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A Geometric Approach to Phase Response Curves and Its Numerical Computation Through the Parameterization Method

  • Alberto Pérez-CerveraEmail author
  • Tere M-Seara
  • Gemma Huguet
Article

Abstract

The phase response curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper, we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.

Keywords

Phase response curves Isochrons Phase equation Parameterization method NHIM Synchronization 

Mathematics Subject Classification

37D10 92B25 65P99 37N30 

Notes

Acknowledgements

This work has been partially funded by the Grants MINECO-FEDER MTM2015-65715-P, MDM-2014-0445, PGC2018-098676-B-100 AEI/FEDER/UE, the Catalan Grant 2017SGR1049, (GH, AP, TS), the MINECO-FEDER-UE MTM-2015-71509-C2-2-R (GH), and the Russian Scientific Foundation Grant 14-41-00044 (TS). GH acknowledges the RyC project RYC-2014-15866. TS is supported by the Catalan Institution for research and advanced studies via an ICREA academia price 2018. AP acknowledges the FPI Grant from project MINECO-FEDER-UE MTM2012-31714. We thank C. Bonet for providing us valuable references to prove Theorem 3.2 and A. Granados for numeric support. We also acknowledge the use of the UPC Dynamical Systems group’s cluster for research computing (https://dynamicalsystems.upc.edu/en/computing/).

References

  1. Bates, P.W., Lu, K., Zeng, C.: Approximately invariant manifolds and global dynamics of spike states. Invent. Math. 174(2), 355–433 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Borisyuk, R.M., Kirillov, A.B.: Bifurcation analysis of a neural network model. Biol. Cybern. 66(4), 319–325 (1992)CrossRefzbMATHGoogle Scholar
  3. Buzsaki, G.: Rhythms of the Brain. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  4. Cabré, X., Fontich, E., De La Llave, R.: The parameterization method for invariant manifolds III: overview and applications. J. Differ. Equ. 218(2), 444–515 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Canadell, M., Haro, A.: Parameterization method for computing quasi-periodic reducible normally hyperbolic invariant tori. In: Advances in Differential Equations and Applications, pp. 85–94. Springer, Berlin (2014)Google Scholar
  6. Canadell, M., Haro, À.: A newton-like method for computing normally hyperbolic invariant tori. In: The Parameterization Method for Invariant Manifolds, pp. 187–238. Springer, Berlin (2016)Google Scholar
  7. Canavier, C.C., Achuthan, S.: Pulse coupled oscillators and the phase resetting curve. Math. Biosci. 226(2), 77–96 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Castejón, O., Guillamon, A., Huguet, G.: Phase-amplitude response functions for transient-state stimuli. J. Math. Neurosci. 3(1), 13 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Castelli, R., Lessard, J.-P., Mireles James, J.D.: Parameterization of invariant manifolds for periodic orbits I: efficient numerics via the floquet normal form. SIAM J. Appl. Dyn. Syst. 14(1), 132–167 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Ermentrout, B.: Type I membranes, phase resetting curves, and synchrony. Neural Comput. 8(5), 979–1001 (1996)CrossRefGoogle Scholar
  11. Ermentrout, B., Terman, D.: Mathematical Foundations of Neuroscience. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  12. Ermentrout, G.B., Kopell, N.: Multiple pulse interactions and averaging in systems of coupled neural oscillators. J. Math. Biol. 29(3), 195–217 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J., 21, 193–226, (1971/1972) Google Scholar
  14. Fenichel, N.: Asymptotic stability with rate conditions. Indiana Univ. Math. J., 23, 1109–1137 (1973/74)Google Scholar
  15. Glass, L., Mackey, M.C.: From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton (1988)zbMATHGoogle Scholar
  16. Glass, L., Winfree, A.T.: Discontinuities in phase-resetting experiments. Am. J. Physiol. Regul. Integr. Comp. Physiol. 246(2), R251–R258 (1984)CrossRefGoogle Scholar
  17. Guckenheimer, J.: Isochrons and phaseless sets. J. Math. Biol. 1(3), 259–273 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Guillamon, A., Huguet, G.: A computational and geometric approach to phase resetting curves and surfaces. SIAM J. Appl. Dyn. Syst. 8(3), 1005–1042 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Haro, À., Canadell, M., Figueras, J.-L., Luque, A., Mondelo, J.-M.: The Parameterization Method for Invariant Manifolds. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
  20. Haro, A., de la Llave, R.: Persistence of normally hyperbolic invariant manifolds, internal communicationGoogle Scholar
  21. Haro, A., de la Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms. Discret. Contin. Dyn. Syst. Ser. B 6(6), 1261 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Haro, A., de La Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity. SIAM J. Appl. Dyn. Syst. 6(1), 142 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Volume 538 of Lecture Notes in Math. Springer, Berlin (1977)Google Scholar
  24. Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks, vol. 126. Springer Science & Business Media, Berlin (2012)Google Scholar
  25. Huguet, G., de la Llave, R.: Computation of limit cycles and their isochrons: fast algorithms and their convergence. SIAM J. Appl. Dyn. Syst. 12(4), 1763–1802 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Mauroy, A., Mezić, I.: On the use of fourier averages to compute the global isochrons of (quasi) periodic dynamics. Chaos Interdiscip. J. Nonlinear Sci. 22(3), 033112 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35(1), 193–213 (1981)CrossRefGoogle Scholar
  28. Nipp, K., Stoffer, D.: Attractive invariant mainfolds for maps: existence, smoothness and continuous dependence on the map. In: Research report/Seminar für Angewandte Mathematik, volume 1992. Eidgenössische Technische Hochschule, Seminar für Angewandte Mathematik (1992)Google Scholar
  29. Nipp, K., Stoffer, D.: Invariant manifolds in discrete and continuous dynamical systems. EMS Tracts in Mathematics 21 (2013)Google Scholar
  30. Oprisan, S.A., Canavier, C.C.: The influence of limit cycle topology on the phase resetting curve. Neural Comput. 14(5), 1027–1057 (2002)CrossRefzbMATHGoogle Scholar
  31. Osinga, H.M., Moehlis, J.: Continuation-based computation of global isochrons. SIAM J. Appl. Dyn. Syst. 9(4), 1201–1228 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Pérez-Cervera, A., Huguet, G., Seara, T.: Computation of Invariant Curves in the Analysis of Periodically Forced Neural Oscillators. Springer, Berlin (2018)CrossRefGoogle Scholar
  33. Rinzel, J., Ermentrout, G.B.: Analysis of Neural Excitability and Oscillations. MIT Press, Cambridge, MA (1989)Google Scholar
  34. Rinzel, J., Huguet, G.: Nonlinear dynamics of neuronal excitability, oscillations, and coincidence detection. Commun. Pure Appl. Math. 66(9), 1464–1494 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Schultheiss, N.W., Prinz, A.A., Butera, R.J.: Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis. Springer Science & Business Media, Berlin (2011)Google Scholar
  36. Smeal, R.M., Ermentrout, G.B., White, J.A.: Phase-response curves and synchronized neural networks. Philos. Trans. R. Soc. Lond. B Biol. Sci. 365(1551), 2407–2422 (2010)CrossRefGoogle Scholar
  37. Wedgwood, K.C., Lin, K.K., Thul, R., Coombes, S.: Phase-amplitude descriptions of neural oscillator models. J. Math. Neurosci. 3(1), 2 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. Wilson, D., Ermentrout, B.: Greater accuracy and broadened applicability of phase reduction using isostable coordinates. J. Math. Biol. 76(1–2), 37–66 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  39. Wilson, D., Moehlis, J.: Extending phase reduction to excitable media: theory and applications. SIAM Rev. 57(2), 201–222 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. Wilson, D., Moehlis, J.: Isostable reduction of periodic orbits. Phys. Rev. E 94(5), 052213 (2016)CrossRefGoogle Scholar
  41. Wilson, H.R., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12(1), 1–24 (1972)CrossRefGoogle Scholar
  42. Winfree, A.: Patterns of phase compromise in biological cycles. J. Math. Biol. 1(1), 73–93 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.BGSMATHBarcelonaSpain

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