{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T11:19:48Z","timestamp":1767179988023,"version":"build-2238731810"},"update-to":[{"DOI":"10.1371\/journal.pcbi.1009799","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T00:00:00Z","timestamp":1644364800000}}],"reference-count":86,"publisher":"Public Library of Science (PLoS)","issue":"1","license":[{"start":{"date-parts":[[2022,1,28]],"date-time":"2022-01-28T00:00:00Z","timestamp":1643328000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S005692\/1"],"award-info":[{"award-number":["EP\/S005692\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000265","name":"Medical Research Council","doi-asserted-by":"publisher","award":["MR\/N013166\/1"],"award-info":[{"award-number":["MR\/N013166\/1"]}],"id":[{"id":"10.13039\/501100000265","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Wellcome Trust and the Royal Society","award":["102857\/Z\/13\/Z"],"award-info":[{"award-number":["102857\/Z\/13\/Z"]}]},{"name":"RS MacDonald Charitable Trust Seedcorn Grant","award":["PWC ref.29"],"award-info":[{"award-number":["PWC ref.29"]}]},{"DOI":"10.13039\/501100015504","name":"Simons Initiative for the Developing Brain","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100015504","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000268","name":"Biotechnology and Biological Sciences Research Council","doi-asserted-by":"publisher","award":["BB\/T007907\/1"],"award-info":[{"award-number":["BB\/T007907\/1"]}],"id":[{"id":"10.13039\/501100000268","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["866386"],"award-info":[{"award-number":["866386"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>One of the main goals of current systems neuroscience is to understand how neuronal populations integrate sensory information to inform behavior. However, estimating stimulus or behavioral information that is encoded in high-dimensional neuronal populations is challenging. We propose a method based on parametric copulas which allows modeling joint distributions of neuronal and behavioral variables characterized by different statistics and timescales. To account for temporal or spatial changes in dependencies between variables, we model varying copula parameters by means of Gaussian Processes (GP). We validate the resulting Copula-GP framework on synthetic data and on neuronal and behavioral recordings obtained in awake mice. We show that the use of a parametric description of the high-dimensional dependence structure in our method provides better accuracy in mutual information estimation in higher dimensions compared to other non-parametric methods. Moreover, by quantifying the redundancy between neuronal and behavioral variables, our model exposed the location of the reward zone in an unsupervised manner (i.e., without using any explicit cues about the task structure). These results demonstrate that the Copula-GP framework is particularly useful for the analysis of complex multidimensional relationships between neuronal, sensory and behavioral variables.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1009799","type":"journal-article","created":{"date-parts":[[2022,1,28]],"date-time":"2022-01-28T13:35:17Z","timestamp":1643376917000},"page":"e1009799","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":6,"title":["Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships"],"prefix":"10.1371","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8529-7250","authenticated-orcid":true,"given":"Nina","family":"Kudryashova","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6975-5888","authenticated-orcid":true,"given":"Theoklitos","family":"Amvrosiadis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8853-2831","authenticated-orcid":true,"given":"Nathalie","family":"Dupuy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3498-6221","authenticated-orcid":true,"given":"Nathalie","family":"Rochefort","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7387-5535","authenticated-orcid":true,"given":"Arno","family":"Onken","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"340","published-online":{"date-parts":[[2022,1,28]]},"reference":[{"issue":"7679","key":"pcbi.1009799.ref001","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1038\/nature24636","article-title":"Fully integrated silicon probes for high-density recording of neural activity","volume":"551","author":"JJ Jun","year":"2017","journal-title":"Nature"},{"key":"pcbi.1009799.ref002","doi-asserted-by":"crossref","DOI":"10.1201\/9781420076851.ch2","volume-title":"Two-photon