Network Plasticity as Bayesian Inference
Publication Date
November 06, 2015
Journal
PLOS Computational Biology
Authors
David Kappel, Stefan Habenschuss, Robert Legenstein & Wolfgang Maass
Volume
11
Issue
11
Pages
e1004485
DOI
https://dx.plos.org/10.1371/journal.pcbi.1004485
Publisher URL
http://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1004485
PubMed
http://www.ncbi.nlm.nih.gov/pubmed/26545099
PubMed Central
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4636322
Europe PMC
http://europepmc.org/abstract/MED/26545099
Web of Science
000365801600009
Scopus
84949292982
Mendeley
http://www.mendeley.com/research/network-plasticity-bayesian-inference
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Mendeley | Further Information

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Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/2416640"], "description"=>"<p><b>A</b>: Illustration of the network architecture. A WTA circuit consisting of ten neurons <b>z</b> receives afferent stimuli from input neurons <b>x</b> (few connections shown for a single neuron in <b>z</b>). <b>B</b>: The STDP learning curve that arises from the likelihood term in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.e027\" target=\"_blank\">Eq (11)</a>. <b>C</b>: Measured STDP curve that results from a related STDP rule for a moderate pairing frequency of 20 Hz, as in [<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.ref041\" target=\"_blank\">41</a>]. (Figure adapted from [<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.ref036\" target=\"_blank\">36</a>]). <b>D, E</b>: Each sensory experience was modeled by 200 ms long spiking activity of 1000 input neurons, that covered some 3D data space with Gaussian tuning curves (the results do not depend on the finite dimension of the data space, we chose 3 dimension for easier visualization). Insets show the firing activity of randomly chosen 50 of the 1000 input neurons for the sample data points marked by green circles. Objects in the environment were represented by Gaussian clusters (ellipses) in this finite dimensional data space. <b>F</b>: During learning phase 1 (3 hours) only samples from SE were presented to the network, in phase 2 (which lasted 1 hour) samples from EE. Shortly after the transition from SE to EE the number of newly formed synaptic connections significantly increases (compare to Fig. 1h in [<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.ref033\" target=\"_blank\">33</a>]). <b>G</b>: Comparison of the survival of synapses for a network with persistent exposure to EE (EE-EE condition) and a network that was returned to SE (EE-SE condition). Newly formed synaptic connections are transient and quickly decay after formation. A significantly larger fraction of synapses persists if the network continuously receives EE inputs (compare to Fig. 2c in [<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.ref033\" target=\"_blank\">33</a>]). The dots show means of measurements taken every 30 minutes, the lines represent two-term exponential fits (<i>r</i><sup>2</sup> = 1). The results in (F, G) show means over 5 trial runs. Error bars indicate STD.</p>", "links"=>[], "tags"=>["Bayesian Inference General results", "Synaptic plasticity", "model"], "article_id"=>1596891, "categories"=>["Uncategorised"], "users"=>["David Kappel", "Stefan Habenschuss", "Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004485.g004", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Adaptation_of_synaptic_connections_to_changing_input_statistics_through_synaptic_sampling_/1596891", "title"=>"Adaptation of synaptic connections to changing input statistics through synaptic sampling.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-06 04:22:37"}
  • {"files"=>["https://ndownloader.figshare.com/files/2416638"], "description"=>"<p><b>A</b>: Illustration of the parametrization of spine motility. Values <i>θ</i> > 0 indicate a functional synaptic connection. <b>B</b>: A Gaussian prior <i>p</i><sub>𝒮</sub>(<i>θ</i>), and a few stochastic sample trajectories of <i>θ</i> according to the synaptic sampling rule <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.e024\" target=\"_blank\">Eq (10)</a>. Negative values of <i>θ</i> (gray area) are interpreted as non-functional connections. Some stable synaptic connections emerge (traces in the upper half), whereas other synaptic connections come and go (traces in lower half). All traces, as well as survival statistics shown in (E, F), are taken from the network simulation described in detail in the next section and <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.s005\" target=\"_blank\">S5 Text</a>. <b>C</b>: The exponential function maps synapse parameters <i>θ</i> to synaptic efficacies <i>w</i>. Negative values of <i>θ</i>, corresponding to (retracted) spines are mapped to a tiny region close to zero in the <i>w</i>-space. <b>D</b>: The Gaussian prior in the <i>θ</i>-space translates to a log-normal distribution in the <i>w</i>-space. The traces from (B) are shown in the right panel transformed into the <i>w</i>-space. Only persistent synaptic connections contribute substantial synaptic efficacies. <b>E, F</b>: The emergent survival statistics of newly formed synaptic connections, (i.e., formed during the preceding 12 hours) evaluated at three different start times throughout learning (blue traces, axes are aligned with start times of the analyses). The survival statistics exhibit in our synaptic sampling model a power-law behavior (red curves, see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.s005\" target=\"_blank\">S5 Text</a>). The time-scale (and exponent of the power-law) depends on the learning rate <i>b</i> in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.e024\" target=\"_blank\">Eq (10)</a>, and can assume any value in our quite general model (shown is <i>b</i> = 10<sup>−4</sup> in (E) and <i>b</i> = 10<sup>−6</sup> in (F)).</p>", "links"=>[], "tags"=>["Bayesian Inference General results", "Synaptic plasticity", "model"], "article_id"=>1596890, "categories"=>["Uncategorised"], "users"=>["David Kappel", "Stefan Habenschuss", "Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004485.g003", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Integration_of_spine_motility_into_the_synaptic_sampling_model_/1596890", "title"=>"Integration of spine motility into the synaptic sampling model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-06 04:22:37"}
