Advertisement

The European Physical Journal Special Topics

, Volume 223, Issue 13, pp 2979–2988 | Cite as

Periodic solutions to a mean-field model for electrocortical activity

Regular Article Nonlinear Partial Differential Equations
Part of the following topical collections:
  1. Advanced Computational and Experimental Techniques in Nolinear Dynamics. Guest Editors: Elbert E.N. Macau and Carlos L. Pando Lambruschini (Eds.)

Abstract

We consider a continuum model of electrical signals in the human cortex, which takes the form of a system of semilinear, hyperbolic partial differential equations for the inhibitory and excitatory membrane potentials and the synaptic inputs. The coupling of these components is represented by sigmoidal and quadratic nonlinearities. We consider these equations on a square domain with periodic boundary conditions, in the vicinity of the primary transition from a stable equilibrium to time-periodic motion through an equivariant Hopf bifurcation. We compute part of a family of standing wave solutions, emanating from this point.

Keywords

Periodic Orbit Hopf Bifurcation European Physical Journal Special Topic Bifurcation Diagram Neutral Stability Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Nunez, R. Srinivasan, Electric Fields of the Brain. The Neurophysics of EEG, 2nd edition (Oxford University Press, Oxford, 2006)Google Scholar
  2. 2.
    D.T.J. Liley, I. Bojak, M.P. Dafilis, L. van Veen, F. Frascoli, B.L. Foster, Modelling phase transitions in the brain, Springer Series in Computational Neuroscience 4 (2009)Google Scholar
  3. 3.
    E. Rolls, T. Webb, G. Deco, Cogn. Neurosci. 36, 2689 (2012)Google Scholar
  4. 4.
    E. Izhikevich, G. Edelman, PNAS 105, 3593 (2008)CrossRefADSGoogle Scholar
  5. 5.
    G. Deco, V.K. Jirsa, P.A. Robinson, M. Breakspear, K. Friston, PLoS One 4, e1000092 (2008)Google Scholar
  6. 6.
    D.A. Pinotsis, M. Leite, K.J. Friston, Frontiers Comp. Neurosci. 7, 1 (2013)Google Scholar
  7. 7.
    H.R. Wilson, J.D. Cowan, Kybernetik 13, 55 (1973)CrossRefMATHGoogle Scholar
  8. 8.
    H.G.E. Meijer, S. Coombes, EPJ Nonlin. Biomed. Phys. 2, 3 (2014)CrossRefGoogle Scholar
  9. 9.
    P.C. Bressloff, J. Phys. A 45 033001 (2012)CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    D.T.J. Liley, P.J. Cadusch, M.P. Dafilis, Network: Comput. Neural Syst. 13, 67 (2002)CrossRefMATHGoogle Scholar
  11. 11.
    I. Bojak, D.T.J. Liley, Phys. Rev E 71, 041902 (2005)CrossRefADSGoogle Scholar
  12. 12.
    K.R. Green, L. van Veen, J. Comp. Sci. 5, 507 (2014)CrossRefGoogle Scholar
  13. 13.
    I. Bojak, D.T.J. Liley, Neurocomputing 70, 2085 (2007)CrossRefGoogle Scholar
  14. 14.
    A. Stepanyants, L.M. Martinez, A.S. Ferecskó, Z.F. Kisvárday, C.F. Stevens, PNAS 106, 3555 (2009)CrossRefADSGoogle Scholar
  15. 15.
    F. Frascoli, L. van Veen, I. Bojak, D. Liley, Physica D 240, 949 (2011)CrossRefMathSciNetADSMATHGoogle Scholar
  16. 16.
    S. Coombes, Neuroimage 52, 731 (2010)CrossRefGoogle Scholar
  17. 17.
    E. Knobloch, M. Silber, Bifurcation and Symmetry, International Series of Numerical Mathematics, Vol. 104, edited by E.L. Allgower, K. Böhmer, M. Golubitsky (Birkhäuser, 1992), p. 241Google Scholar
  18. 18.
    S. Balay, J. Brown, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.M. Curfman, B.F. Smith, H. Zhang, http://www.mcs.anl.gov/petsc, PETSc Web page (2012)
  19. 19.
    Source code available from http://bitbucket.org/kegr/mfm
  20. 20.
    P. Ashwin, K. Böhmer, M. Zhen, Math. Comp. 64, 649 (1995)CrossRefMathSciNetADSMATHGoogle Scholar
  21. 21.
    L.E. Muller, A. Reynaud, F. Chavane, A. Destexhe, BMC Neuroscience 14(Suppl. 1), O8Google Scholar

Copyright information

© EDP Sciences and Springer 2014

Authors and Affiliations

  1. 1.University of Ontario Institute of TechnologyOshawaCanada
  2. 2.INRIA-Nancy Grand Est, team NEUROSYS, Villers-lès-NancyNancyFrance

Personalised recommendations