You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.

Title Stochastic Threshold Models: from single nodes to networks
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Ruediger Thul, Prof Stephen Coombes
Description

The behaviour of excitable systems can often be captured with a simpler threshold description. The Integrate-and-Fire model of a neuron is a great example as is the Fire-Diffuse-Fire model of calcium wave propagation in cardiac cells. The project will use tools from dynamical systems theory (bifurcation theory, nonsmooth vector fields, scientific computation) and stochastic processes (Markov chains, Wiener processes) to analyse models with threshold noise. Beginning with studies of single units the project will build up to understand the collective behaviour of interacting threshold units, with applications in neuroscience (rhythm generation, neural computation) and cardiac dynamics (wave propagation, coherent oscillations and arrhythmias).

Relevant Publications
  • S Coombes and Y Timofeeva (2003) Sparks and waves in a stochastic fire-diffuse-fire model of Calcium release, Physical Review E, 68, 021915
  • R Thul and S Coombes (2010) Understanding cardiac alternans: a piecewise linear modelling framework, Chaos, 20, 045102
  • S Coombes, R Thul, J Laudanski, A R Palmer and C J Sumner (2012) Neuronal spike-train responses in the presence of threshold noise, Frontiers in Life Science, to appear
Other information

Links to Publications

Sparks and waves in a stochastic fire-diffuse-fire model of Calcium release

Understanding cardiac alternans: a piecewise linear modelling framework

Neuronal spike-train responses in the presence of threshold noise

Title The role of space in sub-cellular cardiac alternans
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Ruediger Thul, Prof Stephen Coombes
Description

Cardiac arrhythmia are the leading cause of the death in the UK, killing more people each year than breast cancer, lung cancer and AIDS combined. Among the different kinds of cardiac arrhythmia, atrial fibrillation is the most common one. Here, the smaller chambers of the heart beat so fast that they cannot pump blood anymore, shifting all the blood propelling work of the heart to the larger chambers. This situation is especially problematic under conditions when blood needs to be pumped more quickly (e.g. during exercise) and among the elderly when the heart generally becomes weaker.

One of the precursors of atrial fibrillation are cardiac alternans. Here the heart still exhibits a regular rhythm, but with alternating strength. Only every second heart beat is strong enough to sufficiently contract the heart. A particular form of this arrhythmia are sub-cellular alternans where different parts of a single cell oscillate out of phase. Such a cell does not contract at all, and a group of such cells significantly impairs cardiac contractility. To better understand the emergence of atrial fibrillation and to design treatments, it is vital to gain a comprehensive picture of cardiac alternans.

In this project, we will use a recently developed three dimensional model of an atrial myocyte [2] to investigate the emergence of sub-cellular cardiac alternans. In contrast to earlier work, our approach does not require an ad-hoc compartmentalisation of the cell, but we can work with a realistic representation of the cellular morphology. In turn, this will allow us to better characterise the interaction between different sub-cellular processes that shape cardiac alternans. The challenge is to develop the analysis of alternans for a spatially extended cell model that complements numerical simulations and allows us to predict the onset of alternans more efficiently. Keeping in mind that any drug treatment acts first at the single cell level, our approach will help to identify potential targets for pharmaceutical intervention in cardiac therapy.

Relevant Publications
  • [1] R Thul and S Coombes 2010 Understanding cardiac alternans: a piecewise linear modelling framework, Chaos, Vol 20, 045102
  • [2] R Thul, S Coombes, H L Roderick M D Bootman 2012 Subcellular calcium dynamics in a whole cell model of an atrial myocyte, Proceedings of the National Academy of Sciences, 109, 2150-2155
Other information

Links to Publications

Understanding cardiac alternans: a piecewise linear modelling framework

Subcellular calcium dynamics in a whole cell model of an atrial myocyte

Title Spatio-temporal patterns with piecewise-linear regulatory networks
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Etienne Farcot
Description

A number of fascinating and important biological processes involve
various kinds of spatial patterns: spatial patterns on animal skins, or
the very regular organ arrangements found in plants (called phyllotaxis)
for instance. These patterns often originate at very small scales, and
their onset can only be seen using very recent microscope and image
analysis techniques.
Among several families of models for biological patterning, one of the
simplest is based on the idea that mobile substances (called morphogens)
are acting upstream of their targets, which respond locally to a globally
defined  gradient pattern.
In this project one will consider models where targets are themselves
mobile morphogens, potentially regulating their own input. One will
study the effect of such spatial feedback on patterning. To do so, one will rely
on a class of models which are biologically relevant, tractable analytically,
and not much studied yet in a context with spatial interactions. A class of
models which meet all this criteria is provided by piecewise-linear differential
equations.

Relevant Publications
  • L.G. Morelli et al. Computational Approaches to Developmental Patterning. Science 336, 187 (2012).
  • L. Glass, S.A. Kauffman. The logical analysis of continuous, nonlinear biochemical control networks. Journal of Theoretical Biology 39, 103-129 (1973).
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Title Spine morphogenesis and plasticity
Group(s) Mathematical Medicine and Biology
Proposer(s) Prof Stephen Coombes, Dr Ruediger Thul
Description

Mathematical Neuroscience is increasingly being recognised as a powerful tool to complement neurobiology to understand aspects of the human central nervous system.  The research activity in our group is concerned with developing a sound mathematical description of sub-cellular processes in synapses and dendritic trees.  In particular we are interested in models of dendritic spines [1], which are typically the synaptic contact point for excitatory synapses.  Previous work in our group has focused on voltage dynamics of spine-heads [2].  We are now keen to broaden the scope of this work to include developmental models for spine growth and maintenance, as well as models for synaptic plasticity [3].  Aberrations in spine morphology and density are well known to underly certain brain disorders, including Fragile X syndrome (which can lead to attention deficit and developmental delay) and depression [4].  Computational modelling is an ideal method to do in-silico studies of drug treatments for brain disorders, by modelling their action on spine development and plasticity.  This is an important complementary tool for drug discovery in an area which is struggling to make headway with classical experimental pharmaceutical tools.

