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.

Relevant Publications
  • Coombes, S, R Thul, J Laudanski, A R Palmer, and C J Sumner. 2011. “Neuronal Spike-Train Responses in the Presence of Threshold Noise.” Frontiers in Life Science 5: 91–105.
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Title Spatio-temporal patterns with piecewise-linear regulatory networks
Group(s) Mathematical Medicine and Biology
Proposer(s) Dr Etienne Farcot
Description

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

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

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

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

Relevant Publications
  • [1] Rafael Yuste, 2010, Dendritic spines, MIT Press
  • [2] Y Timofeeva, G J Lord and S Coombes 2006 Spatio-temporal filtering properties of a dendritic cable with active spines, Journal of Computational Neuroscience, Vol 21, 293-306
  • [3] Cian O'Donnell, Matthew F. Nolan, and Mark C. W. van Rossum, 2012, Dendritic Spine Dynamics Regulate the Long-Term Stability of Synaptic Plasticity, The Journal of Neuroscience, 9 November 2011, 31(45):16142-16156
  • [4] R M Henig, 2012, Lifting the black cloud, Scientific American, Mar, p 60-65
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Title 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 cutting-edge 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 cellular-level 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 wide-spread problem in treating cancer is to distinguish indolent (benign) tumours from metastatic-capable 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 quasi-stable 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 state-of-the-art 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.

 

Relevant Publications
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Title Stochastic Neural Network Modelling
Group(s) Mathematical Medicine and Biology
Proposer(s) Prof Stephen Coombes, Dr Ruediger Thul
Description

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Relevant Publications
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Title Neurocomputational models of hippocampus-dependent 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.  

Humans and other animals can readily remember significant places and associated events and return to these places as appropriate. From an experimental point of view, studies of the neuro-psychological mechanisms underlying place learning and navigation offer unique opportunities, because similar tests can be used in rodent models and human participants. Studies in rodent models have led to a detailed understanding of the neuro-psychological mechanisms of place memory, and the importance of the hippocampus for place learning and navigation in humans and other animals is well-established. In this project, we aim to develop quantitative models describing how neurons in the hippocampus and associated brain areas give rise to place learning and navigation, and construct an in silico model for testing ideas about functional mechanisms. The project brings together behavioural neuroscience expertise on hippocampal function and place learning (Bast, Psychology) with expertise in mathematical and computational neuroscience (Coombes, Mathematical Sciences) to understand rapid place learning. A particular emphasis will be on the hippocampal learning-behaviour translation: how place information (as encoded, for example, by hippocampal place cells) is related to decision making processes and, ultimately, translated into motor behaviour (for example, by way of interactions with prefrontal and subcortical circuits). From a mathematical perspective the project will develop new neurocomputational models of hippocampus-dependent place learning and navigation using tools from stochastic optimal control, reinforcement learning theory, dynamical systems and computational neuroscience.

Relevant Publications
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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/apply-online.aspx 

Funding Notes

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

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

Relevant Publications
  • [1] C R Laing 2005 Spiral waves in nonlocal equations, SIAM J. Appl. Dyn. Sys., Vol 4, 588-606.
  • [2] Y Nishiura, T Teramoto, and K-I Ueda. Scattering of traveling spots in dissipative systems. Chaos, 15:047509(1–10), 2005.
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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
capture the inherently low energy of a propagating sound wave. In mosquitoes, the
capture of acoustic energy, and its transduction into neuronal signals, is assisted
by the active mechanical participation of actuators called scolopidia.

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

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

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

Relevant Publications
  • AVITABILE, D, HOMER, M, CHAMPNEYS, AR, JACKSON, JC and ROBERT, D, 2010. Mathematical Modelling Of The Active Hearing Process In Mosquitoes Journal Of The Royal Society Interface. 7(42), 105-122
  • CHAMPNEYS, AR, AVITABILE, D, HOMER, M and SZALAI, R, 2011. The Mechanics Of Hearing: A Comparative Case Study In Bio-Mathematical Modelling Anziam Journal. 52(3), 225-249
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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 force-generating capacity of ASM via a cell-level model of an ASM cell. Previous experimental studies have suggested that the muscle adapts to basal tone in response to application of agonist and is able to regain its contractile ability in response to a second stimulus  over time. This is thought to be due to a transformation in the  cytoskeletal components of the cell enabling it to bear force, thus freeing up subcellular contractile machinery to generate more force. Force adaptation in ASM as a consequence of prolonged exposure to the many spasmogens found in asthmatic airways could be a mechanism contributing to AHR seen in asthma. We will develop and use a cell model in an attempt to either confirm this hypothesis or determine other mechanisms that may give rise to the observed phenomenon of force adaptation.

