A Computational and Geometric Approach to Phase Resetting Curves and Surfaces

This work arises from the purpose of applying new tools in dynamical systems to time problems in biological systems. The main aim of this paper is to develop a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not yet reached the asymptotic state (a limit cycle). That is, we want to extend the computation of the phase resetting curves (PRCs) (the classical tool to compute the phase advancement) to a neighborhood of the limit cycle, obtaining what we call the phase resetting surfaces (PRSs). To achieve this goal we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds and another approach using Lie symmetries), which allows us to describe the isochronous sections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoretical framework applicable, we use the numerical algorithms of the parameterization method and other semianalytical tools to extend invariant manifolds; as a result, we design a numerical scheme to compute both the isochrons and the PRSs of a given oscillator. Finally, to illustrate this algorithm, we apply it to some well-known biological models and we include a discussion on different biological and numerical aspects suggested by these examples.