Measuring Fisher Information Accurately in Correlated Neural Populations
Publication Date
June 01, 2015
Journal
PLOS Computational Biology
Authors
Ingmar Kanitscheider, Ruben Coen Cagli, Adam Kohn & Alexandre Pouget
Volume
11
Issue
6
Pages
e1004218
DOI
https://dx.plos.org/10.1371/journal.pcbi.1004218
Publisher URL
http://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1004218
PubMed
http://www.ncbi.nlm.nih.gov/pubmed/26030735
PubMed Central
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4451760
Europe PMC
http://europepmc.org/abstract/MED/26030735
Web of Science
000357340100008
Scopus
84953259398
Mendeley
http://www.mendeley.com/research/measuring-fisher-information-accurately-correlated-neural-populations
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Mendeley | Further Information

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Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/2090474"], "description"=>"<p>(<b>a</b>) Left: In a typical experiment, the responses of a population of neurons are recorded simultaneously while a visual stimulus is presented. Middle: Population responses (<i>N</i> = 2 in the cartoon) to two different stimulus values (orange vs. green symbols) are collected over many trials. Right: The optimal decoding weights,<i>w</i><sub><i>i</i></sub>, are applied to the population response, <b>r</b>, to obtain an estimate of the stimulus, </p><p></p><p></p><p><mi>θ</mi><mo>^</mo></p><p></p><p></p>. Linear Fisher information corresponds to the inverse of the variance of such estimates, across trials where the same stimulus was presented. (<b>b</b>) Ordinate: Fisher information in a population of <i>N</i> = 50 model neurons, estimated by a linear decoder as illustrated in (<b>a</b>), using cross-validation with early stopping. Abscissa: number of trials per stimulus condition. We ran 200 experiments for each trial. The top line is the information estimated from the training set; the bottom line is the information estimated from the validation set; the two will converge asymptotically. The dashed line is the true information value in the simulated population. The continuous lines represent the mean, the shaded area represents ±1 std across experiments, computed by bootstrap. (<b>c</b>) Fisher information obtained by directly estimating the tuning curves and covariance, and then applying Eq (<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.e030\" target=\"_blank\">11</a>). The continuous lines represent the mean, the shaded area represents ±1 std across experiments, computed by bootstrap. (<b>d</b>) Similar to (<b>c</b>), but after correcting for the estimation bias, according to Eq (<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.e007\" target=\"_blank\">2</a>). The continuous lines represent the mean, the shaded area represents ±1 std across experiments, computed analytically using Eq (<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.e045\" target=\"_blank\">19</a>).<p></p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432551, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.g001", "stats"=>{"downloads"=>1, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Stimulus_decoding_and_Fisher_information_/1432551", "title"=>"Stimulus decoding and Fisher information.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090475"], "description"=>"<p>(<b>a</b>) Mean squared error (MSE) of the cross-validated decoder-based estimate (CV decoder, red) and the bias-corrected direct estimator (BC estimator, blue) calculated from the estimates of <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.g001\" target=\"_blank\">Fig 1</a> (<i>N</i> = 50 neurons). The ordinate axis on the right indicates the corresponding relative error, namely </p><p></p><p></p><p></p><p></p><p></p><p></p><p>MSE</p><p></p><p></p><mo>/</mo><mi>I</mi><p></p><p></p><p></p><p></p>. (<b>b</b>) Relative error for CV decoder (red) and BC estimator (blue), for different population sizes (abscissa). The number of trials was set to 1, 2, or 5 times the number of neurons (indicated at the top of each panel). (<b>c</b>) Relative error for CV decoder (right) and BC estimator (left), for different population sizes (abscissa) and numbers of trials expressed as proportion of the number of neurons (ordinate). (<b>d</b>) Log-ratio of relative errors for BC vs. CV from panel (<b>c</b>). The black contour separates cases in which the CV estimator is more accurate than BC (warm colors) from cases in which BC is more accurate (cold colors).<p></p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432552, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.g002", "stats"=>{"downloads"=>5, "page_views"=>15, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Comparison_of_the_estimation_errors_of_the_decoder_and_the_direct_bias_corrected_method_/1432552", "title"=>"Comparison of the estimation errors of the decoder and the direct bias-corrected method.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090476"], "description"=>"<p>Left: Illustration of the scenario. Decoding weights are optimized for data recorded from population A, then tested on data from population B. Note that the decision boundary derived from A is not optimal for B. (<b>a</b>) Estimate of the Fisher information obtained by decoding (red) or direct estimation with bias correction (blue). The continuous lines represent the mean, the shaded area represents ±1 std across experiments, computed by bootstrap. (<b>b</b>) MSE of the decoder-based estimate (red) and the direct estimator (blue).