Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Collective stochastic coherence in recurrent neuronal networks

Abstract

Recurrent networks of dynamic elements frequently exhibit emergent collective oscillations, which can show substantial regularity even when the individual elements are considerably noisy. How noise-induced dynamics at the local level coexists with regular oscillations at the global level is still unclear. Here we show that a combination of stochastic recurrence-based initiation with deterministic refractoriness in an excitable network can reconcile these two features, leading to maximum collective coherence for an intermediate noise level. We report this behaviour in the slow oscillation regime exhibited by a cerebral cortex network under dynamical conditions resembling slow-wave sleep and anaesthesia. Computational analysis of a biologically realistic network model reveals that an intermediate level of background noise leads to quasi-regular dynamics. We verify this prediction experimentally in cortical slices subject to varying amounts of extracellular potassium, which modulates neuronal excitability and thus synaptic noise. The model also predicts that this effectively regular state should exhibit noise-induced memory of the spatial propagation profile of the collective oscillations, which is also verified experimentally. Taken together, these results allow us to construe the high regularity observed experimentally in the brain as an instance of collective stochastic coherence.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

Figure 1: UP/DOWN oscillations in a cortical network model.
Figure 2: Stochastic coherence in a cortical network model.
Figure 3: Experimental evidence of stochastic coherence in the cortical tissue.
Figure 4: Noise-induced spatial memory of UP-state initiation in the model.
Figure 5: Noise-induced spatial memory of UP-state initiation events in cortical slices.

Similar content being viewed by others

References

  1. Enright, J. T. Temporal precision in circadian systems: a reliable neuronal clock from unreliable components? Science 209, 1542–1545 (1980).

    ADS  Google Scholar 

  2. Garcia-Ojalvo, J., Elowitz, M. B. & Strogatz, S. H. Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proc. Natl Acad. Sci. USA 101, 10955–10960 (2004).

    ADS  MathSciNet  MATH  Google Scholar 

  3. Sanchez-Vives, M. V. & Mattia, M. Slow wave activity as the default mode of the cerebral cortex. Arch. Ital. Biol. 152, 147–155 (2014).

    Google Scholar 

  4. Steriade, M., Nuñez, A. & Amzica, F. A novel slow (<1 Hz) oscillation of neocortical neurons in vivo: depolarizing and hyperpolarizing components. J. Neurosci. 13, 3252–3265 (1993).

    Google Scholar 

  5. Stern, E. A., Kincaid, A. E. & Wilson, C. J. Spontaneous subthreshold membrane potential fluctuations and action potential variability of rat corticostriatal and striatal neurons in vivo. J. Neurophysiol. 77, 1697–1715 (1997).

    Google Scholar 

  6. Shu, Y., Hasenstaub, A. & McCormick, D. A. Turning on and off recurrent balanced cortical activity. Nature 423, 288–293 (2003).

    ADS  Google Scholar 

  7. Sanchez-Vives, M. V. & McCormick, D. A. Cellular and network mechanisms of rhythmic recurrent activity in neocortex. Nature Neurosci. 3, 1027–1034 (2000).

    Google Scholar 

  8. Compte, A. et al. Spontaneous high-frequency (10–80 Hz) oscillations during up states in the cerebral cortex in vitro. J. Neurosci. 28, 13828–13844 (2008).

    Google Scholar 

  9. Mattia, M. & Sanchez-Vives, M. V. Exploring the spectrum of dynamical regimes and timescales in spontaneous cortical activity. Cogn. Neurodynam. 6, 239–250 (2012).

    Google Scholar 

  10. Gluckman, B. J. et al. Stochastic resonance in a neuronal network from mammalian brain. Phys. Rev. Lett. 77, 4098–4101 (1996).

    ADS  Google Scholar 

  11. Deco, G., Martí, D., Ledberg, A., Reig, R. & Sanchez-Vives, M. V. Effective reduced diffusion-models: a data driven approach to the analysis of neuronal dynamics. PLoS Comput. Biol. 5, e1000587 (2009).

    ADS  MathSciNet  Google Scholar 

  12. Steriade, M., Timofeev, I. & Grenier, F. Natural waking and sleep states: a view from inside neocortical neurons. J. Neurophysiol. 85, 1969–1985 (2001).

    Google Scholar 

  13. Parga, N. & Abbott, L. F. Network model of spontaneous activity exhibiting synchronous transitions between up and down states. Front. Neurosci. 1, 57–66 (2007).

    Google Scholar 

  14. Bazhenov, M., Timofeev, I., Steriade, M. & Sejnowski, T. J. Model of thalamocortical slow-wave sleep oscillations and transitions to activated states. J. Neurosci. 22, 8691–8704 (2002).

    Google Scholar 

  15. Pikovsky, A. S. & Kurths, J. Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett. 78, 775–778 (1997).

    ADS  MathSciNet  MATH  Google Scholar 

  16. Lindner, B., Garcia-Ojalvo, J., Neiman, A. & Schimansky-Geier, L. Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004).

    ADS  Google Scholar 

  17. Compte, A., Sanchez-Vives, M. V., McCormick, D. A. & Wang, X.-J. Cellular and network mechanisms of slow oscillatory activity (<1 Hz) and wave propagations in a cortical network model. J. Neurophysiol. 89, 2707–2725 (2003).

    Google Scholar 

  18. Holcman, D. & Tsodyks, M. The emergence of up and down states in cortical networks. PLoS Comput. Biol. 2, e23 (2006).

    ADS  Google Scholar 

  19. Mejias, J. F., Kappen, H. J. & Torres, J. J. Irregular dynamics in up and down cortical states. PLoS ONE 5, e13651 (2010).

    ADS  Google Scholar 

  20. Amzica, F., Massimini, M. & Manfridi, A. Spatial buffering during slow and paroxysmal sleep oscillations in cortical networks of glial cells in vivo. J. Neurosci. 22, 1042–1053 (2002).

    Google Scholar 

  21. McCormick, D. A. Cholinergic and noradrenergic modulation of thalamocortical processing. Trends Neurosci. 12, 215–221 (1989).

    Google Scholar 

  22. Romcy-Pereira, R. N., Leite, J. P. & Garcia-Cairasco, N. Synaptic plasticity along the sleep-wake cycle: implications for epilepsy. Epilepsy Behav. 14, 47–53 (2009).

    Google Scholar 

  23. Han, S. K., Yim, T. G., Postnov, D. E. & Sosnovtseva, O. V. Interacting coherence resonance oscillators. Phys. Rev. Lett. 83, 1771–1774 (1999).

    ADS  Google Scholar 

  24. Balenzuela, P. & García-Ojalvo, J. Role of chemical synapses in coupled neurons with noise. Phys. Rev. E 72, 021901 (2005).

    ADS  Google Scholar 

  25. Bhattacharjee, A., von Hehn, C., Mei, X. & Kaczmarek, L. Localization of the Na-activated K channel slick in the rat central nervous system. J. Comp. Neurol. 484, 80–92 (2005).

    Google Scholar 

  26. Wang, X.-J., Liu, Y., Sanchez-Vives, M. V. & McCormick, D. A. Adaptation and temporal decorrelation by single neurons in the primary visual cortex. J. Neurophysiol. 89, 3279–3293 (2003).

    Google Scholar 

  27. Sanchez-Vives, M. V., Nowak, L. G. & McCormick, D. A. Cellular mechanisms of long-lasting adaptation in visual cortical neurons in vitro. J. Neurosci. 20, 4286–4299 (2000).

    Google Scholar 

  28. Sanchez-Vives, M. V. et al. Inhibitory modulation of cortical up states. J. Neurophysiol. 104, 1314–1324 (2010).

    Google Scholar 

  29. Mattia, M. & Del Giudice, P. Population dynamics of interacting spiking neurons. Phys. Rev. E 66, 051917 (2002).

    ADS  MathSciNet  MATH  Google Scholar 

  30. Satoh, T., Watabe, K. & Eguchi, K. Enhancement during REM sleep of extracellular potassium ion activity in the reticular formation. Brain Res. 174, 180–183 (1979).

    Google Scholar 

  31. Cossart, R., Aronov, D. & Yuste, R. Attractor dynamics of network up states in the neocortex. Nature 423, 283–288 (2003).

    ADS  Google Scholar 

  32. Yamaguchi, T. Cerebral extracellular potassium concentration change and cerebral impedance change in short-term ischemia in gerbil. Bull. Tokyo Med. Dental Univ. 33, 1–8 (1986).

    Google Scholar 

  33. Bazhenov, M., Timofeev, I., Steriade, M. & Sejnowski, T. J. Potassium model for slow (2–3 hz) in vivo neocortical paroxysmal oscillations. J. Neurophysiol. 92, 1116–1132 (2004).

    Google Scholar 

  34. Chialvo, D., Cecchi, G. & Magnasco, M. Noise-induced memory in extended excitable systems. Phys. Rev. E 61, 5654–5657 (2000).

    ADS  Google Scholar 

  35. Baayen, R. H., Davidson, D. J. & Bates, D. M. Mixed-effects modeling with crossed random effects for subjects and items. J. Mem. Lang. 59, 390–412 (2008).

    Google Scholar 

  36. Sagués, F., Sancho, J. & García-Ojalvo, J. Spatiotemporal order out of noise. Rev. Mod. Phys. 79, 829–882 (2007).

    ADS  Google Scholar 

  37. Gang, H., Ditzinger, T., Ning, C. & Haken, H. Stochastic resonance without external periodic force. Phys. Rev. Lett. 71, 807–810 (1993).

    ADS  Google Scholar 

  38. Bulsara, A., Jacobs, E., Zhou, T., Moss, F. & Kiss, L. Stochastic resonance in a single neuron model. J. Theor. Biol. 152, 531–555 (1991).

    Google Scholar 

  39. Longtin, A. Stochastic resonance in neuron models. J. Stat. Phys. 70, 309–327 (1993).

    ADS  MATH  Google Scholar 

  40. Douglass, J. K., Wilkens, L., Pantazelou, E. & Moss, F. Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature 365, 337–340 (1993).

    ADS  Google Scholar 

  41. Levin, J. E. & Miller, J. P. Broadband neural encoding in the cricket cereal sensory system enhanced by stochastic resonance. Nature 380, 165–168 (1996).

    ADS  Google Scholar 

  42. Collins, J. J., Imhoff, T. T. & Grigg, P. Noise-enhanced information transmission in rat SA1 cutaneous mechanoreceptors via aperiodic stochastic resonance. J. Neurophysiol. 76, 642–645 (1996).

    Google Scholar 

  43. Stacey, W. C. & Durand, D. M. Stochastic resonance improves signal detection in hippocampal ca1 neurons. J. Neurophysiol. 83, 1394–1402 (2000).

    Google Scholar 

  44. McDonnell, M. D. & Ward, L. M. The benefits of noise in neural systems: bridging theory and experiment. Nature Rev. Neurosci. 12, 415–426 (2011).

    Google Scholar 

  45. Gu, H., Yang, M., Li, L., Liu, Z. & Ren, W. Experimental observation of the stochastic bursting caused by coherence resonance in a neural pacemaker. Neuro Rep. 13, 1657–1660 (2002).

    Google Scholar 

  46. Manjarrez, E. et al. Internal stochastic resonance in the coherence between spinal and cortical neuronal ensembles in the cat. Neurosci. Lett. 326, 93–96 (2002).

    Google Scholar 

  47. Fries, P. Rhythms for cognition: communication through coherence. Neuron 88, 220–235 (2015).

    Google Scholar 

  48. Barardi, A., Sancristóbal, B. & Garcia-Ojalvo, J. Phase-coherence transitions and communication in the gamma range between delay-coupled neuronal populations. PLoS Comput. Biol. 10, e1003723 (2014).

    ADS  Google Scholar 

  49. Horikawa, Y. Coherence resonance with multiple peaks in a coupled FitzHugh-Nagumo model. Phys. Rev. E 64, 031905 (2001).

    ADS  Google Scholar 

  50. Mountcastle, V. B. Perceptual Neuroscience: The Cerebral Cortex (Harvard Univ. Press, 1998).

    Google Scholar 

  51. Mazzoni, A., Panzeri, S., Logothetis, N. K. & Brunel, N. Encoding of naturalistic stimuli by local field potential spectra in networks of excitatory and inhibitory neurons. PLoS Comput. Biol. 4, e1000239 (2008).

    ADS  MathSciNet  Google Scholar 

  52. Berens, P., Logothetis, N. K. & Tolias, A. S. Local field potentials, BOLD and spiking activity–relationships and physiological mechanisms. Available at http://precedings.nature.com/documents/5216/version/1 (2010).

  53. Buzsáki, G., Anastassiou, C. A. & Koch, C. The origin of extracellular fields and currents - EEG, ECoG, LFP and spikes. Nature Rev. Neurosci. 13, 407–420 (2012).

    Google Scholar 

  54. Reig, R., Mattia, M., Compte, A., Belmonte, C. & Sanchez-Vives, M. V. Temperature modulation of slow and fast cortical rhythms. J. Neurophysiol. 103, 1253–1261 (2010).

    Google Scholar 

Download references

Acknowledgements

We thank A. Compte for useful comments, and our colleagues from CSIC-CNM in Barcelona (X. Villa, R. Villa, G. Gabriel) for providing the recording arrays used in Fig. 5. This work was supported by the Ministerio de Economia y Competividad and FEDER (Spain, projects FIS2012-37655-C02-01, to J.G.-O., and BFU2014-52467-R, to M.V.S.-V.) and EU project CORTICONIC (contract number 600806, to M.V.S.-V.). B.R. was supported by the FPI programme associated to BFU2011-27094 (Spain, Ministerio de Economia y Competividad). J.G.-O. acknowledges support from the ICREA Academia programme and from the Generalitat de Catalunya (project 2014SGR0974).

Author information

Authors and Affiliations

Authors

Contributions

B.S., M.V.S.-V. and J.G.-O. conceived the research. B.S. implemented the mathematical model. B.S. and P.B. analysed the data. B.R. performed the experiments. M.V.S.-V. and J.G.-O. supervised the work. B.S., M.V.S.-V. and J.G.-O. wrote the manuscript. All authors revised and approved the text.

Corresponding authors

Correspondence to Maria V. Sanchez-Vives or Jordi Garcia-Ojalvo.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 727 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sancristóbal, B., Rebollo, B., Boada, P. et al. Collective stochastic coherence in recurrent neuronal networks. Nature Phys 12, 881–887 (2016). https://doi.org/10.1038/nphys3739

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys3739

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing