# Professor Philip Aston

## Professor

Qualifications: BSc, PhD, MIMA, CMath

Email: p.aston@surrey.ac.uk

Phone: Work: 01483 68 2631

Room no: 37 AA 04

## Further information

## Biography

- BSc Mathematics and Computer Science (First class honours), Brunel University, 1979-1983
- PhD, supervisor Prof John Whiteman, Brunel University, 1983-1986
- SERC-funded postdoc working with Prof John Toland and Prof Alastair Spence, University of Bath, 1986-1989
- Department of Mathematics, University of Surrey, 1989 onwards

Further details can be found on my personal web page.

## Research Interests

Bifurcation theory, symmetry, computation of Lyapunov exponents using spatial integration, the dynamics of bouncing balls, pharmacokinetics/pharmacodynamics (PKPD), non-exponential radioactive decay, attractor reconstruction methods for extracting information from physiological time series.

Further details can be found on my personal web page.

## Publications

### Journal articles

- 'Extending Slow Manifolds Near a Degenerate Transcritical Intersection in Three Dimensions'.
Research Perspectives CRM Barcelona: Nonsmooth Dynamics,

[ Status: Accepted ]**Repository URL:**http://epubs.surrey.ac.uk/811071/#### Abstract

Motivated by a problem from pharmacology, we consider a general two parameter slow-fast system in which the critical set consists of a one dimensional manifold and a two dimensional manifold, intersecting transversally at the origin. Using geometric desingularisation, we show that for a subset of the parameter set there is an exchange of stabilities between the attracting components of the critical set and the direction of the continuation can be expressed in terms of the parameters.

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(2016) - 'A Mathematical Analysis of Rebound in a Target-Mediated Drug Disposition Model. II. With Feedback'.
Journal of Mathematical Biology,
**Repository URL:**http://epubs.surrey.ac.uk/812644/#### Abstract

We consider the possibility of free receptor (antigen/cytokine) levels rebounding to higher than the baseline level after the application of an antibody drug using a target-mediated drug disposition model. It is assumed that the receptor synthesis rate experiences homeostatic feedback from the receptor levels. It is shown for a very fast feedback response, that the occurrence of rebound is determined by the ratio of the elimination rates, in a very similar way as for no feedback. However, for a slow feedback response, there will always be rebound. This result is illustrated with an example involving the drug efalizumab for patients with psoriasis. It is shown that slow feedback can be a plausible explanation for the observed rebound in this example.

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(2016) - 'Feasibility of parameter estimation in hepatitis C viral dynamics models'.
Journal of Inverse and Ill-Posed Problems,
**Repository URL:**http://epubs.surrey.ac.uk/810727/#### Abstract

Methodologies are presented for assessing the feasibility of parameter estimation in nonlinear ordinary differential equation (ODE) models. These methods are applied to a recent model for hepatitis C viral dynamics. Subset selection is performed on the model parameters, and maximum likelihood estimation is conducted using available data from the literature.

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(2016) - 'Simulating the influence of plasma protein on measured receptor affinity in biochemical assays reveals the utility of Schild analysis for estimating compound affinity for plasma proteins'.
British Journal of Pharmacology, 172 (21), pp. 5037-5049.doi: 10.1111/bph.13263
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(2015)
- 'The Importance of the Derivative in Sex-Hormone Cycles: A Reason Why Behavioural Measures in Sex-Hormone Studies Are So Mercurial'.
PLoS One, 9 (11) Article number e111891
**Repository URL:**http://epubs.surrey.ac.uk/809291/#### Abstract