The mathematical tools relevant for this project will be drawn from dynamical systems theory, biophysical modelling, statistical physics, and scientific computation.

Relevant Publications
  • [1] Rafael Yuste, 2010, Dendritic spines, MIT Press
  • [2] Y Timofeeva, G J Lord and S Coombes 2006 Spatio-temporal filtering properties of a dendritic cable with active spines, Journal of Computational Neuroscience, Vol 21, 293-306
  • [3] Cian O'Donnell, Matthew F. Nolan, and Mark C. W. van Rossum, 2012, Dendritic Spine Dynamics Regulate the Long-Term Stability of Synaptic Plasticity, The Journal of Neuroscience, 9 November 2011, 31(45):16142-16156
  • [4] R M Henig, 2012, Lifting the black cloud, Scientific American, Mar, p 60-65
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Title Stochastic Neural Network Modelling
Group(s) Mathematical Medicine and Biology
Proposer(s) Prof Stephen Coombes, Dr Ruediger Thul
Description

Large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue. One approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models. An alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies. In this latter case the single neuron model of choice is often a variant of the classic integrate-and-fire model, which is described by a non-smooth dynamical system with a threshold [1]. It has recently been shown [2] that one way to model the variability of neuronal firing is to introduce noise at the threshold level. This project will develop the analysis of networks of synaptically coupled noisy neurons. Importantly it will go beyond standard phase oscillator approaches to treat strong coupling and non-Gaussian noise. One of the main mathematical challenges will be to extend the Master-Stability framework for networks of deterministic limit cycle oscillators to the noisy non-smooth case that is relevant to neural modelling. This work will determine the effect of network dynamics and topology on synchronisation, with potential application to psychiatric and neurological disorders. These are increasingly being understood as disruptions of optimal integration of mental processes sub-served by distributed brain networks [3].

Relevant Publications
  • [1] S Coombes, R Thul and K C A Wedgwood 2012 Nonsmooth dynamics in spiking neuron models, Physica D, DOI: 10.1016/j.physd.2011.05.012
  • [2] S Coombes, R Thul, J Laudanski, A R Palmer and C J Sumner 2011 Neuronal spike-train responses in the presence of threshold noise, Frontiers in Life Science, DOI: 10.1080/21553769.2011.556016
  • [3] J Hlinka and S Coombes 2012 Using computational models to relate structural and functional brain connectivity, European Journal of Neuroscience, to appear
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Title Cell signalling
Group(s) Mathematical Medicine and Biology
Proposer(s) Prof John King
Description

Cell signalling effects have crucial roles to play in a vast range of biological processes, such as in controlling the virulence of bacterial infections or in determining the efficacy of treatments of many diseases. Moreover, they operate over a wide range of scales, from subcellular (e.g. in determining how a particular drug affects a specific type of cell) to organ or population (such as through the quorum sensing systems by which many bacteria determine whether or not to become virulent). There is therefore an urgent need to gain greater quantitative understanding of these highly complex systems, which are well-suited to mathematical study. Experience with the study of nonlinear dynamical systems would provide helpful background for such a project.

Relevant Publications
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Title Modelling DNA Chain Dynamics
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Jonathan Wattis
Description

Whilst the dynamics of the DNA double helix are extremely complicated, a number of well-defined modes of vibration, such as twisting and bending, have been identified. At present the only accurate models of DNA dynamics involve large-scale simulations of molecular dynamics. Such approaches suffer two major drawbacks: they are only able to simulate short strands of DNA and only for extremely short periods (nanoseconds). the aim of this project is to develop simpler models that describe vibrations of the DNA double helix. The resulting systems of equations will be used to simulate the dynamics of longer chains of DNA over long timescales and, hence, allow larger-scale dynamics, such as the unzipping of the double helix, to be studied.

Relevant Publications
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Title Multiscale modelling of vascularised tissue
Group(s) Mathematical Medicine and Biology
Proposer(s) Prof Markus Owen
Description

Most human tissues are perfused by an evolving network of blood vessels which supply nutrients to (and remove waste products from) the cells. The growth of this network (via vasculogenesis and angiogenesis) is crucial for normal embryonic and postnatal development, and its maintenance is essential throughout our lives (e.g. wound healing requires the repair of damaged vessels). However, abnormal remodelling of the vasculature is associated with several pathological conditions including diabetic retinopathy, rheumatoid arthritis and tumour growth.

The phenomena underlying tissue vascularisation operate over a wide range of time and length scales. These features include blood flow in the existing vascular network, transport within the tissue of blood-borne nutrients, cell division and death, and the expression by cells of growth factors such as VEGF, a potent angiogenic factor. We have developed a multiscale model framework for studying such systems, based on a hybrid cellular automaton which couples cellular and subcellular dynamics with tissue-level features such as blood flow and the transport of growth factors. This project will extend and specialise our existing model to focus on particular applications in one of the following areas: wound healing, retinal angiogenesis, placental development, and corpus luteum growth. This work would require a significant element of modelling, numerical simulation and computer programming.

Relevant Publications
Other information
Title Self-similarity in a nanoscale island-growth
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Jonathan Wattis
Description

Molecular Beam Epitaxy is a process by which single atoms are slowly deposited on a surface. These atoms diffuse around the surface until they collide with a cluster or another atom and become part of a cluster. Clusters remain stationary. The distribution of cluster sizes can be measured, and is observed to exhibit self-similarity. Various systems of equations have been proposed to explain the scaling behaviour observed. The purpose of this project is to analyse the systems of differential equations to verify the scalings laws observed and predict the shape of the size-distribution. The relationship of equations with other models of deposition, such as reactions on catalytic surfaces and polymer adsorption onto DNA, will also be explored.

Relevant Publications
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Title Sequential adsorption processes
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Jonathan Wattis
Description

The random deposition of particles onto a surface is a process which arises in many subject areas, and determining its efficiency in terms of the coverage attained is a difficult problem.

In one-dimension the problem can be viewed as how many cars can be parked along a road of a certain length; this problem is similar to a problem in administering gene therapy in which polymers need to be designed to package and deliver DNA into cells.

Here one wishes to know the coverage obtained when one uses a variety of polymer lengths to bind to strands of DNA.

