The Effects of Theta Precession on Spatial Learning and Simplicial Complex Dynamics in a Topological Model of the Hippocampal Spatial Map
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{"title"=>"The Effects of Theta Precession on Spatial Learning and Simplicial Complex Dynamics in a Topological Model of the Hippocampal Spatial Map", "type"=>"journal", "authors"=>[{"first_name"=>"Mamiko", "last_name"=>"Arai", "scopus_author_id"=>"56233475500"}, {"first_name"=>"Vicky", "last_name"=>"Brandt", "scopus_author_id"=>"57032289100"}, {"first_name"=>"Yuri", "last_name"=>"Dabaghian", "scopus_author_id"=>"6603112765"}], "year"=>2014, "source"=>"PLoS Computational Biology", "identifiers"=>{"pmid"=>"24945927", "issn"=>"15537358", "doi"=>"10.1371/journal.pcbi.1003651", "pui"=>"373401389", "isbn"=>"1553-734x", "scopus"=>"2-s2.0-84903397364", "sgr"=>"84903397364"}, "id"=>"3059cd1d-345c-34f9-9f1d-2b413648b0d4", "abstract"=>"Learning arises through the activity of large ensembles of cells, yet most of the data neuroscientists accumulate is at the level of individual neurons; we need models that can bridge this gap. We have taken spatial learning as our starting point, computationally modeling the activity of place cells using methods derived from algebraic topology, especially persistent homology. We previously showed that ensembles of hundreds of place cells could accurately encode topological information about different environments (\"learn\" the space) within certain values of place cell firing rate, place field size, and cell population; we called this parameter space the learning region. Here we advance the model both technically and conceptually. To make the model more physiological, we explored the effects of theta precession on spatial learning in our virtual ensembles. Theta precession, which is believed to influence learning and memory, did in fact enhance learning in our model, increasing both speed and the size of the learning region. Interestingly, theta precession also increased the number of spurious loops during simplicial complex formation. We next explored how downstream readout neurons might define co-firing by grouping together cells within different windows of time and thereby capturing different degrees of temporal overlap between spike trains. Our model's optimum coactivity window correlates well with experimental data, ranging from ∼150-200 msec. We further studied the relationship between learning time, window width, and theta precession. Our results validate our topological model for spatial learning and open new avenues for connecting data at the level of individual neurons to behavioral outcomes at the neuronal ensemble level. Finally, we analyzed the dynamics of simplicial complex formation and loop transience to propose that the simplicial complex provides a useful working description of the spatial learning process.", "link"=>"http://www.mendeley.com/research/effects-theta-precession-spatial-learning-simplicial-complex-dynamics-topological-model-hippocampal", "reader_count"=>38, "reader_count_by_academic_status"=>{"Unspecified"=>1, "Professor > Associate Professor"=>2, "Researcher"=>8, "Student > Doctoral Student"=>3, "Student > Ph. D. Student"=>13, "Student > Postgraduate"=>3, "Student > Master"=>4, "Other"=>2, "Professor"=>2}, "reader_count_by_user_role"=>{"Unspecified"=>1, "Professor > Associate Professor"=>2, "Researcher"=>8, "Student > Doctoral Student"=>3, "Student > Ph. D. Student"=>13, "Student > Postgraduate"=>3, "Student > Master"=>4, "Other"=>2, "Professor"=>2}, "reader_count_by_subject_area"=>{"Unspecified"=>3, "Agricultural and Biological Sciences"=>7, "Philosophy"=>1, "Chemistry"=>1, "Computer Science"=>5, "Biochemistry, Genetics and Molecular Biology"=>2, "Materials Science"=>1, "Mathematics"=>4, "Medicine and Dentistry"=>3, "Neuroscience"=>4, "Physics and Astronomy"=>3, "Psychology"=>2, "Social Sciences"=>1, "Linguistics"=>1}, "reader_count_by_subdiscipline"=>{"Materials Science"=>{"Materials Science"=>1}, "Medicine and Dentistry"=>{"Medicine and Dentistry"=>3}, "Social Sciences"=>{"Social Sciences"=>1}, "Physics and Astronomy"=>{"Physics and Astronomy"=>3}, "Psychology"=>{"Psychology"=>2}, "Mathematics"=>{"Mathematics"=>4}, "Unspecified"=>{"Unspecified"=>3}, "Chemistry"=>{"Chemistry"=>1}, "Neuroscience"=>{"Neuroscience"=>4}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>7}, "Computer Science"=>{"Computer Science"=>5}, "Linguistics"=>{"Linguistics"=>1}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>2}, "Philosophy"=>{"Philosophy"=>1}}, "reader_count_by_country"=>{"Netherlands"=>1, "United States"=>1, "France"=>1}, "group_count"=>0}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/1543004"], "description"=>"<p>As an animal (in experiments, typically a rodent) explores a space, place cells fire in discrete locations that are mapped onto the space as place fields (<b>b</b>, colored ovals). (<b>A</b>) Shown