{"id":"7f1087b5-b907-4416-84ff-b8ae14b62f32","arxiv_id":"2411.14196","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"A moment neural network trained only on the mean firing rate learns to use firing covariance as an unsupervised confidence signal, matching human-level uncertainty reports in a working memory task.","lead":"Working memory is the brain's scratchpad, and people usually know when their memory is shaky. This paper trains a brain-inspired model so that the model's own correlated neural fluctuations, never explicitly trained, become a reliable confidence signal, and the same weights work in a spiking network.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central quantitative claim rests on Eq. (18), which is not a correlation coefficient; until the reported ρ values are recomputed with a normalized measure, 'comparable to human UQ' is unsupported.","rationale":"I read the paper in good faith: the central claim is that a mean-supervised MNN develops an unsupervised covariance signal that tracks its own memory errors with human-comparable reliability, and that this transfers to an SNN. I first considered the reader's weakest assumption, the fidelity of the moment closure at g=3, N=200. That is a real concern because the SNN check in Section 7 is only qualitative. However, Eq. (18) is more immediately load-bearing: every quantitative statement in the paper that the MNN 'captures uncertainty' or is 'comparable to humans' is expressed through this ρ. As written, Eq. (18) is a covariance, not a correlation, so the reported magnitudes are not comparable to the human Pearson correlations cited from Li et al. This is a concrete, fixable issue rather than a fatal flaw, and the central idea may survive a corrected recomputation. I therefore keep the reader's CONDITIONAL verdict unchanged. My agreement is partial because the reader's weakest_assumption was the moment-closure fidelity, whereas I would prioritize the invalid correlation metric as the most load-bearing concern, with the DCR index error as a secondary reproducibility gap.","tokens_in":20449,"tokens_out":3936,"duration_ms":38372,"concrete_test":"Obtain the saved network states or code and recompute Fig. 3e (and the SNN Fig. 5d) using the standard Pearson correlation coefficient, ρ_Pearson = [Σ(κ_i−κ̄)(e_i−ē)] / [sqrt(Σ(κ_i−κ̄)^2) sqrt(Σ(e_i−ē)^2)], with 95% bootstrap confidence intervals. If the normalized correlations remain in the 0.6–0.7 range and the w/ corr vs w/o corr contrast survives, the central claim stands; if they drop or change sign, the comparison to humans and the covariance-based UQ mechanism must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is not the moment closure itself but the metric used to validate the central claim. Methods Eq. (18) defines ρ = (1/L)Σ κ_i e_i − (1/L^2)(Σκ_i)(Σ e_i). This is a sample covariance, not a correlation: it is not normalized by the standard deviations of κ and e, is not dimensionless, and its magnitude depends on the arbitrary scaling of the uncertainty indicators I–IV (determinants and log-determinants of covariance matrices). Every headline number in Figs. 3e, 4b, 4e, 5d–f, and the parameter sweeps in Figs. 6–7 is this ρ. Human performance is quoted as a Pearson correlation of 0.6–0.7 in Li et al.; comparing an unnormalized covariance to a Pearson coefficient is not meaningful. The ablation (w/ corr vs w/o corr) could likewise be inflated or reversed solely by changes in scaling or variance rather than by true association. A second reproducibility gap compounds this: the DCR in Eq. (19) has undefined indices and repeated terms, so the differential-covariance mechanism cannot be checked as written. None of this proves the model's UQ idea is wrong; it means the paper's central quantitative evidence, as written, does not yet establish the claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper trains moment neural networks (MNNs) on a continuous working memory task using a reservoir-computing/low-rank training scheme, with a loss that supervises only the decoded mean. The main claim is that, although covariance is not supervised, the MNN's firing covariance state is nevertheless positively related to trial-to-trial memory error, and that this relationship is comparable to human UQ performance. The paper also claims a mechanistic link between bump width (probabilistic population coding) and covariance-based uncertainty (sampling-based coding) through a differential covariance ratio (DCR), and transfers the trained MNN weights to an LIF spiking network to show that the UQ phenomenon persists. Supplementary theorems S1-S2 are intended to show that supervised mean training can, in an idealized scalar setting, also match or bound the output variance.","tokens_in":20706,"tokens_out":4465,"duration_ms":34473,"significance":"If the central empirical