The Effect of Noise Correlations in Populations of Diversely Tuned Neurons

The Effect of Noise Correlations in Populations of Diversely Tuned Neurons Ecker Berens Tolias Bethge

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The Effect of Noise Correlations in Populations of Diversely Tuned Neurons 2011 Journal of Neuroscience 10.1523/jneurosci.2539-11.2011 Ecker2011-the-effect-of-noise-correlations-in-populati Ecker2011-the-effect-of-noise-correlations-in-populati.txt, the project's staged copy f05895962876c28f5afdc4529e63cdc672ac72a5 not-redistributable neuroscience

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them the belief that reducing noise correlations improves a population code - and the assumption of homogeneous tuning that the belief rests on Quotation withheld because this record is not cleared for redistribution. whether limited-range correlations are detrimental once the population is heterogeneous, large, and has stimulus-dependent variances Quotation withheld because this record is not cleared for redistribution. in heterogeneous populations of more than a few hundred neurons, INCREASING correlations substantially improves encoding accuracy - and with noise entropy held constant, the accuracy is independent of correlation structure and magnitude altogether Quotation withheld because this record is not cleared for redistribution. critique Quotation withheld because this record is not cleared for redistribution. noise correlations in neural population coding Quotation withheld because this record is not cleared for redistribution. simulation simulation Quotation withheld because this record is not cleared for redistribution. mechanisms Raising correlations redistributes noise power out of the high-frequency Fourier components, and because heterogeneous tuning spreads the stimulus signal into exactly those components, their signal-to-noise ratio rises Quotation withheld because this record is not cleared for redistribution. mechanisms Heterogeneous tuning distributes the encoded signal across all spatial frequencies, so the high-frequency channels favoured by strong correlations carry stimulus information Quotation withheld because this record is not cleared for redistribution. mechanisms Attention improves coding by suppressing a shared noise source (raising rates, lowering Fano factors and correlations together), which lowers noise entropy, rather than by decorrelation per se Quotation withheld because this record is not cleared for redistribution. raising correlations moves noise power into a few low-frequency Fourier components and removes it from all the higher ones; a homogeneous population encodes only in the low frequencies and is hurt, a heterogeneous one encodes across all frequencies and is helped Quotation withheld because this record is not cleared for redistribution. and the two effects compete, which is why the dependence on correlation is non-monotonic and why the high frequencies win as the population grows Quotation withheld because this record is not cleared for redistribution. underneath it, correlating neurons at fixed marginal variances lowers the noise entropy, and lower noise entropy shrinks the variance in most directions Quotation withheld because this record is not cleared for redistribution. but lower entropy alone is not enough - it helps only because a heterogeneous population spreads its signal across all the directions the entropy reduction touches Quotation withheld because this record is not cleared for redistribution. Whether noise entropy is equally decisive for scenarios not modelled here (stimulus-dependent or heterogeneous correlations, multi-dimensional stimuli) is untested. what this limits: the noise-entropy conclusion outside the modelled cases: whether it holds FOR OTHER SCENARIOS NOT CONSIDERED EXPLICITLY IN THIS STUDY - STIMULUS-DEPENDENT AND HETEROGENEOUS CORRELATIONS, OR SIMULTANEOUS ENCODING OF MULTIPLE STIMULUS DIMENSIONS - REMAINS TO BE INVESTIGATED Quotation withheld because this record is not cleared for redistribution. The reconciliation with experiments reporting reduced correlations was not worked out in detail against those studies' actual parameters. what this limits: the reconciliation with the experiments reporting reduced correlations: WE DID NOT ASSESS THIS IN DETAIL BY TAKING INTO ACCOUNT ALL RELEVANT PARAMETERS FROM THESE STUDIES Quotation withheld because this record is not cleared for redistribution. Results assume an optimal readout; computationally constrained (e.g. linear) readouts can behave differently. what this limits: every result about what correlations do to a code: OF COURSE, THE EFFECT OF CORRELATIONS CAN BE DIFFERENT FOR OTHER, COMPUTATIONALLY CONSTRAINED READOUTS Quotation withheld because this record is not cleared for redistribution. the model itself stops being valid past a critical population size once correlation coefficients are made variable, because the covariance matrix loses positive definiteness what this limits: the large-population regime the paper is about: THE COVARIANCE MATRIX WILL BE VALID (POSITIVE DEFINITE) ONLY UP TO n ~ 1/(2 sigma)^2 ... FROM A CRITICAL n ON, THE MODEL WILL NOT BE VALID ANY MORE Quotation withheld because this record is not cleared