You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.
Title  Excitability in biology  the role of noisy thresholds 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Ruediger Thul, Prof Stephen Coombes 
Description  Excitability is ubiquitous in biology. Two important examples are the membrane potential of neurons or the dynamics of the intracellular calcium concentration. What characterises excitable systems is the presence of a threshold. For instance, neurons only fire when the membrane potential crosses a critical value. Importantly, the dynamics of excitable systems is often driven by fluctuations such as the opening of ion channels or the binding of hormones to a receptor. A mathematically and computationally appealing approach is to represent this biological noise by a random excitability threshold. This concept has already provided great insights into the dynamics of neurons that process sounds [1]. In this project, we will investigate the role of correlations of the noisy threshold in shaping cellular responses. Our applications will come from neuroscience in the form of single cell and neural field models as well as from cell signalling when we investigate travelling calcium waves. This will help us to understand the emergence of unusual firing patterns in the brain as well as of the wide variety of travelling calcium waves observed in numerous cell types. 
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Title  Spatiotemporal patterns with piecewiselinear regulatory networks 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Etienne Farcot 
Description  A number of fascinating and important biological processes involve 
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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 subcellular 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 spineheads [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 insilico 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. 
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Title  Rare event modelling for the progression of cancer 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Richard Graham, Prof Markus Owen 
Description  Purpose This project will apply cuttingedge mathematical modelling techniques to solve computational and modelling issues in predicting the evolution of cancerous tumours. The project will combine rare event modelling from the physical sciences and cellularlevel models from mathematical biology. The aim is to produce new cancer models with improved biological detail that can be solved on clinically relevant timescales, which can be decades.
Background A widespread problem in treating cancer is to distinguish indolent (benign) tumours from metastaticcapable primary tumours (tumours that can spread to other parts of the body). Although therapies for metastatic disease exist, metastatic disease is a significant cause of death in cancer patients. This problem can lead to misdiagnosis, unnecessary treatment and a lack of clarity on which treatments are most effective.
A predictive mathematical model of cancer development could assist with the above issues. However, as the progression of cancer to metastasis is a rare event, in a direct simulation, virtually all of the computational time is consumed in simulating the quasistable behaviour of the indolent tumour, revealing no information about progression. This generic problem of rare events is common in the physical sciences, where modern techniques have enabled rare events to be simulated and understood. This project will extend these techniques to cancer modelling. The project will build on a stateoftheart spatiotemporal cancer model, which models individual cancer cells in a host tissue, vascular networks and angiogenesis. In this model cells can divide, migrate or die, in response to their microenvironment of cell crowding and cell signalling. To this framework the project will add transitions between cell types, driven by random mutation events and intravasation events.
The project will use a rare event algorithm, forward flux sampling (FFS), to create a statistical map of the transition from indolent cancer to metastatic cancer. In a typical rare event transition the system spends the overwhelming majority of the time close to the start. Consequently, the sampling of the trajectory space is very uneven. Thus, despite a very long simulation the statistical resolution of the mechanism and crossing rate are very poor. FFS solves this problem by dividing the phase space into a series of interfaces that represent sequential advancement towards the rare event. The algorithm logs forward crossings of these interfaces and a series of trajectories are begun at these crossing points. This produces a far more even sampling of the trajectory space and so better statistics of the whole mechanism from a shorter simulation.

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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 integrateandfire model, which is described by a nonsmooth 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 nonGaussian noise. One of the main mathematical challenges will be to extend the MasterStability framework for networks of deterministic limit cycle oscillators to the noisy nonsmooth 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 subserved by distributed brain networks [3]. 
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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 wellsuited to mathematical study. Experience with the study of nonlinear dynamical systems would provide helpful background for such a project. 
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Title  Modelling DNA Chain Dynamics 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Jonathan Wattis 
Description  Whilst the dynamics of the DNA double helix are extremely complicated, a number of welldefined modes of vibration, such as twisting and bending, have been identified. At present the only accurate models of DNA dynamics involve largescale 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 largerscale dynamics, such as the unzipping of the double helix, to be studied. 
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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 bloodborne 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 tissuelevel 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. 
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Title  Selfsimilarity in a nanoscale islandgrowth 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof 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 selfsimilarity. 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 sizedistribution. The relationship of equations with other models of deposition, such as reactions on catalytic surfaces and polymer adsorption onto DNA, will also be explored. 
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Title  Sequential adsorption processes 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof 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 onedimension 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. 
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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 multidisciplinary efforts have been 
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Title  Neurocomputational models of hippocampusdependent place learning and navigation 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes 
Description  This project will be based at the University of Nottingham in the School of Mathematical Sciences and the School of Psychology. 