</p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432553, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.g003", "stats"=>{"downloads"=>1, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Fisher_information_estimates_for_the_general_case_of_a_decoder_trained_and_tested_on_different_response_statistics_/1432553", "title"=>"Fisher information estimates for the general case of a decoder trained and tested on different response statistics.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090477"], "description"=>"<p>Left: Illustration of the scenario. The decoder is trained and tested on shuffled data, i.e. data where correlations have been destroyed by randomly permuting across trials the responses of each neuron independently. The faint lines represent the covariance ellipses of the original data, before shuffling. (<b>a</b>) Estimate of the Fisher information obtained by decoding (red) or direct estimation with bias correction (blue). The continuous lines represent the mean, the shaded area represents ±1 std across experiments, computed by bootstrap. (<b>b</b>) MSE of the decoder-based estimate (red), and the direct estimator (blue).</p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432554, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.g004", "stats"=>{"downloads"=>4, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Fisher_information_estimates_for_an_independent_population_/1432554", "title"=>"Fisher information estimates for an independent population.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090478"], "description"=>"<p>Left: Illustration of the scenario. Decoding weights are optimized for the shuffled data, then tested on the original data. The faint lines on the top plot represent the covariance ellipses of the original data, before shuffling. Note that the decision boundary derived from shuffled data is not necessarily optimal for the original data. (<b>a</b>) Estimate of the Fisher information obtained by decoding (red) or direct estimation with bias correction (blue). The continuous lines represent the mean, the shaded area represents ±1 std across experiments, computed by bootstrap. (<b>b</b>) MSE of the decoder-based estimate (red) and the direct estimator (blue).</p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432555, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.g005", "stats"=>{"downloads"=>3, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Fisher_information_estimates_when_ignoring_correlations_/1432555", "title"=>"Fisher information estimates when ignoring correlations.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090479"], "description"=>"<p>Fisher information in a population of <i>N</i> = 52 macaque V1 neurons, estimated by decoding (red) or direct estimation (blue). Trials were subsampled 100 times without replacement, except for the last point on the abscissa which included all (<i>T</i> = 900) trials. The continuous lines represent the mean, the shaded area represents ±1 std across samples, computed by bootstrap. Stimulus orientations were spaced by 7 deg (top row) and 21 deg (bottom row); in the right column, images were masked by white noise on the pixels (for stimulus details, see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#sec011\" target=\"_blank\">Materials and Methods</a> Section 9). Population-average firing rates were R = 0.7 spikes/trial (top-left); R = 0.7 spikes/trial (bottom-left); R = 2.2 spikes/trial (top-right); and R = 2.1 spikes/trial (bottom-right). Note that for large orientation differences, the stimuli can be more easily discriminated: Using the bias-corrected estimate at <i>T</i> = 900 and the known conversion between Fisher information and percent correct, percent correct with 7 deg separation is 77% (top-left) and 61% (top-right), whereas the corresponding values with 21 deg separation are 92% (bottom-left) and 70% (bottom-right).</p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432556, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.g006", "stats"=>{"downloads"=>6, "page_views"=>23, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Fisher_information_estimated_from_cortical_data_/1432556", "title"=>"Fisher information estimated from cortical data.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090486"], "description"=>"<p>(<b>a,b</b>) Simulated data with model parameters identical to <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.t001\" target=\"_blank\">Table 1</a> and <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.t002\" target=\"_blank\">Table 2</a> except for <i>g</i> = 1 (corresponding to 0.8 spikes per neuron per trial on average). (<b>a</b>) Estimate of the Fisher information obtained by decoding (red) or direct estimation with bias correction (blue). The continuous lines represent the mean, the shaded area represents ±1 std across experiments, computed by bootstrap. (<b>b</b>) MSE of the decoder-based estimate (red) and the direct estimator (blue). (<b>c</b>) Evaluation on the neural data of <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.g006\" target=\"_blank\">Fig 6</a>, in the condition of stimulus orientation spaced by 7 deg, without pixel noise in the image. Each panel corresponds to a different spike count window, reported at the top of the panel, starting 30 ms after stimulus onset. The top-left panel (250ms window) is identical to the top-left panel of <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.g006\" target=\"_blank\">Fig 6</a>. The color code is identical to <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004218#pcbi.1004218.g006\" target=\"_blank\">Fig 6</a>.</p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432558, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.g007", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Comparison_of_estimators_at_low_spike_counts_/1432558", "title"=>"Comparison of estimators at low spike counts.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090487"], "description"=>"<p>Parameters of the input images.</p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432559, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.t001", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Parameters_of_the_input_images_/1432559", "title"=>"Parameters of the input images.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090488"], "description"=>"<p>Parameters of the model filters.</p>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432560, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004218.t002", "stats"=>{"downloads"=>1, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Parameters_of_the_model_filters_/1432560", "title"=>"Parameters of the model filters.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-06-01 03:43:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/2090489", "https://ndownloader.figshare.com/files/2090490", "https://ndownloader.figshare.com/files/2090491", "https://ndownloader.figshare.com/files/2090492", "https://ndownloader.figshare.com/files/2090493", "https://ndownloader.figshare.com/files/2090494", "https://ndownloader.figshare.com/files/2090495", "https://ndownloader.figshare.com/files/2090496"], "description"=>"<div><p>Neural responses are known to be variable. In order to understand how this neural variability constrains behavioral performance, we need to be able to measure the reliability with which a sensory stimulus is encoded in a given population. However, such measures are challenging for two reasons: First, they must take into account noise correlations which can have a large influence on reliability. Second, they need to be as efficient as possible, since the number of trials available in a set of neural recording is usually limited by experimental constraints. Traditionally, cross-validated decoding has been used as a reliability measure, but it only provides a lower bound on reliability and underestimates reliability substantially in small datasets. We show that, if the number of trials per condition is larger than the number of neurons, there is an alternative, direct estimate of reliability which consistently leads to smaller errors and is much faster to compute. The superior performance of the direct estimator is evident both for simulated data and for neuronal population recordings from macaque primary visual cortex. Furthermore we propose generalizations of the direct estimator which measure changes in stimulus encoding across conditions and the impact of correlations on encoding and decoding, typically denoted by <i>I<sub>shuffle</sub></i> and <i>I<sub>diag</sub></i> respectively.</p></div>", "links"=>[], "tags"=>["measure changes", "Correlated Neural Populations Neural responses", "Fisher Information Accurately", "stimulus encoding", "account noise correlations", "population recordings", "reliability measure"], "article_id"=>1432561, "categories"=>["Biological Sciences", "Ecology"], "users"=>["Ingmar Kanitscheider", "Ruben Coen-Cagli", "Adam Kohn", "Alexandre Pouget"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1004218.s001", "https://dx.doi.org/10.1371/journal.pcbi.1004218.s002", "https://dx.doi.org/10.1371/journal.pcbi.1004218.s003", "https://dx.doi.org/10.1371/journal.pcbi.1004218.s004", "https://dx.doi.org/10.1371/journal.pcbi.1004218.s005", "https://dx.doi.org/10.1371/journal.pcbi.1004218.s006", "https://dx.doi.org/10.1371/journal.pcbi.1004218.s007", "https://dx.doi.org/10.1371/journal.pcbi.1004218.s008"], "stats"=>{"downloads"=>7, "page_views"=>10, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Measuring_Fisher_Information_Accurately_in_Correlated_Neural_Populations_/1432561", "title"=>"Measuring Fisher Information Accurately in Correlated Neural Populations", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2015-06-01 03:43:05"}

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  • {"unique-ip"=>"16", "full-text"=>"15", "pdf"=>"4", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"4"}
  • {"unique-ip"=>"8", "full-text"=>"7", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"5"}
  • {"unique-ip"=>"13", "full-text"=>"13", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"8"}
  • {"unique-ip"=>"10", "full-text"=>"11", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"9"}
  • {"unique-ip"=>"17", "full-text"=>"17", "pdf"=>"4", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"10"}
  • {"unique-ip"=>"13", "full-text"=>"10", "pdf"=>"7", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"12"}
  • {"unique-ip"=>"9", "full-text"=>"10", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"2", "year"=>"2020", "month"=>"2"}
  • {"unique-ip"=>"22", "full-text"=>"25", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"3"}
  • {"unique-ip"=>"11", "full-text"=>"11", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"4"}
  • {"unique-ip"=>"14", "full-text"=>"15", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"5"}
  • {"unique-ip"=>"9", "full-text"=>"12", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"6"}
  • {"unique-ip"=>"11", "full-text"=>"13", "pdf"=>"5", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"7"}
  • {"unique-ip"=>"5", "full-text"=>"5", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"8"}
  • {"unique-ip"=>"5", "full-text"=>"3", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"9"}

Relative Metric

{"start_date"=>"2015-01-01T00:00:00Z", "end_date"=>"2015-12-31T00:00:00Z", "subject_areas"=>[]}
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