for redistribution. two of the generality checks are reported without their figures what this limits: the regular-grid idealisation: THE FISHER INFORMATION IS VIRTUALLY UNAFFECTED BY RANDOMLY ASSIGNING PREFERRED DIRECTIONS COMPARED WITH EQUALLY SPACING THEM (DATA NOT SHOWN) Quotation withheld because this record is not cleared for redistribution. whether noise entropy governs accuracy under stimulus-dependent or heterogeneous correlations, or with several stimulus dimensions, is left open what this limits: the same scope as limitations[0] - the scenarios the conclusion has been tested in Quotation withheld because this record is not cleared for redistribution. and the heterogeneity parameter cannot be pushed arbitrarily far - amplitudes are positive, average to one, and firing rates are biophysically capped what this limits: the heterogeneity results, in both directions: J_mean CANNOT BE INCREASED ARBITRARILY ... BECAUSE THE AMPLITUDES ARE CONSTRAINED TO BE POSITIVE, HAVE AN AVERAGE OF 1, AND NEURONS HAVE A MAXIMUM POSSIBLE FIRING RATE, while kappa = 0.25 IS LIKELY TO BE AN UNDERESTIMATE Quotation withheld because this record is not cleared for redistribution. the code that reproduces every figure is released, with the note that it has not been peer reviewed what this limits: the reproduction material: it contains MATLAB CODE TO REPRODUCE ALL FIGURES AND NUMERICAL SIMULATIONS IN THIS PAPER. THIS MATERIAL HAS NOT BEEN PEER REVIEWED Quotation withheld because this record is not cleared for redistribution. the decrease in noise entropy associated with increasing correlations, which is where the improvement in encoding accuracy comes from Quotation withheld because this record is not cleared for redistribution. diagonalisation of a circulant matrix in the Fourier basis, which is what makes very large populations analytically tractable Quotation withheld because this record is not cleared for redistribution. the Cramer-Rao bound and estimator bias, used to check that the measure being compared is attainable at all Quotation withheld because this record is not cleared for redistribution. 408 measured orientation tuning curves, used to set the heterogeneity parameter rather than to test the theory Quotation withheld because this record is not cleared for redistribution. a second analysis holding the TOTAL noise (the noise entropy) constant while varying only the correlation structure eliminates: the amount of noise as the driver of the coding difference - comparing correlated and uncorrelated populations changes two things at once unless the total is pinned · outcome (stated by the tagger, no quotation): held Quotation withheld because this record is not cleared for redistribution. correlation correlation Quotation withheld because this record is not cleared for redistribution. noise entropy noise entropy Quotation withheld because this record is not cleared for redistribution. population code population code Quotation withheld because this record is not cleared for redistribution. attention attention Quotation withheld because this record is not cleared for redistribution. correlations → variances THE TWO CANNOT BE SEPARATED IN THIS STUDY - the DESIGN form / It bounds what the paper's own results can mean, and it lives in a methods paragraph where nothing downstream will find it. And the span adds the part that matters for interpretation: 'neither of both is more important than the other' - the authors decline to rank the two causes they cannot separate. Quotation withheld because this record is not cleared for redistribution. THE FIELD'S CONCLUSION WAS A PROPERTY OF ITS SIMPLIFYING ASSUMPTION: most previous studies use HOMOGENEOUS population models in which neurons differ only in preferred stimulus, and heterogeneity CRITICALLY ALTERS the properties of a population code - 'this has not been sufficiently appreciated' Quotation withheld because this record is not cleared for redistribution. with the belief stated in the form it is usually held - if correlations are unavoidable, Quotation withheld, which is why falling correlations under attention, adaptation and learning are read as better coding Quotation withheld because this record is not cleared for redistribution. AND THE ANSWER REVERSES: in populations with more than a few hundred neurons, INCREASING limited-range correlations can substantially IMPROVE encoding accuracy Quotation withheld because this record is not cleared for redistribution. PUBLISHED PAPERS DISAGREE AND THE PARAMETER THEIR DISAGREEMENT TURNS ON IS NAMED: 'The apparent discrepancy arises from subtle differences in the assumptions that were made about the population activity, such as for instance the noise model (additive vs Poisson-like)' Quotation withheld because this record is not cleared for redistribution. and the two conditions are stated exactly: the information saturates ONLY IF the variances do not depend on the stimulus AND all neurons have identical tuning curves - fail either and it grows linearly Quotation withheld because this record is not cleared for redistribution. A FREE PARAMETER THAT SHOULD NEVER HAVE BEEN FREE: the level of correlations cannot be optimized on its own with tuning functions and variances held fixed, because that drives the model