Ensembles of Spiking Neurons with Noise Support Optimal Probabilistic Inference in a Dynamically Changing Environment
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{"title"=>"Ensembles of Spiking Neurons with Noise Support Optimal Probabilistic Inference in a Dynamically Changing Environment", "type"=>"journal", "authors"=>[{"first_name"=>"Robert", "last_name"=>"Legenstein", "scopus_author_id"=>"6603322416"}, {"first_name"=>"Wolfgang", "last_name"=>"Maass", "scopus_author_id"=>"7005129380"}], "year"=>2014, "source"=>"PLoS Computational Biology", "identifiers"=>{"pui"=>"600310984", "sgr"=>"84908345348", "issn"=>"15537358", "pmid"=>"25340749", "scopus"=>"2-s2.0-84908345348", "doi"=>"10.1371/journal.pcbi.1003859", "isbn"=>"1553-734x"}, "id"=>"7d270a39-ac2b-37f7-b633-ee938d6fb928", "abstract"=>"It has recently been shown that networks of spiking neurons with noise can emulate simple forms of probabilistic inference through \"neural sampling\", i.e., by treating spikes as samples from a probability distribution of network states that is encoded in the network. Deficiencies of the existing model are its reliance on single neurons for sampling from each random variable, and the resulting limitation in representing quickly varying probabilistic information. We show that both deficiencies can be overcome by moving to a biologically more realistic encoding of each salient random variable through the stochastic firing activity of an ensemble of neurons. The resulting model demonstrates that networks of spiking neurons with noise can easily track and carry out basic computational operations on rapidly varying probability distributions, such as the odds of getting rewarded for a specific behavior. We demonstrate the viability of this new approach towards neural coding and computation, which makes use of the inherent parallelism of generic neural circuits, by showing that this model can explain experimentally observed firing activity of cortical neurons for a variety of tasks that require rapid temporal integration of sensory information.", "link"=>"http://www.mendeley.com/research/ensembles-spiking-neurons-noise-support-optimal-probabilistic-inference-dynamically-changing-environ", "reader_count"=>90, "reader_count_by_academic_status"=>{"Unspecified"=>1, "Professor > Associate Professor"=>7, "Researcher"=>22, "Student > Doctoral Student"=>6, "Student > Ph. D. Student"=>34, "Student > Postgraduate"=>4, "Student > Master"=>7, "Other"=>2, "Student > Bachelor"=>5, "Professor"=>2}, "reader_count_by_user_role"=>{"Unspecified"=>1, "Professor > Associate Professor"=>7, "Researcher"=>22, "Student > Doctoral Student"=>6, "Student > Ph. D. Student"=>34, "Student > Postgraduate"=>4, "Student > Master"=>7, "Other"=>2, "Student > Bachelor"=>5, "Professor"=>2}, "reader_count_by_subject_area"=>{"Engineering"=>13, "Unspecified"=>1, "Mathematics"=>3, "Agricultural and Biological Sciences"=>18, "Medicine and Dentistry"=>2, "Neuroscience"=>16, "Physics and Astronomy"=>13, "Psychology"=>4, "Computer Science"=>20}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>13}, "Medicine and Dentistry"=>{"Medicine and Dentistry"=>2}, "Neuroscience"=>{"Neuroscience"=>16}, "Physics and Astronomy"=>{"Physics and Astronomy"=>13}, "Psychology"=>{"Psychology"=>4}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>18}, "Computer Science"=>{"Computer Science"=>20}, "Mathematics"=>{"Mathematics"=>3}, "Unspecified"=>{"Unspecified"=>1}}, "reader_count_by_country"=>{"Canada"=>1, "United States"=>1, "United Kingdom"=>4, "Italy"=>1, "France"=>2, "Australia"=>1, "Switzerland"=>1, "Germany"=>5}, "group_count"=>4}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/1763940"], "description"=>"<p><b>A</b>) The state of a binary random variable is estimated where state 1 transitions to state 2 with some transition rate . <b>B</b>) Estimation is performed by a particle filter circuit without evidence input (: dynamics layer ensembles; : evidence layer ensembles). <b>C</b>) Example for the rate dynamics in layer . Ensemble rate for ensembles (black) and whole layer (gray). While rates in ensembles change due to the prediction of a transition, inhibition keeps the overall firing rate in the layer approximately constant. <b>D</b>) Temporal evolution of estimated posterior probability for state 2 (black) and true posterior (gray) for this example run. <b>E</b>) Circuit estimates of posterior probabilities at time <i>t</i> = 50 ms () in comparison to true posteriors at this time (). Shown are 100 runs (dots) with prior probability for state 1 and transition rate drawn from uniform distributions in [0.1, 0.9] and [0, 30]Hz respectively in each run. The result of the example run from panels C-D is indicated by a cross.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220443, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g007", "stats"=>{"downloads"=>0, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Tracking_of_dynamics_in_ENS_coding_/1220443", "title"=>"Tracking of dynamics in ENS coding.