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.
Toolboxes
We also develop software toolboxes to investigate these phenomena.
This project aims to develop a new physics of the living cell, revealing how physical laws govern the dynamic organisation and self-regulation of life at the cellular scale. How do living cells maintain coherence and function amid continual internal fluctuations? Addressing this fundamental question requires a theoretical framework that captures the inherently time-dependent and complex dynamics of living systems.
The classic Hodgkin-Huxley model remains a landmark in biophysics, showing how physics can explain neuronal activity. Yet it assumes constant membrane voltage - a simplification far from the reality of fluctuating, energy-driven living systems. Recent advances now allow simultaneous measurements of ionic concentrations, pH, cell volume, and ATP production, offering a unique opportunity to develop a more complete physical description of life at the cellular level.
This PhD will integrate these rich experimental datasets with modern theories of nonautonomous dynamical systems and advanced time-series analysis, using tools such as Lancaster’s MODA toolbox. The research will explore phase coherence, synchronisation, and stability in cellular processes, seeking unifying principles that govern both excitable and non-excitable cells.
Working within Lancaster’s internationally recognised Nonlinear and Biomedical Physics Group, the successful candidate will contribute to the theoretical foundations of living matter - developing models that may transform our understanding of how cells, and ultimately the brain, function in health and disease such as cancer or diabetes.
Candidate profile We invite applications from outstanding students (first or upper second class, or equivalent) with a strong background in physics, applied mathematics, computational biology, biophysics, theoretical biology, or other quantitative sciences involving mathematical and computational modelling. A keen interest in the physics and mathematics of living systems and interdisciplinary research is essential.
This interdisciplinary PhD project offers an exciting opportunity to contribute to the development of a novel, noninvasive diagnostic tool for early-stage skin cancer detection -specifically melanoma, the most aggressive form of skin cancer.
By integrating optical sensing, nonlinear dynamics, modern computational techniques, and machine learning, this project aims to build a clinically viable, wearable device capable of distinguishing cancerous tissue from healthy skin with high precision.
Project Overview
Cancerous cells, particularly in melanoma, grow rapidly and often outpace the development of supporting vasculature. This results in distinctive blood flow dynamics that differ significantly from those in healthy or benign skin tissue. Using advanced methods from nonlinear systems analysis, including tools from our open-source platform MODA, we have already demonstrated 100% specificity and 100% sensitivity in differentiating melanoma from atypical naevi and other lesions in initial studies.
The next phase of the research will involve:
Developing a wearable prototype for continuous or point-of-care use,
Applying machine learning to enhance diagnostic accuracy and speed,
Collaborating with a Lancaster-based spin-out company to translate the research into a deployable medical device.
Why Apply?
This PhD will provide training and research experience at the intersection of physics, biomedical engineering, and data science, including:
Nonlinear dynamics and complex systems
Optical instrumentation and signal processing
Machine learning and classification techniques
Clinical translation of physics-based diagnostics
You will work within a supportive, interdisciplinary research environment and contribute to real-world medical innovation with the potential for significant healthcare impact.
Candidate profile
Applicants should hold, or expect to obtain, a first-class or upper second-class degree in physics, applied mathematics, natural sciences, or biomedical engineering (or a closely related discipline). A strong interest in interdisciplinary research, especially at the interface between physics and healthcare technology, is essential.
The lungs and heart can be modelled as coupled nonlinear oscillators whose interaction governs essential physiological rhythms. A prime example is respiratory sinus arrhythmia, where the respiratory cycle modulates the heart’s beat frequency. Classical models have typically employed linear or autonomous oscillator frameworks, which fail to fully capture the inherently time-dependent and nonlinear dynamics of cardio-respiratory coupling observed experimentally.
This project adopts a physics-driven approach by applying the theory of nonautonomous dynamical systems, which explicitly incorporate external time-dependent forcing and parameter variability, to model the cardiorespiratory interaction as a network of coupled oscillators with time-varying coupling strengths.
Importantly, this work will address fundamental theoretical challenges surrounding interactions between nonautonomous oscillators. Unlike autonomous systems, nonautonomous oscillators lack time-invariant phase definitions and exhibit complex stability properties, complicating the analysis of phase synchronisation, intermittency, and long-term behaviour. Developing appropriate phase reduction techniques and rigorous mathematical frameworks to characterise these coupled time-dependent oscillators is a central goal.
You will work with experimental datasets recorded under varying physiological states -awake, anaesthetised, and different ambient conditions - to analyse transient synchronisation and dynamic coupling. Alongside theoretical modelling and numerical simulations,machine learning methods will be employed to identify structure in complex physiological time series, classify coupling regimes, and support model validation.
The ultimate aim is to derive a robust theoretical framework and predictive model that capture the adaptive mechanisms underlying cardio-respiratory synchronisation, providing foundational insight with potential implications for guiding future therapeutic strategies in patients requiring assisted respiration (e.g., asthma or anaesthesia).
Candidates should have a strong background in physics, mathematics, or natural sciences.
Can we build a model of the brain that reflects its true nature — not as an isolated system, but as a living, dynamic network?
This doctoral research project offers the opportunity to develop such a model.
Neurovascular coupling - the dynamic interaction between neurons, astrocytes, and blood vessels - is essential for healthy brain function. As we age, or in conditions such as Alzheimer’s disease and Huntington’s disease, this delicate coordination begins to break down. Understanding how and why this happens remains one of the central challenges in neuroscience.
