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January 2013 - July 2016
Publications
Publications (23)
Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations (DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equations with limit cycle oscillations, there is now a growing interest in the development of analogous approaches for DDEs to understand their response to e...
Networks of coupled nonlinear oscillators can display a wide range of emergent behaviors under the variation of the strength of the coupling. Network equations for pairs of coupled oscillators where the dynamics of each node is described by the evolution of its phase and slowest decaying isostable coordinate have previously been shown to capture bi...
There is enormous interest -- both mathematically and in diverse applications -- in understanding the dynamics of coupled oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of...
Since its inception four decades ago the two-process model introduced by Borbély has provided the conceptual framework to explain sleep wake regulation across many species, including humans. At its core, high level notions of circadian and homeostatic processes are modelled with a low dimensional description in the form of a one dimensional nonauto...
Neural mass models have been used since the 1970s to model the coarse-grained activity of large populations of neurons. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature, and cannot hope to recreate some of the rich repertoire of re...
Neural mass models have been actively used since the 1970s to model the coarse-grained activity of large populations of neurons. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature, and cannot hope to recreate some of the rich reperto...
We introduce an integral model of a two-dimensional neural field that includes a third dimension representing space along a dendritic tree that can incorporate realistic patterns of axodendritic connectivity. For natural choices of this connectivity we show how to construct an equivalent brain-wave partial differential equation that allows for effi...
For coupled oscillator networks with Laplacian coupling the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory this approach has recently been extended to treat more general cluster states. However, the MSF and its generalisations require the de...
For coupled oscillator networks with Laplacian coupling the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory this approach has recently been extended to treat more general cluster states. However, the MSF and its generalisations require the de...
The Nunez model for the generation of electroencephalogram (EEG) signals is naturally described as a neural field model on a sphere with space-dependent delays. For simplicity, dynamical realisations of this model either as a damped wave equation or an integro-differential equation, have typically been studied in idealised one dimensional or planar...
The tools of weakly coupled phase oscillator theory have had a profound
impact on the neuroscience community, providing insight into a variety of
network behaviours ranging from central pattern generation to synchronisation,
as well as predicting novel network states such as chimeras. However, there are
many instances when this theory is expected t...
The Lie group structure of crystals which have uniform continuous distributions of dislocations allows one to construct associated discrete structures—these are discrete subgroups of the corresponding Lie group, just as the perfect lattices of crystallography are discrete subgroups of
$\mathbb{R}^{3}$
, with addition as group operation. We consid...
Crystals which have a uniform distribution of defects are endowed with a Lie
group description which allows one to construct an associated discrete
structure. These structures are in fact the discrete subgroups of the ambient
Lie group. The geometrical symmetries of these structures can be computed in
terms of the changes of generators of the discr...
We outline geometrical and algebraic ideas which appear to lie at the foundation of the theory of defective crystals that was introduced by C. Davini [Arch. Ration. Mech. Anal. 96, 295–317 (1986; Zbl 0623.73002)]. The focus of the paper is on the connection between continuous and discrete models of such crystals, approached by the consideration of...
We find the geometrical symmetries of discrete structures which generalize the perfect lattices of crystallography so as to account for the existence of continuous distributions of defects. Copyright © 2012 John Wiley & Sons, Ltd.
We consider the symmetry of discrete and continuous crystal structures which are compatible with a given choice of dislocation
density tensor. By introducing the notion of a ‘defective point group’ (determined by the dislocation density tensor), we
generalize the notion of Ericksen–Pitteri neighborhoods to this context.
KeywordsCrystals–Defects–Li...
We consider distributions of dislocations in continuum models of crystals which are such that the corresponding dislocation density tensor relates to a particular class of solvable Lie group, and discrete structures which are embedded in these crystals. We provide a canonical form of these structures and, by finding the set of all generators of a c...
Spiral patterns on the surface of a sphere have been seen in laboratory experiments and in numerical simulations of reaction-diffusion equations and convection. We classify the possible symmetries of spirals on spheres, which are quite different from the planar case since spirals typically have tips at opposite points on the sphere. We concentrate...
Using the general theory of Hopf bifurcation with symmetry we study here the example where the group of symmetries is O(3), the rotations and reflections of a sphere. We make some amendments to previously published lists of C-axial isotropy subgroups of O(3) × S1 and list the isotropy subgroups with four-dimensional fixed-point subspaces. We then s...
Bifurcations from spherically symmetric states can occur in many physical and biological systems. These include the development of a spherical ball of cells into an asymmetrical state and the buckling of a sphere under pressure. They also occur in the evolution of reaction–diffusion systems on a spherical surface and in Rayleigh–Benard convection i...
