Dynamics and Synchronization of Weak Chimera States for a Coupled Oscillator System

This thesis is an investigation of chimera states in a network of identical coupled phase oscillators. Chimera states are intriguing phenomena that can occur in systems of coupled identical phase oscillators when synchronized and desynchronized oscillators coexist. We use the Kuramoto model and coupling function of Hansel for a specific system of six oscillators to prove the existence of chimera states. More precisely, we prove analytically there are chimera states in a small network of six phase oscillators previously investigated numerically by Ashwin and Burylko [8]. We can reduce to a two-dimensional system within an invariant subspace, in terms of phase differences. This system is found to have an integral of motion for a specific choice of parameters. Using this we prove there is a set of periodic orbits that is a weak chimera. Moreover, we are able to confirmthat there is an infinite number of chimera states at the special case of parameters, using the weak chimera definition of [8]. We approximate the Poincaré return map for these weak chimera solutions and demonstrate several results about their stability and bifurcation for nearby parameters. These agree with numerical path following of the solutions. We also consider another invariant subspace to reduce the Kuramoto model of six coupled phase oscillators to a first order differential equation. We analyse this equation numerically and find regions of attracting chimera states exist within this invariant subspace. By computing eigenvalues at a nonhyperbolic point for the system of phase differences, we numerically find there are chimera states in the invariant subspace that are attracting within full system.Republic of Iraq, Ministry of Higher Education and Scientific Research

Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators

A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.Comment: 26 pages, 3 figure

Dynamics of Patterns

Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction

Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media

We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with evolution of the local cell density depending on a coupled quasi--static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long--time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.Comment: 30 pages, 21 figure

Brain Dynamics From Mathematical Perspectives: A Study of Neural Patterning

The brain is the central hub regulating thought, memory, vision, and many other processes occurring within the body. Neural information transmission occurs through the firing of billions of connected neurons, giving rise to a rich variety of complex patterning. Mathematical models are used alongside direct experimental approaches in understanding the underlying mechanisms at play which drive neural activity, and ultimately, in understanding how the brain works. This thesis focuses on network and continuum models of neural activity, and computational methods used in understanding the rich patterning that arises due to the interplay between non-local coupling and local dynamics. It advances the understanding of patterning in both cortical and sub-cortical domains by utilising the neural field framework in the modelling and analysis of thalamic tissue – where cellular currents are important in shaping the tissue firing response through the post-inhibitory rebound phenomenon – and of cortical tissue. The rich variety of patterning exhibited by different neural field models is demonstrated through a mixture of direct numerical simulation, as well as via a numerical continuation approach and an analytical study of patterned states such as synchrony, spatially extended periodic orbits, bumps, and travelling waves. Linear instability theory about these patterns is developed and used to predict the points at which solutions destabilise and alternative emergent patterns arise. Models of thalamic tissue often exhibit lurching waves, where activity travels across the domain in a saltatory manner. Here, a direct mechanism, showing the birth of lurching waves at a Neimark-Sacker-type instability of the spatially synchronous periodic orbit, is presented. The construction and stability analyses carried out in this thesis employ techniques from non-smooth dynamical systems (such as saltation methods) to treat the Heaviside nature of models. This is often coupled with an Evans function approach to determine the linear stability of patterned states. With the ever-increasing complexity of neural models that are being studied, there is a need to develop ways of systematically studying the non-trivial patterns they exhibit. Computational continuation methods are developed, allowing for such a study of periodic solutions and their stability across different parameter regimes, through the use of Newton-Krylov solvers. These techniques are complementary to those outlined above. Using these methods, the relationship between the speed of synaptic transmission and the emergent properties of periodic and travelling periodic patterns such as standing waves and travelling breathers is studied. Many different dynamical systems models of physical phenomena are amenable to analysis using these general computational methods (as long as they have the property that they are sufficiently smooth), and as such, their domain of applicability extends beyond the realm of mathematical neuroscience

Light-sheet microscopy: a tutorial

This paper is intended to give a comprehensive review of light-sheet (LS) microscopy from an optics perspective. As such, emphasis is placed on the advantages that LS microscope configurations present, given the degree of freedom gained by uncoupling the excitation and detection arms. The new imaging properties are first highlighted in terms of optical parameters and how these have enabled several biomedical applications. Then, the basics are presented for understanding how a LS microscope works. This is followed by a presentation of a tutorial for LS microscope designs, each working at different resolutions and for different applications. Then, based on a numerical Fourier analysis and given the multiple possibilities for generating the LS in the microscope (using Gaussian, Bessel, and Airy beams in the linear and nonlinear regimes), a systematic comparison of their optical performance is presented. Finally, based on advances in optics and photonics, the novel optical implementations possible in a LS microscope are highlighted.Peer ReviewedPostprint (published version

Large-scale neural dynamics: Simple and complex

We review the use of neural field models for modelling the brain at the large scales necessary for interpreting EEG, fMRI, MEG and optical imaging data. Albeit a framework that is limited to coarse-grained or mean-field activity, neural field models provide a framework for unifying data from different imaging modalities. Starting with a description of neural mass models, we build to spatially extend cortical models of layered two-dimensional sheets with long range axonal connections mediating synaptic interactions. Reformulations of the fundamental non-local mathematical model in terms of more familiar local differential (brain wave) equations are described. Techniques for the analysis of such models, including how to determine the onset of spatio-temporal pattern forming instabilities, are reviewed. Extensions of the basic formalism to treat refractoriness, adaptive feedback and inhomogeneous connectivity are described along with open challenges for the development of multi-scale models that can integrate macroscopic models at large spatial scales with models at the microscopic scale. © 2010 Elsevier Inc