Journal of Computational Neuroscience

, Volume 27, Issue 1, pp 135–160 | Cite as

Spike-time reliability of layered neural oscillator networks

  • Kevin K. Lin
  • Eric Shea-BrownEmail author
  • Lai-Sang Young


We study the reliability of layered networks of coupled “type I” neural oscillators in response to fluctuating input signals. Reliability means that a signal elicits essentially identical responses upon repeated presentations, regardless of the network’s initial condition. We study reliability on two distinct scales: neuronal reliability, which concerns the repeatability of spike times of individual neurons embedded within a network, and pooled-response reliability, which concerns the repeatability of total synaptic outputs from a subpopulation of the neurons in a network. We find that neuronal reliability depends strongly both on the overall architecture of a network, such as whether it is arranged into one or two layers, and on the strengths of the synaptic connections. Specifically, for the type of single-neuron dynamics and coupling considered, single-layer networks are found to be very reliable, while two-layer networks lose their reliability with the introduction of even a small amount of feedback. As expected, pooled responses for large enough populations become more reliable, even when individual neurons are not. We also study the effects of noise on reliability, and find that noise that affects all neurons similarly has much greater impact on reliability than noise that affects each neuron differently. Qualitative explanations are proposed for the phenomena observed.


Spike-time reliability Spiking neural network Neural oscillator Theta neuron Chaos Stochastic dynamics Random dynamical systems 



We thank David Cai, Anne-Marie Oswald, Alex Reyes, and John Rinzel for their helpful discussions of this material. We acknowledge a Career Award at the Scientific Interface from the Burroughs-Wellcome Fund (E.S.-B.), and a grant from the NSF (L.-S.Y.).


  1. Arnold, L. (2003). Random dynamical systems. New York: Springer.Google Scholar
  2. Averbeck, B., Latham, P. E., & Pouget, A. (2006). Neural correlations, population coding and computation. Nature Reviews Neuroscience, 7(5), 358–366, May.PubMedCrossRefGoogle Scholar
  3. Aviel, Y., Mehring, C., Abeles, M., & Horn, D. (2003). On embedding synfire chains in a balanced network. Neural Computation, 15, 1321–1340.PubMedCrossRefGoogle Scholar
  4. Bair, W., Zohary, E., & Newsome, W. T. (2001). Correlated firing in macaque visual area MT: Time scales and relationship to behavior. Journal of Neuroscience, 21(5), 1676–1697.PubMedGoogle Scholar
  5. Banerjee, A. (2006). On the sensitive dependence on initial conditions of the dynamics of networks of spiking neurons. Journal of Computational Neuroscience, 20, 321–348.PubMedCrossRefGoogle Scholar
  6. Banerjee, A., Seriès, P., & Pouget, A. (2008). Dynamical constraints on using precise spike timing to compute in recurrent cortical networks. Neural Computation, 20, 974–993.PubMedCrossRefGoogle Scholar
  7. Baxendale, P. H. (1992). Stability and equilibrium properties of stochastic flows of diffeomorphisms. In Progr. Probab. 27. Boston: Birkhauser.Google Scholar
  8. Bazhenov, M., Rulkov, N., Fellous, J., & Timofeev, I. (2005). Role of network dynamics in shaping spike timing reliability. Physical Review E, 72, 041903.CrossRefGoogle Scholar
  9. Berry, M., Warland, D., & Meister, M. (1997). The structure and precision of retinal spike trains. PNAS, 94, 5411–5416.PubMedCrossRefGoogle Scholar
  10. Bertschlinger, N., & Natschlager, T. (2004). Real-time computation at the edge of chaos in recurrent neural networks. Neural Computation, 16, 1413–1436.CrossRefGoogle Scholar
  11. Borgers, C., Epstein, C., & Kopell, N. (2005). Background gamma rhythmicity and attention in cortical local circuits: A computational study. Journal of Neuroscience, 102, 7002–7007.Google Scholar
  12. Bruno, R. M., & Sakmann, B. (2006). Cortex is driven by weak but synchronously active thalamocortical synapses. Nature, 312, 1622–1627.Google Scholar
  13. Bryant, H. L., & Segundo, J. P. (1976). Spike initiation by transmembrane current: A white-noise analysis. Journal of Physiology, 260, 279–314.PubMedGoogle Scholar
  14. de Reuter van Steveninck, R., Lewen, R., Strong, S., Koberle, R., & Bialek, W. (1997). Reproducibility and variability in neuronal spike trains. Science, 275, 1805–1808.CrossRefGoogle Scholar
  15. Doiron, B., Chacron, M. J., Maler, L., Longtin, A., & Bastian, J. (2003). Inhibitory feedback required for network burst responses to communication but not to prey stimuli. Nature, 421, 539–543.PubMedCrossRefGoogle Scholar
  16. Douglas, E., & Martin, K. (2004). Neuronal circuits of the neocortex. Annual Review of Neuroscience, 27, 419–451.PubMedCrossRefGoogle Scholar
  17. Eckmann, J.-P., & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57, 617–656.CrossRefGoogle Scholar
  18. Ermentrout, G. B. (1996). Type I membranes, phase resetting curves, and synchrony. Neural Computation, 8, 979–1001.PubMedCrossRefGoogle Scholar
  19. Ermentrout, G. B., & Kopell, N. (1984). Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM Journal on Mathematical Analysis, 15, 215–237.CrossRefGoogle Scholar
  20. Faisal, A. A., Selen, L. P. J., & Wolpert, D. M. (2008). Noise in the nervous system. Nature Reviews Neuroscience, 9, 292–303.PubMedCrossRefGoogle Scholar
  21. Hodgkin, A. (1948). The local electric changes associated with repetitive action in a non-medulated axon. Journal of Physiology, 117, 500–544.Google Scholar
  22. Hunter, J., Milton, J., Thomas, P., & Cowan, J. (1998). Resonance effect for neural spike time reliability. Journal of Neurophysiology, 80, 1427–1438.PubMedGoogle Scholar
  23. Johnston, D., & Wu, S. (1997). Foundations of cellular neurophysiology. Cambridge: MIT.Google Scholar
  24. Kandel, E., Schwartz, J., & Jessell, T. (1991). Principles of neural science, 4th edn. New York: McGraw-Hill.Google Scholar
  25. Kara, P., Reinagel, P., & Reid, R. C. (2000). Low response variability in simultaneously recorded retinal, thalamic, and cortical neurons. Neuron, 27, 636–646.CrossRefGoogle Scholar
  26. Kifer, Y. (1986). Ergodic theory of random transformations. Boston: Birkhauser.Google Scholar
  27. Koch, C. (1999). Biophysics of computation: Information processing in single neurons. Oxford: Oxford University Press.Google Scholar
  28. Kunita, H. (1990). Stochastic flows and stochastic differential equations. Cambridge studies in advanced mathematics (Vol. 24). Cambridge: Cambridge University Press.Google Scholar
  29. Lampl, I., Reichova, I., & Ferster, D. S. (1999). Synchronous membrane potential fluctuations in neurons of the cat visual cortex. Neuron, 22, 361–374.PubMedCrossRefGoogle Scholar
  30. Latham, P. E., Richmond, B. J., Nelson, P. G., & Nirenberg, S. (2000). Intrinsic dynamics in neuronal networks. I. theory. Journal of Neurophysiology, 83, 808–827.PubMedGoogle Scholar
