{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:04:39Z","timestamp":1760058279810,"version":"build-2065373602"},"reference-count":84,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,21]],"date-time":"2025-03-21T00:00:00Z","timestamp":1742515200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Abdullah University of Science and Technology (KAUST)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Modeling the brain dependence network is central to understanding underlying neural mechanisms such as perception, action, and memory. In this study, we present a broad range of statistical methods for analyzing dependence in a brain network. Leveraging a combination of classical and cutting-edge approaches, we analyze multivariate hippocampal local field potential (LFP) time series data concentrating on the encoding of nonspatial olfactory information in rats. We present the strengths and limitations of each method in capturing neural dynamics and connectivity. Our analysis begins with exploratory techniques, including correlation, partial correlation, spectral matrices, and coherence, to establish foundational connectivity insights. We then investigate advanced methods such as Granger causality (GC), robust canonical coherence analysis, spectral transfer entropy (STE), and wavelet coherence to capture dynamic and nonlinear interactions. Additionally, we investigate the utility of topological data analysis (TDA) to extract multi-scale topological features and explore deep learning-based canonical correlation frameworks for connectivity modeling. This comprehensive approach offers an introduction to the state-of-the-art techniques for the analysis of dependence networks, emphasizing the unique strengths of various methodologies, addressing computational challenges, and paving the way for future research.<\/jats:p>","DOI":"10.3390\/e27040328","type":"journal-article","created":{"date-parts":[[2025,3,21]],"date-time":"2025-03-21T11:06:48Z","timestamp":1742555208000},"page":"328","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Methods for Brain Connectivity Analysis with Applications to Rat Local Field Potential Recordings"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8886-0587","authenticated-orcid":false,"given":"Anass B.","family":"El-Yaagoubi","sequence":"first","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]},{"given":"Sipan","family":"Aslan","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]},{"given":"Farah","family":"Gomawi","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]},{"given":"Paolo V.","family":"Redondo","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1659-7318","authenticated-orcid":false,"given":"Sarbojit","family":"Roy","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]},{"given":"Malik S.","family":"Sultan","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2375-8634","authenticated-orcid":false,"given":"Mara S.","family":"Talento","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]},{"given":"Francine T.","family":"Tarrazona","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"},{"name":"Department of Mathematics, Ateneo de Manila University, Quezon City 1108, Philippines"}]},{"given":"Haibo","family":"Wu","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0358-9645","authenticated-orcid":false,"given":"Keiland W.","family":"Cooper","sequence":"additional","affiliation":[{"name":"Department of Neurobiology and Behavior, University of California, Irvine, CA 92697, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6793-6984","authenticated-orcid":false,"given":"Norbert J.","family":"Fortin","sequence":"additional","affiliation":[{"name":"Department of Neurobiology and Behavior, University of California, Irvine, CA 92697, USA"}]},{"given":"Hernando","family":"Ombao","sequence":"additional","affiliation":[{"name":"Statistics Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"102414","DOI":"10.1016\/j.arr.2024.102414","article-title":"Unraveling the complexity of human brain: Structure, function in healthy and disease states","volume":"100","author":"Sultana","year":"2024","journal-title":"Ageing Res. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1177\/17456916231191744","article-title":"A Critical Perspective on Neural Mechanisms in Cognitive Neuroscience: Towards Unification","volume":"19","year":"2024","journal-title":"Perspect. Psychol. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Yen, C., Lin, C.L., and Chiang, M.C. (2023). Exploring the Frontiers of Neuroimaging: A Review of Recent Advances in Understanding Brain Functioning and Disorders. Life, 13.","DOI":"10.3390\/life13071472"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1110","DOI":"10.1016\/j.compbiomed.2011.06.020","article-title":"Review of advanced techniques for the estimation of brain connectivity measured with EEG\/MEG","volume":"41","author":"Sakkalis","year":"2011","journal-title":"Comput. Biol. Med."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1146\/annurev-statistics-040522-020722","article-title":"Statistical Brain Network Analysis","volume":"11","author":"Simpson","year":"2024","journal-title":"Annu. Rev. Stat. Its Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1089\/brain.2011.0008","article-title":"Functional and effective connectivity: A review","volume":"1","author":"Friston","year":"2011","journal-title":"Brain Connect."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"144","DOI":"10.3389\/fnsys.2010.00144","article-title":"Connectivity Analysis is Essential to Understand Neurological Disorders","volume":"4","author":"Rowe","year":"2010","journal-title":"Front. Syst. Neurosci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1038\/nrn3801","article-title":"Modern Network Science of Neurological Disorders","volume":"15","author":"Stam","year":"2014","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1038\/nrn3901","article-title":"The Connectomics of Brain Disorders","volume":"16","author":"Fornito","year":"2015","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1037\/0735-7044.100.2.147","article-title":"The Hippocampal Memory Indexing Theory","volume":"100","author":"Teyler","year":"1986","journal-title":"Behav. Neurosci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1038\/nn834","article-title":"Critical Role of the Hippocampus in Memory for Sequences of Events","volume":"5","author":"Fortin","year":"2002","journal-title":"Nat. Neurosci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1785","DOI":"10.1152\/jn.00005.2017","article-title":"The role of the hippocampus in navigation is memory","volume":"117","author":"Eichenbaum","year":"2017","journal-title":"J. Neurophysiol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"10365","DOI":"10.1073\/pnas.1301225110","article-title":"Similarity in form and function of the hippocampus in rodents, monkeys, and humans","volume":"110","author":"Clark","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1547","DOI":"10.1523\/JNEUROSCI.2874-15.2016","article-title":"Nonspatial sequence coding in CA1 neurons","volume":"36","author":"Allen","year":"2016","journal-title":"J. Neurosci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1007\/s12021-024-09686-2","article-title":"Mesoscale Brain Mapping: Bridging Scales and Modalities in Neuroimaging\u2014A Symposium Review","volume":"22","author":"Marchant","year":"2024","journal-title":"Neuroinformatics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"412512","DOI":"10.1155\/2012\/412512","article-title":"Brain Connectivity Analysis: A Short Survey","volume":"2012","author":"Lang","year":"2012","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Shumway, R.H., Stoffer, D.S., and Stoffer, D.S. (2000). Time Series Analysis and Its Applications, Springer.","DOI":"10.1007\/978-1-4757-3261-0"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40810-015-0015-7","article-title":"Coherence a measure of the brain networks: Past and present","volume":"2","author":"Bowyer","year":"2016","journal-title":"Neuropsychiatr. Electrophysiol."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"You, S.D. (2021). Classification of Relaxation and Concentration Mental States with EEG. Information, 12.","DOI":"10.3390\/info12050187"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Newson, J.J., and Thiagarajan, T.C. (2019). EEG Frequency Bands in Psychiatric Disorders: A Review of Resting State Studies. Front. Hum. Neurosci., 12.","DOI":"10.3389\/fnhum.2018.00521"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Brillinger, D.R. (2001). Time Series: Data Analysis and Theory, SIAM.","DOI":"10.1137\/1.9780898719246"},{"key":"ref_22","unstructured":"Talento, M.S.D., Roy, S., and Ombao, H.C. (2024). KenCoh: A Ranked-Based Canonical Coherence. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1080\/00401706.1999.10485670","article-title":"A fast algorithm for the minimum covariance determinant estimator","volume":"41","author":"Rousseeuw","year":"1999","journal-title":"Technometrics"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1080\/01621459.1984.10477105","article-title":"Least median of squares regression","volume":"79","author":"Rousseeuw","year":"1984","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1093\/biomet\/30.1-2.81","article-title":"A new measure of rank correlation","volume":"30","author":"Kendall","year":"1938","journal-title":"Biometrika"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"587","DOI":"10.2307\/3315967","article-title":"Kendall\u2019s tau for serial dependence","volume":"28","author":"Ferguson","year":"2000","journal-title":"Can. J. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jmva.2001.2017","article-title":"The meta-elliptical distributions with given marginals","volume":"82","author":"Fang","year":"2002","journal-title":"J. Multivar. Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.tcs.2004.08.005","article-title":"The complexity of computing the MCD-estimator","volume":"326","author":"Bernholt","year":"2004","journal-title":"Theor. Comput. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1152\/jn.00166.2010","article-title":"A beta oscillation network in the rat olfactory system during a 2-alternative choice odor discrimination task","volume":"104","author":"Kay","year":"2010","journal-title":"J. Neurophysiol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1073\/pnas.97.4.1423","article-title":"The multivariate L 1-median and associated data depth","volume":"97","author":"Vardi","year":"2000","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1111\/j.2517-6161.1995.tb02031.x","article-title":"Controlling the false discovery rate: A practical and powerful approach to multiple testing","volume":"57","author":"Benjamini","year":"1995","journal-title":"J. R. Stat. Soc. Ser. B Methodol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1613\/jair.953","article-title":"SMOTE: Synthetic minority over-sampling technique","volume":"16","author":"Chawla","year":"2002","journal-title":"J. Artif. Intell. Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1080\/16843703.2014.11673322","article-title":"Causal structure learning and inference: A selective review","volume":"11","author":"Kalisch","year":"2014","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1109\/JSTSP.2016.2600023","article-title":"Modeling effective connectivity in high-dimensional cortical source signals","volume":"10","author":"Wang","year":"2016","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Ting, C.M., Seghouane, A.K., and Salleh, S.H. (2016, January 26\u201329). Estimation of high-dimensional connectivity in fmri data via subspace autoregressive models. Proceedings of the 2016 IEEE Statistical Signal Processing Workshop (SSP), Palma de Mallorca, Spain.","DOI":"10.1109\/SSP.2016.7551799"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"116453","DOI":"10.1016\/j.neuroimage.2019.116453","article-title":"Dynamic effective connectivity","volume":"207","author":"Zarghami","year":"2020","journal-title":"Neuroimage"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"5638","DOI":"10.1109\/TSP.2021.3114997","article-title":"Dimension reduction of polynomial regression models for the estimation of Granger causality in high-dimensional time series","volume":"69","author":"Siggiridou","year":"2021","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1146\/annurev-statistics-040120-010930","article-title":"Granger causality: A review and recent advances","volume":"9","author":"Shojaie","year":"2022","journal-title":"Annu. Rev. Stat. Its Appl."},{"key":"ref_39","first-page":"41","article-title":"High-dimensional causal discovery under non-Gaussianity","volume":"107","author":"Wang","year":"2020","journal-title":"Biometrika"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1214\/22-AOAS1702","article-title":"A high-dimensional approach to measure connectivity in the financial sector","volume":"18","author":"Basu","year":"2024","journal-title":"Ann. Appl. Stat."},{"key":"ref_41","first-page":"331","article-title":"The canonical analysis of stationary time series","volume":"2","author":"Brillinger","year":"1969","journal-title":"Multivar. Anal."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1111\/rssb.12076","article-title":"Dynamic functional principal components","volume":"77","author":"Hallin","year":"2015","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1080\/01621459.1999.10473886","article-title":"Detecting common signals in multiple time series using the spectral envelope","volume":"94","author":"Stoffer","year":"1999","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_44","first-page":"424","article-title":"Investigating causal relations by econometric models and cross-spectral methods","volume":"37","author":"Granger","year":"1969","journal-title":"Econom. J. Econom. Soc."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1002\/hipo.20578","article-title":"Cortical efferents of the perirhinal, postrhinal, and entorhinal cortices of the rat","volume":"19","author":"Agster","year":"2009","journal-title":"Hippocampus"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Zhou, W., Qu, A., Cooper, K.W., Fortin, N., and Shahbaba, B. (2024). A model-agnostic graph neural network for integrating local and global information. J. Am. Stat. Assoc., 1\u201314.","DOI":"10.1080\/01621459.2024.2404668"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring information transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","unstructured":"Cover, T.M., and Thomas, J.A. (2012). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/j.neuroimage.2009.12.050","article-title":"The effect of filtering on Granger causality based multivariate causality measures","volume":"50","author":"Florin","year":"2010","journal-title":"Neuroimage"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.jneumeth.2011.08.010","article-title":"Behaviour of Granger causality under filtering: Theoretical invariance and practical application","volume":"201","author":"Barnett","year":"2011","journal-title":"J. Neurosci. Methods"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3293","DOI":"10.1523\/JNEUROSCI.4399-14.2015","article-title":"Granger causality analysis in neuroscience and neuroimaging","volume":"35","author":"Seth","year":"2015","journal-title":"J. Neurosci."},{"key":"ref_52","unstructured":"Redondo, P.V., Huser, R., and Ombao, H. (2023). Measuring information transfer between nodes in a brain network through spectral transfer entropy. arXiv."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1265","DOI":"10.1016\/j.neuroscience.2009.05.051","article-title":"Odorant modulation of neuronal activity and local field potential in sensory-deprived olfactory bulb","volume":"162","author":"Aylwin","year":"2009","journal-title":"Neuroscience"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.1152\/jn.00829.2013","article-title":"Odor-and state-dependent olfactory tubercle local field potential dynamics in awake rats","volume":"111","author":"Carlson","year":"2014","journal-title":"J. Neurophysiol."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"908","DOI":"10.1152\/jn.00261.2013","article-title":"Anesthetic regimes modulate the temporal dynamics of local field potential in the mouse olfactory bulb","volume":"111","author":"Chery","year":"2014","journal-title":"J. Neurophysiol."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Daubechies, I. (1992). Ten Lectures on Wavelets, SIAM.","DOI":"10.1137\/1.9781611970104"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1190\/1.1441328","article-title":"Wave propagation and sampling theory; Part I, Complex signal and scattering in multilayered media","volume":"47","author":"Morlet","year":"1982","journal-title":"Geophysics"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1111\/1467-9868.00231","article-title":"Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum","volume":"62","author":"Nason","year":"2000","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"5240","DOI":"10.1109\/TSP.2014.2343937","article-title":"Estimating Time-Evolving Partial Coherence Between Signals via Multivariate Locally Stationary Wavelet Processes","volume":"62","author":"Park","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_60","unstructured":"Wu, H., Knight, M., and Ombao, H. (2023). Multi-scale wavelet coherence with its applications. arXiv."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of Scaling in Random Networks","volume":"286","author":"Barabasi","year":"1999","journal-title":"Science"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"111","DOI":"10.31887\/DCNS.2018.20.2\/osporns","article-title":"Graph theory methods: Applications in brain networks","volume":"20","author":"Sporns","year":"2018","journal-title":"Dialogues Clin. Neurosci."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Miraglia, F., Vecchio, F., Pappalettera, C., Nucci, L., Cotelli, M., Judica, E., Ferreri, F., and Rossini, P.M. (2022). Brain Connectivity and Graph Theory Analysis in Alzheimer\u2019s and Parkinson\u2019s Disease: The Contribution of Electrophysiological Techniques. Brain Sci., 12.","DOI":"10.3390\/brainsci12030402"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"bhad506","DOI":"10.1093\/cercor\/bhad506","article-title":"Measures of resting-state brain network segregation and integration vary in relation to data quantity: Implications for within and between subject comparisons of functional brain network organization","volume":"34","author":"Han","year":"2024","journal-title":"Cereb. Cortex"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"9164","DOI":"10.1038\/s41467-024-53299-x","article-title":"Measuring the dynamic balance of integration and segregation underlying consciousness, anesthesia, and sleep in humans","volume":"15","author":"Jang","year":"2024","journal-title":"Nat. Commun."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Langer, N., Pedroni, A., and J\u00e4ncke, L. (2013). The Problem of Thresholding in Small-World Network Analysis. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0053199"},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Bordier, C., Nicolini, C., and Bifone, A. (2017). Graph Analysis and Modularity of Brain Functional Connectivity Networks: Searching for the Optimal Threshold. Front. Neurosci., 11.","DOI":"10.3389\/fnins.2017.00441"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1007\/s00429-021-02435-0","article-title":"A hands-on tutorial on network and topological neuroscience","volume":"227","author":"Centeno","year":"2022","journal-title":"Brain Struct. Funct."},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"El-Yaagoubi, A.B., Chung, M.K., and Ombao, H. (2023). Topological Data Analysis for Multivariate Time Series Data. Entropy, 25.","DOI":"10.3390\/e25111509"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"2267","DOI":"10.1109\/TMI.2012.2219590","article-title":"Persistent Brain Network Homology From the Perspective of Dendrogram","volume":"31","author":"Lee","year":"2012","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"109324","DOI":"10.1016\/j.jneumeth.2021.109324","article-title":"Topological signal processing and inference of event-related potential response","volume":"363","author":"Wang","year":"2021","journal-title":"J. Neurosci. Methods"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1399","DOI":"10.1038\/s41467-018-03664-4","article-title":"Towards a new approach to reveal dynamical organization of the brain using topological data analysis","volume":"9","author":"Saggar","year":"2018","journal-title":"Nat. Commun."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1090\/S0273-0979-07-01191-3","article-title":"Barcodes: The persistent topology of data","volume":"45","author":"Ghrist","year":"2008","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_75","first-page":"77","article-title":"Statistical Topological Data Analysis Using Persistence Landscapes","volume":"16","author":"Bubenik","year":"2015","journal-title":"J. Mach. Learn. Res."},{"key":"ref_76","first-page":"1","article-title":"Persistence images: A stable vector representation of persistent homology","volume":"18","author":"Adams","year":"2017","journal-title":"J. Mach. Learn. Res."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"L032007","DOI":"10.1103\/PhysRevResearch.6.L032007","article-title":"Disentangling high-order effects in the transfer entropy","volume":"6","author":"Stramaglia","year":"2024","journal-title":"Phys. Rev. Res."},{"key":"ref_78","doi-asserted-by":"crossref","unstructured":"Herzog, R., Rosas, F.E., Whelan, R., Fittipaldi, S., Santamaria-Garcia, H., Cruzat, J., Birba, A., Moguilner, S., Tagliazucchi, E., and Prado, P. (2022). Genuine high-order interactions in brain networks and neurodegeneration. Neurobiol. Dis., 175.","DOI":"10.1016\/j.nbd.2022.105918"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"10244","DOI":"10.1038\/s41467-024-54472-y","article-title":"Higher-order connectomics of human brain function reveals local topological signatures of task decoding, individual identification, and behavior","volume":"15","author":"Santoro","year":"2024","journal-title":"Nat. Commun."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"El-Yaagoubi, A.B., Chung, M.K., and Ombao, H. (2024, January 10). Topological Analysis of Seizure-Induced Changes in Brain Hierarchy Through Effective Connectivity. Proceedings of the Topology- and Graph-Informed Imaging Informatics: First International Workshop, TGI3 2024, Held in Conjunction with MICCAI 2024, Marrakesh, Morocco.","DOI":"10.1007\/978-3-031-73967-5_13"},{"key":"ref_81","first-page":"4267","article-title":"Neural granger causality","volume":"44","author":"Tank","year":"2021","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_82","unstructured":"Marcinkevi\u010ds, R., and Vogt, J.E. (2021). Interpretable models for granger causality using self-explaining neural networks. arXiv."},{"key":"ref_83","doi-asserted-by":"crossref","unstructured":"Cheng, Y., Yang, R., Xiao, T., Li, Z., Suo, J., He, K., and Dai, Q. (2023). Cuts: Neural causal discovery from irregular time-series data. arXiv.","DOI":"10.1609\/aaai.v38i10.29034"},{"key":"ref_84","first-page":"11525","article-title":"CUTS+: High-dimensional causal discovery from irregular time-series","volume":"38","author":"Cheng","year":"2024","journal-title":"Proc. AAAI Conf. Artif. Intell."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/328\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:58:05Z","timestamp":1760029085000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/328"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,21]]},"references-count":84,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["e27040328"],"URL":"https:\/\/doi.org\/10.3390\/e27040328","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,3,21]]}}}