Can Power-Law Scaling and Neuronal Avalanches Arise from Stochastic Dynamics?
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{"title"=>"Can power-law scaling and neuronal avalanches arise from stochastic dynamics?", "type"=>"journal", "authors"=>[{"first_name"=>"Jonathan", "last_name"=>"Touboul", "scopus_author_id"=>"23029252800"}, {"first_name"=>"Alain", "last_name"=>"Destexhe", "scopus_author_id"=>"7006034474"}], "year"=>2010, "source"=>"PLoS ONE", "identifiers"=>{"pui"=>"358450255", "sgr"=>"77949359925", "issn"=>"19326203", "arxiv"=>"0910.0805", "pmid"=>"20161798", "scopus"=>"2-s2.0-77949359925", "doi"=>"10.1371/journal.pone.0008982", "isbn"=>"10.1371/journal.pone.0008982"}, "id"=>"4e368c01-39c4-3998-9c07-1b400ea5b5a1", "abstract"=>"The presence of self-organized criticality in biology is often evidenced by a power-law scaling of event size distributions, which can be measured by linear regression on logarithmic axes. We show here that such a procedure does not necessarily mean that the system exhibits self-organized criticality. We first provide an analysis of multisite local field potential (LFP) recordings of brain activity and show that event size distributions defined as negative LFP peaks can be close to power-law distributions. However, this result is not robust to change in detection threshold, or when tested using more rigorous statistical analyses such as the Kolmogorov-Smirnov test. Similar power-law scaling is observed for surrogate signals, suggesting that power-law scaling may be a generic property of thresholded stochastic processes. We next investigate this problem analytically, and show that, indeed, stochastic processes can produce spurious power-law scaling without the presence of underlying self-organized criticality. However, this power-law is only apparent in logarithmic representations, and does not survive more rigorous analysis such as the Kolmogorov-Smirnov test. The same analysis was also performed on an artificial network known to display self-organized criticality. In this case, both the graphical representations and the rigorous statistical analysis reveal with no ambiguity that the avalanche size is distributed as a power-law. We conclude that logarithmic representations can lead to spurious power-law scaling induced by the stochastic nature of the phenomenon. This apparent power-law scaling does not constitute a proof of self-organized criticality, which should be demonstrated by more stringent statistical tests.", "link"=>"http://www.mendeley.com/research/powerlaw-scaling-neuronal-avalanches-arise-stochastic-dynamics", "reader_count"=>181, "reader_count_by_academic_status"=>{"Unspecified"=>5, "Professor > Associate Professor"=>14, "Librarian"=>1, "Researcher"=>53, "Student > Doctoral Student"=>6, "Student > Ph. D. Student"=>53, "Student > Postgraduate"=>5, "Student > Master"=>21, "Other"=>4, "Student > Bachelor"=>7, "Lecturer"=>4, "Lecturer > Senior Lecturer"=>1, "Professor"=>7}, "reader_count_by_user_role"=>{"Unspecified"=>5, "Professor > Associate Professor"=>14, "Librarian"=>1, "Researcher"=>53, "Student > Doctoral Student"=>6, "Student > Ph. D. Student"=>53, "Student > Postgraduate"=>5, "Student > Master"=>21, "Other"=>4, "Student > Bachelor"=>7, "Lecturer"=>4, "Lecturer > Senior Lecturer"=>1, "Professor"=>7}, "reader_count_by_subject_area"=>{"Unspecified"=>10, "Agricultural and Biological Sciences"=>48, "Arts and Humanities"=>2, "Philosophy"=>2, "Chemistry"=>1, "Computer Science"=>7, "Earth and Planetary Sciences"=>1, "Engineering"=>13, "Environmental Science"=>1, "Mathematics"=>12, "Medicine and Dentistry"=>10, "Neuroscience"=>24, "Pharmacology, Toxicology and Pharmaceutical Science"=>1, "Physics and Astronomy"=>38, "Psychology"=>5, "Social Sciences"=>6}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>10}, "Social Sciences"=>{"Social Sciences"=>6}, "Physics and Astronomy"=>{"Physics and Astronomy"=>38}, "Psychology"=>{"Psychology"=>5}, "Mathematics"=>{"Mathematics"=>12}, "Unspecified"=>{"Unspecified"=>10}, "Environmental Science"=>{"Environmental Science"=>1}, "Pharmacology, Toxicology and Pharmaceutical Science"=>{"Pharmacology, Toxicology and Pharmaceutical Science"=>1}, "Arts and Humanities"=>{"Arts and Humanities"=>2}, "Engineering"=>{"Engineering"=>13}, "Chemistry"=>{"Chemistry"=>1}, "Neuroscience"=>{"Neuroscience"=>24}, "Earth and Planetary Sciences"=>{"Earth and Planetary Sciences"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>48}, "Computer Science"=>{"Computer Science"=>7}, "Philosophy"=>{"Philosophy"=>2}}, "reader_count_by_country"=>{"Republic of Singapore"=>1, "Hungary"=>1, "United States"=>9, "Japan"=>3, "United Kingdom"=>3, "Belarus"=>1, "Switzerland"=>2, "Portugal"=>1, "India"=>1, "Canada"=>1, "Netherlands"=>1, "Turkey"=>1, "Italy"=>1, "Israel"=>1, "France"=>6, "Nigeria"=>1, "Germany"=>6}, "group_count"=>6}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/864159"], "description"=>"<p>Single-barrier case (A,B) on a log-linear scale (A) and on a log-log scale (B) show a globally linear trend. Excursions (C,D) show exactly the same profile. Simulation parameters: intensity of the process , , , , , maximal value of peaks considered (see text).</p>", "links"=>[], "tags"=>["amplitude", "shot-noise"], "article_id"=>534624, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g008", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Peak_amplitude_distribution_for_the_Shot_Noise_model_/534624", "title"=>"Peak amplitude distribution for the Shot-Noise model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:46:03"}