are seven place cells firing, with some temporal overlap. (<b>B</b>) <i>Top</i>: The seven corresponding place fields, along with a fragment of an animal's trajectory (dashed line). <i>Bottom:</i> The elements of the nerve (a.k.a. Čech) simplicial complex generated by the overlaps among place fields. To form a simplicial complex, each place field center is considered to be a vertex, and each link between vertices is a simplex. Each simplex <i>σ<sub>ij</sub></i> or <i>σ<sub>ijk</sub></i> is labeled to indicate the vertices linked, e.g., σ<sub>617</sub> indicates a link between vertices 6,1 and 7. (<b>C</b>) Persistent homology “barcodes” show the timelines of 0<i>D</i> and 1<i>D</i> loops, respectively: each colored horizontal line represents one 0<i>D</i> loop (top panel) or one 1<i>D</i> loop (bottom panel). The time <i>T<sub>min</sub></i> (dotted red vertical lines) marks the moment when spurious loops (topological ‘noise’) disappear and the correct number of loops persists, in this case one in 0<i>D</i> and one in 1<i>D</i>, indicating that there is one hole in the environment. Thus, <i>T<sub>min</sub></i> is the time after which the correct topological information emerges, which corresponds to the map formation or learning time in this environment, for this particular ensemble of place cells, operating under particular conditions of mean firing rate, mean place field size, number of cells in the population.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "simplicial", "derived"], "article_id"=>1064128, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g001", "stats"=>{"downloads"=>1, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_A_simplicial_complex_can_be_derived_from_place_cell_co_firing_/1064128", "title"=>"A simplicial complex can be derived from place cell co-firing.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 15:12:30"}
  • {"files"=>["https://ndownloader.figshare.com/files/1541575"], "description"=>"<p>(<b>A</b>) The point clouds depict the core of the learning region, as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003651#pcbi-1003651-g002\" target=\"_blank\"><b>Figure 2B</b></a>, for the sake of legibility. In both the <i>θ</i>-off and <i>θ</i>-on cases, note the red circles at the lower boundary of L, indicating longer-persisting loops. (<b>B</b>) Histograms showing the distributions of spurious loop durations in the <i>θ</i>-off and <i>θ</i>-on cases. We performed the Kolmogorov-Smirnov (KS) test to statistically compare the five cases against one another: 0  =  no theta, 1 = a single 8 Hz sinusoidal wave, 4 = a combination of four sinusoids, M = a subcortical EEG signal from a wild-type mouse, and R = a subcortical EEG signal recorded from a wild-type rat. Black squares indicate a significant difference (<i>p</i><0.05) in loop duration between <i>θ</i>-off and all other cases; the <i>p</i>-values for pairwise comparisons between different <i>θ</i>-on cases reveal no significant difference (gray squares, <i>p</i>>0.2). The statistical similarity of all the <i>θ</i>-on distributions enabled us to combine the <i>θ</i>-on loop duration data to obtain better statistics for the histogram. The structure of the histograms is fit by the gamma distribution (blue lines). We used the smooth histogram profile (dashed line) to show that the typical duration of the topological loops in the <i>θ</i>-on cases is about half that of the <i>θ</i>-off case. (<b>C</b>) The typical number of loops in the <i>θ</i>-on cases is ∼40% higher than in the absence of <i>θ</i>-precession. The <i>θ</i>-off histogram is fit by the GEV distribution (red line), while the <i>θ</i>-on histogram is better fit by gamma distribution (blue). The KS diagram on the right shows that while the simulated <i>θ</i>-oscillations produce similar results, the signals recorded from rat and mouse produce statistically different distributions from the simulated signals. (<b>D</b>) The maximal number of spurious loops observed in the <i>θ</i>-on cases, fit by the gamma distributions, is less than half that in the <i>θ</i>-off case.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "precession", "duration", "spurious", "loops", "simplicial"], "article_id"=>1063040, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g003", "stats"=>{"downloads"=>1, "page_views"=>16, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Theta_phase_precession_reduces_the_duration_of_spurious_loops_in_the_simplicial_complex_/1063040", "title"=>"Theta phase precession reduces the duration of spurious loops in the simplicial complex.