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