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Chapter 3 Geometric singular perturbation analysis of neuronal dynamics

Elsevier B.V.
DOI: 10.1016/s1874-575x(02)80024-8
  • Mathematics


Abstract In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which display bursting oscillations. There are, in fact, several different classes of bursting solutions; these have been classified by the geometric properties of how solutions evolve in phase space. We describe several of the bursting classes and then review related rigorous mathematical analysis. We then discuss the dynamics of small networks of neurons. We are primarily interested in whether excitatory or inhibitory synaptic coupling leads to either synchronous or desynchronous rhythms. We demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. Finally, we discuss larger networks of neuronal oscillators involving two distinct cell populations. In particular, we demonstrate how dynamical systems methods can be used to analyze recent models for sleep rhythms and other oscillations generated in the thalamus. The analysis helps to explain the generation of the different thalamic rhythms and the transitions between them, in both of which inhibition plays a crucial role.

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