functional imaging of neuronal activity","author":"F Helmchen","year":"2009"},{"issue":"1","key":"pcbi.1009799.ref003","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.neuron.2007.08.003","article-title":"Imaging large-scale neural activity with cellular resolution in awake, mobile mice","volume":"56","author":"DA Dombeck","year":"2007","journal-title":"Neuron"},{"issue":"6437","key":"pcbi.1009799.ref004","doi-asserted-by":"crossref","first-page":"eaav7893","DOI":"10.1126\/science.aav7893","article-title":"Spontaneous behaviors drive multidimensional, brainwide activity","volume":"364","author":"C Stringer","year":"2019","journal-title":"Science"},{"issue":"10","key":"pcbi.1009799.ref005","doi-asserted-by":"crossref","first-page":"2521","DOI":"10.1016\/j.celrep.2018.08.010","article-title":"The Impact of Visual Cues, Reward, and Motor Feedback on the Representation of Behaviorally Relevant Spatial Locations in Primary Visual Cortex","volume":"24","author":"JM Pakan","year":"2018","journal-title":"Cell reports"},{"key":"pcbi.1009799.ref006","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.conb.2018.04.020","article-title":"Action and learning shape the activity of neuronal circuits in the visual cortex","volume":"52","author":"JM Pakan","year":"2018","journal-title":"Current opinion in neurobiology"},{"key":"pcbi.1009799.ref007","doi-asserted-by":"crossref","DOI":"10.1016\/j.conb.2019.02.002","volume-title":"Towards the neural population doctrine","author":"S Saxena","year":"2019"},{"issue":"2","key":"pcbi.1009799.ref008","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1038\/nn.2731","article-title":"How advances in neural recording affect data analysis","volume":"14","author":"IH Stevenson","year":"2011","journal-title":"Nature Neuroscience"},{"issue":"1-2","key":"pcbi.1009799.ref009","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s10827-009-0195-x","article-title":"CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains","volume":"29","author":"B Staude","year":"2010","journal-title":"Journal of computational neuroscience"},{"issue":"5","key":"pcbi.1009799.ref010","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1038\/nn1228","article-title":"Multiple neural spike train data analysis: state-of-the-art and future challenges","volume":"7","author":"EN Brown","year":"2004","journal-title":"Nature neuroscience"},{"key":"pcbi.1009799.ref011","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1146\/annurev-neuro-070815-013851","article-title":"Correlations and neuronal population information","volume":"39","author":"A Kohn","year":"2016","journal-title":"Annual review of neuroscience"},{"issue":"3","key":"pcbi.1009799.ref012","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1002385","article-title":"State-space analysis of time-varying higher-order spike correlation for multiple neural spike train data","volume":"8","author":"H Shimazaki","year":"2012","journal-title":"PLoS computational biology"},{"issue":"6","key":"pcbi.1009799.ref013","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1016\/j.neunet.2010.05.008","article-title":"Information-theoretic methods for studying population codes","volume":"23","author":"RA Ince","year":"2010","journal-title":"Neural Networks"},{"issue":"6","key":"pcbi.1009799.ref014","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1162\/089976604773717559","article-title":"Nonlinear population codes","volume":"16","author":"M Shamir","year":"2004","journal-title":"Neural computation"},{"issue":"9","key":"pcbi.1009799.ref015","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1038\/s41593-018-0209-y","article-title":"DeepLabCut: markerless pose estimation of user-defined body parts with deep learning","volume":"21","author":"A Mathis","year":"2018","journal-title":"Nature neuroscience"},{"key":"pcbi.1009799.ref016","article-title":"Reward Association Enhances Stimulus-Specific Representations in Primary Visual Cortex","author":"JU Henschke","year":"2020","journal-title":"Current Biology"},{"key":"pcbi.1009799.ref017","doi-asserted-by":"crossref","DOI":"10.1201\/b17116","volume-title":"Dependence modeling with copulas","author":"H Joe","year":"2014"},{"issue":"4","key":"pcbi.1009799.ref018","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1162\/089976604322860659","article-title":"The Shape of Neural Dependence","volume":"16","author":"RL Jenison","year":"2004","journal-title":"Neural Computation"},{"issue":"6","key":"pcbi.1009799.ref019","doi-asserted-by":"crossref","first-page":"68003","DOI":"10.1209\/0295-5075\/88\/68003","article-title":"An