The project will involve the solution of recurrence relations, and differential equations, by a mixture of asymptotic techniques and stochastic simulations.

Relevant Publications
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Title Robustness of biochemical network dynamics with respect to mathematical representation
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Etienne Farcot
Description

In the recent years, a lot of multi-disciplinary efforts have been
devoted to improving our understanding of the dynamics of interactions
between the many types of molecules present in biological cells. This
has led to a widespread viewpoint where networks of genes, proteins and
other biochemical species are considered at once, as complex dynamical
systems from which the global state of cells emerge.
Several mathematical formalisms are used to represent these systems,
from discrete or boolean models to differential equations. One striking
fact, especially regarding models of developmental processes, is that a
number of relevant properties of these networks can be captured
similarly by all these formalisms, like for instance the property of
bistability.
One possible interpretation of this independence with respect to
formalism is that biological regulatory systems are most often extremely
robust.
The project will start by developing parallel models – using different
formalisms – of actual biological networks whose behaviour is known.
Elaborating on these examples the theoretical and practical implications
of this notion of robustness will be explored.

Relevant Publications
Other information
Title Nuclear signalling in eukaryotes: modelling spatio-temporal patterns of intracellular calcium
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Ruediger Thul, Prof Stephen Coombes
Description

Calcium is critically important for a large number of cellular functions, such as muscle contraction, cardiac electrophysiology, secretion, synaptic plasticity, and adaptation in photoreceptors [1].  Mechanisms by which a cell controls its Calcium concentration are of central interest in cell physiology.  The recent use of Calcium specific fluorescent reporter dyes and digital videomicroscopy has begun to reveal the complexity of Calcium dynamics in spatially extended cellular systems.  Calcium signalling in a wide diversity of cell types frequently occurs as Calcium oscillations.  These do not generally occur uniformly throughout the cell but are initiated at a specific site and spread in the form of waves.  The fluorescent imaging of localised elementary Calcium release events has now made it clear that Calcium release is a stochastic process that occurs at spatially discrete sites that are clusters of receptors in the endoplasmic or sarcoplasmic reticulum.  Mathematical modelling is an ideal tool for capturing the details of how intracellular Calcium waves spread throughout a cell and subserve physiological and pathological signals, especially in light of current resolution limitations of imaging technologies.  In particular the stochastic Fire-Diffuse-Fire (FDF) model [2] is an ideal starting point for the development of a computationally economical framework to allow fast simulations of realistic cell geometries with large numbers of release sites.  This project will focus on developing cell models to track waves that allow targeted release of calcium in the nuclear region of eukaryotes.  Nuclear oscillations in calcium are especially important as they can drive downstream responses for gene expression.  The model will be extended to include calcium decoders (such as a nuclear-localised calcium and calmodulin-dependent protein kinase) and developed to model the symbiotic signalling pathway of legumes during root nodulation [3].

This project will mainly draw from the toolbox of computational cell biology (including GPU programming in Python) to address important open problems in cellular calcium signalling in plants.

Relevant Publications
  • [1] M J Berridge 1997 Elementary and global aspects of calcium signalling, Journal of Neurophysiology 499, 291-306.
  • [2] R Thul, S Coombes, H L Roderick M D Bootman 2012 Subcellular calcium dynamics in a whole cell model of an atrial myocyte, Proceedings of the National Academy of Sciences, 109, 2150-2155.
  • [3] W Capoen et al. 2011 Nuclear membranes control symbiotic calcium signaling of legumes, Proceedings of the National Academy of Sciences 108, 14348-14353.
Other information

Link to Publications

Subcellular calcium dynamics in a whole cell model of an atrial myocyte

Title Spirals and auto-soliton scattering: interface analysis in a neural field model
Group(s) Mathematical Medicine and Biology
Proposer(s) Prof Stephen Coombes, Dr Daniele Avitabile
Description

Neural field models describe the coarse grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in 2D, where they are well known to generate rich patterns of spatio-temporal activity. Typical patterns include localised solutions in the form of travelling spots as well as spiral waves [1]. These patterns are naturally defined by the interface between low and high states of neural activity. This project will derive the dimensionally reduced equations of motion for such interfaces from the full nonlinear integro-differential equation defining the neural field.  Numerical codes for the evolution of the interface will be developed, and embedded in a continuation framework for performing a systematic bifurcation analysis.  Weakly nonlinear theory will be developed to understand the scattering of multiple spots that behave as auto-solitons, whilst strong scattering solutions will be investigated using the scattor theory that has previously been developed for multi-component reaction diffusion systems [2].

Relevant Publications
  • [1] C R Laing 2005 Spiral waves in nonlocal equations, SIAM J. Appl. Dyn. Sys., Vol 4, 588-606.
  • [2] Y Nishiura, T Teramoto, and K-I Ueda. Scattering of traveling spots in dissipative systems. Chaos, 15:047509(1–10), 2005.
Other information

S Coombes, H Schmidt and I Bojak 2012 Interface dynamics in planar neural field models, Journal of Mathematical Neuroscience, 2:9

Title STIM-ORAI dependent Calcium oscillations
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Ruediger Thul, Prof Stephen Coombes
Description

Calcium oscillations have long been recognised as a main pathway with which cells translate external stimuli into intracellular responses such as enzyme secretion, neurotransmitter production or cell contraction [1]. Over the last years, it has emerged that calcium oscillations often do not occur uniformly across a cell, but that either different parts of a cell oscillate out of phase with respect to each other or that cellular oscillations actually correspond to traveling calcium waves. The importance of space in shaping intracellular calcium oscillations has recently been highlighted by the discovery of the STIM-ORAI machinery [2]. Here, translocation of intracellular molecules (STIM) to designated areas close to the cell membrane (ORAI) are responsible for initiating and maintaining calcium oscillations.

A large body of experimental data convincingly suggests that most of the information of intracellular calcium oscillations is encoded in their frequency and sometimes in their amplitude. The STIM-ORAI system now shows that only if the cellular calcium oscillations occur through STIM and ORAI certain genes are activated. Intracellular calcium oscillations that look identical but involves different molecular partners fail to initiate a genetic response [3].

