| Summary: | Radial glial cells play an important role during embryonic development in mammals. They are not only important for neural production but help to organise the architecture of the neocortex. Glial cells proliferate during the development of the brain in the embryo, before differentiating to produce neurons at a rate which increases towards the end of embryonic brain development. Glial cells communicate via Adenosine tri-phosphate (ATP) mediated calcium waves, which may have the effect of locally synchronising cell cycles, so that clusters of cells proliferate together, shedding cells in uniform sheets. Hence radial glial cells are not only responsible for the production of most neocortical neurons but also contribute to the architecture of the brain. It has been argued that human developmental disorders which are associated with cortical malfunctions such as infantile epilepsies and mental retardation may involve defects in neuronal production and/or architecture and mathematical modelling may shed some light upon these disorders. This thesis investigates, among other things, the conditions under which radial glial cells' cell cycles become `phase locked', radial glia proliferation and stochastic effects. There are various models for the cell cycle and for intracellular calcium dynamics. As part of our work, we marry two such models to form a model which incorporates the effect of calcium on the cell cycle of a single radial glial cell. Furthermore, with this achieved we consider populations of cells which communicate with each other via the secretion of ATP. Through bifurcation analysis, direct numerical simulation and the application of the theory of weakly coupled oscillators, we investigate and compare the behaviour of two models which differ from each other in the time during the cell cycle at which ATP is released. Our results from this suggest that cell cycle synchronisation is highly dependent upon the timing of ATP release. This in turn suggests that a malfunction in the timing of ATP release may be responsible for some cortical development disorders. We also show how the increase in radial glia proliferation may mostly be down to radial glial cells' ability to recruit quiescent cells onto the cell cycle. Furthermore, we consider models with an additive noise term and through the application of numerical techniques show that noise acts to advance the onset of oscillatory type solutions in both models. We build upon these results and show as a proof of concept how noise may act to enhance radial glia proliferation.
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