Movement Timing and Invariance Arise from Several Geometries
Publication Date
July 10, 2009
Journal
PLOS Computational Biology
Authors
Daniel Bennequin, Ronit Fuchs, Alain Berthoz & Tamar Flash
Volume
5
Issue
7
Pages
e1000426
DOI
https://dx.plos.org/10.1371/journal.pcbi.1000426
Publisher URL
http://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1000426
PubMed
http://www.ncbi.nlm.nih.gov/pubmed/19593380
PubMed Central
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2702097
Europe PMC
http://europepmc.org/abstract/MED/19593380
Web of Science
000269220100020
Scopus
68249106770
Mendeley
http://www.mendeley.com/research/movement-timing-invariance-arise-several-geometries
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Mendeley | Further Information

{"title"=>"Movement timing and invariance arise from several geometries", "type"=>"journal", "authors"=>[{"first_name"=>"Daniel", "last_name"=>"Bennequin", "scopus_author_id"=>"14522045100"}, {"first_name"=>"Ronit", "last_name"=>"Fuchs", "scopus_author_id"=>"35071191500"}, {"first_name"=>"Alain", "last_name"=>"Berthoz", "scopus_author_id"=>"7005666347"}, {"first_name"=>"Tamar", "last_name"=>"Flash", "scopus_author_id"=>"7004131143"}], "year"=>2009, "source"=>"PLoS Computational Biology", "identifiers"=>{"scopus"=>"2-s2.0-68249106770", "sgr"=>"68249106770", "issn"=>"1553734X", "doi"=>"10.1371/journal.pcbi.1000426", "pmid"=>"19593380", "isbn"=>"1553-7358; 1553-734X", "pui"=>"355053822"}, "id"=>"e97c2520-f7b1-3316-9b53-c6c9f0c33efd", "abstract"=>"Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartan's moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limaçons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations.", "link"=>"http://www.mendeley.com/research/movement-timing-invariance-arise-several-geometries", "reader_count"=>84, "reader_count_by_academic_status"=>{"Unspecified"=>2, "Professor > Associate Professor"=>7, "Researcher"=>23, "Student > Doctoral Student"=>1, "Student > Ph. D. Student"=>29, "Student > Postgraduate"=>5, "Student > Master"=>5, "Other"=>2, "Student > Bachelor"=>1, "Lecturer"=>2, "Professor"=>7}, "reader_count_by_user_role"=>{"Unspecified"=>2, "Professor > Associate Professor"=>7, "Researcher"=>23, "Student > Doctoral Student"=>1, "Student > Ph. D. Student"=>29, "Student > Postgraduate"=>5, "Student > Master"=>5, "Other"=>2, "Student > Bachelor"=>1, "Lecturer"=>2, "Professor"=>7}, "reader_count_by_subject_area"=>{"Unspecified"=>3, "Agricultural and Biological Sciences"=>17, "Philosophy"=>1, "Chemistry"=>1, "Computer Science"=>16, "Earth and Planetary Sciences"=>1, "Engineering"=>9, "Medicine and Dentistry"=>10, "Neuroscience"=>8, "Design"=>1, "Physics and Astronomy"=>5, "Psychology"=>11, "Social Sciences"=>1}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>10}, "Social Sciences"=>{"Social Sciences"=>1}, "Physics and Astronomy"=>{"Physics and Astronomy"=>5}, "Psychology"=>{"Psychology"=>11}, "Unspecified"=>{"Unspecified"=>3}, "Design"=>{"Design"=>1}, "Engineering"=>{"Engineering"=>9}, "Chemistry"=>{"Chemistry"=>1}, "Neuroscience"=>{"Neuroscience"=>8}, "Earth and Planetary Sciences"=>{"Earth and Planetary Sciences"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>17}, "Computer Science"=>{"Computer Science"=>16}, "Philosophy"=>{"Philosophy"=>1}}, "reader_count_by_country"=>{"Republic of Singapore"=>1, "Canada"=>1, "Sweden"=>1, "Netherlands"=>1, "Belgium"=>1, "United States"=>1, "Italy"=>2, "Israel"=>2, "France"=>5}, "group_count"=>6}

Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/892102"], "description"=>"<p>The distribution of the functions aggregated over all trials of the same figural form. A point within the triangle gives the values of the , and functions where . The values of function for such a point are equal to the area delineated by the small triangle created by passing lines between this specific point and the two bottom vertices. The values of are equal to the area delineated by the small triangle created by passing lines between this specific point and the left bottom and top vertices. The values of function are equal to the area delineated by the small triangle created be passing lines between this point and the right bottom and top vertices. For example, a point on the triangle's edge marked by is a point where . For a point located at the top vertex and . In the center of the triangle . The color of any point within the large triangle indicates the number of times that that specific combination of function values was found. A white point shows a combination that did not appear in any of the trials. A dark blue point represents a combination occurring many times. Panel (A) contains all the trials of the drawing of cloverleaves. Panel (B) contains all the trials of the drawing of oblate limaçon. Panel (C) contains all the trials of the drawing of asymmetric lemniscate. Panel (D) contains all the trials of the locomotion of cloverleaves. Panel (E) contains all the trials of the locomotion of oblate limaçon. Panel (F) contains all the trials of the locomotion of asymmetric lemniscate.</p>", "links"=>[], "tags"=>["functions"], "article_id"=>562562, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g009", "stats"=>{"downloads"=>1, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Representation_of_the_values_of_the_three_functions_during_the_different_trials_/562562", "title"=>"Representation of the values of the three functions during the different trials.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:42:42"}
  • {"files"=>["https://ndownloader.figshare.com/files/892427"], "description"=>"<p>Log gain factor versus log area.</p><p>Linear regression: , for each of the three speed conditions (Slow, Natural, Fast) and the three subjects (). Presented are the best and , and the values of the coefficient of determination () for the different ellipses. <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000426#pcbi.1000426.s001\" target=\"_blank\">Figure S1</a> shows the data and the regression lines corresponding to table.</p>", "links"=>[], "tags"=>["elliptic", "drawings"], "article_id"=>562887, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.t001", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Statistical_analysis_of_elliptic_drawings_1_/562887", "title"=>"Statistical analysis of elliptic drawings (1).", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2009-07-10 00:48:07"}
  • {"files"=>["https://ndownloader.figshare.com/files/891465"], "description"=>"<p>Log movement time (T) is plotted versus log perimeter (P). The regression lines between log T and Log P are shown for each subject , and and each average speed condition (slow, natural and fast, red, green and black, respectively). Ellipses with different eccentricities are marked by different symbols (circle, cross and plus for narrow, medium eccentricity and circles, respectively). The regression parameters were calculated for all eccentricities together. The parameters of the regression lines are presented in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000426#pcbi-1000426-t003\" target=\"_blank\">Table 3</a>.</p>", "links"=>[], "tags"=>["elliptic", "drawings"], "article_id"=>561929, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g002", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Experimental_data_of_elliptic_drawings_and_regressions_/561929", "title"=>"Experimental data of elliptic drawings and regressions.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:32:09"}
  • {"files"=>["https://ndownloader.figshare.com/files/441815", "https://ndownloader.figshare.com/files/441846", "https://ndownloader.figshare.com/files/441875", "https://ndownloader.figshare.com/files/441903", "https://ndownloader.figshare.com/files/441935", "https://ndownloader.figshare.com/files/441977", "https://ndownloader.figshare.com/files/442018", "https://ndownloader.figshare.com/files/442066", "https://ndownloader.figshare.com/files/442090", "https://ndownloader.figshare.com/files/442138", "https://ndownloader.figshare.com/files/442179", "https://ndownloader.figshare.com/files/442217"], "description"=>"<div><p>Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartan's moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limaçons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations.</p></div>", "links"=>[], "tags"=>["invariance", "geometries"], "article_id"=>147098, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1000426.s001", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s002", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s003", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s004", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s005", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s006", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s007", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s008", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s009", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s010", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s011", "https://dx.doi.org/10.1371/journal.pcbi.1000426.s012"], "stats"=>{"downloads"=>37, "page_views"=>51, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Movement_Timing_and_Invariance_Arise_from_Several_Geometries/147098", "title"=>"Movement Timing and Invariance Arise from Several Geometries", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2009-07-10 01:58:18"}
  • {"files"=>["https://ndownloader.figshare.com/files/892515"], "description"=>"<p>Column <i>Test 1:% significance</i> shows the percentage of trials for which the existence of special segments was found to be statistically significant. This was determined by comparing the success in matching the empirical velocities either with the model for the combination of velocities or their approximation using the mean velocity values. The analysis was performed in the log velocity space.</p><p>Column <i>Test 2:% significance</i> shows the percentage of trials for which the existence of special segments was found to be statistically significant. Here the success in matching the empirical velocities achieved by the model of the combination of geometries was compared with that using the trigonometric approximation. The analysis was performed in the log velocity space.</p>", "links"=>[], "tags"=>["mathematics", "neuroscience/motor systems", "neuroscience/theoretical neuroscience"], "article_id"=>562971, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.t004", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_of_the_tests_for_statistical_significance_of_the_presence_of_intervals_/562971", "title"=>"Results of the tests for statistical significance of the presence of intervals.