Probabilistic Decision Making with Spikes: From ISI Distributions to Behaviour via Information Gain
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{"title"=>"Probabilistic decision making with spikes: From ISI distributions to behaviour via information gain", "type"=>"journal", "authors"=>[{"first_name"=>"Javier A.", "last_name"=>"Caballero", "scopus_author_id"=>"56612904000"}, {"first_name"=>"Nathan F.", "last_name"=>"Lepora", "scopus_author_id"=>"15842160700"}, {"first_name"=>"Kevin N.", "last_name"=>"Gurney", "scopus_author_id"=>"23766890200"}], "year"=>2015, "source"=>"PLoS ONE", "identifiers"=>{"pmid"=>"25923907", "doi"=>"10.1371/journal.pone.0124787", "pui"=>"604050629", "issn"=>"19326203", "sgr"=>"84928799649", "scopus"=>"2-s2.0-84928799649"}, "id"=>"38c24a87-0ec8-36d9-a8a0-d486cc0f7de2", "abstract"=>"Abstract Computational theories of decision making in the brain usually assume that sensory 'evi- dence' is accumulated supporting a number of hypotheses, and that the first accumulator to reach threshold triggers a decision in favour of its associated hypothesis. However, the evi- dence is often assumed to occur as a continuous process whose origins are somewhat ab- stract, with no direct link to the neural signals - action potentials or 'spikes' - that must ultimately form the substrate for decision making in the brain. Here we introduce a new vari- ant of the well-known multi-hypothesis sequential probability ratio test (MSPRT) for decision making whose evidence observations consist of the basic unit of neural signalling - the inter-spike interval (ISI) - and which is based on a new form of the likelihood function. We dub this mechanism s-MSPRT and show its precise form for a range of realistic ISI distribu- tions with positive support. In this way we show that, at the level of spikes, the refractory pe- riod may actually facilitate shorter decision times, and that the mechanism is robust against poor choice of the hypothesized data distribution. We show that s-MSPRT performance is related to the Kullback-Leibler divergence (KLD) or information gain between ISI distribu- tions, through which we are able to link neural signalling to psychophysical observation at the behavioural level. Thus, we find themean information needed for a decision is constant, thereby offering an account of Hick's law (relating decision time to the number of choices). Further, the mean decision time of s-MSPRT shows a power law dependence on the KLD offering an account of Piéron's law (relating reaction time to stimulus intensity). These re- sults show the foundations for a research programme in which spike train analysis can be made the basis for predictions about behavior inmulti-alternative choice tasks.", "link"=>"http://www.mendeley.com/research/probabilistic-decision-making-spikes-isi-distributions-behaviour-via-information-gain", "reader_count"=>25, "reader_count_by_academic_status"=>{"Unspecified"=>2, "Professor > Associate Professor"=>1, "Researcher"=>7, "Student > Doctoral Student"=>1, "Student > Ph. D. Student"=>5, "Student > Master"=>5, "Student > Bachelor"=>3, "Professor"=>1}, "reader_count_by_user_role"=>{"Unspecified"=>2, "Professor > Associate Professor"=>1, "Researcher"=>7, "Student > Doctoral Student"=>1, "Student > Ph. D. Student"=>5, "Student > Master"=>5, "Student > Bachelor"=>3, "Professor"=>1}, "reader_count_by_subject_area"=>{"Engineering"=>2, "Unspecified"=>2, "Medicine and Dentistry"=>2, "Agricultural and Biological Sciences"=>1, "Philosophy"=>1, "Neuroscience"=>6, "Physics and Astronomy"=>1, "Psychology"=>8, "Social Sciences"=>1, "Computer Science"=>1}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>2}, "Medicine and Dentistry"=>{"Medicine and Dentistry"=>2}, "Neuroscience"=>{"Neuroscience"=>6}, "Social Sciences"=>{"Social Sciences"=>1}, "Physics and Astronomy"=>{"Physics and Astronomy"=>1}, "Psychology"=>{"Psychology"=>8}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>1}, "Computer Science"=>{"Computer Science"=>1}, "Unspecified"=>{"Unspecified"=>2}, "Philosophy"=>{"Philosophy"=>1}}, "reader_count_by_country"=>{"United States"=>3}, "group_count"=>4}

Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/2044086"], "description"=>"<p>The left hand column shows the canonical functional form of the distribution in terms of its ‘natural’ parameters. The central column is the evidence contribution <i>L</i><sub><i>i</i></sub>(<i>j</i>), and the right hand column contains expressions for the ‘gains’ in terms of such parameters.</p><p>Analytic expressions for ‘evidence contributions’, <i>L</i><sub><i>i</i></sub>(<i>j</i>) for a range of distributions.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397787, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.t001", "stats"=>{"downloads"=>2, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Analytic_expressions_for_evidence_contributions_L_i_j_for_a_range_of_distributions_/1397787", "title"=>"Analytic expressions for ‘evidence contributions’, <i>L</i><sub><i>i</i></sub>(<i>j</i>) for a range of distributions.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044085"], "description"=>"<p>To ease visualisation, all panels show the exponential of the thresholds; <i>ϕ</i><sub><i>s</i></sub> = exp(<i>θ</i><sub><i>s</i></sub>) <i>ϕ</i><sub><i>u</i></sub> = exp(<i>θ</i><sub><i>u</i></sub>) for s- and u-MSPRT respectively. This yields positive values pertaining to the posterior (rather than negative values for the log-posterior). Panels a and b, respectively, show the lognormal and inverse Gaussian cases in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.g004\" target=\"_blank\">Fig 4a</a> for the parameter set Ω<sub><i>IV</i></sub>. The red lines and symbols are for <i>ϕ</i><sub><i>s</i></sub>, the black lines and symbols for <i>ϕ</i><sub><i>u</i></sub>. In panel c, the box plot labelled ‘lognormal’ shows the median and quartiles (box lines), mean (cross) and one standard deviation of the differences <i>ϕ</i><sub><i>u</i></sub> − <i>ϕ</i><sub><i>s</i></sub> in panels a and b. Other bars show similar quantities for the test distributions of the other MSPRT instantiations used to form <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.g004\" target=\"_blank\">Fig 4a</a>. Panel d is similar to panel c, except it pertains to differences in (exponential) thresholds for results in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.g004\" target=\"_blank\">Fig 4b</a>, with the parameter set Ω<sub><i>FV</i></sub>.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397786, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g011", "stats"=>{"downloads"=>1, "page_views"=>18, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Decision_thresholds_and_their_difference_between_s_MSPRT_and_u_MSPRT_/1397786", "title"=>"Decision thresholds and their difference between s-MSPRT and u-MSPRT.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044080"], "description"=>"<p>Each bar shows the mean decision sample, for <i>N</i> = 10 alternatives, averaged over 950 correct out of 1000 total trials. The parameter set was Ω<sub><i>IV</i></sub>. The pale, patterned bars are for the case when the data is always sampled from an inverse gamma distribution, but inserted into mechanisms which test using the distribution indicated on the <i>x</i> − axis (by definition, the bars have equal height for the inverse gamma). The solid bars are for the case when the tested-for distribution matches the true distribution of ISIs, as indicated on the <i>x</i> − axis. Error bars are at one standard deviation.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397781, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g006", "stats"=>{"downloads"=>0, "page_views"=>16, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Decision_sample_for_u_MSPRT_when_the_distributions_in_the_data_were_not_matched_to_those_tested_for_/1397781", "title"=>"Decision sample for u-MSPRT when the distributions in\nthe data were not matched to those tested for.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044079"], "description"=>"<p>Each bar shows, for the pdf indicated in the legend, the mean decision sample for <i>N</i> = 10 alternatives, averaged over 950 correct out of 1000 total trials. Panel a used parameter sets Ω<sub><i>IV</i></sub>, </p><p></p><p></p><p></p><p><mo>Ω</mo><mo>^</mo></p><p><mi>I</mi><mi>V</mi></p><p></p><mo stretchy=\"false\">(</mo><mn>49</mn><mo>.