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Stochastic Processes in Neuronal Modeling

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Network Science".

Deadline for manuscript submissions: closed (31 October 2021) | Viewed by 18931

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Special Issue Editors

Dr. Enrica Pirozzi

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Guest Editor

Department of Mathematics and Applications "R.Caccioppoli", University of Naples Federico II, Napoli, Italy
Interests: markov processes; semi-markov processes; first passage problems; fractional dynamics; fractional brownian motion; coupled dynamics; queuing theory; leaky integrate-and-fire neuronal models; computational methods for stochastic models; stochastic simulation techniques

Prof. Dr. Eva Löcherbach

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Guest Editor

Université Paris 1 Panthéon Sorbonne, 90 rue de Tolbiac, 75013 Paris, France
Interests: interacting particle systems; stochastic models in neuroscience; longtime behavior of stochastic processes; coupling and perfect simulation; Hawkes processes; chains and processes with memory of variable length

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to publish original research articles covering advances in the theory of stochastic processes and stochastic models for single and networks of neurons. In particular, with the aim of improving upon existing neuronal models or creating new models, continuous and discrete time stochastic processes will be discussed, as well as stochastic differential equations, fractional differential equations, correlated processes, first passage time problems, stochastic optimal controls, mean-field limits, interacting particle systems, statistics of stochastic processes, parameter estimation and simulation techniques.

Potential topics include, but are not limited to:

-Markov and semi-Markov processes

-Time-changed processes

-Markov chains

-Jump processes

-Coupled dynamics

-Fractional processes

-Space-time fractional equations

-Fractional Brownian motion

-Long-range dependence

-Hawkes processes

-Integrate and fire models

-Mean-field limits

-Stochastic optimal control

-Population models

-Computational methods for stochastic models

-Information theory and estimation theory for computational neuroscience

-Large deviations and limit theorems

-Numerical and simulations approaches

Prof. Enrica Pirozzi
Prof. Eva Löcherbach
Guest Editors

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Keywords

Stochastic differential equations
Neuronal models
Fractional dynamics
Long-range dependence
Population models
Information measures
Statistics
Simulation

Published Papers (7 papers)

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Research

11 pages, 265 KiB

Open AccessArticle

Convergence in Total Variation of Random Sums

by Luca Pratelli and Pietro Rigo

Mathematics 2021, 9(2), 194; https://doi.org/10.3390/math9020194 - 19 Jan 2021

Viewed by 1502

Abstract

Let

(X_{n})

be a sequence of real random variables,

(T_{n})

a sequence of random indices, and

(τ_{n})

a sequence of constants such that

τ_{n} \to \infty

. The asymptotic behavior of [...] Read more.

Let

(X_{n})

be a sequence of real random variables,

(T_{n})

a sequence of random indices, and

(τ_{n})

a sequence of constants such that

τ_{n} \to \infty

. The asymptotic behavior of

L_{n} = (1 / τ_{n}) \sum_{i = 1}^{T_{n}} X_{i}

, as

n \to \infty

, is investigated when

(X_{n})

is exchangeable and independent of

(T_{n})

. We give conditions for

M_{n} = \sqrt{τ_{n}} (L_{n} - L) ⟶ M

in distribution, where L and M are suitable random variables. Moreover, when

(X_{n})

is i.i.d., we find constants

a_{n}

and

b_{n}

such that

{sup}_{A \in B (R)} | P (L_{n} \in A) - P (L \in A) | \leq a_{n}

and

{sup}_{A \in B (R)} | P (M_{n} \in A) - P (M \in A) | \leq b_{n}

for every n. In particular,

L_{n} \to L

M_{n} \to M

in total variation distance provided

a_{n} \to 0

b_{n} \to 0

, as it happens in some situations. Full article

(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)

