Extended mean-field theories for networks of real neurons

Christopher W. Lynn; Francesca Mignacco; Luca Di Carlo; William Bialek

REVIEW 3 major objections 4 minor 1 cited by

Extended mean-field theories for networks of real neurons

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Real neural populations are described by a mean-field model that matches the full distribution of activity along one projection, and the inferred potential places these networks near a critical point.

desk verdict Genuinely new mean-field construction for max-entropy models of neural populations, with a solid negative result on naive mean-field, but the near-criticality claim relies on a shuffle control that likely measures projection degradation rather than true distance from criticality. read the letter →

arxiv 2504.15197 v1 pith:JII4SKOA submitted 2025-04-21 physics.bio-ph cond-mat.stat-mechq-bio.NC

Luca Di Carlo , Francesca Mignacco , Christopher W. Lynn , William Bialek This is my paper · ORCID

classification physics.bio-phcond-mat.stat-mechq-bio.NC

keywords maximumentropymodelsmean-fieldtheoryneuralpopulationactivitycriticalitycollectivevariableshippocampusIsingmodelcoding

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the activity of many neurons can be captured by a small number of collective variables, as mean-field theories assume. It shows that the simplest versions—matching the mean and variance of summed activity, or of a few projections—fail on real data, predicting bimodal distributions that are not observed. The authors then construct an extended maximum-entropy model that matches each neuron's mean activity and the full distribution of activity along one projection, and show that this model is at least consistent with data from retina, mouse cortex, and hippocampus. The inferred effective potential places these networks near the boundary of a second-order phase transition, a sign of near-criticality that agrees with other observations. If correct, the work provides a path to analyzing much larger neural recordings with O(N) constraints.

What carries the argument

The central object is the distribution-constrained maximum-entropy model of Eq. (10), $E_\mathrm{dist}(s)=-\sum_n h_n s_n + N U(\varphi)$ with $\varphi=\sum_n W_n s_n$. Its partition function is rewritten exactly using an auxiliary field $z$ as $Z_\mathrm{dist}=2^N \int dz/(2\pi)\int d\varphi \exp[-F_\mathrm{dist}(\varphi;z)]$, with free energy $F_\mathrm{dist}(\varphi;z)=iz\varphi+NU(\varphi)-\sum_n \ln\cosh(h_n+izW_n)$. A saddle-point approximation in $N$ yields the read-off equation: given the single-neuron fields $h_n=\operatorname{atanh}(\langle s_n\rangle_\mathrm{exp})$ from the gauge condition $U'(\varphi_\mathrm{sp})=0$, the potential $U(\varphi)$ is determined by the measured distribution $P_\mathrm{exp}(\varphi)$. The boundary $NU''(\varphi_\mathrm{sp})+\chi_0^{-1}=0$, with $\chi_0=\sum_n W_n^2(1-\langle s_n\rangle^2)$ the independent-neuron variance, marks the second-order phase transition, and its proximity for eigenvector projections is the paper's criticality signature.

What would settle it

If a dataset with substantially longer recordings or a different brain region yields an inferred $U(\varphi)$ for the leading eigenvector projection with $NU''(\varphi_\mathrm{sp})+\chi_0^{-1}$ clearly positive (far from the critical line) while still having strong pairwise correlations, the claim that real networks sit near this critical boundary would be falsified; an even sharper test is to compare the model's predicted third and fourth cumulants of $P(\varphi)$ with direct measurements, since the mean-field saddle point makes definite predictions for these.

Watch

Extended reading notes

Core claim

The central claim is that a generalized mean-field theory—a maximum-entropy model with energy $E_\mathrm{dist}(s)=-\sum_n h_n s_n + N U(\varphi)$, where $\varphi=\sum_n W_n s_n$ is a single projection—can describe real neural population activity, whereas naive mean-field models cannot. Matching only the mean and variance of the summed activity forces the model to a first-order transition and predicts a bimodal activity distribution that the data do not show; matching variances along projections with the largest correlation-matrix eigenvalues runs into the same degeneracy. By instead matching the full distribution $P(\varphi)$ along the leading eigenvector projection, the potential $U(\varphi)$ can be read off from the data via a saddle-point calculation, and it departs significantly from quadratic. Monte Carlo simulation of the resulting model reproduces the projected activity distribution and each neuron's mean activity to good approximation. For the hippocampal population studied, $N U''(\varphi_\mathrm{sp})+\chi_0^{-1}\approx 0$, placing the network on the critical line of a second-order transition; scrambling time bins weakens correlations and moves the system away from this line.

