Research
Systems that evolve continuously with time can often be described by differential equations, and solutions of these can display a variety of behaviours, for instance, they may be at equilibrium, or be periodic in time. A heteroclinic network is a type of solution consisting of a set of dynamical states connected by trajectories. Often the states are equilibrium solutions, but they could equally well be periodic orbits or even chaotic sets. Near a heteroclinic network, the dynamics are typically intermittent: the system spends long periods of time near the states in the network, with rapid transitions between them as the trajectory travels around the network.
Heteroclinic networks can be found in mathematical models describing a diverse range of physical systems; they arise naturally in applications such as population dynamics, fluid mechanics and game theory. Heteroclinic networks are also particularly suited to the modelling of cognitive functions and computation due to the sequential nature of the dynamics as the trajectory explores the network. However, current theory is not yet fully developed for many of these situations, and in particular, physical systems are often affected by the presence of stochastic noise.
Current ongoing research projects are described in brief below.
If you are interested in joining our research group as a student, a list of possible projects is available here.
Heteroclinic cycles in spatially extended systems
Scissors cut Paper, Paper wraps Rock, Rock blunts Scissors: the simple game of Rock–Paper–Scissors provides an appealing model for cyclic dominance between competing populations or strategies in evolutionary game theory and biology. The model has been invoked to explain the repeated growth and decay of three competing strains of microbial organisms and of three colour-morphs of side-blotched lizards. In a mathematical model of three competing species which allows for spatial distribution and mobility, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn; these waves can be organised into spirals, with roughly equal populations of the three species at the core of each spiral, and each species dominating in turn in the spiral arms. In collaboration with Alastair Rucklidge at the University of Leeds in the UK, we are trying to further our understanding of the spiral patterns in the spatially-extended model, in particular, to understand what determines the wavelength, rotation speed and stability of the spiral waves.
Heteroclinic networks as models for neural processes
Heteroclinic networks are particularly suited to the modelling of cognitive functions due to the sequential and intermittent nature of the dynamics as the trajectory explores the network. Some of my recent work with Peter Ashwin (University of Exeter) develops new techniques for designing deterministic systems containing heteroclinic networks of any specified topology. In this project we use these techniques to design heteroclinic networks to model cognitive functions such as cognitive task switching. Switching between cognitive tasks is known to cause an increase in both reaction times and error rates even when a subject is practised in all the different tasks. The interpretation of the analysis of the network will be done in collaboration with experimental psychologists, to make predictions about questions such as: How does the switch cost vary as the relative ease or difficulty of the two tasks are altered? Can we predict the effect on the switch cost of having more than two tasks? Are the two components of the switch cost (delay time and accuracy rate) always related to each other or can they be adjusted independently? Answers to such questions will guide the design of future experiments to test our predictions.
Noisy heteroclinic networks
It is well known that stochastic noise plays a fundamental role in modifying the qualitative behaviour of dynamical systems, but the effects of noise on network attractors are often counter-intuitive and not well understood. Not suprisingly, noise affects the residence times and the transition probabilities of switching between the states in a heteroclinic network. However, a particularly surprising phenomena is the appearance of long-time correlations, or memory, in the sequence of transitions between states when microscopic noise is added to the system. The memory can be understood as a consequence of lift-off, which causes the distribution of trajectories near the network to be assymmetric. WIthin this project, we are trying to answer several questions regarding the properties of noise-induced memory in heteroclinic networks. Specifically: can lift-off affect residence times at states in a network? Are there limits to the length of time for which lift-off induced memory can remain? Can the sequence of states be modelled by an nth order Markov Chain? Is there a limit to the amount of information that can be encoded in the distribution of the coordinates of the trajectory? What are the properties of lift-off if the states in the network are more complicated objects than equilibria?