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.

  • Review Article
  • Published:

The structures and functions of correlations in neural population codes

Nature Reviews Neuroscience volume 23, pages 551–567 (2022)Cite this article

Subjects

Abstract

The collective activity of a population of neurons, beyond the properties of individual cells, is crucial for many brain functions. A fundamental question is how activity correlations between neurons affect how neural populations process information. Over the past 30 years, major progress has been made on how the levels and structures of correlations shape the encoding of information in population codes. Correlations influence population coding through the organization of pairwise-activity correlations with respect to the similarity of tuning of individual neurons, by their stimulus modulation and by the presence of higher-order correlations. Recent work has shown that correlations also profoundly shape other important functions performed by neural populations, including generating codes across multiple timescales and facilitating information transmission to, and readout by, downstream brain areas to guide behaviour. Here, we review this recent work and discuss how the structures of correlations can have opposite effects on the different functions of neural populations, thus creating trade-offs and constraints for the structure–function relationships of population codes. Further, we present ideas on how to combine large-scale simultaneous recordings of neural populations, computational models, analyses of behaviour, optogenetics and anatomy to unravel how the structures of correlations might be optimized to serve multiple functions.

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

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

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

Fig. 1: Different types of population codes and the functions they can perform.
Fig. 2: Noise correlations and information encoding.
Fig. 3: Across-time noise correlations can generate long information consistency timescales.
Fig. 4: Noise correlations for information transmission and behavioural readout.

Similar content being viewed by others

References

  1. Wilson, M. A. & McNaughton, B. L. Dynamics of the hippocampal ensemble code for space. Science 261, 1055–1058 (1993).

    Article  CAS  PubMed  Google Scholar 

  2. Rigotti, M. et al. The importance of mixed selectivity in complex cognitive tasks. Nature 497, 585–590 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  3. Bernardi, S. et al. The geometry of abstraction in the hippocampus and prefrontal cortex. Cell 183, 954–967.e21 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  4. Koren, V., Andrei, A. R., Hu, M., Dragoi, V. & Obermayer, K. Pairwise synchrony and correlations depend on the structure of the population code in visual cortex. Cell Rep. https://doi.org/10.1016/j.celrep.2020.108367 (2020).

    Article  PubMed  Google Scholar 

  5. Battiston, F. et al. The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098 (2021).

    Article  CAS  Google Scholar 

  6. Barlow, H. B. Possible principles underlying the transformation of sensory messages. Sens. Commun. 1, 217–234 (1961).

    Google Scholar 

  7. Attneave, F. Some informational aspects of visual perception. Psychol. Rev. 61, 183–193 (1954).

    Article  CAS  PubMed  Google Scholar 

  8. Olshausen, B. A. & Field, D. J. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381, 607–609 (1996).

    Article  CAS  PubMed  Google Scholar 

  9. Mlynarski, W. F. & Hermundstad, A. M. Efficient and adaptive sensory codes. Nat. Neurosci. 24, 998–1009 (2021).

    Article  CAS  PubMed  Google Scholar 

  10. Vyas, S., Golub, M. D., Sussillo, D. & Shenoy, K. V. Computation through neural population dynamics. Annu. Rev. Neurosci. 43, 249–275 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  11. Panzeri, S., Schultz, S. R., Treves, A. & Rolls, E. T. Correlations and the encoding of information in the nervous system. Proc. Biol. Sci. 266, 1001–1012 (1999).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Pola, G., Thiele, A., Hoffmann, K. P. & Panzeri, S. An exact method to quantify the information transmitted by different mechanisms of correlational coding. Network 14, 35–60 (2003).

    Article  CAS  PubMed  Google Scholar 

  13. Averbeck, B. B., Latham, P. E. & Pouget, A. Neural correlations, population coding and computation. Nat. Rev. Neurosci. 7, 358–366 (2006).

    Article  CAS  PubMed  Google Scholar 

  14. Abbott, L. F. & Dayan, P. The effect of correlated variability on the accuracy of a population code. Neural Comput. 11, 91–101 (1999).

    Article  CAS  PubMed  Google Scholar 

  15. Gawne, T. J. & Richmond, B. J. How independent are the messages carried by adjacent inferior temporal cortical neurons? J. Neurosci. 13, 2758–2771 (1993).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  17. Averbeck, B. B. & Lee, D. Effects of noise correlations on information encoding and decoding. J. Neurophysiol. 95, 3633–3644 (2006).

    Article  PubMed  Google Scholar 

  18. Nogueira, R. et al. The effects of population tuning and trial-by-trial variability on information encoding and behavior. J. Neurosci. 40, 1066–1083 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  19. Zohary, E., Shadlen, M. N. & Newsome, W. T. Correlated neuronal discharge rate and its implications for psychophysical performance. Nature 370, 140–143 (1994).

    Article  CAS  PubMed  Google Scholar 

  20. Ecker, A. S., Berens, P., Tolias, A. S. & Bethge, M. The effect of noise correlations in populations of diversely tuned neurons. J. Neurosci. 31, 14272–14283 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Shamir, M. & Sompolinsky, H. Nonlinear population codes. Neural Comput. 16, 1105–1136 (2004).