  • {"files"=>["https://ndownloader.figshare.com/files/864375"], "description"=>"<p>Results of avalanche-analysis for avalanche-sizes determined using a sequence of avalanches produced by the artificial neural network model, and a smaller set of avalanches corresponding to the typical number of avalanches we have in our experimental simulations. The power-law model provides a very good fit, with high p-value, whereas an exponential law is not a good statistical model of the data in either case.</p>", "links"=>[], "tags"=>["avalanche-analysis"], "article_id"=>534838, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.t005", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_of_avalanche_analysis_for_the_artificial_network_model_8_at_criticality_/534838", "title"=>"Results of avalanche-analysis for the artificial network model [8] at criticality.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-21 02:47:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/863649"], "description"=>"<p>The nLFP avalanche size distributions were computed according to an avalanche analysis (see text). For a high detection threshold, the avalanche distribution is better fit by a power-law (left panels); for a low detection threshold, it is better explained by an exponential distribution (right panels).</p>", "links"=>[], "tags"=>["nlfps"], "article_id"=>534115, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g003", "stats"=>{"downloads"=>2, "page_views"=>15, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Avalanche_analysis_of_nLFPs_in_the_awake_cat_/534115", "title"=>"Avalanche analysis of nLFPs in the awake cat.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:43:09"}
  • {"files"=>["https://ndownloader.figshare.com/files/864466"], "description"=>"<p>Results of avalanche-analysis for avalanches defined from single-electrode LFP peaks, positive and negative, with low and high threshold. Table headers are the same as in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0008982#pone-0008982-t001\" target=\"_blank\">Table 1</a>.</p>", "links"=>[], "tags"=>["avalanche-analysis", "single-electrode", "lfp"], "article_id"=>534934, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.t003", "stats"=>{"downloads"=>1, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_of_avalanche_analysis_for_single_electrode_LFP_peaks_/534934", "title"=>"Results of avalanche-analysis for single-electrode LFP peaks.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-21 02:47:41"}
  • {"files"=>["https://ndownloader.figshare.com/files/864224"], "description"=>"<p>(A,B): single-barrier peaks, on a log-linear scale (A) and on a log-log scale (B), and excursions (C,D), on a log-linear scale (C) and on a log-log scale (D). Both case present the same profile and a globally linear trend for both axis. Simulation parameters: intensity of the process , , , , , maximal value of peaks considered: (see text).</p>", "links"=>[], "tags"=>["amplitude", "ornstein-uhlenbeck"], "article_id"=>534694, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g009", "stats"=>{"downloads"=>1, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Peak_amplitude_distribution_for_the_Ornstein_Uhlenbeck_process_/534694", "title"=>"Peak amplitude distribution for the Ornstein-Uhlenbeck process.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:46:27"}
  • {"files"=>["https://ndownloader.figshare.com/files/864427"], "description"=>"<p>Results of avalanche analysis performed on data obtained from positive (Pos) and negative (Neg) LFP peaks detected with either a Low or High threshold. In this analysis, the avalanche size was the summed amplitude of all LFP peaks within the avalanche. The is the estimated exponential parameter, the KS value corresponds to the Kolmogorov–Smirnov distance. The smaller the KS, the better the fit. The closer to 1 the p-value, the better the fit. The % represents the percentage of data explained by the best fit. We observe that the distributions of these data are better represented by exponential fits, and the positive peaks with a low threshold are not well modeled by either an exponential law or a power law. The log-likelihood ratio test always conclude that a better fit is provided by the exponential law. This test is performed over a common range by fitting the data using the same values of and for both laws. The <i>LLR</i> value is the value of the log-likelihood ratio. It is negative (positive) if the better fit is the exponential (power-law) distribution. The <i>p-val</i> value is the significance log-likelihood ratio (see text). The closer to 0, the more significant the test. <i>Result</i> corresponds to the conclusion of the log likelihood ratio test: <i>Exp</i> indicates that the log likelihood ratio concludes that the exponential fit is better.</p>", "links"=>[], "tags"=>["avalanche-analysis", "lfp"], "article_id"=>534887, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.t001", "stats"=>{"downloads"=>0, "page_views"=>18, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_of_avalanche_analysis_summed_LFP_amplitudes_/534887", "title"=>"Results of avalanche-analysis (summed LFP amplitudes).", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-21 02:47:32"}