In this project, we will develop a spatially extended model for STIM-ORAI induced calcium oscillations that will explain the still unknown mechanisms behind the long periods of calcium oscillations. Introducing the pathway that is responsible for gene activation, we will study the signalling cascade that links calcium oscillations to gene expression with a special emphasis on the emergence of calcium microdomains and exchange mechanisms between the cell cytoplasm and nucleus. Given the importance of STIM-ORAI dependent oscillations in cells of the immune system, our work has direct implications to strengthen human health.

The project will involve model design based on published experimental data and mathematical techniques for partial differential equations, delayed differential equations and stochastic processes.

Relevant Publications
  • [1] Rüdiger Thul, Tomas C. Bellamy, H. Llewelyn Roderick, Martin D. Bootman, and Stephen Coombes. 2008. “Calcium Oscillations” Advances in Experimental Medicine and Biology 641, 1–27
  • [2] Richard S Lewis. 2007. “The Molecular Choreography of a Store-Operated Calcium Channel” Nature 446, 284–287.
  • [3] Pulak Kar, Charmaine Nelson, and Anant B Parekh. 2012. “CRAC Channels Drive Digital Activation and Provide Analog Control and Synergy to Ca(2+)-Dependent Gene Regulation” Current Biology 22, 242–247
Other information

Links to Publications

Calcium Oscillations

Title Modelling signal processing and sexual recognition in mosquitoes: neural computations in insect hearing systems
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Daniele Avitabile, Prof Stephen Coombes
Description

Insects have evolved diverse and delicate morphological structures in order to
capture the inherently low energy of a propagating sound wave. In mosquitoes, the
capture of acoustic energy, and its transduction into neuronal signals, is assisted
by the active mechanical participation of actuators called scolopidia.

When a sound wave reaches the head of a mosquito, the antenna oscillates under the
action of the external pressure field (passive component) and of the force provided by
the mechanical actuators (active component). The latter is particularly
relevant for sexual recognition: when a male mosquito hear the flyby of a female, his
antennal oscillation are greatly amplified by the scolopidia. In other words, the
antenna of a male is tuned very sharply around the frequency and intensity of a
female flyby.

Recent studies have shown that mosquitoes of either sex use both their antenna and
their wing beat to select a partner: understanding how their hearing system works
could help us controlling the population of species that carry viral diseases.

Even though some models of mosquitoes hearing systems have been proposed in the past,
a number of key questions remain unanswered. Where do the mechanical actuators get
their energy? How do they twitch? How is the mechanical motion of the antenna
transformed into an electric signal? Do neurones control the mechanical motion? How
does the brain of a mosquito process the neural information and distinguish various
sources of sound? Is the sexual recognition entirely based on sound perception, or is
it also influenced by olfactory signals? Is the antenna sensitive to sounds from
different directions?

Relevant Publications
  • AVITABILE, D, HOMER, M, CHAMPNEYS, AR, JACKSON, JC and ROBERT, D, 2010. Mathematical Modelling Of The Active Hearing Process In Mosquitoes Journal Of The Royal Society Interface. 7(42), 105-122
  • CHAMPNEYS, AR, AVITABILE, D, HOMER, M and SZALAI, R, 2011. The Mechanics Of Hearing: A Comparative Case Study In Bio-Mathematical Modelling Anziam Journal. 52(3), 225-249
Other information
Title Mechanistic models of airway smooth muscle cells - application to asthma
Group(s) Mathematical Medicine and Biology
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Description

Lung inflammation and airway hyperresponsiveness (AHR) are hallmarks of asthma, but their interrelationship is unclear. Excessive shortening of airway smooth muscle (ASM) in response to bronchoconstrictors is likely an important determinant of AHR. Hypercontractility of ASM could stem from a change in the intrinsic properties of the muscle, or it could be due to extrinsic factors such as chronic exposure of the muscle to inflammatory mediators in the airways with the latter being a possible link between lung inflammation and AHR. The aim of this project will be to investigate the influence of chronic exposure to a contractile agonist on the force-generating capacity of ASM via a cell-level model of an ASM cell. Previous experimental studies have suggested that the muscle adapts to basal tone in response to application of agonist and is able to regain its contractile ability in response to a second stimulus  over time. This is thought to be due to a transformation in the  cytoskeletal components of the cell enabling it to bear force, thus freeing up subcellular contractile machinery to generate more force. Force adaptation in ASM as a consequence of prolonged exposure to the many spasmogens found in asthmatic airways could be a mechanism contributing to AHR seen in asthma. We will develop and use a cell model in an attempt to either confirm this hypothesis or determine other mechanisms that may give rise to the observed phenomenon of force adaptation.

Relevant Publications
  • Adaptation of airway smooth muscle to basal tone relevance to airway hyperresponsiveness, Bosse et al, American Journal of Respiratory Cell Molecular Biology,Vol 40. pp 13–18, 2009
  • The role of contractile unit reorganization in force generation in airway smooth muscle, B S Brook and O E Jensen, Mathematical Medicine and Biology, 2013. doi:10.1093/imammb/dqs031
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Title Computational Cell Biology
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Ruediger Thul, Prof Stephen Coombes
Description

Computational Cell Biology (CSB) uses techniques from nonlinear dynamical systems, partial differential equations and stochastic processes to gain a deeper insight into the reliability and robustness of cellular signalling cascades. By combining analytical and numerical approaches, CSB plays a major role in the discovery and quantitative descriptions of key biological processes. Mathematical models that are tailored to specific biological questions can yield answers that are still out of reach for cutting-edge experimental approaches.

In the current project we will explore how the spatial arrangement of the molecular machinery affects cellular signal transduction. A key feature of cells is to translate external stimuli into cellular responses. Cells in the pancreas produce insulin when extracellular markers indicate high blood sugar levels, neurons in the brain release chemical messengers to their neighbours upon electrical stimulation, and heart cells contract more effectively when they experience a rush of adrenaline. Cells use advanced molecular machinery to trigger the appropriate reaction for a given stimulus. In recent years, convincing evidence has emerged that cells employ spatio-temporal patterns to achieve this task. For instance, complex oscillations and travelling waves of intracellular calcium have been observed, where the frequency of the oscillations and the spatial spread of the waves tightly correlate with the external stimulation.