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Other information  Eligibility/Entry Requirements: We require an enthusiastic graduate with a 1st class degree in Mathematics (or other highly mathematical field such as Physics or Chemistry), preferably at MMath/MSc level, or an equivalent overseas degree (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). Apply: This studentship is available to start from September 2017 and remain open until it is filled. To apply please visit the University Of Nottingham application page: http://www.nottingham.ac.uk/pgstudy/apply/applyonline.aspx Funding NotesSummary: UK/EU students  Tuition Fees paid, and full Stipend at the RCUK rate, which is £14,296 per annum for 2016/17. There will also be some support available for you to claim for limited conference attendance. The scholarship length will be 3 or 3.5, depending on the qualifications and training needs of the successful applicant. 
Title  Spirals and autosoliton 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 spatiotemporal 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 integrodifferential 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 autosolitons, whilst strong scattering solutions will be investigated using the scattor theory that has previously been developed for multicomponent reaction diffusion systems [2]. 
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Other information  S Coombes, H Schmidt and I Bojak 2012 Interface dynamics in planar neural field models, Journal of Mathematical Neuroscience, 2:9 
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 When a sound wave reaches the head of a mosquito, the antenna oscillates under the Recent studies have shown that mosquitoes of either sex use both their antenna and Even though some models of mosquitoes hearing systems have been proposed in the past, 
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Title  Nonsmooth dynamical systems: from nodes to networks 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes, Dr Ruediger Thul 
Description  There is a growing appreciation in the applied mathematics community that many real world systems can be described by nonsmooth dynamical systems. This is especially true of impacting mechanical systems or systems with switches [1]. The latter are ubiquitous in fields ranging from electrical engineering to biology. In a neuroscience context nonsmooth models now pervade the field, with exemplars being low dimensional piecewise linear models of excitable tissue, integrateandfire neurons, and the Heaviside nonlinearity invoked in neural mass models of cortical populations. Despite the relevance and preponderance of such models their mathematical analysis lags behind that of their smooth counterparts. This PhD project will redress this balance, by translating recent advances from nonsmooth dynamical systems to neuroscience as well as developing new approaches. The initial phase of the project will consider the periodic forcing of a nonsmooth node, as a precursor to exploring recurrent network dynamics. The Arnol'd tongue structure will be explored for modelocked states of oscillatory systems, as well as bifurcation diagrams for excitable systems. This will rely heavily on the construction of socalled saltation operators, to ensure the proper propagation of perturbations. Similarly, chaos will be studied using a suitable generalisation of the Liapunov exponent. The subsequent work will address emergent network dynamics, particularly in neural systems with chemical and electrical connections. Explicit analysis at the network level will build upon results at the single node level, with a focus on understanding patterns of synchrony, clustering, and more exotic chimera states [2]. This aspect of the project will first pursue the extension of the Master Stability framework for assessing stability of the synchronous state to treat nonsmooth systems with nonsmooth interactions [3]. The next stage will develop more general techniques, tapping into tools from computational group theory [4], to provide a more complete understanding of the spatiotemporal states that can be generated in realistic neural networks. 
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Title  Pattern formation in biological neural networks with rebound currents 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes 
Description  Waves and patterns in the brain are well known to subserve natural computation. In the case of spatial navigation the geometric firing fields of grid cells is a classic example. Grid cells fire at the nodes of a hexagonal lattice tiling the environment. As an animal approaches the centre of a grid cell firing field, their spiking output increases in frequency. Interestingly the spacing of the hexagonal lattice can range from centimetres to metres and is thought to underly the brain's internal positioning system. The mechanism for controlling this global spatial scale is linked to a local property of neurons within an inhibitory coupled population, namely rebound firing. This arises through the activation of hyperpolarisationactivated channels. For the case of grid cells in the medial enthorinal cortex this gives rise to a socalled I_h current. Many other cells types also utilise rebound currents for firing, and in particular thalamocortical relay cells do so via slow Ttype calcium channels (the I_T current). This gives rise to saltatory lurching waves in thalamic slices. Both of these examples show that rebound currents can contribute significantly to important spatiotemporal brain dynamics. This project will investigate such important phenomenon from a mathematical perspective. One of the most successful approaches to modelling a spiking neuron involves using an integrateandfire process. This couples an ODE model with a reset rule for generating firing events. Almost by definition this precludes analysis using traditional approaches from the theory of smooth dynamical systems. This mathematical challenge is compounded at the network level when recognising that synaptic currents that mediate interactions between neurons are event driven rather than directly state dependent. Fortunately there is a growing appreciation that these mathematical biology challenges can benefit from a crossfertilisation of ideas with those being developed in the engineering community for impact oscillators and piecewise linear systems. This PhD will translate and develop mathematical methodologies from nonsmooth dynamical systems and apply them to two important neurobiological problems. The first being to analytically determine grid cell firing fields in a two dimensional spiking neural field model with an I_h rebound current, and the second to determine lurching wave speed and stability in a firing rate neural field model with an I_T rebound current. As well as mathematical techniques from nonsmooth dynamics, the project will involve large scale simulations of spiking networks, Evans functions for determining wave stability, and require an enthusiasm for learning about neuroscience. 