  • {"files"=>["https://ndownloader.figshare.com/files/864310"], "description"=>"<p>The power-law distribution provides a very good graphical fit (A), whereas the exponential distribution provides a poor fit (B). Data from ref. <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0008982#pone.0008982-Levina1\" target=\"_blank\">[8]</a>.</p>", "links"=>[], "tags"=>["simulated", "neural", "displaying", "self-organized"], "article_id"=>534769, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g010", "stats"=>{"downloads"=>3, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Avalanche_analysis_of_a_simulated_neural_network_displaying_self_organized_criticality_/534769", "title"=>"Avalanche analysis of a simulated neural network displaying self-organized criticality.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:46:53"}
  • {"files"=>["https://ndownloader.figshare.com/files/863951"], "description"=>"<p>The peak distribution is shown on log-linear (A) and logarithmic scale (B).</p>", "links"=>[], "tags"=>["distributions", "lfp", "peaks"], "article_id"=>534413, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g006", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Avalanche_size_distributions_of_negative_LFP_peaks_from_single_channels_/534413", "title"=>"Avalanche-size distributions of negative LFP peaks from single channels.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:44:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/864403"], "description"=>"<p>Results of avalanche-analysis for avalanche-sizes analytically determined for four stochastic processes. Table headers are the same as in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0008982#pone-0008982-t001\" target=\"_blank\">Table 1</a>. The estimated power-law is large because we considered the tail of the distribution, and since the data present an exponential trend, the estimated power-law exponent becomes larger when thresholds are high. Even if the p-value is high, the fit is not realistic and the does not hold for larger intervals. We do not use the log-likelihood ratio since it is defined for samples and does not really make sense for distributions.</p>", "links"=>[], "tags"=>["avalanche-analysis", "thresholded", "stochastic"], "article_id"=>534864, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.t004", "stats"=>{"downloads"=>1, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_of_avalanche_analysis_for_thresholded_stochastic_processes_/534864", "title"=>"Results of avalanche-analysis for thresholded stochastic processes.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-21 02:47:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/863351"], "description"=>"<p>Eight pairs of tungsten electrodes (placement illustrated on top) were inserted in cat cerebral cortex (area 5–7, parietal) as described in detail in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0008982#pone.0008982-Destexhe1\" target=\"_blank\">[10]</a>. The system simultaneously recorded LFPs (left) and multi-unit activity (right) at each pair of electrode.</p>", "links"=>[], "tags"=>["multisite", "lfp", "recordings"], "article_id"=>533809, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g001", "stats"=>{"downloads"=>1, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Simultaneous_multisite_LFP_and_unit_recordings_in_awake_cats_/533809", "title"=>"Simultaneous multisite LFP and unit recordings in awake cats.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:41:28"}
  • {"files"=>["https://ndownloader.figshare.com/files/864023"], "description"=>"<p>A. Sample of the stochastic process and detected peaks. B. Peak size distribution on a log-linear scale. C. Same distribution on a log-log scale. Straight lines indicate the best fit obtained using linear regression.</p>", "links"=>[], "tags"=>["distributions", "thresholded", "poisson", "shot-noise"], "article_id"=>534482, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g007", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Peak_size_distributions_for_the_thresholded_Poisson_shot_noise_process_/534482", "title"=>"Peak-size distributions for the thresholded Poisson shot-noise process.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:45:13"}
