アドホックネットワーク，ユビキタス・ネットワーク

同期現象（シンクロニゼーション）の解明，同期システムへの応用

ネットワークダイナミクス, 粘菌の分散知能の研究

Optimal entrainment with smooth, pulse, and square signals in weakly forced nonlinear oscillators

Hisa-Aki Tanaka
Physica D: Nonlinear Phenomena, 2014.

Keyword

Entrainment, Synchronization, Arnold tongue, Optimization, Holder's inequality

Abstract

A physical limit of entrainability of nonlinear oscillators is considered for an external weak signal (forcing). This limit of entrainability is characterized by the optimization problem maximizing the width of the Arnold tongue (the frequency-locking range versus forcing magnitude) under certain practical constraints. Here we show a solution to this optimization problem, thanks to a direct link to Holder’s inequality. This solution defines an ideal forcing realizing the entrainment limit, and as the result, a fundamental limit of entrainment is clarified as follows. For $1:1$ entrainment, we obtain (i) a construction of the global optimal forcing and a condition for its uniqueness in $L^p$-space with $p > 1$, and (ii) a construction of the global optimal pulse-like forcings in $L^1$-space, and for $m:n$ entrainment ($m≠n$), some informations about the non-existence of the ideal forcing. (iii) In addition, we establish definite algorithms for obtaining the global optimal forcings for $1 < p ≦ ∞$ and these pulse-like forcings for $p = 1$. These theoretical findings are verified by systematic, extensive numerical calculations and simulations.

