[{"day":"01","acknowledgement":"The second author has been partially supported by INdAM through the GNAMPA Research\r\nProject (2017) “Sistemi stocastici singolari: buona posizione e problemi di controllo”. The third\r\nauthor was partly funded by the Austrian Science Fund (FWF) project F 65.","project":[{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"}],"author":[{"full_name":"Flandoli, Franco","last_name":"Flandoli","first_name":"Franco"},{"last_name":"Priola","first_name":"Enrico","full_name":"Priola, Enrico"},{"first_name":"Giovanni A","id":"47491882-F248-11E8-B48F-1D18A9856A87","last_name":"Zanco","full_name":"Zanco, Giovanni A"}],"isi":1,"title":"A mean-field model with discontinuous coefficients for neurons with spatial interaction","publisher":"American Institute of Mathematical Sciences","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","page":"3037-3067","date_created":"2022-03-18T12:33:34Z","publication":"Discrete and Continuous Dynamical Systems","language":[{"iso":"eng"}],"department":[{"_id":"JaMa"}],"type":"journal_article","article_type":"original","article_processing_charge":"No","external_id":{"arxiv":["1708.04156"],"isi":["000459954800003"]},"quality_controlled":"1","volume":39,"keyword":["Applied Mathematics","Discrete Mathematics and Combinatorics","Analysis"],"year":"2019","date_updated":"2023-09-08T11:34:45Z","arxiv":1,"abstract":[{"lang":"eng","text":"Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles."}],"issue":"6","_id":"10878","oa":1,"publication_identifier":{"issn":["1553-5231"]},"status":"public","month":"06","publication_status":"published","doi":"10.3934/dcds.2019126","date_published":"2019-06-01T00:00:00Z","oa_version":"Preprint","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1708.04156"}],"scopus_import":"1","citation":{"apa":"Flandoli, F., Priola, E., &#38; Zanco, G. A. (2019). A mean-field model with discontinuous coefficients for neurons with spatial interaction. <i>Discrete and Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences. <a href=\"https://doi.org/10.3934/dcds.2019126\">https://doi.org/10.3934/dcds.2019126</a>","ista":"Flandoli F, Priola E, Zanco GA. 2019. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical Systems. 39(6), 3037–3067.","mla":"Flandoli, Franco, et al. “A Mean-Field Model with Discontinuous Coefficients for Neurons with Spatial Interaction.” <i>Discrete and Continuous Dynamical Systems</i>, vol. 39, no. 6, American Institute of Mathematical Sciences, 2019, pp. 3037–67, doi:<a href=\"https://doi.org/10.3934/dcds.2019126\">10.3934/dcds.2019126</a>.","chicago":"Flandoli, Franco, Enrico Priola, and Giovanni A Zanco. “A Mean-Field Model with Discontinuous Coefficients for Neurons with Spatial Interaction.” <i>Discrete and Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences, 2019. <a href=\"https://doi.org/10.3934/dcds.2019126\">https://doi.org/10.3934/dcds.2019126</a>.","short":"F. Flandoli, E. Priola, G.A. Zanco, Discrete and Continuous Dynamical Systems 39 (2019) 3037–3067.","ama":"Flandoli F, Priola E, Zanco GA. A mean-field model with discontinuous coefficients for neurons with spatial interaction. <i>Discrete and Continuous Dynamical Systems</i>. 2019;39(6):3037-3067. doi:<a href=\"https://doi.org/10.3934/dcds.2019126\">10.3934/dcds.2019126</a>","ieee":"F. Flandoli, E. Priola, and G. A. Zanco, “A mean-field model with discontinuous coefficients for neurons with spatial interaction,” <i>Discrete and Continuous Dynamical Systems</i>, vol. 39, no. 6. American Institute of Mathematical Sciences, pp. 3037–3067, 2019."},"intvolume":"        39"},{"publication_status":"published","doi":"10.1007/s10959-016-0724-2","oa_version":"Published Version","date_published":"2018-06-01T00:00:00Z","scopus_import":1,"file_date_updated":"2020-07-14T12:44:39Z","ddc":["519"],"citation":{"short":"F. Flandoli, F. Russo, G.A. Zanco, Journal of Theoretical Probability 31 (2018) 789–826.","ama":"Flandoli F, Russo F, Zanco GA. Infinite-dimensional calculus under weak spatial regularity of the processes. <i>Journal of Theoretical Probability</i>. 