To study the dynamic changes in cognition across the human menstrual cycle, twenty, healthy, naturally-cycling women undertook a lateralized spatial figural comparison task on twelve occasions at approximately 3-4 day intervals. Each session was conducted in laboratory conditions with response times, accuracy rates, eye movements, salivary estrogen and progesterone concentrations and Profile of Mood states questionnaire data collected on each occasion. The first two sessions of twelve for the response variables were discarded to avoid early effects of learning thereby providing 10 sessions spread across each participant's complete menstrual cycle. Salivary progesterone data for each participant was utilized to normalize each participant's data to a standard 28 day cycle. Data was analysed categorically by comparing peak progesterone (luteal phase) to low progesterone (follicular phase) to emulate two-session repeated measures typical studies. Neither a significant difference in reaction times or accuracy rates was found. Moreover no significant effect of lateral presentation was observed upon reaction times or accuracy rates although inter and intra individual variance was sizeable. Using a 'phase plane' plot, we demonstrated that hormone concentrations alone cannot be used to predict the response times or accuracy rates. In contrast, we constructed a standard linear model using salivary estrogen, salivary progesterone and their respective derivative values and found these inputs to be very accurate for predicting variance observed in the reaction times for all stimuli and accuracy rates for right visual field stimuli but not left visual field stimuli. The identification of sex-hormone derivatives as predictors of cognitive behaviours is of importance. The finding suggests that there is a fundamental difference between the up-surge and decline of hormonal concentrations where previous studies typically assume all points near the peak of a hormonal surge are the same. How contradictory findings in sex-hormone research may have come about are discussed.

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(2014) - 'A dynamical systems analysis of the data assimilation linked ecosystem carbon (DALEC) models'.
Chaos, 25 (3)
**Repository URL:**http://epubs.surrey.ac.uk/806896/#### Abstract

© 2014 AIP Publishing LLC.Changes in our climate and environment make it ever more important to understand the processes involved in Earth systems, such as the carbon cycle. There are many models that attempt to describe and predict the behaviour of carbon stocks and stores but, despite their complexity, significant uncertainties remain. We consider the qualitative behaviour of one of the simplest carbon cycle models, the Data Assimilation Linked Ecosystem Carbon (DALEC) model, which is a simple vegetation model of processes involved in the carbon cycle of forests, and consider in detail the dynamical structure of the model. Our analysis shows that the dynamics of both evergreen and deciduous forests in DALEC are dependent on a few key parameters and it is possible to find a limit point where there is stable sustainable behaviour on one side but unsustainable conditions on the other side. The fact that typical parameter values reside close to this limit point highlights the difficulty of predicting even the correct trend without sufficient data and has implications for the use of data assimilation methods.

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(2014) - 'Using Computer Based Assessment to Enhance Learning of Matlab Skills'. Mathematics Today, 50 (5), pp. 222-223. . (2014)
- 'Computing the invariant measure and the Lyapunov exponent for one-dimensional maps using a measure-preserving polynomial basis'.
Mathematics of Computation, 83 (288) Article number PII S 0025-5718(2013)02811-6 , pp. 1869-1902.
**Repository URL:**http://epubs.surrey.ac.uk/805803/#### Abstract

We consider a generalisation of Ulam's method for approximating invariant densities of one-dimensional maps. Rather than use piecewise constant polynomials to approximate the density, we use polynomials of degree n which are defined by the requirement that they preserve the measure on n+1 neighbouring subintervals. Over the whole interval, this results in a discontinuous piecewise polynomial approximation to the density. We prove error results where this approach is used to approximate smooth densities. We also consider the computation of the Lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself. Together with using cubic polynomials in the density approximation, this yields a very efficient method for computing highly accurate estimates of the Lyapunov exponent. We illustrate the theoretical findings with some examples.

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(2014) - 'A mathematical analysis of rebound in a target-mediated drug disposition model: I.Without feedback'.
Journal of Mathematical Biology, 68 (6), pp. 1453-1478.
**Repository URL:**http://epubs.surrey.ac.uk/806573/#### Abstract

We consider the possibility of free receptor (antigen/cytokine) levels rebounding to higher than the baseline level after one or more applications of an antibody drug using a target-mediated drug disposition model. Using geometry and dynamical systems analysis, we show that rebound will occur if and only if the elimination rate of the drug-receptor product is slower than the elimination rates of the drug and of the receptor. We also analyse the magnitude of rebound through approximations and simulations and demonstrate that it increases if the drug dose increases or if the difference between the elimination rate of the drug-receptor product and the minimum of the elimination rates of the drug and of the receptor increases. © 2013 Springer-Verlag Berlin Heidelberg.

