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Nonlinear and Biomedical Physics

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We study oscillatory dynamics, theoretically, numerically and from measured data. We have pioneered the understanding of a living system as a collection of time-varying oscillatory processes.

Time-variability is inherent, on account of the system's ability to adjust its characteristic frequencies and adapt to changing circumstances. We have introduced a new class of systems and named them chronotaxic systems (from Chronos - time, and taxis - order). Chronotaxic systems are non-autonomous self-sustained oscillators that possess point attractor. They provide a route to stability in highly complex systems. Until now, such systems have mostly been treated as stochastic, whereas we have recently shown that they can be entirely deterministic. We are currently developing methods for inferring the dynamics of chronotaxic systems from real data.

By applying ideas and methods from nonlinear and stochastic dynamics, we study the fundamental physical properties of living systems. On the nanoscale, we examine ion channels. We treat the permeation of ion channels as a problem in stochastic nonlinear dynamics and electrostatics, illuminating the long-standing conduction-selectivity paradox. That is to say, the issue of how channels can be highly selective for particular species of ion, yet still conduct at an enormous rate, almost as though they were just open holes in the cell membrane.

Oscillations are a universal phenomenon in living systems. We investigate oscillatory behaviour on all scales and levels of complexity - from the cell membrane potential to cardiovascular and brain dynamics. We are especially interested in the influence of the oscillators on each other - their mutual interactions and coupling functions. The resultant modulation and synchronisation phenomena occur in physiology in just the same way that they do for coupled oscillators in physics. Comparison of the model phenomena with physiological data measured for healthy subjects in our laboratory and patients in our collaborating hospitals is illuminating and characterising diverse conditions and diseases.


We also develop software toolboxes to investigate these phenomena.

Key Research

  • Chronotaxic systems - theory and methods for data analyses
  • Networks of oscillators
  • Time-varying Kuramoto model of phase oscillators
  • Analysis of time-varying dynamics
  • Bayesian inference for time-varying systems
  • Nonlinear mode decomposition
  • Ion channels
  • Oscillations in cell membrane potential
  • Applications
  • Cancer as a state of decoupled oscillators at the endothelial level
  • Hypertension and phase coherence between cardiovascular oscillators
  • Cardiovascular and brain dynamics in anaesthesia
  • Cardio-respiratory coupling function as a marker of ageing
  • Spatio-temporal brain dynamics in autism

For full information on the work of the Nonlinear and Biomedical Physics research group, follow this link to access our research portal.

Current PhD Opportunities


  • Ionic Coulomb blockade, conduction and selectivity in biological ion channels


    The research aims to apply ideas from condensed matter physics (quantum dots) to reach an understanding of ion channel conduction and selectivity based on the recently discovered phenomenon of ionic Coulomb blockade also allowing for the effect of dehydration. More generally, we aim to create a statistical theory of the permeation process taking explicit account of non-identical binding sites in the channel, the possibility of more than one ion at a binding site, and the consequences of ions being able to pass each other (i.e. non-single-file conduction).


    • PVE McClintock
    • A Stefanovska
    • DG Luchinsky


    • Biological and Life Sciences Department
    • University of Warwick
    • Rush University (Chicago)
  • Quantitative assessment of Autistic Spectrum Disorder


    Professor Aneta Stefanovska

    Autism Spectrum Disorder (ASD) is a complex neurodevelopmental condition, characterised by persistent deficits in verbal and nonverbal communication as well as restricted and repetitive behaviour, interests and activities. Symptoms most commonly appear in children around the age of 3 or sometimes as early as 1--2 years old. The condition is relatively common, with a reported prevalence of around 1.5%. Current methods of diagnosing ASD rely purely on behavioural observations, and no physiological diagnostic test exits. Hence diagnosis often comes very late, even in adulthood, affecting the individual’s life chances. There is widespread interest in devising training or other remedial action to alleviate the symptoms and (hopefully) normalise the ASD child’s developmental pathway. But for this approach to be effective, there needs to a quantitative physiological method of identifying ASD and monitoring its development.

