Journal of Mathematical Biology

, Volume 66, Issue 1–2, pp 139–161 | Cite as

Limitations of perturbative techniques in the analysis of rhythms and oscillations

  • Kevin K. Lin
  • Kyle C. A. Wedgwood
  • Stephen Coombes
  • Lai-Sang Young
Open Access


Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are “sufficiently weak”, an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of “sticky” phase–space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience.


Oscillators Perturbation theory Phase response curve Neuron models Shear-induced chaos 

Mathematics Subject Classification (2000)

92B25 37N25 


  1. Afraimovich VS, Shilnikov LP (1977) The ring principle in problems of interaction between two self-oscillating systems. J Appl Math Mech 41: 618–627MathSciNetCrossRefGoogle Scholar
  2. Brown E, Moehlis J, Holmes P (2004) On the phase reduction and response dynamics of neural oscillator populations. Neural Computat 16: 673–715MATHCrossRefGoogle Scholar
  3. Cohen, AH, Rossignol, S, Grillner, S (eds) (1988) Neural Control of rhythmic movements in vertebrates. Wiley, New YorkGoogle Scholar
  4. Dayan P, Abbott L (2001) Theoretical neuroscience: computational and mathematical modeling of neural systems. MIT Press, CambridgeMATHGoogle Scholar
  5. Deville REL, Sri Namachchivaya N, Rapti Z (2011) Stability of a stochastic two-dimensional non-Hamiltonian system. SIAM J Appl Math 71: 1458–1476MathSciNetMATHCrossRefGoogle Scholar
  6. Eckmann J-P, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Rev Mod Phys 57: 617–656MathSciNetCrossRefGoogle Scholar
  7. Ermentrout GB, Kopell N (1991) Multiple pulse interactions and averaging in coupled neural oscillators. J Math Biol 29: 195–217MathSciNetMATHCrossRefGoogle Scholar
  8. Ermentrout GB, Rinzel J (1991) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in neuronal modeling: from synapses to networks. Bradford Books, BradfordGoogle Scholar
  9. Ermentrout GB, Terman DH (2010) Mathematical Foundations of Neuroscience. In: Interdisciplinary applied mathematics, vol 35. Springer, BerlinGoogle Scholar
  10. Glass L, Guevara MR, Belair J, Shrier A (1984) Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A 29: 1348–1357MathSciNetCrossRefGoogle Scholar
  11. Glass L, Mackey MC (1988) From clocks to chaos: the rhythms of life. Princeton University Press, New JerseyMATHGoogle Scholar
  12. Golubitsky M, Stewart I, Buono PL, Collins JJ (1999) Symmetry in locomotor central pattern generators and animal gaits. Nature 401: 693–695CrossRefGoogle Scholar
  13. Guckenheimer J (1974) Isochrons and phaseless sets. J Theor Biol 1: 259–273MathSciNetGoogle Scholar
  14. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, BerlinMATHGoogle Scholar
  15. Hale JK (1969) Ordinary Differential Equations. Wiley, New YorkMATHGoogle Scholar
  16. Hoppensteadt FC, Izhikevich EM (1997) Weakly connected neural networks. Springer, BerlinCrossRefGoogle Scholar
  17. Lin KK (2006) Entrainment and chaos in a pulse-driven Hodgkin–Huxley oscillator. SIAM J Appl Dyn Sys 5: 179–204MATHCrossRefGoogle Scholar
  18. Lin KK, Young L-S (2008) Shear-induced chaos. Nonlinearity 21: 899–922MathSciNetMATHCrossRefGoogle Scholar
  19. Lin KK, Young L-S (2010) Dynamics of periodically-kicked oscillators. J Fixed Point Theory Appl 7: 291–312MathSciNetMATHCrossRefGoogle Scholar
  20. Lin KK, Shea-Brown E, Young L-S (2009) Reliability of coupled oscillators. J Nonlinear Sci 19: 630–657MathSciNetCrossRefGoogle Scholar
