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Phase response curves (PRCs) have been extensively used to control the phase of oscillators under perturbations. Their main advantage is the reduction of the whole model dynamics to a single variable (phase) dynamics. However, in some adverse situations (strong inputs, high-frequency stimuli, weak convergence,. . . ), the phase reduction does not provide enough information and, therefore, PRC lose predictive power. To overcome this shortcoming, in the last decade, new contributions have appeared that allow to reduce the system dynamics to the phase plus some transversal variable that controls the deviations from the asymptotic behaviour. We call this setting extended response functions. In particular, we single out the phase response function (PRF, a generalization of the PRC) and the amplitude response function (ARF) that account for the above-mentioned deviations from the oscillating attractor. It has been shown that in adverse situations, the PRC misestimate the phase dynamics whereas the PRF-ARF system provides accurate enough predictions. In this paper, we address the problem of studying the dynamics of the PRF-ARF systems under periodic pulsatile stimuli. This paradigm leads to a two-dimensional discrete dynamical system that we call 2D entrainment map. By using advanced methods to study invariant manifolds and the dynamics inside them, we construct an analytico-numerical method to track the invariant curves induced by the stimulus as two crucial parameters of the system increase (the strength of the input and its frequency). Our methodology also incorporates the computation of Arnold tongues associated to the 2D entrainment map. We apply the method developed to study inner dynamics of the invariant curves of a canonical type II oscillator model. We further compare the Arnold tongues of the 2D map with those obtained with the map induced only by the PRC, which give already noticeable differences. We also observe (via simulations) how high-frequency or strong e, Peer Reviewed, Preprint

In this paper, we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singularity. The method consists in obtaining an asymptotic formula for the difference between these manifolds which turns to be exponentially small respect to the unfolding parameter. The formula obtained is explicit but depends on the so-called Stokes constants, which arise in the study of the original vector field and which corresponds to the so-called inner equation in singular perturbation theory., Peer Reviewed, Postprint (author's final draft)

In this paper we study a beyond all orders phenomenon which appears in the analytic unfoldings of the Hopf-zero singularity. It consists in the breakdown of a two-dimensional heteroclinic surface which exists in the truncated normal form of this singularity at any order. The results in this paper are twofold: on the one hand, we give results for generic unfoldings which lead to sharp exponentially small upper bounds of the difference between these manifolds. On the other hand, we provide asymptotic formulas for this difference by means of the Melnikov function for some non-generic unfoldings., Postprint (author's final draft)

En aquest treball es pretén donar una visió de les aplicacions dels sistemes dinàmics en el camp de la neurociència, així com ampliar les eines que s'utilitzen fins ara. En el primer capítol es descriu el marc teòric on es desenvolupa el treball, introduint els conceptes de fase asimptòtica, isòcrones i corbes de resposta de fase (PRCs), així com la seva aplicació en problemes de sincronització. A continuació, es descriu la implementació numèrica d'algunes d'aquestes eines mitjançant el software matemàtic Sage. A més, també es presenten les contribucions que s'han fet al desenvolupament d'aquest software: l'adaptació d'un integrador numèric (que estarà present a la pròxima versió del Sage), i la creació d'un paquet a partir del programa Brian, un simulador de dinàmiques neuronals (i que es pot baixar des de la pàgina web del Sage i instal·lar per qualsevol usuari). Finalment, en l'últim capítol, s'introdueix una nova eina, les corbes de resposta de fase de segon ordre, que són una extensió de PRCs clàssiques. També es proporcionen exemples numèrics i analítics on aquestes PRCs de segon ordre tenen especial rellevància.. L'objectiu general és aplicar tècniques de sistemes dinàmics per a l'estudi de la dinàmica de fase (corbes de resposta de fase) i, per extensió, propietats de sincronització en oscil.ladors biològics. Paral.lelament, es pretén adaptar els mètodes numèrics involucrats a l'entorn de programació Sage i coordinar-ho el codi amb el programa Brian, destinat a xarxes de neurones i programat també en llenguatge python. En resum, doncs, es vol donar una contribució tant des del punt de vista computacional com teòric a l'estudi d'oscil.ladors biològics. The purpose of this memoir is to expose some applications of dynamical systems in the eld of mathematical biology, providing the basic ideas, presenting and developing computational methods and contributing to extend the mathematical theory in some way."