# 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-
- Joint appointment with the Data Science group at NPL, Teddington, 2018-

### News

### 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, machine learning.

Further details can be found on my personal web page.

### My publications

### Publications

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.

^{+D-}genetic mutation attributable to Brugada syndrome,Heart Rhythm Elsevier

**Background:** Life threatening arrhythmias resulting from genetic mutations are often missed in current ECG analysis. We combined a new method for ECG analysis that uses all the waveform data with machine learning to improve detection of such mutations from short ECG signals in a mouse model.

**Objective:** We sought to detect consequences of Na^{+} channel deficiencies known to compromise action potential conduction in comparisons of Scn5a^{+D-} mutant and wild-type mice using short ECG signals, examining novel and standard features derived from Lead I and II ECG recordings by machine learning algorithms.

**Methods:** Lead I and II ECG signals from anaesthetised wild type and Scn5a^{+D-} mutant mice of length 130s were analysed by extracting various groups of features which were used by machine learning to classify the mice as wild type or mutant. The features used were standard ECG intervals and amplitudes, as well as features derived from attractors generated using the novel Symmetric Projection Attractor Reconstruction method which reformulates the whole signal as a bounded, symmetric two-dimensional attractor. All the features were also combined as a single feature group.

**Results:** Classification of genotype using the attractor features gave higher accuracy than using either the ECG intervals or the intervals and amplitudes. However, the highest accuracy (96%) was obtained using all the features. Accuracies for different subgroups of the data were obtained and compared.

**Conclusion:** Detection of the Scn5a^{+D-} mutation from short mouse ECG signals with high accuracy is possible using our Symmetric Projection Attractor Reconstruction method.