Geometric multi-scale models of the visual cortex

About the Project

Starting from the celebrated brain activity equation of Wilson and Cowan, we will introduce a full multi-scale analysis of the visual cortex V1 from the cellular to the geometric PDE level based on the ideas outlined in a previous joint publication with G. Citti. On the cellular level, this will incorporate dynamical models for neurons and their connections at different levels of sophistication. The next level, obtained by spatial averaging over the activity of neurons, is an activity equation similar to the equation of Wilson and Cowan. A rescaling in space and time exhibits a quasilinear PDE. This emergent PDE can be sub-Riemannian. The project connects continuous PDE models of the visual cortex models with neural connectivity on a cellular level. Depending on the student’s background and interest, one or several of the following questions can be addressed:

1. Effects of noise on the macroscale
2. Effects of different models at cellular level (ODE, spin flip) on the emerging macro-model
3. Connection with machine learning and image processing.

Project partner: Professor Giovanna Citti, Universita di Bologna, Italy

What we offer:

Cardiff's School of Mathematics provides an excellent postgraduate research environment including:

• research facilities at Abacws (co-designed by our PGR community),
• a vibrant research culture including a wide range of research events,
• access to national course centres in mathematics, statistics and Operational Research,
• comprehensive professional skills training offered by Cardiff’s Doctoral Academy.
•  collaborations with other disciplines and with a wide range of external partners (industry and society).

The PGR community is very active in running the SIAM-IMA Student Chapter, a valuable forum for exchange of ideas and public engagement.

This studentship will be undertaken within the Mathematical Analysis research group at the School of Mathematics. The group’s members have wide-ranging interests, and offer a dynamic environment for the student to engage with experts in mathematical analysis and many other fields in mathematics and other disciplines. The group hosts a weekly seminar inviting external speakers.

Completing a PhD in mathematics opens up a range of possible next steps, from continuing an academic career to working in mathematically technical roles in industry.

The 3.5 year programme does not have any funding attached but successful applicants will receive a small Research Training Support Grant to cover costs such as research consumables, training, conferences and travel.

The possible start dates for this PhD are October, January, April or July.

What you bring:

You should have a 1st or upper 2nd class UK Honours degree (or equivalent) and/or a Master’s degree in mathematics or a suitable related subject.

Applicants for whom English is not their first language must demonstrate their proficiency by obtaining an IELTS score of at least 6.5 overall, with a minimum of 5.5 in each skills component, or equivalent qualification.

Cardiff University is committed to supporting and promoting equality and diversity and to creating an inclusive research environment. We particularly welcome applications from underrepresented groups.

How to apply:

Applicants should apply through the Cardiff University online PhD application portal, for a Doctor of Philosophy in Mathematics with your desired entry point.

You must include

• Transcript(s) of your degree(s) (with certified translations if appropriate)
• Degree certificates (with certified translations if appropriate)
• References - these should be current and verifiable
• Personal statement (as part of the university application form, or as a separate attachment, if you prefer) provides a clear explanation of your research interest, preparation undertaken, and an understanding of the project.

Note that the 'register interest' option will not submit an application. Please follow the instructions above and submit your application via our online portal. Incomplete applications will not be considered.

Funding Notes

We welcome application from potential students who have their own sponsorship in place or are in a position to self-fund.

In the Research Proposal section of your application, please specify the project title and supervisors of this project.

In the funding section, please select that you will be self-funding and enter your source of funding.