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| Phase resetting curves (PRCs) constitute a powerful resource in time-control problems in biological processes. They predict the effect of a perturbation on the phase of an oscillator, assuming that it occurs on the attractor. However, factors like the rate of convergence to the oscillator, strong forcing or high stimulation frequency may invalidate the above assumption and raise the question of how is the phase variation away from an attractor. The computation of the phase advancement when we stimulate an oscillator which has not reached yet the asymptotic state (a limit cycle) relies on the concept of isochrons (the sets of points with the same asymptotic phase).
In this talk, we will present efficient algorithms to compute limit cycles and their isochrons for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. We prove analytically the convergence of the algorithms. We will show some examples of the computations we have carried out for some well-known biological models. |
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