claim held as stated, the paper would be a valuable contribution: it offers a biologically motivated MNN framework in which a purely mean-supervised training loss yields an emergent, covariance-based uncertainty signal that transfers to a spiking implementation, plus falsifiable predictions about decorrelation, population size, noise, and heterogeneity. The use of a parametric moment closure derived from LIF neurons, rather than a hand-designed rate model, is a genuine strength, as is the explicit attempt to verify the MNN against an SNN by weight transfer. The theoretical supplement also attempts a nontrivial statement (variance error bounded by mean loss under noise) rather than merely asserting empirical success.","major_comments":[{"comment":"Eq. (18) defines rho as (1/L) sum kappa_i e_i - (1/L^2)(sum kappa_i)(sum e_i). This is a sample covariance, not a correlation coefficient: it is not normalized by the standard deviations of kappa and e, is not dimensionless, and changes magnitude under any positive rescaling of the uncertainty indicators kappa_I through kappa_IV. The headline comparisons to \"human correlation coefficients of 0.6-0.7\" (Sections 3, 5, 9 and Fig. 3e) are therefore not valid as written, since the human benchmarks in Li et al. are Pearson correlations. All figures reporting \"the correlation between uncertainty and error\" or \"correlation coefficient rho\" (Figs. 3e, 4b, 4e, 5d-f, 6a,d,g, 7a,d,g, and the corresponding Supplementary figures) use Eq. (18). The authors should recompute all reported rho values with a normalized measure (e.g., Pearson or Spearman correlation) and either replace the human comparison or explicitly acknowledge that only the sign and relative ordering of the unnormalized quantity are being reported.","section":"Methods, Eq. (18)"},{"comment":"The differential covariance ratio in Eq. (19) is not well defined as written: the indices i, 2N - j, and 2N - i are inconsistent with C being an N x N matrix, and the term C_{i,2N-j} appears twice with no corresponding correct index pattern. Moreover, the claimed sign structure in Eq. (20) is stated as an axiom (\"the differential covariance induced by the drift of the bump exhibits a distinct pattern\") rather than derived from the trained dynamics. Since DCR is used as evidence for the proposed mechanism in Fig. 4e and Fig. 5f, the authors should correct the index definitions, verify the sign pattern empirically from the trained MNN/SNN covariances, and confirm that the reported DCR values are stable under the corrected definition.","section":"Methods, Eq. (19)"},{"comment":"The proof of Theorem S2 contains several inconsistencies that prevent the bound in Eq. (S17) from being checked as written: Eq. (S18) and the surrounding text use m* - y* where the intended argument appears to need m* - m, and the line \"combine with Eq. (S19)\" leading to Eq. (S22) does not follow from the stated Harnack inequality, which is applied at a single point and does not yield the claimed integral inequality. In addition, Theorem S2 is stated for a scalar output while the MNN's decoded uncertainty is a 2x2 covariance; the paper asserts without proof that the result extends to high dimensions. The authors should rewrite the proof with consistent notation, supply the missing justifications for the Harnack step and the extension to vector-valued outputs, or clearly mark this part as a heuristic argument rather than a proved theorem.","section":"S1, Theorem S2"},{"comment":"The ablation that clamps all off-diagonal correlations to zero is reported as a \"significant decrease\" in rho, but with Eq. (18) as the metric, a decrease could be driven purely by a change in the marginal variance of the uncertainty indicators rather than by a change in the strength of association. The claim that \"correlation between neural activities is crucial for UQ\" (Section 5 and Discussion) therefore needs to be re-established with a normalized correlation measure, or with an explicit demonstration that the variance of kappa is not the dominant driver of the reported difference.","section":"Section 5, Fig. 3e and ablation"},{"comment":"The SNN verification is qualitative: Fig. 5 reports that the SNN reproduces the patterns of the MNN, but the paper does not provide quantitative agreement metrics (e.g., how close the SNN's steady-state mean/covariance are to the MNN's fixed points, or how the SNN's rho values and error levels compare numerically with the MNN's). Since the moment closure in Eq. (4) is the load-bearing bridge between the MNN and the LIF SNN, the authors should add quantitative comparisons (e.g., normalized covariance error, rho values, error distributions) or acknowledge more explicitly that the SNN verification only establishes qualitative consistency.","section":"Section 7 and Methods (SNN