</mo><mn>5</mn><mo stretchy=\"false\">)</mo><p></p><p></p>, <p></p><p></p><p></p><p><mo>Ω</mo><mo>^</mo></p><p><mi>I</mi><mi>V</mi></p><p></p><mo stretchy=\"false\">(</mo><mn>66</mn><mo stretchy=\"false\">)</mo><p></p><p></p>, <p></p><p></p><p></p><p><mo>Ω</mo><mo>^</mo></p><p><mi>I</mi><mi>V</mi></p><p></p><mo stretchy=\"false\">(</mo><mn>82</mn><mo>.</mo><mn>5</mn><mo stretchy=\"false\">)</mo><p></p><p></p>, panel b used Ω<sub><i>FV</i></sub>, <p></p><p></p><p></p><p><mo>Ω</mo><mo>^</mo></p><p><mi>F</mi><mi>V</mi></p><p></p><mo stretchy=\"false\">(</mo><mn>49</mn><mo>.</mo><mn>5</mn><mo stretchy=\"false\">)</mo><p></p><p></p>, <p></p><p></p><p></p><p><mo>Ω</mo><mo>^</mo></p><p><mi>F</mi><mi>V</mi></p><p></p><mo stretchy=\"false\">(</mo><mn>66</mn><mo stretchy=\"false\">)</mo><p></p><p></p>, <p></p><p></p><p></p><p><mo>Ω</mo><mo>^</mo></p><p><mi>F</mi><mi>V</mi></p><p></p><mo stretchy=\"false\">(</mo><mn>82</mn><mo>.</mo><mn>5</mn><mo stretchy=\"false\">)</mo><p></p><p></p> (see <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#sec019\" target=\"_blank\">Methods</a>). Each group of bars relates to one parameter set with its <i>μ</i><sub>0</sub> indicated on the <i>x</i> − axis (Ω<sub><i>IV</i></sub>, Ω<sub><i>FV</i></sub> have <i>μ</i><sub>0</sub> = 33). For the case of Ω<sub><i>FV</i></sub> and any <p></p><p></p><p></p><p><mo>Ω</mo><mo>^</mo></p><p><mi>F</mi><mi>V</mi></p><p></p><mo stretchy=\"false\">(</mo><p><mi>μ</mi><mn>0</mn></p><mo stretchy=\"false\">)</mo><p></p><p></p>, the gamma and exponential distributions are identical and so not reported separately.<p></p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397780, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g005", "stats"=>{"downloads"=>0, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Decision_sample_in_s_MSPRT_against_mean_ISI_for_range_of_pdfs_and_parameter_sets_/1397780", "title"=>"Decision sample in s-MSPRT against mean ISI for range of pdfs and parameter sets.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044076"], "description"=>"<p>The trial is for an s-MSPRT using the gamma distribution, with 4 choices, under the parameterisation set Ω<sub><i>IV</i></sub> (see <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#sec019\" target=\"_blank\">Methods</a>). Panel a, shows the spike rasters of the 4 spike trains as small vertical line markers, with that of the preferred channel in red. This panel also shows the accumulated evidence <i>y</i><sub><i>k</i></sub>(<i>T</i>) as a red line graph. Panel b shows <i>y</i><sub><i>k</i></sub>(<i>T</i>) of all four hypotheses with the preferred hypothesis in bold red. Panel c shows the posteriors with the preferred hypothesis in black and the others in gray. The threshold is shown by the horizontal dashed line, and was chosen to give a 5% error rate.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397777, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g002", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Time_course_of_signals_in_a_single_trial_of_s_MSPRT_/1397777", "title"=>"Time course of signals in a single trial of s-MSPRT.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044075"], "description"=>"<p>Panel a shows the general MSPRT where all the <i>C</i> data streams contribute to all of the <i>N</i> likelihoods and thus posteriors, which are then evaluated at a termination stage. Panel b only shows the effective components after all simplifications have been applied.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397776, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g001", "stats"=>{"downloads"=>0, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_MSPRT_in_schematic_form_/1397776", "title"=>"MSPRT in schematic form.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044087"], "description"=>"<p>The Neuron tag is that used in the original data repository; the asterisk denotes the data used for the histogram in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.g009\" target=\"_blank\">Fig 9</a>. ‘Coherence’ is the percentage of the dots moving coherently in the random dot stimulus. ‘Direction’ (0 or 1) indicates overall stimulus direction: direction ‘1’ is the preferred direction of the neuron (across its entire tuning curve) while direction ‘0’ is its opposite. ‘Number of ISIs’ is the size of our sample, resulting from pooling data across the ‘number of trials’ indicated in the adjacent column.