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...

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Veröffentlicht in:SIAM journal on applied dynamical systems Jg. 8; H. 3; S. 1005 - 1042
Hauptverfasser: Guillamon, Antoni, Huguet, Gemma
Format: Journal Article Verlag
Sprache:Englisch
Veröffentlicht: Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
Schlagworte:
34 Ordinary differential equations
34C Qualitative theory
37 Dynamical systems and ergodic theory
37N Applications
Chaos theory
Classificació AMS
Differentiable dynamical systems
Dynamical systems
Equacions diferencials i integrals
Equacions diferencials ordinàries
Ergodic theory
Matemàtiques i estadística
Neighborhoods
Numerical analysis
Oscillators
Sistemes dinàmics
Àrees temàtiques de la UPC
ISSN:1536-0040, 1536-0040
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Abstract 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.
AbstractList Peer Reviewed
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.
Author Huguet, Gemma
Guillamon, Antoni
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10.1162/neco.2006.18.4.817
10.1152/ajpregu.1994.267.4.R1118
10.1142/S0218127405013861
10.1016/S0006-3495(81)84782-0
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10.1006/jcph.1998.6059
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10.3934/dcdsb.2006.6.1261
10.1111/j.1432-1033.1968.tb00175.x
10.1016/j.jde.2004.03.011
ContentType Journal Article
Publication
Contributor Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
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Copyright [Copyright] © 2009 Society for Industrial and Applied Mathematics
Attribution-NonCommercial-NoDerivs 3.0 Spain info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/3.0/es
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SubjectTerms 34 Ordinary differential equations
34C Qualitative theory
37 Dynamical systems and ergodic theory
37N Applications
Chaos theory
Classificació AMS
Differentiable dynamical systems
Dynamical systems
Equacions diferencials i integrals
Equacions diferencials ordinàries
Ergodic theory
Matemàtiques i estadística
Neighborhoods
Numerical analysis
Oscillators
Sistemes dinàmics
Àrees temàtiques de la UPC
Title A Computational and Geometric Approach to Phase Resetting Curves and Surfaces
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