# Modelling and Analytics for Medicine and Life sciences Doctoral Training Centre

The MAML doctoral training programme focuses on innovative modelling, simulation and data analysis approaches for the biomedical sciences, working across disciplines to study real-world problems in medicine and biology.

Maintaining a healthy society creates major challenges across the biomedical sciences in areas including ageing, cancer, drug resistance, chronic disease and mental health.

Addressing such challenges necessitates continuing development and implementation of a raft of new mathematical approaches and their integration with experimental and clinical science.

The programme will equip a cohort of graduate students with fit-for-purpose methodologies to tackle these applications.

Students will apply mathematical approaches (from areas such as dynamic modelling, informatics, network theory, scientific computation and uncertainty quantification) to research projects at the forefront of biomedical and life sciences identified through well-established collaborations with both academic and industrial partners.

MAML students will be provided with an excellent training environment within the Centre for Mathematical Medicine and Biology and their collaborative departments. Students will undertake tailored training, complemented by broadening, soft-skills, wet-lab (where appropriate) and student-led activities. There will also be opportunities for training and exchanges with world-leading partners.

## High-quality funded PhD opportunities

The programme will fund six fully-funded 3.5 year PhD scholarships, to start in September 2017.

Successful applicants will receive a stipend (£14,553 per annum for 2017/8) for up to 3.5 years, tuition fees and a Research Training Support Grant. Fully funded studentships are available for UK applicants. EU applicants who are able to confirm that they have been resident in the UK for a minimum of 3 years prior to the start date of the programme may be eligible for a full award, and may apply for a fees only award if they are not able to prove they satisfy the 3 years of residency criterion.

Students will be based within the School of Mathematical Sciences, and co-supervised by one or more academics from partner schools.

## Applications

Please apply by using our application form. Applicants for the MAML programme should have at least a 2:1 degree in mathematics, statistics or a similarly quantitative discipline (such as physics, engineering, or computer science).

Referee forms and guidance notes for referees are also available. Applicants are also required to complete the Equal Opportunities Form.

Completed applications and references should be submitted by Midnight GMT **Friday 28 April 2017**. Further application deadlines will be announced in due course.

## Available projects

For queries in relation to a particular project, please contact the supervisors associated with that project.

**Supervisors:** Dr Bindi Brook (Mathematical Sciences), Professor Markus Owen (Mathematical Sciences)

The precisely orchestrated migration of leukocytes (white blood cells of the immune system) is a key feature of all immune and inflammatory responses, including those that occur in infectious diseases. Rapid leukocyte transport around the body is facilitated by fluid delivery in the blood and lymphatic vessels.

However, their guidance to key destinations in tissues, lymph nodes or other tissue spaces is driven by gradients in a family of small secreted proteins called chemokines. Despite major advances in understanding chemokine function, it is still unclear how chemokine gradients are formed, maintained and regulated in tissues.

In addition to molecular diffusion, chemokine binding to extra-cellular matrix (ECM) components is likely to play a key role. Interstitial fluid flow will also contribute to gradient formation, and in the case of chemokine production near blood or lymphatic vessels, the transmural movement of fluid is likely to advect chemokines further into tissues than would be possible by pure diffusion. ‘Atypical’ chemokine receptors (ACKRs), a small family of molecules that scavenge and destroy extracellular chemokines are also likely to play a critical role in establishing, stabilizing and regulating chemokine gradients. The type of leukocyte migration induced depends on chemokine context, with soluble chemokine gradients directing chemotactic cell movement (migration up concentration gradients), while immobilized chemokine gradients induce integrin-dependent haptotaxis (migration up adhesion gradients).

The mechanisms that set up these gradients therefore include diffusion, advection (fluid movement), cell-mediated scavenging, and selective binding to extracellular matrix (ECM), some of which may be modified during inflammation. The aim of this project will be to develop mathematical models of chemokine gradient development during an immune or inflammatory response. The models will be developed in collaboration with immunologists based at the University of Glasgow (Profs Nibbs and Graham) [1], and a bioengineer at Imperial College London (Prof James Moore) who will be quantifying chemokine transport dynamics using a novel microfluidic platform to obtain a better understanding of chemokine transport and distribution in interstitial tissues around lymphatic vessels.