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Title  Mechanistic models of airway smooth muscle cells  application to asthma 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Bindi Brook 
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 forcegenerating capacity of ASM via a celllevel 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. 
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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. 
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Title  Multiscale modelling of cell signalling and mechanics in tissue development and cancer 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof John King, Dr Reuben O'Dea 
Description  Cells respond to their physical environment through mechanotransduction, the translation of mechanical forces into biochemical signals; evoked cell phenotypic changes can lead to an altered cell microenvironment, creating a developmental feedback. Interplay between such mechanosentive pathways and other inter and intracellular signalling mechanisms determines cell differentiation and, ultimately, tissue development. Such developmental mechanisms have key relevance to the initiation and development of cancer, a disease of such inherent complexity (involving the interaction of a variety of processes across disparate spatiotemporal scales, from intracellular signalling cascades to tissuelevel mechanics) that, despite a wealth of theoretical and experimental studies, it remains a leading cause of mortality and morbidity: in the UK, more than one in three people will develop some form of cancer. There is therefore an urgent need to gain greater quantitative understanding of these highly complex systems, which are wellsuited to mathematical study. 
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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 
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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 
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Title  Bottomup development of multiscale models of airway remodelling in asthma: from cell to tissue. 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Bindi Brook, Dr Reuben O'Dea 
Description  Airway remodelling in asthma has until recently been associated almost exclusively with inflammation over long timescales. Current experimental evidence suggests that bronchoconstriction (as a result of airway smooth muscle contraction) itself triggers activation of proremodelling growth factors that causes airway smooth muscle growth over much shorter timescales. This project will involve the coupling of subcellular mechanotransduction signalling pathways to biomechanical models of airway smooth muscle cells and extracellular matrix proteins with the aim of developing a tissuelevel biomechanical description of the resultant growth in airway smooth muscle. The mechanotransduction 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 bottomup and generate a tissue level description of biological growth will require the combination of agentbased 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 timescales present. While this study will be aimed specifically at airway remodelling, the methodology developed will have application in multiscale models of vascular remodelling and tissue growth in artificially engineered tissues. Initially models will be informed by data from ongoing 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. 
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Title  From molecular dynamics to intracellular calcium waves 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Ruediger Thul, Prof Stephen Coombes 
Description  Intracellular calcium waves are at the centre of a multitude of cellular processes. Examples include the generation of a heartbeat or the beginning of life when egg cells are fertilised. A key driver of intracellular calcium waves are ion channels, which are large molecules that control the passage of calcium ions across a cell. Importantly, these ion channels display stochastic behaviour such as random opening and closing. A key challenge in mathematical physiology and computational biology is to link this molecular stochasticity to travelling calcium waves. In this project, we will use a firediffusefire (FDF) model of intracellular calcium waves and couple it to Markov chains of ion channels. Traditionally, simulating large numbers of Markov chains is computationally expensive. Our goal is to derive an effective description for the stochastic ion channel dynamics. This will allow us to incorporate the molecular fluctuations from the ion channels into the FDF model without having to evolve Markov chains. This will put us in an ideal position to answer current questions in cardiac dynamics (How does an irregular heart beat emerge, leading to a potentially lifethreatening condition?) as well as to elucidate fundamental concepts in cell signalling.

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Title  Waves on a folded brain 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Daniele Avitabile, Prof Stephen Coombes 
Description  The human brain has a wonderfully folded cortex with regions of both negative and positive curvature at gyri and sulci respectively. As the state of the brain changes waves of electrical activity spread and scatter through this complicated surface geometry. This project will focus on the mathematical modelling of realistic cortical tissue and the analysis of wave propagation and scattering using techniques from dynamical systems theory and scientific computation. 
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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 microenvironment 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. 
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Title  Next generation neural field models on spherical domains 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Rachel Nicks 
Description  The number of neurons in the brain is immense (of the order of 100 billion). A popular approach to modelling such cortical systems is to use neural field models which are mathematically tractable and which capture the large scale dynamics of neural tissue without the need for detailed modelling of individual neurons. Neural field models have been used to interpret EEG and brain imaging data as well as to investigate phenomena such as hallucinogenic patterns, shortterm (working) memory and binocular rivalry. A typical formulation of a neural field equation is an integrodifferential equation for the evolution of the activity of populations of neurons within a given domain. Neural field models are nonlinear spatially extended pattern forming systems. That is, they can display dynamic behaviour including spatially and temporally periodic patterns beyond a Turing instability in addition to localised patterns of activity. The majority of research on neural field models has been restricted to the line or planar domains, however the cortical white matter system is topologically close to a sphere. It is relevant to study neural field models as pattern forming systems on spherical domains, particularly as the periodic boundary conditions allow for natural generation (via interference) of the standing waves observed in EEG signals. This project will build on recent developments in neural field theory, focusing in particular on extending to spherical geometry the neural field equations arising from “Next generation neural mass models” (which incorporate a description of the evolution of synchrony within the system). Techniques from dynamical systems theory, including linear stability analysis, weakly nonlinear analysis, symmetric bifurcation theory and numerical simulation will be used to consider the global and local patterns of activity that can arise in these models. 
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Title  Exploiting network symmetries for analysis of dynamics on neural networks 

Group(s)  Mathematical Medicine and Biology, Industrial and Applied Mathematics 
Proposer(s)  Dr Rachel Nicks, Prof Stephen Coombes, Dr Paul Matthews 
Description  Networks of interacting dynamical systems occur in a huge variety of applications including gene regulation networks, food webs, power networks and neural networks where the interacting units can be individual neurons or brain centres. The challenge is to understand how emergent network dynamics results from the interplay between local dynamics (the behaviour of each unit on its own), and the nature and structure of the interactions between the units. Recent work has revealed that real complex networks can exhibit a large number of symmetries. Network symmetries can be used to catalogue the possible patterns of synchrony which could be present in the network dynamics, however which of these exist and are stable depends on the local dynamics and the nature of the interactions between units. Additionally, the more symmetry a network has the more possible patterns of synchrony it may possess. Computational group theory can be used to automate the process of identifying the spatial symmetries of synchrony patterns resulting in a catalogue of possible network cluster states. This project will extend current methods for analysing dynamics on networks of (neural) oscillators through automating the process of determining possible phase relations between oscillators in large networks in addition to spatial symmetries. This will be used to investigate dynamics on coupled networks of simplified (phaseamplitude reduced or piecewiselinear) neuron and neural population models. We will also consider the effect on the network dynamics of introducing delays in the coupling between oscillators which will give a more realistic representation of interactions in real world networks. 
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Title  Analysing and interpreting neuroimaging data using mathematical frameworks for network dynamics 

Group(s)  Mathematical Medicine and Biology, Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes 
Description  Modern noninvasive probes of human brain activity, such as magnetoencephalography (MEG), give high temporal resolution and increasingly improved spatial resolution. With such a detailed picture of the workings of the brain, it becomes possible to use mathematical modelling to establish increasingly complete mechanistic theories of spatiotemporal neuroimaging signals. There is an ever expanding toolkit of mathematical techniques for addressing the dynamics of oscillatory neural networks allowing for the analysis of the interplay between local population dynamics and structural network connectivity in shaping emergent spatial functional connectivity patterns. This project will be primarily mathematical in nature, making use of notions from nonlinear dynamical systems and network theory, such as coupledoscillator theory and phaseamplitude network dynamics. Using experimental data and data from the output of dynamical systems on networks with appropriate connectivities, we will obtain insights on structural connectivity (the underlying network) versus functional connectivity (constructed from similarity of real time series or from timeseries output of oscillator models on networks). The project will focus in particular on developing techniques for the analysis of dynamics on “multilayer networks” to better understand functional connectivity within and between frequency bands of neural oscillations. This project will be in collaboration with Dr Matt Brookes from the Nottingham MEG group. 
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Title  Mathematical modelling of macromolecular capillary permeability 

Group(s)  Mathematical Medicine and Biology, Scientific Computation 
Proposer(s)  Dr Reuben O'Dea, Dr Matthew Hubbard 
Description  The primary function of blood vessels is to transport molecules to tissues. In diseases such as cancer and diabetes this transport, particularly of large molecules such as albumin, can be an order of magnitude higher than normal. The project is to model transient flow of macromolecules across the vascular wall in physiology and pathology. With additional supervision from Dr Kenton Arkill and Professor David Bates (Medicine), the doctoral student will join a team that includes medical researchers, biophysicists and mathematicians acquiring structural and functional data. Detailed microscale models of vascular wall hydrodynamics and transport properties will be employed; in addition, powerful multiscale homogenisation techniques will be exploited that enable permeability and convection parameters on the nanoscale to be linked through the microscale into translatable information on the tissue scale. Computational simulations will be used to investigate and understand the model behaviour, including, for example, stochastic and multiphysics effects in the complex diffusionconvection nanoscale environment. The project will afford a great opportunity to form an information triangle where modelling outcomes will determine physiological experiments to feedback to the model. Furthermore, the primary results will inform medical researchers on potential molecular therapeutic targets. 
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Title  Optimising experiments for developing ion channel models 

Group(s)  Mathematical Medicine and Biology, Statistics and Probability 
Proposer(s)  Dr Gary Mirams, Dr Simon Preston 
Description  Background: in biological systems ion channel proteins sit in cell membranes and selectively allow the passage of particular types of ions, creating currents. Ion currents are important for many biological processes, for instance: regulating ionic concentrations within cells; passing signals (such as nerve impulses); or coordinating contraction of muscle (skeletal muscle and also the heart, diaphragm, gut, uterus etc.). Mathematical ion channel electrophysiology models have been used for thousands of studies since their development by Hodgkin & Huxley in 1952 [1], and are the basis for whole research fields, such as cardiac modelling and brain modelling [2]. It has been suggested that there are problems in identifying which set of equations is most appropriate as an ion channel model. Often it appears different structures and/or parameter values could fit the training data equally well, but they may make different predictions in new situations [3]. Eligibility/Entry Requirements: this PhD will suit a graduate with a 1st class degree in Mathematics (or other highly mathematical field such as Physics), ideally at the MMath/MSc level, or an equivalent overseas degree. Prior knowledge of biology is not essential. 
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Other information  Please see Gary Mirams' research homepage for more information. 
Title  Multiscale models for growing tissues – simulation and analysis. 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Donald Brown 
Description  A fundamental barrier to advancing the understanding of biological tissue growth lies in its inherently multiscale nature: interactions between processes acting at disparate scales can profoundly influence the emergent dynamics. A unified description of such phenomena requires reconciling insight obtained by theoretical or experimental study at one scale with observations at another. For example, diseases that manifest at the organ scale often arise through the interaction of microscopic events at the cellular scale; moreover, the resulting macroscale changes influence the microscopic dynamics. [1] Effective equations governing an active poroelastic medium. J Collis, DL Brown, ME Hubbard, RD O'Dea Proceedings of the Royal Society A 473 (2198) 
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Title  Partitioneddomain concurrent multiscale modelling 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Kris van der Zee 
Description  Partitioneddomain concurrent multiscale modelling 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 multiscalemodeling type is known as partitioneddomain concurrent modelling. This type addresses problems that require a finescale 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 finescale model. Unfortunately, it is far from trivial to develop a working multiscale model for a particular problem. Challenges for students: Depending on the interest of the student, several of these issues (or others) can be addressed. 
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Title  Phasefield modelling of evolving interfaces 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Kris van der Zee 
Description  Phasefield modelling of evolving interfaces Evolving interfaces are ubiquitous in nature, think of the melting of the polar ice caps, the separation of oil and water, or the growth of cancerous tumours. Two mathematical descriptions exist to model evolving interfaces: those with sharpinterface descriptions, such as parametric and levelset methods, and those with diffuseinterface descriptions, commonly referred to as phasefield models. Challenges for students: Depending on the interest of the student, one of these issues (or others) can be addressed. 
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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 110 millimetres. The data is therefore highdimensional 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 intersubject variability in anatomical and functional brain structures is quite large. 
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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 longterm 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. 
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Title  Modelling hospital superbugs 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Prof Philip O'Neill, Dr Theodore Kypraios 
Description  The spread of socalled 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, the way in which different strains interact with each other, and the information contained in data on highresolution data (e.g. whole genome sequences). 
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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. 
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Title  StructuredPopulation 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 pseudolikelihood and Markov chain Monte Carlo methods). 
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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, Dr Theodore Kypraios 
Description  Dataanalysis for reallife 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 the latest methods in computational statistics (e.g. Markov Chain Monte Carlo (MCMC) methods, Approximate Bayesian Computation, Sequential Monte Carlo methods etc). 
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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, Dr Theodore Kypraios 
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 goodnessoffit of epidemic models to data, and methods for comparing different models. 
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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. 
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