Vivien Kirk

When Shil'nikov meets Hopf in excitable systems

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 on a characteristic C-shaped curve of wave speed versus parameter. This C lies to the left of a U-shaped 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 CUsystems, thereby illustrating various of the 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.

Noisy heteroclinic networks

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.

Models of Calcium Signalling

This book discusses the ways in which mathematical, computational, and modelling methods can be used to help understand the dynamics of intracellular calcium. The concentration of free intracellular calcium is vital for controlling a wide range of cellular processes, and is thus of great physiological importance. However, because of the complex ways in which the calcium concentration varies, it is also of great mathematical interest. This book presents the general modelling theory as well as a large number of specific case examples, to show how mathematical modelling can interact with experimental approaches, in an interdisciplinary and multifaceted approach to the study of an important physiological control mechanism.

A mechanism for switching near a heteroclinic network

We describe an example of a robust 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 linearized about one of the equilibria common to all cycles in the network. We construct and use return maps to investigate the asymptotic stability of the network, and show that switching is ubiquitous near the network. Some of the unstable manifolds involved in the network are two-dimensional; we develop a technique to account for all trajectories on those manifolds. A simple numerical example illustrates the rich dynamics that can result from the interplay between the various cycles in the network.

Effect of a refractory period on the entrainment of pulse-coupled integrate-and-fire oscillators

Abstract It was shown by Mirollo and Strogatz [SIAM J. Appl. Math. 50 (1990) 1645] that a population of globally coupled, identical, integrate-and-fire oscillators will almost always become entrained. We find that the inclusion of a refractory period in the cycle of each oscillator can result in open sets of initial configurations evolving to asynchronous states.

Understanding and Distinguishing Three-Time-Scale Oscillations: Case Study in a Coupled Morris--Lecar System

Many physical systems feature interacting components that evolve on disparate time scales. Significant insights about the dynamics of such systems have resulted from grouping time scales into two classes and exploiting the time scale separation between classes through the use of geometric singular perturbation theory. It is natural to expect, however, that some dynamic phenomena cannot be captured by a two-time-scale decomposition. In this work, we are motivated by applications in neural dynamics to focus on a model consisting of a pair of Morris--Lecar systems coupled so that there are three time scales in the full system. We demonstrate that two approaches previously developed in the context of geometric singular perturbation theory for the analysis of two-time-scale systems extend naturally to the three-time-scale setting, where they complement each other nicely. Our analysis explains the dynamic mechanisms underlying solution features in the three-time-scale model. By comparison with certain two-time-...

The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various phase space variables, but details of the mechanisms underlying the complicated dynamics have not previously been investigated. We identify global bifurcations that induce the onset of chaotic dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis of approximate return maps, locate the global bifurcations in parameter space. We find there is a threshold in the size of certain symmetry-breaking terms, below which there can be no persistent switching. Our results are illustrated by a numerical example.

On the dynamical structure of calcium oscillations

Significance Oscillations in the concentration of free cytosolic calcium are an important control mechanism in many cell types. However, we still have little understanding of how some cells can exhibit calcium oscillations with a period of less than a second, whereas other cells have oscillations with a period of hundreds of seconds. Here, we show that one common type of calcium oscillation has a dynamic structure that is independent of the period. We thus hypothesize that cells control their oscillation period by varying the rate at which their critical internal variables move around this common dynamic structure and that this rate can be controlled by the rate at which calcium activates calcium release from the endoplasmic/sarcoplasmic reticulum. Oscillations in the concentration of free cytosolic Ca2+ 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 Ca2+ 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 Ca2+ oscillation period can be controlled by modulation of the rate of activation by Ca2+ 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.

Effects of quasi-steady-state reduction on biophysical models with oscillations.

Many biophysical models have the property that some variables in the model evolve much faster than others. A common step in the analysis of such systems is to simplify the model by assuming that the fastest variables equilibrate instantaneously, an approach that is known as quasi-steady state reduction (QSSR). QSSR is intuitively satisfying but is not always mathematically justified, with problems known to arise, for instance, in some cases in which the full model has oscillatory solutions; in this case, the simplified version of the model may have significantly different dynamics to the full model. This paper focusses on the effect of QSSR on models in which oscillatory solutions arise via one or more Hopf bifurcations. We first illustrate the problems that can arise by applying QSSR to a selection of well-known models. We then categorize Hopf bifurcations according to whether they involve fast variables, slow variables or a mixture of both, and show that Hopf bifurcations that involve only slow variables are not affected by QSSR, Hopf bifurcations that involve fast and slow variables (i.e., singular Hopf bifurcations) are generically preserved under QSSR so long as a fast variable is kept in the simplified system, and Hopf bifurcations that primarily involve fast variables may be eliminated by QSSR. Finally, we present some guidelines for the application of QSSR if one wishes to use the method while minimising the risk of inadvertently destroying essential features of the original model.

Traveling waves in a simplified model of calcium dynamics

We analyze traveling wave propagation in a simplified model of intracellular calcium dynamics. Despite its simplicity, the model is thought to capture fundamental features of wave propagation in ca...