2018;31(2):789-826. doi:<a href=\"https://doi.org/10.1007/s10959-016-0724-2\">10.1007/s10959-016-0724-2</a>","ieee":"F. Flandoli, F. Russo, and G. A. Zanco, “Infinite-dimensional calculus under weak spatial regularity of the processes,” <i>Journal of Theoretical Probability</i>, vol. 31, no. 2. Springer, pp. 789–826, 2018.","chicago":"Flandoli, Franco, Francesco Russo, and Giovanni A Zanco. “Infinite-Dimensional Calculus under Weak Spatial Regularity of the Processes.” <i>Journal of Theoretical Probability</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s10959-016-0724-2\">https://doi.org/10.1007/s10959-016-0724-2</a>.","apa":"Flandoli, F., Russo, F., &#38; Zanco, G. A. (2018). Infinite-dimensional calculus under weak spatial regularity of the processes. <i>Journal of Theoretical Probability</i>. Springer. <a href=\"https://doi.org/10.1007/s10959-016-0724-2\">https://doi.org/10.1007/s10959-016-0724-2</a>","ista":"Flandoli F, Russo F, Zanco GA. 2018. Infinite-dimensional calculus under weak spatial regularity of the processes. Journal of Theoretical Probability. 31(2), 789–826.","mla":"Flandoli, Franco, et al. “Infinite-Dimensional Calculus under Weak Spatial Regularity of the Processes.” <i>Journal of Theoretical Probability</i>, vol. 31, no. 2, Springer, 2018, pp. 789–826, doi:<a href=\"https://doi.org/10.1007/s10959-016-0724-2\">10.1007/s10959-016-0724-2</a>."},"intvolume":"        31","date_updated":"2021-01-12T06:49:09Z","abstract":[{"lang":"eng","text":"Two generalizations of Itô formula to infinite-dimensional spaces are given.\r\nThe first one, in Hilbert spaces, extends the classical one by taking advantage of\r\ncancellations when they occur in examples and it is applied to the case of a group\r\ngenerator. The second one, based on the previous one and a limit procedure, is an Itô\r\nformula in a special class of Banach spaces having a product structure with the noise\r\nin a Hilbert component; again the key point is the extension due to a cancellation. This\r\nextension to Banach spaces and in particular the specific cancellation are motivated\r\nby path-dependent Itô calculus."}],"issue":"2","year":"2018","_id":"1215","status":"public","oa":1,"month":"06","has_accepted_license":"1","date_created":"2018-12-11T11:50:45Z","page":"789-826","publication":"Journal of Theoretical Probability","type":"journal_article","language":[{"iso":"eng"}],"department":[{"_id":"JaMa"}],"publist_id":"6119","article_processing_charge":"Yes (via OA deal)","pubrep_id":"712","quality_controlled":"1","volume":31,"acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). The second named author benefited partially from the support of the “FMJH Program Gaspard Monge in Optimization and Operations Research” (Project 2014-1607H). He is also grateful for the invitation to the Department of Mathematics of the University of Pisa. The third named author is grateful for the invitation to ENSTA.","day":"01","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"author":[{"last_name":"Flandoli","first_name":"Franco","full_name":"Flandoli, Franco"},{"full_name":"Russo, Francesco","last_name":"Russo","first_name":"Francesco"},{"id":"47491882-F248-11E8-B48F-1D18A9856A87","last_name":"Zanco","first_name":"Giovanni A","full_name":"Zanco, Giovanni A"}],"file":[{"date_updated":"2020-07-14T12:44:39Z","file_name":"IST-2016-712-v1+1_s10959-016-0724-2.pdf","date_created":"2018-12-12T10:17:13Z","creator":"system","checksum":"47686d58ec21c164540f1a980ff2163f","file_id":"5266","access_level":"open_access","content_type":"application/pdf","relation":"main_file","file_size":671125}],"publisher":"Springer","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","title":"Infinite-dimensional calculus under weak spatial regularity of the processes"}]