Relevant Publications
  • C.P. Fall, E.S. Marland, J.N. Wagner, J.J. Tyson, Computational Cell Biology, Springer, 2002
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Title Synchronisation and propagation in human cortical networks
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Reuben O'Dea
Description

Around 25% of the 50million epilepsy sufferers worldwide are not responsive to antiepileptic medication; improved understanding of this disorder has the potential to improve diagnosis, treatment and patient outcomes. The idea of modelling the brain as a complex network is now well established. However, the emergence of pathological brain states via the interaction of large interconnected neuronal populations remains poorly understood. Current theoretical study of epileptic seizures is flawed by dynamical simulation on inadequate network models, and by the absence of customised network measures that capture pathological connectivity patterns.

This project aims to address these deficiencies via improved computational models with which to investigate thoroughly the influence of the geometry and connectivity of the human brain on epileptic seizure progression and initiation, and the development of novel network measures with which to characterise epileptic brains. Such investigations will be informed by exhaustive patient datasets (such as recordings of neural activity in epilepsy patients and age-matched controls), and will be used to study (i) improved diagnostic strategies, (ii) the influence of treatment strategies on seizure progression and initiation, and (iii) the identification of key sites of epilepsy initiation.

Relevant Publications
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Title Multiscale analysis of growing deformable tissues
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Reuben O'Dea Notice: Undefined index: pmzbsb in /maths/www/html/postgraduate/projects/index.php on line 479 Notice: Undefined index: pmzbsb in /maths/www/html/postgraduate/projects/index.php on line 479 ,
Description

Biological tissue is distinguished from materials described historically by continuum mechanical theory by its ability to grow and remodel adaptively, regulated by complex processes occurring within autonomous discrete cells. Current continuum approaches are wholly unable to represent comprehensively this detail and, in general, employ ad hoc modelling strategies, applied at the macroscale, with little or no rigorous consideration of the underlying dynamics; effective descriptions obtained via multiscale analyses have not addressed adequately the combination of (i) discrete cell behaviour (including population expansion), and (ii) micromechanics (the latter consideration incorporating the influence of relevant tissue microstructure and mechanical properties on macroscale tissue dynamics).

This project will seek to address this deficiency, by developing improved mathematical representations of a wide class of deformable, adaptively remodelling materials, accommodating the influence both of microscale (cell-scale) and mesoscale (tissue structure and mechanics) effects on corresponding macroscale (tissue-scale) formulations, exploiting a combination of multiscale asymptotic homogenisation techniques, mathematical modelling and detailed numerical simulation. Collaboration with experimental experts at the University of Nottingham and Keele University will inform and validate the models developed.

Relevant Publications
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Title Patterns of synchrony in discrete models of gene networks
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Etienne Farcot
Description

One of the greatest challenges of biology is to decipher the relation between genotype and
phenotype. One core difficulty in this task is that this relation is not a map; the proteins which
are produced thanks to the information contained in the genome are themselves used to
control which parts of the genome are being used in a given situation. To understand the effect
of this feedback between genes and their product it is crucial to consider the dynamics of this
process.
The term 'gene network' refers to a set of genes which regulate each other; understanding
the dynamics of gene networks is thus crucial to decipher the genotype-phenotype relation. Mathematical
models of gene networks have been proposed since the 1960's, among which the class of Boolean
models has proved very successful. Because of the discrete nature of these models, the effect of time
is often described using representations inspired by manufactured computing device, where the
genes are updated in parallel, or in series. However, the updating scheme of genes could in principle
be much more general. In this project, one will investigate the effect of such a generalization. One
will consider arbitrary update schemes, both deterministic and stochastic, notably in relation to
the dynamics of continuous models of gene networks.

Relevant Publications
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Title Cell cycle desynchronization in growing tissues
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Etienne Farcot
Description

A very general phenomenon is the fact that coupled oscillators tend to naturally
synchronize [1]. This simple fact takes many forms observable in real life:
synchronization of applause after a concert, of neural cells, of flashing fireflies, and
many other. A complete understanding of this phenomenon, depending on the particular
dynamics of individual oscillators or the nature of their coupling, is still an on-going
topic of mathematical research.
However general, the synchronization of coupled oscillators is not a universal rule. In this
project, one will study a situation where it indeed does not seem to occur: the division cycles
of cells in a growing tissue does not seem to be synchronized, as observed in recent
data form plant tissues. A plausible explanation is that the divisions of cell induce a change in
the coupling between cells, which is mostly due to physical or chemical exchanges between
neighbouring cells. Relying on simplified representations, one will consider the effect of growth,
whereby the coupling structure of a system changes in time, on synchronization in populations
of oscillators.

Relevant Publications
  • [1] S.H. Strogatz and I. Stewart. Coupled oscillators and biological synchronization. Scientific American 269 (6) 102-109 (1993).
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Title Bottom-up development of multi-scale models of airway remodelling in asthma: from cell to tissue.
Group(s) Mathematical Medicine and Biology
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Description

Airway remodelling in asthma has until recently been associated almost exclusively with inflammation over long time-scales. Current experimental evidence suggests that broncho-constriction (as a result of airway smooth muscle contraction) itself triggers activation of pro-remodelling growth factors that causes airway smooth muscle growth over much shorter time-scales. This project will involve the coupling of sub-cellular mechano-transduction signalling pathways to biomechanical models of airway smooth muscle cells and extra-cellular matrix proteins with the aim of developing a tissue-level biomechanical description of the resultant growth in airway smooth muscle. 

The mechano-transduction pathways and biomechanics of airway smooth muscle contraction are extremely complex. The cytoskeleton and contractile machinery within the cell and ECM proteins surrounding it are thought to rearrange dynamically (order of seconds). The cell is thought to adapt its length (over 10s of seconds). To account for all these processes from the bottom-up and generate a tissue level description of biological growth will require the combination of agent-based models to biomechanical models governed by PDEs. The challenge will be to come up with suitably reduced models with elegant mathematical descriptions that are still able to reproduce observed experimental data on cell and tissue scales, as well as the different time-scales present.

While this study will be aimed specifically at airway remodelling, the methodology developed will have application in multi-scale models of vascular remodelling and tissue growth in artificially engineered tissues. Initially models will be informed by data from on-going experiments in Dr Amanda Tatler's lab in Respiratory Medicine but there will also be the opportunity to design new experiments based on model results.

Relevant Publications
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Title Modelling macrophage extravasation and phenotype selection
Group(s) Mathematical Medicine and Biology
Proposer(s) Prof Markus Owen
Description

Macrophages are a type of white blood cell, a vital component of the immune system, and play a complex role in tumour growth and other diseases. Macrophage precursors, called monocytes, are produced in the bone marrow and enter the blood, before leaving the bloodstream (extravasating). Monocyte extravasation requires adhesion to, and active movement through, the blood vessel wall, both of which are highly regulated processes. Once in the tissue, monocytes begin to differentiate into macrophages, and it has become clear that the tissue micro-environment is a crucial determinant of macrophage function [1]. A spectrum of phenotypes have been identified: at one end, macrophages produce a variety of signals that are beneficial to a tumour, including those that promote the formation of new blood vessels and suppress inflammation. At the other end of the scale, inflammation is promoted and appropriately stimulated macrophages can kill tumour cells.

This project will consider in some detail the mechanisms that regulate monocyte extravasation and macrophage phenotype selection. Initially, mathematical models will be formulated as systems of ordinary differential equations describing transitions between monocyte subpopulations (for example, those fully adherent to the vessel wall, and those that are actively moving through the wall), regulated by various signalling and adhesion molecules. Research on phenotype selection will determine whether the dynamics can be manipulated by subsequent external intervention. For example, if the system is bistable, it may be possible to force a switch from a deleterious to a beneficial phenotype. Relevant signal transduction pathways will be modelled in detail, and the law of mass-action will be used to derive systems of ODEs. Where possible, model reductions based on a separation of timescales will be used to simplify the system, and analytical and numerical approaches will be used to characterise steady state structure and bifurcations as tissue conditions vary.

Relevant Publications
  • A Mantovani, S Sozzani, M Locati, P Allavena, and A Sica: Macrophage polarization: tumor-associated macrophages as a paradigm for polarized M2 mononuclear phagocytes. Trends Immunol., 23:549--555, (2002).
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Title On the dynamics of the Lighthouse model for spiking neural networks
Group(s) Mathematical Medicine and Biology, Industrial and Applied Mathematics
Proposer(s) Prof Stephen Coombes
Description

One of the holy grails of the theoretical neuroscience community is to develop a tractable model of neural tissue. This must necessarily involve a single cell model, capable of generating spikes of activity (so-called action-potentials), that when connected into a synaptic network can generate the rich repertoire of behaviour seen in a real nervous system. For all of the popular conductance-based single neurons models, and also the simpler integrate-and-fire variety, the understanding of network dynamics has proved elusive. In essence this is because we have not yet developed an appropriate mathematical framework to understand the neurodynamics of spiking networks. To date progress in this area has been restricted to firing rate neural models, which cannot adequately capture known spike-train correlations. Interestingly, the recently proposed Lighthouse model of Hermann Haken is a candidate single neuron model that may allow a bridge to be built between spiking neuron models and firing rate descriptions. Indeed in the limit of slow synaptic interactions it may be shown to reduce to the oft-studied Amari firing rate model. Importantly the Lighthouse model is sufficiently simple that it may also be analysed at the network level, even for fast synaptic responses. Hence, a comprehensive study of a network of synaptically coupled Lighthouse neurons may pave the way for the development of a specific exactly soluble neurodynamics. This may also shed light on how best to develop a more general approach valid for more detailed models of coupled spiking neurons. This project will pursue the study of the Lighthouse network using techniques from dynamical systems theory and statistical physics, building upon emerging techniques and principles from the physics of complex systems. As it will closely focus on the generation of realistic spike-train correlations from a mathematical model it will benefit enormously from locally available multi-electrode array data collected from both in-vitro and in-vivo neuronal ensembles.

Relevant Publications
  • C.C. Chow & S. Coombes 2006 Existence and wandering of bumps in a spiking neural network model, SIAM Journal on Applied Dynamical Systems, Vol 5, 552-574
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Title Shear induced chaos in neuronal networks
Group(s) Mathematical Medicine and Biology, Mathematical Medicine and Biology
Proposer(s) Prof Stephen Coombes
Description

Shear induced chaos has recently been shown to be an important mechanism for determing the response of conductance based models of single neurons to time-dependent (typically periodic) input [i].  This Phd project will develop a natural phase-amplitude coordinate system [ii] for describing reduced networks of synaptically interacting neurons.  Network states, including phase-locking, synchrony, heteroclinic cycles, and routes to chaos, will be analysed using techniques from dynamical systems theory (both analytical and numerical) to understand fundamental aspects of information processing within the central nervous system including network reliability in the presence of shear.

Relevant Publications
Other information
  1. K K Lin, K C A Wedgwood, S Coombes and L-S Young 2013 Limitations of perturbative techniques in the analysis of rhythms and oscillations, Journal of Mathematical Biology, Vol 66, 139-161
  2. K C A Wedgwood, K K Lin, R Thul and S Coombes 2013 Phase-amplitude descriptions of neural oscillator models, Journal of Mathematical Neuroscience, 3:2.
Title Pain matrices and their analysis: a combined neuroimaging, statistical and modelling analysis
Group(s) Mathematical Medicine and Biology, Mathematical Medicine and Biology
Proposer(s) Prof Stephen Coombes, Dr Theodore Kypraios
Description

Scientific background

There is increasing evidence to suggest that chronic pain is a disease that can alter brain function. In particular neuroimaging studies have demonstrated structural remapping and functional reorganisation of brain circuits under various pain conditions. In parallel, preclinical models have demonstrated that chronic pain causes long-term neuroplasticity. For a recent review see [1].