    Article  PubMed  Google Scholar 

  22. Josic, K., Shea-Brown, E., Doiron, B. & de la Rocha, J. Stimulus-dependent correlations and population codes. Neural Comput. 21, 2774–2804 (2009).

    Article  PubMed  Google Scholar 

  23. Azeredo da Silveira, R. & Rieke, F. The geometry of information coding in correlated neural populations. Annu. Rev. Neurosci. 44, 403–424 (2021).

    Article  CAS  PubMed  Google Scholar 

  24. Gray, C. M., Konig, P., Engel, A. K. & Singer, W. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338, 334–337 (1989).

    Article  CAS  PubMed  Google Scholar 

  25. deCharms, R. C. & Merzenich, M. M. Primary cortical representation of sounds by the coordination of action-potential timing. Nature 381, 610–613 (1996).

    Article  CAS  PubMed  Google Scholar 

  26. Franke, F. et al. Structures of neural correlation and how they favor coding. Neuron 89, 409–422 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  27. Dan, Y., Alonso, J. M., Usrey, W. M. & Reid, R. C. Coding of visual information by precisely correlated spikes in the lateral geniculate nucleus. Nat. Neurosci. 1, 501–507 (1998).

    Article  CAS  PubMed  Google Scholar 

  28. Zylberberg, J., Cafaro, J., Turner, M. H., Shea-Brown, E. & Rieke, F. Direction-selective circuits shape noise to ensure a precise population code. Neuron 89, 369–383 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. 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 

  30. Romo, R., Hernandez, A., Zainos, A. & Salinas, E. Correlated neuronal discharges that increase coding efficiency during perceptual discrimination. Neuron 38, 649–657 (2003).

    Article  CAS  PubMed  Google Scholar 

  31. Reich, D. S., Mechler, F. & Victor, J. D. Independent and redundant information in nearby cortical neurons. Science 294, 2566–2568 (2001).

    Article  CAS  PubMed  Google Scholar 

  32. Pillow, J. W. et al. Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature 454, 995–999 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  33. Graf, A. B., Kohn, A., Jazayeri, M. & Movshon, J. A. Decoding the activity of neuronal populations in macaque primary visual cortex. Nat. Neurosci. 14, 239–245 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  34. Rupasinghe, A. et al. Direct extraction of signal and noise correlations from two-photon calcium imaging of ensemble neuronal activity. eLife https://doi.org/10.7554/eLife.68046 (2021).

    Article  PubMed  PubMed Central  Google Scholar 

  35. Rothschild, G., Nelken, I. & Mizrahi, A. Functional organization and population dynamics in the mouse primary auditory cortex. Nat. Neurosci. 13, 353–360 (2010).

    Article  CAS  PubMed  Google Scholar 

  36. Kwon, S. E., Tsytsarev, V., Erzurumlu, R. S. & O’Connor, D. H. Organization of orientation-specific whisker deflection responses in layer 2/3 of mouse somatosensory cortex. Neuroscience 368, 46–56 (2018).

    Article  CAS  PubMed  Google Scholar 

  37. Bartolo, R., Saunders, R. C., Mitz, A. R. & Averbeck, B. B. Information-limiting correlations in large neural populations. J. Neurosci. 40, 1668–1678 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  38. Kafashan, M. et al. Scaling of sensory information in large neural populations shows signatures of information-limiting correlations. Nat. Commun. 12, 473 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  39. Petersen, R. S., Panzeri, S. & Diamond, M. E. Population coding of stimulus location in rat somatosensory cortex. Neuron 32, 503–514 (2001).

    Article  CAS  PubMed  Google Scholar 

  40. Chen, Y. P., Lin, C. P., Hsu, Y. C. & Hung, C. P. Network anisotropy trumps noise for efficient object coding in macaque inferior temporal cortex. J. Neurosci. 35, 9889–9899 (2015).

    Article  CAS  PubMed  Google Scholar 

  41. Adibi, M., McDonald, J. S., Clifford, C. W. & Arabzadeh, E. Adaptation improves neural coding efficiency despite increasing correlations in variability. J. Neurosci. 33, 2108–2120 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  42. Rumyantsev, O. I. et al. Fundamental bounds on the fidelity of sensory cortical coding. Nature 580, 100–105 (2020).

    Article  CAS  PubMed  Google Scholar 

  43. Sanayei, M. et al. Perceptual learning of fine contrast discrimination changes neuronal tuning and population coding in macaque V4. Nat. Commun. 9, 4238 (2018).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  44. Tremblay, S., Pieper, F., Sachs, A. & Martinez-Trujillo, J. Attentional filtering of visual information by neuronal ensembles in the primate lateral prefrontal cortex. Neuron 85, 202–215 (2015).

    Article  CAS  PubMed  Google Scholar 

  45. Moreno-Bote, R. et al. Information-limiting correlations. Nat. Neurosci. 17, 1410–1417 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  46. Kanitscheider, I., Coen-Cagli, R. & Pouget, A. Origin of information-limiting noise correlations. Proc. Natl Acad. Sci. USA 112, E6973–E6982 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  48. Okun, M. et al. Diverse coupling of neurons to populations in sensory cortex. Nature 521, 511–515 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  49. Minces, V., Pinto, L., Dan, Y. & Chiba, A. A. Cholinergic shaping of neural correlations. Proc. Natl Acad. Sci. USA 114, 5725–5730 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  50. Shamir, M. & Sompolinsky, H. Implications of neuronal diversity on population coding. Neural Comput. 18, 1951–1986 (2006).