verification)"},{"comment":"The claim that the theorems in the Supplement support the MNN's UQ mechanism is overstated. The theorems concern an abstract scalar system with input noise and do not involve recurrent dynamics, a ring manifold, a trained weight matrix, or the specific MNN moment closure. For example, the MNN's covariance is not a simple function of input noise; it is shaped by the recurrent structure. The authors should either connect the theorems to the MNN architecture explicitly, or soften the claims in Section 6 and Section 8 that these theorems explain the empirical UQ results.","section":"Section 6 and Supplementary S1"}],"minor_comments":[{"comment":"The paper reports all rho values only as qualitative statements (\"strongly positively correlated\", \"comparable\", \"significant decrease\") without error bars, confidence intervals, or numerical values in the main text. Given that each test is repeated 10 times, reporting the mean and standard deviation of rho (or a bootstrap CI) for the key figures would substantially strengthen the claims.","section":"Throughout"},{"comment":"The description of the human experiment in Li et al. states that participants \"first being presented with a cue\" and later \"report the remembered feature and indicate their uncertainty by adjusting the length of an arc\". It would be helpful to clarify that the human UQ measure is a confidence-interval report, not a confidence rating, and to state the exact human correlation value and its error bar as reported by Li et al.","section":"Section 2"},{"comment":"The moment activation functions in Eq. (4) include the term (∂phi_mu/∂bar_mu)_{ii}(∂phi_mu/∂bar_mu)_{jj} for the off-diagonal covariance, but the notation is not fully defined: the partial derivative of phi_mu with respect to the vector bar_mu is a Jacobian, and the meaning of the double index is unclear. The authors should define this derivative explicitly.","section":"Methods, Eq. (4)"},{"comment":"The inference step sets sigma_xi ~ Unif[0,1] independently per instance, but the paper does not state how this noise is incorporated into Eq. (14) (it appears only in the text as \"adding external noise ... with sigma_xi\"). The precise dependence of bar_mu on xi_t should be written out, since the uncertainty indicators are functions of the covariance only and the role of this input noise is otherwise unexplained.","section":"Methods, Network inference"},{"comment":"Theorem S1 is stated for a loss that vanishes exactly (L = 0), but the paper later acknowledges that \"minimizing the training loss to zero is impossible\" and Theorem S2 provides the approximate version. The logical relationship between S1 and S2 should be stated more explicitly (S1 as an idealized exact case, S2 as the practical bound).","section":"Supplementary S1"},{"comment":"The claim that \"the precision of our model also reaches comparable level reported in [18, 40, 14]\" is not supported by any numerical comparison of error distributions or psychophysical metrics. Either provide the quantitative comparison or soften the sentence.","section":"Section 5 and Fig. 3"}],"recommendation":"major_revision","confidential_remarks":"The paper's core idea (unsupervised covariance as an emergent UQ signal in an MNN) is interesting and the MNN-to-SNN transfer is a useful methodological contribution. However, the central quantitative evidence is currently built on an unnormalized covariance that is compared to human Pearson correlations, and the DCR definition is not checkable as written. These are fixable within the paper's scope, but they are load-bearing issues that affect every headline figure. I would not recommend reject: the empirical phenomenon may well survive re-analysis with a normalized correlation, and the MNN framework plus weight transfer is a genuine contribution. The main risk is that once rho is normalized, the \"comparable to human\" claim and the ablation effect may change in magnitude."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper has a genuinely interesting core idea—train a moment neural network on a working memory task with a loss on the mean only, and the covariance ends up tracking the network's own memory errors. That is worth reading about. The second thing: the quantitative evidence for it, as written, is not valid. The paper's Eq. (18) defines rho as a sample covariance, not a correlation coefficient, and every headline number in Figs 3e, 4b, 5d-f, and the parameter sweeps is this rho. Comparing that to the human Pearson correlation of 0.6-0.7 is apples-to-oranges.