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

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.

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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).

    CAS  PubMed  Article  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).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

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

    CAS  Article  PubMed  Google Scholar 

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

    CAS  PubMed  Article  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).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

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

    CAS  PubMed  Article  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).

    CAS  PubMed  Article  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).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  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).

    CAS  PubMed  Article  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).

    CAS  PubMed  Article  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).

    CAS  PubMed  PubMed Central  Article  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).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  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).

    PubMed  PubMed Central  Article  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).

    CAS  PubMed  Article  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).

    CAS  PubMed  Article  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  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).

    CAS  PubMed  Article  Google Scholar 

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

    CAS  PubMed  Article  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).

    PubMed  Article  Google Scholar 

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

    PubMed Central  Article  CAS  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  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).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  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).

    PubMed  PubMed Central  Article  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).

    CAS  PubMed  Article  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  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).

    PubMed  Article  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).

    CAS  PubMed  Article  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).

    CAS  PubMed  Article  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).

    CAS  PubMed  Article  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).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

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

    CAS  PubMed  Article  Google Scholar 

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

    PubMed  Article  Google Scholar 

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

    CAS  PubMed  Article  PubMed Central  Google Scholar 

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

    CAS  PubMed  Article  Google Scholar 

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

    PubMed  Article  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).

    CAS  Article  PubMed  Google Scholar 

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

    CAS  PubMed  Article  Google Scholar 

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

    CAS  PubMed  PubMed Central  Article  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).

    CAS  PubMed  Article  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).

    CAS  PubMed  PubMed Central  Article  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 

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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.

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

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Correspondence to Ashok Litwin-Kumar or Brent Doiron.

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The authors declare no competing financial interests.

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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

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