Waves and Oscillations in Networks of Coupled Neurons

  • B Ermentrout
Part of the Lecture Notes in Physics book series (LNP, volume 671)


Neural systems are characterized by the interactions of thousands of individual cells called neurons. Individual neurons vary in their properties with some of them spontaneously active and others active only when given a sufficient perturbation. In this note, I will describe work that has been done on the mathematical analysis of waves and synchronous oscillations in spatially distributed networks of neurons. These classes of behavior are observed both in vivo (that is, in the living brain) and in vitro (isolated networks, such as slices of brain tissue.) We focus on these simple behaviors rather than on the possible computations that networks of neurons can do (such as filtering sensory inputs and producing precise motor output) mainly because they are mathematically tractable. The chapter is organized as follows. First, I will introduce the kinds of equations that are of interest and from these abstract some simplified models. I will consider several different types of connectivity – from “all-to-all” to spatially organized. Typically (although not in every case), each individual neuron is represented by a scalar equation for its dynamics. These individuals can be coupled together directly or indirectly and in spatially discrete or continuous arrays.


Hopf Bifurcation Coupling Function Spiral Wave Couple Neuron Gastric Mill 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Ayers, A.I. Selverston, Synaptic perturbation and entrainment of gastric mill rhythm of the spiny lobster, J. Neurophysiol. 51 (1984), no. 1, 113–25.PubMedGoogle Scholar
  2. 2.
    E.N. Best, Null space in the Hodgkin-Huxley Equations. A critical test, Biophys. J. 27 (1979), no.1, 87–104.PubMedGoogle Scholar
  3. 3.
    P.C. Bressloff, Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons, J. Math. Biol. 40 (2000), no. 2, 169–198.CrossRefPubMedGoogle Scholar
  4. 4.
    E. Brown, J. Moehlis, and P. Holmes. On the phase reduction and response dynamics of neural oscillator populations, Neural Computation 16 (2004), 673–715.CrossRefPubMedGoogle Scholar
  5. 5.
    C.C. Canavier, R.J. Butera, R.O. Dror, D.A. Baxter, J.W. Clark, J.H. Byrne, Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation, Biol. Cybern. 77 (1997), no. 6, 367–80.CrossRefPubMedGoogle Scholar
  6. 6.
    X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997), no. 1, 125–160.Google Scholar
  7. 7.
    B. Ermentrout, The analysis of synaptically generated traveling waves, J. Computat. Neurosci. 5 (1998), 191–208.CrossRefGoogle Scholar
  8. 8.
    B. Ermentrout, Neural networks as spatio-temporal pattern-forming systems, Reports on Progress in Physics 61 (1998), 353–430.CrossRefGoogle Scholar
  9. 9.
    B. Ermentrout, N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math. 46 (1986), no. 2, 233-253.CrossRefGoogle Scholar
  10. 10.
    P. Goel, B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D 163 (2002), no. 3-4, 191–216.CrossRefGoogle Scholar
  11. 11.
    D. Golomb, B. Ermentrout, Continuous and lurching waves in a neuronal network with delay and spatially decaying connectivity, Proc. Nat. Acad. Sci. U.S.A. 99:13480-13485 (1999).CrossRefGoogle Scholar
  12. 12.
    D. Golomb, B. Ermentrout, Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity, Network Comp. Neural 11 (2000), no. 3, 221–246.CrossRefGoogle Scholar
  13. 13.
    F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks, Springer-Verlag, New York, 1997.Google Scholar
  14. 14.
    E.M. Izhikevich, Class 1 neural excitability, conventional synapses, weakly connected networks, and mathematical foundations of pulse-coupled models, IEEE Trans. on Neural Networks 10 (1999), 499–507.CrossRefGoogle Scholar
  15. 15.
    E.M. Izhikevich, Neural excitability, spiking, and bursting, Int. J. of Bifurcation and Chaos 10 (2000), 1171–1266.CrossRefGoogle Scholar
  16. 16.
    S.A. Oprisan, A.A. Prinz and C.C. Canavier, Phase resetting and phase locking in hybrid circuits of one model and one biological neuron, Biophys J. 87 (2004), no. 4, 2283–98.CrossRefPubMedGoogle Scholar
  17. 17.
    R. Osan, J. Rubin, and B. Ermentrout, Regular traveling waves in a one-dimensional network of theta neurons, SIAM J. Appl. Math. 62 (2002), 1197–1221.CrossRefGoogle Scholar
  18. 18.
    J.E. Paullet, B. Ermentrout, Stable rotating waves in two-dimensional discrete active media, SIAM J. Appl. Math. 54 (1994), 1720–1744.CrossRefGoogle Scholar
  19. 19.
    A.D. Reyes, E.E. Fetz, Effects of transient depolarizing potentials on the firing rate of cat neocortical neurons, J. Neurophysiol. 69 (1993) 1673 1683.PubMedGoogle Scholar
  20. 20.
    J. Rubin, A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium, Disc. Cont. Dyn. Sys. A 4 (2004), 925–940.Google Scholar
  21. 21.
    R. Stoop, K. Schindler and L.A. Bunimovich, Neocortical networks of pyramidal neurons: from local locking and chaos to macroscopic chaos and synchronization, Nonlinearity 13 (2000), 1515–1529.CrossRefGoogle Scholar
  22. 22.
    D.H. Terman, B. Ermentrout, A.C. Yew, Propagating activity patterns in thalamic neuronal networks, SIAM J. Appl. Math. 61 (2001), no. 5, 1578–1604.CrossRefGoogle Scholar
  23. 23.
    A.T. Winfree, The Geometry of Biological Time, Springer, New York, 1980.Google Scholar
  24. 24.
    D.L. Ypey, W.P. Van Meerwijk, G. de Bruin, Suppression of pacemaker activity by rapid repetitive phase delay, Biol. Cybern. 45 (1982), no. 3, 187–94.CrossRefPubMedGoogle Scholar

Authors and Affiliations

  • B Ermentrout
    • 1
  1. 1.Department of MathematicsUniversity of PittsburghPittsburgh

Personalised recommendations