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Travelling waves of localised high neuronal activity have been observed in experimental preparations of cortical and thalamic brain slices. In vivo, EEG rhythms and brain activity during epileptic seizures are striking phenomena of large-scale synchronisation of neuronal populations. Using mathematical description of neural tissue by integro-differential equations linking the intrinsic local population dynamics with the nonlocal synaptic coupling between populations one is able to achieve similar type of dynamics. I am studying pattern formation in such nonlocal neural field models by adapting techniques that have been developed in physics for local PDE equations.
We have undertaken this analysis for a general 1D model with any time-dependent connectivity allowing us to study models with various types of delays and adaptation. We formulated also the full PDE form and studied instabilities in a 2D multi-population model with axonal delays which is applicable to EEG simulation. Currently we are looking at approaches for understanding localised solutions in 1D and 2D fields. Solutions of models incorporating dynamic threshold have characteristics very similar to dispersive solitons found in various physical systems and we are adapting techniques from those fields to handle integral equations.

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Results for more complicated neural field models have not been that common although they are both biologically more realistic and can produce more interesting dynamic patterns. We are interested in models incorporating space-dependent delays (axonal or dendritic) in the biologically relevant regime of local inhibition-distal excitation connectivity, and models involving various types of adaptation, including dynamic thresholds. The additional features allow interesting solutions even for one-population reductions of the models. We have studied globally periodic solutions by Turing-Hopf instability theory, both linear and weakly non-linear (deriving amplitude equations).