    Article  PubMed  Google Scholar 

  51. Wilke, S. D. & Eurich, C. W. Representational accuracy of stochastic neural populations. Neural Comput. 14, 155–189 (2002).

    Article  PubMed  Google Scholar 

  52. Cohen, M. R. & Maunsell, J. H. Attention improves performance primarily by reducing interneuronal correlations. Nat. Neurosci. 12, 1594–1600 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  53. Gutnisky, D. A. & Dragoi, V. Adaptive coding of visual information in neural populations. Nature 452, 220–224 (2008).

    Article  CAS  PubMed  Google Scholar 

  54. Jeanne, J. M., Sharpee, T. O. & Gentner, T. Q. Associative learning enhances population coding by inverting interneuronal correlation patterns. Neuron 78, 352–363 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  55. Ruff, D. A. & Cohen, M. R. Attention can either increase or decrease spike count correlations in visual cortex. Nat. Neurosci. 17, 1591–1597 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  56. Downer, J. D., Niwa, M. & Sutter, M. L. Task engagement selectively modulates neural correlations in primary auditory cortex. J. Neurosci. 35, 7565–7574 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  57. Gu, Y. et al. Perceptual learning reduces interneuronal correlations in macaque visual cortex. Neuron 71, 750–761 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  58. Nigam, S., Pojoga, S. & Dragoi, V. Synergistic coding of visual information in columnar networks. Neuron 104, 402–411.e4 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  59. Ni, A. M., Ruff, D. A., Alberts, J. J., Symmonds, J. & Cohen, M. R. Learning and attention reveal a general relationship between population activity and behavior. Science 359, 463–465 (2018).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  60. Valente, A. et al. Correlations enhance the behavioral readout of neural population activity in association cortex. Nat. Neurosci. 24, 975–986 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  61. Umakantha, A. et al. Bridging neuronal correlations and dimensionality reduction. Neuron 109, 2740–2754.e12 (2021).

    Article  CAS  PubMed  Google Scholar 

  62. Ganmor, E., Segev, R. & Schneidman, E. Sparse low-order interaction network underlies a highly correlated and learnable neural population code. Proc. Natl Acad. Sci. USA 108, 9679–9684 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  63. Granot-Atedgi, E., Tkacik, G., Segev, R. & Schneidman, E. Stimulus-dependent maximum entropy models of neural population codes. PLoS Comput. Biol. 9, e1002922 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  64. Ohiorhenuan, I. E. et al. Sparse coding and high-order correlations in fine-scale cortical networks. Nature 466, 617–621 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  65. Montani, F. et al. The impact of high-order interactions on the rate of synchronous discharge and information transmission in somatosensory cortex. Philos. Trans. R. Soc. A Phys. Eng. Sci. 367, 3297–3310 (2009).

    Article  Google Scholar 

  66. Giusti, C., Pastalkova, E., Curto, C. & Itskov, V. Clique topology reveals intrinsic geometric structure in neural correlations. Proc. Natl Acad. Sci. USA 112, 13455–13460 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  67. Froudarakis, E. et al. Population code in mouse V1 facilitates readout of natural scenes through increased sparseness. Nat. Neurosci. 17, 851–857 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  68. Chelaru, M. I. et al. High-order interactions explain the collective behavior of cortical populations in executive but not sensory areas. Neuron https://doi.org/10.1016/j.neuron.2021.09.042 (2021).

    Article  PubMed  Google Scholar 

  69. Yu, S. et al. Higher-order interactions characterized in cortical activity. J. Neurosci. 31, 17514–17526 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  70. Gardner, R. J. et al. Toroidal topology of population activity in grid cells. Nature 602, 123–128 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  71. Cayco-Gajic, N. A., Zylberberg, J. & Shea-Brown, E. Triplet correlations among similarly tuned cells impact population coding. Front. Comput. Neurosc. https://doi.org/10.3389/fncom.7015.00057 (2015).

    Article  Google Scholar 

  72. Panzeri, S., Brunel, N., Logothetis, N. K. & Kayser, C. Sensory neural codes using multiplexed temporal scales. Trends Neurosci. 33, 111–120 (2010).

    Article  CAS  PubMed  Google Scholar 

  73. Gold, J. I. & Shadlen, M. N. The neural basis of decision making. Annu. Rev. Neurosci. 30, 535–574 (2007).

    Article  CAS  PubMed  Google Scholar 

  74. Andersen, R. A. & Cui, H. Intention, action planning, and decision making in parietal-frontal circuits. Neuron 63, 568–583 (2009).

    Article  CAS  PubMed  Google Scholar 

  75. Bisley, J. W. & Goldberg, M. E. Attention, intention, and priority in the parietal lobe. Annu. Rev. Neurosci. 33, 1–21 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  76. Curtis, C. E. & Lee, D. Beyond working memory: the role of persistent activity in decision making. Trends Cogn. Sci. 14, 216–222 (2010).

