Huguet Casades, Gemma
Total activity: 11
Department
Department of Applied Mathematics I
E-mail
gemma.huguetupc.edu
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Scientific and technological production
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1 to 11 of 11 results
  • Nonlinear dynamics of neuronal excitability, oscillations, and coincidence detection

     Rinzel, John; Huguet Casades, Gemma
    Communications on pure and applied mathematics
    Date of publication: 2013-09
    Journal article

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    We review some widely studied models and firing dynamics for neuronal systems, both at the single cell and network level, and dynamical systems techniques to study them. In particular, we focus on two topics in mathematical neuroscience that have attracted the attention of mathematicians for decades: single-cell excitability and bursting. We review the mathematical framework for three types of excitability and onset of repetitive firing behavior in single-neuron models and their relation with Hodgkin's classification in 1948 of repetitive firing properties. We discuss the mathematical dissection of bursting oscillations using fast/slow analysis and demonstrate the approach using single-cell and mean-field network models. Finally, we illustrate the properties of Type III excitability in which case repetitive firing for constant or slow inputs is absent. Rather, firing is in response only to rapid enough changes in the stimulus. Our case study involves neuronal computations for sound localization for which neurons in the auditory brain stem perform extraordinarily precise coincidence detection with submillisecond temporal resolution

  • Phase-amplitude response functions for transient-state stimuli

     Castejon i Company, Oriol; Guillamon Grabolosa, Antoni; Huguet Casades, Gemma
    Journal of Mathematical Neuroscience
    Date of publication: 2013-08
    Journal article

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    The phase response curve (PRC) is a powerful tool to study the effect of a perturbation on the phase of an oscillator, assuming that all the dynamics can be explained by the phase variable. 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 concept of isochrons turns out to be crucial to answer this question; from it, we have built up Phase Response Functions (PRF) and, in the present paper, we complete the extension of advancement functions to the transient states by defining the Amplitude Response Function (ARF) to control changes in the transversal variables. Based on the knowledge of both the PRF and the ARF, we study the case of a pulse-train stimulus, and compare the predictions given by the PRC-approach (a 1D map) to those given by the PRF-ARF-approach (a 2D map); we observe differences up to two orders of magnitude in favor of the 2D predictions, especially when the stimulation frequency is high or the strength of the stimulus is large. We also explore the role of hyperbolicity of the limit cycle as well as geometric aspects of the isochrons. Summing up, we aim at enlightening the contribution of transient effects in predicting the phase response and showing the limits of the phase reduction approach to prevent from falling into wrong predictions in synchronization problems.

  • Fast iteration of cocycles over rotations and computation of hyperbolic bundles

     Huguet Casades, Gemma; de la Llave Canosa, Rafael; Sire, Yannick
    Discrete and continuous dynamical systems. Series A
    Date of publication: 2013
    Journal article

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    We present numerical algorithms that use small requirements of storage and operations to compute the iteration of cocycles over a rotation. We also show that these algorithms can be used to compute efficiently the stable and unstable bundles and the Lyapunov exponents of the cocycle.

  • Computation of limit cycles and their isochrons: Fast algorithms and their convergence

     Huguet Casades, Gemma; de la Llave Canosa, Rafael
    SIAM journal on applied dynamical systems
    Date of publication: 2013
    Journal article

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    We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) 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. The algorithms are efficient in the sense that if we discretize the functions using N points, a Newton step requires O(N) storage and O(N log N) operations in Fourier discretization or O(N) operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem also shows that the isochrons are analytic and depend analytically on the base point. Moreover, it establishes smooth dependence of the solutions on parameters and provides efficient algorithms to compute perturbative expansions with respect to external parameters. We include a discussion on the numerical implementation of the algorithms as well as numerical results for representative examples.

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    A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems  Open access

     Delshams i Valdes, Amadeu; Huguet Casades, Gemma
    Date: 2010-07
    Report

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    In this paper we consider a representative a priori unstable Hamiltonian system with 2 + 1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform explicitly the computations along the proof, which contribute to present in an easily understandable way the geometric mechanism of diffusion. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.

  • DINAMICA ASOCIADA A CONEXIONES ENTRE OBJETOS INVARIANTES, ASTRODINÁMICA, NEUROCIENCIA Y OTRAS APLICACIONES

     Blazquez Sanz, David; Olive Farre, Maria del Carme; Gomez-Ullate Oteiza, David; Morales Ruiz, Juan Jose; Canalias Vila, Elisabet; Martinez-seara Alonso, Maria Teresa; Baldoma Barraca, Inmaculada Concepcion; Fedorov, Yury; Garcia Taberner, Laura; Guardia Munarriz, Marcel; Guillamon Grabolosa, Antoni; Gonchenko, Marina; Gutiérrez Serrés, Pere; Lazaro Ochoa, Jose Tomas; Luque Jiménez, Alejandro; Martin De La Torre, Pablo; Masdemont Soler, Josep Joaquim; Olle Torner, Maria Mercedes; Pacha Andujar, Juan Ramon; Pantazi, Chara; Puig Sadurni, Joaquim; Ramirez Ros, Rafael; Roldan Gonzalez, Pablo; Villanueva Castelltort, Jordi; Tamarit, Anna; Olivé Farré, Carme; Morales Ruíz, Juan José; Larreal Barreto, Oswaldo José; Huguet Casades, Gemma; Gómez-Ullate Oteiza, David; Acosta Humanez, Primitivo Belen; Benita Bordes, Jose Manuel; Basak, Inna; Blázquez Sanz, David; de la Llave Canosa, Rafael; de la Rosa Ibarra, Abraham; Delshams i Valdes, Amadeu
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  • SISTEMES DINAMICS DE LA UPC