Load-bearing premise

The conclusion that real networks are near-critical depends on treating independently shuffled time bins as a faithful way to weaken correlations while preserving everything else; if shuffling also changes single-neuron statistics or higher-order structure in ways the model does not control, the observed drift away from the critical line in Fig. 4B would not measure distance from criticality.

Editorial extensions

If this is right

Networks of real neurons are not in the regime where naive mean-field approximations hold; matching only low-order moments of collective variables is insufficient.
The distribution-constrained model reproduces the projected activity distribution and per-neuron mean activities, so it provides a consistent, parameter-free description once $U(\varphi)$ is read off from data.
Neural populations appear poised near a second-order critical point, as measured by the distance $NU''(\varphi_\mathrm{sp})+\chi_0^{-1}$ from the critical line.
Weakening correlations by shuffling time bins moves the inferred system away from criticality, implying the near-critical placement is tied to the real correlation structure.
Matching the distribution along a single projection reduces the entropy by roughly 5% of the independent entropy, and about 200 projections each contribute above the per-neuron independent entropy, motivating a multi-projection extension with $O(N)$ constraints.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the one-projection model is right, the dominant eigenvector of the correlation matrix defines a single collective variable that summarizes much of the population's structure; this could be tested by asking whether other projections become redundant for predicting higher-order statistics.
The shuffling proxy suggests a concrete experimental test: interventions that reduce pairwise correlations (e.g., by decoupling inputs) should systematically increase the distance from the critical line in the $U''$–$\chi_0^{-1}$ plane.
The success of a one-dimensional latent-field model is consistent with a low-dimensional dynamical manifold; an extension to several projections could reveal whether additional collective modes are needed to capture the remaining entropy, which would sharpen the picture of criticality in the brain.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Here's my read on arXiv:2504.15197. The genuinely new piece is the maximum entropy model that matches the full distribution of activity along a single projection, with the potential U(φ) read off from the data via a saddle-point argument. That construction is a real extension of earlier population and projection models, and the negative result on naive mean-field is solid: the exact integral representation in Eq (11) shows why the naive self-consistent equations fail, and the bimodality of the free energy in Fig 2 is a clear qualitative failure. The Monte Carlo check in Fig 3B is also a genuine sanity check, even if it is a consistency check rather than an independent prediction, since U(φ) is fit to the same distribution.

My main worry is the criticality claim. The paper argues that shuffling time bins weakens correlations and that the model moves away from the critical line in Fig 4B. But the construction is fully repeated on each shuffled dataset, including re-estimating the projection as the top eigenvector of the shuffled correlation matrix. As correlations weaken, that eigenvector becomes increasingly dominated by sampling noise, and Fig 4A shows that random projections sit far from the critical line. So the observed trajectory in Fig 4B could reflect a degrading projection rather than a physical change in proximity to criticality. That is not a controlled comparison. On top of that, there are no error bars anywhere in Fig 4, so the separation you see is not quantified.

There are smaller soft spots. The model in Eq (10) is fit to P_exp(φ), so the reproduction of P(φ) is a consistency check, not a prediction; the paper is honest about this ("at least consistent"). Some derivations are deferred to a companion paper [24], which makes it harder to check details. And no code is released.

That said, the core mean-field construction appears sound and is an interesting step toward O(N) constraints for large populations. The criticality observation is suggestive but should be treated with caution until the shuffle control is cleaned up. I would send this to peer review. The referee should ask for a proper control (e.g., fixing the projection direction from the original data and only shuffling the activity, or comparing to random projections with matching eigenvalue statistics), plus error bars.