    Article  PubMed  PubMed Central  Google Scholar 

  77. Shadlen, M. N. & Newsome, W. T. Neural basis of a perceptual decision in the parietal cortex (area LIP) of the rhesus monkey. J. Neurophysiol. 86, 1916–1936 (2001).

    Article  CAS  PubMed  Google Scholar 

  78. Wang, X. J. Decision making in recurrent neuronal circuits. Neuron 60, 215–234 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  79. Inagaki, H. K., Fontolan, L., Romani, S. & Svoboda, K. Discrete attractor dynamics underlies persistent activity in the frontal cortex. Nature 566, 212–217 (2019).

    Article  CAS  PubMed  Google Scholar 

  80. Li, N., Daie, K., Svoboda, K. & Druckmann, S. Robust neuronal dynamics in premotor cortex during motor planning. Nature 532, 459–464 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  81. Li, N., Chen, T. W., Guo, Z. V., Gerfen, C. R. & Svoboda, K. A motor cortex circuit for motor planning and movement. Nature 519, 51–56 (2015).

    Article  CAS  PubMed  Google Scholar 

  82. Hanks, T. D. et al. Distinct relationships of parietal and prefrontal cortices to evidence accumulation. Nature 520, 220–223 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  83. Crowe, D. A., Averbeck, B. B. & Chafee, M. V. Rapid sequences of population activity patterns dynamically encode task-critical spatial information in parietal cortex. J. Neurosci. 30, 11640–11653 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  84. Pastalkova, E., Itskov, V., Amarasingham, A. & Buzsaki, G. Internally generated cell assembly sequences in the rat hippocampus. Science 321, 1322–1327 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  85. Baeg, E. H. et al. Dynamics of population code for working memory in the prefrontal cortex. Neuron 40, 177–188 (2003).

    Article  CAS  PubMed  Google Scholar 

  86. Harvey, C. D., Coen, P. & Tank, D. W. Choice-specific sequences in parietal cortex during a virtual-navigation decision task. Nature 484, 62–68 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  87. Morcos, A. S. & Harvey, C. D. History-dependent variability in population dynamics during evidence accumulation in cortex. Nat. Neurosci. 19, 1672–1681 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  88. Runyan, C. A., Piasini, E., Panzeri, S. & Harvey, C. D. Distinct timescales of population coding across cortex. Nature 548, 92–96 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  89. Akrami, A., Kopec, C. D., Diamond, M. E. & Brody, C. D. Posterior parietal cortex represents sensory history and mediates its effects on behaviour. Nature 554, 368–372 (2018).

    Article  CAS  PubMed  Google Scholar 

  90. Zariwala, H. A., Kepecs, A., Uchida, N., Hirokawa, J. & Mainen, Z. F. The limits of deliberation in a perceptual decision task. Neuron 78, 339–351 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  91. Mazurek, M. E. & Shadlen, M. N. Limits to the temporal fidelity of cortical spike rate signals. Nat. Neurosci. 5, 463–471 (2002).

    Article  CAS  PubMed  Google Scholar 

  92. Piasini, E. et al. Temporal stability of stimulus representation increases along rodent visual cortical hierarchies. Nat. Commun. 12, 4448 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  93. Gold, J. I. & Shadlen, M. N. Neural computations that underlie decisions about sensory stimuli. Trends Cogn. Sci. 5, 10–16 (2001).

    Article  PubMed  Google Scholar 

  94. Beck, J. M., Wei, J. M., Pitkow, X., Peter, E. L. & Pouget, A. Not noisy, just wrong: the role of suboptimal inference in behavioral variability. Neuron 74, 30–39 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  95. Panzeri, S., Harvey, C. D., Piasini, E., Latham, P. E. & Fellin, T. Cracking the neural code for sensory perception by combining statistics, intervention, and behavior. Neuron 93, 491–507 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  96. Wu, S., Nakahara, H. & Amari, S. Population coding with correlation and an unfaithful model. Neural Comput. 13, 775–797 (2001).

    Article  CAS  PubMed  Google Scholar 

  97. Latham, P. E. & Nirenberg, S. Synergy, redundancy, and independence in population codes, revisited. J. Neurosci. 25, 5195–5206 (2005).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  98. Nirenberg, S., Carcieri, S. M., Jacobs, A. L. & Latham, P. E. Retinal ganglion cells act largely as independent encoders. Nature 411, 698–701 (2001).

    Article  CAS  PubMed  Google Scholar 

  99. Karpas, E. M., Kiani, R. O. & Schneidman, E. Strongly correlated spatiotemporal encoding and simple decoding in the prefrontal cortex. bioRxiv https://doi.org/10.1101/693192 (2019).

    Article  Google Scholar 

  100. Salinas, E. & Sejnowski, T. J. Correlated neuronal activity and the flow of neural information. Nat. Rev. Neurosci. 2, 539–550 (2001).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  101. Stringer, C., Michaelos, M., Tsyboulski, D., Lindo, S. E. & Pachitariu, M. High-precision coding in visual cortex. Cell 184, 2767 (2021).

