A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock-Paper-Scissors

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models

Excitable networks for finite state computation with continuous time recurrent neural networks

This is the final version. Available on open access from Springer via the DOI in this recordContinuous time recurrent neural networks (CTRNN) are systems of coupled ordinary differential equations that are simple enough to be insightful for describing learning and computation, from both biological and machine learning viewpoints. We describe a direct constructive method of realising finite state input-dependent computations on an arbitrary directed graph. The constructed system has an excitable network attractor whose dynamics we illustrate with a number of examples. The resulting CTRNN has intermittent dynamics: trajectories spend long periods of time close to steady-state, with rapid transitions between states. Depending on parameters, transitions between states can either be excitable (inputs or noise needs to exceed a threshold to induce the transition), or spontaneous (transitions occur without input or noise). In the excitable case, we show the threshold for excitability can be made arbitrarily sensitive.Royal Society Te ApārangiEngineering and Physical Sciences Research Council (EPSRC)London Mathematical Laborator

How do parents experience being asked to enter a child in a randomised controlled trial?

<p>Abstract</p> <p>Background</p> <p>As the number of randomised controlled trials of medicines for children increases, it becomes progressively more important to understand the experiences of parents who are asked to enrol their child in a trial. This paper presents a narrative review of research evidence on parents' experiences of trial recruitment focussing on qualitative research, which allows them to articulate their views in their own words.</p> <p>Discussion</p> <p>Parents want to do their best for their children, and socially and legally their role is to care for and protect them yet the complexities of the medical and research context can challenge their fulfilment of this role. Parents are simultaneously responsible for their child and cherish this role yet they are dependent on others when their child becomes sick. They are keen to exercise responsibility for deciding to enter a child in a trial yet can be fearful of making the 'wrong' decision. They make judgements about the threat of the child's condition as well as the risks of the trial yet their interpretations often differ from those of medical and research experts. Individual pants will experience these and other complexities to a greater or lesser degree depending on their personal experiences and values, the medical situation of their child and the nature of the trial. Interactions at the time of trial recruitment offer scope for negotiating these complexities if practitioners have the flexibility to tailor discussions to the needs and situation of individual parents. In this way, parents may be helped to retain a sense that they have acted as good parents to their child whatever decision they make.</p> <p>Summary</p> <p>Discussing randomised controlled trials and gaining and providing informed consent is challenging. The unique position of parents in giving proxy consent for their child adds to this challenge. Recognition of the complexities parents face in making decisions about trials suggests lines for future research on the conduct of trials, and ultimately, may help improve the experience of trial recruitment for all parties.</p

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

Stability of heteroclinic cycles in ring graphs

Networks of interacting nodes connected by edges arise in almost every branch of scientific inquiry. The connectivity structure of the network can force the existence of invariant subspaces, which would not arise in generic dynamical systems. These invariant subspaces can result in the appearance of robust heteroclinic cycles, which would otherwise be structurally unstable. Typically, the dynamics near a stable heteroclinic cycle is non-ergodic: mean residence times near the fixed points in the cycle are undefined, and there is a persistent slowing down. In this paper, we examine ring graphs with nearest-neighbor or nearest-m-neighbor coupling and show that there exist classes of heteroclinic cycles in the phase space of the dynamics. We show that there is always at least one heteroclinic cycle that can be asymptotically stable, and, thus, the attracting dynamics of the network are expected to be non-ergodic. We conjecture that much of this behavior persists in less structured networks and as such, non-ergodic behavior is somehow typical

Excitable networks for finite state computation with continuous time recurrent neural networks

This is the author accepted manuscript. The final version is available from Springer via the DOI in this recordContinuous time recurrent neural networks (CTRNN) are systems of coupled ordinary differential equations that are simple enough to be insightful for describing learning and computation, from both biological and machine learning viewpoints. We describe a direct constructive method of realising finite state input-dependent computations on an arbitrary directed graph. The constructed system has an excitable network attractor whose dynamics we illustrate with a number of examples. The resulting CTRNN has intermittent dynamics: trajectories spend long periods of time close to steady-state, with rapid transitions between states. Depending on parameters, transitions between states can either be excitable (inputs or noise needs to exceed a threshold to induce the transition), or spontaneous (transitions occur without input or noise). In the excitable case, we show the threshold for excitability can be made arbitrarily sensitive.Royal Society Te ApārangiEngineering and Physical Sciences Research Council (EPSRC)London Mathematical Laborator

Numerical continuation of spiral waves in heteroclinic networks of cyclic dominance

Heteroclinic-induced spiral waves may arise in systems of partial differential equations that exhibit robust heteroclinic cycles between spatially uniform equilibria. Robust heteroclinic cycles arise naturally in systems with invariant subspaces and their robustness is considered with respect to perturbations that preserve these invariances. We make use of particular symmetries in the system to formulate a relatively low-dimensional spatial two-point boundary-value problem in Fourier space that can be solved efficiently in conjunction with numerical continuation. The standard numerical set-up is formulated on an annulus with small inner radius, and Neumann boundary conditions are used on both inner and outer radial boundaries. We derive and implement alternative boundary conditions that allow for continuing the inner radius to zero and so compute spiral waves on a full disk. As our primary example, we investigate the formation of heteroclinic-induced spiral waves in a reaction-diffusion model that describes the spatiotemporal evolution of three competing populations in a two-dimensional spatial domain--much like the Rock-Paper-Scissors game. We further illustrate the efficiency of our method with the computation of spiral waves in a larger network of cyclic dominance between five competing species, which describes the so-called Rock-Paper-Scissors-Lizard-Spock game

Spatiotemporal stability of periodic travelling waves in a heteroclinic-cycle model

We study a Rock-Paper-Scissors model for competing populations that exhibits travelling waves in one spatial dimension and spiral waves in two spatial dimensions. A characteristic feature of the model is the presence of a robust heteroclinic cycle that involves three saddle equilibria. The model also has travelling fronts that are heteroclinic connections between two equilibria in a moving frame of reference, but these fronts are unstable. However, we find that large-wavelength travelling waves can be stable in spite of being made up of three of these unstable travelling fronts. In this paper, we focus on determining the essential spectrum (and hence, stability) of large-wavelength travelling waves in a cyclic competition model with one spatial dimension. We compute the curve of transitions from stability to instability with the continuation scheme developed by Rademacher et al. (2007 Physica D 229 166-83). We build on this scheme and develop a method for computing what we call belts of instability, which are indicators of the growth rate of unstable travelling waves. Our results from the stability analysis are verified by direct simulation for travelling waves as well as associated spiral waves. We also show how the computed growth rates accurately quantify the instabilities of the travelling waves