Friday, 13 April 2018

Incorporating stress-assisted diffusion in cardiac models

Oxford Mathematician Ricardo Ruiz Baier, in collaboration mainly with the biomedical engineer Alessio Gizzi from Campus Bio-Medico, Rome, have come up with a new class of models that couple diffusion and mechanical stress and which are specifically tailored to the study of cardiac electromechanics. 

Cardiac tissue is a complex multiscale medium constituted by highly interconnected units (cardiomyocytes, the cardiac cells) which have remarkable structural and functional properties. Cardiomyocytes are excitable and deformable cells. Inside them, plasma membrane proteins and intracellular organelles all depend on the current mechanical state of the (macroscopic) tissue. Special structures, such as ion channels or gap junctions, rule the passage of charged particles throughout the cell as well as between different cells and their behaviour can be described by reaction-diffusion systems. All these mechanisms work in synchronisation to conform the coordinated contraction and pumping function of the heart.

During the cardiac cycle, mechanical deformation undoubtedly affects the electrical impulses that modulate muscle contraction, and also modifies the properties of the substrate where the electrical wave propagates. These multiscale interactions are commonly referred to as the mechano-electric feedback (MEF). Theoretical and clinical studies have been contributing to the systematic investigation of MEF effects for over a century; however, several open questions still remain. For example, and focusing on the cellular level, it is still now not completely understood what is the effective contribution of stretch-activated ion channels and what is the most appropriate way to describe them. In addition, and focusing on the organ scale, the clinical relevance of MEF in patients with heart diseases remains an open issue, specifically in relation to how MEF mechanisms translate into ECGs.

The idea of coupling mechanical stress directly as a mechanism to modify diffusive properties has been exploited for several decades by focusing on the context of dilute solutes in a solid, but remarkable similarities exist between these fundamental processes and the propagation of voltage membrane within cardiac tissue. Indeed, on a macroscopically rigid matrix, the propagating membrane voltage can be regarded as a continuum field undergoing slow diffusion.  

The approach described above basically generalizes Fick's diffusion by using the classical Euler's axioms of continuously distributed matter. An important part of the project, now under development, deals with the stability of the governing partial differential equations, the existence and uniqueness of weak solutions, and the formulation of mixed-primal and fully mixed discretisations needed to compute numerical solutions in an accurate, robust, and efficient manner. Some of the challenges involved relate to strong nonlinearities, heterogeneity, anisotropy, and the very different spatio-temporal scales present in the model. The construction and analysis of the proposed models and methods requires advanced techniques from abstract mathematics, the interpretation of the obtained solutions necessitates a clear understanding of the underlying bio-physical mechanisms, and the implementation (carried out exploiting modern computational architectures) depends on sophisticated tools from computer science. 
Other applications of a similar framework are encountered in quite different scenarios, for instance in the modelling of lithium ion batteries. Oxford visiting student Bryan Gomez (from Concepcion, Chile, co-supervised by Ruiz Baier and Gabriel Gatica) is currently looking at the fixed-point solvability and regularity of weak solutions, as well as the construction and analysis of finite element methods tailored for this kind of coupled problems (see also a different perspective focusing on homogenisation and asymptotic analysis, carried out by Oxford Mathematicians Jon Chapman, Alain Goriely, and Colin Please). 

Friday, 13 April 2018

How do node attributes mix in large-scale networks? Oxford Mathematics Research investigates

In this collaboration with researchers from the University of Louvain, Renaud Lambiotte from Oxford Mathematics explores the mixing of node attributes in large-scale networks.

A central theme of network science is the heterogeneity present in real-life systems. Take an element, called a node, and its number of connections, called its degree, for instance. Many systems do not have a characteristic degree for the nodes, as they are made of a few highly connected nodes, i.e. hubs, and a majority of poorly connected nodes. Networks are also well-known to be small-world in a majority of contexts, as a few links are typically sufficient to connect any pair of nodes. For instance, the Erdős number of Renaud Lambiotte is 3, as he co-authored a paper with Vincent D. Blondel, who co-authored with Harold S. Shapiro, who co-authored with Paul Erdős. 3 links are sufficient to reach Paul Erdős in the co-authorship network.




