@article{10878,
  abstract     = {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.},
  author       = {Flandoli, Franco and Priola, Enrico and Zanco, Giovanni A},
  issn         = {1553-5231},
  journal      = {Discrete and Continuous Dynamical Systems},
  keywords     = {Applied Mathematics, Discrete Mathematics and Combinatorics, Analysis},
  number       = {6},
  pages        = {3037--3067},
  publisher    = {American Institute of Mathematical Sciences},
  title        = {{A mean-field model with discontinuous coefficients for neurons with spatial interaction}},
  doi          = {10.3934/dcds.2019126},
  volume       = {39},
  year         = {2019},
}

@article{8513,
  author       = {Kaloshin, Vadim and Saprykina, Maria},
  issn         = {1553-5231},
  journal      = {Discrete & Continuous Dynamical Systems - A},
  number       = {2},
  pages        = {611--640},
  publisher    = {American Institute of Mathematical Sciences (AIMS)},
  title        = {{Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits}},
  doi          = {10.3934/dcds.2006.15.611},
  volume       = {15},
  year         = {2006},
}