\n\nWhat is genuinely valuable: the observation that covariance emerges without any uncertainty supervision, the four uncertainty metrics, the ablation that removes correlations, and the transfer to a spiking network. The theoretical theorems in the supplement are close to formal identities once you assume a perfect mean fit, but they do show a route from mean supervision to variance learning. The testable predictions about noise and heterogeneity are concrete. Credit where due: the authors are not fitting the covariance to error; the correlation is an emergent property, so the circularity concern is mild.\n\nNow the soft spots, in order of severity. First, Eq. (18) must be replaced by a normalized correlation coefficient (or the unnormalized values must not be compared to human Pearson values). Without that, the central claim is unsupported. Second, Eq. (19) the DCR has a repeated term and invalid indices; as written it is undefined. That's a reproducibility issue for the mechanism claim. Third, no code or data is released, so the reported numbers cannot be checked; given the metric problem, this is more than a formality. Fourth, the SNN validation is qualitative—the authors say 'comparable' but give no quantitative comparison. Fifth, the supplementary theorems have notational slips (e.g., y* in the derivative of the mean loss) that make the proofs hard to trust. These are all fixable, but the paper as submitted does not support its own headline.\n\nWho is this for? Computational neuroscientists working on uncertainty representation and WM models. A reader should take the qualitative story seriously and the numbers with a grain of salt. I would not cite the quantitative claims until fixed.\n\nRecommendation: send it to peer review. The idea is significant and the flaws are repairable; a good referee can separate the signal from the noise. But the paper should not be accepted in this form, and ideally the code/data should be posted before or during revision.","headline":"Interesting idea, broken evaluation metric: the paper's claim of human-comparable UQ rests on an unnormalized covariance written as a correlation coefficient.","tokens_in":21289,"tokens_out":3170,"would_cite":false,"duration_ms":29211,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A recurrent circuit trained to report a remembered feature in its mean firing rate spontaneously develops a firing-covariance readout that tracks its own memory error, about as well as human confidence reports do, and the signal transfers…","keywords":["uncertainty quantification","working memory","moment neural network","firing covariance","differential covariance","spiking neural network","reservoir computing","confidence"],"falsifier":"Simulate the full LIF spiking network with the transferred weights at $g=3$, $N=200$ across many trials, estimate spike-count covariance directly, and compare its leading eigenvectors and the decoded uncertainty-error correlation with the MNN's predictions; if the spiking covariance's informative component is much weaker or differently oriented than predicted, the uncertainty readout is an artifact of the moment closure. A complementary check is to decorrelate the recorded spike counts (clamp off-diagonal covariances to zero) and see whether the uncertainty-error correlation vanishes, as it does in the MNN's ablation.","tokens_in":20159,"feed_emoji":"🧠","tokens_out":8566,"duration_ms":74748,"temperature":0.7,"pith_summary":"Working memory is error-prone, yet people have a calibrated sense of how wrong their memory is; this paper asks what neural signal carries that sense. The authors use a moment neural network (MNN), which tracks both the mean firing rate and the pairwise firing covariance of a leaky integrate-and-fire population, and train it with a loss that supervises only the mean. They find that the covariance, through its nonlinear coupling to the mean, develops into a readout of the network's own error: larger decoded covariance means larger memory errors, at a correlation comparable to the human confidence-error correlation in the task of ref. [14]. Transferring the trained weights to a spiking neural network reproduces the effect, and the analysis ties the signal to a specific covariance structure (differential covariance) induced by random drift of the memory bump. The paper's broader claim is that noise and heterogeneity are not liabilities but necessary ingredients of this uncertainty mechanism.","feed_headline":"Firing variability flags working-memory mistakes","feed_subtitle":"Mean-only training still yields a covariance signal that tracks memory error, matching humans and surviving in spiking circuits.","key_machinery":"The central object is the moment neural network, a pair of coupled equations (Eq. (2)) for the mean firing rate $\\mu$ and firing covariance $C$, with moment activations $\\phi_\\mu$ and $\\phi_C$ (Eq. (4)) derived from the diffusion approximation of leaky integrate-and-fire neurons; these activations define how input-current statistics (mean $\\bar{\\mu} = W\\mu + \\mu_s$, covariance $\\bar{C} = W C W^{\\top} + \\sigma_s^2 I$) map