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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.

We introduce cycle-linear hybrid automata (CLHA) and show how they can be used to efficiently model dynamical systems that exhibit nonlinear, pseudo-periodic behavior. CLHA are based on the observation that such systems cycle through a fixed set of operating modes, although the dynamics and duration of each cycle may depend on certain computational aspects of past cycles. CLHA are constructed around these modes such that the per-cycle, per-mode dynamics are given by a time-invariant linear system of equations; the parameters of the system are dependent on a deformation coefficient computed at the beginning of each cycle as a function of memory units. Viewed over time, CLHA generate a very intuitive, linear approximation of the entire phase space of the original, nonlinear system. We show how CLHA can be used to efficiently model the action potential of various types of excitable cells and their adaptation to pacing frequency.