### References

- Immune regulation by atypical chemokine receptors. Nibbs and Graham. Nature Reviews Immunology 13:815-829, 2013.

**Supervisors:** Professor Stephen Coombes (School of Mathematical Sciences), Dr Rachel Nicks (School of Mathematical Sciences), Dr Matthew Brookes (Sir Peter Mansfield Imaging Centre)

Modern non-invasive probes of human brain activity, such as magneto-encephalography, 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 spatio-temporal 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 coupled-oscillator theory and phase-amplitude 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 time-series output of oscillator models on networks). The project will focus in particular on developing techniques for the analysis of dynamics on “multi-layer networks” to better understand functional connectivity within and between frequency bands of neural oscillations.

### Relevant papers

- P Ashwin, S Coombes and R Nicks (2016) Mathematical frameworks for network dynamics in neuroscience. Journal of Mathematical Neuroscience. 6:2.
- J Hlinka and S Coombes (2012) Using Computational Models to Relate Structural and Functional Brain Connectivity, European Journal of Neuroscience, Vol 36, 2137—2145
- M J Brookes, P K Tewarie, B A E Hunt, S E Robson, L E Gascoyne, E B Liddle, P F Liddle and P G Morris (2016) A multi-layer network approach to MEG connectivity analysis, NeuroImage 132, 425-438

**Supervisors:** Dr Gary Mirams and Dr Simon Preston (School of Mathematical Sciences)

Background: in biological systems ion channel proteins sit in cell membranes and selectively allow the passage of particular types of ions, creating currents. Ion currents are important for many biological processes, for instance: regulating ionic concentrations within cells; passing signals (such as nerve impulses); or co-ordinating contraction of muscle (skeletal muscle and also the heart, diaphragm, gut, uterus etc.). Mathematical ion channel electrophysiology models have been used for thousands of studies since their development by Hodgkin & Huxley in 1952 [1], and are the basis for whole research fields, such as cardiac modelling and brain modelling [2]. It has been suggested that there are problems in identifying which set of equations is most appropriate as an ion channel model. Often it appears different structures and/or parameter values could fit the training data equally well, but may make different predictions in new situations [3].

Aim: we have been developing novel experimental designs to provide more information about ion channel behaviour from shorter experiments. We would like to improve our techniques – to describe the ion current and also to characterise drug binding to ion channels (which can physically block them and reduce the current that flows to zero, sometimes leading to fatal heart rhythm changes). It is difficult to measure the rate at which drug/ion channel binding occurs and whether it occurs when the channels are open, closed, or both. These factors may be crucial in determining whether novel pharmaceutical compounds are likely to have side effects or not, and there is a need to develop efficient ways to measure them.

Approach: this project will involve computational biophysical modelling (efficient numerical solution of nonlinear ODE systems); the application of statistical techniques to quantify our uncertainty in model parameters and model equations/structure; and some wet-lab laboratory electrophysiology experiments. We will design more information-rich experiments to reduce our uncertainty in the models we develop [4] and work closely with labs to test out experiments we design and improve them.

### Relevant publications

- A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,”
*J. Physiol.*, vol. 117, pp. 500–544, 1952. - D. Noble, A. Garny, and P. J. Noble, “How the Hodgkin – Huxley equations inspired the Cardiac Physiome Project,” vol. 11, pp. 2613–2628, 2012.
- M. Fink and D. Noble, “Markov models for ion channels : versatility versus identifiability and speed,”
*Philos. Trans. A. Math. Phys. Eng. Sci.*, vol. 367, no. 1896, pp. 2161–79, Jun. 2009. - G. R. Mirams, P. Pathmanathan, R. A. Gray, P. Challenor, and R. H. Clayton, “White paper: Uncertainty and variability in computational and mathematical models of cardiac physiology.,”
*J. Physiol.*, Mar. 2016.

**Supervisors:** Dr Kenton Arkill (Medicine), Dr Reuben O’Dea (Maths), Professor David Bates (Medicine), Dr Matthew Hubbard (Maths)

The primary function of blood vessels is to transport molecules to tissues. In diseases such as cancer and diabetes this transport, particularly of large molecules such as albumin, can be an order of magnitude higher than normal.

The project is to model transient flow of macromolecules across the vascular wall in physiology and pathology. The doctoral student will join a team that includes medical researchers, biophysicists and mathematicians acquiring structural and functional data.

Detailed microscale models of vascular wall hydrodynamics and transport properties will be employed; in addition, powerful multiscale homogenisation techniques will be exploited that enable permeability and convection parameters on the nanoscale to be linked through the microscale into translatable information on the tissue scale. Computational simulations will be used to investigate and understand the model behaviour, including, for example, stochastic and multiphysics effects in the complex diffusion-convection nanoscale environment. The project will afford a great opportunity to form an information triangle where modelling outcomes will determine physiological experiments to feedback to the model. Furthermore, the primary results will inform medical researchers on potential molecular therapeutic targets.

**Supervisors:** Dr Ruediger Thul (Mathematical Sciences), Dr Tom Bellamy (Life Sciences) and Dr Theo Kypraios (Mathematical Sciences).

It is often thought that when cells are genetically identical, they behave in the same manner. However, recent experiments have convincingly demonstrated that this is far from the truth.

For example, stimulating the same cell multiple times with the same stimulus may lead to very different responses. Given that these responses are crucial for healthy well-being (e.g. insulin release after eating cake) a key question in cell signalling is to understand and predict these heterogeneous cell dynamics.

In this project, we will use stochastic point-processes to describe such inhomogeneous cellular behaviour and use Bayesian methods to fit them to data. As a test case, we will investigate the dynamics of one of the most important cellular messengers: intracellular calcium. More precisely, we will study so-called calcium spike sequences and aim to ascertain what statistical features best describe the observed heterogeneity. The insights that we will gain will be both of practical and conceptual importance. As for the former, a key goal in computational cell physiology is to model cell behaviour in a computationally inexpensive, yet accurate manner. A Bayesian approach offers a natural way for doing this in conjunction with the quantification of uncertainty of the dynamics of the process. In terms of conceptual advances, the statistical details of calcium spike sequences may point towards the mechanisms that generate them. In turn, this is crucial information when we need to tailor signalling cascades for biomedical applications. The project will offer opportunities to explore either or both of these avenues in more detail.

The project will involve a significant element of computing.

**Supervisors:** Dr Jonathan Wattis (Maths) and Professor Andy Salter (Biosciences).

The aim of this project is to produce detailed models of certain fat metabolism pathways in the liver. Metabolic Syndrome (MetSyn) represents a group of metabolic abnormalities associated with insulin resistance and leading to increased risk of developing cardiovascular disease (CVD) and type 2 diabetes. A further consequence of MetSyn is accumulation of lipid inside liver cells (Non-alcoholic Fatty Liver Disease , NAFLD) which can ultimately lead to cirrhosis, cancer or liver failure.

A greater understanding of the pathophysiology of MetSyn should help provide potential interventions to prevent NAFLD and more serious liver degeneration. There are many pathways which influence hepatic lipid accumulation. Existing models of fluxes through these pathways in the fasted and fed states have been fitted to healthy subjects, as well as a range of patients with varying degrees of insulin resistance. However, the longer term influences of the amount and type of food consumed on the accumulation of liver fat remain to be effectively modelled.

We will start with the production of VLDL in the liver and the role of specific enzymes , including stearoyl coenzyme A desaturase (SCD) and diacylglycerol acyltransferase enzymes (DGAT) . SCD regulates the conversation of saturated to monounsaturated fatty acids and appears critical to the subsequent formation of triacylglycerol (TAG). DGAT are involved in the synthesis of TAG from diacylgycerol in the cytosol and endoplasmic reticulum, and the relative activity of different isoforms may be a key factor in whether TAG is stored intracellularly or secreted within VLDL. Other areas where more detailed modelling is required is the breakdown of chylomicrons into remnant particles and the delivery of dietary TAG to the liver within these remnants. This means that ingested fats are adsorbed slowly and give rise to an input of fat over a prolonged period after eating. As these particles are digested their size decreases and their composition also changes, as TAG and cholesterol are removed separately. A main challenge in the project is to convert results on the dynamics which occur between one meal and the next into an understanding of the longer timescale of decades over which fat accumulates in the liver. Thus we will aim to investigate the effects of allowing certain parameters to vary slowly over various significantly longer timescales. Such modelling may help to identify key points in the development of NAFLD and suggest potential nutritional/pharmaceutical interventions to slow, or even reverse, the process.

