{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:18:10Z","timestamp":1760149090628,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T00:00:00Z","timestamp":1688601600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Engineering and Physical Sciences Research Council","award":["EP\/S005692\/1","MR\/N013166\/1"],"award-info":[{"award-number":["EP\/S005692\/1","MR\/N013166\/1"]}]},{"name":"Medical Research Council","award":["EP\/S005692\/1","MR\/N013166\/1"],"award-info":[{"award-number":["EP\/S005692\/1","MR\/N013166\/1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Coordinated activity in neural populations is crucial for information processing. Shedding light on the multivariate dependencies that shape multineuronal responses is important to understand neural codes. However, existing approaches based on pairwise linear correlations are inadequate at capturing complicated interaction patterns and miss features that shape aspects of the population function. Copula-based approaches address these shortcomings by extracting the dependence structures in the joint probability distribution of population responses. In this study, we aimed to dissect neural dependencies with a C-Vine copula approach coupled with normalizing flows for estimating copula densities. While this approach allows for more flexibility compared to fitting parametric copulas, drawing insights on the significance of these dependencies from large sets of copula densities is challenging. To alleviate this challenge, we used a weighted non-negative matrix factorization procedure to leverage shared latent features in neural population dependencies. We validated the method on simulated data and applied it on copulas we extracted from recordings of neurons in the mouse visual cortex as well as in the macaque motor cortex. Our findings reveal that neural dependencies occupy low-dimensional subspaces, but distinct modules are synergistically combined to give rise to diverse interaction patterns that may serve the population function.<\/jats:p>","DOI":"10.3390\/e25071026","type":"journal-article","created":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T00:34:30Z","timestamp":1688603670000},"page":"1026","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Discovering Low-Dimensional Descriptions of Multineuronal Dependencies"],"prefix":"10.3390","volume":"25","author":[{"given":"Lazaros","family":"Mitskopoulos","sequence":"first","affiliation":[{"name":"School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7387-5535","authenticated-orcid":false,"given":"Arno","family":"Onken","sequence":"additional","affiliation":[{"name":"School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.conb.2019.02.002","article-title":"Towards the neural population doctrine","volume":"55","author":"Saxena","year":"2019","journal-title":"Curr. Opin. Neurobiol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1146\/annurev-neuro-092619-094115","article-title":"Computation through neural population dynamics","volume":"43","author":"Vyas","year":"2020","journal-title":"Annu. Rev. Neurosci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1038\/s41593-021-00980-9","article-title":"Large-scale neural recordings call for new insights to link brain and behavior","volume":"25","author":"Urai","year":"2022","journal-title":"Nat. Neurosci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1818","DOI":"10.1038\/nprot.2012.106","article-title":"LOTOS-based two-photon calcium imaging of dendritic spines in vivo","volume":"7","author":"Chen","year":"2012","journal-title":"Nat. Protoc."},{"key":"ref_5","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":"Jun","year":"2017","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1146\/annurev.neuro.29.051605.113024","article-title":"Complete functional characterization of sensory neurons by system identification","volume":"29","author":"Wu","year":"2006","journal-title":"Annu. Rev. Neurosci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/j.pneurobio.2011.08.002","article-title":"The neuronal encoding of information in the brain","volume":"95","author":"Rolls","year":"2011","journal-title":"Prog. Neurobiol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1146\/annurev-statistics-041715-033733","article-title":"Computational neuroscience: Mathematical and statistical perspectives","volume":"5","author":"Kass","year":"2018","journal-title":"Annu. Rev. Stat. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.conb.2021.07.003","article-title":"Building population models for large-scale neural recordings: Opportunities and pitfalls","volume":"70","author":"Hurwitz","year":"2021","journal-title":"Curr. Opin. Neurobiol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1038\/nature09178","article-title":"Sparse coding and high-order correlations in fine-scale cortical networks","volume":"466","author":"Ohiorhenuan","year":"2010","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"17514","DOI":"10.1523\/JNEUROSCI.3127-11.2011","article-title":"Higher-order interactions characterized in cortical activity","volume":"31","author":"Yu","year":"2011","journal-title":"J. Neurosci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Shimazaki, H., Amari, S.I., Brown, E.N., and