Singularly perturbed phase response curves for relaxation oscillators

[en] We exploit a novel geometric method to construct the global isochrones of relaxation oscillators and the associated phase response curve. This method complements the classical infinitesimal (local) phase response curve approach by constructively predicting the finite phase response curve near the singular limit of infinite timescale separations between the oscillator variables. We illustrate the power of our construction on the FitzHugh-Nagumo model of neuronal spike generation. Because of its global and constructive nature, not requiring extensive numerical simulations, the proposed approach is particularly suited to control design applications.

Disciplines :

Engineering, computing & technology: Multidisciplinary, general & others

Author, co-author :

Sacré, Pierre ; Université de Liège - ULiège > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Robotique intelligente

Franci, Alessio ; Université de Liège - ULiège > Département d'électricité, électronique et informatique (Institut Montefiore) > Brain-Inspired Computing

Language :

English

Title :

Singularly perturbed phase response curves for relaxation oscillators

Publication date :

2016

Event name :

55th IEEE Conference on Decision and Control (CDC)

Event organizer :

IEEE

Event place :

Las Vegas, United States - Nevada

Event date :

from 12-12-2016 to 14-12-2016

Audience :

International

Main work title :

Proceedings of the 55th IEEE Conference on Decision and Control (CDC)

Publisher :

IEEE

Peer reviewed :

Peer reviewed

Available on ORBi :

since 26 November 2018

Statistics

Number of views

47 (6 by ULiège)

Number of downloads

130 (2 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

A. T. Winfree, The Geometry of Biological Time, 1st ed., ser. Biomathematics. New York, NY: Springer-Verlag, 1980, vol. 8.
L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life. Princeton, NJ: Princeton University Press, 1988.
D. V. Efimov, P. Sacré, and R. Sepulchre, "Controlling the phase of an oscillator: A phase response curve approach," in Proc. 48th IEEE Conf. Decision and Control and 28th Chinese Control Conf., Shanghai, China, Dec. 2009, pp. 7692-7697.
P. Danzl, J. Hespanha, and J. Moehlis, "Event-based minimum-time control of oscillatory neuron models: phase randomization, maximal spike rate increase, and desynchronization," Biol. Cybern., vol. 101, no. 5-6, pp. 387-399, Dec. 2009.
A. Mauroy, P. Sacré, and R. Sepulchre, "Kick synchronization versus diffusive synchronization," in Proc. 51st IEEE Conf. Decision and Control, Maui, HI, Dec. 2012, pp. 7171-7183.
F. Dörfler and F. Bullo, "Synchronization in complex networks of phase oscillators: A survey," Automatica, vol. 50, no. 6, pp. 1539-1564, 2014.
P. Sacré and R. Sepulchre, "Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems," IEEE Control Syst. Mag., vol. 34, no. 2, pp. 50-74, Apr. 2014.
E. M. Izhikevich, "Phase equations for relaxation oscillators," SIAM J. Appl. Math., vol. 60, no. 5, pp. 1789-1804, 2000.
N. Fenichel, "Persistence and smoothness of invariant manifolds for flows," Indiana Univ. Math. J., vol. 21, no. 3, pp. 193-226, 1972.
M. Krupa and P. Szmolyan, "Relaxation oscillation and canard explosion," J. Differential Equations, vol. 174, no. 2, pp. 312-368, Aug. 2001.
P. Sacré, "Systems analysis of oscillator models in the space of phase response curves," Ph.D. dissertation, University of Liège, Belgium, Sep. 2013.
J. Cannon and N. Kopell, "The leaky oscillator: Properties of inhibitionbased rhythms revealed through the singular phase response curve," SIAM J. Applied Dynamical Systems, vol. 14, no. 4, pp. 1930-1977, 2015.
A. Franci, G. Drion, and R. Sepulchre, "Modeling the modulation of neuronal bursting: A singularity theory approach," SIAM J. Applied Dynamical Systems, vol. 13, no. 2, pp. 798-829, 2014.
E. M. Izhikevich, "Synchronization," in Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge, MA: The MIT Press, 2007, ch. 10.
R. FitzHugh, "Impulses and physiological states in theoretical models of nerve membrane," Biophys. J., vol. 1, no. 6, pp. 445-466, Jul. 1961.
J. Nagumo, S. Arimoto, and S. Yoshizawa, "An active pulse transmission line simulating nerve axon," Proc. IRE, vol. 50, no. 10, pp. 2061-2070, Oct. 1962.
P. Sacré and R. Sepulchre, "Sensitivity analysis of circadian entrainment in the space of phase response curves," in A Systems Theoretic Approach to Systems and Synthetic Biology II: Analysis and Design of Cellular Systems, V. V. Kulkarni, G.-B. Stan, and K. Raman, Eds. Springer Netherlands, 2014, pp. 59-81.
D. Somers and N. Kopell, "Rapid synchronization through fast threshold modulation," Biol. Cybern., vol. 68, no. 5, pp. 393-407, Mar. 1993.