In theory, physiological changes at the single-unit, multi-unit, and circuitry levels can be used as predictors of pain, and ultimately to guide targeted neuromodulation of specific brain regions for therapeutic purposes. The Pain Imaging group at Nottingham is developing circuit level imaging biomarkers (using MRI techniques) to track such physiological changes. The complementary statistical techniques for prediction (and identification of brain states associated with pain) and computational modelling that would allow in-silico design of pain therapies are skill sets that exist within the School of Mathematical Sciences. Thus Nottingham is well positioned to develop multidisciplinary research into the mechanisms of pain-related phenomena in the brain that can offer insights into novel approaches for the diagnosis, monitoring, and management of persistent pain.

Aims and objectives

In light of recent breakthroughs in the statistical analysis of brain network signals [2] and computational models of interacting neuronal populations [3], as well as locally available data sets from the Pain Imaging group, our aim is to equip a PhD student with multi-disciplinary skills for understanding how humans experience pain.  The objective is for them to develop a novel systems perspective of pain as a complex multidimensional experience that can be understood with the modern tools of applied mathematics and statistics.   

Although activation patterns may vary, the regions most consistently reported to have increased blood-oxygen-level-dependent signals associated with experimentally induced pain include the thalamus, somatosensory cortex, anterior cingulate cortex, prefrontal cortex, insula, and the cerebellum, forming a so-called pain matrix.  We will develop network models of this system of interacting neural populations building on recent work in [3].  This will allow us to explore the mechanisms for the emergence of functional connectivity associated with normal activation of the ‘pain matrix’, and dysfunctional connectivity associated with the experience of chronic pain. The transition between the two states will be studied, with a particular focus on the dependence of the functional connectivity patterns on the dynamics of a sub-population, the dynamics of synaptic currents, and plasticity of interconnections (and of course disturbances in each, mimicking various forms of sensitisation, channelopathies, sub-circuit over-activation, etc.). The development of an in silicomodel will also allow the design of restorative stimulation protocols, such as via deep brain stimulation or patient-controlled real-time feedback, to alleviate pain. The mathematical challenge will be to understand how a dysfunctional `pain matrix' state induced within the model environment can be coaxed back to a normal activation pattern.

Statistical methods will be developed to decode neuroimaging signals and predict a sensory pain experience on the basis of spatially correlated fMRI voxels. Exponential random graph models (ERGMs) will allow us to gain deeper insights into the complex neurobiological interactions and changes that occur in many disorders. Although ERGMs have been extensively utilised in social science to analyse highly complex networks, it is only until recently that they have been successfully used to study brain networks using resting fMRI data showing some very promising results [2].

Training

The student will do a laboratory rotation in the Pain Imaging group, to appreciate the data sets that are available to work with. Training on Neuroimaging data acquisition and analysis will be provided by participation at the MSc Translatianal Neuroimaging (Course director: D Auer)

The student will learn about advanced techniques in Computational Neuroscience by attending the course G14TNS Theoretical Neuroscience (School of Mathematical Sciences). The student will also learn about advanced statistical computational techniques such as Markov Chain Monte Carlo (MCMC) by attending the course G14CST and courses from the Academy for PhD Training in Statistics (APTS).

Relevant Publications
  • [1] C Y Saab 2012 Pain-related changes in the brain: diagnostic and therapeutic potentials, Trends in Neurosciences, Vol 35, 629-637
  • [2] S Sala, T Kypraios and F Massimo 2012 Exponential Random Graph Models (ERGMs) for Resting State fMRI Connectivity. In preparation.
  • [3] J Hlinka and S Coombes 2012 Using computational models to relate structural and functional brain connectivity, European Journal of Neuroscience, Vol 36, 2137–2145
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Title Nonlinear dynamics in the heart - the role of atrial myocytes
Group(s) Mathematical Medicine and Biology, Scientific Computation
Proposer(s) Dr Ruediger Thul
Description

A human heart beats more than a billion times during the average lifespan, and is required to do so with great fidelity. The ventricular (larger) chambers of the heart are responsible for generating the force that propels blood to the lungs and body [1]. Under sedentary conditions, the atrial (smaller) chambers make only a minor contribution to blood pumping. However, during periods of increased hemodynamic demand, such as exercise, atrial contraction increases to enhance the amount of blood within the ventricles before they contract. This `atrial kick' is believed to account for up to 30% extra blood pumping capacity. Deterioration of atrial myocytes, i.e. muscle cells, with ageing causes the loss of this blood pumping reserve, thereby increasing frailty in the elderly. Atrial kick is also lost during atrial fibrillation (AF), the most common form of cardiac arrhythmia. The stagnation of blood within the atrial chambers during AF can cause thrombus formation, leading to thromboembolism. Approximately 15% of all strokes occur in people with AF. As shown in numerous reports, the genesis and maintenance of AF is causally linked to the dysregulation of calcium signalling, which is bidirectionally coupled to the membrane potential of the cell [2-4].

In this project, we will investigate how changes in the membrane potential lead to changes in the intracellular calcium concentration, which in turn feeds back to the temporal evolution of the membrane potential. We will employ a recently developed three-dimensional model of an atrial myocyte with a biologically realistic distribution of calcium release sites. Through detailed numerical simulations we will achieve a better understanding of how the specific morphology of atrial myocytes impacts on the membrane driven generation of calcium transients, and how clinically relevant pathologies like early after depolarisation and delayed after depolarisation are shaped through the interaction of localised calcium transients near the plasma membrane and the membrane potential itself.

Relevant Publications
  • D. M. Bers, Nature 415, 198–205 (2002)
  • M. D. Bootman, I. Smyrnias, R. Thul, S. Coombes, and H. L. Roderick, Biochim. Biophys. Acta 1813, 922–934 (2011)
  • J. T. Koivumäki, T. Korhonen, and P. Tavi, PLoS Comput. Biol. 7, e1001067– (2011)
  • E. Grandi, S. V. Pandit, N. Voigt, A. J. Workman, D. Dobrev, J. Jalife, and D. M. Bers, Circ. Res. 109, 1055–1066 (2011)
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Title Partitioned-domain concurrent multiscale modelling
Group(s) Scientific Computation, Mathematical Medicine and Biology
Proposer(s) Dr Kris van der Zee
Description

Partitioned-domain concurrent multiscale modelling
(Or- How does one get cheap, but accurate, models?)

