MathsNet: Research Student Directory

Name Davis, Joshua
Group Mathematical Medicine and Biology
Room A14, Math Sci
Telephone +44 (0) 115 82 32068 [direct dial]
Thesis Title Analysis and dynamics of multiple- spike waves in neural networks
Thesis Abstract As a result of modern imaging technologies, waves and bumps of neuronal activity have been experimentally verified at a variety of spatial scales in the cortex. Spatially localised bumps of activity are known to be involved in mechanisms of orientation tuning in the visual cortex, the rat head direction system, and working memory. In the turtle visual cortex, the presentation of stimuli has been shown to evoke propagating waves of activity. Also, numerous mental processes including sleep and binocular rivalry are characterised through waves, as well as neurological disorders such as epilepsy and migraines. Simulations of a discrete spiking network of integrate-and-fire network are shown to exhibit a rich variety of bump and wave states. In particular we find a family of coherent pulsating wave states composed of multiple synchronously firing neurons spread across the spatial domain. A continuum assumption is then taken to construct analytical solutions of such waves solely in terms the mean wave speed and firing times. The stability of the multiple spike waves is analysed by perturbing around firing times and an eigenvalue problem is solved and tested in the discrete regime. Numerical continuation is used to gain insight into the bifurcation structure of such waves investigated in terms of the parameters governing synaptic efficacy and connectivity. It is shown that multiple spike waves destabilise via a sequence of Hopf bifurcations. In addition, composite wave solutions are also found that match up in simulation to multiple spike pulses that have combined and coalesced. The latter part of the project involves simulating waves in coupled networks of neurons with more biophysically realistic single neuron models. The Fitz-Hugh Nagumo, Izhikevich, QIF, Moris-Lecar and the Hodgkin-Huxley models are among these. Models with inhomogeneities in connectivity and system parameters are hypothesised to be play a role in pathological waves. To study waves in heterogeneous stochastic neural networks we plan to use coarse-graining numerical techniques, in conjunction with GPU computations.
School contact information School of Mathematical Sciences
University of Nottingham
University Park
Nottingham, NG7 2RD
Tel: +44 (0) 115 951 4949
Fax: +44 (0) 115 951 4951