Modelling the alternative splicing of tissue growth regulators and its implications for tumour growth

**Supervisors**: Professor Markus Owen (School of Mathematical Sciences), Professor David Bates (Division of Cancer and Stem Cells, School of Medicine)

Normal and pathological tissue growth is regulated by diverse growth factors and related molecules, many of which are produced in cells via the transcription of associated genes and translation of mRNA to protein. In many cases, *alternative splicing*, regulated by *splicing factors,* leads to different isoforms of proteins, which can have different effects. This is particularly pertinent to angiogenesis, the process whereby new blood vessels are produced from existing ones, which is crucial in cancer and also diseases such as diabetic retinopathy.

Different isoforms of Vascular Endothelial Growth Factor (VEGF), whose balance is regulated by alternative splicing, can promote or inhibit angiogenesis. In fact, the relevant splicing factors seem to regulate alternative splicing of families of genes controlling cell death, growth factor signaling, the cell cycle, invasion and immune responses. Thus it important to consider the overall effect of splicing factors in the context of a whole tissue where all these processes are modulated.

This project will focus on mathematical modelling of the various aspects of alternative growth factor splicing, regulation of angiogenesis, and tumour growth, with the following objectives:

- model splicing control at network level;
- model the implications for tissue growth of altered splicing control
- couple O1 and O2 to predict the efficacy of interventions that modulate alternative splicing in cancer.

This will require the develop and application of advanced mathematical and computational techniques to make the link from molecules to cells to tissues. A significant challenge is to use a blend of mathematical and statistical approaches to allow the translation of varied experimental data and knowledge into tractable parameterised mathematical frameworks that combine dynamics over a range of scales.

This project would also involve co-operation with Exonate, a biopharmaceutical company focussed on the discovery and development of small molecule drugs that modulate alternative mRNA splicing to address diseases of high unmet medical need. Exonate will provide relevant data and scientific input, and also contribute to the student training, for example by through hosting them within the company on secondment.

### References

- M R Owen et al. Cancer Res 71(8) 2826-37 (2011)

Understanding how variability in cellular oxidative stress networks impacts tissue sensitivity

**Supervisors: **Dr Etienne Farcot, Dr Simon Preston (School of Mathematical Sciences) and Dr Alistair Middleton (Unilever)

Unilever is a large multinational company which produces a wide range of personal care, homecare, food and refreshment products. In toxicological science, there has been significant shift towards developing a mechanistic understanding of how certain compounds cause toxicity in order determine safe levels of exposure.

For many ingredients, cells can adapt to alterations in biological pathways provided the exposure level is sufficiently low [1]. Examples include biological stress response pathways, of which there are approximately ten in humans, including oxidative stress, ER stress and DNA damage [2-5]. Models of these pathways have appeared in the literature, and are often composed of systems of nonlinear ordinary differential equations which are parameterised using *in vitro* data (i.e. data measured in the lab) using a single cell type. For example, HepG2 is a cell line that is often used *in vitro* as a surrogate for liver cells. However, it is likely (1) that responses will vary between different tissue types (liver cells may be more sensitive than skin cells, for example), and (2) that this variability is underpinned in large part by variations in the abundance of different stress response network components. While there is data available on the variability between different tissues types, both in terms of sensitivity to different compounds and abundance, a systematic attempt at understanding whether one can combine abundance data and mathematical models to understand variations in sensitivity in still lacking. To this end we propose to use an existing mathematical model of oxidative stress, together with the available experimental data to explore this. A key aspect of the work will be to draw on techniques from Bayesian statistics to explore uncertainties in the data and model predictions [6].

This project will be based in the School of Mathematical Sciences, with regular interactions with the industrial supervisor.