</p><p>Data sets from [<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.ref031\" target=\"_blank\">31</a>] used to generate ISI distributions.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397788, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.t002", "stats"=>{"downloads"=>0, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Data_sets_from_31_used_to_generate_ISI_distributions_/1397788", "title"=>"Data sets from [31] used to generate ISI distributions.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044084"], "description"=>"<p>The black closed circles show the goodness of fit statistics for each of the first five data sets in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.t002\" target=\"_blank\">Table 2</a>. The red closed squares are for the data set from [<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.ref027\" target=\"_blank\">27</a>]. The mean for all six data sets is shown by the large open circles which also has a line plot; the number of significant fits at a level of 0.05 is noted per pdf next to such circles. Panels a and b are for the Kolomogorov-Smirnov, and Anderson-Darling tests respectively (note log y-axis in the latter). The ordering of the results from left to right preserves rank order of the mean statistic and is the same for both tests.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397785, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g010", "stats"=>{"downloads"=>0, "page_views"=>12, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Goodness_of_fit_of_selected_pdfs_to_ISI_data_/1397785", "title"=>"Goodness of fit of selected pdfs to ISI data.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044082"], "description"=>"<p>Panel a is a direct counterpart of <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.g004\" target=\"_blank\">Fig 4a</a>. The decision samples for s-MSPRT are shown as solid lines and the predictions from <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.e098\" target=\"_blank\">Eq 27</a> shown as solid symbols. Panel b shows results of the virtual experiment derived from <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.e103\" target=\"_blank\">Eq 30</a> (blue symbols) and a best fit power law (solid black line).</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397783, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g008", "stats"=>{"downloads"=>0, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Hick_8217_s_and_Pi_233_ron_8217_s_laws_from_conservation_of_information_/1397783", "title"=>"Hick’s and Piéron’s laws from conservation of information.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044083"], "description"=>"<p>The ISIs in the grey histogram (identical in each panel) were recorded by [<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.ref031\" target=\"_blank\">31</a>] from the MT neuron with tag e093 (<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.t002\" target=\"_blank\">Table 2</a>). The few ISIs lying farther than four standard deviations beyond the mean were not plotted for clarity. Overlaid as a solid blue line in each panel is the best-fit pdf from a set of theoretical distributions: Gaussian, gamma, inverse gamma, inverse Gaussian, lognormal and exponential</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397784, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g009", "stats"=>{"downloads"=>0, "page_views"=>21, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Fitting_pdfs_to_an_ISI_histogram_/1397784", "title"=>"Fitting pdfs to an ISI histogram.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044081"], "description"=>"<p>The KLD in all cases is <i>D</i>(<i>f</i><sub>*</sub>‖<i>f</i><sub>0</sub>) (see text). Panels a, b are for decision with the parameter sets Ω<sub><i>IV</i></sub>, Ω<sub><i>FV</i></sub>, respectively, and both use <i>N</i> = 10. The data points are the open symbols and the dashed lines, best fit power laws (nonlinear least squares). In panel c, the data points shown in blue symbols correspond to all the decision samples in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.g005\" target=\"_blank\">Fig 5</a>. The solid line is the best fit power law.