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Phase description of spiking neuron networks with global electric and synaptic coupling

Dipanjan Roy*, Anandamohan Ghosh†, and Viktor K. Jirsa‡

  • Theoretical Neuroscience Group, Institut des Sciences du Mouvement, UMR6233 CNRS and Université de la Méditerranée, 163 Avenue de Luminy, F-13288 Marseille, France

  • *dipanjan.roy@etumel.univmed.fr
  • †anandamohan.ghosh@univmed.fr
  • ‡viktor.jirsa@univmed.fr
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Phys. Rev. E 83, 051909 – Published 11 May, 2011

DOI: https://doi.org/10.1103/PhysRevE.83.051909

Abstract

Phase models are among the simplest neuron models reproducing spiking behavior, excitability, and bifurcations toward periodic firing. However, coupling among neurons has been considered only using generic arguments valid close to the bifurcation point, and the differentiation between electric and synaptic coupling remains an open question. In this work we aim to address this question and derive a mathematical formulation for the various forms of coupling. We construct a mathematical model based on a planar simplification of the Morris-Lecar model. Based on geometric arguments we then derive a phase description of a network of the above oscillators with biologically realistic electric coupling and subsequently with chemical coupling under fast synapse approximation. We demonstrate analytically that electric and synaptic coupling are differently expressed on the level of the network’s phase description with qualitatively different dynamics. Our mathematical analysis shows that a breaking of the translational symmetry in the phase flows is responsible for the different bifurcations paths of electric and synaptic coupling. Our numerical investigations confirm these findings and show excellent correspondence between the dynamics of the full network and the network’s phase description.

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      References (45)

      1. C. Morris and H. Lecar, Biophys. J. 35, 193 (1981).
      2. D. A. McCormick, B. W. Connors, J. W. Lighthall, and D. A. Prince, J. Neurophysiol. 54, 782 (1985).
      3. A. Kaske and N. Bertschinger, Biol. Cybern. 92, 3 (2005).
      4. B. S. Gutkin, G. B. Ermentrout, and A. D. Reyes, J. Neurophysiol. 94, 1623 (2005).
      5. B. Naundorf, F. Wolf, and M. Volgushev, Nature (London) 440, 7087 (2006).
      6. E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience) (MIT Press, Cambridge, MA, 2006), p. 457.
      7. J. L. Hindmarsh and R. M. Rose, Proc. R. Soc. Lond. B 221, 87 (1984).
      8. X. J. Wang, Neuroscience 59, 21 (1994).
      9. P. F. Pinsky and J. Rinzel, J. Comput. Neurosci. 1, 39 (1994).
      10. J. C. Brumberg and B. S. Gutkin, Brain Res. 1171, 122 (2007).
      11. A. Ghosh, D. Roy, and V. K. Jirsa, Phys. Rev. E 80, 041930 (2009).
      12. G. B. Ermentrout and N. Kopell, SIAM J. Appl. Math. 46, 359 (1986).
      13. E. M. Izhikevich, Int. J. Bifurcation Chaos 10, 1171 (2000).
      14. B. S. Gutkin and G. B. Ermentrout, Information Processing in Cells and Tissues (Plenum Press, New York, 1998), p. 57.
      15. B. S. Gutkin, J. Jost, and H. C. Tuckwell, Europhys. Lett. 81, 20005 (2008).
      16. D. Pazo and E. Montbrio, Phys. Rev. E 73, 055202(R) (2006).
      17. G. B. Ermentrout, Neural Comput. 6, 679 (1994).
      18. F. C. Hoppendsteadt and E. M. Izhikevich, Weakly Connected Neural Networks (Springer-Verlag, New York, 1997).
      19. E. S. Brown, J. Mohelis, and P. Holmes, Neural Comput. 16, 4 (2004).
      20. S. Coombes, Phys. Rev. E 67, 041910 (2003).
      21. T. B. Kepler, L. F. Abbott, and E. Marder, Biol. Cybern. 66, 381 (1992).
      22. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, New York, 1984).
      23. R. M. Smeal, G. B. Ermentrout, and J. A. White, Phil. Trans. R. Soc. B 365, 2407 (2010).
      24. V. Jirsa and A. McIntosh, Handbook of Brain Connectivity Understanding complex systems Springer complexity. Nonlinear Dynamics (Springer, Berlin, 2007).
      25. A. Tonnelier and W. Gerstner, Phys. Rev. E 67, 021908 (2003).
      26. S. Coombs, SIAM J. Appl. Dyn. Syst. 7, 1101 (2008).
      27. D. T. W. Chik, S. Coombes, and Z. D. Wang, Phys. Rev. E 70, 011908 (2004).
      28. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry, and Engineering (Westview Press, 2000), p. 498.
      29. C. G. Assisi, V. K. Jirsa, and J. A. S. Kelso, Phys. Rev. Lett. 94, 018106 (2005).
      30. M. Dhamala, V. K. Jirsa, and M. Ding, Phys. Rev. Lett. 92, 028101 (2004).
      31. M. Dhamala, V. K. Jirsa, and M. Ding, Phys. Rev. Lett. 92, 074104 (2004).
      32. X. J. Wang and J. Rinzel, Neural Comput. 4, 84 (1992).
      33. D. Golomb and J. Rinzel, Phys. Rev. E 48, 4810 (1993).
      34. E. C. Keen and A. J. Hudspeth, Proc. Natl. Acad. Sci. 103, 14 (2006).
      35. H. Parnas and I. Parnas, J. Membr. Biol. 142, 267 (1994).
      36. B. L. Sabatini and W. G. Regher, Nature (London) 384, 170 (1996).
      37. F. Gabbiani, J. Midtgarred, and T. Knöpfel, J. Neurophysiol. 72, 999 (1994).
      38. E. J. Doedl, [http://indy.cs.concordia.ca/auto/].
      39. R. H. Clewley, W. E. Sherwood, M. D. LaMar, and J. M. Guckenheimer, [http://pydstool.sourceforge.net].
      40. M. Bazhenov, R. Huerta, M. I. Rabinovich, and T. Sejnowskic, Physica D 116, 3 (1998).
      41. M. Bazhenov, N. F. Rulkov, and I. Timofeev, J. Neurophysiol. 100, 1562 (2008).
      42. C. Koch, Biophysics of Computation (Oxford University Press, New York, 1999).
      43. W. Gerstner and W. Kistler, Spiking Neuron Models: An Introduction (Cambridge University Press, Cambridge, 2002).
      44. A. L. Hodgkin and A. F. Huxley, J. Physiol. 117, 500 (1952).
      45. N. Kopell and G. B. Ermentrout, Proc. Natl. Acad. Sci. USA 101, 15482 (2004).

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      Phys. Rev. E 83, 051909– Published 11 May, 2011
      • Received 24 July 2010

      ©2011 American Physical Society

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