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(2014) - 'Erratum: Alternating period-doubling cascades (Nonlinearity (2013) 26 (2553))'.
Nonlinearity, 26 (9)
**Repository URL:**http://epubs.surrey.ac.uk/806574/
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(2013) - 'Alternating Period-Doubling Cascades'.
Nonlinearity, 26, pp. 2553-2576.
**Repository URL:**http://epubs.surrey.ac.uk/794551/#### Abstract

We consider period-doubling cascades in two-dimensional iterated maps. We define forward and backward period-doubling bifurcations, and use these concepts to describe an alternating period-doubling cascade in which forward and backward period-doubling bifurcations alternate. By tracking the eigenvalues of a typical map throughout such cascades we show that two-dimensional maps may give rise to two qualitatively different alternating period-doubling cascades. We apply renormalisation theory to one class of alternating period-doubling cascades, and derive universal spatial scalings for such cascades from fixed points of the appropriate renormalisation operator. We also derive universal parameter scalings for these cascades from the eigenvalues of the linearisation of the renormalisation operator, and provide the corresponding eigenfunctions. The theory is illustrated by an example.

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(2013) - 'Reply to the comment by Cleanthes A. Nicolaides'.
EPL, 101 (4)
**Repository URL:**http://epubs.surrey.ac.uk/768197/
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(2013) - 'Is radioactive decay really exponential?'.
Europhysics Letters, 97 (5) Article number 52001
**Repository URL:**http://epubs.surrey.ac.uk/293545/#### Abstract

Radioactive decay of an unstable isotope is widely believed to be exponential. This view is supported by experiments on rapidly decaying isotopes but is more difficult to verify for slowly decaying isotopes. The decay of 14C can be calibrated over a period of 12550 years by comparing radiocarbon dates with dates obtained from dendrochronology. It is well known that this approach shows that radiocarbon dates of over 3000 years are in error, which is generally attributed to past variation in atmospheric levels of 14C. We note that predicted atmospheric variation (assuming exponential decay) does not agree with results from modelling, and that theoretical quantum mechanics does not predict exact exponential decay. We give mathematical arguments that non-exponential decay should be expected for slowly decaying isotopes and explore the consequences of non-exponential decay. We propose an experimental test of this prediction of non-exponential decay for 14C. If confirmed, a foundation stone of current dating methods will have been removed, requiring a radical reappraisal both of radioisotope dating methods and of currently predicted dates obtained using these methods.

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(2012) - 'Mathematical analysis of the pharmacokinetic-pharmacodynamic (PKPD) behaviour of monoclonal antibodies: predicting in vivo potency.'.
J Theor Biol, England: 281 (1), pp. 113-121.
**Repository URL:**http://epubs.surrey.ac.uk/39605/#### Abstract

We consider the relationship between the target affinity of a monoclonal antibody and its in vivo potency. The dynamics of the system is described mathematically by a target-mediated drug disposition model. As a measure of potency, we consider the minimum level of the free receptor following a single bolus injection of the ligand into the plasma compartment. From the differential equations, we derive two expressions for this minimum level in terms of the parameters of the problem, one of which is valid over the full range of values of the equilibrium dissociation constant K(D) and the other which is valid only for a large drug dose or for a small value of K(D). Both of these formulae show that the potency achieved by increasing the association constant k(on) can be very different from the potency achieved by decreasing the dissociation constant k(off). In particular, there is a saturation effect when decreasing k(off) where the increase in potency that can be achieved is limited, whereas there is no such effect when increasing k(on). Thus, for certain monoclonal antibodies, an increase in potency may be better achieved by increasing k(on) than by decreasing k(off).