Who is this for? People working on maximum entropy models of neural data and mean-field approaches to collective variables. It's a short, idea-rich paper, and even if the criticality claim doesn't fully hold up, the mean-field construction is worth having in the literature.

Best, [Name]

Referee Report

3 major / 4 minor

Summary. The paper studies mean-field descriptions of binary neural population activity. It first shows that the naive mean-field model, obtained by matching only the mean and variance of the summed population activity (Eq 7), fails against real data: measured susceptibilities exceed the naive bound (Fig 1), and the exact solution develops two nearly degenerate free-energy minima that would predict bimodal activity, contradicting the data (Fig 2). The authors then propose a maximum-entropy model (Eq 10) that matches each neuron's mean activity and the full distribution of activity along one projection. Using a saddle-point approximation, they derive a relation that lets them read off the effective potential U(phi) from the empirical projection distribution, and they verify by Monte Carlo that the model reproduces that distribution and the per-neuron means (Fig 3). Finally, they derive a criticality condition (Eq 32) and present evidence, including a time-bin shuffling control, that real hippocampal networks lie near this critical line (Fig 4). The central claims are that naive mean-field theories fail, that the extended distribution-matching theory is consistent with real data, and that the inferred potentials place real networks near criticality.

Significance. If fully supported, the paper would provide a practical path to maximum-entropy models with O(N) constraints for very large neural recordings, and would add quantitative evidence for criticality in hippocampal population activity. The paper has clear strengths: the exact integral representation of the population model (Eq 11), the saddle-point derivation in Eqs (22)-(26), and the crisp demonstration that the naive model fails not only in its mean-field limit but also through its qualitatively wrong bimodal prediction. The Monte Carlo check in Fig 3 is a useful sanity check. However, the validation of the central positive claim is partially circular, and the criticality evidence relies on a shuffle control that confounds weakened correlations with re-estimated projection directions and on point estimates without error bars. These issues are addressable in revision and do not invalidate the core construction, but they currently prevent the paper from fully establishing its headline conclusions.

major comments (3)

[Distributions of projections, Eq (10), Fig 3B] The Monte Carlo validation is a consistency check rather than an independent test. The potential U(phi) is read off from the empirical distribution P_exp(phi) via Eqs (30)-(31), and the fields h_n are set to atanh(<s_n>_exp) in Eq (28), so the model is forced to reproduce the per-neuron means and the projection distribution. Reproducing these same quantities in the Monte Carlo, as in Fig 3B, therefore does not demonstrate that real networks lie in the regime where the mean-field approximation has predictive power. To support the claim, the authors should test statistics that were not used to set the parameters, for example the pairwise correlations <s_n s_m>, the distribution along a second projection different from the one used to construct the model, or higher-order moments of the projection. Please add such a test or soften the claim about the validity of the approximation.
[Fig 4B and text on shuffling a fraction of time bins] The shuffle control confounds weakened correlations with re-estimated projection directions. The text says that for each shuffled data set the construction is repeated, which means the projection W is recomputed as the leading eigenvector of the shuffled correlation matrix via Eq (19). As correlations weaken, this eigenvector degrades toward a sampling-noise-dominated direction, and Fig 4A shows that random projections sit far from the critical line. The observed separation from criticality in Fig 4B could therefore reflect the changing projection rather than a changing physical distance from criticality. The paper should validate the shuffle proxy, for instance by fixing W at the real-data projection and shuffling only the activity, or by using surrogate data with controlled pairwise correlations and known projection directions. Without such a control, the abstract's 'signs of criticality' are not established.
[Figs 1 and 4, including Eq (16) boundary] The key quantitative comparisons lack error bars. In Fig 1, the claim that real neurons exceed the naive mean-field bound relies on point estimates of mu and chi for each data set; sampling variability over the finite recording time is not shown. In Fig 4, the distances from the critical line NU''+chi0^-1=0 and the separation between real and shuffled trajectories are not quantified. Bootstrap over time bins (or equivalent) is needed to assess whether the data actually sit near the critical line and whether the shuffle control produces a statistically significant movement. This is directly load-bearing for the near-criticality claim.

minor comments (4)