  • {"files"=>["https://ndownloader.figshare.com/files/2416646", "https://ndownloader.figshare.com/files/2416647", "https://ndownloader.figshare.com/files/2416648", "https://ndownloader.figshare.com/files/2416650", "https://ndownloader.figshare.com/files/2416651", "https://ndownloader.figshare.com/files/2416652", "https://ndownloader.figshare.com/files/2416653"], "description"=>"<div><p>General results from statistical learning theory suggest to understand not only brain computations, but also brain plasticity as probabilistic inference. But a model for that has been missing. We propose that inherently stochastic features of synaptic plasticity and spine motility enable cortical networks of neurons to carry out probabilistic inference by sampling from a posterior distribution of network configurations. This model provides a viable alternative to existing models that propose convergence of parameters to maximum likelihood values. It explains how priors on weight distributions and connection probabilities can be merged optimally with learned experience, how cortical networks can generalize learned information so well to novel experiences, and how they can compensate continuously for unforeseen disturbances of the network. The resulting new theory of network plasticity explains from a functional perspective a number of experimental data on stochastic aspects of synaptic plasticity that previously appeared to be quite puzzling.</p></div>", "links"=>[], "tags"=>["Bayesian Inference General results", "Synaptic plasticity", "model"], "article_id"=>1596893, "categories"=>["Uncategorised"], "users"=>["David Kappel", "Stefan Habenschuss", "Robert Legenstein", "Wolfgang Maass"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1004485.s001", "https://dx.doi.org/10.1371/journal.pcbi.1004485.s002", "https://dx.doi.org/10.1371/journal.pcbi.1004485.s003", "https://dx.doi.org/10.1371/journal.pcbi.1004485.s004", "https://dx.doi.org/10.1371/journal.pcbi.1004485.s005", "https://dx.doi.org/10.1371/journal.pcbi.1004485.s006", "https://dx.doi.org/10.1371/journal.pcbi.1004485.s007.tar"], "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Network_Plasticity_as_Bayesian_Inference_/1596893", "title"=>"Network Plasticity as Bayesian Inference", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2015-11-06 04:22:37"}
  • {"files"=>["https://ndownloader.figshare.com/files/2416643"], "description"=>"<p><b>A</b>: A spike-based generative neural network (illustrated at the bottom) received simultaneously spoken and handwritten representations of the same digit (and for tests only spoken digits, see (B)). Stimulus examples for spoken and written digit <i>2</i> are shown at the top. These inputs are presented to the network through corresponding firing rates of “auditory” (<b>x</b><sub><i>A</i></sub>) and “visual” (<b>x</b><sub><i>V</i></sub>) input neurons. Two populations <b>z</b><sub><i>A</i></sub> and <b>z</b><sub><i>V</i></sub> of 40 neurons, each consisting of four WTA circuits like in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.g004\" target=\"_blank\">Fig 4</a>, receive exclusively auditory or visual inputs. In addition, arbitrary lateral excitatory connections between these “hidden” neurons are allowed. <b>B</b>: Assemblies of hidden neurons emerge that encode the presented digit (<i>1</i> or <i>2</i>). Top panel shows PETH of all neurons from <b>z</b><sub><i>V</i></sub> for stimulus <i>1</i> (left) and <i>2</i> (right) after learning, when only an auditory stimulus is presented. Neurons are sorted by the time of their highest average firing. Although only auditory stimuli are presented, it is possible to reconstruct an internally generated “guessed” visual stimulus that represents the same digit (bottom). <b>C</b>: First three PCA components of the temporal evolution of a subset <b><i>θ</i></b>′ of network parameters <b><i>θ</i></b> (see text). Two major lesions were applied to the network. In the first lesion (transition to red) all neurons that significantly encode stimulus <i>2</i> were removed from the population <b>z</b><sub><i>V</i></sub>. In the second lesion (transition to green) all currently existing synaptic connections between neuron in <b>z</b><sub><i>A</i></sub> and <b>z</b><sub><i>V</i></sub> were removed, and not allowed to regrow. After each lesion the network parameters <b><i>θ</i></b>′ migrate to a new manifold. <b>D</b>: The generative reconstruction performance of the “visual” neurons <b>z</b><sub><i>V</i></sub> for the test case when only an auditory stimulus is presented was tracked throughout the whole learning session, including lesions <i>1</i> and <i>2</i> (bottom panel). After each lesion the performance strongly degrades, but reliably recovers. Insets show at the top the synaptic weights of neurons in <b>z</b><sub><i>V</i></sub> at 4 time points <i>t</i><sub>1</sub>, …, <i>t</i><sub>4</sub>, projected back into the input space like in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004485#pcbi.1004485.g004\" target=\"_blank\">Fig 4E</a>. Network diagrams in the middle show ongoing network rewiring for synaptic connections between the hidden neurons <b>z</b><sub><i>A</i></sub> and <b>z</b><sub><i>V</i></sub>. Each arrow indicates a functional connection between two neurons. To keep the figure uncluttered only subsets of synapses are shown (1% randomly drawn from the total set of possible lateral connections). Connections at time <i>t</i><sub>2</sub> that were already functional at time <i>t</i><sub>1</sub> are plotted in gray. The neuron whose parameter vector <b><i>θ</i></b>′ is tracked in (C) is highlighted in red. The text under the network diagrams shows the total number of functional connections between hidden neurons at the time point.