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 03:42:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/1541572"], "description"=>"<p>(<b>A</b>) Point clouds representing the mean learning times <i>T<sub>min</sub></i> computed for the <i>θ</i>-off case (left) and the maps driven by a <i>θ</i>–signal recorded in rat (right). Each point corresponds to a place cell ensemble with a specific number of place cells, <i>N</i>, the mean ensemble firing rate, <i>f</i>, the mean ensemble place field size <i>s</i>. Dark blue circles represent those ensembles that form correct topological maps most rapidly and reliably; as the color shades from blue through green, yellow, and red, the learning times increase and map formation becomes less reliable (see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003651#pcbi.1003651-Dabaghian1\" target=\"_blank\">[4]</a>, Methods). The rat <i>θ</i>–signal enlarges the learning region L and speeds map formation. (<b>B</b>) To zero in on the effect of <i>θ</i> precession on the quality of learning, these clouds depict only the maps that converged at least 7 out of 10 times (<i>ρ</i> ≥0.7), and for which the variance of the learning times, <i>ξ</i> =  Δ<i>T<sub>min</sub></i>/<i>T<sub>min</sub></i>, did not exceed 30% of the mean value. Even in this more rigorously defined core of L, with ensembles that already function well, the <i>θ</i>–signal has a pronounced effect. (<b>C</b>) Histograms of the minimal times obtained in the <i>θ</i>-off (left) and the <i>θ</i>-driven case (right), fit by the GEV distribution. The blue dot marks the mode of the distribution; m in the center of each panel gives the value of the mode. All convergent maps are included. The typical learning time <i>T<sub>min</sub></i> in the <i>θ</i>-on case is about half as long as in the <i>θ</i>-off case. (<b>D</b>) The same histograms obtained for the core of L (<i>ρ</i> ≥0.7, <i>ξ</i>≤0.3). The typical learning time <i>T<sub>min</sub></i> in the <i>θ</i>-on case is about 6.5 minutes, whereas without <i>θ</i> it is 15% longer. (<b>E</b>) One of the major effects of <i>θ</i> phase precession is to reduce the variability of the learning times. The histograms show that the typical value of the relative variation <i>ξ</i> in the <i>θ</i>-on case is less than half that of the <i>θ</i>-off case, i.e., that repeated simulations of the <i>θ</i>-driven maps more reliably reproduce similar learning time values.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "precession", "enlarges"], "article_id"=>1063038, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g002", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Theta_precession_both_enlarges_the_learning_region_and_reduces_mean_learning_times_/1063038", "title"=>"Theta precession both enlarges the learning region and reduces mean learning times.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 03:42:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/1543013"], "description"=>"<p>(A) The distribution of the stabilization window widths <i>w<sub>s</sub></i> across all convergent maps (any finite <i>ρ</i> and <i>ξ</i> values) in <i>θ</i>-off and <i>θ</i>-on cases. (B) Statistical distribution of <i>w<sub>s</sub></i> in <i>θ</i>-off and <i>θ</i>-on cases and show a 15% statistically significant difference in the typical value of the <i>w<sub>s</sub></i> for the <i>θ</i>-on cases.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "optimal", "width", "larger"], "article_id"=>1064137, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g006", "stats"=>{"downloads"=>1, "page_views"=>14, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_optimal_window_width_is_slightly_larger_than_a_single_952_period_/1064137", "title"=>"The optimal window width is slightly larger than a single <i>θ</i>-period.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 15:12:30"}
  • {"files"=>["https://ndownloader.figshare.com/files/1543010"], "description"=>"<p>(<b>A</b>) Dependence of learning time, <i>T<sub>min</sub></i>, on window width, <i>w</i>, for the ensemble <i>N</i> = 350, <i>f<sub>max</sub></i> = 28 Hz, and <i>s</i> = 23 cm, in the <i>θ</i>-off and in the <i>θ</i>-on cases. The radius of the circles indicates the percentage of times the map converges on correct information (the larger the radius, the greater percentage of convergence). In both cases, the first convergence to the correct signature occurs as the window widens to about <i>w<sub>o</sub></i> = 0.2 <i>θ</i>-periods (one <i>θ</i>-period is approximately 125 msec). At this “opening” value the learning time is about 300 mins, which is about 60 times higher than the typical value obtained for a time window of two <i>θ</i>-periods. The learning time <i>T<sub>min</sub></i> at the “opening” values of <i>w</i> is highly sensitive to variations of <i>w</i>: as <i>w</i> changes from 0.2 to 0.3 <i>θ-</i>periods, the learning time <i>T<sub>min</sub></i> changes by over 300%. As the integration time <i>w</i> increases, the dependence <i>T<sub>min</sub></i>(<i>w</i>) rapidly drops off until it plateaus at about <i>w<sub>s</sub></i>∼1.5 <i>θ</i>-periods, where it stabilizes. As <i>w</i> increases further, the learning time <i>T<sub>min</sub></i> does not change significantly, but learning becomes less and less reliable, i.e., the likelihood of the spatial map converging to accurate topological information drops. Finally, the neuronal ensemble fails to encode the correct topological information at <i>w</i> ??? 4.5 <i>θ</i>-periods. (<b>B</b>) The dependence of the learning time on <i>w</i> shown above suggests that <i>T<sub>min</sub></i> is inversely proportional to a power of the window width, <i>T<sub>min</sub></i> = <i>C/w<sup>α</sup></i>, where <i>α</i> and <i>C</i> are constants. To test this hypothesis, we selected the maps that converged for at least 19 out of 24 values of <i>w</i>, and computed the product <i>T<sub>min</sub> w<sup>α</sup></i> for 12 values of <i>α</i> taken from the interval 1<<i>α</i><2 (See <b>Supplemental Figure 7</b>). The results show that the product <i>T<sub>min</sub> w<sup>α</sup></i> remains bounded for the entire range of window sizes. While in the <i>θ</i>-off case the variation of the product <i>T<sub>min</sub> w<sup>α</sup></i> remains large, the <i>θ</i>-on case it is nearly constant, which suggests that a nearly hyperbolic relationship <i>T<sub>min</sub> w<sup>α</sup></i> = <i>C</i> is more tight in the <i>θ</i>-on case.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "theta", "precession", "depends"], "article_id"=>1064135, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g005", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_effect_of_theta_precession_on_learning_time_depends_on_window_width_/1064135", "title"=>"The effect of theta precession on learning time depends on window width.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 15:12:30"}
  • {"files"=>["https://ndownloader.figshare.com/files/1541581"], "description"=>"<p>(A) The distribution of the stabilization window widths <i>w<sub>s</sub></i> across all convergent maps (any finite <i>ρ</i> and <i>ξ</i> values) in <i>θ</i>-off and <i>θ</i>-on cases. (B) Statistical distribution of <i>w<sub>s</sub></i> in <i>θ</i>-off and <i>θ</i>-on cases and show a 15% statistically significant difference in the typical value of the <i>w<sub>s</sub></i> for the <i>θ</i>-on cases.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "optimal", "width", "larger"], "article_id"=>1063047, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g006", "stats"=>{"downloads"=>1, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_optimal_window_width_is_slightly_larger_than_a_single_952_period_/1063047", "title"=>"The optimal window width is slightly larger than a single <i>θ</i>-period.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 03:42:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/1543007"], "description"=>"<p>(<b>A</b>) The point clouds depict the core of the learning region, as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003651#pcbi-1003651-g002\" target=\"_blank\"><b>Figure 2B</b></a>, for the sake of legibility. In both the <i>θ</i>-off and <i>θ</i>-on cases, note the red circles at the lower boundary of L, indicating longer-persisting loops. (<b>B</b>) Histograms showing the distributions of spurious loop durations in the <i>θ</i>-off and <i>θ</i>-on cases. We performed the Kolmogorov-Smirnov (KS) test to statistically compare the five cases against one another: 0  =  no theta, 1 = a single 8 Hz sinusoidal wave, 4 = a combination of four sinusoids, M = a subcortical EEG signal from a wild-type mouse, and R = a subcortical EEG signal recorded from a wild-type rat. Black squares indicate a significant difference (<i>p</i><0.05) in loop duration between <i>θ</i>-off and all other cases; the <i>p</i>-values for pairwise comparisons between different <i>θ</i>-on cases reveal no significant difference (gray squares, <i>p</i>>0.2). The statistical similarity of all the <i>θ</i>-on distributions enabled us to combine the <i>θ</i>-on loop duration data to obtain better statistics for the histogram. The structure of the histograms is fit by the gamma distribution (blue lines). We used the smooth histogram profile (dashed line) to show that the typical duration of the topological loops in the <i>θ</i>-on cases is about half that of the <i>θ</i>-off case. (<b>C</b>) The typical number of loops in the <i>θ</i>-on cases is ∼40% higher than in the absence of <i>θ</i>-precession. The <i>θ</i>-off histogram is fit by the GEV distribution (red line), while the <i>θ</i>-on histogram is better fit by gamma distribution (blue). The KS diagram on the right shows that while the simulated <i>θ</i>-oscillations produce similar results, the signals recorded from rat and mouse produce statistically different distributions from the simulated signals. (<b>D</b>) The maximal number of spurious loops observed in the <i>θ</i>-on cases, fit by the gamma distributions, is less than half that in the <i>θ</i>-off case.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "precession", "duration", "spurious", "loops", "simplicial"], "article_id"=>1064131, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g003", "stats"=>{"downloads"=>4, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Theta_phase_precession_reduces_the_duration_of_spurious_loops_in_the_simplicial_complex_/1064131", "title"=>"Theta phase precession reduces the duration of spurious loops in the simplicial complex.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 15:12:30"}
  • {"files"=>["https://ndownloader.figshare.com/files/1543019", "https://ndownloader.figshare.com/files/1543020", "https://ndownloader.figshare.com/files/1543021", "https://ndownloader.figshare.com/files/1543022", "https://ndownloader.figshare.com/files/1543023", "https://ndownloader.figshare.com/files/1543024", "https://ndownloader.figshare.com/files/1543025", "https://ndownloader.figshare.com/files/1543026", "https://ndownloader.figshare.com/files/1543027"], "description"=>"<div><p>Learning arises through the activity of large ensembles of cells, yet most of the data neuroscientists accumulate is at the level of individual neurons; we need models that can bridge this gap. We have taken spatial learning as our starting point, computationally modeling the activity of place cells using methods derived from algebraic topology, especially persistent homology. We previously showed that ensembles of hundreds of place cells could accurately encode topological information about different environments (“learn” the space) within certain values of place cell firing rate, place field size, and cell population; we called this parameter space the learning region. Here we advance the model both technically and conceptually. To make the model more physiological, we explored the effects of theta precession on spatial learning in our virtual ensembles. Theta precession, which is believed to influence learning and memory, did in fact enhance learning in our model, increasing both speed and the size of the learning region. Interestingly, theta precession also increased the number of spurious loops during simplicial complex formation. We next explored how downstream readout neurons might define co-firing by grouping together cells within different windows of time and thereby capturing different degrees of temporal overlap between spike trains. Our model's optimum coactivity window correlates well with experimental data, ranging from ∼150–200 msec. We further studied the relationship between learning time, window width, and theta precession. Our results validate our topological model for spatial learning and open new avenues for connecting data at the level of individual neurons to behavioral outcomes at the neuronal ensemble level. Finally, we analyzed the dynamics of simplicial complex formation and loop transience to propose that the simplicial complex provides a useful working description of the spatial learning process.</p></div>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "theta", "precession", "spatial", "simplicial", "topological", "hippocampal"], "article_id"=>1064143, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1003651.s001", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s002", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s003", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s004", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s005", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s006", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s007", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s008", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s009"], "stats"=>{"downloads"=>13, "page_views"=>15, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_Effects_of_Theta_Precession_on_Spatial_Learning_and_Simplicial_Complex_Dynamics_in_a_Topological_Model_of_the_Hippocampal_Spatial_Map_/1064143", "title"=>"The Effects of Theta Precession on Spatial Learning and Simplicial Complex Dynamics in a Topological Model of the Hippocampal Spatial Map", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2014-06-19 15:12:30"}
  • {"files"=>["https://ndownloader.figshare.com/files/1541571"], "description"=>"<p>As an animal (in experiments, typically a rodent) explores a space, place cells fire in discrete locations that are mapped onto the space as place fields (<b>b</b>, colored ovals). (<b>A</b>) Shown are seven place cells firing, with some temporal overlap. (<b>B</b>) <i>Top</i>: The seven corresponding place fields, along with a fragment of an animal's trajectory (dashed line). <i>Bottom:</i> The elements of the nerve (a.k.a. Čech) simplicial complex generated by the overlaps among place fields. To form a simplicial complex, each place field center is considered to be a vertex, and each link between vertices is a simplex. Each simplex <i>σ<sub>ij</sub></i> or <i>σ<sub>ijk</sub></i> is labeled to indicate the vertices linked, e.g., σ<sub>617</sub> indicates a link between vertices 6,1 and 7. (<b>C</b>) Persistent homology “barcodes” show the timelines of 0<i>D</i> and 1<i>D</i> loops, respectively: each colored horizontal line represents one 0<i>D</i> loop (top panel) or one 1<i>D</i> loop (bottom panel). The time <i>T<sub>min</sub></i> (dotted red vertical lines) marks the moment when spurious loops (topological ‘noise’) disappear and the correct number of loops persists, in this case one in 0<i>D</i> and one in 1<i>D</i>, indicating that there is one hole in the environment. Thus, <i>T<sub>min</sub></i> is the time after which the correct topological information emerges, which corresponds to the map formation or learning time in this environment, for this particular ensemble of place cells, operating under particular conditions of mean firing rate, mean place field size, number of cells in the population.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "simplicial", "derived"], "article_id"=>1063037, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g001", "stats"=>{"downloads"=>2, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_A_simplicial_complex_can_be_derived_from_place_cell_co_firing_/1063037", "title"=>"A simplicial complex can be derived from place cell co-firing.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 03:42:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/1541585", "https://ndownloader.figshare.com/files/1541586", "https://ndownloader.figshare.com/files/1541587", "https://ndownloader.figshare.com/files/1541588", "https://ndownloader.figshare.com/files/1541589", "https://ndownloader.figshare.com/files/1541590", "https://ndownloader.figshare.com/files/1541591", "https://ndownloader.figshare.com/files/1541592", "https://ndownloader.figshare.com/files/1541593"], "description"=>"<div><p>Learning arises through the activity of large ensembles of cells, yet most of the data neuroscientists accumulate is at the level of individual neurons; we need models that can bridge this gap. We have taken spatial learning as our starting point, computationally modeling the activity of place cells using methods derived from algebraic topology, especially persistent homology. We previously showed that ensembles of hundreds of place cells could accurately encode topological information about different environments (“learn” the space) within certain values of place cell firing rate, place field size, and cell population; we called this parameter space the learning region. Here we advance the model both technically and conceptually. To make the model more physiological, we explored the effects of theta precession on spatial learning in our virtual ensembles. Theta precession, which is believed to influence learning and memory, did in fact enhance learning in our model, increasing both speed and the size of the learning region. Interestingly, theta precession also increased the number of spurious loops during simplicial complex formation. We next explored how downstream readout neurons might define co-firing by grouping together cells within different windows of time and thereby capturing different degrees of temporal overlap between spike trains. Our model's optimum coactivity window correlates well with experimental data, ranging from ∼150–200 msec. We further studied the relationship between learning time, window width, and theta precession. Our results validate our topological model for spatial learning and open new avenues for connecting data at the level of individual neurons to behavioral outcomes at the neuronal ensemble level. Finally, we analyzed the dynamics of simplicial complex formation and loop transience to propose that the simplicial complex provides a useful working description of the spatial learning process.</p></div>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "theta", "precession", "spatial", "simplicial", "topological", "hippocampal"], "article_id"=>1063051, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1003651.s001", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s002", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s003", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s004", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s005", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s006", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s007", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s008", "https://dx.doi.org/10.1371/journal.pcbi.1003651.s009"], "stats"=>{"downloads"=>15, "page_views"=>18, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_Effects_of_Theta_Precession_on_Spatial_Learning_and_Simplicial_Complex_Dynamics_in_a_Topological_Model_of_the_Hippocampal_Spatial_Map_/1063051", "title"=>"The Effects of Theta Precession on Spatial Learning and Simplicial Complex Dynamics in a Topological Model of the Hippocampal Spatial Map", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2014-06-19 03:42:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/1541579"], "description"=>"<p>(<b>A</b>) Dependence of learning time, <i>T<sub>min</sub></i>, on window width, <i>w</i>, for the ensemble <i>N</i> = 350, <i>f<sub>max</sub></i> = 28 Hz, and <i>s</i> = 23 cm, in the <i>θ</i>-off and in the <i>θ</i>-on cases. The radius of the circles indicates the percentage of times the map converges on correct information (the larger the radius, the greater percentage of convergence). In both cases, the first convergence to the correct signature occurs as the window widens to about <i>w<sub>o</sub></i> = 0.2 <i>θ</i>-periods (one <i>θ</i>-period is approximately 125 msec). At this “opening” value the learning time is about 300 mins, which is about 60 times higher than the typical value obtained for a time window of two <i>θ</i>-periods. The learning time <i>T<sub>min</sub></i> at the “opening” values of <i>w</i> is highly sensitive to variations of <i>w</i>: as <i>w</i> changes from 0.2 to 0.3 <i>θ-</i>periods, the learning time <i>T<sub>min</sub></i> changes by over 300%. As the integration time <i>w</i> increases, the dependence <i>T<sub>min</sub></i>(<i>w</i>) rapidly drops off until it plateaus at about <i>w<sub>s</sub></i>∼1.5 <i>θ</i>-periods, where it stabilizes. As <i>w</i> increases further, the learning time <i>T<sub>min</sub></i> does not change significantly, but learning becomes less and less reliable, i.e., the likelihood of the spatial map converging to accurate topological information drops. Finally, the neuronal ensemble fails to encode the correct topological information at <i>w</i> ??? 4.5 <i>θ</i>-periods. (<b>B</b>) The dependence of the learning time on <i>w</i> shown above suggests that <i>T<sub>min</sub></i> is inversely proportional to a power of the window width, <i>T<sub>min</sub></i> = <i>C/w<sup>α</sup></i>, where <i>α</i> and <i>C</i> are constants. To test this hypothesis, we selected the maps that converged for at least 19 out of 24 values of <i>w</i>, and computed the product <i>T<sub>min</sub> w<sup>α</sup></i> for 12 values of <i>α</i> taken from the interval 1<<i>α</i><2 (See <b>Supplemental Figure 7</b>). The results show that the product <i>T<sub>min</sub> w<sup>α</sup></i> remains bounded for the entire range of window sizes. While in the <i>θ</i>-off case the variation of the product <i>T<sub>min</sub> w<sup>α</sup></i> remains large, the <i>θ</i>-on case it is nearly constant, which suggests that a nearly hyperbolic relationship <i>T<sub>min</sub> w<sup>α</sup></i> = <i>C</i> is more tight in the <i>θ</i>-on case.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "theta", "precession", "depends"], "article_id"=>1063045, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g005", "stats"=>{"downloads"=>1, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_effect_of_theta_precession_on_learning_time_depends_on_window_width_/1063045", "title"=>"The effect of theta precession on learning time depends on window width.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 03:42:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/1543009"], "description"=>"<p>Distributions of the (<b>A</b>) 1<i>D</i> and (<b>B</b>) 2<i>D</i> connectivity indexes, <i>η<sub>1</sub></i> and the <i>η<sub>2</sub></i>, aross the ensembles that form correct maps at least 70% of the time (convergence rate <i>ρ</i> ≥0.7) and have low relative variability (<i>ξ</i><0.3, as in <b>Figs. 2</b> and <b>3</b>) in the <i>θ</i>-off and the <i>θ</i>-on cases. The distribution of the <i>η<sub>1</sub></i> and the <i>η<sub>2</sub></i> values over the learning region L indicates that the normalized number of 1D and 2D simplices scales with the number of combinatorially possible connections in the place cell ensemble. Correspondingly, the structure of the normalized connectivity in the temporal simplicial complexes can be seen in the cross-sections of the learning region (third column; notice that the normalized connectivity increases with a rise in both the mean ensemble firng rate and the mean ensemble place field size, in both the <i>1D</i> and <i>2D</i> cases). In the ensembles with high firing rates and low spatial selectivity, up to 25% of place cell pairs and up to 8% of place cell triplets are coactive. The KS test shows that more simplices make for inefficient learning whether or not there is theta precession, though there is a difference between the <i>θ</i>-off and the <i>θ</i>-on cases when considering the <i>1D</i> and <i>2D</i> connectivity indices together.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "simplices", "correlates", "inefficient"], "article_id"=>1064133, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g004", "stats"=>{"downloads"=>2, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Too_many_simplices_correlates_with_inefficient_learning_/1064133", "title"=>"Too many simplices correlates with inefficient learning.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 15:12:30"}
