# Dr Philip Aston

## Reader

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.

Further details can be found on my personal web page.

## Publications

### Journal articles

- 'COMPUTING THE INVARIANT MEASURE AND THE LYAPUNOV EXPONENT FOR ONE-DIMENSIONAL MAPS USING A MEASURE-PRESERVING POLYNOMIAL BASIS'. AMER MATHEMATICAL SOC
MATHEMATICS OF COMPUTATION, 83 (288) Article number PII S 0025-5718(2013)02811-6 , pp. 1869-1902.
#### 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.

.
(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.
#### 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.

.
(2014) - 'Erratum: Alternating period-doubling cascades (Nonlinearity (2013) 26 (2553))'. Nonlinearity, 26 (9), pp. 2745-2745. . (2013)
- 'Alternating Period-Doubling Cascades'. IOP
Nonlinearity, 26, pp. 2553-2576.
#### 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.

.
(2013) - 'A mathematical analysis of rebound in a target-mediated drug disposition model: I.Without feedback.'.
J Math Biol, 68 (6), pp. 1453-1478.
**Full text**is available at: http://epubs.surrey.ac.uk/792107/#### 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) - 'Reply to the comment by Cleanthes A. Nicolaides'. EPL, 101 (4) . (2013)
- 'Is radioactive decay really exponential?'. Institute of Physics
Europhysics Letters, 97 (5) Article number 52001
**Full text**is available at: 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.

.
(2012) - 'Mathematical analysis of the pharmacokinetic-pharmacodynamic (PKPD) behaviour of monoclonal antibodies: predicting in vivo potency.'. Elsevier
J Theor Biol, England: 281 (1), pp. 113-121.
**Full text**is available at: 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).

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