    In a recent project in collaboration with Blackpool Victoria Hospital, we investigated the network couplings between brain waves recorded by electroencephalograph (EEG) in very young children, to seek possible differences between ASD children and normally developing children. We indeed found characteristic differences, and we believe that these can provide the basis of a novel instrument for the detection and quantitative assessment of ASD

    The project will be carried out in collaboration with MyMind, a life science company based in Vienna, Austria. MyMind is developing an EEG-based neurofeedback device Brain Hero®, which is designed for convenient home-use and provides personalized therapy for ASD. The quantitative way of assessing the condition will be based on novel methods for data analysis developed at Lancaster and now available in the MODA toolbox https://github.com/luphysics/MODA.

    This interdisciplinary enterprise will develop algorithms and software for Brain Hero®. Data being recorded in Switzerland, Portugal and Austria will be made available by the company. The PhD research will therefore involve theory, data analysis and computation.

    The applicant will be expected to have a first or upper second-class degree in physics, applied mathematics, natural sciences, computational neuroscience, or biomedical engineering.

    Interested candidates should contact Professor Aneta Stefanovska for further information.

  • Networks of coupled nonautonomous oscillators to model neurovascular coupling in the brain


    Professor Aneta Stefanovska

    Neurovascular coupling is essential for the functioning of the brain. Recent studies show that its efficiency changes with ageing or dementia. However, an efficient model of the interactions between the vasculature, astrocytes, and the neurones in the brain is still missing. Current models are mainly based on linear approaches and use a large number of differential equations to describe flows and concentrations of metabolites in relevant compartments of the brain. Such models are based on closed-system assumptions and focus on relationships between magnitudes of physical quantities involved. This project will investigate the potential advantages of models based on networks of phase oscillators that do not include any closed-system assumptions. Coupled nonautonomous phase oscillators will be used to represent the metabolic processes occurring within brain cells. Within the context of the model to be developed, the interaction between astrocytes and neurons and their changes with ageing and dementia will be investigated. The modelling will be tested by comparison with recent experimental studies in healthy subjects of different age, as well as with studies in subjects with Huntington’s and Alzheimer’s diseases.

    The applicant will be expected to have a first or upper second-class degree in physics, applied mathematics, natural sciences, computational neuroscience, or biomedical engineering.

    Interested candidates should contact Professor Aneta Stefanovska for further information.

  • Non-invasive evaluation of endothelial function


    Professor Aneta Stefanovska

    By bringing together optics, modern computational facilities, the growing understanding of nonlinear oscillators and their mutual interactions, and wireless connectivity, it is planned to create a novel diagnostic instrument to determine the health of the human endothelium – the inner lining of all the blood vessels. In each individual, the endothelium occupies an area equal to a football pitch making it a major, albeit under-appreciated, organ of the body. It orchestrates the dynamics of blood circulation including the continuous distribution and exchange of nutrients and oxygen with all the cells of the body and the removal of waste products. Recently, the health state of the endothelium has been shown to play a crucial role in determining the severity of Covid-19. Although the health of the endothelium is of crucial importance for general health, it has been extremely difficult to measure up to now. So, the new “endotheliometer” is likely to be valuable to GPs and other health professionals.

    This interdisciplinary project will be based on novel methods for data analysis developed at Lancaster now available in the MODA toolbox https://github.com/luphysics/MODA.

    The applicant will be expected to have a first or upper second-class degree in physics, applied mathematics, natural sciences, or biomedical engineering.

    Interested candidates should contact Professor Aneta Stefanovska for further information.

  • Is the resting membrane potential really resting?


    Professor Aneta Stefanovska

    The famous Hodgkin-Huxley model describes an action potential in the axon of a neurone. It is an excellent example of how, by combining experiment and theory, physics can help resolve important questions in biology. It is arguably still the most realistic model of a living system. However, it assumes that the voltage across the membrane is constant, and to fulfil this condition in the experiments the voltage was clamped. In reality, however, the voltage continuously fluctuates in living cells, and the physics behind the fluctuations of the membrane potential therefore needs to be revisited. Recent advances in technology now enable the simultaneous recording of ionic concentrations, PH, cell volume, and production of the ATP that fuels the operation of ion pumps in the membrane.

    This project aims to propose a new physics of the living cell by combining the experimental data obtained from simultaneous measurements, time-series analysis using novel methods developed at Lancaster now available in the MODA toolbox https://github.com/luphysics/MODA, and the new physics of nonautonomous dynamical systems. Phase coherence and synchronization will be analysed to assess stability of interactions, to characterise the normal and dysfunctional states a cell, and to build the new model.