    Article  CAS  PubMed  Google Scholar 

  102. Zhao, Y., Yates, J. L., Levi, A. J., Huk, A. C. & Park, I. M. Stimulus-choice (mis)alignment in primate area MT. PLoS Comput. Biol. 16, e1007614 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  103. Pica, G. P., et al. In Advances in Neural Information Processing Systems (NeurIPS). (ed. Luxburg G. I., et al.) 3686–3696 (Curran Associates, Inc., 2020).

  104. Koch, C., Rapp, M. & Segev, I. A brief history of time (constants). Cereb. Cortex 6, 93–101 (1996).

    Article  CAS  PubMed  Google Scholar 

  105. Diesmann, M., Gewaltig, M. O. & Aertsen, A. Stable propagation of synchronous spiking in cortical neural networks. Nature 402, 529–533 (1999).

    Article  CAS  PubMed  Google Scholar 

  106. Zandvakili, A. & Kohn, A. Coordinated neuronal activity enhances corticocortical communication. Neuron 87, 827–839 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  107. Alonso, J. M., Usrey, W. M. & Reid, R. C. Precisely correlated firing in cells of the lateral geniculate nucleus. Nature 383, 815–819 (1996).

    Article  CAS  PubMed  Google Scholar 

  108. Softky, W. R. & Koch, C. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13, 334–350 (1993).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  109. Reyes, A. D. Synchrony-dependent propagation of firing rate in iteratively constructed networks in vitro. Nat. Neurosci. 6, 593–599 (2003).

    Article  CAS  PubMed  Google Scholar 

  110. Losonczy, A. & Magee, J. C. Integrative properties of radial oblique dendrites in hippocampal CA1 pyramidal neurons. Neuron 50, 291–307 (2006).

    Article  CAS  PubMed  Google Scholar 

  111. Ariav, G., Polsky, A. & Schiller, J. Submillisecond precision of the input-output transformation function mediated by fast sodium dendritic spikes in basal dendrites of CA1 pyramidal neurons. J. Neurosci. 23, 7750–7758 (2003).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  112. London, M. & Hausser, M. Dendritic computation. Annu. Rev. Neurosci. 28, 503–532 (2005).

    Article  CAS  PubMed  Google Scholar 

  113. Polsky, A., Mel, B. W. & Schiller, J. Computational subunits in thin dendrites of pyramidal cells. Nat. Neurosci. 7, 621–627 (2004).

    Article  CAS  PubMed  Google Scholar 

  114. Golding, N. L., Staff, N. P. & Spruston, N. Dendritic spikes as a mechanism for cooperative long-term potentiation. Nature 418, 326–331 (2002).

    Article  CAS  PubMed  Google Scholar 

  115. Smith, S. L., Smith, I. T., Branco, T. & Hausser, M. Dendritic spikes enhance stimulus selectivity in cortical neurons in vivo. Nature 503, 115–120 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  116. Schmidt-Hieber, C. et al. Active dendritic integration as a mechanism for robust and precise grid cell firing. Nat. Neurosci. 20, 1114–1121 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  117. Wilson, D. E., Whitney, D. E., Scholl, B. & Fitzpatrick, D. Orientation selectivity and the functional clustering of synaptic inputs in primary visual cortex. Nat. Neurosci. 19, 1003–1009 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  118. Ackels, T. et al. Fast odour dynamics are encoded in the olfactory system and guide behaviour. Nature 593, 558–563 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  119. Zylberberg, J., Pouget, A., Latham, P. E. & Shea-Brown, E. Robust information propagation through noisy neural circuits. PLoS Comput. Biol. 13, e1005497 (2017).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  120. Histed, M. H. & Maunsell, J. H. Cortical neural populations can guide behavior by integrating inputs linearly, independent of synchrony. Proc. Natl Acad. Sci. USA 111, E178–E187 (2014).

    Article  CAS  PubMed  Google Scholar 

  121. Shahidi, N., Andrei, A. R., Hu, M. & Dragoi, V. High-order coordination of cortical spiking activity modulates perceptual accuracy. Nat. Neurosci. 22, 1148–1158 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  122. Balaguer-Ballester, E., Nogueira, R., Abofalia, J. M., Moreno-Bote, R. & Sanchez-Vives, M. V. Representation of foreseeable choice outcomes in orbitofrontal cortex triplet-wise interactions. PLoS Comput. Biol. https://doi.org/10.1371/journal.pcbi.1007862 (2020).

    Article  PubMed  PubMed Central  Google Scholar 

  123. Zylberberg, J. & Shea-Brown, E. Input nonlinearities can shape beyond-pairwise correlations and improve information transmission by neural populations. Phys. Rev. E https://doi.org/10.1103/PhysRevE.92.062707 (2015).

    Article  Google Scholar 

  124. Emiliani, V., Cohen, A. E., Deisseroth, K. & Hausser, M. All-optical interrogation of neural circuits. J. Neurosci. 35, 13917–13926 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  125. Rickgauer, J. P., Deisseroth, K. & Tank, D. W. Simultaneous cellular-resolution optical perturbation and imaging of place cell firing fields. Nat. Neurosci. 17, 1816–1824 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  126. Marshel, J. H. et al. Cortical layer-specific critical dynamics triggering perception. Science https://doi.org/10.1126/science.aaw5202 (2019).