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2009-07-10 00:49:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/892392"], "description"=>"<p>The PLL is: if , if ; the regular Power Law is , .</p><p>For each of the three eccentricities and three subjects the table presents the best and for the PLL and the best and for the regular power law, plus the ratios of the sums of square errors () for the two linear regressions for the different ellipses. Also presented are the probabilities that the PLL model is better than the power law model, computed according to the equation: where and is the number of trials.</p>", "links"=>[], "tags"=>["elliptic", "drawings", "velocity", "euclidian", "piecewise", "linear", "compared"], "article_id"=>562844, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.t002", "stats"=>{"downloads"=>1, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Statistical_analysis_of_elliptic_drawings_2_Log_Velocity_versus_log_Euclidian_curvature_piecewise_linear_law_PLL_compared_to_the_regular_power_law_/562844", "title"=>"Statistical analysis of elliptic drawings (2): Log Velocity versus log Euclidian curvature, piecewise linear law (PLL) compared to the regular power law.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2009-07-10 00:47:24"}
  • {"files"=>["https://ndownloader.figshare.com/files/891556"], "description"=>"<p>(A) Akaike's Information Criterion () scores averaged across subjects and repetitions for the 4 models for all drawn shapes. Also shown are the standard deviations (SDs)of the AIC scores. The lower the score is, the better is the model. Red bars show the scores of the model of the combination of geometries (); green bars, scores of the constrained minimum jerk model (); yellow bars, the constant equi-affine speed model (); cyan bars, scores of the constant affine velocity model (). (B) Brown bars show the average probabilities (averaged across subjects and repetitions) that the combined model is better than the minimum-jerk model for the different shapes. Standard deviations are also shown. The probabilities were calculated according to the equation , where is the differences in scores between the two models. In both figure panels, the cloverleaves are marked by in the order of ascending speed. The limaçon and the lemniscate are marked by and by , respectively, according to the ascending ratios of perimeters of the large to the small loops.</p>", "links"=>[], "tags"=>["scores", "models", "figural"], "article_id"=>562015, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g003", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Drawing_data_scores_for_the_4_models_and_for_the_different_figural_forms_/562015", "title"=>"Drawing data: scores for the 4 models and for the different figural forms.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:33:35"}
  • {"files"=>["https://ndownloader.figshare.com/files/892455"], "description"=>"<p>The mean and SD values of the functions of the combinations. The function represents the influence of affine geometry in the combination of geometries. The function represents the influence of the equi-affine geometry in the combination of geometries. The function represents the influence of Euclidian geometry in the combination of geometries.</p>", "links"=>[], "tags"=>["functions", "figural"], "article_id"=>562916, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.t006", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_mean_values_of_the_functions_for_the_different_figural_forms_/562916", "title"=>"The mean values of the functions for the different figural forms.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2009-07-10 00:48:36"}
  • {"files"=>["https://ndownloader.figshare.com/files/891944"], "description"=>"<p>Summary of the scores for all 4 models for all the figural forms for both drawing and locomotion data. The bars represent mean scores ±SD averaged over all subjects and trials. Red, score obtained for the model of the combination of geometries (); green, score of the constrained minimum-jerk model (); yellow, score for the constant equi-affine velocity model (); cyan, score for the constant affine velocity model (). For the marking of the different forms see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000426#pcbi-1000426-g003\" target=\"_blank\">Figure 3</a>.</p>", "links"=>[], "tags"=>["scores", "models", "figural"], "article_id"=>562405, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g007", "stats"=>{"downloads"=>2, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_scores_of_the_4_models_for_the_different_figural_forms_/562405", "title"=>"The scores of the 4 models for the different figural forms.