to a DEGENERATE, biologically implausible solution Quotation withheld because this record is not cleared for redistribution. stated against its own model too - MOST MODELS USED IN PREVIOUS STUDIES, INCLUDING OUR ABOVE MODEL, are not well constrained, because they permit the noise entropy to be made arbitrarily small Quotation withheld because this record is not cleared for redistribution. AND ONCE THE RIGHT THING IS HELD CONSTANT, THE QUANTITY THE WHOLE LITERATURE MEASURES DROPS OUT: with noise entropy fixed, the information depends ONLY on amplitude variability and is INDEPENDENT of both the structure and the magnitude of correlations Quotation withheld because this record is not cleared for redistribution. with the conclusion drawn in the field's own terms - accuracy is set by the amount of heterogeneity and the overall noise entropy, and the specific noise structure is NOT AS IMPORTANT AS COMMONLY ASSUMED Quotation withheld because this record is not cleared for redistribution. A BOUND IS ONLY A BOUND UNTIL YOU SHOW IT IS TIGHT: Fisher information gives the Cramer-Rao bound, the bound is not always attainable, and COMPARING LOOSE UPPER BOUNDS IS MEANINGLESS Quotation withheld because this record is not cleared for redistribution. so the measure is validated before it is used - the maximum-likelihood estimator comes within 5% of the bound for every population above 64 neurons Quotation withheld because this record is not cleared for redistribution. THE SAME MODEL GIVES THE OPPOSITE ANSWER UNDER A DIFFERENT DECODER: the OLE does not improve with correlations at any population size, so under a linear readout REDUCING correlations increases accuracy Quotation withheld because this record is not cleared for redistribution. and a standard interpretation is refused on the way - the linear part of the Fisher information is often called the portion readable by linear methods, and that notion is PROBLEMATIC because the linear estimator is BIASED Quotation withheld because this record is not cleared for redistribution. TWO QUANTITIES THAT MOVE TOGETHER AND CANNOT BE APPORTIONED: removing one common noise source changes correlations by 50% and variances by 5%, and CONSIDERING EACH ONE IN ISOLATION IS NOT MEANINGFUL Quotation withheld because this record is not cleared for redistribution. with the published comparison called out by name - the relative importance of correlations was assessed using HIGHLY SUBOPTIMAL POOLING RULES, and conclusions from those do not generalize Quotation withheld because this record is not cleared for redistribution. THE MODEL SILENTLY STOPS BEING VALID PAST A SIZE: with variable correlation coefficients the covariance matrix is positive definite only up to a critical n, beyond which the model is not valid any more Quotation withheld because this record is not cleared for redistribution. AND THE EXPLANATION THAT WOULD SAVE THE OLD READING IS TESTED AND REJECTED ON ITS OWN TERMS: it would need the read-out population to be at most a few hundred cells, which - given the number in a single cortical column - seems rather unlikely Quotation withheld because this record is not cleared for redistribution. and the claim is bounded in its final sentence - reducing correlations improving encoding is NOT A GENERAL PRINCIPLE but true only under certain conditions, with the assumptions about population size and readout named as what can change it Quotation withheld because this record is not cleared for redistribution. claim_type: critique a second reading reaching the same value. Recorded: critique — note: a second reading reaching the same value from a different sentence. Not applied - a person decides. Quotation withheld because this record is not cleared for redistribution. method: an analytic expression for the inverse covariance obtained by diagonalising a circulant correlation matrix in the Fourier basis, which makes populations of tens of thousands of neurons tractable a second reading DIFFERING from the recorded value. Recorded: simulation — note: a second reading, verbatim-verified, disagreeing with the value on the record. Not applied - a person decides. Quotation withheld because this record is not cleared for redistribution. phenomenon: limited-range noise correlations, and the experimental finding that they fall under attention, adaptation and learning a second reading DIFFERING from the recorded value. Recorded: noise correlations in neural population coding — note: a second reading, verbatim-verified, disagreeing with the value on the record. Not applied - a person decides. Quotation withheld because this record is not cleared for redistribution. extends → Sompolinsky et al., 2001; Shamir and Sompolinsky, 2006 Quotation withheld because this record is not cleared for redistribution. extends → Shamir and Sompolinsky (2006) Quotation withheld because this record is not cleared for redistribution. contradicts → Cohen and Maunsell, 2009; Mitchell et al., 2009 Quotation withheld because this record is not cleared for redistribution. replicates → Zohary et al., 1994; Abbott and Dayan, 1999; Sompolinsky et al., 2001; Wilke and Eurich, 2002 Quotation withheld because this record is not cleared for redistribution. contradicts → the general principle that reducing noise correlations improves a population code Quotation withheld because this record is not cleared for redistribution. explains-away → the disagreement among earlier theoretical studies, as arising from different unstated noise-model and tuning assumptions Quotation withheld because this record is not cleared for redistribution. critiques → the practice of treating correlation strength as a free parameter while holding tuning and variances fixed, which admits a degenerate noise-free optimum differs in: THE AXIS IS WHETHER CORRELATION LEVEL CAN BE VARIED INDEPENDENTLY. 