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397782, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g007", "stats"=>{"downloads"=>0, "page_views"=>20, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Mean_decision_sample_for_s_MSPRT_against_KLD_/1397782", "title"=>"Mean decision sample for s-MSPRT against KLD.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044078"], "description"=>"<p>Each panel shows mean decision sample as a function of the number of choices for a range of pdfs (see legend) and for the two alternative mechanisms: s-MSPRT (solid lines) and u-MSPRT (solid circles). Panels a and b are for the parameter sets Ω<sub><i>IV</i></sub> and Ω<sub><i>FV</i></sub> respectively (see <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#sec019\" target=\"_blank\">Methods</a>). In the case of Ω<sub><i>FV</i></sub>, the gamma and exponential distributions are identical and so not reported separately. All data points are the mean of 950 correct, out of 1000 total trials (see text for inverse gamma based s-MSPRT). Error bars are omitted for clarity and are small; the standard error of the mean is typically 2% of the mean.</p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397779, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g004", "stats"=>{"downloads"=>0, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Mean_decision_samples_against_number_of_choices_for_a_range_of_pdfs_parameter_sets_and_mechanisms_/1397779", "title"=>"Mean decision samples against number of choices for a range of pdfs, parameter sets, and mechanisms.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/2044077"], "description"=>"<p>Panels a-d show the lognormal, gamma, inverse Gaussian, and inverse gamma pdfs respectively, for the independent variance parameter set Ω<sub><i>IV</i></sub> (see <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#sec019\" target=\"_blank\">Methods</a>). The ‘preferred’ and ‘null’ density functions (<i>f</i><sub>*</sub>, <i>f</i><sub>0</sub>) are in red and black respectively. The plots are for ISIs from 1 to 100 ms. For infinitesimal ISIs, the lognormal, inverse Gaussian and inverse gamma tend to zero; for the the gamma the pdf grows up to a bound as the ISI tends to zero. Panels e-f are the corresponding contributions <i>L</i><sub><i>i</i></sub>(<i>j</i>) to the accumulated ‘evidence’ <i>y</i><sub><i>i</i></sub>(<i>T</i>) (see <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.e019\" target=\"_blank\">Eq 9</a>) and the separate components therein (see <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0124787#pone.0124787.t001\" target=\"_blank\">Table 1</a>). <i>L</i><sub><i>i</i></sub>(<i>j</i>) itself is shown in red, the constant term </p><p></p><p></p><p><mi>g</mi><mn>0</mn><mi>D</mi></p><p></p><p></p> (<i>D</i> = <i>L</i>, <i>γ</i>, <i>S</i>, <i>M</i>) by the solid black line, and non-constant terms by dashed-grey and broken-black lines. The horizontal dashed grey line indicates 0 on the <i>y</i>-axis.<p></p>", "links"=>[], "tags"=>["spike train analysis", "decision time", "MSPRT", "power law dependence", "Information Gain Computational theories", "choice", "evidence", "ISI distributions", "KLD"], "article_id"=>1397778, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Javier A. Caballero", "Nathan F. Lepora", "Kevin N. Gurney"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0124787.g003", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_two_parameter_families_of_pdfs_top_and_their_evidence_contributions_L_i_j_bottom_/1397778", "title"=>"The two-parameter families of pdfs (top) and their ‘evidence contributions’ <i>L</i><sub><i>i</i></sub>(<i>j</i>) (bottom).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-04-29 02:57:14"}

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Relative Metric

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