Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0305438

Towards the Modeling of Neuronal Firing by Gaussian Processes

E. Di Nardo·A. G. Nobile·E. Pirozzi·L. M. Ricciardi
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper focuses on the outline of some computational methods for the approximate solution of the integral equations for the neuronal firing probability density and an algorithm for the generation of sample-paths in order to construct histograms estimating the firing densities. Our results originate from the study of non-Markov stationary Gaussian neuronal models with the aim to determine the neuron's firing probability density function. A parallel algorithm has been implemented in order to simulate large numbers of sample paths of Gaussian processes characterized by damped oscillatory covariances in the presence of time dependent boundaries. The analysis based on the simulation procedure provides an alternative research tool when closed-form results or analytic evaluation of the neuronal firing densities are not available.

Explore related subjects

math.PRq-bio

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

E. Di Nardo, A. G. Nobile, E. Pirozzi, L. M. Ricciardi. 2003-05-30. Towards the Modeling of Neuronal Firing by Gaussian Processes. https://arxiv.org/abs/math/0305438

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantitative Propagation of Chaos for Singular Interacting Particle Systems Driven by Fractional Brownian Motion

We consider interacting systems particle driven by i.i.d. fractional Brownian motions, subject to irregular, possibly distributional, pairwise interactions. We show propagation of chaos and mean field convergence to the law of the associated McKean--Vlasov equation, as the number of particles $N\to\infty$, with quantitative sharp rates of order $N^{-1/2}$. Our results hold for a wide class of possibly time-dependent interactions, which are only assumed to satisfy a Besov-type regularity, related to the Hurst parameter $H\in (0,+\infty)\setminus \mathbb{N}$ of the driving noises. In particular, as $H$ decreases to $0$, interaction kernels of arbitrary singularity can be considered, a phenomenon frequently observed in regularization by noise results. Our proofs rely on a combinations of Sznitman's direct comparison argument with stochastic sewing techniques.

math.PR↗
arXiv2026

Power rate of convergence of discrete curves: framework and applications

We develop a framework for power-law rates of convergence of random discrete curves to Schramm Loewner Evolution (SLE). We show that a power-law rate of convergence of a suitable martingale observable gives a power-law rate of convergence of the interface to SLE, provided the discrete curves satisfy the Kemppainen-Smirnov crossing condition; the proof uses known estimates on the derivative of the SLE map. We apply the framework to the exploration process of critical site percolation on the triangular lattice, obtaining a power-law rate of convergence to SLE$_6$.

math.PR↗
arXiv2026

Refining Concentration Bounds for Gaussian Quadratic Chaos with Applications

This paper studies concentration of measure for Gaussian quadratic chaos in the non-asymptotic regime. We begin by tightening the classic Hanson-Wright inequality, where we increase its absolute constant $κ$ from 0.125 to at least 0.1457 in the symmetric case. We present a sharper version of Laurent-Massart inequality (LMI), which increases $κ$ from the largest available value of $1-\frac{\sqrt{3}}{2}\approx 0.134$ to $\frac{9-\sqrt{17}}{32}\approx 0.152$ in the positive-semidefinite case. We develop a sequence of inequalities indexed by $m\ge1$ that involve Schatten norms of the underlying matrix. The case $m=1$ recovers HWI. The bounds undergo a phase transition in the sense that if the tail parameter is smaller than a critical threshold $τ_c$, then $m=1$ is the tightest, whereas if it is larger than $τ_c$, then $m=\infty$ is the tightest. This leads to a novel bound called the $m_\infty$ bound. All HWI, LMI and the $m_\infty$ inequality rely on Markov's inequality. By avoiding Markov's inequality, we introduce the strong and weak $χ^2$ inequalities. We provide a list of concentration bounds that depend only on the operator norm in the positive-definite case, namely, the $m_\infty$ bound, the weak $χ^2$ bound, relaxed versions of HWI and LMI, and the so-called large-deviations bound. The sharpest among these is shown to be always either the $m_\infty$ bound or the weak $χ^2$ bound. By utilizing conditioning, modified versions of $m_\infty$, $χ^2$, and HW inequalities of various orders are proposed, all of which can be significantly tighter than the original inequalities. By leveraging the Ruben-Kotz expansion for the distribution of a positive-definite quadratic form, we introduce the Ruben-Kotz inequalities (RKIs) for general symmetric quadratic forms. A key advantage of RKIs is that they provide an explicit upper bound on the approximation gap.

math.PR↗