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References

  1. A.S. Pikovsky, M.G. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.
  2. A.T. Winfree, The Geometry of Biological Time, Springer-Verlag, New York, 1980.
  3. L. Glass, Cardiac arrhythmias and circle maps: A classical problem, Chaos 1 (1991) 13--19.
  4. J.C. Jackson, J.F. Windmill, V.G. Pook, D. Robert, Synchrony through twicefrequency forcing for sensitive and selective auditory processing, Proc. Natl. Acad. Sci. 106 (2009) 10177--10182.
  5. B. van der Pol, Forced oscillations in a circuit with nonlinear resistance (reception with reactive triode), Philos. Mag. 3 (13) (1927) 65--80.
  6. R.A. Adler, Study of locking phenomena in oscillators, Proc. IRE 34 (6) (1946) 351--357.
  7. K. Takano, M. Motoyoshi, M. Fujishima, 4.8 GHz CMOS frequency multiplier with subharmonic pulse-injection locking, 2007 IEEE Asian Solid-State Circuits Conference (2007) 336--338.
  8. J. Moehlis, E. Shea-Brown, H. Rabitz, Optimal inputs for phase models of spiking neurons, ASME J. Comput. Nonlinear Dynam. 1 (4) (2006) 358--367.
  9. I. Dasanayake, J.-S. Li, Optimal design of minimum-power stimuli for phase models of neuon oscillators, Phys. Rev. E 83 (2011) 061916.
  10. X.L. Feng, C.J. White, A. Hajimiri, M.L. Roukes, A self-sustaining ultrahighfrequency nanoelectromechanical oscillator, Nat. Nanotechnol. 3 (2008) 342--346.
  11. M.C. Cross, A. Zumdieck, R. Lifshitz, J.L. Rogers, Synchronization by nonlinear frequency pulling, Phys. Rev. Lett. 93 (2004) 224101.
  12. K. Kawasaki, Y. Akiyama, K. Komori, M. Uno, H. Takeuchi, T. Itagaki, Y. Hino, Y. Kawasaki, K. Ito, A. Hajimiri, A millimeter-wave intra-connect solution, ISSCC Dig. Tech. Pap. (2010) 414--415.
  13. O. Mondragon-Palomino, T. Danino, J. Selimkhanov, L. Tsimring, J. Hasty, Entrainment of a population of synthetic genetic oscillators, Science 333 (2011) 1315--1319.
  14. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, 1984.
  15. F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks, Springer, New York, 1997.
  16. A.E. Granada, H. Herzel, How to achieve fast entrainment? The timescale to synchronization, PLoS One 4 (2009) e7057.
  17. A. Zlotnik, Y. Chen, I.Z. Kiss, H.-A. Tanaka, J.-S. Li, Optimal waveform for fast entrainment of weakly forced nonlinear oscillators, Phys. Rev. Lett. 111 (2013) 024102.
  18. N. Bagheri, J. Stelling, F.J. Doyle, Circadian phase resetting via single and multiple control targets, PLoS Comput. Biol. 4 (2008) e1000104.
  19. D. Forger, D. Paydarfar, Starting, stopping, and resetting biological oscillators: In search of optimum perturbations, J. Theoret. Biol. 230 (2004) 521.
  20. D. Lebiedz, S. Sager, H.G. Bock, P. Lebiedz, Annihilation of limit-cycle oscillations by identification of critical perturbing stimuli via mixed-integer optimal control, Phys. Rev. Lett. 95 (2005) 108303.
  21. V. Gintautas, A.W. Hubler, Resonant forcing of nonlinear systems of differential equations, Chaos 18 (3) (2008) 033118.
  22. A. Nabi, J. Moehlis, Time optimal control of spiking neurons, J. Math. Biol. 64 (2012) 981--1004.
  23. J. Ritt, Evaluation of entrainment of a nonlinear neural oscillator to white noise, Phys. Rev. E 68 (2003) 041915.
  24. J. Feng, H.C. Tuckwell, Optimal control of neuronal activity, Phys. Rev. Lett. 91 (2003) 018101.
  25. T. Harada, H.-A. Tanaka, M.J. Hankins, I.Z. Kiss, Optimal waveform for the entrainment of a weakly forced oscillator, Phys. Rev. Lett. 105 (2010) 088301.
  26. I.M. Gelf, S.V. Fomin, Calculus of Variations, Dover, New York, 2000, (Chapter 1).
  27. D.E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall, New Jersey, 1970.
  28. B. Ermentrout, Type I membranes, phase resetting curves, and synchrony, Neural Comput. 8 (1996) 988.
  29. W. Rudin, Real and Complex Analysis, third ed., McGraw-Hill, New York, 1987, pp. 63--65.
  30. H.-A. Tanaka, Synchronization limit of weakly forced nonlinear oscillators, Fast Track Communication in J. Phys. A. (2014) Accepted.
  31. J.R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, revised ed., Wiley, Chichester, 1999.
  32. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, second ed., Cambridge University Press, Cambridge, 1988, (Chapter 7).
  33. I.Z. Kiss, Y. Zhai, private communications (2012).
  34. Y. Kawamura, H. Nakao, K. Arai, H. Kori, Y. Kuramoto, Collective phase sensitivity, Phys. Rev. Lett. 101 (2008) 024101.
  35. H. Kori, Y. Kawamura, H. Nakao, K. Arai, Y. Kuramoto, Collective-phase description of coupled oscillators with general network structure, Phys. Rev. E 80 (2009) 036207.
  36. Y. Kawamura, H. Nakao, Y. Kuramoto, Collective phase description of globally coupled excitable elements, Phys. Rev. E 84 (2011) 046211.
  37. T.W. Ko, G.B. Ermentrout, Phase-response curves of coupled oscillators, Phys. Rev. E 79 (2009) 016211.
  38. Z. Levnajic, A. Pikovsky, Phase resetting of collective rhythm in ensembles of oscillators, Phys. Rev. E 82 (2010) 056202.
  39. A. Abouzeid, G.B. Ermentrout, Type-II phase resetting curve is optimal for stochastic synchrony, Phys. Rev. E 80 (2009) 011911.
  40. S. Hata, K. Arai, R.F. Galan, H. Nakao, Optimal phase response curves for stochastic synchronization of limit-cycle oscillators by common Poisson noise, Phys. Rev. E 84 (2011) 016229.
  41. D.S. Goldobin, J.-n. Teramae, H. Nakao, G.B. Ermentrout, Dynamics of limit cycle oscillators subject to general noise, Phys. Rev. Lett. 105 (2010) 154101.
  42. H. Nakao, T. Yanagita, Y. Kawamura, Phase description of stable limit-cycle solutions in reaction-diffusion systems, Procedia IUTAM 5 (2012) 227--233.
  43. Z.P. Kilpatrick, G.B. Ermentrout, Response of traveling waves to transient inputs in neural fields, Phys. Rev. E 85 (2012) 021910.
  44. T. Hikihara, Y. Okamoto, Y. Ueda, An experimental spatio-temporal state transition of coupled magneto-elastic system, Chaos 7 (1997) 810--816.
  45. M. Sato, B.E. Hubbard, A.J. Sievers, B. Ilic, D.A. Czaplewski, H.G. Craighead, Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array, Phys. Rev. Lett. 90 (2003) 044102.
  46. M. Sato, B.E. Hubbard, A.J. Sievers, B. Ilic, H.G. Craighead, Optical manipulation of intrinsic localized vibrational energy in cantilever arrays, Europhys. Lett. 66 (2004) 318--323.
  47. R.S. MacKay, S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7 (1994) 1623--1643.
  48. M. Sato, A.J. Sievers, Experimental and numerical exploration of intrinsic localized modes in an atomic lattice, J. Biol. Phys. 35 (2009) 57--72.