'we suggest that Quotation withheld.' The earlier optimum is real inside its own assumption and vanishes outside it - the same paper whose homogeneity assumption reverses a sign, making its objection twice about what is HELD CONSTANT rather than what is measured. Quotation withheld because this record is not cleared for redistribution. critiques → the attention studies' attribution of improved coding to reduced correlations, since the pooling rules used were suboptimal and correlations and variances moved together Quotation withheld because this record is not cleared for redistribution. supports → the use of Fisher information here specifically, having shown the maximum-likelihood estimator comes within 5% of the bound Quotation withheld because this record is not cleared for redistribution. distinguishes-from → the linear-readout case, in which the original conclusion does hold differs in: THE AXIS IS WHETHER FISHER INFORMATION CAPTURES THE DEPENDENCE. 'Quotation withheld.' The distinction is between a quantity and the tool used to measure it - the linear-readout conclusion holds, and it holds because the measure happens to be adequate there. Same paper whose homogeneity assumption reverses a sign elsewhere in this corpus. Quotation withheld because this record is not cleared for redistribution. attention —causes→ correlation attention correlation status: REPORTED OF THE LITERATURE, and the INTERPRETATION placed on it is what this paper goes on to question · direction on the measure: reduces · strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. Fisher information —operationalised-as→ encoding accuracy Fisher information encoding accuracy strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. correlation —causes→ encoding accuracy correlation encoding accuracy strength of the relation: 1 · note: THE ANSWER REVERSES: in populations of more than a few hundred neurons, INCREASING limited range correlations can substantially IMPROVE encoding accuracy Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the span says reduced correlations do not necessarily improve coding; causes is loose. Quotation withheld because this record is not cleared for redistribution. noise entropy —causes→ encoding accuracy noise entropy encoding accuracy strength of the relation: 1 · note: and for constant noise entropy in the large-population limit, accuracy is INDEPENDENT of both structure and magnitude of correlations Quotation withheld because this record is not cleared for redistribution. correlation —causes→ population code correlation population code status: THE FIELD'S BELIEF, not a result of this paper - and the paper's own finding is that it holds in only one of two regimes · strength of the relation: 1 · note: the belief in the form it is held - Quotation withheld Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the span reports a common belief that reducing correlations improves a code; causes states it as a finding. Quotation withheld because this record is not cleared for redistribution. heterogeneity —contrasted-with→ correlation heterogeneity correlation strength of the relation: 1 · note: asymptotically the relative performance depends only on c 0 and on the degree of heterogeneity - the more heterogeneous, the higher Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the span says performance depends on correlation and heterogeneity; contrasted-with is loose. Quotation withheld because this record is not cleared for redistribution. amplitude —distinct-from→ correlation amplitude correlation strength of the relation: 1 · note: THE QUANTITY THE WHOLE LITERATURE MEASURES DROPS OUT - with entropy fixed the information depends ONLY on amplitude variability, and NONE of the correlation-structure parameters enter Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the span says J mean is independent of correlations; distinct-from 'amplitude' is loose. Quotation withheld because this record is not cleared for redistribution. correlation —causes→ noise entropy correlation noise entropy holds when: at fixed marginal variances - the paper's own condition, and the claim is false without it · direction on the measure: reduces · strength of the relation: 1 · note: correlating at fixed marginal variances LOWERS the noise entropy, and lower entropy shrinks the variance in most directions Quotation withheld because this record is not cleared for redistribution. noise entropy —causes→ population code noise entropy population code strength of the relation: 1 · note: but lower entropy ALONE is not enough - it helps only because the heterogeneous population spreads its signal across the directions the reduction touches Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but a lower noise entropy 'should lead to an improvement in coding accuracy'; 'causes population code' is loose. Quotation withheld because this record is not cleared for redistribution. correlation —causes→ population code correlation population code holds when: in the low-correlation regime - below c_min, where the behaviour matches a homogeneous population · direction on the measure: reduces · strength of the relation: 1 · note: ONE OF A PAIR. The same relation carries the OPPOSITE sign above c_min; see the companion edge. Neither edge is the paper's claim on its own. Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the span states a regime-conditional claim (below c_min); the sign holds only in that regime. Quotation withheld because this record is not cleared for redistribution. correlation —causes→ population code correlation population code holds when: in the high-correlation regime - above c_min · direction on the measure: increases · strength of the relation: 1 · note: ONE OF A PAIR. The same relation carries the OPPOSITE sign below c_min; see the companion edge. c_min falls with population size and converges to zero, so for large populations this is the regime that applies. Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the span states a regime-conditional claim (the high-correlation regime). Quotation withheld because this record is not cleared for redistribution. Fisher information Fisher information Quotation withheld because this record is not cleared for redistribution. maximum likelihood estimation maximum likelihood estimation Quotation withheld because this record is not cleared for redistribution. Cramér-Rao bound Cramér-Rao bound Quotation withheld because this record is not cleared for redistribution. mean squared error mean squared error Quotation withheld because this record is not cleared for redistribution. lognormal distribution lognormal distribution Quotation withheld because this record is not cleared for redistribution. Fisher information Fisher information Quotation withheld because this record is not cleared for redistribution. maximum-likelihood maximum-likelihood Quotation withheld because this record is not cleared for redistribution. Fano factor Fano factor Quotation withheld because this record is not cleared for redistribution. In heterogeneous populations of more than a few hundred neurons, increasing rather than decreasing limited-range noise correlations substantially improves encoding accuracy. Quotation withheld because this record is not cleared for redistribution. The accuracy gain from stronger correlations is attributable to the drop in noise entropy that accompanies them when marginal distributions are held fixed. Quotation withheld because this record is not cleared for redistribution. Once noise entropy is held constant, encoding accuracy in large populations no longer depends on the structure or magnitude of the correlations at all. Quotation withheld because this record is not cleared for redistribution. Fisher information stops growing with population size only under the restrictive combination of stimulus-independent variances and identical tuning curves. Quotation withheld because this record is not cleared for redistribution. The maximally detrimental correlation level decreases in inverse proportion to population size. Quotation withheld because this record is not cleared for redistribution. The maximum-likelihood estimator comes close enough to the Cramér–Rao bound that comparing Fisher information across conditions is legitimate. Quotation withheld because this record is not cleared for redistribution. For a suboptimal linear readout the conclusion reverses: the OLE gains nothing from stronger correlations even in large populations. Quotation withheld because this record is not cleared for redistribution. IN HOMOGENEOUS POPULATIONS limited range correlations are highly detrimental, which is why falling correlations under attention, adaptation and learning are read as better coding Quotation withheld because this record is not cleared for redistribution. AND THE ANSWER REVERSES: in populations with more than a few hundred neurons, INCREASING the level of limited range correlations can substantially improve encoding accuracy Quotation withheld because this record is not cleared for redistribution. with the improvement traced to a decrease in noise entropy, and - for constant entropy in the large-population limit - accuracy INDEPENDENT of both structure and magnitude of correlations Quotation withheld because this record is not cleared for redistribution. THE CONCLUSION WAS A PROPERTY OF THE SIMPLIFYING ASSUMPTION: most previous studies use homogeneous models, and heterogeneity CRITICALLY ALTERS a population code - 'this has not been sufficiently appreciated' Quotation withheld because this record is not cleared for redistribution. with the belief stated as it is held - it is GENERALLY BELIEVED that reducing correlations improves any population code WHOSE RESULT THIS IS: studies find - THE SUBJECT OF THIS RESULT IS AN INDEFINITE STUDY, NOT THIS PAPER. 'Previous studies reported', 'studies in guinea pig showed', 'recent studies have reported' - somebody else's work under a common noun rather than a name, with a citation in the sentence and no first-person marker. The quote is verbatim and the claim is true; only the attribution is wrong. FOUND BY MEASURING MY OWN CUE'S RECALL: the first sweep required a NAMED author as the subject ('Sperling (1960) demonstrated') and found 17. Reading eleven citation-bearing findings on 'Plakke2016' by hand showed four the cue could not see - 'A study examining recognition found', 'studies of typical human subjects have demonstrated', 'A follow up study found' - so the named cue's 17 was a floor and not a count. A CUE'S FALSE POSITIVES TELL YOU NOTHING ABOUT ITS FALSE NEGATIVES, and the only way to find these was to read a record's whole layer. Quotation withheld because this record is not cleared for redistribution. A NUMBER OF AUTHORS REPORTED DIFFERENT RESULTS, and the apparent discrepancy arises from subtle differences in the assumptions that were made about the population activity, such as the noise model, additive against Poisson-like Quotation withheld because this record is not cleared for redistribution. and the two conditions are stated exactly - the information saturates ONLY IF variances are stimulus-independent AND all tuning curves identical Quotation withheld because this record is not cleared for redistribution. TWO REGIMES, with the boundary falling as 1/n, so that for large populations increasing correlations is almost always advantageous Quotation withheld because this record is not cleared for redistribution. asymptotically the relative performance depends only on the same-preference correlation coefficient and on the degree of heterogeneity Quotation withheld because this record is not cleared for redistribution. and heterogeneity cannot be pushed arbitrarily far, while the value used is likely an UNDERESTIMATE because real populations vary in more than amplitude Quotation withheld because this record is not cleared for redistribution. A FREE PARAMETER THAT SHOULD NEVER HAVE BEEN FREE: optimising correlations with tuning and variances held fixed reaches a DEGENERATE, biologically implausible solution Quotation withheld because this record is not cleared for redistribution. said against its own model too - MOST MODELS USED IN PREVIOUS STUDIES, INCLUDING OUR ABOVE MODEL, are not well constrained Quotation withheld because this record is not cleared for redistribution. AND THE QUANTITY THE WHOLE LITERATURE MEASURES DROPS OUT: with noise entropy fixed the information depends only on amplitude variability, and no correlation-structure parameter enters Quotation withheld because this record is not cleared for redistribution. so accuracy is set by heterogeneity and overall noise entropy, and the specific noise structure is NOT AS IMPORTANT AS COMMONLY ASSUMED Quotation withheld because this record is not cleared for redistribution. A BOUND IS ONLY A BOUND UNTIL YOU SHOW IT IS TIGHT: the responses here are neither independent nor identically distributed, and COMPARING LOOSE UPPER BOUNDS IS MEANINGLESS Quotation withheld because this record is not cleared for redistribution. so it is checked before use - the maximum-likelihood estimator is within 5% of the bound for every population above 64 neurons Quotation withheld because this record is not cleared for redistribution. THE SAME MODEL GIVES THE OPPOSITE ANSWER UNDER A DIFFERENT DECODER: under a linear readout, reducing pairwise correlations increases accuracy Quotation withheld because this record is not cleared for redistribution. and a standard interpretation is refused - calling the linear Fisher term the portion readable by linear methods is PROBLEMATIC, because that estimator is biased Quotation withheld because this record is not cleared for redistribution. because the readout mechanism is unknown, the paper quantifies the MAXIMUM extractable information and states that a linear readout would give different conclusions Quotation withheld because this record is not cleared for redistribution. TWO QUANTITIES THAT MOVE TOGETHER AND CANNOT BE APPORTIONED: removing one common noise source gives a 50% change in correlation and a 5% change in variance, and considering each in isolation IS NOT MEANINGFUL Quotation withheld because this record is not cleared for redistribution. with the published comparison named - the relative importance of correlations was assessed using HIGHLY SUBOPTIMAL POOLING RULES Quotation withheld because this record is not cleared for redistribution. THE MODEL SILENTLY STOPS BEING VALID PAST A SIZE: with variable correlation coefficients the covariance matrix is positive definite only up to a critical n Quotation withheld because this record is not cleared for redistribution. the regular-grid assumption on preferred directions is checked and found not to matter - DATA NOT SHOWN Quotation withheld because this record is not cleared for redistribution. with generality checked against random tuning widths and against triples drawn from real V1 tuning curves Quotation withheld because this record is not cleared for redistribution. and the heterogeneity parameter SET from 408 measured macaque V1 tuning curves rather than tested against them Quotation withheld because this record is not cleared for redistribution. the mechanism in Fourier