Optimal Waveform for Fast Entrainment of Weakly Forced Nonlinear Oscillators

Anatoly Zlotnik, Yifei Chen, István Z. Kiss, Hisa-Aki Tanaka and Jr-Shin Li
Physical Review Letters, 2013.

Abstract

For many biological and engineered systems, a central function or design goal is to abbreviate the time required to synchronize a rhythmic process to an external forcing signal. We present a theory for deriving the input that effectively minimizes the average transient time required to entrain a phase model, which enables a practical technique for constructing fast entrainment waveforms for general nonlinear oscillators. This result is verified in numerical simulations using the Hodgkin-Huxley neuron model, and in experiments on an oscillatory electrochemical system.

Download PDF

Figures at a glance

References

  1. S. H. Strogatz, Nonlinear Dynamics and Chaos (Perseus Books Group, Cambridge, MA, 2001), 1st ed.
  2. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Science (Cambridge University Press, New York, 2001).
  3. T. Naqvi and D. Winter, U.S. Patent No. 8 112 150 (2012).
  4. J. Waterhouse, T. Reilly, G. Atkinson, and B. Edwards, Lancet 369, 1117 (2007).
  5. C. R. McClung, Adv. Genet. 74, 105 (2011).
  6. X. Feng, C. White, A. Hajimiri, and M. Roukes, Nat. Nanotechnol. 3, 342 (2008).
  7. H. Moehlis, E. Brown, and H. Rabitz, J. Comp. Nonlin. Dyn. 1, 358 (2006).
  8. I. Dasanayake and J.-S. Li, Phys. Rev. E 83, 061916 (2011).
  9. A. Zlotnik and J.-S. Li, in Proceedings of the 2011 ASME Dynamic Systems and Control Conference, Arlington, VA, 2011 (ASME, New York, 2011), Vol. 1, pp. 479-484 [http://proceedings.asmedigitalcollection.asme.org/proceeding.aspx?articleid=1638452].
  10. T. Harada, H.-A. Tanaka, M. J. Hankins, and I. Z. Kiss, Phys. Rev. Lett. 105, 088301 (2010).
  11. A. Zlotnik and J.-S. Li, J. Neural Eng. 9, 046015 (2012).
  12. A. Granada and H. Herzel, PLoS One 4, e7057 (2009).
  13. S. Hata, K. Arai, R. F. Gala´n, and H. Nakao, Phys. Rev. E84, 016229 (2011).
  14. B. Ermentrout, Neural Comput. 8, 979 (1996).
  15. S. H. Strogatz, Physica (Amsterdam) 143D, 1 (2000).
  16. F. Hoppensteadt and E. Izhikevich, Phys. Rev. Lett. 82, 2983 (1999).
  17. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, New York, 1984).
  18. I. Z. Kiss, Y. Zhai, and J. L. Hudson, Science 296, 1676 (2002).
  19. S. Nakata, K. Miyazaki, S. Izuhara, H. Yamaoka, and D. Tanaka, J. Phys. Chem. A 113, 6876 (2009).
  20. E. Brown, J. Moehlis, and P. Holmes, Neural Comput. 16, 673 (2004).
  21. D. Efimov and T. Raissi, in Proceedings of the 8th IFAC Symposium on Nonlinear Control Systems Bologna, 2010 (IFAC, Laxenberg, Austria, 2010), pp. 332-337 [http://www.ifac-papersonline.net/Detailed/43863.html].
  22. W. Govaerts andB. Sautois, NeuralComput. 18, 817 (2006).
  23. B. Ermentrout, A Guide to XPPAUT for Researchers and Students (SIAM, Philadelphia, 2002).
  24. I. Kornfeld, S. Fomin, and Y. Sinai, Ergodic Theory: Differentiable Dynamical Systems, Grundlehren der Mathematischen Wissenschaften Vol. 245 (Springer-Verlag, Berlin, 1982).
  25. F. Hoppensteadt and E. Izhikevich, Weakly Connected Neural Networks (Springer-Verlag, New Jersey, 1997).
  26. A. Hodgkin and A. Huxley, J. Physiol. 117, 500 (1952) [http://www.ncbi.nlm.nih.gov/pubmed/12991237].
  27. O. Lev, A. Wolfberg, L.M. Pismen, and M. Sheintuch, J. Phys. Chem. 93, 1661 (1989).
  28. I. Z. Kiss, Y. M. Zhai, and J. L. Hudson, Phys. Rev. Lett. 94, 248301 (2005).
  29. D. Andrieux and P. Gaspard, J. Chem. Phys. 128, 154506 (2008).