to our understanding of many important cellular processes. Interestingly, it appears that the signal may not be carried by the absolute oscillation frequency but rather by a change in frequency (13), leading to a signaling system that is robust to intercellular variability, even within the same cell type. A concept similar to that of the Ca 2+ toolbox is important in the mathematical modeling of Ca 2+ dynamics. Models try to extract fundamental mechanisms, omitting less important details so that the basic skeleton-the basic toolbox components-can become clear. In the construction of such skeleton models, the concept of dynamical structure becomes important. The behavior of a model can be qualitatively described by a set of bifurcations and attracting or repelling sets, and this description is essentially independent of the exact model equations and parameters used to realize the underlying dynamical structure (in that there can be many different equations and parameters that have the same dynamical structure).One important question is how cells can generate Ca 2+ oscillations of widely differing periods, even though they appear to be using the same elements o...

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 codimension-two heteroclinic cycle occurs when there are two codimension-one heteroclinic connections, with the connection from the periodic orbit to the equilibrium corresponding to a tangency between the two relevant manifolds. The results are restricted to R 3 , the lowest possible dimension in which such a heteroclinic cycle can occur, but are expected to be applicable to systems of higher dimension as well.A geometric analysis is used to partially unfold the dynamics near such a heteroclinic cycle by constructing a leading-order expression for the Poincaré map in a full neighbourhood of the cycle in both phase and parameter space. Curves of orbits homoclinic to the equilibrium are located in a generic parameter plane, as are curves of homoclinic tangencies to the periodic orbit. Moreover, it is shown how curves of folds of periodic orbits, which have different asymptotics near the homoclinic bifurcation of the equilibrium and the homoclinic bifurcation of the periodic orbit, are glued together near the codimension-two point.A simple global assumption is made about the existence of a pair of codimension-two heteroclinic cycles corresponding to a first and last tangency of the stable manifold of the equilibrium and the unstable manifold of the periodic orbit. Under this assumption, it is shown how the locus of homoclinic orbits to the equilibrium should oscillate in the parameter space, a phenomenon known as homoclinic snaking. Finally, we present several numerical examples of systems that arise in applications, which corroborate and illustrate our theory.Key words: global bifurcation, Shil'nikov analysis, heteroclinic cycle, homoclinic orbit, homoclinic tangency, snaking.AMS subject classifications: 34C23, 34C37, 37C29, 37G20.1. Introduction. We are interested in the dynamics near a heteroclinic cycle connecting a hyperbolic equilibrium solution and a hyperbolic periodic orbit, referred to here as an EP-cycle. Specifically, we consider dynamics in R 3 , and assume that the equilibrium (denoted E) has a one-dimensional unstable manifold W u (E), while the periodic orbit (denoted P ) has a two-dimensional unstable manifold W u (P ). In this case, a heteroclinic connection from E to P generically will be of codimension one, while a connection from P to E generically will be of codimension zero. We denote by EP1-cycle an EP-cycle in R 3 where the connections from E to P and from P to E are both of the appropriate generic type. Such a heteroclinic cycle as a whole is then a codimension-one object, i.e., it occurs at isolated points in a generic one-parameter family of vector fields. In a two-parameter setting, EP1-cycles will occur on onedimensional curves in the two-par...

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 near one of the saddle points, noise induced stability of a cycle, and intermittent switching between cycles. All three responses are explained by examining the size of the stable and unstable eigenvalues at the equilibria.

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