    Article  PubMed  PubMed Central  Google Scholar 

  127. Carrillo-Reid, L., Han, S., Yang, W., Akrouh, A. & Yuste, R. Controlling visually guided behavior by holographic recalling of cortical ensembles. Cell 178, 447–457.e5 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  128. Dalgleish, H. W. et al. How many neurons are sufficient for perception of cortical activity? eLife https://doi.org/10.7554/eLife.58889 (2020).

    Article  PubMed  PubMed Central  Google Scholar 

  129. Packer, A. M., Russell, L. E., Dalgleish, H. W. & Hausser, M. Simultaneous all-optical manipulation and recording of neural circuit activity with cellular resolution in vivo. Nat. Methods 12, 140–146 (2015).

    Article  CAS  PubMed  Google Scholar 

  130. Pegard, N. C. et al. Three-dimensional scanless holographic optogenetics with temporal focusing (3D-SHOT). Nat. Commun. 8, 1228 (2017).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  131. Forli, A. et al. Two-photon bidirectional control and imaging of neuronal excitability with high spatial resolution in vivo. Cell Rep. 22, 3087–3098 (2018).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  132. Chettih, S. N. & Harvey, C. D. Single-neuron perturbations reveal feature-specific competition in V1. Nature 567, 334–340 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  133. Daie, K., Svoboda, K. & Druckmann, S. Targeted photostimulation uncovers circuit motifs supporting short-term memory. Nat. Neurosci. 24, 259–265 (2021).

    Article  CAS  PubMed  Google Scholar 

  134. Robinson, N. T. M. et al. Targeted activation of hippocampal place cells drives memory-guided spatial behavior. Cell 183, 2041–2042 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  135. Gill, J. V. et al. Precise holographic manipulation of olfactory circuits reveals coding features determining perceptual detection. Neuron 108, 382–393.e5 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  136. Chong, E. et al. Manipulating synthetic optogenetic odors reveals the coding logic of olfactory perception. Science https://doi.org/10.1126/science.aba2357 (2020).

    Article  PubMed  PubMed Central  Google Scholar 

  137. Hansen, B. J., Chelaru, M. I. & Dragoi, V. Correlated variability in laminar cortical circuits. Neuron 76, 590–602 (2012).

    Article  CAS  PubMed  Google Scholar 

  138. Olshausen, B. A. & Field, D. J. Sparse coding of sensory inputs. Curr. Opin. Neurobiol. 14, 481–487 (2004).

    Article  CAS  PubMed  Google Scholar 

  139. Simoncelli, E. P. & Olshausen, B. A. Natural image statistics and neural representation. Annu. Rev. Neurosci. 24, 1193–1216 (2001).

    Article  CAS  PubMed  Google Scholar 

  140. Pinto, L., Tank, D. W. & Brody, C. D. Multiple timescales of sensory-evidence accumulation across the dorsal cortex. bioRxiv https://doi.org/10.1101/2020.12.28.424600 (2021).

    Article  PubMed  PubMed Central  Google Scholar 

  141. Lynn, C. W. & Bassett, D. S. The physics of brain network structure, function and control. Nat. Rev. Phys. 1, 318–332 (2019).

    Article  Google Scholar 

  142. Sporns, O., Tononi, G. & Kotter, R. The human connectome: a structural description of the human brain. PLoS Comput. Biol. 1, e42 (2005).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  143. Kuan, A. T. et al. Dense neuronal reconstruction through X-ray holographic nano-tomography. Nat. Neurosci. 23, 1637–1643 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  144. Ocker, G. K. et al. From the statistics of connectivity to the statistics of spike times in neuronal networks. Curr. Opin. Neurobiol. 46, 109–119 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  145. Rosenbaum, R., Smith, M. A., Kohn, A., Rubin, J. E. & Doiron, B. The spatial structure of correlated neuronal variability. Nat. Neurosci. 20, 107–114 (2017).

    Article  CAS  PubMed  Google Scholar 

  146. Goris, R. L., Movshon, J. A. & Simoncelli, E. P. Partitioning neuronal variability. Nat. Neurosci. 17, 858–865 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  147. Kohn, A., Coen-Cagli, R., Kanitscheider, I. & Pouget, A. Correlations and neuronal population information. Annu. Rev. Neurosci. 39, 237–256 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  148. Ostojic, S., Brunel, N. & Hakim, V. How connectivity, background activity, and synaptic properties shape the cross-correlation between spike trains. J. Neurosci. 29, 10234–10253 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  149. Stringer, C. et al. Spontaneous behaviors drive multidimensional, brainwide activity. Science 364, aav7893 (2019).

    Article  CAS  Google Scholar 

  150. Verhoef, B. E. & Maunsell, J. H. R. Attention-related changes in correlated neuronal activity arise from normalization mechanisms. Nat. Neurosci. 20, 969–977 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  151. Sadeh, S. & Clopath, C. Theory of neuronal perturbome in cortical networks. Proc. Natl Acad. Sci. USA 117, 26966–26976 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  152. Khan, A. G. et al. Distinct learning-induced changes in stimulus selectivity and interactions of GABAergic interneuron classes in visual cortex. Nat. Neurosci. 21, 851–859 (2018).