[Fig 2 caption and axis labels] The axis labels in Fig 2 appear garbled in the manuscript text (the string 'omega_exp epsilon h omega vartheta' is repeated); please ensure the printed figure has clear, readable labels for the free energy f(psi) and the auxiliary field psi.
[Eq (16)] Equation (16) has a 0/0 limit at mu=0; please state the limiting value chi_max(0) and specify the domain of mu for which the expression is meaningful.
[Shuffle procedure description] The shuffle procedure is described only verbally. Please specify the number of shuffles, whether the fraction of scrambled time bins is applied per neuron independently with replacement, and how the leading eigenvector is computed for each shuffled data set.
[Abstract wording] The phrase 'at least consistent' in the abstract is vague. Please specify whether consistency means agreement within sampling error with the constrained statistics or agreement with unconstrained observables as well.

Circularity Check

1 steps flagged · score 4.0 of 10

The Fig 3B 'reproduction' of P(φ) is a consistency check because U(φ) is read off from Pexp(φ); the criticality finding has independent content, but the shuffle control is an unvalidated proxy with no error bars.

fitted input called prediction [Section 'Distributions of projections'; Eqs (10), (28), (30); Fig 3B]
"If we substitute into Eq (30) we see that all terms except a factor e^{−NU(φ)} are determined, so we can “read off” U(φ) from the measured distribution of activity Pexp(φ) along the projection. ... Figure 3B shows that we correctly reproduce the distribution of activity along the projection; the inset shows that we recover the correct mean activities as well."

By Eq (10), U(φ) is defined as the potential that makes the model match the observed projected-activity distribution: h_n are fixed by Eq (28) and then U is read off from Pexp(φ) via the saddle-point expression Eq (30), so in that approximation P_model(φ) = Pexp(φ) by construction. The Monte Carlo simulation of Eq (10) therefore tests the saddle-point/numerical implementation, but the agreement in Fig 3B is not an independent prediction of the projected-activity distribution. The mean activities are also close to being enforced by the gauge choice and Eq (28), though the full model can shift them slightly.

full rationale

The paper has substantial independent content: the failure of the naive population model (Fig 1) and of the projection model (Eq 9) are falsifiable results obtained by fitting low-order statistics and then checking higher-order structure; the near-critical condition NU''(φsp)+χ0^{−1}=0 is derived, not imposed, and evaluating it on the fitted potential yields a nontrivial finding; and the entropy reductions in Fig 5 are consequences of the construction rather than fits to those entropy values. The genuinely circular component is the validation of the distribution-constrained model: since U(φ) is read off from Pexp(φ), the Monte Carlo 'reproduction' of P(φ) in Fig 3B is a self-consistency check of the saddle-point approximation rather than an independent prediction. That check does have independent content — the full Monte Carlo could disagree with the saddle-point read-off — but the agreement on the projected distribution is built into the choice of U. The shuffle control in Fig 4B is a robustness concern (the projection is re-estimated on each shuffled set and no error bars are shown), but it is not a circular reduction: the control does not define its own outcome. The in-preparation self-citation [24] is used for the careful numerical construction of h*(λ) and for a fuller account; this is missing support rather than load-bearing circularity, since the essential equations are stated in the text. Overall score 4: partial circularity in the 'reproduction' claim, while the criticality and failure-of-naive-models findings remain independent.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central construction fits one field per neuron and a full potential function to the data; the only genuinely emergent, non-fitted quantity is the proximity of the inferred curvature to the critical line, which is a derived property of the fitted potential.

free parameters (4)

per-neuron fields h_n = atanh(<s_n>_exp), one field per neuron (N=1416 for hippocampus)
In Eq (10), the fields are set to reproduce the measured mean activity of each neuron; all model predictions, including the criticality measure, depend on these fitted values.
effective potential U(phi) = Read off from measured P_exp(phi) via Eqs (30)-(31); shown in Fig 3A
U(phi) is chosen to match the observed distribution of activity along the selected projection. It is a fitted function, not a theoretical prediction, and its curvature determines the proximity to the critical line.
projection coefficients W_n = Normalized leading eigenvector of the measured correlation matrix (Eqs 17-19)
The choice of projection defines the collective variable whose distribution is matched; the paper selects the maximum-eigenvalue mode for the main consistency check.
population-model fields h and lambda = Values matching measured mu_exp and chi_exp (Fig 2 inset)
Used to show that the naive mean-field population model develops two nearly degenerate minima; this demonstration is central to the paper's negative result, although the model itself is rejected.

