[{"year":"2019","citation":{"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>.","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.","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>","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>.","short":"F. Flandoli, E. Priola, G.A. Zanco, Discrete and Continuous Dynamical Systems 39 (2019) 3037–3067."},"date_updated":"2023-09-08T11:34:45Z","external_id":{"arxiv":["1708.04156"],"isi":["000459954800003"]},"isi":1,"day":"01","arxiv":1,"doi":"10.3934/dcds.2019126","abstract":[{"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.","lang":"eng"}],"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.","volume":39,"scopus_import":"1","_id":"10878","issue":"6","author":[{"full_name":"Flandoli, Franco","last_name":"Flandoli","first_name":"Franco"},{"full_name":"Priola, Enrico","last_name":"Priola","first_name":"Enrico"},{"full_name":"Zanco, Giovanni A","last_name":"Zanco","first_name":"Giovanni A","id":"47491882-F248-11E8-B48F-1D18A9856A87"}],"department":[{"_id":"JaMa"}],"article_processing_charge":"No","date_created":"2022-03-18T12:33:34Z","publication_status":"published","intvolume":"        39","title":"A mean-field model with discontinuous coefficients for neurons with spatial interaction","quality_controlled":"1","page":"3037-3067","publisher":"American Institute of Mathematical Sciences","article_type":"original","type":"journal_article","date_published":"2019-06-01T00:00:00Z","publication_identifier":{"issn":["1553-5231"]},"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1708.04156","open_access":"1"}],"status":"public","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publication":"Discrete and Continuous Dynamical Systems","project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"}],"oa_version":"Preprint","month":"06","keyword":["Applied Mathematics","Discrete Mathematics and Combinatorics","Analysis"],"language":[{"iso":"eng"}]},{"quality_controlled":"1","page":"611-640","language":[{"iso":"eng"}],"publisher":"American Institute of Mathematical Sciences (AIMS)","article_type":"original","_id":"8513","publication":"Discrete & Continuous Dynamical Systems - A","issue":"2","author":[{"id":"FE553552-CDE8-11E9-B324-C0EBE5697425","full_name":"Kaloshin, Vadim","orcid":"0000-0002-6051-2628","last_name":"Kaloshin","first_name":"Vadim"},{"full_name":"Saprykina, Maria","last_name":"Saprykina","first_name":"Maria"}],"date_created":"2020-09-18T10:48:43Z","article_processing_charge":"No","publication_status":"published","oa_version":"None","intvolume":"        15","title":"Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits","month":"05","volume":15,"status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","extern":"1","citation":{"ista":"Kaloshin V, Saprykina M. 2006. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete &#38; Continuous Dynamical Systems - A. 15(2), 611–640.","mla":"Kaloshin, Vadim, and Maria Saprykina. “Generic 3-Dimensional Volume-Preserving Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits.” <i>Discrete &#38; Continuous Dynamical Systems - A</i>, vol. 15, no. 2, American Institute of Mathematical Sciences (AIMS), 2006, pp. 611–40, doi:<a href=\"https://doi.org/10.3934/dcds.2006.15.611\">10.3934/dcds.2006.15.611</a>.","short":"V. Kaloshin, M. Saprykina, Discrete &#38; Continuous Dynamical Systems - A 15 (2006) 611–640.","ieee":"V. Kaloshin and M. Saprykina, “Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits,” <i>Discrete &#38; Continuous Dynamical Systems - A</i>, vol. 15, no. 2. American Institute of Mathematical Sciences (AIMS), pp. 611–640, 2006.","chicago":"Kaloshin, Vadim, and Maria Saprykina. “Generic 3-Dimensional Volume-Preserving Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits.” <i>Discrete &#38; Continuous Dynamical Systems - A</i>. American Institute of Mathematical Sciences (AIMS), 2006. <a href=\"https://doi.org/10.3934/dcds.2006.15.611\">https://doi.org/10.3934/dcds.2006.15.611</a>.","ama":"Kaloshin V, Saprykina M. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. <i>Discrete &#38; Continuous Dynamical Systems - A</i>. 2006;15(2):611-640. doi:<a href=\"https://doi.org/10.3934/dcds.2006.15.611\">10.3934/dcds.2006.15.611</a>","apa":"Kaloshin, V., &#38; Saprykina, M. (2006). Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. <i>Discrete &#38; Continuous Dynamical Systems - A</i>. American Institute of Mathematical Sciences (AIMS). <a href=\"https://doi.org/10.3934/dcds.2006.15.611\">https://doi.org/10.3934/dcds.2006.15.611</a>"},"year":"2006","date_updated":"2021-01-12T08:19:48Z","type":"journal_article","date_published":"2006-05-01T00:00:00Z","publication_identifier":{"issn":["1553-5231"]},"day":"01","doi":"10.3934/dcds.2006.15.611"}]