18 pages, 550 KiB

Open AccessArticle

Analysis of the Past Lifetime in a Replacement Model through Stochastic Comparisons and Differential Entropy

by Antonio Di Crescenzo, Patrizia Di Gironimo and Suchandan Kayal

Mathematics 2020, 8(8), 1203; https://doi.org/10.3390/math8081203 - 22 Jul 2020

Cited by 7 | Viewed by 1646

Abstract

A suitable replacement model for random lifetimes is extended to the context of past lifetimes. At a fixed time u an item is planned to be replaced by another one having the same age but a different lifetime distribution. We investigate the past lifetime of this system, given that at a larger time t the system is found to be failed. Subsequently, we perform some stochastic comparisons between the random lifetimes of the single items and the doubly truncated random variable that describes the system lifetime. Moreover, we consider the relative ratio of improvement evaluated at

x \in (u, t)

, which is finalized to measure the goodness of the replacement procedure. The characterization and the properties of the differential entropy of the system lifetime are also discussed. Finally, an example of application to the firing activity of a stochastic neuronal model is provided. Full article

(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)

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13 pages, 1624 KiB

Open AccessArticle

Revealing Spectrum Features of Stochastic Neuron Spike Trains

by Simone Orcioni, Alessandra Paffi, Francesca Apollonio and Micaela Liberti

Mathematics 2020, 8(6), 1011; https://doi.org/10.3390/math8061011 - 20 Jun 2020

Cited by 6 | Viewed by 3739

Abstract

Power spectra of spike trains reveal important properties of neuronal behavior. They exhibit several peaks, whose shape and position depend on applied stimuli and intrinsic biophysical properties, such as input current density and channel noise. The position of the spectral peaks in the frequency domain is not straightforwardly predictable from statistical averages of the interspike intervals, especially when stochastic behavior prevails. In this work, we provide a model for the neuronal power spectrum, obtained from Discrete Fourier Transform and expressed as a series of expected value of sinusoidal terms. The first term of the series allows us to estimate the frequencies of the spectral peaks to a maximum error of a few Hz, and to interpret why they are not harmonics of the first peak frequency. Thus, the simple expression of the proposed power spectral density (PSD) model makes it a powerful interpretative tool of PSD shape, and also useful for neurophysiological studies aimed at extracting information on neuronal behavior from spike train spectra. Full article

(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)

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Figure 1

24 pages, 331 KiB

Open AccessArticle

A Semi-Markov Leaky Integrate-and-Fire Model

by Giacomo Ascione and Bruno Toaldo

Mathematics 2019, 7(11), 1022; https://doi.org/10.3390/math7111022 - 29 Oct 2019

Cited by 11 | Viewed by 1945

Abstract

In this paper, a Leaky Integrate-and-Fire (LIF) model for the membrane potential of a neuron is considered, in case the potential process is a semi-Markov process. Semi-Markov property is obtained here by means of the time-change of a Gauss-Markov process. This model has some merits, including heavy-tailed distribution of the waiting times between spikes. This and other properties of the process, such as the mean, variance and autocovariance, are discussed. Full article

(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)

12 pages, 392 KiB

Open AccessArticle

On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

by Mario Abundo and Enrica Pirozzi

Mathematics 2019, 7(10), 991; https://doi.org/10.3390/math7100991 - 18 Oct 2019

Cited by 12 | Viewed by 3581

Abstract

We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided. Full article

(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)

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18 pages, 299 KiB

Open AccessArticle

Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

by Pierre Hodara and Ioannis Papageorgiou

Mathematics 2019, 7(6), 518; https://doi.org/10.3390/math7060518 - 06 Jun 2019

Cited by 4 | Viewed by 2081

Abstract

We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework. Full article

(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)

13 pages, 374 KiB

Open AccessArticle

Retrieving a Context Tree from EEG Data

by Aline Duarte, Ricardo Fraiman, Antonio Galves, Guilherme Ost and Claudia D. Vargas

Mathematics 2019, 7(5), 427; https://doi.org/10.3390/math7050427 - 14 May 2019

Cited by 2 | Viewed by 3341

Abstract

It has been repeatedly conjectured that the brain retrieves statistical regularities from stimuli. Here, we present a new statistical approach allowing to address this conjecture. This approach is based on a new class of stochastic processes, namely, sequences of random objects driven by [...] Read more.

(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)

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