Because of their small-worldness, it is often implicitly assumed that node attributes (for instance, the age or gender of an individual in a social network) are homogeneously mixed in a network and that different regions exhibit the same behaviour. The contribution of this work is to show that this not the case in a variety of systems. Here, the authors focus on assortativity, a network analogue of correlation used to describe how the presence and absence of edges co-varies with the properties of nodes. The authors design a method to characterise the heterogeneity and local variations of assortativity within a network. The left-hand figure (please click to enlarge) for instance, illustrates an analogy to the classical Anscombe’s quartet, with 5 networks having the same number of nodes, number of links and average assortativity, but different local mixing patterns. The method developed by the authors is based on the notion of random walk with restart and allows them to define localized metrics of assortativity in the network. The method is tested on various biological, ecological and social networks, and reveals rich mixing patterns that would be obscured by summarising assortativity with a single statistic. As an example, the right-hand figure shows the local assortativity of gender in a sample of Facebook friendships. One observes that different regions of the graph exhibit strikingly different patterns, confirming that a single variable, e.g. global assortativity, would provide a poor description of the system.

For a more detailed description of the work please click here.

Tuesday, 10 April 2018

Ada Lovelace - the Making of a Computer Scientist. The latest book from Oxford Mathematics

Our latest book features the remarkable story of Ada Lovelace, often considered the world’s first computer programmer, as told in a new book co-written by Oxford Mathematicians Christopher Hollings and Ursula Martin together with colleague Adrian Rice from Randolph-Macon College.

A sheet of apparent doodles of dots and lines lay unrecognised in the Bodleian Library until Ursula Martin spotted what it was - a conversation between Ada Lovelace and Charles Babbage about finding patterns in networks, a very early forerunner of the sophisticated computer techniques used today by the likes of Google and Facebook. It is just one of the remarkable mathematical images to be found in the new book, 'Ada Lovelace: The Making of a Computer Scientist'.

Ada, Countess of Lovelace (1815–1852) was the daughter of poet Lord Byron and his highly educated wife, Anne Isabella. Active in Victorian London's social and scientific elite alongside Mary Somerville, Michael Faraday and Charles Dickens, Ada Lovelace became fascinated by the computing machines devised by Charles Babbage.  A table of mathematical formulae sometimes called the ‘first programme’ occurs in her 1843 paper about his most ambitious invention, his unbuilt ‘Analytical Engine.’

Ada Lovelace had no access to formal school or university education but studied science and mathematics from a young age. This book uses previously unpublished archival material to explore her precocious childhood: her ideas for a steam-powered flying horse, pages from her mathematical notebooks, and penetrating questions about the science of rainbows. A remarkable correspondence course with the eminent mathematician Augustus De Morgan shows her developing into a gifted, perceptive and knowledgeable mathematician, not afraid to challenge her teacher over controversial ideas.

 “Lovelace’s far sighted remarks about whether the machine might think, or compose music, still resonate today,” said Professor Martin. “This book shows how Ada Lovelace, with astonishing prescience, learned the maths she needed to understand the principles behind modern computing.”

Ada Lovelace: The Making of a Computer Scientist, by Christopher Hollings, Ursula Martin and Adrian Rice will be launched on 16th April 2018 by Bodleian Library Publishing, in partnership with the Clay Mathematics Institute.  

The page of doodles is on display until February 2019 as part of the Bodleian Library’s exhibition 'Sappho to Suffrage: women who dared.'

Ursula Martin will be speaking at the Hay Festival and Edinburgh Book Festival.

Monday, 9 April 2018

The contact-free knot - Oxford Mathematics Research explains

Knots are widespread, universal physical structures, from shoelaces to Celtic decoration to the many variants familiar to sailors. They are often simple to construct and aesthetically appealing, yet remain topologically and mechanically quite complex.