Multiscale modeling is an active area of research in all scientific disciplines. The main aim is to address problems involving phenomena at disparate length and/or time scales that span several orders of magnitude! An important multiscale-modeling type is known as partitioned-domain concurrent modelling. This type addresses problems that require a fine-scale model in only a small part of the domain, while a coarse model is employed in the remainder of the domain. By doing this, significant computational savings are obtained compared to a full fine-scale model. Unfortunately, it is far from trivial to develop a working multiscale model for a particular problem.

Challenges for students:
* How can one couple, e.g., discrete (particle) systems with continuum (PDE) models?
* Or a fine-scale PDE with a coarse-scale PDE?
* How can one decide on the size and location of the fine-scale domain?
* Is it possible to proof the numerically observed efficiency of concurrent multiscale models? 
* Can the multiscale methodology be applied to biological growth phenomena (e.g., tumours) where one couples cell-based (agent-based) models with continuum PDE models?

Depending on the interest of the student, several of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group!

Relevant Publications
  • J.T. Oden, S. Prudhomme, A. Romkes, and P.T. Bauman, Multiscale modeling of physical phenomena: Adaptive control of models, SIAM J. Sci. Comput. 28 (2006), pp. 2359-2389
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Title Statistical analysis of neuroimaging data
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Dr Christopher Brignell
Description

The activity of neurons within the brain can be detected by function magnetic resonance imaging (fMRI) and magnetoencephalography (MEG).   The techniques record observations up to 1000 times a second on a 3D grid of points separated by 1-10 millimetres.  The data is therefore high-dimensional and highly correlated in space and time.  The challenge is to infer the location, direction and strength of significant underlying brain activity amongst confounding effects from movement and background noise levels.  Further, we need to identify neural activity that are statistically significant across individuals which is problematic because the number of subjects tested in neuroimaging studies is typically quite small and the inter-subject variability in anatomical and functional brain structures is quite large.

Relevant Publications
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Title Identifying fibrosis in lung images
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Dr Christopher Brignell
Description

Many forms of lung disease are characterised by excess fibrous tissue developing in the lungs.  Fibrosis is currently diagnosed by human inspection of CT scans of the affected lung regions.  This project will develop statistical techniques for objectively assessing the presence and extent of lung fibrosis, with the aim of identifying key factors which determine long-term prognosis.  The project will involve developing statistical models of lung shape, to perform object recognition, and lung texture, to classify healthy and abnormal tissue.  Clinical support and data for this project will be provided by the School of Community Health Sciences.

Relevant Publications
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Title Modelling hospital superbugs
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Philip O'Neill
Description

The spread of so-called superbugs such as MRSA within healthcare settings provides one of the major challenges to patient welfare within the UK. However, many basic questions regarding the transmission and control of such pathogens remain unanswered. This project involves stochastic modelling and data analysis using highly detailed data sets from studies carried out in hospital, addressing issues such as the effectiveness of patient isolation, the impact of different antibiotics, and the way in which different strains interact with each other.

Relevant Publications
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Title Modelling of Emerging Diseases
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Frank Ball
Description

When new infections emerge in populations (e.g. SARS; new strains of influenza), no vaccine is available and other control measures must be adopted. This project is concerned with addressing questions of interest in this context, e.g. What are the most effective control measures? How can they be assessed? The project involves the development and analysis of new classes of stochastic models, including intervention models, appropriate for the early stages of an emerging disease.

Relevant Publications
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Title Structured-Population Epidemic Models
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Frank Ball
Description

The structure of the underlying population usually has a considerable impact on the spread of the disease in question. In recent years the Nottingham group has given particular attention to this issue by developing, analysing and using various models appropriate for certain kinds of diseases. For example, considerable progress has been made in the understanding of epidemics that are propogated among populations made up of households, in which individuals are typcially more likely to pass on a disease to those in their household than those elsewhere. Other examples of structured populations include those with spatial features (e.g. farm animals placed in pens; school children in classrooms; trees planted in certain configurations), and those with random social structure (e.g. using random graphs to describe an individual's contacts). Projects in this area are concerned with novel advances in the area, including developing and analysing appropriate new models, and methods for statistical inference (e.g. using pseudo-likelihood and Markov chain Monte Carlo methods).

Relevant Publications
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Title Bayesian Inference for Complex Epidemic Models
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Philip O'Neill
Description

Data-analysis for real-life epidemics offers many challenges; one of the key issues is that infectious disease data are usually only partially observed. For example, although numbers of cases of a disease may be available, the actual pattern of spread between individuals is rarely known. This project is concerned with the development and application of methods for dealing with these problems, and involves using Markov Chain Monte Carlo (MCMC) techniques.

Relevant Publications
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Title Bayesian model choice assessment for epidemic models
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Philip O'Neill
Description

During the last decade there has been a significant progress in the area of parameter estimation for stochastic epidemic models. However, far less attention has been given to the issue of model adequacy and assessment, i.e. the question of how well a model fits the data. This project is concerned with the development of methods to assess the goodness-of-fit of epidemic models to data.

Relevant Publications
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Title Epidemics on random networks
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Frank Ball
Description

There has been considerable interest recently in models for epidemics on networks describing social contacts.  In these models one first constructs an undirected random graph, which gives the network of possible contacts, and then spreads a stochastic epidemic on that network.  Topics of interest include: modelling clustering and degree correlation in the network and analysing their effect on disease dynamics; development and analysis of vaccination strategies, including contact tracing; and the effect of also allowing for casual contacts, i.e. between individuals unconnected in the network.  Projects in this area will address some or all of these issues.

Relevant Publications
  • Ball F G and Neal P J (2008) Network epidemic models with two levels of mixing. Math Biosci 212, 69-87.
  • Ball F G, Sirl D and Trapman P (2009) Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv Appl Prob 41, 765-796.
  • Ball F G, Sirl D and Trapman P (2010) Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math Biosci 224, 53-73.
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