# PhD studentship opportunities

Breaking news: PhD studentships (scholarships) available for UK and EU* students in the Department of Engineering Mathematics at the University of Bristol for immediate start.

Due to a number of students withdrawing for personal reasons, there are now several PhD studentships available that must be taken up by September 2016. I am looking for motivated and able  students who are interested in doing research at the intersection of Applied Mathematics and Engineering/Science (my interests are quite broad!). In particular, I have three different topics that I’m actively pursuing at the moment all of which feature numerical computation/numerical analysis:

• Nonlinear dynamics of stochastic differential equations — I’m interested in investigating how the tools and concepts of nonlinear dynamics and bifurcation theory can be applied to stochastic differential equations arising from various application areas (e.g., neuroscience or climate science).
• Control-based continuation — numerical continuation is a very effective tool for investigating the nonlinear behaviour and bifurcations of mathematical models, and control-based continuation is a means for applying this tool to physical experiments (engineered systems or, hopefully, biological systems) without the need for a mathematical model. Research in this area requires a very interesting mix of numerical analysis, control theory, system identification and the theory of stochastic processes (I don’t expect students to have a background in all of these subjects!).
• Equation-free methods and agent-based modelling — equation-free methods are a means for obtaining a macroscopic model from microscopic simulations. They have been used for many physical processes previously and I am interested in how they can be extended to agent-based models, such as models of Zebrafish locomotion to investigate the dynamics of shoaling (Zebrafish are just one example).

I have also worked extensively with delay differential equations and, though I don’t have any active work in this area at the moment, I’m happy to solicit project suggestions in this area. All of these projects are very open ended and I’m happy to work with students to tailor the projects to their own interests.

There is no deadline for these studentships, though obviously it’s better to apply sooner rather than later. If you are interested, get in touch for more information.

* EU students are eligible provided they have been resident in the UK for at least 3 years. See the EPSRC website for more details.

# Data-driven stochastic modelling of Zebrafish locomotion

Ever wondered how you can write down equations that govern the behaviour of a moving creature? Let’s start simple and try something like a fish. (OK, a fish isn’t that simple…) It turns out that Zebrafish are quite a convenient animal to study and one of our collaborators (Maurizio Porfiri from New York University, USA) has great facilities for the fish to happily swim around in while being tracked by video cameras.

With the data from Maurizio’s lab, we have constructed a mathematical model of Zebrafish locomotion based on stochastic differential equations (SDEs), that is equations that govern the rates of change of speed and of turning speed (the differential bit) and are driven by random noise (the stochastic bit). The randomness is there to provide a mathematical description of all the seemingly erratic behaviours that the Zebrafish exhibit which we don’t really understand — we can match the statistics of the randomness to the statistics of the erratic behaviour but we can’t say anything about the higher level thought (if that is an appropriate description for a fish!) that is generating it.

The SDEs are relatively simple (if you are used to such things) and take the form
$$\begin{gathered} \mathrm{d}U = -\theta_u(U – \mu_u – U^*)\mathrm{d}t + \sigma_u\mathrm{d}W_1(t) \\ \mathrm{d}\Omega = -\theta_\omega(\Omega – \Omega^*)\mathrm{d}t + \sigma_\omega\mathrm{d}W_2(t) \end{gathered}$$
where $U$ is the speed and $\Omega$ is the turning speed. This is an Ito stochastic differential equation (though perhaps it should be Stratonovich…). This is in effect a mean-reverting process or Ornstein-Uhlenbeck process. In short, the speed is tracking the value $\mu_u + U^*$ with a time scale $\theta_u$ where $\mu_u$ is the natural speed of the Zebrafish and $U^*$ is a modification to the natural speed due to external factors (e.g., getting close to a wall or other Zebrafish). The same description holds for the turning speed equation (except that the natural turning speed is zero and so not included). The terms $U^*$, $\Omega^*$ and $\sigma_u$ are not actually constants but functions of the fish position relative to other fish and any obstacles.

Take a read of the article and see what you think! (The article is open access to all.)

If you like this, a follow on post will provide a nice graphical interface for playing with these equations (written in Python).

Reference: Data-driven stochastic modelling of zebrafish locomotion Adam Zienkiewicz, David A.W. Barton, Maurizio Porfiri and Mario di Bernardo, Journal of Mathematical Biology 71(5) 2015 pp. 1081–1105. DOI: 10.1007/s00285-014-0843-2 (open access).