Understanding emerging behaviours in networks of pulse-coupled oscillators

About the Project

These projects are open to students worldwide, but have no funding attached. Therefore, the successful applicant will be expected to fund tuition fees at the relevant level (home or international) and any applicable additional research costs. Please consider this before applying.

Pulse-coupled oscillators (PCOs) have been successfully used to model the dynamics of neurons in the brain, pacemaker cells in the heart, flashing of gregarious fireflies, and chirping of crickets. Research on PCOs has mainly focused on understanding the emergence of synchronised states (i.e., the self-adjustment of rhythms to a common rhythm), chaos (i.e., nonperiodic behaviour with sensitive dependence on initial conditions), or chimeras (i.e., a state where synchrony and asynchrony behaviours coexist). Because of the nonlinearities present in most PCO models, previous analyses have been restricted to numerical simulations or analyses using the Thermodynamic limit (i.e., making the system size tend to infinity so that it can be treated as a continuum).

In this PhD project we aim to analyse the dynamics emerging from a mathematically tractable PCO model (Physica D, 182, 254-273 (2003), Eur. Phys. J. D 62, 51-56 (2011), Europhysics Letters 94 (6), 60007 (2011), Eur. Phys. J. Spec. Top. 223(13), 2819-2829 (2014), Eur. Phys. J. Spec. Top. 225(13), 2487-2506 (2016)). Initially, this model was designed based on a dual RC circuit that switched between a state where the capacitor charged and a state where the capacitor discharged, during which the circuit flashed a light. Through these pulses of light, the circuits interact by accelerating the charging state (i.e., making the charge shorter) or decelerating the discharging state (i.e., making the discharge longer) due to voltage increase in the capacitor. This means that the original model was excitable during the charging state and inhibitory during the discharging state. However, real-world systems such as neurons, pacemaker cells, fireflies, and crickets are typically either excitable or inhibitory. Moreover, PCO require determining which oscillators are coupled to which other oscillators, which means defining a network of interactions that can even be time-varying.

Consequently, we first aim to extend this PCO model to make it completely excitable (i.e., the action of a pulse can only shorten the charging or discharging states) or completely inhibitory (i.e., the action of any pulse can only lengthen the charging and discharging states). Then, we will study the emerging dynamics in each of these extensions, including synchronisation, oscillation death, and chaos, and their dependence on the network structure.

This is a great opportunity for a computationally minded and mathematically strong student with an interest in non-linear dynamics and graph theory/complex networks in an active research field of applied mathematics with a broad inter-disciplinary scope. The student choosing this project will have some freedom to choose the type of emerging dynamic to analyse according to their interest, as well as whether to focus more on the computational aspect of the project or the mathematical aspects.

Informal enquiries can be made by contacting Dr N Rubido (nicolas.rubidoobrer@abdn.ac.uk).

Decisions will be based on academic merit. The successful applicant should have, or expect to obtain, a UK Honours Degree at 2.1 (or equivalent) in physics or mathematics

We encourage applications from all backgrounds and communities, and are committed to having a diverse, inclusive team.

Application Procedure:

Formal applications can be completed online: https://www.abdn.ac.uk/pgap/login.php.

You should apply for Degree of Doctor of Philosophy in Physics to ensure your application is passed to the correct team for processing.

Please clearly note the name of the lead supervisor and project titleon the application form. If you do not include these details, it may not be considered for the project.

Your application must include: A personal statement, an up-to-date copy of your academic CV, and clear copies of your educational certificates and transcripts.

Please note: you do not need to provide a research proposal with this application.

If you require any additional assistance in submitting your application or have any queries about the application process, please don't hesitate to contact us at researchadmissions@abdn.ac.uk

Funding Notes

This is a self-funding project open to students worldwide. Our typical start dates for this programme are February or October.

Fees for this programme can be found here Finance and Funding | Study Here | The University of Aberdeen.

References

• Ávila, G. R., Guisset, J. L., & Deneubourg, J. L. (2003). Synchronization in light-controlled oscillators. Physica D: Nonlinear Phenomena, 182(3-4), 254-273.
• Rubido, N., Cabeza, C., Kahan, S., Ramírez Ávila, G. M., & Marti, A. C. (2011). Synchronization regions of two pulse-coupled electronic piecewise linear oscillators. The European Physical Journal D, 62(1), 51-56.
• Ávila, G. R., Deneubourg, J. L., Guisset, J. L., Wessel, N., & Kurths, J. (2011). Firefly courtship as the basis of the synchronization-response principle. Europhysics Letters, 94(6), 60007.
• García, R. A., Rubido, N., Marti, A. C., & Cabeza, C. (2014). The role of intermediaries in the synchronization of pulse-coupled oscillators. The European Physical Journal Special Topics, 223(13), 2819-2829.
• Ramírez-Ávila, G. M., & Kurths, J. (2016). Unraveling the primary mechanisms leading to synchronization response in dissimilar oscillators. The European Physical Journal Special Topics, 225(13), 2487-2506.