</p>", "links"=>[], "tags"=>["Bayesian Inference General results", "Synaptic plasticity", "model"], "article_id"=>1596892, "categories"=>["Uncategorised"], "users"=>["David Kappel", "Stefan Habenschuss", "Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004485.g005", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Inherent_compensation_for_network_perturbations_/1596892", "title"=>"Inherent compensation for network perturbations.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-06 04:22:37"}
  • {"files"=>["https://ndownloader.figshare.com/files/2416636"], "description"=>"<p><b>A</b>: The training set, consisting of five samples of a handwritten <i>1</i>. Below a cartoon illustrating the network architecture of the restricted Boltzmann machine (RBM), composed of a layer of 784 visible neurons <b>x</b> and a layer of 9 hidden neurons <b>z</b>. <b>B</b>: Examples from the test set. It contains many different styles of writing that are not part of the training set. <b>C</b>: Evolution of 50 randomly selected synaptic weights throughout learning (on the training set). The weight histogram (right) shows the distribution of synaptic weights at the end of learning. 80 histogram bins were equally spaced between -4 and 4. <b>D</b>: Performance of the network in terms of log likelihood on the training set (blue) and on the test set (red) throughout learning. Mean values over 100 trial runs are shown, shaded area indicates std. The performance on the test set initially increases but degrades for prolonged learning. <b>E</b>: Evolution of 50 weights for the same network but with a bimodal prior. The prior <i>p</i><sub>𝒮</sub>(<i>w</i>) is indicated by the blue curve. Most synaptic weights settle in the mode around 0, but a few larger weights also emerge and stabilize in the larger mode. Weight histogram (green) as in (C). <b>F</b>: The log likelihood on the test set maintains a constant high value throughout the whole learning session, compare to (D).</p>", "links"=>[], "tags"=>["Bayesian Inference General results", "Synaptic plasticity", "model"], "article_id"=>1596889, "categories"=>["Uncategorised"], "users"=>["David Kappel", "Stefan Habenschuss", "Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004485.g002", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Priors_for_synaptic_weights_improve_generalization_capability_/1596889", "title"=>"Priors for synaptic weights improve generalization capability.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-06 04:22:37"}
  • {"files"=>["https://ndownloader.figshare.com/files/2416632"], "description"=>"<p><b>A, B, C</b>: Illustration of ML learning for two parameters <b><i>θ</i></b> = (<i>θ</i><sub>1</sub>,<i>θ</i><sub>2</sub>) of a neural network 𝒩. <b>A</b>: 3D plot of an example likelihood function. For a fixed set of inputs <b>x</b> it assigns a probability density (amplitude on z-axis) to each parameter setting <b><i>θ</i></b>. <b>B</b>: This likelihood function is defined by some underlying neural network</p>", "links"=>[], "tags"=>["Bayesian Inference General results", "Synaptic plasticity", "model"], "article_id"=>1596886, "categories"=>["Uncategorised"], "users"=>["David Kappel", "Stefan Habenschuss", "Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004485.g001", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Maximum_likelihood_ML_learning_vs_synaptic_sampling_/1596886", "title"=>"Maximum likelihood (ML) learning vs. synaptic sampling.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-06 04:22:37"}

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  • {"unique-ip"=>"6", "full-text"=>"2", "pdf"=>"5", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"12"}
  • {"unique-ip"=>"13", "full-text"=>"11", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"2", "year"=>"2020", "month"=>"2"}
  • {"unique-ip"=>"17", "full-text"=>"19", "pdf"=>"8", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"2", "supp-data"=>"7", "cited-by"=>"0", "year"=>"2020", "month"=>"3"}
  • {"unique-ip"=>"11", "full-text"=>"13", "pdf"=>"5", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"2", "cited-by"=>"0", "year"=>"2020", "month"=>"4"}
  • {"unique-ip"=>"11", "full-text"=>"10", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"5"}
  • {"unique-ip"=>"8", "full-text"=>"6", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"6"}
  • {"unique-ip"=>"3", "full-text"=>"3", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"1", "cited-by"=>"0", "year"=>"2020", "month"=>"7"}
  • {"unique-ip"=>"9", "full-text"=>"8", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"6", "cited-by"=>"0", "year"=>"2020", "month"=>"8"}
  • {"unique-ip"=>"9", "full-text"=>"9", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"10"}

Relative Metric

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

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