     Lazaro Ochoa, Jose Tomas; Roldan Gonzalez, Pablo; Olle Torner, Maria Mercedes; Gutiérrez Serrés, Pere; Masdemont Soler, Josep Joaquim; Gomez-Ullate Oteiza, David; Luque Jiménez, Alejandro; Larreal Barreto, Oswaldo José; Huguet Casades, Gemma; Pacha Andujar, Juan Ramon; Ramirez Ros, Rafael; Martinez-seara Alonso, Maria Teresa; Guillamon Grabolosa, Antoni; Martin De La Torre, Pablo; Canalias Vila, Elisabet; Puig Sadurni, Joaquim; Benita Bordes, Jose Manuel; Morales Ruiz, Juan Jose; Guardia Munarriz, Marcel; Olive Farre, Maria del Carme; de la Rosa Ibarra, Abraham; Baldoma Barraca, Inmaculada Concepcion; Blazquez Sanz, David; Fedorov, Yury; Acosta Humanez, Primitivo Belen; Gonchenko, Marina; Pantazi, Chara; de la Llave Canosa, Rafael; Villanueva Castelltort, Jordi; Delshams i Valdes, Amadeu
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  • Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems

     Delshams i Valdes, Amadeu; Huguet Casades, Gemma
    Nonlinearity
    Date of publication: 2009
    Journal article

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    A computational and geometric approach to phase resetting curves and surfaces  Open access

     Guillamon Grabolosa, Antoni; Huguet Casades, Gemma
    SIAM journal on applied dynamical systems
    Date of publication: 2009
    Journal article

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  • The role of hyperbolic invariant objects: From Arnold diffusion to biological clocks  Open access

     Huguet Casades, Gemma
    Defense's date: 2008-10-16
    School of Mathematics and Statistics (FME), Universitat Politècnica de Catalunya
    Theses

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    El marc d'aquesta tesi són els objectes invariants hiperbòlics (tors amb bigotis, cicles límit, NHIM,. . .), que constitueixen, per aquesta tesi, els objectes essencials per a l'estudi de diversos problemes des de la difusió d'Arnold fins als rellotges biològics. Treballem en tres temes diferents des d'un enfocament tant teòric com numèric, amb una especial atenció per a les aplicacions, especialment en neurobiologia:· Existència de difusió d'Arnold per a sistemes Hamiltonians a priori inestables· Algorismes numèrics ràpids per al càlcul de tors invariants i els "bigotis" associats, per a sistemes Hamiltonians utilitzant el mètode de la parametrització.· Càlcul d'isòcrones i corbes de resposta de fase (PRC) en sistemes neurobiològics usant el mètode de la parametrització.En la primera part de la tesi, hem considerat el cas d'un sistema Hamiltonià a priori inestable amb 2+1/2 graus de llibertat sotmès a una pertorbació de tipus general. "A priori inestable" significa que el sistema no pertorbat presenta un punt d'equilibri hiperbòlic amb una òrbita homoclínica associada. El resultat principal d'aquesta part de la tesi és que per a un conjunt genèric de pertorbacions prou regulars, el sistema presenta el fenòmen de la difusió d'Arnold, és a dir, existeixen trajectòries la variable acció de les quals experimenta un canvi d'ordre 1. La demostració es basa en un estudi detallat de les zones ressonants i els objectes invariants generats en elles, i ofereix una descripció completa de la geografia de les ressonàncies generades per una pertorbació genèrica.En la segona part d'aquest memòria, desenvolupem mètodes numèrics eficients que requereixen poca memòria i operacions per al càlcul de tors invariants i els "bigotis" associats en sistemes Hamiltonians (aplicacions simplèctiques i camps vectorials Hamiltonians).En particular, això inclou els objectes invariants involucrats en el mecanisme de la difusió d'Arnold, estudiat en el capítol anterior. Els algorismes es basen en el mètode de la parametrització i segueixen de prop demostracions recents del teorema KAM que no usen variables acció-angle. Donem detalls de la implementació numèrica que hem dut a terme i mostrem alguns exemples.En la darrera part de la tesi relacionem problemes de temps en sistemes biològics amb algunes eines conegudes de sistemes dinàmics. En particular, usem el mètode de la parametrització i les simetries de Lie per a calcular numèricament les isòcrones i les corbes de resposta de fase (PRC) associades a oscil·ladors i ho apliquem a diversos models biològics ben coneguts. A més a més, aconseguim estendre el càlcul de PRCs en un entorn de l'oscil·lador. Les PRCs són útils per a l'estudi de la sincronització d'oscil·ladors acoblats i una eina bàsica en biologia experimental (ritmes circadians, acoblament sinàptic i elèctric de neurones,. . . ).

  • Effective computation of esochronous sections using symmetries

     Huguet Casades, Gemma
    Non lineal 2004
    Presentation's date: 2004-06-01
    Presentation of work at congresses

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