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Title  Excitability in biology  the role of noisy thresholds 

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

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes 
Description  This project will be based at the University of Nottingham in the School of Mathematical Sciences and the School of Psychology. 
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Other information  Eligibility/Entry Requirements: We require an enthusiastic graduate with a 1st class degree in Mathematics (or other highly mathematical field such as Physics or Chemistry), preferably at MMath/MSc level, or an equivalent overseas degree (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). Apply: This studentship is available to start from September 2017 and remain open until it is filled. To apply please visit the University Of Nottingham application page: http://www.nottingham.ac.uk/pgstudy/apply/applyonline.aspx Funding NotesSummary: UK/EU students  Tuition Fees paid, and full Stipend at the RCUK rate, which is £14,296 per annum for 2016/17. There will also be some support available for you to claim for limited conference attendance. The scholarship length will be 3 or 3.5, depending on the qualifications and training needs of the successful applicant. 
Title  Spirals and autosoliton scattering: interface analysis in a neural field model 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes, Dr Daniele Avitabile 
Description  Neural field models describe the coarse grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in 2D, where they are well known to generate rich patterns of spatiotemporal activity. Typical patterns include localised solutions in the form of travelling spots as well as spiral waves [1]. These patterns are naturally defined by the interface between low and high states of neural activity. This project will derive the dimensionally reduced equations of motion for such interfaces from the full nonlinear integrodifferential equation defining the neural field. Numerical codes for the evolution of the interface will be developed, and embedded in a continuation framework for performing a systematic bifurcation analysis. Weakly nonlinear theory will be developed to understand the scattering of multiple spots that behave as autosolitons, whilst strong scattering solutions will be investigated using the scattor theory that has previously been developed for multicomponent reaction diffusion systems [2]. 
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Other information  S Coombes, H Schmidt and I Bojak 2012 Interface dynamics in planar neural field models, Journal of Mathematical Neuroscience, 2:9 
Title  Modelling signal processing and sexual recognition in mosquitoes: neural computations in insect hearing systems 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Daniele Avitabile, Prof Stephen Coombes 
Description  Insects have evolved diverse and delicate morphological structures in order to When a sound wave reaches the head of a mosquito, the antenna oscillates under the Recent studies have shown that mosquitoes of either sex use both their antenna and Even though some models of mosquitoes hearing systems have been proposed in the past, 
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Title  Nonsmooth dynamical systems: from nodes to networks 

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

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

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

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Title  Waves on a folded brain 

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

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Ruediger Thul, Prof Stephen Coombes 
Description  A key role of cells is to translate external signals into appropriate cellular responses. For example, when cells that line blood vessels experience weak stimulation, they initiate the expression of certain genes, while for strong stimuli, they begin to move. How cells accomplish such diverse responses is still an open question. What has transpired, though, is that so called microdomains are vital for cellular decisionmaking. Microdomains are small parts of a cell where molecular mediators and switches are concentrated in close proximity. This is advantageous since cell signalling intrinsically relies on molecules interacting, and if they are close to each other, chances are higher that signal transduction is successful. In many cases, these signalling pathways rely on small molecules that diffuse through the microdomain and hence can carry information from one molecular partner to the next. To appreciate the full potential of the signalling micordomains, it is crucial to have a comprehensive understanding of the dynamics of these diffusible messengers. In this project, we will use a combination of semianalytical and numerical techniques to develop three dimensional models of signalling microdomains. In particular, we will investigate how the intracellular calcium concentration changes in space and time within microdomains, and how these changes affect signal transduction. Gaining deeper insights into microdomains is key for understanding for understanding healthy physiology such as fertilisation and muscle contraction as well as diseases such as immunodeficiency and neurological disorders. The model will be informed by experiments conducted at Oxford and Penn State University. 
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Title  Exploiting network symmetries for analysis of dynamics on neural networks 

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

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