Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Area 1
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amari S.: Dynamics of pattern formation in lateral-inhibition type neural fields. Biolog. Cybern. 27, 77–87 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aronson D.G., Weinberger H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bressloff P.C., Folias S.E.: Front bifurcations in an excitatory neural network. SIAM J. Appl. Math. 65, 131–151 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen X.: Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differ. Equ. 2, 125–160 (1997)

    MATH  Google Scholar 

  6. Coombes S.: Waves, bumps, and patterns in neural field theories. Biol. Cybern. 93, 91–108 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  10. Evans J.W.: Nerve axon equations. V. The stable and the unstable impulse 24, 1169–1190 (1975)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  14. Hutt A.: Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay. Phys. Rev. E 70, 052902 (2004)

    Article  Google Scholar 

  15. Hutt A., Atay F.M.: Analysis of nonlocal neural fields for both general and gamma-distributed connectivities. Phys. D 203, 30–54 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Laing C.R.: Spiral waves in nonlocal equations. SIAM J. Appl. Dyn. Syst. 4, 588–606 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Li Y.-X.: Tango waves in a bidomain model of fertilization calcium waves. Phys. D 186, 27–49 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yanagida E.: Stability of fast travelling pulse solutions of the FitzHugh–Nagumo equations. J. Math. Biol. 22, 81–104 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  22. Yanagida E.: Stability of travelling front solutions of the FitzHugh–Nagumo equations. Int. J. Math. Comput. Modell. 12, 289–301 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zhang L.: On stability of traveling wave solutions in synaptically coupled neuronal networks. Differ. Int. Equ. 16, 513–536 (2003)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  25. Zhang L.: Dynamics of neuronal waves. Math. Zeit. 255, 283–321 (2007)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo, 152-8551, Japan

    Eiji Yanagida

  2. Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, PA, 18015, USA

    Linghai Zhang

Authors
  1. Eiji Yanagida
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Linghai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eiji Yanagida.

About this article

Cite this article

Yanagida, E., Zhang, L. Speeds of traveling waves in some integro-differential equations arising from neuronal networks. Japan J. Indust. Appl. Math. 27, 347–373 (2010). https://doi.org/10.1007/s13160-010-0021-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-010-0021-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation