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.

Slow dynamics and high variability in balanced cortical networks with clustered connections

Nature Neuroscience volume 15, pages 1498–1505 (2012)Cite this article

Subjects

Abstract

Anatomical studies demonstrate that excitatory connections in cortex are not uniformly distributed across a network but instead exhibit clustering into groups of highly connected neurons. The implications of clustering for cortical activity are unclear. We studied the effect of clustered excitatory connections on the dynamics of neuronal networks that exhibited high spike time variability owing to a balance between excitation and inhibition. Even modest clustering substantially changed the behavior of these networks, introducing slow dynamics during which clusters of neurons transiently increased or decreased their firing rate. Consequently, neurons exhibited both fast spiking variability and slow firing rate fluctuations. A simplified model shows how stimuli bias networks toward particular activity states, thereby reducing firing rate variability as observed experimentally in many cortical areas. Our model thus relates cortical architecture to the reported variability in spontaneous and evoked spiking activity.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

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

Figure 1: Comparison between uniform and clustered network connectivity and dynamics.
Figure 2: Marginal and pairwise spiking statistics for neurons in clustered and uniform networks.
Figure 3: Heterogeneous clustered network.
Figure 4: Effect of increased clustered connection probability.
Figure 5: Emergence of bistability in simplified model.
Figure 6: Effects of stimulation.
Figure 7: Effect of stimulation on spiking variability and dynamics.
Figure 8: Effect of stimulation on inhibitory neuron spiking variability.

References

  1. Britten, K.H., Shadlen, M.N., Newsome, W.T. & Movshon, J.A. Responses of neurons in macaque MT to stochastic motion signals. Vis. Neurosci. 10, 1157–1169 (1993).

    Article  CAS  PubMed  Google Scholar 

  2. London, M., Roth, A., Beeren, L., Hausser, M. & Latham, P.E. Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex. Nature 466, 123–127 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  3. Shadlen, M.N. & Newsome, W.T. Noise, neural codes and cortical organization. Curr. Opin. Neurobiol. 4, 569–579 (1994).

    Article  CAS  PubMed  Google Scholar 

  4. van Vreeswijk, C. & Sompolinsky, H. Chaotic balanced state in a model of cortical circuits. Neural Comput. 10, 1321–1371 (1998).

    Article  CAS  PubMed  Google Scholar 

  5. Vogels, T.P. & Abbott, L.F. Signal propagation and logic gating in networks of integrate-and-fire neurons. J. Neurosci. 25, 10786–10795 (2005).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  7. Destexhe, A., Rudolph, M. & Pare, D. The high-conductance state of neocortical neurons in vivo. Nat. Rev. Neurosci. 4, 739–751 (2003).

    Article  CAS  PubMed  Google Scholar 

  8. Kohn, A. & Smith, M.A. Stimulus dependence of neuronal correlation in primary visual cortex of the macaque. J. Neurosci. 25, 3661–3673 (2005).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Churchland, M.M. et al. Stimulus onset quenches neural variability: a widespread cortical phenomenon. Nat. Neurosci. 13, 369–378 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  10. Churchland, A.K. et al. Variance as a signature of neural computations during decision making. Neuron 69, 818–831 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  11. Arieli, A., Sterkin, A., Grinvald, A. & Aertsen, A. Dynamics of ongoing activity: Explanation of the large variability in evoked cortical responses. Science 273, 1868–1871 (1996).

    Article  CAS  PubMed  Google Scholar 

  12. Tsodyks, M., Kenet, T., Grinvald, A. & Arieli, A. Linking spontaneous activity of single cortical neurons and the underlying functional architecture. Science 286, 1943–1946 (1999).

