Japan Journal of Industrial and Applied Mathematics

, Volume 27, Issue 3, pp 347–373

Speeds of traveling waves in some integro-differential equations arising from neuronal networks

Original Paper Area 1
  • 108 Downloads

Abstract

We study traveling waves of some integro-differential equations arising from synaptically coupled neuronal networks. We investigate the influence of synaptic couplings and parameter values on the propagation of waves, and derive lower and upper bounds of traveling speeds. We also compare speeds of waves for various types of synaptic couplings.

Keywords

Integro-differential equations Traveling waves Speed index functions Synaptic couplings 

Mathematics Subject Classification (2000)

45G10 92B20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amari S.: Dynamics of pattern formation in lateral-inhibition type neural fields. Biolog. Cybern. 27, 77–87 (1977)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aronson D.G., Weinberger H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Atay F.M., Hutt A.: Stability and bifurcations in neural fields with finite propagation speed and general connectivity. SIAM J. Appl. Math. 65, 644–666 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bressloff P.C., Folias S.E.: Front bifurcations in an excitatory neural network. SIAM J. Appl. Math. 65, 131–151 (2004)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen X.: Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differ. Equ. 2, 125–160 (1997)MATHGoogle Scholar
  6. 6.
    Coombes S.: Waves, bumps, and patterns in neural field theories. Biol. Cybern. 93, 91–108 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coombes S., Owen M.R.: Evans functions for integral neural field equations with Heaviside firing rate function. SIAM J. Appl. Dyn. Syst. 3, 574–600 (2004)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Enculescu M.: A note on traveling fronts and pulses in a firing rate model of a neuronal network. Phys. D 196, 362–386 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ermentrout G.B., McLeod J.B.: Existence and uniqueness of travelling waves for a neural network. Proc. R. Soc. Edinburgh 123A, 461–478 (1993)MathSciNetGoogle Scholar
  10. 10.
    Evans J.W.: Nerve axon equations. V. The stable and the unstable impulse 24, 1169–1190 (1975)MATHGoogle Scholar
  11. 11.
    Folias S.E., Bressloff P.C.: Stimulus-locked traveling waves and breathers in an excitatory neural network. SIAM J. Appl. Math. 65, 2067–2092 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fife P.C., McLeod J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rational Mech. Anal. 65, 335–361 (1977)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Huang X., Troy W.C., Yang Q., Ma H., Laing C.R., Schiff S.J., Wu J.-Y.: Spiral waves in disinhibited mammalian neocortex. J. Neurosci. 24, 9897–9902 (2004)CrossRefGoogle Scholar
  14. 14.
    Hutt A.: Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay. Phys. Rev. E 70, 052902 (2004)CrossRefGoogle Scholar
  15. 15.
    Hutt A., Atay F.M.: Analysis of nonlocal neural fields for both general and gamma-distributed connectivities. Phys. D 203, 30–54 (2005)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Laing C.R.: Spiral waves in nonlocal equations. SIAM J. Appl. Dyn. Syst. 4, 588–606 (2005)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Li Y.-X.: Tango waves in a bidomain model of fertilization calcium waves. Phys. D 186, 27–49 (2003)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pinto, D.J., Ermentrout, G.B.: Spatially structured activity in synaptically coupled neuronal networks. I. Traveling fronts and pulses, II. Lateral inhibition and standing pulses, SIAM J. Appl. Math. 62, I. 206–225, II. 226–243 (2001)Google Scholar
  19. 19.
    Sandstede B.: Evans functions and nonlinear stability of travelling waves in neuronal network models. Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2693–2704 (2007)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Terman D.H., Ermentrout G.B., Yew A.C.: Propagating activity patterns in thalamic neuronal networks. SIAM J. Appl. Math. 61, 1578–1604 (2001)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Yanagida E.: Stability of fast travelling pulse solutions of the FitzHugh–Nagumo equations. J. Math. Biol. 22, 81–104 (1985)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Yanagida E.: Stability of travelling front solutions of the FitzHugh–Nagumo equations. Int. J. Math. Comput. Modell. 12, 289–301 (1989)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Zhang L.: On stability of traveling wave solutions in synaptically coupled neuronal networks. Differ. Int. Equ. 16, 513–536 (2003)MATHGoogle Scholar
  24. 24.
    Zhang L.: Traveling waves of a singularly perturbed system of integral-differential equations arising from neuronal networks. J. Dyn. Differ. Equ. 17, 489–522 (2005)MATHCrossRefGoogle Scholar
  25. 25.
    Zhang L.: Dynamics of neuronal waves. Math. Zeit. 255, 283–321 (2007)MATHCrossRefGoogle Scholar
  26. 26.
    Zhang L.: How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks? SIAM J. Appl. Dyn. Syst. 6, 597–644 (2007)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of MathematicsLehigh UniversityBethlehemUSA

Personalised recommendations