    Article  CAS  PubMed  Google Scholar 

  153. Poort, J. et al. Learning enhances sensory and multiple non-sensory representations in primary visual cortex. Neuron 86, 1478–1490 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  154. Bittner, S. R. et al. Population activity structure of excitatory and inhibitory neurons. PLoS One 12, e0181773 (2017).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  155. Economo, M. N. et al. Distinct descending motor cortex pathways and their roles in movement. Nature 563, 79–84 (2018).

    Article  CAS  PubMed  Google Scholar 

  156. Chen, J. L., Carta, S., Soldado-Magraner, J., Schneider, B. L. & Helmchen, F. Behaviour-dependent recruitment of long-range projection neurons in somatosensory cortex. Nature 499, 336–340 (2013).

    Article  CAS  PubMed  Google Scholar 

  157. Huda, R. et al. Distinct prefrontal top-down circuits differentially modulate sensorimotor behavior. Nat. Commun. 11, 6007 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  158. Itokazu, T. et al. Streamlined sensory motor communication through cortical reciprocal connectivity in a visually guided eye movement task. Nat. Commun. 9, 338 (2018).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  159. Sofroniew, N. J., Flickinger, D., King, J. & Svoboda, K. A large field of view two-photon mesoscope with subcellular resolution for in vivo imaging. eLife https://doi.org/10.7554/eLife.14472 (2016).

    Article  PubMed  PubMed Central  Google Scholar 

  160. Jun, J. J. et al. Fully integrated silicon probes for high-density recording of neural activity. Nature 551, 232–236 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  161. Urai, A. E., Doiron, B., Leifer, A. M. & Churchland, A. K. Large-scale neural recordings call for new insights to link brain and behavior. Nat. Neurosci. 25, 11–19 (2022).

    Article  CAS  PubMed  Google Scholar 

  162. Kohn, A. et al. Principles of corticocortical communication: proposed schemes and design considerations. Trends Neurosci. 43, 725–737 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  163. Semedo, J. D., Gokcen, E., Machens, C. K., Kohn, A. & Yu, B. M. Statistical methods for dissecting interactions between brain areas. Curr. Opin. Neurobiol. 65, 59–69 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  164. Semedo, J. D., Zandvakili, A., Machens, C. K., Yu, B. M. & Kohn, A. Cortical areas interact through a communication subspace. Neuron 102, 249–259.e4 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  165. Schneidman, E., Berry, M. J. II, Segev, R. & Bialek, W. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature 440, 1007–1012 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  166. Onken, A., Grunewalder, S., Munk, M. H. & Obermayer, K. Analyzing short-term noise dependencies of spike-counts in macaque prefrontal cortex using copulas and the flashlight transformation. PLoS Comput. Biol. 5, e1000577 (2009).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  167. Berkes, P., Wood, F. & Pillow, J. Characterizing neural dependencies with copula models. Adv. Neural Inf. Process. Syst. 21, 129–136 (2009).

    Google Scholar 

  168. Safaai, H., Onken, A., Harvey, C. D. & Panzeri, S. Information estimation using nonparametric copulas. Phys. Rev. E https://doi.org/10.1103/PhysRevE.98.053302 (2018).

    Article  PubMed  PubMed Central  Google Scholar 

  169. Nienborg, H. & Cumming, B. G. Decision-related activity in sensory neurons reflects more than a neuron’s causal effect. Nature 459, 89–92 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  170. Moreno-Bote, R. & Drugowitsch, J. Causal inference and explaining away in a spiking network. Sci. Rep. 5, 17531 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  171. Peron, S. et al. Recurrent interactions in local cortical circuits. Nature 579, 256–259 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  172. Cossell, L. et al. Functional organization of excitatory synaptic strength in primary visual cortex. Nature 518, 399–403 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  173. Lee, W. C. et al. Anatomy and function of an excitatory network in the visual cortex. Nature 532, 370–374 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  174. Carrillo-Reid, L. & Yuste, R. Playing the piano with the cortex: role of neuronal ensembles and pattern completion in perception and behavior. Curr. Opin. Neurobiol. 64, 89–95 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

Download references

Acknowledgements

The authors thank members of their laboratories for helpful discussions, and B. Babadi, C. Becchio, G. Bondanelli, J. Drugowitsch, M. Histed, G. Iurilli, S. Lemke, J. Maunsell and E. Piasini for feedback. This work was supported by the US National Institutes of Health (NIH) grants DP1 MH125776 (C.D.H.); the US National Institute of Neurological Disorders and Stroke (NINDS) R01 NS089521 (C.D.H.); the Brain Research Through Advancing Innovative Neurotechnologies (BRAIN) Initiative R01 NS108410 (C.D.H. and S.P.), U19 NS107464 (S.P.), R01 NS109961 (S.P.); and the Fondation Bertarelli (S.P.).

Author information

Authors and Affiliations

  1. Department of Excellence for Neural Information Processing, Center for Molecular Neurobiology (ZMNH), University Medical Center Hamburg-Eppendorf (UKE), Hamburg, Germany

    Stefano Panzeri

  2. Istituto Italiano di Tecnologia, Rovereto, Italy

    Stefano Panzeri & Monica Moroni

  3. Department of Neurobiology, Harvard Medical School, Boston, MA, USA

    Houman Safaai & Christopher D. Harvey

Authors
  1. Stefano Panzeri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monica Moroni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Houman Safaai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christopher D. Harvey
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed to all aspects of the article.