    Article  CAS  PubMed  Google Scholar 

  13. Churchland, M.M. et al. Neural variability in premotor cortex provides a signature of motor preparation. J. Neurosci. 26, 3697–3712 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  14. Luczak, A., Bartho, P. & Harris, K.D. Spontaneous events outline the realm of possible sensory responses in neocortical populations. Neuron 62, 413–425 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  15. Renart, A. et al. The asynchronous state in cortical circuits. Science 327, 587–590 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Song, S., Sjöström, P.J., Reigl, M., Nelson, S. & Chklovskii, D.B. Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS Biol. 3, e68 (2005).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  17. Perin, R., Berger, T.K. & Markram, H. A synaptic organizing principle for cortical neuronal groups. Proc. Natl. Acad. Sci. USA 108, 5419–5424 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  18. Yoshimura, Y., Dantzker, J.L.M. & Callaway, E.M. Excitatory cortical neurons form fine-scale functional networks. Nature 433, 868–873 (2005).

    Article  CAS  PubMed  Google Scholar 

  19. Ko, H. et al. Functional specificity of local synaptic connections in neocortical networks. Nature 473, 87–91 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  20. Yassin, L. et al. An embedded subnetwork of highly active neurons in the neocortex. Neuron 68, 1043–1050 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Amit, D.J. & Brunel, N. Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cereb. Cortex 7, 237–252 (1997).

    Article  CAS  PubMed  Google Scholar 

  22. Wang, X.-J. Probabilistic decision making by slow reverberation in cortical circuits. Neuron 36, 955–968 (2002).

    Article  CAS  PubMed  Google Scholar 

  23. Renart, A., Moreno-Bote, R., Wang, X.-J. & Parga, N. Mean-driven and fluctuation-driven persistent activity in recurrent networks. Neural Comput. 19, 1–46 (2007).

    Article  PubMed  Google Scholar 

  24. Roudi, Y. & Latham, P.E. A balanced memory network. PLoS Comput. Biol. 3, e141 (2007).

    Article  PubMed Central  CAS  Google Scholar 

  25. Deco, G. & Hugues, E. Neural network mechanisms underlying stimulus driven variability reduction. PLoS Comput. Biol. 8, e1002395 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  26. Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002).

    Article  Google Scholar 

  27. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.-U. Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006).

    Article  Google Scholar 

  28. Oswald, A.-M.M., Doiron, B., Rinzel, J. & Reyes, A.D. Spatial profile and differential recruitment of GABAB modulate oscillatory activity in auditory cortex. J. Neurosci. 29, 10321–10334 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Fino, E. & Yuste, R. Dense inhibitory connectivity in neocortex. Neuron 69, 1188–1203 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  30. Hromádka, T., Deweese, M.R. & Zador, A.M. Sparse representation of sounds in the unanesthetized auditory cortex. PLoS Biol. 6, e16 (2008).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  31. Tolhurst, D.J., Movshon, J.A. & Dean, A.F. The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vision Res. 23, 775–785 (1983).

    Article  CAS  PubMed  Google Scholar 

  32. Cohen, M.R. & Kohn, A. Measuring and interpreting neuronal correlations. Nat. Neurosci. 14, 811–819 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  33. Brunel, N. & Wang, X.-J. What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation-inhibition balance. J. Neurophysiol. 90, 415–430 (2003).

    Article  PubMed  Google Scholar 

  34. Teich, M.C., Heneghan, C., Lowen, S.B., Ozaki, T. & Kaplan, E. Fractal character of the neural spike train in the visual system of the cat. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 14, 529–546 (1997).

    Article  CAS  PubMed  Google Scholar 

  35. Sompolinsky, H. & van Vreeswijk, C. Irregular activity in large networks of neurons. in Les Houches Lectures LXXX: Methods and Models in Neurophysics (eds. Chow, C.C., Gutkin, B., Hansel, D., Meunier, C. & Dalibard, J.) 341–402 (Elsevier, London, 2005).

  36. Hänggi, P., Talkner, P. & Borkovec, M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251–341 (1990).

    Article  Google Scholar 

  37. Rajan, K., Abbott, L.F. & Sompolinsky, H. Stimulus-dependent suppression of chaos in recurrent neural networks. Phys. Rev. E 82, 011903 (2010).

    Article  CAS  Google Scholar 

  38. Mitchell, J.F., Sundberg, K.A. & Reynolds, J.H. Differential attention-dependent response modulation across cell classes in macaque visual area V4. Neuron 55, 131–141 (2007).