assumptions (4)

domain assumption The saddle-point (Laplace) approximation is accurate at the finite population sizes used.
Eqs (26) and (30) discard O(1/N) terms; the paper validates this with Monte Carlo for one projection and one dataset, but the entropy expression Eq (33) and criticality analysis rely on the approximation throughout.
domain assumption Measured means and distributions are faithful estimates of the underlying stationary distribution.
The analysis treats <s_n>_exp and P_exp(phi) as ground truth; finite recording duration and sampling noise are only partially represented (grey bands in Fig 3B).
domain assumption Maximum entropy with the chosen constraints is the right model class for neural population activity.
This is the field-standard Jaynes construction used in Eqs (1)-(10); the paper does not question it.
ad hoc to paper Shuffling a fraction of time bins independently for each neuron produces a valid proxy for weaker correlations.
Fig 4B uses this surrogate to argue real networks are closer to criticality; the assumption is introduced for this analysis and not independently justified.

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Cite this review

Pith. "Pith review of Extended mean-field theories for networks of real neurons." pith.science (2026). https://pith.science/paper/JII4SKOA

@misc{pith2026250415197,
  author       = {Pith},
  title        = {Pith review of: Extended mean-field theories for networks of real neurons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JII4SKOA}},
  note         = {Machine review of arXiv:2504.15197}
}

read the original 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.

Figures

Figures reproduced from arXiv: 2504.15197 by the authors.

**Figure 3.** FIG. 3: Maximum entropy model that matches the mean ac [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** FIG. 5: Difference between the entropy of the independent [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: Approach to a critical point in matching the distri [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Reference graph

Works this paper leans on

30 extracted references · 27 canonical work pages · cited by 1 Pith paper

[24]

Di Carlo, F

L. Di Carlo, F. Mignacco, C. W. Lynn, and W. Bialek (2025), in preparation

work page 2025
[1]

Meshulam and W

L. Meshulam and W. Bialek, arXiv preprint arXiv:2409.00412 (2024). 6

arXiv 2024
[2]

J. J. Hopfield, Proceedings of the National Academy of Sciences (USA) 79, 2554 (1982)

work page 1982
[3]

D. H. Ackley, G. E. Hinton, and T. J. Sejnowski, Cogni- tive Science 9, 147 (1985)

work page 1985
[4]

J. J. Hopfield and D. W. Tank, Biological Cybernetics 52, 141 (1985)

work page 1985
[5]

J. J. Hopfield and D. W. Tank, Science 233, 625 (1986)

work page 1986
[6]

Schneidman, M

E. Schneidman, M. J. Berry II, R. Segev, and W. Bialek, Nature 440, 1007 (2006)

work page 2006
[7]

E. T. Jaynes, Physical Review 106, 620 (1957)

work page 1957

Show all 30 references

[8]

E. T. Jaynes, Proceedings of the IEEE 70, 939 (1982)

1982
[9]

Bialek, A

W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Sil- vestri, M. Viale, and A. M. Walczak, Proceedings of the National Academy of Sciences (USA) 109, 4786 (2012)

2012
[10]

Cavagna, L

A. Cavagna, L. Del Castello, S. Dey, I. Giardina, S. Melillo, L. Parisi, and M. Viale, Physical Review E 92, 012705 (2015)

2015
[11]

M. Okun, P. Yger, S. L. Marguet, F. Gerard-Mercier, A. Benucci, S. Katzner, L. Busse, M. Carandini, and K. D. Harris, Journal of Neuroscience 32, 17108 (2012)

2012
[12]