  • {"files"=>["https://ndownloader.figshare.com/files/863493"], "description"=>"<p>Top: detection of negative LFP peaks. The LFP signal is shown together with the detected nLFPs (circles). Middle: nLFP-based wave-triggered average (WTA) of unit activity, showing that the negative peaks were associated with an increase of neuronal firing. Bottom: rasters of nLFP activity. The same procedure is compared for high threshold (left panels) and low threshold (right panels).</p>", "links"=>[], "tags"=>["peaks", "potentials", "neuronal"], "article_id"=>533955, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g002", "stats"=>{"downloads"=>6, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Detection_of_negative_peaks_in_local_field_potentials_and_their_relation_with_neuronal_activity_/533955", "title"=>"Detection of negative peaks in local field potentials and their relation with neuronal activity.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:42:18"}
  • {"files"=>["https://ndownloader.figshare.com/files/863861"], "description"=>"<p>A. Shuffled peaks obtained from randomizing the timing of nLFP peaks. B. The WTA indicates that shuffling removes the relationship between nLFPs and neural activity C. Scaling of avalanche peak size distribution, showing similar behavior as for nLFPs (compare with <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0008982#pone-0008982-g003\" target=\"_blank\">Fig. 3</a>).</p>", "links"=>[], "tags"=>["shuffled", "lfp"], "article_id"=>534326, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g005", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Avalanche_analysis_of_shuffled_negative_LFP_peaks_/534326", "title"=>"Avalanche analysis of shuffled negative LFP peaks.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:44:21"}
  • {"files"=>["https://ndownloader.figshare.com/files/863763"], "description"=>"<p>A. Detection of positive LFP peaks using identical procedures as for nLFPs. B. The WTA indicates no relation between pLFPs and unit activity. C. Scaling of avalanche size distribution, showing similar behavior as observed for nLFPs (compare with <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0008982#pone-0008982-g003\" target=\"_blank\">Fig. 3</a>).</p>", "links"=>[], "tags"=>["lfp", "peaks"], "article_id"=>534225, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.g004", "stats"=>{"downloads"=>5, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Avalanche_analysis_of_positive_LFP_peaks_in_the_awake_cat_/534225", "title"=>"Avalanche analysis of positive LFP peaks in the awake cat.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-21 02:43:48"}
  • {"files"=>["https://ndownloader.figshare.com/files/864492"], "description"=>"<p>Results of avalanche analysis for avalanche size defined as the number of LFP peaks within the avalanche, for both positive and negative events. Table headers are the same as in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0008982#pone-0008982-t001\" target=\"_blank\">Table 1</a>. The indicates that the fit is not statistically significant.</p>", "links"=>[], "tags"=>["avalanche", "lfp"], "article_id"=>534955, "categories"=>["Mathematics", "Medicine"], "users"=>["Jonathan Touboul", "Alain Destexhe"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0008982.t002", "stats"=>{"downloads"=>3, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_of_the_avalanche_size_analysis_number_of_LFP_peaks_/534955", "title"=>"Results of the avalanche size analysis (number of LFP peaks).", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-21 02:47:51"}

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  • {"unique-ip"=>"11", "full-text"=>"10", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"9"}
  • {"unique-ip"=>"22", "full-text"=>"12", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"11", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"10"}

Relative Metric

{"start_date"=>"2010-01-01T00:00:00Z", "end_date"=>"2010-12-31T00:00:00Z", "subject_areas"=>[{"subject_area"=>"/Computer and information sciences", "average_usage"=>[375, 666, 837, 949, 1062, 1171, 1233, 1318, 1399, 1470, 1550, 1608, 1701, 1771, 1835, 1908, 1973, 2046, 2091, 2160, 2220, 2285, 2359, 2455, 2537, 2610, 2687, 2778, 2836, 2908, 2974, 3021, 3082, 3169, 3224, 3300, 3382, 3469, 3550, 3640, 3728, 3765, 3814, 3858, 3915, 3952, 3991, 4067, 4107]}, {"subject_area"=>"/Computer and information sciences/Neural networks", "average_usage"=>[385, 711, 886, 982, 1085, 1183, 1228, 1315, 1421, 1482, 1529, 1565, 1602, 1665, 1826, 1879, 1911, 1965, 2035, 2110, 2190, 2272, 2382, 2446, 2476, 2547, 2626, 2707, 2779, 2881, 2995, 3041, 3092, 3126, 3187, 3263, 3365, 3493, 3550, 3639, 3683, 3747, 3821, 3891, 3940, 3998, 4090, 4146, 4192]}, {"subject_area"=>"/Engineering and technology", "average_usage"=>[318, 573, 707, 808, 912, 1021, 1093, 1150, 1214, 1302, 1370, 1431, 1494, 1568, 1629, 1698, 1775, 1829, 1889, 1954, 2007, 2060, 2128, 2170, 2208, 2247, 2311, 2357, 2410, 2453, 2525, 2602, 2660, 2715, 2764, 2818, 2863, 2905, 2970, 3018, 3072, 3127, 3179, 3241, 3298, 3341, 3384, 3433, 3474]}]}
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