Knots are also common in biopolymers such as DNA and proteins, with significant and often detrimental effects, and biological mechanisms also exist for 'unknotting'.

There are numerous types of questions when studying knots. From a topological standpoint, fundamental issues include knot classification and equivalence of different knot descriptions. In continuum mechanics and elasticity, a knot is a physical structure with finite thickness, and aspects of interest include the strength, stability, equilibrium shape, and dynamic behaviour of a knotted filament. Such aspects are strongly connected to points/regions of self-contact, at which distant points push against each other.

Consider a simple hand-held experiment: take a strip of paper or flexible wire, tie it into a standard but loose knot (an open trefoil), and you will observe 2 isolated points of self-contact surrounding an interval of self-contact. Now add twist by rotating the ends, change the end-to-end distance by bringing your hands closer or further apart, and combine with small transverse displacements, i.e. shifting the end. For certain materials and with a little finesse, all points of contact can be removed.

Such configurations – contact-free, knotted, and mechanically stable – have never been described before, and Oxford Mathematician Derek Moulton and colleagues sought to understand and characterise them in terms of the underlying geometry and mechanics. To do so, they turned to the Kirchhoff equations for elastic rods, a set of 18 nonlinear differential equations that describe the balance of forces and moments as well as the geometrical shape of a thin and long elastic material. These equations admit an incredibly rich and non-unique solution space. A small modification to these equations yields the 'ribbon equations', more appropriate for a strip of paper and with a similarly complex solution space.

The goal was to find configurations within this solution space that satisfy the conditions of being contact-free, mechanically stable, and knotted. This was a bit like finding a needle in a haystack, but after applying some numerical tricks they showed that in fact such configurations exist as theoretical solutions of the full nonlinear 18D system; they then categorised the space of 'good knots' in terms of the 3 experimental measures: end-rotation, end-displacement, and end-shift. The numerical study was complemented with an asymptotic analysis of a perturbed 'double ring' solution; the idea being that knotted solutions can be found in the neighbourhood of a planar circle that overlaps itself exactly once.

The analysis suggests that the transverse displacement is a necessary component for generating contact-free knots. While the researchers only considered the "simplest" trefoil knot, they conjecture that toroidal knots of increasing genus can be stabilised in a contact-free state.

For a fuller explanation of the team's work please click here.

Thursday, 5 April 2018

The Oxford Maths Festival 28-29 April 2018

Bringing together talks, workshops, hands-on activities and walking tours, the Oxford Maths Festival is an extravaganza of all the wonderful curiosities mathematics holds. Board games, sport, risk and the wisdom of crowds courtesy of Marcus du Sautoy are all on the menu.

Over two days you can immerse yourself in a wide range of events, with something for everyone, no matter what your age or prior mathematical experience. 

All events are free to attend. Some require pre-booking. For the entire programme, please click here.

Tuesday, 3 April 2018

Alain Goriely and Mike Giles made SIAM Fellows

Oxford Mathematicians Alain Goriely and Mike Giles have been made Fellows of the Society for Industrial and Applied Mathematics (SIAM). Alain is recognised for his "contributions to nonlinear elasticity and theories of biological growth" while Mike receives his Fellowship for his "contributions to numerical analysis and scientific computing, particularly concerning adjoint methods, stochastic simulation, and Multilevel Monte Carlo."

Alain is Professor of Mathematical Modelling in the University of Oxford where he is Director of the Oxford Centre for Industrial and Applied Mathematics (OCIAM) and Co-Director of the International Brain Mechanics and Trauma Lab (IBMTL). He is an applied mathematician with broad interests in mathematics, mechanics, sciences, and engineering. His current research also include the modelling of new photovoltaic devices, the modelling of cancer and the mechanics of the human brain. He is author of the recently published Applied Mathematics: A Very Short Introduction. Alain is also the founder of the successful Oxford Mathematics Public Lecture series. You can watch his recent Public Lecture, 'Can Mathematics Understand the Brain' here.

