2020/05/31 by Massimiliano Tamborrino, Petr Lánský, Petr Lansky
Biochemistry, Genetics and Molecular Biology · Mathematics · Neuroscience · Physics and Astronomy · #Algorithm #Applied mathematics #Artificial intelligence #Computer science #Convergence (economics) #Diffusion #Diffusion and Search Dynamics #Diffusion process #Gaussian #Gaussian noise #Gaussian process #Inverse #Inverse Gaussian distribution #Limit (mathematics) #Mathematical analysis #Mathematics #Neural dynamics and brain function #Noise (video) #Ornstein–Uhlenbeck process #Physics #Quantum mechanics #Shot noise #Statistical physics #Statistics #Stochastic process #math.PR #stochastic dynamics and bifurcation
paper · pdf · doi:10.1016/j.physd.2021.132845
20 pages, 3 figures
arxiv created 2020/11/26 · openalex publication_date 2021/01/19 · arxiv updated 2021/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Shot noise processes have been extensively studied due to their mathematical properties and their relevance in several applications. Here, we consider nonnegative shot noise processes and prove their weak convergence to Lévy-driven Ornstein-Uhlenbeck (OU), whose features depend on the underlying jump distributions. Among others, we obtain the OU-Gamma and OU-Inverse Gaussian processes, having gamma and inverse gaussian processes as background Lévy processes, respectively. Then, we derive the necessary conditions guaranteeing the diffusion limit to a Gaussian OU process, show that they are not met unless allowing for negative jumps happening with probability going to zero, and quantify the error occurred when replacing the shot noise with the OU process and the non-Gaussian OU processes. The results offer a new class of models to be used instead of the commonly applied Gaussian OU processes to approximate synaptic input currents, membrane voltages or conductances modelled by shot noise in single neuron modelling.