Tkaˇ cik, O

G. Tkaˇ cik, O. Marre, T. Mora, D. Amodei, M. J. Berry II, and W. Bialek, Journal of Statistical Mechanics Theory and Experiment 2013(3), P03011 (2013)

2013
[13]

Parisi, Statistical Field Theory (Frontiers in Physics, Addison-Wesley, 1988)

G. Parisi, Statistical Field Theory (Frontiers in Physics, Addison-Wesley, 1988)

1988
[14]

Sethna, Statistical Mechanics: Entropy, Order Param- eters, and Complexity (Oxford University Press, Oxford, 2021)

J. Sethna, Statistical Mechanics: Entropy, Order Param- eters, and Complexity (Oxford University Press, Oxford, 2021)

2021
[15]

S. A. Kivelson, J. M. Jiang, and J. Chang, Statistical Mechanics of Phases and Phase Transitions (Princeton University Press, Princeton, 2024)

2024
[16]

Sahani, Ph.D

M. Sahani, Ph.D. thesis, California Institute of Technol- ogy, Pasdena, CA (1999)

1999
[17]

M. R. Whiteway and D. A. Butts, Current Opinion in Neurobiology 58, 86 (2019)

2019
[18]

J. P. Cunningham and B. M. Yu, Nature Neuroscience 17, 1500 (2014)

2014
[19]

J. A. Gallego, M. G. Perich, L. E. Miller, and S. A. Solla, Neuron 94, 978 (2017)

2017
[20]

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 595, 80 (2021)

2021
[21]

Cocco, R

S. Cocco, R. Monasson, and V. Sessak, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 83 (2011)

2011
[22]

D. J. Amit, H. Gutfreund, and H. Sompolinsky, Annals of Physics 173, 30 (1987)

1987
[23]

Krotov and J

D. Krotov and J. J. Hopfield, in Advances in Neu- ral Information Processing Systems , edited by D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett (Curran Associates, Inc., 2016), vol. 29, pp. 1172–1180

2016
[25]

Tkaˇ cik, O

G. Tkaˇ cik, O. Marre, D. Amodei, E. Schneidman, W. Bialek, and M. J. Berry, PLoS Computational Bi- ology 10, e1003408 (2014)

2014
[26]

Allen Institute MindScope Program Allen Brain Ob- servatory, Neuropixels visual coding (dataset) 2019 , https://brain-map.org/explore/circuits

2019
[27]

J. L. Gauthier and D. W. Tank, Neuron 99, 179 (2018)

2018
[28]

Meshulam, J

L. Meshulam, J. L. Gauthier, C. D. Brody, D. W. Tank, and W. Bialek, Physical Review Letters 123, 178103 (2019)

2019
[29]

Meshulam, J

L. Meshulam, J. L. Gauthier, C. D. Brody, D. W. Tank, and W. Bialek, arXiv preprint arXiv:2112.14735 (2021)

2021 arXiv
[30]

C. W. Lynn, Q. Yu, R. Pang, S. E. Palmer, and W. Bialek (2023), URL http://arxiv.org/abs/2402.00007

2023 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [24]

Di Carlo, F

L. Di Carlo, F. Mignacco, C. W. Lynn, and W. Bialek (2025), in preparation

work page 2025

[2] [1]

Meshulam and W

L. Meshulam and W. Bialek, arXiv preprint arXiv:2409.00412 (2024). 6

arXiv 2024

[3] [2]

J. J. Hopfield, Proceedings of the National Academy of Sciences (USA) 79, 2554 (1982)

work page 1982

[4] [3]

D. H. Ackley, G. E. Hinton, and T. J. Sejnowski, Cogni- tive Science 9, 147 (1985)

work page 1985

[5] [4]

J. J. Hopfield and D. W. Tank, Biological Cybernetics 52, 141 (1985)

work page 1985

[6] [5]

J. J. Hopfield and D. W. Tank, Science 233, 625 (1986)

work page 1986

[7] [6]

Schneidman, M

E. Schneidman, M. J. Berry II, R. Segev, and W. Bialek, Nature 440, 1007 (2006)

work page 2006

[8] [7]

E. T. Jaynes, Physical Review 106, 620 (1957)

work page 1957