### Professor Philip Aston

## Academic and research departments

Department of Mathematics, Centre for Mathematical and Computational Biology.### 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

### Research

### Research interests

My research interests include bifurcation theory, symmetry, computation of Lyapunov exponents using spatial integration, the dynamics of bouncing balls, pharmacokinetics/pharmacodynamics (PKPD), non-exponential radioactive decay, mathematical models of hepatitis C infection and attractor reconstruction methods for extracting information from physiological time series.

Further details can be found on my personal web page.

### My publications

### Publications

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.

HRV Methods for Analysing Blood Pressure Data, Computing in Cardiology 41 pp. 437-440

Our new approach uses all the available data and so can

detect changes in the shape of the waveform.

applied to analyse arterial blood pressure and photoplethysmogram

signals. This study extends this novel technique

to ECG signals. We show that the method gives high

accuracy in identifying gender from ECG signals, performing

significantly better than the same classification by

interval measures.

inconsistencies between the models and known behaviour of the infection. A new model for HCV

infection is proposed, based on various dynamical processes that occur during the infection that are

described in the literature. This new model is analysed and three steady state branches of solutions

are found when there is no stem cell generation of hepatocytes. Unusually, the branch of infected

solutions that connects the uninfected branch and the pure infection branch can be found analytically

and always includes a limit point. When the action of stem cells is included, the bifurcation between

the pure infection and infected branches unfolds, leaving a single branch of infected solutions. It

is shown that this model can generate various viral load profiles that have been described in the

literature which is confirmed by fitting the model to four viral load datasets. Suggestions for possible

changes in treatment are made based on the model.

When a superball is thrown forwards but with backspin, it is observed to reverse both direction and spin for a few bounces before settling to bouncing motion in one direction. The bouncing ball is modelled by a two-dimensional iterated map in terms of the horizontal velocity and spin immediately after each bounce. The asymptotic motion of this system is easily determined. However, of more interest is the transient behaviour. The two-dimensional linear map is reduced to a one-dimensional nonlinear map and this map is used to determine the number of direction and spin reversals that can occur for given initial conditions. The model is then generalised to describe a ball bouncing up and down a single step or successively down a staircase.

We study a codimension two steady-state/steady-state mode interaction with **D**_{4} symmetry, where the center manifold is three-dimensional. Primary branches of equilibria undergo secondary Hopf bifurcations to periodic solutions which undergo further bifurcations leading to chaotic dynamics. This is not an exponentially small effect, and the chaos obtained in simulations using DsTool is large-scale, in contrast to the "weak" chaos associated with Shilnikov theory.

Moreover, there is an abundance of *symmetric chaotic attractors* and *symmetry-increasing bifurcations*. The local bifurcation studied in this paper is the simplest (in terms of dimension of the center manifold and codimension of the bifurcation) in which such phenomena have been identified. Numerical investigations demonstrate that the symmetric chaos is part of the local codimension two bifurcation. The two-dimensional parameter space is mapped out in detail for a specific choice of Taylor coefficients for the center manifold vector field. We use AUTO to compute the transitions involving periodic solutions, Lyapunov exponents to determine the chaotic region, and symmetry detectives to determine the symmetries of the various attractors.

When a superball is thrown forwards but with backspin, it is observed to reverse both direction and spin for a few bounces before settling to bouncing motion in one direction. The bouncing ball is modelled by a two-dimensional iterated map in terms of the horizontal velocity and spin immediately after each bounce. The asymptotic motion of this system is easily determined. However, of more interest is the transient behaviour. The two-dimensional linear map is reduced to a one-dimensional non-linear map and this map is used to determine the number of direction and spin reversals that can occur for given initial conditions. The model is then generalised to describe a ball bouncing up and down a single step or successively down a staircase.

We study a codimension two steady-state/steady-state mode interaction with D-4 symmetry, where the center manifold is three-dimensional. Primary branches of equilibria undergo secondary Hopf bifurcations to periodic solutions which undergo further bifurcations leading to chaotic dynamics. This is not an exponentially small effect, and the chaos obtained in simulations using DsTool is large-scale, in contrast to the "weak" chaos associated with Shilnikov theory. Moreover, there is an abundance of symmetric chaotic attractors and symmetry-increasing bifurcations. The local bifurcation studied in this paper is the simplest ( in terms of dimension of the center manifold and codimension of the bifurcation) in which such phenomena have been identified. Numerical investigations demonstrate that the symmetric chaos is part of the local codimension two bifurcation. The two-dimensional parameter space is mapped out in detail for a specific choice of Taylor coefficients for the center manifold vector field. We use AUTO to compute the transitions involving periodic solutions, Lyapunov exponents to determine the chaotic region, and symmetry detectives to determine the symmetries of the various attractors.

Techniques from singular perturbation theory such as asymptotic analysis, geometric singular perturbation theory and geometric desingularisation work well for the TMDD model where the underlying critical manifold consists of two two-dimensional submanifolds. However, the known techniques can not be applied to the dimerisation problem, as the underlying critical manifold is degenerate and consists of a two-dimensional and a three-dimensional submanifold.

The intersection of these manifolds is important for the Dimerisation problem as there is a type of rebound in the dimerisation problem occurring at the specified bifurcation point.

Motivated by the dimerisation problem, we consider a general two parameter slow-fast system in which the critical manifold consists of a one-dimensional and a two-dimensional submanifold. These submanifolds intersect transversally at the origin.

Using geometric desingularisation, we show that for a particular subset of parameters the continuation of the slow manifold connects the attracting components of the critical set. We also show that the direction of this continuation on the two-dimensional manifold can be expressed in terms of the model parameters.

This method is then applied to the dimerisation problem in order to understand and approximate the rebound.

However, they typically report averaged or low frequency summary data such as heart rate

and systolic, mean and diastolic blood pressure. In doing so, a potential wealth of information

contained in the high fidelity waveform data is discarded, data which has long been known to

contain useful information on cardiovascular performance.

Here we summarise a new mathematical method, attractor reconstruction, which enables the

quantification of arterial waveform shape and variability in real-time. The method can handle

long streams of non-stationary data and does not require preprocessing of the raw physiological

data by the end user. Whilst the detailed mathematical proofs have been described elsewhere

(Aston et al., 2018), the authors were motivated to write a summary of the method and its

potential utility for biomedical researchers, physiologists and clinician readers.

Here we illustrate how this new method may supplement and potentially enhance the sensitivity of detecting cardiovascular disturbances, to aid with biomedical research and clinical

decision making.

the treatment of chronic Hepatitis C virus infection.We introduce a mathematical model for

Hepatitis C dynamics treated with the direct acting antiviral drug, telaprevir, alongside

traditional interferon and ribavirin treatments to understand how this combination therapy

affects the viral load of patients exhibiting different types of response.We use sensitivity and

identifiability techniques to determine which model parameters can be best estimated from

viral load data. Parameter estimationwith these best estimable parameters is then performed

to give patient-specific fits of the model to partial virologic response, sustained virologic

response and breakthrough patients.