    The model will help integrate existing biological knowledge about individual components of the cell. It will provide unifying principles of functioning for both excitable and nonexecutable cells, and will thus pave new ways of modelling the brain in health and disease.

    The applicant will be expected to have a first or upper second-class degree in physics, applied mathematics, natural sciences, or computational biology.

    Interested candidates should contact Professor Aneta Stefanovska for further information

  • Nonlinear wave interactions and the origin of ocean rogue waves


    Professor Aneta Stefanovska, Physics

    Dr Suzana Ilic, Environmental Science

    Professor Peter McClintock, Physics

    Occasionally, rogue waves – with wave heights much larger than those of their neighbours – appear on the ocean and can sometimes overwhelm even the largest vessels e.g. supertankers. Their origins are still a mystery, but a theory suggesting that their creation mechanism involves nonlinear interactions between smaller, conventional, usually wind-blown, surface waves is the best candidate to explain their formation. To seek experimental evidence in order to test this idea, experiments have been carried out in the Marintek wave basin in Trondheim, Norway. The result is a large volume of time series data, some of which shows clear evidence of rogue waves, but which has yet to be analysed. The PhD project is to analyse the Marintek data using state-of-the-art time-series analysis methods, many of which have been developed at Lancaster and are available in MODA toolbox https://github.com/luphysics/MODA in order to investigate the hydrodynamic conditions under which rogue waves are created. In particular, evidence will be sought for the growth of rogue waves through nonlinear mutual phase interactions between smaller waves. It is a challenging problem involving spatio-temporal dynamics, but it is clear that the results could be extremely important.

    The applicant will be expected to have a first or upper second-class degree in physics, applied mathematics, or other appropriate qualification.

    Interested candidates should contact Professor Aneta Stefanovska for further information.

  • Chronotaxic dynamics on a superfluid surface?


    Professor Aneta Stefanovska

    Dr Dmitry Zmeev

    Professor Peter McClintock

    The 2-D electron system that can be created on the surface of superfluid He-4 has some remarkable properties. The electrons can move freely, without dissipation, over the interface between the vacuum above and a surface that is almost perfect. There is compelling evidence suggesting that, under the right conditions, this system exhibits chronotaxic dynamics – a phenomenon previously associated exclusively with biological systems.

    The identification of this new class of non-autonomous oscillatory dynamical systems by the Lancaster group represented a major advance in the understanding of time-varying dynamics. These are oscillators whose characteristic frequencies vary in time, in contrast to e.g. the simple pendulum and many other familiar physical oscillators. Chronotaxic systems can be regarded as one manifestation of the thermodynamically open systems that abound in nature, and especially in biology. In collaboration with scientists at Riken in Japan, we have identified seemingly chronotaxic behaviour of the currents recorded for the 2-dimensional electron gas on the superfluid surface.

    The aim of this PhD research project is to explain the physical origin of the oscillations of variable frequency observed in the experiments, and to provide a theoretical model of the experimental results, thus expanding and generalising the theory of chronotaxic non-autonomous dynamical systems.

    The applicant will be expected to have a first or upper second-class degree in physics, applied mathematics, natural sciences, or engineering.

    Interested candidates should contact Professor Aneta Stefanovska for further information.

  • Biomimetic theory, design and control of ionic nanodevices


    Professor Aneta Stefanovska

    Professor Peter McClintock

    Dr Dmitri Luchinsky

    For a billion years, life has been crucially dependent on ion channels for selective control of the fluxes of ions into and out of biological cells, with evolution fine-tuning each kind of channel to be optimal in its particular role. Very recently, humans have fabricated artificial channels and pores from solid state materials, aiming to emulate and extend many of the functions of biological channels in more robust formats. A whole new sub-nanoscale technology has started to develop, with applications to e.g. fuel cells, water desalination, gas and isotope separation, lithium extraction, DNA sequencing, water pumps, field effect ionic transistors, and “blue energy” harvesting.

    Not surprisingly, artificial channels are still, in general, much less efficient than biological ones. For example, they are less selective for particular ionic species and the fluxes they pass tend to be smaller. They are difficult to design, partly because there is still no satisfactory general theory of how an ion permeates through a channel. Hence design usually relies on experiments and heavy-duty molecular dynamics simulations coupled with trial-and-error – which is slow, and therefore expensive, because the parameter space is huge.