Mike is Professor of Scientific Computing in the University of Oxford. After working at MIT and the Oxford University Computing Laboratory on computational fluid dynamics applied to the analysis and design of gas turbines, he moved into computational finance and research on Monte Carlo methods for a variety of applications. His research focuses on improving the accuracy, efficiency and analysis of Monte Carlo methods. He is also interested in various aspects of scientific computing, including high performance parallel computing and has been working on the exploitation of GPUs (graphics processors) for a variety of financial, scientific and engineering applications.

Monday, 19 March 2018

Knots and surfaces - the fascinating topology of n-manifolds

Oxford Mathematician Andras Juhasz discusses and illustrates his latest research into knot theory.

"We can only see a small part of Space, even with the help of powerful telescopes. This looks like 3-dimensional coordinate space, but globally it might have a more complicated shape. An n-dimensional manifold, or n-manifold in short, is a space that locally looks like the standard n-dimensional coordinate space, whose points we can describe with n real coordinates. Topology considers such spaces up to continuous or smooth deformations, as if they were made out of rubber.

The only connected 1-manifolds are the real line and the circle. 2-dimensional manifolds are also called surfaces. The closed oriented (or 2-sided) surfaces are the sphere, the surface of a doughnut (the torus), or the surface of a doughnut with several holes. The number of holes is called the genus of  the surface, and is an example of a topological invariant: an algebraic object (e.g., a number, polynomial, or vector space) assigned to a space that is unchanged by deformations. We have already seen that we live in a 3-manifold, and, if we add the time dimension, in a 4-dimensional spacetime.


2-manifolds (please view all films in Chrome, Firefox or Explorer):


Genus                 0                                     1                                               2


The theory of 1- and 2-manifolds is classical. Surprisingly, dimensions greater than 4 are simpler than dimensions 3 and 4, due to the fact that there is enough space to perform a certain topological trick that allows one to reduce the classification problem to algebra. The focus of modern topology is hence in dimensions 3 and 4.  While 3-manifold topology is closely related to geometry, the theory of smooth 4-manifolds is more analytical. In dimension 4, the difference between smooth and continuous deformations becomes essential. For example, there is just one 4-manifold that looks like 4-dimensional coordinate space up to continuous deformation, but infinitely many of these are smoothly different.

A knot is a circle embedded in 3-space, up to deformation. (Topologically, a knot on a string is always trivial, as one can just pull one end along the string itself until the knot disappears.) A link is a collections of knots that link with each other (hence the name). These play an important role in low-dimensional topology, since every 3- and 4-manifold can be described by a link whose components are each labelled by an integer.

Deformation of an unknot:





Knot Floer homology is an invariant of links defined independently by Ozsváth-Szabó and Rasmussen in 2002. It assigns a finite-dimensional vector space to every link, and contains important geometric information.

Two links are cobordant if they can be connected by a surface in 4-space. If we think of the fourth coordinate as time, each time slice gives a (possibly singular) link. As time varies from say 0 to 1, we obtain a movie of links. In a recent paper published in Advances in Mathematics, I have shown that a link cobordism induces a linear map on knot Floer homology. This can be used to understand the possible surfaces links can bound in 4-space, which is closely related to the topology of smooth 4-manifolds".

A link cobordism:





Thursday, 15 March 2018

Understanding plasma-liquid interactions

Oxford Mathematician John Allen, Professor Emeritus of Engineering Science, talks about his work on the electrohydrodynamic stability of a plasma-liquid interface. His collaborators are Joshua Holgate and Michael Coppins at Imperial College.

 '"The study of plasma-liquid interactions is an increasingly important topic in the field of plasma science and technology with applications in nanoparticle synthesis, catalysis of chemical reactions, material processing, water treatment, sterilization and plasma medicine. This particular work is motivated by the plasma-liquid interactions inherent in magnetic confinement fusion devices, such as tokamaks, either due to melt damage of the metal walls or in new liquid metal divertor concepts. The ejection of molten droplets has been observed in both cases and is of considerable concern to the operation of a successful fusion device. Understanding the stability of the liquid metal surface is a critical issue.

