A three-scale model of spatio-temporal bursting

Geometric singular perturbation theory; Reaction diffusion; Singularity theory; Spatio-temporal bursting; Bifurcation diagram; Reaction diffusion equations; Spatio temporal; Time scale separation; Universal unfolding; Analysis; Modeling and Simulation

Abstract :

[en] We study spatio-temporal bursting in a three-scale reaction diffusion equation organized by the winged cusp singularity. For large time-scale separation the model exhibits traveling bursts, whereas for large space-scale separation the model exhibits standing bursts. Both behaviors exhibit a common singular skeleton, whose geometry is fully determined by persistent bifurcation diagrams of the winged cusp. The modulation of spatio-temporal bursting in such a model naturally translates into paths in the universal unfolding of the winged cusp.

Disciplines :

Mathematics

Author, co-author :

Franci, Alessio ; Université de Liège - ULiège > Département d'électricité, électronique et informatique (Institut Montefiore) > Systèmes et modélisation ; Department of Mathematics, Universidad Nacional Autónoma de México (UNAM), Ciudad de México, Mexico

Sepulchre, Rodolphe; Department of Engineering, University of Cambridge, Cambridge, United Kingdom

Language :

English

Title :

A three-scale model of spatio-temporal bursting

Publication date :

2016

Journal title :

SIAM Journal on Applied Dynamical Systems

eISSN :

1536-0040

Publisher :

Society for Industrial and Applied Mathematics Publications

Volume :

Issue :

Pages :

2143 - 2175

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

http://epubs.siam.org/doi/pdf/10.1137/15M1046101

Funders :

UNAM - Universidad Nacional Autónoma de México [MX]
ERC - European Research Council [BE]

Funding text :

The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet 670645 and from DGAPA-Universidad Nacional Autónoma de México under the PAPIIT Grant IA105816.

Available on ORBi :

since 26 May 2023

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Bibliography

P. CARDIN, P. DA SILVA, AND M. TEIXEIRA, Three time scale singular perturbation problems and nonsmooth dynamical systems, Quart. Appl. Math., 72(2014), pp. 673-687.
G. A. CARPENTER, A geometric approach to singular perturbation problems with applications to nerve impulse equations, J. Differential Equations, 23(1977), pp. 335-367.
P. CARTER AND B. SANDSTEDE, Fast pulses with oscillatory tails in the Fitzhugh-Nagumo system, SIAM J. Math. Anal., 47(2015), pp. 3393-3441.
M. DESROCHES, J. GUCKENHEIMER, B. KRAUSKOPF, C. KUEHN, H. M. OSINGA, AND M. WECHSELBERGER, Mixed-mode oscillations with multiple time scales, SIAM Rev., 54(2012), pp. 211-288.
J. D. DOCKERY, Existence of standing pulse solutions for an excitable activator-inhibitory system, J. Dynam. Differential Equations, 4(1992), pp. 231-257.
A. DOELMAN, P. VAN HEIJSTER, AND T. J. KAPER, Pulse dynamics in a three-component system: Existence analysis, J. Dynam. Differential Equations, 21(2009), pp. 73-115.
G. DRION, A. FRANCI, R. DETHIER, AND J. SEPULCHRE, Dynamic input conductances shape neuronal spiking, Eneuro, 2(2015), e0031, doi:10.1523/ENEURO.0031-14.2015.
G. B. ERMENTROUT AND D. H. TERMAN, Mathematical Foundations of Neuroscience, Interdiscip. Appl. Math. 35, Springer, New York, 2010.
R. FITZHUGH, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1(1961), pp. 445-466.
A. FRANCI, G. DRION, AND R. SEPULCHRE, An organizing center in a planar model of neuronal excitability, SIAM J. Appl. Dyn. Syst., 11(2012), pp. 1698-1722.
A. FRANCI, G. DRION, AND R. SEPULCHRE, Modeling the modulation of neuronal bursting: A singularity theory approach, SIAM J. Appl. Dyn. Syst., 13(2014), pp. 798-829.
D. GOLOMB AND Y. AMITAI, Propagating neuronal discharges in neocortical slices: Computational and experimental study, J. Neurophys., 78(1997), pp. 1199-1211.
M. GOLUBITSKY AND D. G. SCHAEFFER, Singularities and groups in bifurcation theory, Appl. Math. Sci. 51, Springer, New York, 1985.
C. JONES, Geometric singular perturbation theory, in Dynamical Systems, Springer Lecture Notes in Math. 1609, Springer, Berlin, 1995, pp. 44-120.
C. K. R. T. JONES, Stability of the travelling wave solution of the Fitzhugh-Nagumo system, Trans. Amer. Math. Soc., 286(1984), pp. 431-469.
C. K. R. T. JONES AND N. KOPELL, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 108(1994), pp. 64-88.
C. K. R. T. JONES AND J. E. RUBIN, Existence of standing pulse solutions to an inhomogeneous reactiondiffusion system, J. Dynam. Differential Equations, 10(1998), pp. 1-35.
M. KRUPA, N. POPOVIĆ, AND N. KOPELL, Mixed-mode oscillations in three time-scale systems: A prototypical example, SIAM J. Appl. Dyn. Syst., 7(2008), pp. 361-420.
C. KUEHN AND P. SZMOLYAN, Multiscale geometry of the Olsen model and non-classical relaxation oscillations, J. Nonlinear Sci., 25(2015), pp. 583-629.
E. LEE AND D. TERMAN, Uniqueness and stability of periodic bursting solutions, J. Differential Equations, 158(1999), pp. 48-78.
J. M. LEE, Introduction to Smooth Manifolds, 2nd ed., Grad. Texts In Math., Springer, New York, 2013.
D. LINARO, A. CHAMPNEYS, M. DESROCHES, AND M. STORACE, Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-Rose burster, SIAM J. Appl. Dyn. Syst., 11(2012), pp. 939-962.
E. MAEDA, H. P. ROBINSON, AND A. KAWANA, The mechanisms of generation and propagation of synchronized bursting in developing networks of cortical neurons, J. Neurosci., 15(1995), pp. 6834-6845.
P. NAN, Y. WANG, V. KIRK, AND J. E. RUBIN, Understanding and distinguishing three-time-scale oscillations: Case study in a coupled Morris-Lecar system, SIAM J. Appl. Dyn. Syst., 14(2015), pp. 1518-1557.
S. A. NEWMAN AND H. L. FRISCH, Dynamics of skeletal pattern formation in developing chick limb, Science, 205(1979), pp. 662-668.
J. E. RUBIN, Stability, bifurcations and edge oscillations in standing pulse solutions to an inhomogeneous reaction-diffusion system, Proc. Roy. Soc. Edinburgh Sect. A, 129(1999), pp. 1033-1079.
J. SU, J. RUBIN, AND D. TERMAN, Effects of noise on elliptic bursters, Nonlinearity, 17(2003), pp. 133-157.
T. VO, R. BERTRAM, AND M. WECHSELBERGER, Multiple geometric viewpoints of mixed mode dynamics associated with pseudo-plateau bursting, SIAM J. Appl. Dyn. Syst., 12(2013), pp. 789-830.