to output spike statistics. The training loss (Eq. (13)) supervises only $W_{\\mathrm{out}}^{\\top}\\mu$ against the encoded feature, leaving $C$ unsupervised. Two auxiliary constructs carry the argument: the decoded covariance $\\hat{C}_z = W_{\\mathrm{out}}^{\\top} C W_{\\mathrm{out}}$, from which four entropy-based uncertainty metrics are computed, and the differential covariance ratio (DCR, Eq. (19)), which isolates the covariance component parallel to the tuning-curve derivative that shrinks coding information. The paper's supplementary theorems (Thm. S1, Thm. S2) use the backward uniqueness property of parabolic equations to show, in a simplified scalar system, that a mean-only loss can force the model to learn the ground-truth output variance, and that variance error is bounded by mean loss under higher training noise.","core_discovery":"The central claim is that uncertainty quantification in working memory can be accomplished by firing covariance alone, without any supervised training signal for that covariance. After reservoir-computing training with the mean-only loss of Eq. (13), an MNN's fixed points show that trial-to-trial error in the decoded feature is positively correlated with uncertainty readouts computed from the decoded covariance $\\hat{C}_z = W_{\\mathrm{out}}^{\\top} C W_{\\mathrm{out}}$; the correlations are comparable to the 0.6–0.7 range reported for humans in ref. [14]. The mechanism proposed is that error arises from random drift of the bump, and this drift injects a differential covariance component into the neural covariance; a differential covariance ratio (DCR, Eq. (19)) constructed from the sign structure of the covariance tracks all four uncertainty metrics, and the bump width is simultaneously coupled to the covariance, linking probabilistic population coding to sampling-based coding. Clamping off-diagonal correlations to zero abolishes the uncertainty signal, showing that the correlations are causal. The same weights, run as a spiking LIF network, give qualitatively similar tuning, covariance patterns, error distributions, and uncertainty-error correlations, which the paper takes as evidence that the MNN is a faithful substitute for the SNN and that the UQ mechanism is spike-compatible.","pith_inferences":["Editorial inference: if covariance calibration emerges from mean-only error signals, then artificial systems could attach confidence to their outputs without ever supervising variance; the covariance of the last-layer activations after mean-squared-error training could be monitored as a free uncertainty signal.","Editorial inference: the differential-covariance story implies that any perturbation that drifts the memory bump, such as a distractor stimulus or a brief current injection, should inflate the decoded uncertainty readout even when the mean bump looks unchanged; a direct perturbation experiment could test this causality.","Editorial inference: the MNN-to-SNN transfer is qualitative, so a quantitative comparison of the spike-count covariance eigenvectors between the LIF simulation and the MNN's predicted covariance would sharpen the claim and could reveal the heterogeneity level at which the moment closure begins to break down."],"forward_implications":["A mean-only training objective is sufficient for a network to acquire a calibrated uncertainty readout: the covariance self-organizes through mean-covariance coupling rather than being fitted to an explicit variance loss.","Correlated variability is causally necessary for UQ in this circuit: when pairwise correlations are clamped to zero, the correlation between uncertainty and error drops sharply.","The mechanism is realizable in spike-based computation: weights trained in the differentiable MNN transfer to an LIF spiking network and preserve the uncertainty-error correlation, giving the paper's claimed first spike-based implementation of uncertainty quantification in working memory.","Noise and heterogeneity are functionally beneficial: increasing training noise, population size, and, within a range, heterogeneity improves both coding precision and UQ, which yields testable predictions about how individual differences in neural heterogeneity relate to confidence calibration."],"supporting_citations":[{"why":"Supplies the human working-memory task, the confidence-arc protocol, and the human uncertainty-error correlation (0.6–0.7) that the model's UQ performance is benchmarked against.","marker":"[14]"},{"why":"Introduces the moment neural network model that couples mean firing rate and firing covariance through moment activations.","marker":"[33]"},{"why":"Derives the LIF-based moment activation functions used in Eq. (4), the model's nonlinearity.","marker":"[64]"},{"why":"Gives the low-rank-plus-quenched-noise connection structure and reservoir-computing training procedure