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(2011) - 'The bouncing motion of a superball between a horizontal floor and a vertical wall'.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS, 46 (1), pp. 204-221.
**Repository URL:**http://epubs.surrey.ac.uk/39608/
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(2011) - 'Period-doubling/symmetry-breaking mode interactions in iterated maps'.
PHYSICA D-NONLINEAR PHENOMENA, 238 (19), pp. 1992-2002.
**Repository URL:**http://epubs.surrey.ac.uk/39607/
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(2009) - 'The dynamics of a bouncing superball with spin'.
DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, 22 (3), pp. 291-322.
**Repository URL:**http://epubs.surrey.ac.uk/39609/
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(2007) - 'Lyapunov exponents of symmetric attractors'.
NONLINEARITY, 19 (10), pp. 2455-2466.
**Repository URL:**http://epubs.surrey.ac.uk/39604/
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(2006)

### Conference papers

- 'MEASUREMENT OF CARDIOVASCULAR STATE USING ATTRACTOR RECONSTRUCTION ANALYSIS'. IEEE
2015 23RD EUROPEAN SIGNAL PROCESSING CONFERENCE (EUSIPCO), Nice, FRANCE: 23rd European Signal Processing Conference (EUSIPCO), pp. 444-448.
**Repository URL:**http://epubs.surrey.ac.uk/809633/
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(2015) - 'Comparison of Attractor Reconstruction and
HRV Methods for Analysing Blood Pressure Data'.
Computing in Cardiology, Computing in Cardiology 2014 41, pp. 437-440.
**Repository URL:**http://epubs.surrey.ac.uk/807809/#### Abstract

Many methods have been proposed for analysing high frequency blood pressure or ECG data. We review a recently proposed new approach for analysing such data based on attractor reconstruction and compare it to heart rate variability that analyses the beat-to-beat intervals. Our new approach uses all the available data and so can detect changes in the shape of the waveform.

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(2014)

### Books

- UK Success Stories in Industrial Mathematics. Switzerland : Springer
**Repository URL:**http://epubs.surrey.ac.uk/810039/#### Abstract

This publication showcases the work of UK mathematicians and statisticians by describing industrial problems that have been successfully solved, together with a summary of the financial and/or societal impact that arose from the work. The articles are grouped by sector, and include contributions to climate modelling, engineering and health. The articles are based on Impact Case Studies that were submitted to the Research Excellence Framework (REF2014), a UK government sponsored exercise that assessed the research quality within UK universities. There are many publications in the realm of 'popular mathematics' as well as a vast research literature that underpins this. This work is aimed at a middle ground between these two. Articles contain some mathematical detail, but the emphasis is on telling the story of a successful collaboration between academia and industry and on the results obtained. The book is therefore accessible to a wide readership with interest in the applications of mathematics and statistics to problems of industrial importance and to those interested in how mathematics and statistics research affects our everyday lives and leads to economic and societal benefits.

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(2016) - Nonlinear Mathematics and Its Applications. Cambridge, UK : Cambridge University Press
**Repository URL:**http://epubs.surrey.ac.uk/810038/#### Abstract

At the University of Surrey in 1995 an EPSRC Spring School was held in Applied Nonlinear Mathematics for postgraduate students in mathematics, engineering, physics or biology; this volume contains the bulk of the lectures given there by a team of internationally distinguished scientists. The aim of the school was to introduce students to current topics of research interest at an appropriate level. The majority of the courses are in the area of nonlinear dynamics with application to fluid dynamics, boundary layer transition, driven oscillators and waves. However, there are also lectures considering problems in nonlinear elasticity and mathematical biology. The articles have all been edited so that the book forms a coherent and accessible account of recent advances in nonlinear mathematics.

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(1996)

### Patents

- Delay Coordinate Analysis of Periodic Data.
Article number PCT/GB2015/050429
**Repository URL:**http://epubs.surrey.ac.uk/808863/#### Abstract

In embodiments of the present invention, periodic data is analysed by obtaining (SI) a vector of delay coordinates for each one of a plurality of samples of the periodic data in a time window, and transforming (S2) each of the vectors into a coordinate system comprising a plurality of predefined vectors, to obtain a projection of an attractor of the periodic data along one of the predefined vectors. The periodic data may be physiological data. Information representing one or more characteristics of the obtained attractor, which is of diagnostic value, is then displayed (S3) to enable a diagnosis.

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