terms - correlations move noise power into a few low frequencies and out of all higher ones, and the two effects COMPETE Quotation withheld because this record is not cleared for redistribution. correlating neurons at fixed marginal variances lowers the noise entropy, which shrinks the variance in most directions Quotation withheld because this record is not cleared for redistribution. but lower entropy alone is not enough - it helps only because a heterogeneous population spreads its signal across the directions the reduction touches Quotation withheld because this record is not cleared for redistribution. THE ESCAPE ROUTE IS TESTED AND REJECTED ON ITS OWN TERMS: it would need the read-out population to be at most a few hundred cells, which given a single cortical column seems RATHER UNLIKELY Quotation withheld because this record is not cleared for redistribution. and the claim is bounded in its last sentence - NOT A GENERAL PRINCIPLE but true only under certain conditions Quotation withheld because this record is not cleared for redistribution. with what is left open named: stimulus-dependent and heterogeneous correlations, and several stimulus dimensions at once Quotation withheld because this record is not cleared for redistribution. and the code that reproduces every figure released, with the note that it HAS NOT BEEN PEER REVIEWED Quotation withheld because this record is not cleared for redistribution. limited range correlation structure limited range correlation structure Quotation withheld because this record is not cleared for redistribution. homogeneous population model homogeneous population model Quotation withheld because this record is not cleared for redistribution. relative J mean relative J mean Quotation withheld because this record is not cleared for redistribution. low-correlation regime low-correlation regime Quotation withheld because this record is not cleared for redistribution. noise entropy noise entropy Quotation withheld because this record is not cleared for redistribution. degenerate degenerate Quotation withheld because this record is not cleared for redistribution. bound bound Quotation withheld because this record is not cleared for redistribution. heterogeneity heterogeneity Quotation withheld because this record is not cleared for redistribution. J mean J mean Quotation withheld because this record is not cleared for redistribution. J cov J cov Quotation withheld because this record is not cleared for redistribution. population size population size Quotation withheld because this record is not cleared for redistribution. correlation coefficient correlation coefficient Quotation withheld because this record is not cleared for redistribution. amplitude variability amplitude variability Quotation withheld because this record is not cleared for redistribution. Fisher information Fisher information Quotation withheld because this record is not cleared for redistribution. correlation correlation Quotation withheld because this record is not cleared for redistribution. noise entropy noise entropy Quotation withheld because this record is not cleared for redistribution. size size Quotation withheld because this record is not cleared for redistribution. amplitude amplitude Quotation withheld because this record is not cleared for redistribution. κ = 0.25 — degree of tuning-amplitude heterogeneity used throughout, taken from 408 measured macaque V1 orientation tuning curves Quotation withheld because this record is not cleared for redistribution. within 5% — how close the maximum-likelihood estimator gets to the Cramér–Rao bound for populations larger than 64 neurons Quotation withheld because this record is not cleared for redistribution. ∼0.5 bits — entropy reduction produced by correlating two neurons at r = 0.8 while leaving their marginals unchanged Quotation withheld because this record is not cleared for redistribution. 1.29 — marginal standard deviation needed to restore the independent population's entropy once a correlation of 0.8 is imposed Quotation withheld because this record is not cleared for redistribution. 4096 — number of simulated heterogeneous population realizations generated per population size for the maximum-likelihood analysis Quotation withheld because this record is not cleared for redistribution. 5% — how close the maximum-likelihood estimator comes to the bound above 64 neurons - the check that makes every later comparison of Fisher information meaningful Quotation withheld because this record is not cleared for redistribution. 50% ... 5% — the change in correlation and in variance produced by removing ONE common noise source - two numbers from one cause, which is why neither can be credited with the effect Quotation withheld because this record is not cleared for redistribution. 408 — measured macaque V1 tuning curves used to SET the heterogeneity parameter, so the model's key input is data and its test is not Quotation withheld because this record is not cleared for redistribution. a few hundred — the largest read-out population that would rescue the old interpretation - stated so that it can be judged implausible against a cortical column Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1523/jneurosci.2539-11.2011.