Global Phase-Amplitude Description of Oscillatory Dynamics via the Parameterization Method

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F20%3A00532247" target="_blank" >RIV/67985807:_____/20:00532247 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.1063/5.0010149" target="_blank" >http://dx.doi.org/10.1063/5.0010149</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1063/5.0010149" target="_blank" >10.1063/5.0010149</a>

Result language

angličtina

Original language name

Global Phase-Amplitude Description of Oscillatory Dynamics via the Parameterization Method

Original language description

In this paper, we use the parameterization method to provide a complete description of the dynamics of an n-dimensional oscillator beyond the classical phase reduction. The parameterization method allows us, via efficient algorithms, to obtain a parameterization of the attracting invariant manifold of the limit cycle in terms of the phase-amplitude variables. The method has several advantages. It provides analytically a Fourier–Taylor expansion of the parameterization up to any order, as well as a simplification of the dynamics that allows for a numerical globalization of the manifolds. Thus, one can obtain the local and global isochrons and isostables, including the slow attracting manifold, up to high accuracy, which offer a geometrical portrait of the oscillatory dynamics. Furthermore, it provides straightforwardly the infinitesimal phase and amplitude response functions, that is, the extended infinitesimal phase and amplitude response curves, which monitor the phase and amplitude shifts beyond the asymptotic state. Thus, the methodology presented yields an accurate description of the phase dynamics for perturbations not restricted to the limit cycle but to its attracting invariant manifold. Finally, we explore some strategies to reduce the dimension of the dynamics, including the reduction of the dynamics to the slow stable submanifold. We illustrate our methods by applying them to different three-dimensional single neuron and neural population models in neuroscience. We extend the applications of the parameterization method to compute the full set of phase-amplitude coordinates for high-dimensional oscillators. We use the Floquet normal form and automatic differentiation techniques to drastically reduce the computational cost of the required calculations. Our methods provide an analytical expression for the local isochrons, isostables, and infinitesimal phase and amplitude response functions in a neighborhood of the limit cycle, while allowing for the globalization of these objects and functions to the full basin of attraction. We illustrate our methodology by applying it to relevant single neuron and neural population models in neuroscience. Moreover, we perform a perturbation study of a single neuron model and study the scope of validity of different dynamical reductions, namely, the slow-manifold reduction and the phase reduction. Our results provide an efficient methodology that allows for a geometrical understanding of the dynamics of high-dimensional oscillators.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10102 - Applied mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2020

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Chaos

ISSN

1054-1500

e-ISSN

—

Volume of the periodical

30

Issue of the periodical within the volume

8

Country of publishing house

US - UNITED STATES

Number of pages

30

Pages from-to

083117

UT code for WoS article

000560034000001

EID of the result in the Scopus database

2-s2.0-85090112462