information-theoretic approach to statistical dependence: Copula information","volume":"88","author":"RS Calsaverini","year":"2009","journal-title":"EPL (Europhysics Letters)"},{"issue":"3","key":"pcbi.1009799.ref020","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1038\/nrn2578","article-title":"Extracting information from neuronal populations: information theory and decoding approaches","volume":"10","author":"RQ Quiroga","year":"2009","journal-title":"Nature Reviews Neuroscience"},{"issue":"2","key":"pcbi.1009799.ref021","first-page":"182","article-title":"Pair-copula constructions of multiple dependence","volume":"44","author":"K Aas","year":"2009","journal-title":"Insurance: Mathematics and economics"},{"key":"pcbi.1009799.ref022","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/978-3-642-12465-5_4","volume-title":"Copula theory and its applications","author":"C Czado","year":"2010"},{"key":"pcbi.1009799.ref023","unstructured":"Onken A, Panzeri S. Mixed Vine Copulas as Joint Models of Spike Counts and Local Field Potentials. In: Proceedings of the 30th International Conference on Neural Information Processing Systems. NIPS\u201916. Red Hook, NY, USA: Curran Associates Inc.; 2016. p. 1333\u20131341."},{"issue":"11","key":"pcbi.1009799.ref024","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1000577","article-title":"Analyzing short-term noise dependencies of spike-counts in macaque prefrontal cortex using copulas and the flashlight transformation","volume":"5","author":"A Onken","year":"2009","journal-title":"PLoS computational biology"},{"issue":"23","key":"pcbi.1009799.ref025","doi-asserted-by":"crossref","first-page":"8745","DOI":"10.1523\/JNEUROSCI.5041-14.2015","article-title":"Copula regression analysis of simultaneously recorded frontal eye field and inferotemporal spiking activity during object-based working memory","volume":"35","author":"M Hu","year":"2015","journal-title":"Journal of Neuroscience"},{"issue":"9","key":"pcbi.1009799.ref026","doi-asserted-by":"crossref","first-page":"2025","DOI":"10.1162\/NECO_a_00631","article-title":"A Semiparametric Bayesian Model for Detecting Synchrony Among Multiple Neurons","volume":"26","author":"B Shahbaba","year":"2014","journal-title":"Neural Computation"},{"key":"pcbi.1009799.ref027","first-page":"129","volume-title":"Advances in neural information processing systems","author":"P Berkes","year":"2009"},{"key":"pcbi.1009799.ref028","unstructured":"Safaai H, Wang A, Panzeri S, Harvey C. Characterizing information processing of parietal cortex projections using vine copulas. In: Bernstein Conference 2019. American Physical Society; 2019. Available from: https:\/\/abstracts.g-node.org\/abstracts\/f80ac63f-88fc-4203-9c2b-a279bb9e201a."},{"issue":"3","key":"pcbi.1009799.ref029","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1038\/nn.4242","article-title":"The mechanics of state-dependent neural correlations","volume":"19","author":"B Doiron","year":"2016","journal-title":"Nature neuroscience"},{"key":"pcbi.1009799.ref030","unstructured":"Lopez-Paz D, Hern\u00e1ndez-Lobato JM, Zoubin G. Gaussian process vine copulas for multivariate dependence. In: International Conference on Machine Learning; 2013. p. 10\u201318."},{"key":"pcbi.1009799.ref031","first-page":"1736","volume-title":"Advances in Neural Information Processing Systems","author":"JM Hern\u00e1ndez-Lobato","year":"2013"},{"key":"pcbi.1009799.ref032","doi-asserted-by":"crossref","first-page":"e63705","DOI":"10.7554\/eLife.63705","article-title":"Spatial modulation of visual responses arises in cortex with active navigation","volume":"10","author":"EM Diamanti","year":"2021","journal-title":"Elife"},{"issue":"3","key":"pcbi.1009799.ref033","doi-asserted-by":"crossref","DOI":"10.1523\/ENEURO.0052-18.2018","article-title":"A tutorial for information theory in neuroscience","volume":"5","author":"NM Timme","year":"2018","journal-title":"eneuro"},{"key":"pcbi.1009799.ref034","first-page":"229","article-title":"Fonctions de reprtition an dimensions et leursmarges","volume":"8","author":"A Sklar","year":"1959","journal-title":"Publ Inst Statis Univ Paris"},{"key":"pcbi.1009799.ref035","doi-asserted-by":"crossref","first-page":"053302","DOI":"10.1103\/PhysRevE.98.053302","article-title":"Information estimation using nonparametric copulas","volume":"98","author":"H Safaai","year":"2018","journal-title":"Phys Rev E"},{"issue":"3","key":"pcbi.1009799.ref036","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1177\/1471082X1101200302","article-title":"Maximum likelihood estimation of mixed C-vines with application to exchange rates","volume":"12","author":"C Czado","year":"2012","journal-title":"Statistical Modelling"},{"key":"pcbi.1009799.ref037","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.jmva.2016.07.003","article-title":"Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas","volume":"151","author":"T Nagler","year":"2016","journal-title":"Journal of Multivariate Analysis"},{"issue":"7207","key":"pcbi.1009799.ref038","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1038\/nature07140","article-title":"Spatio-temporal correlations and visual signalling in a complete neuronal population","volume":"454","author":"JW Pillow","year":"2008","journal-title":"Nature"},{"issue":"1","key":"pcbi.1009799.ref039","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/ncomms12190","article-title":"Accurate spike estimation from noisy calcium signals for ultrafast three-dimensional imaging of large neuronal populations in vivo","volume":"7","author":"T Deneux","year":"2016","journal-title":"Nature communications"},{"key":"pcbi.1009799.ref040","doi-asserted-by":"crossref","unstructured":"Mahuas G, Isacchini G, Marre O, Ferrari U, Mora T. A new inference approach for training shallow and deep generalized linear models of noisy interacting neurons. arXiv preprint arXiv:200606497. 2020;.","DOI":"10.1101\/2020.06.11.145904"},{"key":"pcbi.1009799.ref041","unstructured":"Macke JH, Buesing L, Cunningham JP, Yu BM, Shenoy KV, Sahani M. Empirical models of spiking in neural populations. In: Advances in Neural Information Processing Systems 24: 25th conference on Neural Information Processing Systems (NIPS 2011); 2012. p. 1350\u20131358."},{"issue":"5","key":"pcbi.1009799.ref042","doi-asserted-by":"crossref","first-page":"862","DOI":"10.1016\/j.neuron.2012.02.011","article-title":"Imaging calcium in neurons","volume":"73","author":"C Grienberger","year":"2012","journal-title":"Neuron"},{"issue":"1","key":"pcbi.1009799.ref043","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-018-21640-2","article-title":"FISSA: A neuropil decontamination toolbox for calcium imaging signals","volume":"8","author":"SW Keemink","year":"2018","journal-title":"Scientific reports"},{"issue":"21","key":"pcbi.1009799.ref044","doi-asserted-by":"crossref","first-page":"5195","DOI":"10.1523\/JNEUROSCI.5319-04.2005","article-title":"Synergy, redundancy, and independence in population codes, revisited","volume":"25","author":"PE Latham","year":"2005","journal-title":"Journal of Neuroscience"},{"key":"pcbi.1009799.ref045","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.csda.2016.02.014","article-title":"A fast and objective multidimensional kernel density estimation method: fastKDE","volume":"101","author":"TA O\u2019Brien","year":"2016","journal-title":"Computational Statistics & Data Analysis"},{"issue":"7","key":"pcbi.1009799.ref046","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1038\/nn.2842","article-title":"Measuring and interpreting neuronal correlations","volume":"14","author":"MR Cohen","year":"2011","journal-title":"Nature neuroscience"},{"issue":"7345","key":"pcbi.1009799.ref047","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1038\/nature09880","article-title":"Functional specificity of local synaptic connections in neocortical networks","volume":"473","author":"H Ko","year":"2011","journal-title":"Nature"},{"key":"pcbi.1009799.ref048","unstructured":"Mehler DMA, Kording KP. The lure of misleading causal statements in functional connectivity research. arXiv preprint arXiv:181203363. 2018;."},{"issue":"14","key":"pcbi.1009799.ref049","doi-asserted-by":"crossref","first-page":"3661","DOI":"10.1523\/JNEUROSCI.5106-04.2005","article-title":"Stimulus dependence of neuronal correlation in primary visual cortex of the macaque","volume":"25","author":"A Kohn","year":"2005","journal-title":"Journal of Neuroscience"},{"issue":"10","key":"pcbi.1009799.ref050","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1038\/nn.3807","article-title":"Information-limiting correlations","volume":"17","author":"R Moreno-Bote","year":"2014","journal-title":"Nature neuroscience"},{"key":"pcbi.1009799.ref051","doi-asserted-by":"crossref","first-page":"e53268","DOI":"10.7554\/eLife.53268","article-title":"Cortical State Transitions and Stimulus Response Evolve along Stiff and Sloppy Parameter Dimensions, Respectively","volume":"9","author":"A Ponce-Alvarez","year":"2020","journal-title":"eLife"},{"issue":"1","key":"pcbi.1009799.ref052","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13634-020-00675-6","article-title":"A survey of Monte Carlo methods for parameter estimation","volume":"2020","author":"D Luengo","year":"2020","journal-title":"EURASIP Journal on Advances in Signal Processing"},{"key":"pcbi.1009799.ref053","doi-asserted-by":"crossref","first-page":"066138","DOI":"10.1103\/PhysRevE.69.066138","article-title":"Estimating mutual information","volume":"69","author":"A Kraskov","year":"2004","journal-title":"Phys Rev E"},{"key":"pcbi.1009799.ref054","doi-asserted-by":"crossref","unstructured":"Gao W, Oh S, Viswanath P. Demystifying Fixed k-Nearest Neighbor Information Estimators; 2016.","DOI":"10.1109\/ISIT.2017.8006732"},{"key":"pcbi.1009799.ref055","unstructured":"Belghazi MI, Baratin A, Rajeswar S, Ozair S, Bengio Y, Courville A, et al. MINE: Mutual Information Neural Estimation; 2018."},{"issue":"6","key":"pcbi.1009799.ref056","doi-asserted-by":"crossref","first-page":"68003","DOI":"10.1209\/0295-5075\/88\/68003","article-title":"An information-theoretic approach to statistical dependence: Copula information","volume":"88","author":"RS Calsaverini","year":"2009","journal-title":"EPL (Europhysics Letters)"},{"key":"pcbi.1009799.ref057","unstructured":"Song J, Ermon S. Understanding the Limitations of Variational Mutual Information Estimators; 2019."},{"key":"pcbi.1009799.ref058","doi-asserted-by":"crossref","unstructured":"Holmes C, Nemenman I. Estimation of mutual information for real-valued data with error 737 bars and controlled bias. arXiv. doi. arXiv preprint arXiv:190309280. 2019;738.","DOI":"10.1101\/589929"},{"key":"pcbi.1009799.ref059","unstructured":"Lin X, Sur I, Nastase SA, Divakaran A, Hasson U, Amer MR. Data-efficient mutual information neural estimator. arXiv preprint arXiv:190503319. 2019;."},{"issue":"4","key":"pcbi.1009799.ref060","doi-asserted-by":"crossref","first-page":"490","DOI":"10.3390\/e22040490","article-title":"Limitations to Estimating Mutual Information in Large Neural Populations","volume":"22","author":"J M\u00f6lter","year":"2020","journal-title":"Entropy"},{"key":"pcbi.1009799.ref061","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.conb.2017.08.001","article-title":"Maximum entropy models as a tool for building precise neural controls","volume":"46","author":"C Savin","year":"2017","journal-title":"Current opinion in neurobiology"},{"issue":"03","key":"pcbi.1009799.ref062","doi-asserted-by":"crossref","first-page":"P03011","DOI":"10.1088\/1742-5468\/2013\/03\/P03011","article-title":"The simplest maximum entropy model for collective behavior in a neural network","volume":"2013","author":"G Tka\u010dik","year":"2013","journal-title":"Journal of Statistical Mechanics: Theory and Experiment"},{"issue":"4","key":"pcbi.1009799.ref063","doi-asserted-by":"crossref","DOI":"10.1523\/ENEURO.0160-15.2016","article-title":"A tractable method for describing complex couplings between neurons and population rate","volume":"3","author":"C Gardella","year":"2016","journal-title":"eneuro"},{"issue":"1","key":"pcbi.1009799.ref064","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1162\/NECO_a_00910","article-title":"The Population Tracking Model: A Simple, Scalable Statistical Model for Neural Population Data","volume":"29","author":"C O\u2019Donnell","year":"2017","journal-title":"Neural computation"},{"key":"pcbi.1009799.ref065","doi-asserted-by":"crossref","unstructured":"Hurwitz C, Kudryashova N, Onken A, Hennig MH. Building population models for large-scale neural recordings: opportunities and pitfalls; 2021.","DOI":"10.1016\/j.conb.2021.07.003"},{"key":"pcbi.1009799.ref066","doi-asserted-by":"crossref","unstructured":"Prince LY, Bakhtiari S, Gillon CJ, Richards BA. Parallel inference of hierarchical latent dynamics in two-photon calcium imaging of neuronal populations. bioRxiv. 2021;.","DOI":"10.1101\/2021.03.05.434105"},{"key":"pcbi.1009799.ref067","volume-title":"Advances in Neural Information Processing Systems","author":"A Paszke","year":"2017"},{"key":"pcbi.1009799.ref068","first-page":"7576","article-title":"GPyTorch: Blackbox matrix-matrix Gaussian process inference with GPU acceleration","author":"J Gardner","year":"2018","journal-title":"Advances in Neural Information Processing Systems"},{"key":"pcbi.1009799.ref069","unstructured":"Kleinegesse S, Gutmann MU. Bayesian experimental design for implicit models by mutual information neural estimation. In: International Conference on Machine Learning. PMLR; 2020. p. 5316\u20135326."},{"key":"pcbi.1009799.ref070","first-page":"3496","article-title":"Gaussian process based nonlinear latent structure discovery in multivariate spike train data","volume":"30","author":"A Wu","year":"2017","journal-title":"Adv Neur In"},{"key":"pcbi.1009799.ref071","article-title":"Non-Reversible Gaussian Processes for Identifying Latent Dynamical Structure in Neural Data","volume":"33","author":"V Rutten","year":"2020","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"2","key":"pcbi.1009799.ref072","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.jmva.2004.06.003","article-title":"Asymptotic efficiency of the two-stage estimation method for copula-based models","volume":"94","author":"H Joe","year":"2005","journal-title":"Journal of Multivariate Analysis"},{"key":"pcbi.1009799.ref073","first-page":"567","volume-title":"Artificial Intelligence and Statistics","author":"M Titsias","year":"2009"},{"key":"pcbi.1009799.ref074","first-page":"5184","volume-title":"Advances in Neural Information Processing Systems","author":"CA Cheng","year":"2017"},{"key":"pcbi.1009799.ref075","first-page":"168","volume-title":"Artificial Intelligence and Statistics","author":"D Hern\u00e1ndez-Lobato","year":"2016"},{"key":"pcbi.1009799.ref076","first-page":"1648","volume-title":"Advances in Neural Information Processing Systems","author":"J Hensman","year":"2015"},{"key":"pcbi.1009799.ref077","first-page":"14648","volume-title":"Advances in Neural Information Processing Systems","author":"K Wang","year":"2019"},{"key":"pcbi.1009799.ref078","first-page":"351","volume-title":"Artificial Intelligence and Statistics","author":"J Hensman","year":"2015"},{"key":"pcbi.1009799.ref079","unstructured":"Wilson A, Nickisch H. Kernel interpolation for scalable structured Gaussian processes (KISS-GP). In: International Conference on Machine Learning; 2015. p. 1775\u20131784."},{"key":"pcbi.1009799.ref080","volume-title":"Gaussian processes for machine learning","author":"CK Williams","year":"2006"},{"issue":"Mar","key":"pcbi.1009799.ref081","first-page":"867","article-title":"A widely applicable Bayesian information criterion","volume":"14","author":"S Watanabe","year":"2013","journal-title":"Journal of Machine Learning Research"},{"issue":"6","key":"pcbi.1009799.ref082","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1007\/s11222-013-9416-2","article-title":"Understanding predictive information criteria for Bayesian models","volume":"24","author":"A Gelman","year":"2014","journal-title":"Statistics and computing"},{"issue":"1-4","key":"pcbi.1009799.ref083","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1023\/A:1016725902970","article-title":"Probability density decomposition for conditionally dependent random variables modeled by vines","volume":"32","author":"T Bedford","year":"2001","journal-title":"Annals of Mathematics and Artificial intelligence"},{"key":"pcbi.1009799.ref084","first-page":"1031","article-title":"Vines: A new graphical model for dependent random variables","author":"T Bedford","year":"2002","journal-title":"Annals of Statistics"},{"key":"pcbi.1009799.ref085","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.jmva.2012.02.001","article-title":"Beyond simplified pair-copula constructions","volume":"110","author":"EF Acar","year":"2012","journal-title":"Journal of Multivariate Analysis"},{"issue":"5","key":"pcbi.1009799.ref086","doi-asserted-by":"crossref","first-page":"1296","DOI":"10.1016\/j.jmva.2009.12.001","article-title":"On the simplified pair-copula construction\u2014simply useful or too simplistic?","volume":"101","author":"IH Haff","year":"2010","journal-title":"Journal of Multivariate Analysis"}],"updated-by":[{"DOI":"10.1371\/journal.pcbi.1009799","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T00:00:00Z","timestamp":1644364800000}}],"container-title":["PLOS Computational Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1009799","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T13:45:55Z","timestamp":1644414355000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1009799"}},"subtitle":[],"editor":[{"given":"Robin A A","family":"Ince","sequence":"first","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,1,28]]},"references-count":86,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1,28]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pcbi.1009799","relation":{},"ISSN":["1553-7358"],"issn-type":[{"value":"1553-7358","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,28]]}}}