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Biological oscillations, despite their vast range of periodicities, can be described mathematically by stable limit cycles. Therefore, a general theory characterizing the effects of perturbations applied to such limit cycles allows predictions of qualitative features of a particular oscillation subject to perturbation. In this chapter, we summarize this topological approach and discuss ways in which the theory breaks down, mainly for neuronal and cardiac oscillators. In particular, we describe experimental and computational studies that demonstrate apparent discontinuities in the response to perturbations, and others where there is not a rapid return to the limit cycle following a perturbation. Finally, we discuss differences between the topological and the excitability-type descriptions of neuronal oscillators.
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