The geometry of a dendritic spine influences the dynamics of calcium in the spine and is regulated during synaptic plasticity. For instance, a moderate rise in calcium can cause elongation, while a very large increase in calcium causes fast shrinkage and the eventual collapse of a spine. In this book, we presented computational models for the calcium mediated spine-stem restructuring. This expansion and shrinkage depends on the frequency of the synaptic input to a spine as well as the activation of the calcium channels located on the spine head membrane. We are using computational studies to investigate the changes in spine density and structure for a variety of synaptic inputs of different frequencies. In particular, we are using the models to investigate the mechanisms underlying changes in spine density and morphology and the role of spine plasticity in long-term depression (LTD) and long-term potentiation (LTP). Finally, for the integration of our system, we also presented two new algorithms based on spectral collocation method for these types of problems and compare the results with the conventional finite difference methods.

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