### References

- Shah, I., Setzer, R. W., Jack, J., Houck, K. A., Judson, R. S., Knudsen, T. B., ... & Thomas, R. S. (2016). Using ToxCast™ data to reconstruct dynamic cell state trajectories and estimate toxicological points of departure. Environmental health perspectives, 124(7), 910
- Khalil, H. S., Goltsov, A., Langdon, S. P., Harrison, D. J., Bown, J., & Deeni, Y. (2015). Quantitative analysis of NRF2 pathway reveals key elements of the regulatory circuits underlying antioxidant response and proliferation of ovarian cancer cells. Journal of biotechnology, 202, 12-30
- Erguler, K., Pieri, M., & Deltas, C. (2013). A mathematical model of the unfolded protein stress response reveals the decision mechanism for recovery, adaptation and apoptosis. BMC systems biology, 7(1), 16
- Li, Z., Sun, B., Clewell, R. A., Andersen, M. E., & Zhang, Q. (2013). Dose response modeling of etoposide-induced DNA damage response. Toxicological sciences, kft259
- Simmons, S. O., Fan, C. Y., & Ramabhadran, R. (2009). Cellular stress response pathway system as a sentinel ensemble in toxicological screening. Toxicological sciences, kfp140
- Girolami, M. (2008). Bayesian inference for differential equations. Theoretical Computer Science, 408(1), 4-16

**Supervisors:** Professor Theo Kypraios (School of Mathematical Sciences), Professor Stam Sotiropoulos (School of Medicine)

This project will capitalise on advances and data offered by the cornerstone Human Connectome Project (HCP) (www.humanconnectome.org), for which the principal supervisor (SS) has been a major contributor. We will build novel computational methodology for estimating connections using complementary MRI and MEG through state-of-the-art inference techniques. In particular, we will develop models for estimating network structure from multimodal data and we will explore causal interactions. Using data-driven exploratory analysis, we will then identify latent associations between brain organisation and function. This will further allow the extraction of summary imaging-derived measures with certain contextual relevance that could comprise potential markers for subsequently exploring pathology-induced abnormalities and dysfunction. For instance, we will identify predictive behavioral traits of psychiatric disorders and explore their associations with estimated connectivity.

**Supervisors:** Professor Steve Coombes (School of Mathematical Sciences), Dr Chris Sumner (MRC Institute of Hearing Research), Patrick May (Leibniz Institute, Magdeburg), Dr Katrin Krumbholz (MRC Institute of Hearing Research)

Understanding how the underlying sensory processing by neural networks in the human brain gives rise to sensory perception is a difficult problem.

It is straightforward to measure human perception via behavioural experiments, but even accessing, let alone understanding, the underlying signals in the brain is difficult. Non-invasive measures such as EEG and fMRI allow monitoring of aggregate responses, but they are dramatically limited in their resolution, either spatially or temporally. Relating non-invasive responses back to underlying neural activity necessitates solving an inverse problem.

Moreover, our knowledge about how individual neurons behave can only be drawn from animal experiments. There is currently no principled way for inferring the behaviour of underlying neural circuits from non-invasive measurements.

This project will address the problem of sensory perception by developing a novel computational model of one part of the brain: the auditory cortex. This model will have the power to simulate individual spiking neurons, large populations of neurons, and far-field electrical signals (EEG, MEG) that are normally accessible in humans. This forward-model will allow the testing of hypotheses about the possible ways in which neural activity could give rise to the non-invasive observations, which in turn, are linked to results from behavioural experiments.

This will be achieved by bringing together two existing models: a large scale firing rate model of multiple auditory cortical fields, constrained by all the known anatomy (May et al. 2015); and a modelling framework which allows a rigorous abstraction from spiking models of neurons to neural field models (Byrne et al. in press). These field models can capture the dynamics of extended regions of the brain (Coombes 2010), be projected onto surfaces, and folded in the manner of the cortical surface. From these, far-field potentials (EEG, MEG) can be predicted. Thus for the first time we aim to provide a computational model that can predict, in a principled way, non-invasive measurements from the responses of single neurons. This will function as a platform for theoretically linking various measures of neural activity to sound perception.