Previously-studied instabilities of liquid metal surfaces in tokamaks include a Kelvin-Helmholtz instability due to plasma flow across the metal surface, a Rayleigh-Taylor instability driven by the j × B force due to a current flowing in the metal, a Rayleigh-Plateau instability of the liquid metal rim around a cathode arc spot crater, and droplet emission from bursting bubbles which are formed by liquid boiling or absorption of gases from the plasma. However, none of these studies considers the effect of the strong electric fields and ion flows in the sheath region between the plasma and the liquid surface despite the observations of electrical effects such as arcing, which cause considerable damage to the tokamak wall, and enhanced droplet emission rates from electrically-biased surfaces. Furthermore electrostatic breakup has been identified as an important process for liquid droplets in plasmas.

Instabilities driven by electric fields, i.e. electrohydrodynamic (EHD) instabilities, at the interface between a conducting liquid and vacuum, were originally studied by Melcher and subsequently by Taylor and McEwan. Melcher’s marginal stability criterion was invoked by Bruggeman et al. in order to explain the filamentary structure of a glow discharge over a water cathode and, additionally, to explain the instability of an electrolytic water solution cathode from an earlier experiment. Earlier evidence for EHD instabilities of the plasma-liquid interface appears in an experiment on unrelated work where an arc spot occasionally formed on an electrically-isolated mercury pool which was in contact with the plasma. Another EHD effect, the deformation of a liquid surface into a Taylor cone, has recently been used to form the cathode of a corona discharge.

Our work investigates the EHD stability of a plasma-liquid interface with a linear perturbation analysis. Melcher’s stability criterion is found to apply to short-wavelength perturbations of the surface. However the fast-moving ions in the sheath provide a significant pressure on the liquid surface which can overcome the electric stress for long-wavelength perturbations. This effect has been neglected in previous studies and provides an overall increase in the critical voltage which must be applied to the surface in order to make it unstable. This effect is encouraging for the ongoing development of new plasma-liquid technologies."

Thursday, 15 March 2018

John Ball wins Leonardo da Vinci Award

Oxford Mathematician John Ball has won the European Academy of Sciences Leonardo da Vinci award. The award is given annually for outstanding lifetime scientific achievement. In the words of the Committee,  "through a research career spanning more than 45 years, Professor Ball has made groundbreaking and highly significant contributions to the mathematical theory of elasticity, the calculus of variations, and the mathematical analysis of infinite-dimensional dynamical systems."

Sir John Ball FRS is Sedleian Professor of Natural Philosophy in the University of Oxford and Director of the Oxford Centre for Nonlinear Partial Differential Equations. He is a Fellow of The Queen's College.



Monday, 12 March 2018

Oxford Summer School on Economic Networks 25-29 June 2018 - register by 15 March

The Oxford Summer School on Economic Networks, hosted by Oxford Mathematics and the Institute of New Economic Thinking, aims to bring together graduate students from a range of disciplines (maths, statistics, economics, policy, geography, development, ..) to learn about the techniques, applications and impact of network theory in economics and development. 
We look forward to welcoming a large number of world renowned experts in economic networks and complexity science. Confirmed speakers for the 2018 edition include Fernando Vega-Redondo, Mihaela van der Schaar, Rama Cont, Doyne Farmer, Pete Grindrod, Renaud Lambiotte, Elsa Arcaute and Taha Yasseri. Tutorials and lectures include social networks, games and learning, financial networks, economic complexity and urban systems.
Alongside a rigorous academic schedule, the summer school also includes a walking tour of the historic university and city centre, a punting trip on the river Cherwell and a dinner in one of Oxford's historic colleges.
The deadline for applications is March 15th - more information is available here. Please contact us at with any questions.