Most existing models rely on large systems of differential equations that traditionally treat the brain as a closed physical system, assuming limited exchange of matter or energy with its surroundings. While mathematically robust, these assumptions can restrict the models’ ability to capture the brain’s complex, open, and dynamic metabolic interactions.
This project will take a fundamentally different approach. It aims to develop a biologically motivated model based on coupled nonautonomous oscillators. These oscillators will represent key metabolic processes within neurons and astrocytes, allowing the model to reflect the brain’s inherently dynamic and interactive nature. By avoiding closed-system constraints, the model will better capture the evolving patterns of metabolic activity in both health and disease.
Beyond theoretical insights, this model holds potential for practical applications, including the development of devices for noninvasive evaluation of neurovascular unit function. Such tools could transform diagnosis and monitoring of ageing and neurodegenerative diseases.
The model will be tested and refined using recent experimental data from healthy individuals of different ages, as well as from individuals with Alzheimer’s disease and Huntington’s disease.
Who should apply
We are seeking a motivated and capable individual with a strong academic background in physics, mathematics, natural sciences, or computational neuroscience. A keen interest in modelling complex biological systems and working across disciplinary boundaries is essential.
For a billion years, life has depended 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 the functions of biological channels in more robust formats. This rapidly-developing sub-nanoscale technology has applications to e.g. fuel cells, water desalination, gas and isotope separation, lithium extraction, DNA sequencing, field effect ionic transistors, and “blue energy” harvesting.
Artificial channels are difficult to design, but we propose a biomimeticapproach that learns from Nature. It builds on our discovery of Coulomb blockade in biological ion channels, on our statistical physics theory of permeation, and on our recent and ongoing numerical simulations of pores and channels in artificial membranes. The project will develop theory and numerical tools to predict and control their free energy landscapes, selectivity and conductivity. The new understanding will be applied to nano-pumps, nano-sensors, and energy-harvesting nanodevices.
We seek a student with enthusiasm for theoretical physics with interdisciplinary applications, with some prior experience of computational and numerical work. They will be expected to have a first or upper second-class degree in physics, applied mathematics or natural sciences, or the equivalent.
Turbulence remains one of the most important unsolved problems in physics. At ultra-low temperatures, quantum turbulence (QT) in superfluid helium offers a clean platform to study turbulence through quantised vortices - identical, discrete structures that allow for detailed, fundamental insights.
This EPSRC-funded project involves two experimental approaches:
A torsional oscillator setup to study vortex pinning and critical velocities
A levitated superconducting sphere, moved through superfluid ⁴He to explore QT generation
The student will analyse large experimental datasets using our nonlinear dynamics toolbox, MODA, and apply machine learning methods to identify and classify turbulent states. These skills have wide relevance across physics and beyond.
Candidate Profile
Applicants should hold, or expect to obtain, a first or upper second-class degree in physics, applied mathematics, or a related field. Prior interest or experience in fluid dynamics, data analysis, or machine learning is beneficial.
Electrons confined to the surface of superfluid helium exhibit remarkable properties, including frictionless motion along an interface that is nearly atomically smooth. Recent work by the Lancaster group has shown that, under specific conditions, this system exhibits chronotaxic dynamics - a form of non-autonomous oscillatory behaviour previously observed only in biological systems.
The discovery of this new class of time-varying dynamical systems marks a significant advance in the understanding of complex oscillators. Unlike classical oscillators such as the simple pendulum, chronotaxic systems have characteristic frequencies that evolve over time. These systems serve as examples of thermodynamically open systems commonly found in nature, especially within biological contexts.
This PhD project will investigate the physical origins of the variable-frequency oscillations observed experimentally. The candidate will develop theoretical models to explain the data, extending the theory of chronotaxic non-autonomous dynamical systems and exploring potential links to quantum computing.
Applicants should hold a first or upper second-class degree in physics, applied mathematics, natural sciences, or a related discipline.
Rogue waves - rare and extreme ocean waves that appear without warning - remain one of the most compelling unsolved problems in ocean physics. These unusually large waves can damage even the largest vessels and offshore structures, but the precise mechanisms behind their formation are still under debate.
This PhD project investigates a leading hypothesis: that rogue waves arise from phase coherence - a synchronised alignment of wave phases that focuses energy into a single, large wave. You will explore this hypothesis by analysing a unique experimental dataset from a controlled wave basin, where rogue-wave-like events have been captured. Additional real-world ocean data will also be available.
You will apply advanced nonlinear time-series analysis tools, including the MODA toolbox developed at Lancaster, alongside machine learning techniques such as unsupervised clustering and neural networks, to detect early indicators of rogue wave formation. The project also includes numerical simulations of wave propagation and interaction, allowing you to test the role of phase coherence in virtual environments.
In addition to advancing our understanding of the physics of extreme ocean waves, the project explores how these mechanisms could be applied to wave energy harvesting - with the goal of optimising systems that capture and convert ocean wave energy, particularly during high-intensity events.
This interdisciplinary project is ideal for students with a background in physics, applied mathematics, natural sciences, or engineering, and an interest in nonlinear dynamics, ocean physics, data science, machine learning, or renewable energy.
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
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.