  31. Le Jan, Y. (1987). Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Annales de l’Institut Henri Poincaré Probabilités et Statistiques, 23(1), 111–120.Google Scholar
  32. Ledrappier, F., & Young, L.-S. (1988). Entropy formula for random transformations. Probability Theory and Related Fields, 80, 217–240.CrossRefGoogle Scholar
  33. Lin, K. K., Shea-Brown, E., & Young, L.-S. (2009a). Reliability of coupled oscillators. Journal of Nonlinear Science (in press).Google Scholar
  34. Lin, K. K., Shea-Brown, E., & Young, L.-S. (2009b). Reliability of layered neural oscillator networks. Communications in Mathematical Sciences (in press).Google Scholar
  35. Lu, T., Liang, L., & Wang, X. (2001). Temporal and rate representations of time-varying signals in the auditory cortex of awake primates. Nature Neuroscience, 4, 1131–1138.PubMedCrossRefGoogle Scholar
  36. Maei, H. R., & Latham, P. E. (2005). Can randomly connected networks exhibit long memories? Preprint, Gatsby Computational Neuroscience Unit.Google Scholar
  37. Mainen, Z., & Sejnowski, T. (1995). Reliability of spike timing in neocortical neurons. Science, 268, 1503–1506.PubMedCrossRefGoogle Scholar
  38. Mazurek, M., & Shadlen, M. (2002). Limits to the temporal fidelity of cortical spike rate signals. Nature Neuroscience, 5, 463–471.PubMedGoogle Scholar
  39. Murphy, G., & Rieke, F. (2007). Network variability limits stimulus-evoked spike timing precision in retinal ganglion cells. Neuron, 52, 511–524.CrossRefGoogle Scholar
  40. Pakdaman, K., & Mestivier, D. (2001). External noise synchronizes forced oscillators. Physical Review E, 64, 030901–030904.CrossRefGoogle Scholar
  41. Perkel, D., & Bullock, T. (1968). Neural coding. Neurosciences Research Program Bulletin, 6, 221–344.Google Scholar
  42. Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization: A universal concept in nonlinear sciences. Cambridge: Cambridge University Press.Google Scholar
  43. Reyes, A. (2003). Synchrony-dependent propagation of firing rate in iteratively constructed networks in vitro. Nature Neuroscience, 6, 593–599.PubMedCrossRefGoogle Scholar
  44. Rieke, F., Warland, D., de Ruyter van Steveninck, R., & Bialek, W. (1996). Spikes: Exploring the neural code. Cambridge: MIT.Google Scholar
  45. Rinzel, J., & Ermentrout, G. B. (1998). Analysis of neural excitability and oscillations. In C. Koch, & I. Segev (Eds.), Methods in neuronal modeling (pp. 251–291). Cambridge: MIT.Google Scholar
  46. Ritt, J. (2003). Evaluation of entrainment of a nonlinear neural oscillator to white noise. Physical Review E, 68, 041915–041921.CrossRefGoogle Scholar
  47. Seriès, P., Latham, P. E., & Pouget, A. (2004). Tuning curve sharpening for orientation selectivity: Coding efficiency and the impact of correlations. Nature Neuroscience, 7, 1129–1135.PubMedCrossRefGoogle Scholar
  48. Shadlen, M. N., & Newsome, W. T. (1998). The variable discharge of cortical neurons: Implications for connectivity, computation, and information coding. Journal of Neuroscience, 18, 3870–3896.PubMedGoogle Scholar
  49. Shepard, G. (2004). The synaptic organization of the brain. Oxford: Oxford University Press.CrossRefGoogle Scholar
  50. Teramae, J., & Fukai, T. (2007). Reliability of temporal coding on pulse-coupled networks of oscillators. arXiv:0708.0862v1 [nlin.AO].
  51. Teramae, J., & Tanaka, D. (2004). Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators. Physical Review Letters, 93, 204103–204106.PubMedCrossRefGoogle Scholar
  52. Terman, D., Rubin, J., Yew, A., & Wilson, C. J. (2002). Activity patterns in a model for the subthalamopallidal network of the basal ganglia. Journal of Neuroscience, 22, 2963–2976.PubMedGoogle Scholar
  53. van Vreeswijk, C., & Sompolinsky, H. (1996). Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science, 274, 1724–1726.PubMedCrossRefGoogle Scholar
  54. van Vreeswijk, C., & Sompolinsky, H. (1998). Chaotic balanced state in a model of cortical circuits. Neural Computation, 10, 1321–1371.PubMedCrossRefGoogle Scholar
  55. Vogels, T., & Abbott, L. (2005). Signal propagation and logic gating in networks of integrate-and-fire neurons. Journal of Neuroscience, 25, 10786–10795.PubMedCrossRefGoogle Scholar
  56. Winfree, A. (2001). The geometry of biological time. New York: Springer.Google Scholar
  57. Zhou, C., & Kurths, J. (2003). Noise-induced synchronization and coherence resonance of a hodgkin-huxley model of thermally sensitive neurons. Chaos, 13, 401–409.PubMedCrossRefGoogle Scholar
  58. Zohary, E., Shadlen, M. N., & Newsome, W. T. (1994). Correlated neuronal discharge rate and its implication for psychophysical performance. Nature, 370, 140–143.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Kevin K. Lin
    • 1
  • Eric Shea-Brown
    • 2
    Email author
  • Lai-Sang Young
    • 3
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations