Stochastic Computations in Cortical Microcircuit Models
Publication Date
November 14, 2013
PLOS Computational Biology
Stefan Habenschuss, Zeno Jonke & Wolfgang Maass
Publisher URL
PubMed Central
Europe PMC
Web of Science
Loading … Spinner

Mendeley | Further Information

{"title"=>"Stochastic Computations in Cortical Microcircuit Models", "type"=>"journal", "authors"=>[{"first_name"=>"Stefan", "last_name"=>"Habenschuss", "scopus_author_id"=>"55699123400"}, {"first_name"=>"Zeno", "last_name"=>"Jonke", "scopus_author_id"=>"55938036600"}, {"first_name"=>"Wolfgang", "last_name"=>"Maass", "scopus_author_id"=>"7005129380"}], "year"=>2013, "source"=>"PLoS Computational Biology", "identifiers"=>{"issn"=>"1553734X", "isbn"=>"1553-7358 (Electronic)\\r1553-734X (Linking)", "pmid"=>"24244126", "scopus"=>"2-s2.0-84888268117", "doi"=>"10.1371/journal.pcbi.1003311", "sgr"=>"84888268117", "pui"=>"370341758"}, "id"=>"186ea14f-6514-398b-8a89-2a4e97f14ab7", "abstract"=>"Experimental data from neuroscience suggest that a substantial amount of knowledge is stored in the brain in the form of probability distributions over network states and trajectories of network states. We provide a theoretical foundation for this hypothesis by showing that even very detailed models for cortical microcircuits, with data-based diverse nonlinear neurons and synapses, have a stationary distribution of network states and trajectories of network states to which they converge exponentially fast from any initial state. We demonstrate that this convergence holds in spite of the non-reversibility of the stochastic dynamics of cortical microcircuits. We further show that, in the presence of background network oscillations, separate stationary distributions emerge for different phases of the oscillation, in accordance with experimentally reported phase-specific codes. We complement these theoretical results by computer simulations that investigate resulting computation times for typical probabilistic inference tasks on these internally stored distributions, such as marginalization or marginal maximum-a-posteriori estimation. Furthermore, we show that the inherent stochastic dynamics of generic cortical microcircuits enables them to quickly generate approximate solutions to difficult constraint satisfaction problems, where stored knowledge and current inputs jointly constrain possible solutions. This provides a powerful new computing paradigm for networks of spiking neurons, that also throws new light on how networks of neurons in the brain could carry out complex computational tasks such as prediction, imagination, memory recall and problem solving.", "link"=>"", "reader_count"=>162, "reader_count_by_academic_status"=>{"Unspecified"=>2, "Professor > Associate Professor"=>7, "Researcher"=>58, "Student > Doctoral Student"=>5, "Student > Ph. D. Student"=>51, "Student > Postgraduate"=>8, "Student > Master"=>11, "Other"=>3, "Student > Bachelor"=>6, "Lecturer"=>2, "Professor"=>9}, "reader_count_by_user_role"=>{"Unspecified"=>2, "Professor > Associate Professor"=>7, "Researcher"=>58, "Student > Doctoral Student"=>5, "Student > Ph. D. Student"=>51, "Student > Postgraduate"=>8, "Student > Master"=>11, "Other"=>3, "Student > Bachelor"=>6, "Lecturer"=>2, "Professor"=>9}, "reader_count_by_subject_area"=>{"Unspecified"=>5, "Agricultural and Biological Sciences"=>41, "Veterinary Science and Veterinary Medicine"=>1, "Chemistry"=>1, "Computer Science"=>28, "Energy"=>1, "Engineering"=>22, "Environmental Science"=>1, "Mathematics"=>5, "Medicine and Dentistry"=>2, "Neuroscience"=>26, "Physics and Astronomy"=>16, "Psychology"=>12, "Social Sciences"=>1}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>2}, "Social Sciences"=>{"Social Sciences"=>1}, "Physics and Astronomy"=>{"Physics and Astronomy"=>16}, "Psychology"=>{"Psychology"=>12}, "Mathematics"=>{"Mathematics"=>5}, "Unspecified"=>{"Unspecified"=>5}, "Environmental Science"=>{"Environmental Science"=>1}, "Engineering"=>{"Engineering"=>22}, "Chemistry"=>{"Chemistry"=>1}, "Neuroscience"=>{"Neuroscience"=>26}, "Energy"=>{"Energy"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>41}, "Computer Science"=>{"Computer Science"=>28}, "Veterinary Science and Veterinary Medicine"=>{"Veterinary Science and Veterinary Medicine"=>1}}, "reader_count_by_country"=>{"United States"=>3, "Japan"=>1, "United Kingdom"=>2, "Belarus"=>1, "Switzerland"=>4, "Russia"=>1, "Spain"=>1, "New Zealand"=>1, "Belgium"=>2, "Taiwan"=>1, "Brazil"=>1, "Italy"=>2, "Australia"=>1, "France"=>2, "Germany"=>6}, "group_count"=>6}

Scopus | Further Information

{"@_fa"=>"true", "link"=>[{"@_fa"=>"true", "@ref"=>"self", "@href"=>""}, {"@_fa"=>"true", "@ref"=>"author-affiliation", "@href"=>",affiliation"}, {"@_fa"=>"true", "@ref"=>"scopus", "@href"=>""}, {"@_fa"=>"true", "@ref"=>"scopus-citedby", "@href"=>""}], "prism:url"=>"", "dc:identifier"=>"SCOPUS_ID:84888268117", "eid"=>"2-s2.0-84888268117", "dc:title"=>"Stochastic Computations in Cortical Microcircuit Models", "dc:creator"=>"Habenschuss S.", "prism:publicationName"=>"PLoS Computational Biology", "prism:issn"=>"1553734X", "prism:eIssn"=>"15537358", "prism:volume"=>"9", "prism:issueIdentifier"=>"11", "prism:pageRange"=>nil, "prism:coverDate"=>"2013-11-01", "prism:coverDisplayDate"=>"November 2013", "prism:doi"=>"10.1371/journal.pcbi.1003311", "citedby-count"=>"29", "affiliation"=>[{"@_fa"=>"true", "affilname"=>"Technische Universitat Graz", "affiliation-city"=>"Graz", "affiliation-country"=>"Austria"}], "pubmed-id"=>"24244126", "prism:aggregationType"=>"Journal", "subtype"=>"ar", "subtypeDescription"=>"Article", "article-number"=>"e1003311", "source-id"=>"4000151810", "openaccess"=>"1", "openaccessFlag"=>true}


  • {"url"=>"", "share_count"=>0, "like_count"=>0, "comment_count"=>0, "click_count"=>0, "total_count"=>0}

Journal Comments | Further Information



  • {"month"=>"11", "year"=>"2013", "pdf_views"=>"224", "xml_views"=>"18", "html_views"=>"1705"}
  • {"month"=>"12", "year"=>"2013", "pdf_views"=>"174", "xml_views"=>"9", "html_views"=>"1060"}
  • {"month"=>"1", "year"=>"2014", "pdf_views"=>"94", "xml_views"=>"4", "html_views"=>"318"}
  • {"month"=>"2", "year"=>"2014", "pdf_views"=>"43", "xml_views"=>"0", "html_views"=>"185"}
  • {"month"=>"3", "year"=>"2014", "pdf_views"=>"38", "xml_views"=>"2", "html_views"=>"223"}
  • {"month"=>"4", "year"=>"2014", "pdf_views"=>"47", "xml_views"=>"1", "html_views"=>"232"}
  • {"month"=>"5", "year"=>"2014", "pdf_views"=>"71", "xml_views"=>"0", "html_views"=>"222"}
  • {"month"=>"6", "year"=>"2014", "pdf_views"=>"41", "xml_views"=>"5", "html_views"=>"153"}
  • {"month"=>"7", "year"=>"2014", "pdf_views"=>"34", "xml_views"=>"0", "html_views"=>"167"}
  • {"month"=>"8", "year"=>"2014", "pdf_views"=>"21", "xml_views"=>"4", "html_views"=>"105"}
  • {"month"=>"9", "year"=>"2014", "pdf_views"=>"17", "xml_views"=>"0", "html_views"=>"111"}
  • {"month"=>"10", "year"=>"2014", "pdf_views"=>"29", "xml_views"=>"1", "html_views"=>"168"}
  • {"month"=>"11", "year"=>"2014", "pdf_views"=>"23", "xml_views"=>"1", "html_views"=>"151"}
  • {"month"=>"12", "year"=>"2014", "pdf_views"=>"17", "xml_views"=>"2", "html_views"=>"94"}
  • {"month"=>"1", "year"=>"2015", "pdf_views"=>"36", "xml_views"=>"0", "html_views"=>"103"}
  • {"month"=>"2", "year"=>"2015", "pdf_views"=>"29", "xml_views"=>"0", "html_views"=>"96"}
  • {"month"=>"3", "year"=>"2015", "pdf_views"=>"75", "xml_views"=>"0", "html_views"=>"120"}
  • {"month"=>"4", "year"=>"2015", "pdf_views"=>"29", "xml_views"=>"1", "html_views"=>"71"}
  • {"month"=>"5", "year"=>"2015", "pdf_views"=>"32", "xml_views"=>"0", "html_views"=>"126"}
  • {"month"=>"6", "year"=>"2015", "pdf_views"=>"25", "xml_views"=>"0", "html_views"=>"100"}
  • {"month"=>"7", "year"=>"2015", "pdf_views"=>"41", "xml_views"=>"0", "html_views"=>"102"}
  • {"month"=>"8", "year"=>"2015", "pdf_views"=>"24", "xml_views"=>"0", "html_views"=>"111"}
  • {"month"=>"9", "year"=>"2015", "pdf_views"=>"31", "xml_views"=>"0", "html_views"=>"142"}
  • {"month"=>"10", "year"=>"2015", "pdf_views"=>"14", "xml_views"=>"0", "html_views"=>"74"}
  • {"month"=>"11", "year"=>"2015", "pdf_views"=>"23", "xml_views"=>"0", "html_views"=>"119"}
  • {"month"=>"12", "year"=>"2015", "pdf_views"=>"12", "xml_views"=>"0", "html_views"=>"142"}
  • {"month"=>"1", "year"=>"2016", "pdf_views"=>"36", "xml_views"=>"0", "html_views"=>"112"}
  • {"month"=>"2", "year"=>"2016", "pdf_views"=>"19", "xml_views"=>"0", "html_views"=>"77"}
  • {"month"=>"3", "year"=>"2016", "pdf_views"=>"20", "xml_views"=>"0", "html_views"=>"86"}
  • {"month"=>"4", "year"=>"2016", "pdf_views"=>"29", "xml_views"=>"0", "html_views"=>"86"}
  • {"month"=>"5", "year"=>"2016", "pdf_views"=>"8", "xml_views"=>"0", "html_views"=>"80"}
  • {"month"=>"6", "year"=>"2016", "pdf_views"=>"20", "xml_views"=>"0", "html_views"=>"72"}
  • {"month"=>"7", "year"=>"2016", "pdf_views"=>"18", "xml_views"=>"0", "html_views"=>"55"}
  • {"month"=>"8", "year"=>"2016", "pdf_views"=>"14", "xml_views"=>"0", "html_views"=>"67"}
  • {"month"=>"9", "year"=>"2016", "pdf_views"=>"29", "xml_views"=>"0", "html_views"=>"67"}
  • {"month"=>"10", "year"=>"2016", "pdf_views"=>"15", "xml_views"=>"0", "html_views"=>"70"}
  • {"month"=>"11", "year"=>"2016", "pdf_views"=>"22", "xml_views"=>"0", "html_views"=>"72"}
  • {"month"=>"12", "year"=>"2016", "pdf_views"=>"34", "xml_views"=>"0", "html_views"=>"70"}
  • {"month"=>"1", "year"=>"2017", "pdf_views"=>"29", "xml_views"=>"1", "html_views"=>"83"}
  • {"month"=>"2", "year"=>"2017", "pdf_views"=>"23", "xml_views"=>"0", "html_views"=>"62"}
  • {"month"=>"3", "year"=>"2017", "pdf_views"=>"35", "xml_views"=>"0", "html_views"=>"73"}
  • {"month"=>"4", "year"=>"2017", "pdf_views"=>"19", "xml_views"=>"0", "html_views"=>"56"}
  • {"month"=>"5", "year"=>"2017", "pdf_views"=>"37", "xml_views"=>"1", "html_views"=>"69"}
  • {"month"=>"6", "year"=>"2017", "pdf_views"=>"8", "xml_views"=>"0", "html_views"=>"85"}
  • {"month"=>"7", "year"=>"2017", "pdf_views"=>"31", "xml_views"=>"0", "html_views"=>"84"}
  • {"month"=>"8", "year"=>"2017", "pdf_views"=>"13", "xml_views"=>"2", "html_views"=>"36"}
  • {"month"=>"9", "year"=>"2017", "pdf_views"=>"14", "xml_views"=>"1", "html_views"=>"41"}
  • {"month"=>"10", "year"=>"2017", "pdf_views"=>"10", "xml_views"=>"2", "html_views"=>"36"}
  • {"month"=>"11", "year"=>"2017", "pdf_views"=>"17", "xml_views"=>"1", "html_views"=>"50"}
  • {"month"=>"12", "year"=>"2017", "pdf_views"=>"13", "xml_views"=>"0", "html_views"=>"28"}
  • {"month"=>"1", "year"=>"2018", "pdf_views"=>"13", "xml_views"=>"0", "html_views"=>"62"}
  • {"month"=>"2", "year"=>"2018", "pdf_views"=>"14", "xml_views"=>"0", "html_views"=>"39"}
  • {"month"=>"3", "year"=>"2018", "pdf_views"=>"11", "xml_views"=>"1", "html_views"=>"14"}
  • {"month"=>"4", "year"=>"2018", "pdf_views"=>"9", "xml_views"=>"0", "html_views"=>"12"}
  • {"month"=>"5", "year"=>"2018", "pdf_views"=>"21", "xml_views"=>"1", "html_views"=>"16"}
  • {"month"=>"6", "year"=>"2018", "pdf_views"=>"20", "xml_views"=>"0", "html_views"=>"14"}
  • {"month"=>"7", "year"=>"2018", "pdf_views"=>"12", "xml_views"=>"3", "html_views"=>"16"}
  • {"month"=>"8", "year"=>"2018", "pdf_views"=>"14", "xml_views"=>"1", "html_views"=>"24"}
  • {"month"=>"9", "year"=>"2018", "pdf_views"=>"26", "xml_views"=>"0", "html_views"=>"18"}
  • {"month"=>"10", "year"=>"2018", "pdf_views"=>"16", "xml_views"=>"2", "html_views"=>"12"}
  • {"month"=>"11", "year"=>"2018", "pdf_views"=>"15", "xml_views"=>"1", "html_views"=>"17"}
  • {"month"=>"12", "year"=>"2018", "pdf_views"=>"15", "xml_views"=>"0", "html_views"=>"24"}
  • {"month"=>"1", "year"=>"2019", "pdf_views"=>"16", "xml_views"=>"0", "html_views"=>"17"}
  • {"month"=>"2", "year"=>"2019", "pdf_views"=>"14", "xml_views"=>"0", "html_views"=>"16"}
  • {"month"=>"3", "year"=>"2019", "pdf_views"=>"25", "xml_views"=>"3", "html_views"=>"17"}
  • {"month"=>"4", "year"=>"2019", "pdf_views"=>"12", "xml_views"=>"0", "html_views"=>"12"}
  • {"month"=>"5", "year"=>"2019", "pdf_views"=>"13", "xml_views"=>"0", "html_views"=>"14"}


  • {"files"=>[""], "description"=>"<p><b>A</b>. Data-based cortical microcircuit template from Cereb. Cortex (2007) 17: 149-162 <a href=\"\" target=\"_blank\">[30]</a>; reprinted by permission of the authors and Oxford University Press. <b>B</b>. A small instantiation of this model consisting of 10 network neurons and 2 additional input neurons . Neurons are colored by type (blue:input, black:excitatory, red:inhibitory). Line width represents synaptic efficacy. The synapse from neuron 8 to 7 is removed for the simulation described in E. <b>C</b>. Notions of network state considered in this article. Markov states are defined by the exact timing of all recent spikes within some time window , shown here for . Simple states only record which neurons fired recently (0 = no spike, 1 = at least one spike within a short window , with throughout this figure). <b>D</b>. Empirically measured stationary distribution of simple network states. Shown is the marginal distribution for a subset of three neurons 2,7,8 (their spikes are shown in C in black), under two different input conditions (input pattern 1: firing at and at , input pattern 2: at and at ). The distribution for each input condition was obtained by measuring the relative time spent in each of the simple states (0,0,0), …, (1,1,1) in a single long trial (). The zero state (0,0,0) is not shown. <b>E</b>. Effect of removing one synapse, from neuron 8 to neuron 7, on the stationary distribution of network states (input pattern 1 was presented). <b>F</b>. Illustration of trial-to-trial variability in the small cortical microcircuit (input pattern 1). Two trials starting from identical initial network states are shown. Blue bars at the bottom of each trial mark periods where the subnetwork of neurons 2,7,8 was in simple state (1,1,1) at this time . Note that the “blue” initial Markov state is shown only partially: it is actually longer and comprises all neurons in the network (as in panel C, but with ). <b>G</b>. Two trials starting from a different (“red”) initial network state. Red bars denote periods of state (1,1,1) for “red” trials. <b>H</b>. Convergence to the stationary distribution in this small cortical microcircuit is fast and independent of the initial state: This is illustrated for the relative frequency of simple state (1,1,1) within the first after input onset. The blue/red line shows the relative frequency of simple state (1,1,1) at each time estimated from many () “blue”/“red” trials. The relative frequency of simple state (1,1,1) rapidly converges to its stationary value denoted by the symbol (marked also in panels D and E). The relative frequency converges to the same value regardless of the initial state (blue/red).</p>", "links"=>[], "tags"=>["states", "stationary", "distributions", "cortical", "microcircuit"], "article_id"=>851688, "categories"=>["Biological Sciences", "Information And Computing Sciences"], "users"=>["Stefan Habenschuss", "Zeno Jonke", "Wolfgang Maass"], "doi"=>"", "stats"=>{"downloads"=>5, "page_views"=>18, "likes"=>0}, "figshare_url"=>"", "title"=>"Network states and stationary distributions of network states in a cortical microcircuit model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-11-14 04:48:46"}
  • {"files"=>[""], "description"=>"<p><b>A</b>. Typical spike response of the microcircuit model based on <a href=\"\" target=\"_blank\">[30]</a> comprising 560 stochastic point neurons. Spikes of inhibitory neurons are indicated in red. <b>B</b>. Fast convergence of a marginal for a representative layer 5 neuron (frequency of “on”-state, with ) to its stationary value, shown for two different initial Markov states (blue/red). Statistics were obtained for each initial state from trials. <b>C</b>. Gelman-Rubin convergence diagnostic was applied to the marginals of all single neurons (simple states, ). In all neurons the Gelman-Rubin value drops to a value close to 1 within a few , suggesting generally fast convergence of single neuron marginals (shown are 20 randomly chosen neurons; see panel E for a summary of all neurons). The shaded area below 1.1 indicates a range where one commonly assumes that convergence has taken place. <b>D</b>. Convergence speed of pairwise spike coincidences (simple states (1,1) of two neurons, 20 randomly chosen pairs of neurons) is comparable to marginal convergence. <b>E</b>. Summary of marginal convergence analysis for single neurons in C: Mean (solid) and worst (dashed line) marginal convergence of all 560 neurons. Mean/worst convergence is reached after a few . <b>F</b>. Convergence analysis was applied to networks of different sizes (500–5000 neurons). Mean and worst marginal convergence of single neurons are hardly affected by network size. <b>G</b>. Convergence properties of populations of neurons. Dotted: multivariate Gelman-Rubin analysis was applied to a subpopulation of 30 neurons (5 neurons were chosen randomly from each pool). Solid: convergence of a “random readout” neuron which receives spike inputs from 500 randomly chosen neurons in the microcircuit. It turns out that the convergence speed of such a generic readout neuron is even slightly faster than for neurons within the microcircuit (compare with panel E). A remarkable finding is that in all these cases the network size does not affect convergence speed.</p>", "links"=>[], "tags"=>["convergence", "marginals", "neurons", "quantities", "cortical", "microcircuit"], "article_id"=>851689, "categories"=>["Biological Sciences", "Information And Computing Sciences"], "users"=>["Stefan Habenschuss", "Zeno Jonke", "Wolfgang Maass"], "doi"=>"", "stats"=>{"downloads"=>2, "page_views"=>18, "likes"=>0}, "figshare_url"=>"", "title"=>"Fast convergence of marginals of single neurons and more complex quantities in a cortical microcircuit model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-11-14 04:48:46"}
  • {"files"=>[""], "description"=>"<p>Convergence properties for single neurons (as in <a href=\"\" target=\"_blank\">Figure 2C</a>) in different network architectures were assessed using univariate Gelman-Rubin analysis. Typical network activity is shown on the left, convergence speed on the right (solid: mean marginal, dashed: worst marginal). <b>A</b>. Small cortical column model from <a href=\"\" target=\"_blank\">Figure 1</a> (input neurons not shown). <b>B</b>. Network with sparse activity (20 neurons). <b>C</b>. Network with stereotypical trajectories (50 neurons, inhibitory neurons not shown). Despite strongly irreversible dynamics, convergence is only slightly slower. <b>D</b>. Network with bistable dynamics (two competing populations, each comprising 10 neurons). Convergence is slower in this circuit due to low-frequency switching dynamics between two attractors.</p>", "links"=>[], "tags"=>["convergence"], "article_id"=>851690, "categories"=>["Biological Sciences", "Information And Computing Sciences"], "users"=>["Stefan Habenschuss", "Zeno Jonke", "Wolfgang Maass"], "doi"=>"", "stats"=>{"downloads"=>0, "page_views"=>4, "likes"=>0}, "figshare_url"=>"", "title"=>"Impact of network architecture and network dynamics on convergence speed.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-11-14 04:48:46"}
  • {"files"=>[""], "description"=>"<p><b>A</b>. A network with a built-in stereotypical trajectory is stimulated with a background oscillation. The oscillation (top) is imposed on the neuronal thresholds of all neurons. The trajectories produced by the network (bottom) become automatically synchronized to the background rhythm. The yellow shading marks the three neurons for which the analysis in panels B and C was carried out. The two indicated time points (green and purple lines) mark the two phases for which the phase-specific stationary distributions are considered in panels B and D ( and into the cycle, with phase-specific distributions and , respectively). <b>B</b>. The empirically measured distributions of network states are observed to differ significantly at two different phases of the oscillation (phases marked in panel A). Shown is for each phase the phase-specific marginal distribution over 3 neurons (4, 5 and 6), using simple states with . The zero state (0,0,0) is not shown. The empirical distribution for each phase was obtained from a single long run, by taking into account the network states at times , etc., with cycle length . <b>C</b>. Illustration of convergence to phase-specific stationary distributions. Shown is the relative frequency of subnetwork state (1,1,0) on the subset of neurons 4,5 and 6 over time, when the network is started from two different initial states (red/blue). In each case, the state frequency quickly approaches a periodic limit cycle. <b>D</b>. Convergence to phase-specific stationary distributions takes place within a few cycles of the underlying oscillation. Shown is the multivariate Gelman-Rubin convergence analysis to the phase-specific stationary distribution for two different phases. <b>E</b>. Bi-stable network under the influence of a background oscillation. <b>F</b>. In response to the periodic stimulation, transitions between the two attractors (modes) become concentrated around a specific phase of the distribution.</p>", "links"=>[], "tags"=>["phase-specific", "stationary", "distributions", "states", "periodic"], "article_id"=>851691, "categories"=>["Biological Sciences", "Information And Computing Sciences"], "users"=>["Stefan Habenschuss", "Zeno Jonke", "Wolfgang Maass"], "doi"=>"", "stats"=>{"downloads"=>1, "page_views"=>3, "likes"=>0}, "figshare_url"=>"", "title"=>"Emergence of phase-specific stationary distributions of network states in the presence of periodic network input.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-11-14 04:48:46"}
  • {"files"=>[""], "description"=>"<p><b>A</b>. A “hard” Sudoku puzzle with 26 given numbers (left). The solution (right) is defined uniquely by the set of givens and the additional constraints that each digit must appear only once in each row, column and 3×3 subgrid. <b>B</b>. An implementation of the constraints of the Sudoku game in a spiking neural network consists of overlapping WTA circuits. WTA circuits are ubiquitous connection motifs in cortical circuits <a href=\"\" target=\"_blank\">[29]</a>. A WTA circuit can be modeled by a set of stochastically spiking output neurons that are subject to lateral inhibition (left). The same pyramidal cell can be part of several such WTA motifs (right). In the Sudoku example, each digit in a Sudoku field is associated with four pyramidal cells which vote for this digit when they emit a spike. Each such pyramidal cell participates in four WTA motifs, corresponding to the constraints that only one digit can be active in each Sudoku field, and that a digit can appear only once in each row, column and 3×3 subgrid. <b>C</b>. A typical network run is shown during the last before the correct solution was found to the Sudoku from panel A (the total solve time was approximately in this run, see panel D for statistics of solve times). The network performance (fraction of cells with correct values) over time is shown at the top. The spiking activity is shown for 3 (out of the 81) WTA motifs associated with the 3 colored Sudoku fields in A and B. In each of these WTA motifs there are 36 pyramidal cells (9 digits and 4 pyramidal cells for each digit). Spikes are colored green for those neurons which code for the correct digit in each Sudoku field (6, 8 and 4 in the example). <b>D</b>. Histogram of solve times (the first time the correct solution was found) for the Sudoku from panel A. Statistics were obtained from 1000 independent runs. The sample mean is . <b>E</b>. Average network performance for this Sudoku converges quickly during the first five seconds to a value of , corresponding to % correctly found digits (average taken over 1000 runs; shaded area: standard deviations). Thereafter, from all possible configurations the network spends most time in good approximate solutions. The correct solution occurs particularly often, on average approximately 2% of the time (not shown).</p>", "links"=>[], "tags"=>["structured", "interactions", "stochastically", "firing", "excitatory", "inhibitory"], "article_id"=>851692, "categories"=>["Biological Sciences", "Information And Computing Sciences"], "users"=>["Stefan Habenschuss", "Zeno Jonke", "Wolfgang Maass"], "doi"=>"", "stats"=>{"downloads"=>2, "page_views"=>72, "likes"=>0}, "figshare_url"=>"", "title"=>"Solving Sudoku, a constraint satisfaction problem, through structured interactions between stochastically firing excitatory and inhibitory neurons.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-11-14 04:48:46"}
  • {"files"=>[""], "description"=>"<p>Number of randomly chosen neurons per pool for readout neuron in <a href=\"\" target=\"_blank\">Figure 2G</a>.</p>", "links"=>[], "tags"=>["randomly", "chosen", "neurons", "readout", "neuron"], "article_id"=>851693, "categories"=>["Biological Sciences", "Information And Computing Sciences"], "users"=>["Stefan Habenschuss", "Zeno Jonke", "Wolfgang Maass"], "doi"=>"", "stats"=>{"downloads"=>1, "page_views"=>8, "likes"=>0}, "figshare_url"=>"", "title"=>"Number of randomly chosen neurons per pool for readout neuron in Figure 2G.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-11-14 04:48:46"}

PMC Usage Stats | Further Information

  • {"unique-ip"=>"12", "full-text"=>"13", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"7"}
  • {"unique-ip"=>"14", "full-text"=>"16", "pdf"=>"6", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"8"}
  • {"unique-ip"=>"14", "full-text"=>"24", "pdf"=>"5", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"9"}
  • {"unique-ip"=>"38", "full-text"=>"62", "pdf"=>"5", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"10"}
  • {"unique-ip"=>"21", "full-text"=>"38", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"11"}
  • {"unique-ip"=>"16", "full-text"=>"28", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"2", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"12"}
  • {"unique-ip"=>"12", "full-text"=>"19", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"1"}
  • {"unique-ip"=>"17", "full-text"=>"15", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"2"}
  • {"unique-ip"=>"10", "full-text"=>"12", "pdf"=>"6", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"3"}
  • {"unique-ip"=>"8", "full-text"=>"6", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"4"}
  • {"unique-ip"=>"7", "full-text"=>"9", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"5"}
  • {"unique-ip"=>"8", "full-text"=>"8", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"1", "year"=>"2015", "month"=>"6"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"7"}
  • {"unique-ip"=>"14", "full-text"=>"10", "pdf"=>"4", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"4", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"8"}
  • {"unique-ip"=>"7", "full-text"=>"39", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"9"}
  • {"unique-ip"=>"4", "full-text"=>"3", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"10"}
  • {"unique-ip"=>"17", "full-text"=>"12", "pdf"=>"7", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"1", "year"=>"2015", "month"=>"11"}
  • {"unique-ip"=>"6", "full-text"=>"7", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"12"}
  • {"unique-ip"=>"5", "full-text"=>"3", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"4", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"1"}
  • {"unique-ip"=>"8", "full-text"=>"5", "pdf"=>"4", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"2"}
  • {"unique-ip"=>"4", "full-text"=>"4", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"3"}
  • {"unique-ip"=>"3", "full-text"=>"4", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"4"}
  • {"unique-ip"=>"10", "full-text"=>"10", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"5"}
  • {"unique-ip"=>"2", "full-text"=>"1", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"6"}
  • {"unique-ip"=>"7", "full-text"=>"7", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"7"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"8"}
  • {"unique-ip"=>"6", "full-text"=>"4", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"9"}
  • {"unique-ip"=>"3", "full-text"=>"1", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"10"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"11"}
  • {"unique-ip"=>"5", "full-text"=>"6", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"12"}
  • {"unique-ip"=>"6", "full-text"=>"4", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"1"}
  • {"unique-ip"=>"4", "full-text"=>"8", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"2", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"2"}
  • {"unique-ip"=>"2", "full-text"=>"2", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"3"}
  • {"unique-ip"=>"6", "full-text"=>"5", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"4"}
  • {"unique-ip"=>"2", "full-text"=>"2", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"5"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"6"}
  • {"unique-ip"=>"6", "full-text"=>"7", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"8"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"9"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"10"}
  • {"unique-ip"=>"6", "full-text"=>"5", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"11"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"12"}
  • {"unique-ip"=>"2", "full-text"=>"2", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"1"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"3"}
  • {"unique-ip"=>"8", "full-text"=>"8", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"1"}
  • {"unique-ip"=>"6", "full-text"=>"6", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"4"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"5"}
  • {"unique-ip"=>"16", "full-text"=>"24", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"6"}
  • {"unique-ip"=>"1975", "full-text"=>"5072", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"7"}
  • {"unique-ip"=>"2", "full-text"=>"2", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"8"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"9"}
  • {"unique-ip"=>"8", "full-text"=>"5", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"10"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"4", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"11"}
  • {"unique-ip"=>"4", "full-text"=>"6", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"12"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"2"}
  • {"unique-ip"=>"4", "full-text"=>"3", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"3"}
  • {"unique-ip"=>"2", "full-text"=>"2", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"4"}

Relative Metric

{"start_date"=>"2013-01-01T00:00:00Z", "end_date"=>"2013-12-31T00:00:00Z", "subject_areas"=>[{"subject_area"=>"/Biology and life sciences", "average_usage"=>[269, 466, 588, 697, 800, 896, 988, 1076, 1165, 1254, 1340, 1417]}, {"subject_area"=>"/Biology and life sciences/Physiology", "average_usage"=>[256, 449, 572, 676, 775, 866, 955, 1041, 1127, 1213, 1290, 1370, 1437]}, {"subject_area"=>"/Engineering and technology", "average_usage"=>[254, 440, 558, 675, 785, 883, 972, 1061, 1154, 1248, 1336, 1414, 1489]}, {"subject_area"=>"/Engineering and technology/Electrical engineering", "average_usage"=>[309, 409, 531, 718, 817, 898, 984, 1044, 1093, 1148, 1189, 1276, 1324, 1455]}, {"subject_area"=>"/Medicine and health sciences", "average_usage"=>[264, 460, 584, 692, 794, 887, 978, 1067, 1154, 1241, 1328, 1408, 1474]}, {"subject_area"=>"/Physical sciences", "average_usage"=>[254, 431, 547, 651, 748, 842, 932, 1017, 1098, 1178, 1259, 1336, 1404]}]}
Loading … Spinner
There are currently no alerts