  • {"files"=>["https://ndownloader.figshare.com/files/1541578"], "description"=>"<p>Distributions of the (<b>A</b>) 1<i>D</i> and (<b>B</b>) 2<i>D</i> connectivity indexes, <i>η<sub>1</sub></i> and the <i>η<sub>2</sub></i>, aross the ensembles that form correct maps at least 70% of the time (convergence rate <i>ρ</i> ≥0.7) and have low relative variability (<i>ξ</i><0.3, as in <b>Figs. 2</b> and <b>3</b>) in the <i>θ</i>-off and the <i>θ</i>-on cases. The distribution of the <i>η<sub>1</sub></i> and the <i>η<sub>2</sub></i> values over the learning region L indicates that the normalized number of 1D and 2D simplices scales with the number of combinatorially possible connections in the place cell ensemble. Correspondingly, the structure of the normalized connectivity in the temporal simplicial complexes can be seen in the cross-sections of the learning region (third column; notice that the normalized connectivity increases with a rise in both the mean ensemble firng rate and the mean ensemble place field size, in both the <i>1D</i> and <i>2D</i> cases). In the ensembles with high firing rates and low spatial selectivity, up to 25% of place cell pairs and up to 8% of place cell triplets are coactive. The KS test shows that more simplices make for inefficient learning whether or not there is theta precession, though there is a difference between the <i>θ</i>-off and the <i>θ</i>-on cases when considering the <i>1D</i> and <i>2D</i> connectivity indices together.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "simplices", "correlates", "inefficient"], "article_id"=>1063044, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g004", "stats"=>{"downloads"=>1, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Too_many_simplices_correlates_with_inefficient_learning_/1063044", "title"=>"Too many simplices correlates with inefficient learning.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 03:42:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/1543005"], "description"=>"<p>(<b>A</b>) Point clouds representing the mean learning times <i>T<sub>min</sub></i> computed for the <i>θ</i>-off case (left) and the maps driven by a <i>θ</i>–signal recorded in rat (right). Each point corresponds to a place cell ensemble with a specific number of place cells, <i>N</i>, the mean ensemble firing rate, <i>f</i>, the mean ensemble place field size <i>s</i>. Dark blue circles represent those ensembles that form correct topological maps most rapidly and reliably; as the color shades from blue through green, yellow, and red, the learning times increase and map formation becomes less reliable (see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003651#pcbi.1003651-Dabaghian1\" target=\"_blank\">[4]</a>, Methods). The rat <i>θ</i>–signal enlarges the learning region L and speeds map formation. (<b>B</b>) To zero in on the effect of <i>θ</i> precession on the quality of learning, these clouds depict only the maps that converged at least 7 out of 10 times (<i>ρ</i> ≥0.7), and for which the variance of the learning times, <i>ξ</i> =  Δ<i>T<sub>min</sub></i>/<i>T<sub>min</sub></i>, did not exceed 30% of the mean value. Even in this more rigorously defined core of L, with ensembles that already function well, the <i>θ</i>–signal has a pronounced effect. (<b>C</b>) Histograms of the minimal times obtained in the <i>θ</i>-off (left) and the <i>θ</i>-driven case (right), fit by the GEV distribution. The blue dot marks the mode of the distribution; m in the center of each panel gives the value of the mode. All convergent maps are included. The typical learning time <i>T<sub>min</sub></i> in the <i>θ</i>-on case is about half as long as in the <i>θ</i>-off case. (<b>D</b>) The same histograms obtained for the core of L (<i>ρ</i> ≥0.7, <i>ξ</i>≤0.3). The typical learning time <i>T<sub>min</sub></i> in the <i>θ</i>-on case is about 6.5 minutes, whereas without <i>θ</i> it is 15% longer. (<b>E</b>) One of the major effects of <i>θ</i> phase precession is to reduce the variability of the learning times. The histograms show that the typical value of the relative variation <i>ξ</i> in the <i>θ</i>-on case is less than half that of the <i>θ</i>-off case, i.e., that repeated simulations of the <i>θ</i>-driven maps more reliably reproduce similar learning time values.</p>", "links"=>[], "tags"=>["Computational biology", "computational neuroscience", "neuroscience", "Learning and memory", "precession", "enlarges"], "article_id"=>1064129, "categories"=>["Biological Sciences"], "users"=>["Mamiko Arai", "Vicky Brandt", "Yuri Dabaghian"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003651.g002", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Theta_precession_both_enlarges_the_learning_region_and_reduces_mean_learning_times_/1064129", "title"=>"Theta precession both enlarges the learning region and reduces mean learning times.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-06-19 15:12:30"}

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