used to train the MNN.","marker":"[31]"},{"why":"Provides empirical evidence that differential covariance reduces coding precision in working-memory networks, which the paper's UQ mechanism builds on.","marker":"[34]"},{"why":"Earlier first-two-moment rate network whose weights cannot be transferred to an SNN, the contrast that motivates the direct MNN-to-SNN weight transfer.","marker":"[32]"},{"why":"Defines differential covariance as covariance along the tuning-curve derivative direction, the quantity the DCR indicator measures.","marker":"[60]"},{"why":"Supplies the backward-uniqueness result for parabolic equations used in Thm. S1 to show that mean matching at positive noise implies function matching at zero noise.","marker":"[66]"}],"fun_headline_variants":["Covariance signals flag working-memory errors","Noise boosts working memory, not a bug","Spiking network shows uncertainty via covariance","Moment neural nets trace memory uncertainty","Uncertainty readout from firing variability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the moment-closure equations (Eq. (2) and Eq. (4)), derived from the diffusion approximation of LIF neurons, faithfully reproduce the trial-to-trial firing covariance of the full spiking network at the trained heterogeneous regime ($g=3$, $N=200$), a premise checked only qualitatively by the SNN transfer in Section 7.","fun_headline_variants_meta":{"raw":{"variants":["Covariance signals flag working-memory errors","Noise boosts working memory, not a bug","Spiking network shows uncertainty via covariance","Moment neural nets trace memory uncertainty","Uncertainty readout from firing variability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000242,"raw_usage":{"total_tokens":1535,"prompt_tokens":962,"completion_tokens":573,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":507}},"tokens_in":578,"tokens_out":573,"duration_ms":5819,"temperature":1.0,"reasoning_tokens":507,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:26:47.878144+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the full LIF spiking network with the transferred weights at $g=3$, $N=200$ across many trials, estimate spike-count covariance directly, and compare its leading eigenvectors and the decoded uncertainty-error correlation with the MNN's predictions; if the spiking covariance's informative component is much weaker or differently oriented than predicted, the uncertainty readout is an artifact of the moment closure. A complementary check is to decorrelate the recorded spike counts (clamp off-diagonal covariances to zero) and see whether the uncertainty-error correlation vanishes, as it does in the MNN's ablation.","supporting_citations":[{"cited_title":"Joint representation of working memory and uncertainty in human cortex","cited_arxiv_id":null,"evidence_quote":"Supplies the human working-memory task, the confidence-arc protocol, and the human uncertainty-error correlation (0.6–0.7) that the model's UQ performance is benchmarked against."},{"cited_title":"On a gaussian neuronal field model","cited_arxiv_id":null,"evidence_quote":"Introduces the moment neural network model that couples mean firing rate and firing covariance through moment activations."},{"cited_title":"Dynamics of moment neuronal networks","cited_arxiv_id":null,"evidence_quote":"Derives the LIF-based moment activation functions used in Eq. (4), the model's nonlinearity."},{"cited_title":"Learning to represent continuous variables in heterogeneous neural networks","cited_arxiv_id":null,"evidence_quote":"Gives the low-rank-plus-quenched-noise connection structure and reservoir-computing training procedure used to train the MNN."},{"cited_title":"Self-organization of nonlinearly coupled neural fluctuations into synergistic population codes","cited_arxiv_id":null,"evidence_quote":"Provides empirical evidence that differential covariance reduces coding precision in working-memory networks, which the paper's UQ mechanism builds on."},{"cited_title":"Cortical-like dynamics in recurrent circuits optimized for sampling-based probabilistic inference","cited_arxiv_id":null,"evidence_quote":"Earlier first-two-moment rate network whose weights cannot be transferred to an SNN, the contrast that motivates the direct MNN-to-SNN weight transfer."},{"cited_title":"Correlations and neuronal population information","cited_arxiv_id":null,"evidence_quote":"Defines differential covariance as covariance along the tuning-curve derivative direction, the quantity the DCR indicator measures."},{"cited_title":"Backward uniqueness for general parabolic operators in the whole space","cited_arxiv_id":null,"evidence_quote":"Supplies the backward-uniqueness result for parabolic equations used in Thm. S1 to show that mean matching at positive noise implies function matching at zero noise."}],"review_version":1}