- 08-206 Antoni Guillamon, Gemma Huguet
- A computational and geometric approach to Phase Resetting Curves and Surfaces
Oct 31, 08
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Abstract. This work arises from the purpose of relating time problems in biological systems with some known tools in dynamical systems. More precisely, how are the phase resetting curves (PRCs) around a limit cycle $\gamma$ of a vector field $X$ related to the fact that $X$ is the infinitesimal generator of a Lie symmetry ($[Y,X] = \mu Y$). We show how the time variables involved in the Lie symmetry provide a natural way (a kind of normal form) to express the vector field around $\gamma$, similar to action-angle variables for integrable systems. In addition, the knowledge of the orbits of $Y$ gives a trivial way to compute the PRC, not only on $\gamma$, but also in a neighborhood of it, thus obtaining what we call phase resetting surfaces (PRSs). However, the aim of the paper is not only to state relationships among different concepts, but also to perform the effective computation of these symmetries. The numerical scheme is based on the theoretical ground of the so-called parameterization method to compute invariant manifolds (the orbits of $Y$) in a neighborhood of $\gamma$. Limit cycles in biological (more specifically, neuroscience) models encompass numerical problems that are often neglected or underestimated; we present a discussion about them and give general solutions whenever it is possible. Finally, we use all theoretical and numerical results to compute both the PRCs and PRSs and the isochronous sections of limit cycles for well-known biological models. In this part of the paper, we also explore how the PRCs evolve (in the parameter space) between different bifurcation values.