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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 IntegrateandFire model of a neuron is a great example as is the FireDiffuseFire 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 

Other information  Links to Publications Sparks and waves in a stochastic firediffusefire model of Calcium release Understanding cardiac alternans: a piecewise linear modelling framework Neuronal spiketrain responses in the presence of threshold noise 
Title  The role of space in subcellular 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 subcellular 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 subcellular cardiac alternans. In contrast to earlier work, our approach does not require an adhoc 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 subcellular 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 

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  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  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)  Dr 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)  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 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)  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 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  Nuclear signalling in eukaryotes: modelling spatiotemporal 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 FireDiffuseFire (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 nuclearlocalised calcium and calmodulindependent 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 

Other information  Link to Publications Subcellular calcium dynamics in a whole cell model of an atrial myocyte 
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]. 
Relevant Publications 

Other information  S Coombes, H Schmidt and I Bojak 2012 Interface dynamics in planar neural field models, Journal of Mathematical Neuroscience, 2:9 
Title  STIMORAI 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 STIMORAI 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 STIMORAI 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 STIMORAI 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 STIMORAI 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. 
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Other information  Links to Publications 
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  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  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 cuttingedge 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 spatiotemporal 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. 
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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  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  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  Neuronal dynamics of perceptual inference 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes 
Description  Inference of the world around us is made by processing sensory signals in the brain and relating them to memories of previous experience. The study of this process has generated a number of candidate frameworks, with perhaps the most popular being ‘Bayesian cognition'. This powerful statistical description posits that the central nervous system of animals is capable of integrating prior probabilities with new sensory data in an optimal way to make perceptual decisions^{1}. How this process could be realised in dynamic circuits of neurons is as yet unclear^{2}. Additionally, while there are some spectacular experimental data on the capability of humans to accomplish such Bayesoptimal computations, strong evidence only comes from a limited set of experiments^{3}, other evidence often assuming Bayesian algorithms a priori^{4} and there is an influential literature demonstrating the failures of human information processing to incorporate prior probabilities^{5}. This project will probe the mechanisms involved in perceptual inference in a multidisciplinary way mixing techniques from cognitive psychology and neuroimaging with those from mathematical neuroscience.
1) Knill, D.C. & Pouget, A. The Bayesian brain: the role of uncertainty in neural coding and computation. Trends Neurosci 27, 712719, (2004). 2) Pouget, A., Beck, J.M., Ma, W. J. & Latham, P. E. Probabilistic brains: knowns and unknowns. Nat Neurosci 16, 11701178, (2013). 3) Ernst, M.O. & Banks, M.S. Humans integrate visual and haptic information in a statistically optimal fashion. Nature 415, 429433, doi:10.1038/415429a (2002). 4) Vossel, S., Bauer, M. et al. Cholinergic stimulation enhances bayesian belief updating in the deployment of spatial attention. J Neurosci 34, 1573515742, (2014). 5) Kahneman, D. & Tversky, A. On the study of statistical intuitions. Cognition 11, 123141 (1982). 6) Coombes, S. Largescale neural dynamics: simple and complex. Neuroimage 52, 731739, (2010). 7) Wenzlaff, H., Bauer, M., Maess, B. & Heekeren, H.R. Neural characterization of the speedaccuracy tradeoff in a perceptual decisionmaking task. J Neurosci 31, 12541266, (2011). 8) Coombes, S., Thul, R. & Wedgwood, K.C.A. Nonsmooth dynamics in spiking neuron models. Physica D 241, 20422057, (2012). 9) Bastos, A.M. et al. Canonical microcircuits for predictive coding. Neuron 76, 695711, (2012). 10) Friston, K.J. et al. LFP and oscillationswhat do they tell us? Curr Opin Neurobiol 31, 16, (2015). 11) Bauer, M., et al.. Attentional modulation of alpha/beta and gamma oscillations reflect functionally distinct processes. J Neurosci 34, 1611716125, (2014). 12) Adams, R.A., et al. The computational anatomy of psychosis 
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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 (socalled actionpotentials), 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 conductancebased single neurons models, and also the simpler integrateandfire 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 spiketrain 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 oftstudied 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 spiketrain correlations from a mathematical model it will benefit enormously from locally available multielectrode array data collected from both invitro and invivo neuronal ensembles. 
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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 timedependent (typically periodic) input [i]. This Phd project will develop a natural phaseamplitude coordinate system [ii] for describing reduced networks of synaptically interacting neurons. Network states, including phaselocking, 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. 
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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 longterm neuroplasticity. For a recent review see [1]. In theory, physiological changes at the singleunit, multiunit, 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 insilico 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 painrelated 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 multidisciplinary 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 bloodoxygenleveldependent signals associated with experimentally induced pain include the thalamus, somatosensory cortex, anterior cingulate cortex, prefrontal cortex, insula, and the cerebellum, forming a socalled 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 subpopulation, the dynamics of synaptic currents, and plasticity of interconnections (and of course disturbances in each, mimicking various forms of sensitisation, channelopathies, subcircuit overactivation, etc.). The development of an in silicomodel will also allow the design of restorative stimulation protocols, such as via deep brain stimulation or patientcontrolled realtime 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). 
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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 [24]. 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 threedimensional 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. 
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Title  The mathematics of mechanoperception 

Group(s)  Mathematical Medicine and Biology, Scientific Computation 
Proposer(s)  Prof Stephen Coombes, Dr Daniele Avitabile 
Description  Perception is, in general, an active process. Animals use purposive control of their sense organs (e.g., eyes, fingers, whiskers) to obtain sensory information to guide current behavioural goals (‘active sensation’). Rodents actively move use whiskers to sense the world around them and, since these whisker movements can be precisely imaged in an awake, behaving animal, it is an ideal system in which to investigate active sensing. Stereotypical patterns of 'whisking' allow the animal to transduce mechanical forces arising upon impact of their whiskers with objects, textures and surfaces, into patterns of spiking activity that their brain uses to form a representation of the real world [1]. However, surprisingly little is known about how the mechanics of the whisker and its dynamics upon impact can lead to a behaviourally useful tactile sensation. 
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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  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 
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, and the way in which different strains interact with each other. 
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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 
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 Markov Chain Monte Carlo (MCMC) techniques. 
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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 goodnessoffit of epidemic models to data. 
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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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