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:40:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/892202"], "description"=>"<p>The values of all the possible constant coefficients for the equation: . A point within a triangle describes the values of , and where . The values of equal the area delineated by the small triangle created by passing lines between this specific point and the two bottom vertices, where the area of the large triangle is equal to 1. The values of are equal to the area delineated by the small triangle created by passing lines between this specific point to the left bottom and top vertices. The values of are equal to the area delineated by the small triangle created by passing lines between the point to the right bottom and top vertices. For example, at a point on the edge of the triangle marked by . For a point located on the top vertex and . In the center of the triangle . The color of a point represents the value of the score for the corresponding combination of the values; the darker the color, the higher the value of the scores. The red points are those with the highest score. This value is given in red beside each triangle. Panel (A) contain the data of the locomotion of oblate limaçon. Panel (B) contain the data of the drawing of oblate limaçon. Panel (C) contain the data of the locomotion of asymmetric lemniscate. Panel (D) contain the data of the drawing of asymmetric lemniscate.</p>", "links"=>[], "tags"=>["coefficient", "combinations"], "article_id"=>562662, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g010", "stats"=>{"downloads"=>1, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_score_for_all_the_coefficient_combinations_for_equation_13_/562662", "title"=>"The score for all the coefficient combinations for equation 13.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:44:22"}
  • {"files"=>["https://ndownloader.figshare.com/files/892010"], "description"=>"<p>The mean values of the and functions averaged over trials and subjects, summarized over the templates of the different figural forms. In panel (A) the values of the functions are aggregated and in panel (B) they are displayed separately with their corresponding SDs. For the marking of the different forms see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000426#pcbi-1000426-g003\" target=\"_blank\">Figure 3</a>.</p>", "links"=>[], "tags"=>["functions", "figural"], "article_id"=>562472, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g008", "stats"=>{"downloads"=>1, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_mean_values_of_the_functions_for_the_different_figural_forms_/562472", "title"=>"The mean values of the functions for the different figural forms.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:41:12"}
  • {"files"=>["https://ndownloader.figshare.com/files/891635"], "description"=>"<p>(A) Akaike's Information Criterion () scores, averaged across subjects and repetitions, for the 4 models for all the shapes of the locomotion data. SDs of the scores are also shown. The figure in the upper row in panel (A) shows the scores for the (all data), while the figure in the lower row in panel (A) shows the scores for the data. The lower the score is, the better is the model. Red bars show the scores of the combination of geometries model (); green bars, scores of the constrained minimum-jerk model (); yellow bars, the constant equi-affine speed model (); cyan bars, the scores of the constant affine velocity model (). (B) Brown bars show the average probabilities (averaged across subjects and repetitions) that the combined model is better than the minimum-jerk model for the different shapes. SDs are also shown. The probabilities were calculated according to the equation , where is the differences in scores between the two models. The figure on the left (panel (B)) shows the results for the , while panel (B) on the right shows the results for the . In both figure panels, the cloverleaves are marked by . The limaçon and the lemniscate are marked by and by , respectively, according to the ascending ratios of perimeters of the large to the small loops.</p>", "links"=>[], "tags"=>["scores", "models", "figural"], "article_id"=>562093, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g004", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Locomotion_data_scores_for_the_4_models_and_for_the_different_figural_forms_/562093", "title"=>"Locomotion data: scores for the 4 models and for the different figural forms.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:34:53"}
  • {"files"=>["https://ndownloader.figshare.com/files/891817"], "description"=>"<p>Every row shows an example of the second repetition of a locomotion trial. First row, walking around a cloverleaf. Second row, walking along an oblate limaçon. Third row, walking around an asymmetric lemniscate. Panels (A), (D) and (G) show the paths drawn by the subject. The colors on the paths represent the Euclidian curvature; Blue, segments with a relatively low curvature (∼0); red, segments with a higher curvature (∼0.75). Color scale is shown in the panel. Panels (B), (E) and (H), the velocity profiles of the drawing. Red, experimental velocity profile; blue, the velocity profile predicted by the model of the combination of geometries. Panels (C), (F) and (I) show values of the functions. Red area, value of the function; green area, value of the function; blue area, value of the function. The values are aggregated one above the other such that their sum equals 1.</p>", "links"=>[], "tags"=>["locomotion"], "article_id"=>562280, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g006", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Examples_from_the_locomotion_experiments_/562280", "title"=>"Examples from the locomotion experiments.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:38:00"}
  • {"files"=>["https://ndownloader.figshare.com/files/892288"], "description"=>"<p>The red dots represent the experimentally measured ratios of movement durations versus experimentally measured ratios of movement lengths . The black line represents the function for one of the set of constant <i>B</i>-s in the region of high scores.</p>", "links"=>[], "tags"=>["experimentally", "ratios", "durations", "euclidian"], "article_id"=>562745, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g011", "stats"=>{"downloads"=>1, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Examples_of_the_experimentally_measured_ratios_of_movement_durations_versus_the_experimentally_measured_ratios_of_Euclidian_lengths_/562745", "title"=>"Examples of the experimentally measured ratios of movement durations versus the experimentally measured ratios of Euclidian lengths.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:45:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/891336"], "description"=>"<p>Empirical values (blue) of the pairs are compared to the piecewise regression lines of the PLL: for (red line), and for (green line), versus the regression line of the regular Power Law ( black). For all . For this trajectory , and .</p>", "links"=>[], "tags"=>["elliptic", "comparing", "piecewise", "linear", "velocity", "euclidian"], "article_id"=>561786, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g001", "stats"=>{"downloads"=>0, "page_views"=>22, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_An_example_of_elliptic_segmentation_Comparing_the_piecewise_linear_law_PLL_of_log_versus_log_versus_the_regular_Power_Law_where_is_the_velocity_and_is_the_Euclidian_curvature_/561786", "title"=>"An example of elliptic segmentation: Comparing the piecewise linear law (PLL) of log versus log , versus the regular Power Law, where is the velocity and is the Euclidian curvature.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:29:46"}
  • {"files"=>["https://ndownloader.figshare.com/files/892351"], "description"=>"<p>The equation for the linear regression is: . The best and are presented for each of the three subjects and three speeds, as well as the goodness of fit of the data points to the linear regression expressed by: (). Note that the values assess how much the approximation of the data provided by the linear regression is better than the mean value of the data. For subject the scores are low because the slopes () are close to zero. Hence, the linear approximations are not better than the mean values of the data points. For further details see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000426#pcbi-1000426-g002\" target=\"_blank\">Figure 2</a>.</p>", "links"=>[], "tags"=>["elliptic", "drawings"], "article_id"=>562811, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.t003", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Statistical_analysis_of_elliptic_drawings_3_Log_time_versus_log_perimeter_/562811", "title"=>"Statistical analysis of elliptic drawings (3): Log time versus log perimeter.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2009-07-10 00:46:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/892484"], "description"=>"<p>The means and SDs of the scores for the pure equi-affine and affine geometries, minimum-jerk and the combination of the Euclidian, equi-afine and affine geometries. Remark: the scores of non-linear functions can be negative. In this case we say that the mean value of the data points matches the data better than the values predicted by the tested model.</p>", "links"=>[], "tags"=>["scores", "models", "figural"], "article_id"=>562940, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.t005", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_scores_of_the_4_models_for_the_various_figural_forms_/562940", "title"=>"The scores of the 4 models for the various figural forms.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2009-07-10 00:49:00"}
  • {"files"=>["https://ndownloader.figshare.com/files/891718"], "description"=>"<p>Every row shows an example of the second repetition of a drawing trial. First row, drawing of a cloverleaf; second row, drawing of an oblate limaçon; third row, drawing of an asymmetric lemniscate. Panels (A), (D) and (G) show the paths drawn by the subject. The colors marked on the paths represent the Euclidian curvature. Blue segments have relatively low curvature (∼0), red segments have a higher curvature (∼0.75). Color scale is shown at the top of the panel. Panels (B), (E) and (H) show the velocity profiles of the drawing. Red, experimental velocity profile; blue, velocity profile predicted by the model of the combination of geometries. Panels (C), (F) and (I) show values of the functions. Red area, value of the function; green area value of the function; blue area, value of the function. The values are aggregated one above the other such that their sum equals 1.</p>", "links"=>[], "tags"=>["mathematics", "neuroscience/motor systems", "neuroscience/theoretical neuroscience"], "article_id"=>562174, "categories"=>["Neuroscience", "Mathematics"], "users"=>["Daniel Bennequin", "Ronit Fuchs", "Alain Berthoz", "Tamar Flash"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000426.g005", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Examples_from_the_drawing_experiment_/562174", "title"=>"Examples from the drawing experiment.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2009-07-10 00:36:14"}

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Relative Metric

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