### References

- Coombes S (2010). Large-scale neural dynamics: Simple and complex, NeuroImage, Vol 52, 731–739.
- May PJ, Westö J, Tiitinen H. (2015). Computational modelling suggests that temporal integration results from synaptic adaptation in auditory cortex. Eur J Neurosci. 41:615-30.
- Á Byrne, M J Brookes and S Coombes 2016 (in press). A mean field model for movement induced changes in the beta rhythm, Journal of Computational Neuroscience.

**Supervisors:** Professor John King (School of Mathematical Sciences), Professor Alison Goodall (University of Leicester)

The coagulation cascade is a complex network of protein reactions that is central to the formation of blood clots. While clots are essential to prevent bleeding when formed inappropriately they can lead to heart attacks and strokes. The assessment of coagulation is key in diagnosing thrombotic conditions but routine assays do not presently correlate with an individual’s risk of thrombosis. The aims of this project are to develop a definitive mathematical model of the coagulation cascade, validate the models against experimental data describing coagulation profiles in both healthy individuals and patients with cardiovascular disease and utilise the models to furnish understanding of the key drivers of the differences seen in the data to provide insights into therapeutic markers.

**Supervisors:**Professor Markus Owen (School of Mathematical Sciences), Ruman Rahman, Stuart Smith, Dorothee Auer

Glioblastoma (GBM) is a very aggressive cancer of the brain. Although standard cancer therapies such as surgery, radiation and chemotherapy often reduce the size of the tumour temporarily, they rarely act as a cure; average survival after diagnosis is around 18 months. Spatial heterogeneity in brain cancer is thought to play an important role in progression and therapeutic response - certain regions may be more aggressive or harbour cells that are chemo- or radio-therapy resistant. We are building up a dataset on brain-cancer heterogeneity using samples from multiple regions of patients’ tumours, integrated with MRI scans, to give us clinically relevant information such as areas at most risk of recurrence following surgery to remove a brain tumour. The ability to identify at risk areas may open up new surgical strategies, local drug delivery strategies or new focused radiotherapy strategies.

A crucial ingredient here is to be able to forecast GBM growth using mathematical models based on the local biological aggressiveness of the cancer (reflecting its heterogeneity) and the micro milieu. Such models could be used to predict which parts of the tumour margin are most “dangerous”. Furthermore, slower-growing low-grade gliomas can progress to GBM, and understanding and predicting this progression could be crucial. Using datasets that characterise genetic spatial intra-tumour heterogeneity in low grade glioma, and comparing with GBM data, we hope to identify any putative low grade glioma signatures that may predict progression to GBM. This project will therefore blend mathematical modelling and data analysis, in collaboration with partners in brain cancer research, neurosurgery and radiology, to make key contributions to personalised medicine for brain cancer.

**Supervisors:** Dr Daniele Avitabile (School of Mathematical Sciences), Professor Stamatios Sotiropoulos (School of medicine), Professor Stephen Coombes (School of Mathematical Sciences), Professor Paul Houston (School of Mathematical Sciences)

The large number of neurons forming the cortex are intricately connected and, to a first approximation, can be modelled as a continuum in space. Neural field models, which make this assumption, have been used to model large-scale neural activity observed in electroencephalogram and magnetoencephalogram neuroimaging studies. Spatio-temporal patterns in these models are relevant to understand epileptic seizures, visual hallucinations and short-term working memory (see [1,2] and references therein).

When neural fields are posed on flat surfaces, analytical progress can be made to understand the origin of a wide variety of activity patterns (stripes, localised spots, hexagons, travelling waves, spiral waves).

Our brain, however, is not flat. Sculped on the cortical surface are characteristic bumps and grooves, known as gyri and sulci, respectively. This heterogeneity is not only geometrical: neurons have a heterogeneous density and a heterogeneous synaptic wiring (see image above, where densely connected regions are coloured in red).

