In this thesis we use tools provided by dynamical systems theory to analyse transient behaviour of neural cells. We study models in Hodgkin-Huxley formalism that have a natural time-scale separation, which allows us to apply concepts of geometrical singular perturbation theory. The analysis presented in this thesis is applied to physiologically-realistic models represented by high-dimensional systems with multiple time scales. We construct a model of hippocampal pyramidal neurons and study after-depolarisation (ADP), which is a hallmark of excitability and precursor of transient bursting. Through careful analysis of the model we investigate the contribution of particular ionic currents to the excitability behaviour of the model. Furthermore using model simulations and experimental data we define ADP mathematically. Based on the essence of ADP and transient bursting we perform a reduction of the model, which enables an in-depth study of these phenomena. To understand spike adding during transient bursting we use a two-point boundary value formulation of the model, inspired by the experimental protocol. We show that the spikes are added through a canard-like transition, during which the orbit segment traces unstable sheets of a critical manifold up to a jump point. Our analysis suggests at least two mechanisms of spike adding: one is organised by a fold of the critical manifold and the other due to the presence of an additional unstable equilibria of the full system. The results of this study extend the definition of an excitability threshold and show that spikes can be added' through a continuous deformation of an orbit segment, not by a discontinuous abrupt process. We also apply the ideas of geometrical singular perturbation theory to study periodic bursting in a pituitary cell model. We use nullclines to investigate the nature of plateau bursting . taking place below the branch of attracting equilibria in the fast subsystem. Moreover, we continue orbit segments in order to compute the stable manifolds of the branch of saddle equilibria, which plays a role of separatrix in this system. We show that seemingly premature termination of the active phase of the plateau burst is related to the orbit crossing this separatrix before reaching the end of the stable equilibrium branch. Finally, we use numerical continuation to compute onsets of ADP and a spike in transient bursting. We show that these on sets correspond to extrema of slow variables of the full system. In our boundary value problem formation the on sets are detected as folds, which allows further continuation in two parameters to establish the boundaries of different model behaviours. This new technique is a form of parameter sensitivity analysis and, in principle, could be applied to other models.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:559074 |
| Date | January 2012 |
| Creators | Nowacki, Jakub |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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