Corresponding authors

Correspondence to Stefano Panzeri or Christopher D. Harvey.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Reviews Neuroscience thanks V. Dragoi and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Glossary

Population code

The features and patterns of activity of neural populations that are used to perform key information-processing computations, such as encoding information and/or transmitting information.

Space of neural population activity

A space where each dimension represents the activity of a neuron and each point is a population vector.

Functional interactions

The statistical relationships between the activity of different neurons, often quantified as correlations between the activity of different neurons.

Pseudo-population

Collections of activity of non-simultaneously measured neurons, either because they were recorded at a different time or from different experiments, or because they were created by trial shuffling.

Efficient coding theories

Theories that postulate that the properties of neurons in sensory areas are designed to maximize the information that these neurons carry about sensory stimuli with naturalistic features.

Signal correlations

The correlations of the trial-averaged neural responses across different stimuli.

Noise correlations

The correlated trial-to-trial variability of the activity of different neurons or of different neural populations over repeated presentations of the same stimulus.

Signal–noise angle

The angle between the noise axis and the signal axis.

Signal axis

The axis in neural population activity space of the largest stimulus-related variations, which in linear cases is measured as the axis that connects the trial-averaged population responses to the different stimuli.

Noise axis

The axis of largest variation in neural population activity for a fixed stimulus.

Trial shuffling

An analytical procedure to remove the effect of noise correlations by combining responses of neurons taken from different trials to a given stimulus.

Redundant neuron pairs

Pairs of neurons that together carry less information than the sum of the information carried by the two neurons in each pair, owing to the information-limiting effect of noise and signal correlations.

Synergistic neuron pairs

Pairs of neurons that together carry more information than the sum of the information carried by the two neurons in each pair, owing to the information-enhancing effect of noise correlations.

Redundant hubs

Neurons with high probability of having redundant interactions with other neurons.

Synergistic hubs

Neurons with high probability of having synergistic interactions with other neurons.

Population-wise correlations

Correlated variability of an entire population of neurons, usually measured applying dimensionality-reduction techniques to the population covariance matrix.

Across-neuron noise correlations

The noise correlation between the time-averaged activity of two different neurons or two different neural populations, quantifying the similarity of the time-averaged neural or population responses across trials with the same stimulus.

Across-time noise correlations

The noise correlation between the population activity vector of the same population at different times, quantifying the similarity of the population responses at different times across trials with the same stimulus.

Persistent activity

The activity of individual cells whose firing rate remains sustained over an entire task period, for example, during working memory or decision-making tasks.

Ramping activity

The activity of individual cells whose firing rate decreases or increases constantly over time during a task, for example, to reflect the accumulation of evidence to make a decision.

Attractor states

Set of values of population vectors towards which the activity of a neuronal network is attracted during its temporal evolution.

Posterior parietal cortex

(PPC). A region of cortex considered to be at the interface of sensation and action and to participate in evidence accumulation for decision-making, movement planning, spatial navigation and other processes.

Population vector

Vector in the space of neural population activity whose components represent the activity of individual neurons in the population.

Information consistency timescale

The correlation across time of the instantaneous stimulus or choice signal (for example, the posterior probability of stimulus or choice given the observation of a population vector at a specific time).

Optimal stimulus-discrimination boundary

The plane (or surface) in the high-dimensional space of population activity that optimally separates responses elicited by different sensory stimuli, and that thus serves as an indication of how to extract sensory information from neural activity optimally.

Coincidence detection

Spike-generation mechanism that, because of the neuron’s short integration time constant, requires the near-simultaneous occurrence of several input action potentials to generate an output action potential.

Consistent information encoding

When different elements of a population code (for example, the activity of different pools of neurons) all signal the presence of the same stimulus.

Across-time encoding consistency

When population activity at a given time signals the same stimulus as the population activity at another time.

Across-neuron encoding consistency

When the activity of separate neuronal pools in the same time window signals the same stimulus.

Feature amplification motifs

Motifs of cells with similar tuning that functionally excite one another to increase the signal contained in the neural population as revealed by anatomical connections or influence mapping.

Two-photon patterned optogenetics

The use of light-sculpting, such as with a spatial light modulator, and two-photon excitation to create arbitrary spatial and temporal patterns of light to photostimulate neurons with approximately single-cell resolution.

Influence mapping

The process of measuring how spikes added by two-photon-patterned optogenetic perturbation to one or a few neurons causally affect the spiking of neighbouring neurons.

Multi-objective optimization

An optimization procedure that minimizes multiple cost functions simultaneously.

Retrograde labelling

Methods based on dyes or viruses that are taken up by axons and transported back to a neuron’s cell body.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panzeri, S., Moroni, M., Safaai, H. et al. The structures and functions of correlations in neural population codes. Nat Rev Neurosci 23, 551–567 (2022). https://doi.org/10.1038/s41583-022-00606-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41583-022-00606-4

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