This project will use methods from dynamical systems and computational science to develop a theory for the evolution of synaptic activity on folded brains. We will address the following questions:

- Tractography and neuroimaging techniques provide us with a detailed map of gyri, sulci and neural wiring. Can we incorporate heterogeneities into neural fields?
- We expect that curvature plays an important role in pattern selection [3]. What is the effect of the gyrification on neural activity? If a pattern of cortical activity is observed in a model of a “flat brain”, will it persist on a “curved brain”? Are cortical waves accelerated/decelerated by curvature and heterogeneities?
- The analytical techniques used to study patterns in flat cortices have a numerical counterpart on curved surfaces [4]. Owing to recent developments in neural field theory, it has now become possible to track and analyse patterns numerically and predict whether they will be observable in experiments [5]. Can we develop robust and efficient algorithms to perform bifurcation analysis on generic folded cortices?

### References

- S Coombes, P beim Graben and R Potthast (2014) Tutorial on Neural Field Theory, Neural Fields, Ed. S Coombes, P beim Graben, R Potthast and J J Wright, Springer Verlag
- P C Bressloff (2012). Spatiotemporal dynamics of continuum neural fields. Journal of Physics A: Mathematical and Theoretical, 45(3), 033001
- S Visser, R Nicks, O Faugeras and S Coombes (2017) Standing and travelling waves in a spherical brain model: the Nunez model revisited, Physica D, to appear
- D Avitabile, P Matthews, R Nicks, O Smith (2017). Patterns of cortical activity in neural fields posed on spherical domains. Preprint
- J Rankin, D Avitabile, J Baladron, G Faye, D J B Lloyd (2014) Continuation of localized coherent structures in nonlocal neural field equations. SIAM Journal on Scientific Computing 36 (1), B70-B93

**Supervisors:** Dr Daniele Avitabile (School of Mathematical Sciences), Professor Alan Johnston (School of Psychology), Professor Stephen Coombes (School of Mathematical Sciences)

When we are exposed to a visual stimulus (right), the visual input from the retina is relayed from the lateral geniculate nucleus (LGN) to the visual cortex.

The primary visual cortex (V1) contains neurons which respond to the stimulus’ orientation.

Neural field models are now in common usage in mathematical neuroscience to describe the coarse-grained activity of cortical tissue [1]. For mathematical convenience they often assume that anatomical connectivity is homogenous. However, this is far from the truth. For example, in the primary visual cortex (V1) it is known that there are maps reflecting the fact that neurons respond preferentially to stimuli with particular features. The classic example is that of orientation preference (OP), whereby cells respond preferentially to lines and edges of a particular orientation. The OP map changes continuously as a function of cortical location, except at singularities or pinwheels. The underlying periodicity in the microstructure of V1 is approximately 1mm, the domain of which corresponds to the so-called cortical hypercolumn. Other anatomical evidence suggests that longer-range, patchy horizontal connections link neurons in different hypercolumns provided that they have similar orientation preferences. This project will consider a field of hypercolumns that respects this biological reality. The mathematical model will be that of an integro-differential equation for V1 activity, with V1 viewed as a fiber bundle that associates to every point of the cortex (or retina by the retino-cortical map) a copy of the unit circle [2].

The project will focus on combining realistic retino-cortical maps [3] with next generation neural field models [4] and state-of the art numerical methods [5] to understand not only mechanisms for visual illusions, but also basic notions of how biological tissue can perform visual computations for image completion. The project will involve a mix of high performance scientific computation, nonlinear dynamics, differential geometry, and an enthusaism for learning about visual neuroscience.

### References

- S Coombes, P Beim Graben and R Potthast, 2014. Tutorial on Neural Field Theory, Neural Fields, Ed. S Coombes, P beim Graben, R Potthast and J J Wright, Springer Verlag
- P C Bressloff and J D Cowan, 2003. The functional geometry of local and horizontal connections in a model of V1. Journal of Physiology-Paris, 97:221Ð236.
- A Johnston 1989 The geometry of the topographic map in striate cortex. Vision Research, 29, 1493–1500
- A Byrne, D Avitabile and S Coombes, 2017. A next generation neural field model: The evolution of synchrony within patterns and waves, preprint
- J Rankin, D Avitabile, J Baladron, G Faye, DJB Lloyd, 2014. Continuation of localized coherent structures in nonlocal neural field equations. SIAM Journal on Scientific Computing 36 (1), B70-B93