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Cycle-linear hybrid automata (CLHAs), a new model of excitable cells that efficiently and accurately captures action-potential morphology and other typical excitable-cell characteristics such as refractoriness and restitution, is introduced. Hybrid automata combine discrete transition graphs with continuous dynamics and emerge in a natural way during the (piecewise) approximation process of any nonlinear system. CLHAs are a new form of hybrid automata that exhibit linear behaviour on a per-cycle basis but whose overall behaviour is appropriately nonlinear. To motivate the need for this modelling formalism, first it is shown how to recast two recently proposed models of excitable cells as hybrid automata: the piecewise-linear model of Biktashev and the nonlinear model of Fenton-Karma. Both of these models were designed to efficiently approximate excitable-cell behaviour. We then show that the CLHA closely mimics the behaviour of several classical highly nonlinear models of excitable cells, thereby retaining the simplicity of Biktashev-s model without sacrificing the expressiveness of Fenton-Karma. CLHAs are not restricted to excitable cells; they can be used to capture the behaviour of a wide class of dynamic systems that exhibit some level of periodicity plus adaptation.