Abstract:
Physiological models typically consist of suites of variables evolving on timescales that may differ by several orders of magnitude. Such models can sometimes be simplified by the use of one or more methods that reduce the dimension of the model. One common technique is to remove one or more fast variables by quasi-steady-state reduction (QSSR). However, QSSR is known to alter or destroy important characteristics of some models [27]. For example, the onset of oscillatory behaviour, which is an important feature in many physiological models, may be delayed or removed completely from some models by the use of QSSR. Zhang et al. [27] discusses some conditions under which the use of QSSR is mathematically justified, and specifically suggests that so-called singular Hopf bifurcations will be preserved under QSSR. It is hypothesized that, at a singular Hopf bifurcation, the quasi-steady-state manifold and the centre manifold become linearly aligned in the singular limit, which makes it possible for a singular Hopf bifurcation to persist. In this thesis, we analyze the effects of QSSR on four different models with multiple timescales: the Hodgkin-Huxley model, the Class II Atri model, the Hindmarsh-Rose model and the Chay-Keizer model. We especially focus on how the reduction influences the Hopf bifurcations and resulting periodic orbits in each system. We compare our numerical results with the conjectures in Zhang et al. [27] and investigate for consistency. As the conjectures suggest, singular Hopf bifurcations in the models are preserved under QSSR. However, we find some inconsistencies between our results and the conjectures in Zhang et al. [27]. Our numerical computations show that the quasi-steady-state manifold and the centre manifold do not become linearly aligned at most of the singular Hopf bifurcations that we examined in this thesis. Based on our numerical results for the four models of interest, we propose some directions of enquiry that might help the development of a general theory of QSSR.