Gr\u00fcn, S. (2012). State-space analysis of time-varying higher-order spike correlation for multiple neural spike train data. PLoS Comput. Biol., 8.","DOI":"10.1371\/journal.pcbi.1002385"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1016\/j.physa.2016.12.002","article-title":"Higher-order correlations in common input shapes the output spiking activity of a neural population","volume":"471","author":"Montangie","year":"2017","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1038\/370140a0","article-title":"Correlated neuronal discharge rate and its implications for psychophysical performance","volume":"370","author":"Zohary","year":"1994","journal-title":"Nature"},{"key":"ref_15","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":"Brown","year":"2004","journal-title":"Nat. Neurosci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1038\/nn.3807","article-title":"Information-limiting correlations","volume":"17","author":"Beck","year":"2014","journal-title":"Nat. Neurosci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1146\/annurev-neuro-070815-013851","article-title":"Correlations and neuronal population information","volume":"39","author":"Kohn","year":"2016","journal-title":"Annu. Rev. Neurosci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1038\/s41583-022-00606-4","article-title":"The structures and functions of correlations in neural population codes","volume":"23","author":"Panzeri","year":"2022","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Onken, A., Gr\u00fcnew\u00e4lder, S., Munk, M.H., and Obermayer, K. (2009). Analyzing short-term noise dependencies of spike-counts in macaque prefrontal cortex using copulas and the flashlight transformation. PLoS Comput. Biol., 5.","DOI":"10.1371\/journal.pcbi.1000577"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Kudryashova, N., Amvrosiadis, T., Dupuy, N., Rochefort, N., and Onken, A. (2022). Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships. PLoS Comput. Biol., 18.","DOI":"10.1371\/journal.pcbi.1009799"},{"key":"ref_21","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":"Pillow","year":"2008","journal-title":"Nature"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"660","DOI":"10.1162\/neco.2006.18.3.660","article-title":"The costs of ignoring high-order correlations in populations of model neurons","volume":"18","author":"Michel","year":"2006","journal-title":"Neural Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Jaworski, P., Durante, F., and H\u00e4rdle, W.K. (2012). Copulae in Mathematical and Quantitative Finance: Proceedings of the Workshop Held in Cracow, 10\u201311 July 2012, Springer.","DOI":"10.1007\/978-3-642-35407-6"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1162\/089976604322860659","article-title":"The shape of neural dependence","volume":"16","author":"Jenison","year":"2004","journal-title":"Neural Comput."},{"key":"ref_25","first-page":"129","article-title":"Characterizing neural dependencies with copula models","volume":"21","author":"Berkes","year":"2008","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_26","first-page":"85117","article-title":"Modeling short-term noise dependence of spike counts in macaque prefrontal cortex","volume":"21","author":"Onken","year":"2008","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.brainres.2011.08.064","article-title":"Detecting dependencies between spike trains of pairs of neurons through copulas","volume":"1434","author":"Sacerdote","year":"2012","journal-title":"Brain Res."},{"key":"ref_28","first-page":"910122","article-title":"Mixed vine copulas as joint models of spike counts and local field potentials","volume":"29","author":"Onken","year":"2016","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1515\/demo-2017-0008","article-title":"Inference for copula modeling of discrete data: A cautionary tale and some facts","volume":"5","author":"Faugeras","year":"2017","journal-title":"Depend. Model."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"475","DOI":"10.2143\/AST.37.2.2024077","article-title":"A primer on copulas for count data","volume":"37","author":"Genest","year":"2007","journal-title":"ASTIN Bull. J. IAA"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/j.spl.2018.02.040","article-title":"A generic approach to nonparametric function estimation with mixed data","volume":"137","author":"Nagler","year":"2018","journal-title":"Stat. Probab. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.insmatheco.2007.02.001","article-title":"Pair-copula constructions of multiple dependence","volume":"44","author":"Aas","year":"2009","journal-title":"Insur. Math. Econ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1111\/j.1541-0420.2008.01058.x","article-title":"Joint regression analysis of correlated data using Gaussian copulas","volume":"65","author":"Song","year":"2009","journal-title":"Biometrics"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1002\/sim.4087","article-title":"Copula-based regression models for a bivariate mixed discrete and continuous outcome","volume":"30","author":"Wu","year":"2011","journal-title":"Stat. Med."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1080\/01621459.2011.644501","article-title":"Estimation of copula models with discrete margins via Bayesian data augmentation","volume":"107","author":"Smith","year":"2012","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1080\/01621459.2012.682850","article-title":"Pair copula constructions for multivariate discrete data","volume":"107","author":"Panagiotelis","year":"2012","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s00181-015-0913-3","article-title":"Mixed data kernel copulas","volume":"48","author":"Racine","year":"2015","journal-title":"Empir. Econ."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1848","DOI":"10.3150\/15-BEJ798","article-title":"Probit transformation for nonparametric kernel estimation of the copula density","volume":"23","author":"Geenens","year":"2017","journal-title":"Bernoulli"},{"key":"ref_39","unstructured":"Schallhorn, N., Kraus, D., Nagler, T., and Czado, C. (2017). D-vine quantile regression with discrete variables. arXiv."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1515\/demo-2017-0007","article-title":"Nonparametric estimation of simplified vine copula models: Comparison of methods","volume":"5","author":"Nagler","year":"2017","journal-title":"Depend. Model."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.3389\/fnins.2022.910122","article-title":"Mixed vine copula flows for flexible modelling of neural dependencies","volume":"16","author":"Mitskopoulos","year":"2022","journal-title":"Front. Neurosci."},{"key":"ref_42","unstructured":"Durkan, C., Bekasov, A., Murray, I., and Papamakarios, G. (2019, January 8\u201314). Neural spline flows. Proceedings of the 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, BC, Canada."},{"key":"ref_43","unstructured":"Rezende, D., and Mohamed, S. (2015, January 6\u201311). Variational inference with normalizing flows. Proceedings of the International Conference on Machine Learning, Lille, France."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1038\/44565","article-title":"Learning the parts of objects by non-negative matrix factorization","volume":"401","author":"Lee","year":"1999","journal-title":"Nature"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1016\/j.neuron.2018.01.004","article-title":"Motor cortex embeds muscle-like commands in an untangled population response","volume":"97","author":"Russo","year":"2018","journal-title":"Neuron"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2447","DOI":"10.1016\/S0167-8655(03)00089-8","article-title":"Introducing a weighted non-negative matrix factorization for image classification","volume":"24","author":"Guillamet","year":"2003","journal-title":"Pattern Recognit. Lett."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Zhou, Q., Feng, Z., and Benetos, E. (2019). Adaptive noise reduction for sound event detection using subband-weighted NMF. Sensors, 19.","DOI":"10.3390\/s19143206"},{"key":"ref_48","first-page":"229","article-title":"Fonctions de repartition an dimensions et leurs marges","volume":"8","author":"Sklar","year":"1959","journal-title":"Publ. Inst. Stat. Univ. Paris"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1214\/aos\/1031689016","article-title":"Vines\u2014A new graphical model for dependent random variables","volume":"30","author":"Bedford","year":"2002","journal-title":"Ann. Stat."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Czado, C. (2019). Analyzing Dependent Data with Vine Copulas, Springer. Lecture Notes in Statistics.","DOI":"10.1007\/978-3-030-13785-4"},{"key":"ref_51","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":"Haff","year":"2010","journal-title":"J. Multivar. Anal."},{"key":"ref_52","first-page":"281","article-title":"Random search for hyper-parameter optimization","volume":"13","author":"Bergstra","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1093\/mnras\/225.1.155","article-title":"A multidimensional version of the Kolmogorov\u2014Smirnov test","volume":"225","author":"Fasano","year":"1987","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1214\/08-AOAS227","article-title":"Bi-cross-validation of the SVD and the nonnegative matrix factorization","volume":"3","author":"Owen","year":"2009","journal-title":"Ann. Appl. Stat."},{"key":"ref_55","unstructured":"Nelsen, R.B. (2007). An Introduction to Copulas, Springer Science & Business Media."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1866","DOI":"10.1016\/j.cub.2020.03.018","article-title":"Reward association enhances stimulus-specific representations in primary visual cortex","volume":"30","author":"Henschke","year":"2020","journal-title":"Curr. Biol."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1038\/nature11129","article-title":"Neural population dynamics during reaching","volume":"487","author":"Churchland","year":"2012","journal-title":"Nature"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/7\/1026\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:07:00Z","timestamp":1760126820000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/7\/1026"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,6]]},"references-count":57,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["e25071026"],"URL":"https:\/\/doi.org\/10.3390\/e25071026","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,7,6]]}}}