  21. Lin KK, Shea-Brown E, Young L-S (2009) Reliability of layered neural oscillator networks. Commun Math Sci 7: 239–247MathSciNetMATHGoogle Scholar
  22. Lin KK, Shea-Brown E, Young L-S (2009) Spike-time reliability of layered neural oscillator networks. J Comput Neurosci 27: 135–160MathSciNetCrossRefGoogle Scholar
  23. Lu K, Wang Q, Young L-S (2012) Strange attractors for periodically forced parabolic equations. Mem Am Math Soc (to appear)Google Scholar
  24. Ly C, Ermentrout GB (2011) Analytic approximations of statistical quantities and response of noisy oscillators. Physica D 240: 719–731MATHCrossRefGoogle Scholar
  25. May RM (1972) Limit cycles in predator–prey communities. Science 177: 900–902CrossRefGoogle Scholar
  26. Medvedev GS (2011) Synchronization of coupled limit cycles. J Nonlinear Sci 21: 441–464MathSciNetMATHCrossRefGoogle Scholar
  27. Mirollo RE, Strogatz SH (1990) Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 50: 1645–1662MathSciNetMATHCrossRefGoogle Scholar
  28. Netoff TI, Acker CD, Bettencourt JC, White JA (2005) Beyond two-cell networks: experimental measurement of neuronal responses to multiple synaptic inputs. J Comput Neurosci 18: 287–295CrossRefGoogle Scholar
  29. Oprisan SA, Thirumalai V, Canavier CC (2003) Dynamics from a time series: can we extract the phase resetting curve from a time series?. Biophys J 84: 2919–2928CrossRefGoogle Scholar
  30. Ott W, Stenlund M (2010) From limit cycles to strange attractors. Commun Math Phys 296: 215–249MathSciNetMATHCrossRefGoogle Scholar
  31. Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  32. Şuvak Ö, Demir A (2011) On phase models for oscillators. IEEE Trans Comput Aided Des Integrated Circ Syst 30: 972–985CrossRefGoogle Scholar
  33. Thul R, Bellamy TC, Roderick HL, Bootman MD, Coombes S (2008) Calcium oscillations. In: Maroto M, Monk N (eds) Cellular oscillatory mechanisms, advances in experimental medicine and biology. Springer, BerlinGoogle Scholar
  34. Wang Q, Ott W (2011) Dissipative homoclinic loops of two-dimensional maps and strange attractors with one direction of instability. Commun Pure Appl Math 64: 1439–1496MathSciNetMATHGoogle Scholar
  35. Wang Q, Young L-S (2001) Strange attractors with one direction of instability. Commun Math Phys 218: 1–97MathSciNetMATHCrossRefGoogle Scholar
  36. Wang Q, Young L-S (2002) From invariant curves to strange attractors. Commun Math Phys 225: 275–304MathSciNetMATHCrossRefGoogle Scholar
  37. Wang Q, Young L-S (2003) Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Commun Math Phys 240: 509–529MathSciNetMATHGoogle Scholar
  38. Wang Q, Young L-S (2008) Toward a theory of rank one attractors. Ann Math 167: 349–480MathSciNetMATHCrossRefGoogle Scholar
  39. Winfree A (2000) Geometry of biological time, 2nd edn. Springer, BerlinGoogle Scholar
  40. Young L-S (2002) What are SRB measures, and which dynamical systems have them?. J Stat Phys 108: 733–754MATHCrossRefGoogle Scholar
  41. Zaslavsky G (1978) The simplest case of a strange attractor. Phys Lett 69A: 145–147MathSciNetGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Kevin K. Lin
    • 1
  • Kyle C. A. Wedgwood
    • 2
  • Stephen Coombes
    • 2
  • Lai-Sang Young
    • 3
  1. 1.Department of Mathematics and Program in Applied MathematicsUniversity of ArizonaTucsonUSA
  2. 2.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations