Lotka–Volterra Like Dynamics in Phase Oscillator Networks

Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 20)

Abstract

Many neural processes, including higher order cognitive functions, are characterized by metastable rather than attractor dynamics. Generalized Lotka–Volterra equations provide a suitable model that can give rise to switching dynamics between metastable equilibria and we review some results how such models can predict bounds on neural processing capacity. In this context generalized Lotka–Volterra system describes the interaction of macroscopic quantities rather than individual neural oscillators. We indicate a connection between generalized Lotka–Volterra equations and the mean field dynamics of networks of multiple identical populations of phase oscillators. This relationship not only gives insight into the global dynamics of phase oscillator populations but also hints at how spatiotemporal dynamics of synchronization can arise in phase oscillator networks.

Keywords

Lotka–Volterra equation Phase oscillators Synchronization Networks 

Notes

Acknowledgements

The author is indebted to M.I. Rabinovich for his guidance and support over the years. Moreover, he would like to thank P. Ashwin and E.A. Martens for many helpful discussions. This work has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement no. 626111.

References

  1. 1.
    Abeles, M., Bergman, H., Gat, I., Meilijson, I., Seidemann, E., Tishby, N., Vaadia, E.: Cortical activity flips among quasi-stationary states. Proc. Natl. Acad. Sci. USA 92 (19), 8616–8620 (1995)CrossRefGoogle Scholar
  2. 2.
    Abrams, D.M., Mirollo, R.E., Strogatz, S.H., Wiley, D.A.: Solvable model for chimera states of coupled oscillators. Phys. Rev. Lett. 101 (8), 084103 (2008)CrossRefGoogle Scholar
  3. 3.
    Acebrón, J., Bonilla, L., Pérez Vicente C., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77 (1), 137–185 (2005)CrossRefGoogle Scholar
  4. 4.
    Afraimovich, V.S., Rabinovich, M.I., Varona, P.: Heteroclinic contours in neural ensembles and the winnerless competition principle. Int. J. Bifurcation Chaos 14 (4), 1195–1208 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Afraimovich, V.S., Zhigulin, V.P., Rabinovich, M.I.: On the origin of reproducible sequential activity in neural circuits. Chaos 14 (4), 1123–1129 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ashwin, P., Burylko, O.: Weak chimeras in minimal networks of coupled phase oscillators. Chaos 25, 013106 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bick, C.: Isotropy of angular frequencies and weak chimeras with broken symmetry. J. Nonlinear Sci. 27 (2), 605–626 (2017). http://doi.org/10.1007/s00332-016-9345-2 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bick, C., Ashwin, P.: Chaotic weak chimeras and their persistence in coupled populations of phase oscillators. Nonlinearity 29 (5), 1468–1486 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bick, C., Martens, E.A.: Controlling chimeras. New J. Phys. 17 (3), 033030 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bick, C., Rabinovich, M.I.: Dynamical origin of the effective storage capacity in the brain’s working memory. Phys. Rev. Lett. 103 (21), 218101 (2009)CrossRefGoogle Scholar
  11. 11.
    Bick, C., Rabinovich, M.I.: On the occurrence of stable heteroclinic channels in Lotka-Volterra models. Dyn. Syst. 25 (1), 97–110 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Friston, K.J.: Transients, metastability, and neuronal dynamics. NeuroImage 5 (2), 164–171 (1997)CrossRefGoogle Scholar
  13. 13.
    Golubitsky, M., Stewart, I.: The Symmetry Perspective. Progress in Mathematics, vol. 200. Birkhäuser, Basel (2002)Google Scholar
  14. 14.
    Komarov, M., Pikovsky, A.: Effects of nonresonant interaction in ensembles of phase oscillators. Phys. Rev. E 84 (1), 16210 (2011)CrossRefGoogle Scholar
  15. 15.
    Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer Series in Synergetics, vol. 19. Springer, Berlin (1984)Google Scholar
  16. 16.
    Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5 (4), 380–385 (2002)Google Scholar
  17. 17.
    Laing, C.R.: Chimera states in heterogeneous networks. Chaos 19 (1), 013113 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Laing, C.R.: The dynamics of chimera states in heterogeneous Kuramoto networks. Physica D 238 (16), 1569–1588 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Laing, C.R.: Disorder-induced dynamics in a pair of coupled heterogeneous phase oscillator networks. Chaos 22 (4), 043104 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Martens, E.A.: Bistable chimera attractors on a triangular network of oscillator populations. Phys. Rev. E 82 (1), 016216 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Martens, E.A., Panaggio, M.J., Abrams, D.M.: Basins of attraction for chimera states. New J. Phys. 18 (2), 022002 (2016)CrossRefGoogle Scholar
  22. 22.
    Martens, E.A., Bick, C., Panaggio, M.J.: Chimera states in two populations with heterogeneous phase-lag. Chaos 26 (9), 094819 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    May, R.M., Leonard, W.J.: Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29 (2), 243–253 (1975)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Miller, G.: The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychol. Rev. 63, 81–97 (1956)CrossRefGoogle Scholar
  25. 25.
    Montbrió, E., Kurths, J., Blasius, B.: Synchronization of two interacting populations of oscillators. Phys. Rev. E 70 (5), 056125 (2004)CrossRefGoogle Scholar
  26. 26.
    Nowotny, T., Rabinovich, M.I.: Dynamical origin of independent spiking and bursting activity in neural microcircuits. Phys. Rev. Lett. 98 (12), 1–4 (2007)CrossRefGoogle Scholar
  27. 27.
    Orosz, G., Moehlis, J., Ashwin, P.: Designing the dynamics of globally coupled oscillators. Prog. Theor. Phys. 122 (3), 611–630 (2009)CrossRefMATHGoogle Scholar
  28. 28.
    Ott, E., Antonsen, T.M.: Low dimensional behavior of large systems of globally coupled oscillators. Chaos 18 (3), 037113 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ott, E., Antonsen, T.M.: Long time evolution of phase oscillator systems. Chaos 19 (2), 023117 (2009)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ott, E., Hunt, B.R., Antonsen, T.M.: Comment on “Long time evolution of phase oscillator systems” [Chaos 19, 023117 (2009)]. Chaos 21 (2), 025112 (2011)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Panaggio, M.J., Abrams, D.M.: Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28 (3), R67–R87 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Pikovsky, A., Rosenblum, M.: Partially integrable dynamics of hierarchical populations of coupled oscillators. Phys. Rev. Lett. 101 (26), 1–4 (2008)CrossRefGoogle Scholar
  33. 33.
    Rabinovich, M.I., Varona, P., Selverston, A., Abarbanel, H.D.I.: Dynamical principles in neuroscience. Rev. Mod. Phys. 78 (4), 1213–1265 (2006)CrossRefGoogle Scholar
  34. 34.
    Rabinovich, M.I., Huerta, R., Laurent, G.: Transient dynamics for neural processing. Science 321 (5885), 48–50 (2008)CrossRefGoogle Scholar
  35. 35.
    Rabinovich, M.I., Afraimovich, V.S., Bick, C., Varona, P.: Information flow dynamics in the brain. Phys. Life Rev. 9 (1), 51–73 (2012)CrossRefGoogle Scholar
  36. 36.
    Rabinovich, M.I., Simmons, A.N., Varona, P.: Dynamical bridge between brain and mind. Trends Cogn. Sci. 19 (8), 453–461 (2015)CrossRefGoogle Scholar
  37. 37.
    Sakaguchi, H., Kuramoto, Y.: A soluble active rotater model showing phase transitions via mutual entertainment. Prog. Theor. Phys. 76 (3), 576–581 (1986)CrossRefGoogle Scholar
  38. 38.
    Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143 (1–4), 1–20 (2000)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Tognoli, E., Scott Kelso, J.A.: The metastable brain. Neuron 81 (1), 35–48 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Centre for Systems, Dynamics and Control and Department of MathematicsUniversity of ExeterExeterUK

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