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

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

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

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

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

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

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

Relevant Publications
  • [1] S.H. Strogatz and I. Stewart. Coupled oscillators and biological synchronization. Scientific American 269 (6) 102-109 (1993).
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Title Bottom-up development of multi-scale models of airway remodelling in asthma: from cell to tissue.
Group(s) Mathematical Medicine and Biology
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 time-scales. Current experimental evidence suggests that broncho-constriction (as a result of airway smooth muscle contraction) itself triggers activation of pro-remodelling growth factors that causes airway smooth muscle growth over much shorter time-scales. This project will involve the coupling of sub-cellular mechano-transduction signalling pathways to biomechanical models of airway smooth muscle cells and extra-cellular matrix proteins with the aim of developing a tissue-level biomechanical description of the resultant growth in airway smooth muscle. 

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

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

Relevant Publications
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Title 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 intra-cellular 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 spatio-temporal scales, from intracellular signalling cascades to tissue-level 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 well-suited to mathematical study.

This project will develop a predictive framework, coupling key signalling pathways to cell- and tissue-level mechanics, to elucidate key developmental mechanisms and their interaction. Investigations will include both multiscale computational approaches, and asymptotic methods for model reduction and analysis. Importantly, model development, analysis and experimental validation will be enabled via close collaboration with Dr Robert Jenkins (Francis Crick Institute, a multidisciplinary biomedical discovery institute dedicated to understanding the scientific mechanisms of living things), thereby ensuring the relevance of the investigations undertaken.

Experience of mathematical/numerical techniques for ODEs and PDEs, the study of nonlinear dynamical systems, or mathematical biology more generally would be an advantage; prior knowledge of the relevant biology is not required.

Relevant Publications
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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 fire-diffuse-fire (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 life-threatening condition?) as well as to elucidate fundamental concepts in cell signalling.

 

Relevant Publications
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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.

In more detail the project will consider models of neural activity represented by non-local integro-differential equations posed on both idealised and human realistic cortical structures.  The former will allow the development of analytical tools to understand the role of tissue heterogeneity and disorder in sculpting wave dynamics, such as the recently developed interface approach [1].  The latter will extend this so-called neural field approach [2] using cortical meshes from human connectome databases, making extensive use of spectral and finite element methods.  

This applied mathematical project will be facilitated by interaction with colleagues from the Sir Peter Mansfield Imaging Centre. As well as exposing the PhD student to rich neuroimaging data-sets collected locally using cutting edge magnetoencephalography techniques, the project will contribute to our understanding of cortical waves in the functioning of the human brain.

Relevant Publications
  • [1] S Coombes, H Schmidt and I Bojak 2012 Interface dynamics in planar neural field models, Journal of Mathematical Neuroscience, 2:9.
  • [2] S Coombes, H Schmidt and D Avitabile 2014 Spots: Breathing, drifting and scattering in a neural field model, Neural Fields, Ed. S Coombes, P beam Graben, R Pottiest and J J Wright, Springer Verlag
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Title Modelling macrophage extravasation and phenotype selection
Group(s) Mathematical Medicine and Biology
Proposer(s) Prof Markus Owen
Description

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

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

Relevant Publications
  • A Mantovani, S Sozzani, M Locati, P Allavena, and A Sica: Macrophage polarization: tumor-associated macrophages as a paradigm for polarized M2 mononuclear phagocytes. Trends Immunol., 23:549--555, (2002).
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Title 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 decisions1.  How this process could be realised in dynamic circuits of neurons is as yet unclear2. Additionally, while there are some spectacular experimental data on the capability of humans to accomplish such Bayes-optimal computations, strong evidence only comes from a limited set of experiments3, other evidence often assuming Bayesian algorithms a priori4 and there is an influential literature demonstrating the failures of human information processing to incorporate prior probabilities5. This project will probe the mechanisms involved in perceptual inference in a multi-disciplinary way mixing techniques from cognitive psychology and neuroimaging with those from mathematical neuroscience.  

A PhD student will approach this question on two frontiers: Phenomenologically, they will use advanced psychophysical techniques and mathematical modelling 6 to study how humans integrate prior probabilities and sensory information more generally (i.e. investigate the parameter space where the Bayesian observer model applies) and measure neuronal activity during such processes noninvasively7. Computationally, they will then investigate a new class of forward models for the generation of brain rhythms based on mean-field reductions of synaptically interacting nonlinear integrate-and-fire systems8.  These are ideally suited for studying the phase-amplitude coupling of brain rhythms in hierarchical cortical networks that have been reported in human and animal studies on the integration of prior information9,10.  The implementation of a Bayesian machine in this architecture will shed light on the debate about how higher frequency 'gamma oscillations' can communicate sensory feed-forward information, whilst top-down feedback is mediated by lower frequency 'alpha-' or 'beta-' band oscillations11.  The PhD student will be actively involved not only in model development and delivery but the acquisition of psychophysical data.  Initial experiments will be used to validate the model, progressing through to the design of new experiments to test model-generated hypotheses about the breakdown of computation in mental illness.  In particular the project will explore the role of NMDA vs. AMPA glutamatergic receptors in subserving neurodynamics for Bayesian cognition and their disturbance in schizophrenia12.

The project will be jointly supervised by Dr Markus Bauer (School of Psychology) and Professor Stephen Coombes (School of Mathematical Sciences).

 

1) Knill, D.C. & Pouget, A. The Bayesian brain: the role of uncertainty in neural coding and computation. Trends Neurosci 27, 712-719, (2004). 2) Pouget, A., Beck, J.M., Ma, W. J. & Latham, P. E. Probabilistic brains: knowns and unknowns. Nat Neurosci 16, 1170-1178, (2013).

3) Ernst, M.O. & Banks, M.S. Humans integrate visual and haptic information in a statistically optimal fashion. Nature 415, 429-433, 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, 15735-15742, (2014).

5) Kahneman, D. & Tversky, A. On the study of statistical intuitions. Cognition 11, 123-141 (1982).

6) Coombes, S. Large-scale neural dynamics: simple and complex. Neuroimage 52, 731-739, (2010).

7) Wenzlaff, H., Bauer, M., Maess, B. & Heekeren, H.R. Neural characterization of the speed-accuracy tradeoff in a perceptual decision-making task. J Neurosci 31, 1254-1266, (2011).

8) Coombes, S., Thul, R. & Wedgwood, K.C.A. Nonsmooth dynamics in spiking neuron models. Physica D 241, 2042-2057, (2012).

9) Bastos, A.M. et al. Canonical microcircuits for predictive coding. Neuron 76, 695-711, (2012).

10) Friston, K.J. et al. LFP and oscillations-what do they tell us? Curr Opin Neurobiol 31, 1-6, (2015).

11) Bauer, M., et al.. Attentional modulation of alpha/beta and gamma oscillations reflect functionally distinct processes. J Neurosci 34, 16117-16125, (2014).

12) Adams, R.A., et al. The computational anatomy of psychosis

Relevant Publications
Other information
Title Partitioned-domain concurrent multiscale modelling
Group(s) Scientific Computation, Mathematical Medicine and Biology
Proposer(s) Dr Kris van der Zee
Description

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

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

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

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

Relevant Publications
  • J.T. Oden, S. Prudhomme, A. Romkes, and P.T. Bauman, Multiscale modeling of physical phenomena: Adaptive control of models, SIAM J. Sci. Comput. 28 (2006), pp. 2359-2389
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Title Phase-field modelling of evolving interfaces
Group(s) Scientific Computation, Mathematical Medicine and Biology
Proposer(s) Dr Kris van der Zee
Description

Phase-field modelling of evolving interfaces
(Or – How does one effectively model and simulate interfacial phenomena?)

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 sharp-interface descriptions, such as parametric and level-set methods, and those with diffuse-interface descriptions, commonly referred to as phase-field models.

Challenges for students:
* Can one develop a phase-field model for a particular interfacial phenomenon?
* What are the foundational laws underpinning phase-field models?
* What is the connection between sharp-interface models and phase-field models?
* Can one design stable time-stepping schemes for phase-field models?
* Or efficient adaptive spatial discretisation methods?

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

Relevant Publications
  • H. GOMEZ, K.G. VAN DER ZEE, Computational Phase-Field Modeling, in Encyclopedia of Computational Mechanics, Second Edition, E. Stein, R. de Borst and T.J.R. Hughes, eds., to appear
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Title Statistical analysis of neuroimaging data
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Dr Christopher Brignell
Description

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

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

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

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

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

Relevant Publications
  • Kypraios, T., O'Neill, P. D., Huang, S. S., Rifas-Shiman, S. L. and Cooper, B. S. (2010) Assessing the role of undetected colonization and isolation precautions in reducing Methicillin-Resistant Staphylococcus aureus transmission in intensive care units. BMC Infectious Diseases 10(29).
  • Worby, C., Jeyaratnam, D., Robotham, J. V., Kypraios, T., O.Neill, P. D., De Angelis, D., French, G. and Cooper, B. S. (2013) Estimating the effectiveness of isolation and decolonization measures in reducing transmission of methicillin-resistant Staphylococcus aureus in hospital general wards. American Journal of Epidemiology 177 (11), 1306-1313.
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Title Modelling of Emerging Diseases
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Frank Ball
Description

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

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

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

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

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

Relevant Publications
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Title Bayesian model choice assessment for epidemic models
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Philip O'Neill, 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 goodness-of-fit of epidemic models to data, and methods for comparing different models.

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

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

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