    Article  CAS  PubMed  Google Scholar 

  39. Yoshimura, Y. & Callaway, E.M. Fine-scale specificity of cortical networks depends on inhibitory cell type and connectivity. Nat. Neurosci. 8, 1552–1559 (2005).

    Article  CAS  PubMed  Google Scholar 

  40. Hofer, S.B. et al. Differential connectivity and response dynamics of excitatory and inhibitory neurons in visual cortex. Nat. Neurosci. 14, 1045–1052 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  42. Izhikevich, E.M., Gally, J.A. & Edelman, G.M. Spike-timing dynamics of neuronal groups. Cereb. Cortex 14, 933–944 (2004).

    Article  PubMed  Google Scholar 

  43. Abeles, M. et al. Cortical activity flips among quasi-stationary states. Proc. Natl. Acad. Sci. USA 92, 8616–8620 (1995).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  44. Ecker, A.S. et al. Decorrelated neuronal firing in cortical microcircuits. Science 327, 584–587 (2010).

    Article  CAS  PubMed  Google Scholar 

  45. Morrison, A., Aertsen, A. & Diesmann, M. Spike-timing-dependent plasticity in balanced random networks. Neural Comput. 19, 1437–1467 (2007).

    Article  PubMed  Google Scholar 

  46. Gentet, L.J., Avermann, M., Matyas, F., Staiger, J.F. & Petersen, C.C.H. Membrane potential dynamics of GABAergic neurons in the barrel cortex of behaving mice. Neuron 65, 422–435 (2010).

    Article  CAS  PubMed  Google Scholar 

  47. Ferezou, I. et al. Spatiotemporal dynamics of cortical sensorimotor integration in behaving mice. Neuron 56, 907–923 (2007).

    Article  CAS  PubMed  Google Scholar 

  48. Han, F., Caporale, N. & Dan, Y. Reverberation of recent visual experience in spontaneous cortical waves. Neuron 60, 321–327 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  49. Ma, W.J., Beck, J.M., Latham, P.E. & Pouget, A. Bayesian inference with probabilistic population codes. Nat. Neurosci. 9, 1432–1438 (2006).

    Article  CAS  PubMed  Google Scholar 

  50. Buesing, L., Bill, J., Nessler, B. & Maass, W. Neural dynamics as sampling: a model for stochastic computation in recurrent networks of spiking neurons. PLoS Comput. Biol. 7, e1002211 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  51. Fruchterman, T.M.J. & Reingold, E.M. Graph drawing by force-directed placement. Softw. Pract. Exp. 21, 1129–1164 (1991).

    Article  Google Scholar 

  52. Glauber, R. Time-dependent statistics of the Ising model. J. Math. Phys. 4, 294–307 (1963).

    Article  Google Scholar 

Download references

Acknowledgements

We thank R. da Silveira and H. Sompolinsky for discussions that helped shape an early version of this work. We also thank B. Yu and J. de la Rocha for comments. B.D. was supported by the US National Science Foundation (NSFDMS-1121784) and a Sloan research fellowship. A.L.-K. was supported by the US National Defense Science & Engineering Graduate Fellowship program.

Author information

Authors and Affiliations

  1. Program for Neural Computation, Carnegie Mellon University and University of Pittsburgh, Pittsburgh, Pennsylvania, USA

    Ashok Litwin-Kumar

  2. Center for the Neural Basis of Cognition, Pittsburgh, Pennsylvania, USA

    Ashok Litwin-Kumar & Brent Doiron

  3. Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

    Brent Doiron

Authors
  1. Ashok Litwin-Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brent Doiron
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.L.-K. and B.D. conceived the study and wrote the manuscript. A.L.-K. performed the simulations and analyzed the data.

Corresponding authors

Correspondence to Ashok Litwin-Kumar or Brent Doiron.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Text and Figures

Supplementary Figures 1–3 (PDF 934 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Litwin-Kumar, A., Doiron, B. Slow dynamics and high variability in balanced cortical networks with clustered connections. Nat Neurosci 15, 1498–1505 (2012). https://doi.org/10.1038/nn.3220

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nn.3220

This article is cited by

Search

Advanced 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