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1764064", "https://ndownloader.figshare.com/files/1764065", "https://ndownloader.figshare.com/files/1764066", "https://ndownloader.figshare.com/files/1764067", "https://ndownloader.figshare.com/files/1764068", "https://ndownloader.figshare.com/files/1764069"], "description"=>"<div><p>It has recently been shown that networks of spiking neurons with noise can emulate simple forms of probabilistic inference through “neural sampling”, i.e., by treating spikes as samples from a probability distribution of network states that is encoded in the network. Deficiencies of the existing model are its reliance on single neurons for sampling from each random variable, and the resulting limitation in representing quickly varying probabilistic information. We show that both deficiencies can be overcome by moving to a biologically more realistic encoding of each salient random variable through the stochastic firing activity of an ensemble of neurons. The resulting model demonstrates that networks of spiking neurons with noise can easily track and carry out basic computational operations on rapidly varying probability distributions, such as the odds of getting rewarded for a specific behavior. We demonstrate the viability of this new approach towards neural coding and computation, which makes use of the inherent parallelism of generic neural circuits, by showing that this model can explain experimentally observed firing activity of cortical neurons for a variety of tasks that require rapid temporal integration of sensory information.</p></div>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220474, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1003859.s001", "https://dx.doi.org/10.1371/journal.pcbi.1003859.s002", "https://dx.doi.org/10.1371/journal.pcbi.1003859.s003", "https://dx.doi.org/10.1371/journal.pcbi.1003859.s004", "https://dx.doi.org/10.1371/journal.pcbi.1003859.s005", "https://dx.doi.org/10.1371/journal.pcbi.1003859.s006"], "stats"=>{"downloads"=>162, "page_views"=>59, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Ensembles_of_Spiking_Neurons_with_Noise_Support_Optimal_Probabilistic_Inference_in_a_Dynamically_Changing_Environment_/1220474", "title"=>"Ensembles of Spiking Neurons with Noise Support Optimal Probabilistic Inference in a Dynamically Changing Environment", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1763938"], "description"=>"<p><b>A</b>) Represented random variables. <b>Aa</b>) Evidence integration is performed for a random variable with 16 hidden states corresponding to direction-color pairs. Values of the random variable are depicted as circles. Observations accessible to the monkey in one example state are shown as boxes. <b>Ab</b>) The action readout layer infers a color-independent random variable by marginalization over color in each direction. <b>B</b>) Circuit structure. The circuit on the top approximates evidence integration through particle filtering (top gray box; : dynamics layer ensembles; : evidence layer ensembles)) on the random variable indicated in panel (Aa). An action readout layer (bottom gray box; ensembles <i>X</i>) receives feed-forward projections from the particle filter circuit. <b>C</b>) Spike rasters from simulations for afferent neurons (Ca) and neurons in the action readout layer (Cb). Each line corresponds to the output of one neuron. Afferent neurons are ordered by feature selectivity (e.g., top neurons code the presence of the fixation cross). Action readout neurons are ordered by preferred movement direction. See also <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g001\" target=\"_blank\">Figure 1</a>.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220442, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g006", "stats"=>{"downloads"=>1, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Particle_filtering_in_ENS_coding_for_the_ambiguous_target_task_/1220442", "title"=>"Particle filtering in ENS coding for the ambiguous target task.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1763918"], "description"=>"<p><b>A</b>) Circuit with ensembles (indicated by red and blue neurons respectively) and neurons per ensemble. Neurons in layer receive synaptic connections from neurons in layer and update the represented distribution according to evidence input from afferent neurons (green). Lateral inhibition (magenta; see panel C) stabilizes activity in this layer. Neurons project back to layer . For task class B (evidence integration; static random variable ), only connections between neurons that code for the same hidden state are necessary and layer simply copies the distribution represented by layer , see <i>Task class A</i> and <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g003\" target=\"_blank\">Figure 3</a> (in contrast to <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g003\" target=\"_blank\">Figure 3</a>, the copying ensembles are plotted above ensembles in order to avoid a cluttered diagram). For task class C (Bayesian filtering; random variable with time-independent dynamics), implements changes of the represented distribution due to the dynamics of the random variable and is potentially fully connected to . Neurons in layer disinhibit neurons in layer (double-dot connections; see panel B). Disinhbition and lateral inhibition is indicated by shortcuts as defined in B, C. Arrows indicate efferent connections. A schematic overview of the circuit is shown in the inset. <b>B</b>) Disinhibition : neurons excite an interneuron (purple) which inhibits the inhibitory drive to some neuron . As a graphical shortcut, we draw such disinhibitory influence as a connection with two circles (inset) <b>C</b>) Lateral inhibition: Pyramidal cells (blue) excite a pool of inhibitory neurons (magenta) which feed back common inhibition . The graphical shortcut for lateral inhibition is shown in the inset.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220439, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g004", "stats"=>{"downloads"=>0, "page_views"=>10, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Particle_filter_circuit_architecture_for_task_classes_B_and_C_/1220439", "title"=>"Particle filter circuit architecture for task classes B and C.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1763907"], "description"=>"<p>A binary random variable is represented in ENS coding through neurons . The posterior for a binary variable is represented by neurons . Each variable is represented by ensembles, one for each possible state (indicated by neuron color), and neurons per ensemble. The two layers are connected in an all-to-all manner. Arrows indicate efferent connections (i.e., outputs in ENS coding). The architecture is summarized in the inset.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220436, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g003", "stats"=>{"downloads"=>0, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Computations_in_ENS_coding_in_a_feed_forward_circuit_architecture_/1220436", "title"=>"Computations in ENS coding in a feed forward circuit architecture.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1764005"], "description"=>"<p><b>A</b>) Two-chamber maze (black lines) and transitions between states in various contexts. States are arranged on a 10×10 grid in the maze (crossing points of gray lines). Light gray lines indicate bidirectional state transitions with low transition rates (0.1 Hz). Dark gray arrows indicate transitions with high transition rates (3.5 Hz). Context is defined by movement direction (right, down, left, up). Colored circles indicate sensory evidence. Each color stands for one afferent neuron with a Gaussian spatial receptive field. The circle indicates the STD of the Gaussian. Note that the southern chambers give rise to identical observations. Observations are truncated at the height of the opening between the chambers such that no observations are experienced in the most northern parts. <b>B</b>) Network estimate of posterior probability (see color bar on the right for color code) for one trajectory through the maze (white trace; dot denotes current position) at different times. Spatial layout as in A. Various phases of the trajectory are shown: Uninformative prior knowledge (<i>t</i> = 10 ms), ambiguous estimates (<i>t</i> = 300 ms); disambiguation (<i>t</i> = 600, 900 ms); states without evidence (<i>t</i> = 1500, 2400 ms); unambiguous state estimation based on ambiguous evidence (<i>t</i> = 3300, 3800 ms).</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220465, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g010", "stats"=>{"downloads"=>2, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Self_localization_through_particle_filtering_in_ENS_coding_/1220465", "title"=>"Self-localization through particle filtering in ENS coding.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1763989"], "description"=>"<p>Extended circuit with ensembles (indicated by red and blue neurons respectively) and neurons per ensemble and two possible contexts. Ensembles in layer are duplicated for each context. These neurons receive context information via disinhibition from context neurons (yellow; only connections from context 1 shown for clarity). Disinhbition and lateral inhibition indicated by shortcuts as defined in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g004\" target=\"_blank\">Figure 4B, C</a>. Arrows indicate efferent connections. A schematic overview of the circuit is shown in the inset.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220463, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g009", "stats"=>{"downloads"=>0, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Particle_filter_circuit_architecture_for_task_class_D_/1220463", "title"=>"Particle filter circuit architecture for task class D.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1763890"], "description"=>"<p><b>A)</b> Task structure. After an initial fixation (center-hold time; CHT), the spatial cue (SC) is shown in the form of two color markers at one of eight possible locations and displaced from each other by 180 degrees. They mark two potentially rewarded movement directions. After a memory epoch (MEM), the color cue (CC) is shown at the fixation cross. The rewarded movement direction is defined by the direction of matching color in the color cue (time periods of simulation indicated). <b>B)</b> Firing activity of neurons in dorsal premotor cortex during the task. Before the spatial cue is shown, neurons are diffusely active. As the spatial cue is shown, neurons with preferred directions consistent with the spatial cue increase the firing rate and others are silenced. This circuit behavior is retained during the memory epoch. As the color cue is presented, neurons with consistent preferred directions increase their firing rates. <b>C)</b> Simulation result for a circuit that performs evidence integration in ENS coding (activity smoothed; horizontal axis: time). Neurons are ordered by their preferred direction. Panel B modified with permission from <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi.1003859-Cisek2\" target=\"_blank\">[52]</a>.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220432, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g001", "stats"=>{"downloads"=>3, "page_views"=>12, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Representation_of_a_belief_in_dorsal_premotor_cortex_PMd_in_the_ambiguous_target_task_/1220432", "title"=>"Representation of a belief in dorsal premotor cortex (PMd) in the ambiguous target task.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1763921"], "description"=>"<p><b>A</b>) The state of a binary random variable that gives rise to two possible observations is estimated. Both observations occur more frequently in state 1 (indicated by sharpness of arrows). <b>B</b>) Estimation is performed by a particle filtering circuit with evidence input (: dynamics layer ensembles; : evidence layer ensembles). <b>C</b>) An evidence spike is observed at times 20 ms and 25 ms in evidence neuron and respectively. <b>D</b>) Example for the rate dynamics in layer . Ensemble rate for ensemble (black) and whole layer (gray). The input leads to a transient increase in the ensemble rate. Inhibition recovers baseline activity. The ensemble rate for state 1 undergoes a transient and a sustained activity increase. <b>E</b>) Temporal evolution of estimated posterior probability for state 1 (black) in comparison to true posterior (gray) for this example run. <b>F</b>) Posterior probability at <i>t</i> = 45 ms () for state 1 of the circuit in comparison to true posterior at this time (). Each dot represents one out of 100 runs with prior probabilities and observation likelihoods drawn independently in each run (see <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#s4\" target=\"_blank\">Methods</a></i>). The results of the example run from panels A–E is indicated by a cross.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220440, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g005", "stats"=>{"downloads"=>0, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Evidence_integration_through_particle_filtering_in_ENS_coding_/1220440", "title"=>"Evidence integration through particle filtering in ENS coding.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1764050"], "description"=>"<p>Description of frequently used variables for easy reference. In general, capital letters refer to ensembles and lower case letters to neurons in these ensembles.</p><p>Notation.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220473, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.t003", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Notation_/1220473", "title"=>"Notation.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1764047"], "description"=>"<p>Here we have defined and . denotes lateral inhibition and disinhibition. is an arbitrary constant. In task class B (evidence integration), for , leading to for and .</p><p>Particle filter circuit equations for task classes B and C.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220471, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.t001", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Particle_filter_circuit_equations_for_task_classes_B_and_C_/1220471", "title"=>"Particle filter circuit equations for task classes B and C.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1764048"], "description"=>"<p>Here we have defined and . denotes lateral inhibition and disinhibition. is an arbitrary constant.</p><p>Particle filter circuit equations for task class D.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220472, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.t002", "stats"=>{"downloads"=>4, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Particle_filter_circuit_equations_for_task_class_D_/1220472", "title"=>"Particle filter circuit equations for task class D.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1764044"], "description"=>"<p><b>A</b>) Represented random variables. <b>Aa</b>) Dynamics of a random variable that codes the current phase in a trial of the ambiguous target task (CHT: fixation; SC: spatial cue; MEM: memory cue; CC: color cue). Possible observations in each phase are indicated in boxes. <b>Ab</b>) Context-dependent Bayesian filtering is performed for a random variable with 16 hidden states corresponding to direction-color pairs as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g006\" target=\"_blank\">Figure 6A</a>a. Gray lines indicate context-dependent transitions. All-to-all transitions are possible in the fixation phase (CHT). There are no transitions in other phases of a trial. <b>Ac</b>) The action readout layer infers a color-independent random variable by marginalization over color in each direction. <b>B</b>) Circuit structure. The circuit on the top (ensembles and ) performs Bayesian filtering on the random variable indicated in panel (Aa). It provides context for another particle filter circuit (middle gray box; ensembles and ) that generates a belief about the random variable indicated in Ab. An action readout layer is added (bottom gray box; ensembles ). <b>C</b>) Spike rasters from a simulation of two successive trials for afferent neurons (Ca), neurons in the particle filter circuit for the phase in the trial (Cb), and neurons in the action readout layer (Cc). Neurons in Cb are coding for the current phase of the trial (ordered from bottom to top: CHT, SC, MEM, and CC). Neuron ordering in Ca and Cc as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g006\" target=\"_blank\">Figure 6</a>.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220470, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g011", "stats"=>{"downloads"=>0, "page_views"=>16, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Context_dependent_Bayesian_filtering_in_two_successive_trials_of_the_ambiguous_target_task_/1220470", "title"=>"Context-dependent Bayesian filtering in two successive trials of the ambiguous target task.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1763961"], "description"=>"<p><b>Aa</b>) State diagram of the Markov chain for the dynamics of the hidden random variable (bottom) and state-dependent firing rates of afferent neurons (top). Colors indicate the value of the hidden state. <b>Ab</b>) Actual hidden state over time is indicated by color in correspondence with colors in panel Aa. <b>Ac</b>) Spike trains of afferent neurons. Each line corresponds to the output of one afferent neuron ordered according to panel Aa. <b>Ad</b>) Network response to the input in panel Ac. Neurons are ordered according to their preferred state from state 1 (top neurons) to 5 (bottom neurons). <b>Ae</b>) Network belief (estimated posterior state probability) derived from network activity. Rows ordered by state as neurons in panel Ad. Hot color indicates high probability of the state. Note the uncertainty when state 2 or 3 is entered. <b>Af</b>) Summary of network performance (“model”; fraction of incorrect state estimates) in comparison with the optimal Bayesian filter (“opt”), a network with jittered synaptic efficacies (“jit”), and the optimal decision based on the most recent observation only (“inp”). Bars are means and errorbars STDs over 20 state and observation sequences (12 seconds each). <b>B</b>) Particle filtering for task class D. <b>Ba</b>) As panel Aa but with context. Dark gray arrows in the state diagram indicate transitions in context A. In context B, the transitions from states 2 to 4 and 3 to 5 are interchanged (light gray arrows). <b>Bb</b>) As panel Ab. Background shading indicates context (context A: white; context B: gray). <b>Bc–Be</b>) Actual hidden state, input spikes, networks spikes, and network belief; see panels Ac–Ae. <b>Bf</b>) Summary of network performance. “opt” shows performance of the optimal context-dependent Bayesian filter and “mix” a Bayesian filter where the transition rates are the mean rates over contexts A and B. The spiking network performs significantly better than the mixed Bayesian filter (paired t-test, <i>p</i><0.001).</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220449, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g008", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Particle_filtering_for_task_class_C_A_and_task_class_D_B_in_the_ENS_code_/1220449", "title"=>"Particle filtering for task class C (A) and task class D (B) in the ENS code.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/1763901"], "description"=>"<p><b>A</b>) Sample-based representations of probability distributions. True distribution of a random variable (green) and approximated distribution (yellow) based on 20 samples (top) and 200 samples (bottom) <b>B</b>) Interpretation of the spiking activity of two neuronal ensembles as samples from a probability distribution over a temporally changing random variable . Shown is an example for a random variable with two possible states. Black lines in the top traces indicate action potentials in two ensembles (5 neurons per state shown). Traces above spikes show EPSP-filtered versions of these spikes (red: state 1; blue: state 2). Bottom plot: Estimated probabilities for state 1 (red) and state 2 (blue) according to eq. (1) based on the spiking activity of 10 neurons per state.</p>", "links"=>[], "tags"=>["probability", "information", "Noise Support Optimal Probabilistic Inference", "spiking neurons", "sampling", "model"], "article_id"=>1220434, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Robert Legenstein", "Wolfgang Maass"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1003859.g002", "stats"=>{"downloads"=>1, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Spikes_as_samples_from_probability_distributions_/1220434", "title"=>"Spikes as samples from probability distributions.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-10-23 17:26:10"}

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