    We therefore propose a different approach, building on our 2015 discovery of Coulomb blockade in biological ion channels, on our new statistical physics theory of the ionic permeation process, and on our recent and ongoing numerical simulations of pores and channels in artificial membranes. There is probably a great deal to learn from how Nature has “designed” biological channels through evolution over hundreds of millions of years, so that biomimetic approaches are likely to be useful in understanding and designing artificial channels.

    The aims of the project are to develop theory and numerical tools that enable the prediction and control of free energy landscapes, selectivity and conductivity of artificial nanodevices. These methods will be applied to the design and optimization of nano-pumps, nano-sensors, and energy harvesting nanodevices. It is expected that the successful applicant will use molecular dynamics and Brownian dynamics simulations to verify and validate the results obtained.

    We are looking for a student with enthusiasm for theoretical physics and some prior experience of computational and numerical work.

    Interested candidates should contact Professor Aneta Stefanovska for further information.

Postgraduate Training

The Nonlinear and Biomedical Physics group runs training workshops for postgraduate students throughout the year, covering both subject-specific and more general research skills.

The form and content of the workshops are determined through dialogue with the PhD students so that the most effective training can be provided. Although the events are designed to meet the needs of students in Nonlinear and Biomedical Physics, they are also available to other postgraduate students on request, as well as to MPhys project students working within the group. Additional training is offered by the Faculty of Science and Technology, ISS, and the Library.

Our students attend a variety of scientific conferences, for which they receive support in the preparation of posters and oral presentations. They also have the opportunity to develop their presentation skills via participation in the Department’s outreach programme. They play an important role in working with the A-level and internship students that visit our group during the summer.

Recent tutorials, workshops and conferences include:

  • Nonlinear time series analysis methods (October 2015) – Professor A. Stefanovska
  • Ionic Coulomb blockade oscillations and the physical origins of permeation, selectivity, and their mutation transformations in biological ion channels (October 2015) – Professor PVE McClintock (chair)
  • Biological Oscillations ESGCO–2016 Conference (10-14 April 2016) [Covered talks on topics including biological ion channels, cellular, cardiovascular and brain dynamics, data analysis methods, theories of coupled oscillators and networks, non-autonomous dynamics] - Professor A. Stefanovska (chair)
  • Reconstructing non-autonomous dynamics (November 2015) – Professor A. Stefanovska
  • Introduction to MatLab and wavelet analysis tutorial November 2015– Dr G. Lancaster
  • Inverse approaches to dynamical systems tutorial February 2016 – Dr G. Lancaster
  • Physics of living systems (February 2016)– Professor A. Stefanovska

Our students are part of two Horizon 2020 Marie Skłodowska-Curie training networks:

  • Complex Oscillatory Systems: Modelling and Analysis (COSMOS): Innovative Training Network – European Joint Doctorate
  • Critical Transitions in Complex Systems (CRITICS)

Students have the possibility of attending summer schools organised by both networks:

  • First COSMOS school and workshop, Florence, Italy, November 2015.
  • Second COSMOS school and workshop, Aberdeen, UK, 27th June - 6th July 2016.
  • Workshop on Critical Transitions in Complex Systems, Copenhagen, Denmark, 4-9 September 2016.

as well as a variety of national and international summer schools relevant to their projects.

Nonlinear and Biomedical Physics Toolboxes

Here you can find GitHub links for the numerical toolbox MODA and algorithms developed by the Nonlinear and Biomedical Physics group at Lancaster University for analysing time-series data, either measured or numerically generated.


MODA is a user-friendly toolbox, written both in MatLab and in Python. It is designed for analysing time-series that result from multiscale oscillatory dynamics. It encompasses non-autonomous dynamics in which frequencies vary in time. MODA provides time-frequency spectra and enables detection of instantaneous frequencies. It includes an algorithm to detect high harmonics of time-varying frequencies.

MODA also contains algorithms for the investigation of interactions between oscillatory processes, including wavelet phase coherence and phase shifts, wavelet bispectral analysis, and coupling functions obtained through dynamical Bayesian inference.

MODA includes methods for univariate and simultaneously recorded/generated multivariate time-series.

Other downloads

These are algorithms and toolboxes that can be run individually. Some of them are included in MODA. They are all written in MatLab.