Supervisor: Stephen Coombes
School of Mathematical Sciences

Description:

A network of globally coupled quadratic integrate-and-fire neurons with conductance based synapses has recently been shown to admit to an exact mean field description using the Ott-Antonsen (OA) ansatz [1,2]. The resulting neuronal population model is ideally suited as a model for understanding local field potentials. This PhD project will initially treat finite-size effects around the asynchronous state using a Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy with an appropriate moment-closure approximation [3]. It will then move on to consider a path-integral formalism to derive a perturbation expansion in the inverse system size to determine the evolution of system covariances [4]. The final part of the project will consider generalisations of the OA ansatz to allow the treatment of more general nonlinear integrate-and-fire neurons, and in particular piece-wise linear models with adaptation [5].

Requirements:

A strong quantitative background in mathematics or theoretical physics, and keen interest in statistical physics and nonlinear dynamical systems as applied to problems in neuroscience.

• S Coombes and Á Byrne 2019 Next generation neural mass models, Nonlinear Dynamics in Computational Neuroscience, PoliTO Springer Series, Ed. A Torcini and F Corinto, Springer, 1-16
• Á Byrne, M J Brookes and S Coombes 2017 A mean field model for movement induced changes in the beta rhythm, Journal of Computational Neuroscience, Vol 43, 143-158
• E J Hildebrand, M A Buice and C C Chow 2007. Kinetic theory of coupled oscillators. Physical Review Letters, 98:054101
• S Qiu and C C Chow Finite-size effects for spiking neural networks with spatially dependent coupling, Physical Review E, Vol 98, 062414
• R Nicks, L Chambon and S Coombes 2018 Clusters in nonsmooth oscillator networks, Physical Review E, Vol 97, 032213 

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